The first evidence of charmed hadron production in neutrino interactions was produced in 1974 throughthe observation of opposite sign dimuons. Since that time, it has been assessed that high energy neutrinointeractions make charmed hadron production at the level of a few percent and therefore they representa powerful tool to study charm physics. Moreover, unlike colliding beams, neutrino scattering producescharmed hadrons also via peculiar processes like quasi-elastic and diffractive productions which makesthem an unique tool for exclusive charm studies.
Since its discovery, charmed hadron production has been studied basically in two ways: dilepton studieswith both calorimeter and bubble chamber techniques on one side and nuclear emulsion experiments withthe visual observation of charmed hadron decays on the other side.
In this paper, after a brief introduction of the formalism to describe the inclusive neutrino-nucleonscattering (Section 2), we briefly discuss the theoretical aspects of neutrino charm-production (Section 3)and the main features of the experiments where the charm has been looked for (Section 4). In Section 5 wefocus our attention on the charm-production rate, both for the inclusive process and for exclusive channelssuch as quasi-elastic, diffractive and associated charm-production. Finally, in Section 6 we report on thedetermination of production characteristics of charmed particles in neutrino interactions. In particular,we focus on the impact of charm studies, e.g. the new measurement of the semi-muonic decay branchingratio, on the extraction of the CKM matrix elements. The measurement of strange-quark distributionfunctions is also given and tests of thes-s asymmetries are discussed.
2. The inclusive cross-section for neutrino scattering
2.1. Kinematics
The tree-level Feynman diagram for charged-current (CC) neutrino-nucleon (N ) deep-inelastic scat-tering (DIS) is shown inFig. 1. A neutrino (anti-neutrino) with incoming four-momentumk1 scatters offfrom a quark (anti-quark) in the nucleon via the exchange of aW+ (W−) boson with four-momentumq.The process, in the laboratory frame, can be studied by measuring the energy,E, and the direction,
230 G. De Lellis et al. / Physics Reports 399 (2004) 227–320
Fig. 1. The tree-level Feynman diagram for deep-inelastic neutrino scattering.
and, of the outgoing muon, and the energy and the angles of the outgoing hadrons,Ehad, had, had.By choosing thez-axis along the neutrino direction, we can write
k1 = (E,0,0, E) , (1)
whereE = E + Ehad. The remaining 4-vectors are then given by
k2 = (E, E sin cos, E sin sin, E cos) ,
p = (M,0,0,0) ,q = k1 − k2 ,
whereM indicates the nucleon mass.A neutrino interaction is usually described through the following Lorentz invariant quantities
where the equivalence indicates the expressions of the invariants in the laboratory frame.
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 231
Fig. 2. Examples of kinematic regions accessible to some of the high statistics neutrino experiments, taken from Ref.[1].
To complete this brief discussion we give, for a vanishing lepton mass, the allowed physical region ofthe kinematical variables in Eq. (2)
0 E
(1+ 2Mx/E),
0 y1
(1+ 2Mx/E),
0Q22MEx
(1+ 2Mx/E),
0 x 1 . (3)
The kinematic range in thex andQ2 plane is shown inFig. 2 for four high-statistics deep-inelasticneutrino scattering experiments.
2.2. The neutrino cross-section
The interaction we wish to describe is the charged-current scattering of neutrinos off a nucleon, shownin Fig. 1. The final state consists of a muon and a hadronic shower, the details of which will not concernus since all hadronic final states are integrated over. Schematically, we are considering the process
(k1)+N(p)→ (k2)+X(p′) , (4)
232 G. De Lellis et al. / Physics Reports 399 (2004) 227–320
wherek1, p, k2 andp′ are the four-momenta.X corresponds to a set ofn final hadrons, each carrying thefour-momentump′
j (with j ∈ 1, . . . , n); in such a way
p′ =n∑j=1
p′j . (5)
The differential cross-section is written as
d = d|M|2
, (6)
where the terms entering the previous expression are:
• d is the density of final states per incident particle; it measures the phase space volume of the finalstate;
• |M|2 is the square of the invariant amplitude of the process considered;• is the incident particle flux, the normalization of the initial state.
Without entering the detail of the derivation of any of the terms in Eq. (6), we only briefly discuss theinvariant amplitudeM. In the framework of the Standard Model, the leptons are point-like particles andtheir couplings to theW bosons are fixed by the gauge groupSU(2)×U(1). The matrix elements of thehadronic current,J , between the initial nucleon (we are assuming unpolarized nucleon) and the finalhadronic state can be parametrized asH , thus we write
M = √2GF × (k2)
(1− 5)(k1)︸ ︷︷ ︸ × 1
1+Q2/M2W
× 〈X|J|p〉︸ ︷︷ ︸= √
2GF × L × 1
1+Q2/M2W
× H ,
(7)
whereGF is the Fermi constant. Squaring this amplitude and using the well established rules for cross-sections we write
d2()N
dx dy= G2
FME
(1+Q2/M2W)
2
[y2xF
()N1 + (1− y)F ()N
2 ±(1− y
2
)yxF
()N3
], (8)
where the sign of the last term is+ for neutrinos,− for anti-neutrinos. The structure functionsFi dependon the Lorentz-invariantx andQ2 and take into account the fact that the hadronic tensorH can bewritten as
H =∑X
〈p|J |X〉〈X|J |p〉 = −g
MF
()N1 + pp
M2 F()N2 − iε pq
2M2 F()N3 , (9)
where only the Lorentz structures relevant for the neutrino (anti-neutrino) nucleon deep-inelastic cross-section are reported.1 The definitions of the kinematic variables such asx, Q2, y and, as well as thetransformations between different sets of variables, are given in Section 2.1.
1 There are, in general, six independent Lorentz structures for the hadronic tensor[1,2]. The structure function correspondingto the missing ones are either function of the ones reported in Eq. (9) or give a negligible contribution to the cross-section dueto the smallness of the lepton mass.
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 233
2.3. The parton model
In the previous section we have discussed the neutrino-nucleon cross-section as general as possibleand we have not given any physical interpretation of the structure functionsFi , since it requires a modelfor nucleons. Historically, the experimental data on electron-nucleon scattering at SLAC[3,4] lead to thedevelopment, due to Feynman, of the Parton Model (PM)[5]. The idea is that the hadrons are made ofquasi-free point-like constituents, the partons. In theinfinite momentum frame(IMF), a frame in which thenucleon momentum is very large, the parton mass and any momentum transverse to the nucleon directioncan be neglected. Hence,x represents the fraction of the nucleon momentum carried by the parton and theinclusive cross-sections can be obtained by summing incoherently cross-sections of individual partons.A byproduct of the PM is the confirmation of the (earlier) Bjorken scaling hypothesis[6]
limx fixedQ2→∞
Fi(x,Q2)= Fi(x) . (10)
The identification between partons and quarks[7,8], in the so-called quark parton model (QPM), camefrom the experimental confirmation of the Callan–Gross[9] relation
F2 = 2xF 1 , (11)
which is a consequence of the assumption that the partons are spin 1/2 particles.We shall see, in the next section, that the QCD, once the identification between partons and quarks
is done, justifies the scaling of the structure functions. In fact, QCD predicts that the scaling of thestructure functions is violated by logarithmic corrections, which can be calculated, and the predictionsare confirmed by experimental results.
In the framework of the QPM, the connection between expressions for the neutrino-parton scatteringcross-section and the structure functions appearing in the neutrino-hadron scattering cross-section canbe obtained introducing the parton (quark) density functions. If we defineqh(x)dx as the probability offinding in a hadronh a quarkq carrying a fractionx to x + dx of the hadron momentum, we can writethe cross-section in Eq. (8) as
d()h(P, q)
dEd=∑q
∫ 1
0dx
d()q(xP , q)
dEd(qh(x)+ qh(x)) , (12)
whereP is the momentum carried by the hadronh.The structure functions can then be directly related to the parton probability densities through the
relations2
F()h1,3 (x)=
∑q
∫ 1
0
d
F
()q1,3
(x
)(qh()+ qh()) , (13)
F()h2 (x)=
∑q
∫ 1
0dF ()q
2
(x
)(qh()+ qh()) . (14)
2 See next section for a discussion on the decomposition in Eqs. (13) and (14).
234 G. De Lellis et al. / Physics Reports 399 (2004) 227–320
The functionsF ()q1,2,3 (x) are, in the QPM, proportional to -functions of(1− x). Therefore,
where we have explicitly written the flavours of the partons and the indexv means the valence quarkdistributions
qv(x)= q(x)− q(x) q ∈ u, d . (18)
It is worth noting thatN indicates that the interaction occurs off an hypothetical entity corresponding tothe average of a free neutron and a free proton: i.e.N = (n+ p)/2.
2.4. Evolution equations in QCD
We have briefly seen that the Bjorken scaling property of nucleon structure functions can be explainedin the PM by assuming free point-like, spin 1/2 constituents that build up the nucleon. These constituentsare the spin 1/2 quarks, originally postulated by Gell-Mann and Zweig to explain hadron spectroscopy[7,8]. The theory that describes the interactions between quarks is called Quantum ChromoDynamics(QCD) [10–12]. A lot of reviews are devoted to this argument and we address the interested reader tothem (see for example[13,14]). In this section we shall only mention that in the framework of QCD thestrength of the interaction, the coupling constant, goes asymptotically to zero as the distance betweenquarks (or equivalently as the energy scaleQ2 goes to infinity) goes to zero (Asymptotic freedom[15,16]).Moreover, there are theoretical indications that the confinement, i.e. all observed hadronic particles arecolorless, is predicted by the QCD.
We briefly discuss in the following two fundamental features of the QCD: the factorization theorem[17], and the evolution[18–20].
The factorization theorem allows the writing of the structure functions as a direct generalization of theones in the PM, Eqs. (13) and (14),
F()h1,3 (x,Q2)=
∑q=f,f ,g
∫ 1
0
d
F
()q1,3
(x
,Q2
2 ,2f
2 , s(2)
)qh(, f ,
2) , (19)
F()h2 (x,Q2)=
∑q=f,f ,g
∫ 1
0dF ()q
2
(x
,Q2
2 ,2f
2 , s(2)
)qh(, f ,
2) . (20)
where the functionsFi depend on the renormalization scale,, which is necessary in any renormalizablequantum field theory, and on the factorization scalef . This represents the separation between the short-distance (perturbative) effects and the long-distance (non-perturbative) effects.Fi represent perturbativeeffects, while quarks (q, q) and gluon (g) distribution functions include non-perturbative effects. Theycan be obtained by comparing theoretical expressions to the experimental data (see discussion at the end
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 235
of this section). The structure functions naturally develop a dependence onQ2 that is responsible for theobserved breaking of the Bjorken scaling.
Finally, the Fi depend on the running coupling constants() which, at next-to-leading order,is given by
s()= 4
0 ln(2/2)
[1− 21
20
ln(ln(2/2))
ln(2/2)
], (21)
where0 = 11− 2nf /3, 1 = 51− 19nf /3 andnf the number of quarks with mass smaller than thescale, i.e. the active flavours. The dimensional parameter is the so-called “QCD parameter”. Theexpression in Eq. (21) is an approximate solution of the equation
s()=−2s()
[1
(s()
4
)+ 2
(s()
4
)2]
(22)
in powers of 1/ ln(2/2), where the coefficient of(1/ ln(2/2))n is a polynomial in ln(ln(2/2)).Contributions of the order of(1/ ln(2/2))3 are neglected in Eq. (21) and is fixed in such a way thatthe term proportional to 1/(ln(2/2))2 is absent. Moreover, must change as the flavour thresholds arecrossed. For a detailed review we refer to[21] and references therein.
Obviously, observable quantities can depend neither on the renormalization scale nor on the factor-ization scale,f .3 Therefore, for any observable there exists a renormalization group equation
dObs
d= 0 . (23)
Looking at the structure functions in Eqs. (19) and (20), one understands that the evolution equations for theparton distributions should exist. These are the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP)equations[18–20]usually written as integral-differential equations
2 d
d2 qh(x, , 2)=
∑q ′=q,q,G
∫ 1
x
d
Pqq ′
(x
, s(
2)
)q ′h(, , 2) , (24)
where the evolution kernels (the splitting functionsPij ) are perturbatively calculable and their first orderis O(s). If we introduce the following combinations of parton density functions
non-singlet: qNS(x, t)=∑q
[q(x, t)− q(x, t)] , (25)
singlet : qS(x, t)=∑q
[q(x, t)+ q(x, t)] , (26)
3 Hereafter, without loss of generality, we put = f .
236 G. De Lellis et al. / Physics Reports 399 (2004) 227–320
and withg(x, t) indicate the gluon distribution function, the DGLAP equations can be rewritten as
dqNS(x, t)
dt= s(t)
2
∫ 1
x
dy
yPqq
(x
y
)qNS(x, t) , (27)
dqS(x, t)
dt= s(t)
2
∫ 1
x
dy
y
[Pqq
(x
y
)qS(y, t)+ 2nf Pqg
(x
y
)g(y, t)
], (28)
dg(x, t)
dt= s(t)
2
∫ 1
x
dy
y
[Pgq
(x
y
)qS(y, t)+ Pgg
(x
y
)g(y, t)
], (29)
wheret = ln(Q2).In order to make the connection between Eqs. (27)–(29) and the scaling structure functionsFi defined
before, we rewrite the structure functions in terms of the singlet and non-singlet quark distributions andgluon distribution at the renormalization scale2 ≡ Q2, introducing the coefficient functionsCi
2xF 1(x,Q2)= x
∫ 1
x
dy
[Cq1
(x
y,Q2
)qS(y,Q
2)+ Cg1
(x
y,Q2
)g(y,Q2)
], (30)
F2(x,Q2)= x
∫ 1
x
dy
[Cq2
(x
y,Q2
)qS(y,Q
2)+ Cg2
(x
y,Q2
)g(y,Q2)
], (31)
xF 3(x,Q2)= x
∫ 1
x
dy Cq3
(x
y,Q2
)qNS(y,Q
2) . (32)
If we switch off the strong interaction, the coefficient functions reduce to
Cqi ∝
(1− x
y
)and C
gi = 0 , (33)
which imply
2xF 1(x,Q2)= F2(x,Q
2)= xqS(x,Q2) ,
F3(x,Q2)= xqNS(x,Q
2) . (34)
As we have seen, the QCD predicts theQ2 dependence of the structure functions but cannot giveinformation on their shapes due to the non-perturbative nature of the parton distribution functions. Nev-ertheless, by taking into account the evolution equations we can compare different experimental data,at differentQ2 and probes, to have reliable information on the quark and gluon distribution functions[22–24].
3. Theoretical aspects of neutrino charm-production
The neutrino deep-inelastic scattering is an interesting tool to have information on the parton densi-ties in the nucleon. Furthermore, the neutrino scattering is a way to study the heavy quark production
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 237
due to the light-to-heavy quark charged-current. In particular, charm-production can be used to study thestrange content of the nucleon and the element of the Cabibbo–Kobayashi–Maskawa matrix [25,26]|Vcd |.Indeed, in the case of neutrino scattering the underlying process is a neutrino scattering off as (≡ ss)4 ord (≡ dv + ds) quark producing a charm quark that fragments into a charmed hadron. The analogousprocess with an incident anti-neutrino proceeds through the scattering off as (≡ ss) or a d (≡ ds).Being thes → c transition Cabibbo favoured, charm-production induced by anti-neutrino is compara-ble to the one induced by neutrinos allowing for the study of sea-quark distributions. Furthermore, byproper combinations of neutrino and anti-neutrino data it is possible to isolate the valence contribution ofquarkd and extract|Vcd |.
Differently from the case ofu, d, s quarks, neglecting the mass of the charm is a crude approximation.In this respect, the first theoretical attempts to include finite charm quark mass effects in neutrino-inducedproduction is due to Barnett et al.[27,28]. This formalism is referred to as ‘slow rescaling’ and consistsin a redefinition of thex variable.
A comprehensive analysis of lepto-production of heavy quark at leading order (LO) and next-to-leadingorder (NLO) was given in Refs.[29–33].At the same order, but in the more conventionalMS factorizationscheme, the neutrino charm-production was studied in[34,35]. The two approaches differ in a technicaldetail regarding the mass of the strange quark, but the last one is almost always used in the analyses ofdeep-inelastic scattering data (see for example,[22,23]). The LO contribution for charm-production inN → −cX comes from theW+s′ → c subprocess, where
s′ ≡ |Vcs |2s + |Vcd |2u+ d2
(35)
andVcs andVcd are Cabibbo–Kobayashi–Maskawa matrix elements[25,26]. At NLO theW+g → cs
subprocess together with theW+s′ → gcprocess should be taken into account. Denoting the contributionsof all above processes in theMS factorization scheme to the structure functionsFi(x,Q2) (cf. Eq. (8))byFci (x,Q
2) (i ∈ 1,2,3), the semi-inclusive NLO isoscalar cross-sections for the neutrino productionof charm are given by (cf. Eq. (8))
d2(N → −cX)dx dy
= G2FME
(1+Q2/M2W)
2
[y2xF c1 + (1− y) F c2 +
(1− y
2
)yxF c3
], (36)
d2(N → +cX)dx dy
= G2FME
(1+Q2/M2W)
2
[y2xF c1 + (1− y) F c2 −
(1− y
2
)yxF c3
]. (37)
Following the notation in Ref.[34], the structure functionsFci5 appearing in Eqs. (36), (37) can be
written in terms ofFci : Fci ≡ Fc
1,Fc3 ≡ 2Fc3,Fc2 ≡ 2Fc
2. At NLO theFci have the following structure
(cf. Eqs. (19)–(20))
Fci (x,Q
2)= s′(, 2)+ s(2)
2
∫ 1
d′
′[Hqi (
′, 2, )s′(
′, 2
)+Hgi (′, 2, )g
(
′, 2
)], (38)
4 The suffixs stands for sea component.5Analogously can be obtainedF c
i.
238 G. De Lellis et al. / Physics Reports 399 (2004) 227–320
where
= Q2
Q2 +m2c
, = x(
1+ m2c
Q2
), (39)
and the expressions forHq,gi can be found in[34,36]. In Eq. (39)mc is a parameter related to the mass ofthe charm quark. For a detailed discussion on the meaning of the parametermc we refer to Section 6.4.1.
The processes described in the previous section are not directly accessible by experiments. The dif-ferential cross-sections for charm quark production induced by neutrinos are reported in Eqs. (36)–(37).However, experiments can only detect (directly or by means of leptons produced in semi-leptonic charmdecays) charmed hadrons not charmed quarks. Therefore, the fragmentation process, in which a quarkbecomes a hadron, is an interesting subject in which perturbative and non-perturbative phenomena enterin the game. As in the analysis of inclusive processes, the study of the fragmentation of an heavy quarkcan be carried out by using the factorization theorem. In that approach, the cross-sections for charmedhadron (C) production can be written, at LO, as
d(N → −CX)dx dy dz
= d(N → −cX)d dy
∑h
fhDhc (z) ,
d(N → +CX)dx dy dz
= d(N → +cX)d dy
∑h
fhDhc (z) , (40)
where the non-perturbative quantityDhc (z) represents the probability distribution for a charmed quarkto fragment into a charmed hadronh (=D0,D+,D+
s ,+c ) with a momentum which is the fractionz of
the charm momentum. Moreover,fh is the mean multiplicity of the hadronh in neutrino production ofcharm. So,Dhc (z) andfh are normalized as∫ 1
0dzDhc (z)=
∫ 1
0dzDhc (z)= 1,
∑h
fh =∑h
fh = 1 . (41)
At NLO the expressions for the semi-inclusive cross-sections have a more complex structure involvinga new convolution between the coefficient functionsHq,gi (they depend also on the variablez) and thefragmentation functionDhc (z). The cross-sections in Eq. (40) can be obtained in term of the NLO structurefunctions[35] as
Fci (x, z,Q
2)=Fci (LO)+ s(2)
2Fci (NLO) ,
Fci (LO)=
∑h
s′(, 2)Dhc (z) ,
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 239
Fci (NLO)=
∑h
∫ 1
d′
′∫ 1
max(z,min)
d
[Hqi (
′, , 2, )s′(
′, 2
)+Hgi (′, , 2, )g
(
′, 2
)]Dhc
(z
), (42)
wheremin = (1 − )′/(1 − ′). The fragmentation functionDhc incorporates the long-distance, non-perturbative physics of the hadronization process. Therefore, theDhc is not calculable in perturbationtheory and can be only phenomenologically guessed. Among the most used parametrizations we onlymention a few
Kartvelishvili et al.[37]:
D(z) ∝ z(1− z) , (43)
Peterson et al.[38]:
D(z) ∝ 1
z
(1− 1
z− εP
1− z)−2
, (44)
Collins and Spiller[39]:
D(z) ∝(
1− zz
+ εC(2− z)1− z
)(1+ z2)
(1− 1
z− εC
1− z)−2
, (45)
where , εP and εC are non-perturbative parameters to be fitted from experimental data. A moredetailed discussion about the determination of the above non-perturbative parameters can be found inSection 6.4.2.
Parton fragmentation functions are analogous to the parton distributions, and in both cases the scalingviolation is obtained when the QCD corrections are taken into account. The evolution of the partonfragmentation functions with the scaleQ2 is governed by DGLAP-like equations; symbolically
Q2
Q2 Dj(z,Q2)= s
2
∑j
∫ 1
z
d
Pji(, s)Dj
(z
,Q2
), (46)
wherePji are the splitting functions which, at lowest order, are the same as those in deep-inelasticscattering. The higher order terms are, instead, different[40]. Any analysis of the experimental data oncharm-production should take into account that the parameters appearing in Eqs. (43), (44) and (45) arescale dependent and the scaling is governed by the evolution equations in Eq. (46).
3.3. Quasi-elastic charm-production
The simplest exclusive charm-production reaction is the quasi-elastic process where ad-valence quarkis changed into ac-quark, thus transforming the target nucleon into a charmed baryon. Explicitly, thequasi-elastic reactions are
n→ −+c (2285) , (47)
p → −++c (2455) , (48)
240 G. De Lellis et al. / Physics Reports 399 (2004) 227–320
n→ −+c (2455) , (49)
p → −++c (2520) , (50)
n→ −+c (2520) . (51)
In literature there are two different kind of approaches to describe the processes (47)–(51). The first one,discussed in Refs.[41–46], is essentially a quark model approach based on theSU(4) flavour symmetry.The large mass splittings between charmed and non-charmedSU(4)multiplet members make it clear thatthe symmetry is badly broken. Within the context of the quark model the neutrino production of charmedbaryons affords a very nice way to study such a basicSU(4)-breaking mechanism, since the neutrinocurrent induces a transition from a light non-charmed to a heavy charmed quark.
In these models theSU(4) breaking effects are incorporated by introducing quark mass breaking andtheir results should be used only atq2=0, where the quark model has been quite successful in accountingfor the photo-excitation feature of the low-lying baryon resonances. Forq2>0 the quark model resultsbecome increasingly model-dependent and tend to be unreliable[47].
The second kind of approach[48] is based on the Bloom–Gilman[49,50]local duality inN scatteringmodified on the basis of QCD[51,52].
3.3.1. TheSU(4) quark modelsTheSU(4) based models differ among them for the way how theSU(4) breaking effects are treated.
Among these models[41–46], we follow the approach discussed in[46] and finally we compare it withother calculations.
The model in Ref.[46] is based on a quark model[53] and takes into account theSU(4) breakingeffects due to the quark mass differencemc −m, wherem is the mass of light quarks. For details on thecalculations we refer to[46], here we only summarize the main features of this model. It includes:
• explicit treatment of the hadron center of mass motion;• semi-relativistic approximation of velocities: non-relativistic approximation to internal quark veloc-
ities and relativistic treatment of hadron velocities, including Lorentz contraction and quark spinboosts;
• form factors coming from the combined effect of the overlap of the spatial wave functions and ofquark structure, assumed to be dominated by the corresponding vector and axial mesons.
The first model based onSU(4) was proposed by Finjord and Ravndal[41]. They used the quark modelof Feynman et al.[54] and nucleon-like form factorsG(q2) = 1/(1 + |q2|/0.71 GeV2). As noted byseveral authors[42,46], such form factors are probably too steep. Furthermore, the overlap integraleffects (see Ref.[46] for details) are not taken into account, unlike the mass effect in the quark current.The model proposed by Lee and Shrock[42] assumed dipole-like form factors dominated by theD
mesons. Consequently, it has the same basic assumptions as[46] concerning these form factors, but doesnot consider the effects ofSU(4) breaking on the quark currents and the spatial overlap.
The predicted form factors for the different models are shown inFig. 3 and numerical results forquasi-elastic charm-production are given inTable 1.
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 241
Fig. 3. Comparison of the form factors in the various models. The solid line is for Orsay model charge transition form factor.The dotted line is for the Lee and Schrock model. The dashed is for Avilez, Kobayashi and Korner invariant form factor. Thedashed–dotted is for Finjoird and Rayndal.
Table 1Predicted quasi-elastic charm-production cross-section assuming a neutrino energy of 10 GeV
(10−40cm2)\ Model F.R.[41] S.L. [42] A.K.K. [43–45] A.G.Y.O. [46] K. [48]
3.3.2. The duality model approachBloom and Gilman[49,50]showed that the behaviour of deep-inelastic scattering is connected to the
behaviour of the nucleon-resonance electro-production. By means of the finite-energy sum rules methods,they found that the observed structure functions of inelastic scattering, in the scaling regions, and thestructure functions measured at lower energy, where the effects of resonances dominate, are very similarif they are averaged over an appropriate kinematic interval. This result comes from the theoretical idea ofduality, developed before QCD. The same approach was developed by[51,52]in the framework of QCDwhere the asymptotic freedom property allows to evaluate perturbatively the high energy behaviour ofthe structure functions. In particular, in Refs.[49–52] the authors were able to show that the observedstructure functionF ph
i is related to the one,F thi , evaluated in perturbative QCD provided that an average
on the whole kinematic range 0 x 1 (global duality)∫ 1
0dx[F ph
i (x,Q2)− F th
i (x,Q2)] 0 , (52)
242 G. De Lellis et al. / Physics Reports 399 (2004) 227–320
Fig. 4. Diagram for neutrino induced diffractiveD+s meson production.
and over some vicinity of well definite resonances,x0 x x1 (local duality)∫ x1
x0
dx[F phi (x,Q
2)− F thi (x,Q
2)] 0 , (53)
is performed. The approach in Ref.[48] is based on the local duality and takes into account part of thepower corrections toF th
i by replacing the Bjorkenx variable with
(x,Q2)= Q2/M
+√2 +Q2
[1+ M2
0
Q2
(1+ M2
0
Q2 +M20
)](54)
whereM0 is the scale of non-perturbative, higher twist, corrections. The model adoptsSU(4) symmetryto fix the form factors describing nucleon transitionsN → +
c ,+c ,
++c atQ2=0. Moreover, to estimate
theF thi the author neglects theQ2 dependence of the parton distribution functions which are assumed
to be
f (x) ∝ x−1/2(1− x)4 . (55)
The numerical results are reported inTable 1.
3.4. Diffractive charm-production
In charged-current interactions, theW boson canfluctuateinto a charmed meson. The on-shell meson isproduced by scattering off a nucleon without breaking up the recoiling partner. The diffractive productionmechanism (N → −D()+s N ) is shown schematically inFig. 4. The same mechanism applies toD()
production, but theD()s one is Cabibbo favoured by a factor|Vcs/Vcd |2 ∼ 20. The process was for the
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 243
(b)(a)
D
-
µ+ µ+
WW-
s-Ds
-
c c
s
s s
_ _νµ νµ
NNNN
Fig. 5. Two of the five contributing diagrams to the +N → + +N +D−s process.
first time studied in Ref.[55] in the framework of vector meson dominance (VMD) which was adoptedand its predictions were compared with the existing experimental data[56–59]. The theoretical studyof this kind of process received much attention in particular after the proof of the factorization theorem[60,61]. In Refs.[60,61] it was shown that every contribution to the amplitude for hard exclusive mesonproduction can be written as a convolution of a skewed parton distribution (SPD)[61,62]and a hard part.Until now there only exists the calculation of theDs production cross-section[63,64]. Here we shallfollow the approach in Ref.[63] where the reaction
(k)+N(p)→ +(k′)+N(p′)+D−s (q
′) (56)
is studied. The analysis was performed for largeQ2 compared to the Pomeron (seeFig. 4) invariant mass,t and the squared masses of the involved particles. The differential cross-section of the process is given by
d
dx dQ2 dt= e2
4(4)3 sin2 W
x
Q2(Q2 +M2W)
2
(1− Q2
2xp · k)∑
s′|T |2 , (57)
whereT is the amplitude for the subprocess
W−∗L (q)+N(p)→ D−
s (q′)+N(p′) . (58)
At leading order,T is obtained evaluating the sum of three diagrams involving a gluon SPD (seeFig. 5a)and diagrams obtained by an interchange of the order of the vertexes) and two diagrams with a contri-bution of the (polarized and unpolarized) strange quark SPD (Fig.5b plus one diagram with a changedorder of vertexes). The explicit expression ofT can be found in[63].The numerical evaluation of thecross-section needs
• theDs distribution amplitude,Ds , which is modelled following Refs.[65,66] and the asymptoticform
Ds (z)= fDs6z(1− z) (59)
with fDs = 270 MeV;• the skewed parton distributions, which are parametrized by combining the model in Ref.[61] with the
parametrizations of theusualparton distribution in[67].
The results are plotted inFig. 6 where the variablet = (p − p′)2 has been integrated over the rangetmin=M2x2/(1−x)<− t <2 GeV2 and the neutrino energy fixed atE =34 GeV. The plot of d/dx is
244 G. De Lellis et al. / Physics Reports 399 (2004) 227–320
Fig. 6. The differential cross-section for exclusiveD−s production as a function ofxBj ≡ x andQ2, respectively. The dotted lines
show the contribution stemming from the gluon SPD. The results obtained for the asymptotic form of the distribution amplitudeD−s
are plotted with dashed lines.
obtained by integratingQ2 from 6 GeV2 to the value given by the constrainty <1. The plot of d/dQ2
is obtained integratingx over the interval [0.1,0.75] and by taking into account the same kinematicalconstraint. The solid lines correspond to the form ofD−
sgiven in Refs.[65,66], while the dashed lines
correspond to the asymptotic form in Eq. (59). To explicitly show the dominance of the two gluons ex-change, the dotted lines correspond to the case of negligible strange quark SPD content of the nucleon.The integral over all variablesQ2, x, andt , on the kinematical regions discussed before, gives a valuefor the total cross-section of = 2.2 × 10−5 pb= 2.2 × 10−41cm2 for a monochromatic anti-neutrinobeam of 34 GeV. This result is very similar to the one obtained in Ref.[64], where only the graph inFig. 5a has been considered and smallerx values have been taken into account.
3.5. Associated charm-production
The associated charm-production process consists of the creation of a charm–anticharm pair. Thesmall cross-section of neutrino interactions together with the requirement for production of two charmedhadrons in the final state makes the associated charm-production in neutrino interactions very rare and,therefore, difficult to observe. It can be observed both in charged- and neutral-current (NC) neutrinointeractions.
3.5.1. Associated charm-production inCC interactionsIn the standard model, associated charm-production in neutrino CC interactions occurs via the diagrams
shown inFig. 7.
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 245
Fig. 7. Feynman diagrams of the partonic process contributing to weak charged-current production of a charmed quark pair inlowest order QCD.
Let us consider the weak charged-current production of a charmed quark pair(cc) off a nucleon
(k)+N(P )→ (k′)+ cc(P ′)+X , (60)
where is the incident neutrino of four-momentumk and its weak charged–current partner of momen-tumk′. P ′ is the four-momentum of the gluon. The differential cross-section depends on seven variables.A possible choice is
Q2 =−q2, x = Q2
2P · q , y = P · qP · k ,
z= P · P ′
P · q , M2cc = P ′2, P 2
T = |P′T |2, , (61)
whereq = k − k′. The transverse momentum(P′T ) of the pair and the azimuthal angle() between
P′T andkT are measured in the laboratory frame taking the direction of the transferred momentumq as
polar axis.The differential cross-section of the process described in Eq. (60) can be written as
d6N→ccX
dx dy dz dM2cc dP 2
T d=∑q
∫dx d (x − x)qN(,Q
2)d6q→ccX
dx dy dz dM2cc dP 2
T d(62)
where is the parton cross-section andqN(,Q2) is theq-parton density in the nucleon,N , with mo-mentum fraction. The corresponding partonic process (seeFig. 7) is
(k)+ q(p1)→ (k′)+ q ′(p2)+ c(p3)+ c(p4) (63)
wherep1 andp2 denote the four-momenta ofu and d quarks, respectively, andP ′ = p3 + p4. Wesummarize here the main steps to calculate the partonic cross-section following, among all available
246 G. De Lellis et al. / Physics Reports 399 (2004) 227–320
E (GeV)
Lo
g(σ
)
-43
-42
-41
-40
-39
-38
100 200 300
Fig. 8. Cross-section (cm2/nucleon) for the charmed quark pair production as a function of the energy. The charm quark massis set atmc = 1.25 GeV. Logarithmic scale is used.
theoretical calculations (see for instance[69,70]), the one in Ref.[68]. The invariants are
x = Q2
2p1 · q = x
, y = p1 · q
p1 · k = y, z= p1 · P ′
p1 · q = z , (64)
andM2cc, P
2T and are the same as in Eq. (61). Moreover, by neglecting light quark and lepton masses
we have
k2 = k′2 = p21 = p2
2 = 0, p23 = p2
4 =m2c , (65)
once integrate overP 2T , the parton cross-section can be written as
d5
dM2cc dx dy dz d
= G2F
4(4)4yLM
, (66)
where, the leptonic tensor for neutrino scattering is
L = 8[kk′ − 12gQ
2 + ikq] .
while, the hadronic tensor has, in general, 16 independent form factors. In the massless lepton limit only9 of the 16 independent form factors give contribution to the cross-section. They have been evaluated atthe orders in QCD, and their expressions can be found in Ref.[68].
In the following, as in the original paper, we assume that once the charmed quark pair is producedit converts with unit probability into either a charmed particle pair or a bound charmonium state. Therelative ratio should be phenomenologically determined.
Fig. 8shows the total cross-section per nucleon for charmed quark pair production in charged-currentneutrino scattering off isoscalar nuclei as predicted by the above-described model. The charmed quarkmass is chosen to be 1.25 GeV[71] and the scale2 = 4m2
c .
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 247
The valence quark parametrization in Ref.[72] has been assumed for the parton density andthe sea quark contribution has been neglected for simplicity. The Cabibbo factor is also neglected, i.e.cos2 C 1, while for the running coupling constant of QCD leading order formula with three flavourshas been assumed
s = 4
9 ln(M2/2)
with = 0.5 GeV2.After the confirmation of the early experimental results on same-sign dimuons by another experiment
[73], an ad hoc model to describe the data was proposed[74]. A scheme with a large non-perturbativeprobability for a quark jet, produced in a charged-current interaction, to fragment in acc pair was invoked.The fragmentation functionDu→cc was adjusted in such a way to account for the data. A couple of yearslater, a strong argument against this model was presented[75] which, at the same time, resolved thesame-sign dilepton puzzle.
In fact, it was pointed out that quark jet fragmentation would have induced charm pair produc-tion in hadron collisions as well, while the results of a Fermilab beam-dump experiment[76] limitedthe cross-section of such a charm source to 30b. By translating this limit into a dimuon signal, theupper limit
(−−)/(−) 10−4
was set.In spite of that, the kinematic characteristics of the dimuon events were compatible[73] with the
predictions of the leading-order gluon bremsstrahlung diagrams ofFig. 7. Thus the interpretation interms of such a process was not dismissed. Alternatively, a comprehensive list[75] of uncertainties in thecalculation of the same-sign dimuon cross-section was presented. The main contributions come from thethreshold parametermc, the choice of structure function parametrization, the scale of running couplingsand the fragmentationc → D, which amounted to a factor of about 60. This study thus explained theapparent disagreement between theory and experimental results. In doing so, the same-sign dileptonpuzzle was solved and the hypothesis of charmed quark pair generation process confirmed.
3.5.2. Associated charm-production in NC interactionsThe associated charm-production in NC interactions occurs via the so-calledZ0-gluon fusion
mechanism as illustrated inFig. 9. The general form for the neutrino-nucleon differential cross-sectionis given in Eq. (8). Here we will consider the contribution to the structure functionFi due to the sub-
processZ0g → cc. We indicate withF (Z0)
i the corresponding structure functions, which, at LO, aregiven by[77]
F(Z0)1 (x,Q2)=
∫ 1/a
x
dy
yg
(x
y, s
)f1(y,Q
2) , (67)
F(Z0)2 (x,Q2)=
∫ 1/a
x
dy
y
x
yg
(x
y, s
)f1(y,Q
2) , (68)
F(Z0)3 (x,Q2)= 0 , (69)
248 G. De Lellis et al. / Physics Reports 399 (2004) 227–320
Fig. 9.Z0-gluon fusion process.
wherea = 1 + 4m2c/Q
2, s = Q2(1 − y)/y, andg(x/y, s) indicates the gluon distribution function.For pair-production of equal-mass quarks the expressions forf1 andf2 are
f1(y,Q2)= s(s)
4q+−v
[(1− 2y)2 + 4y(1− y) q−
q+m2c
Q2
]+[1− 2y(1− y)+ 4y
(1− y − y 2m2
c
Q2
)q−q+m2c
Q2
]L
, (70)
f2(y,Q2)= s(s)
yq+
v
[−1
2+ y(1− y)
(4− 2m2
c
Q2
)]+ 1
2
[1− 2y(1− y)+ (1+ 2y − 6y2)
2m2c
Q2 − 2m2c
Q2
q−q+
− 8m4c
Q4 y2]L
, (71)
wherev = √1− (4m2
c/Q2)(y/(1− y)), L = ln(1 + v)/(1 − v), the vector (V) and axial vector (A)
couplings entervia q± = V 2 ± A2 = (12 − 4
3 sinW)2 ± 1
4. Note that, in Ref.[77], the leading orderapproximation fors(s) is used.
3.6. Charmonium production
In Section 3.5 we discussed the production of two charmed hadrons in NC and CC interactions throughthe creation of a charmed quark-antiquark pair. Here we discuss the case when the charmed quark-antiquark pair formscharmoniumstates.
TheJ/ state can be produced either directly or via radiative or hadronic decays of heavier charmonia,such ascJ and′ mesons, whereJ =0,1,2 (Fig.10). Theoretical calculations of the cross-section of thedirectJ/ production by neutrinos were made in the framework of QCD-basedZ0-gluon fusion[78–80]and Vector Dominance Model (VDM)[78,79,81]. In the VDM only the vector coupling,gV, of theZ0
boson to thec-quark contributes, while in theZ0-gluon fusion approach both the vector and the axial
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 249
Fig. 10. Feynman diagrams for production of charmonium states in weak neutral-current interactions: (a)Z0-gluon fusion model,(b) VDM model (diffraction).
vector couplings are at work. Moreover, one would expect the QCD-basedZ0-gluon fusion mechanismto dominate because of the numerical smallness of vectorial coupling to the charm (gcV) in comparisonwith the value of axial one,gcA (g
cA/g
cV)
2 ≈ 7. On the other hand, there are no predictions for the indirectJ/ production rate in this approach. Recently, the Non-relativistic QCD (NRQCD) approach[82] hasbeen used to evaluate diffractive[83] and non-diffractive[84] J/ production. The results of NRQCD[83] confirm the results of VDM calculations[78] that predict comparable contributions of direct andindirectJ/ production.
In the following we briefly discuss the cross-section calculation both in the QCD-basedZ0-gluonfusion and VDM models, and in the non-relativistic QCD approach.
3.6.1. Charmonium production through the boson–gluon fusion mechanismThe production of theJ/ in neutrino NC interactions through the boson–gluon fusion mechanism is
illustrated inFig. 10.For largeQ2, where QCD can be applied and we expect that the inelasticJ/ production increases
with respect to elastic production, free charmed quarks are produced through the processZ0g → cc.Local duality relates this inclusive charm-production to inclusive charmonium production by identifyingJ/ with a fixed fraction of freecc production in the invariant mass interval 2mc <Mcc <2mD. Such anapproach has been successfully applied, by considering also theg → cc process, to hadronic[85,86],muon[87–90]and photo[91,92] production ofJ/. Following the calculation in Ref.[78], the cross-section forJ/ production in neutrino NC interactions is given by
d2(N → J/N)
d dQ2 = 9G2F
4(4)3
[1+
(1− 8
3sin2 W
)2]Q2
(1− Q2
2M
)× 1+ (1− /E)2
f (E, ,Q2)(,Q2) , (72)
250 G. De Lellis et al. / Physics Reports 399 (2004) 227–320
where
f (E, ,Q2)= 1+ 16m2c
Q2
1
1+ (1− 83 sin2 W)
2
1− /E
1+ (1− /E)2(73)
and
(,Q2)= 1
n
3
4
9sm2D
m4c
(1−m2c/m
2D)
3/2
1−Q2/2M
g()
(1+Q2/4m2c)
2 , (74)
with = (Q2 +M2J/)/2M andg() is the gluon distribution function of the nucleon. The integern is
associated with the number of charmonium states in the mass range 2mc to 2mD [78].
3.6.2. The diffractive charmonium production in the VDMThe charmonium production cross-section for neutral-current interactions in the framework of VDM
has been derived in Ref.[78], where it is shown that in the limitQ2>2 the diffractive production ofJ/can be written as
d2(N → J/N)
d dQ2 = 9G2F(1− 8
3 sin2 W)2
4(4)3Q2
(1− Q2
2M
)1+ (1− /E)2
(,Q2) . (75)
The Vector Dominance hypothesis implies that
(,Q2)= ()
(1
1+Q2/M2J/
)2
, (76)
and, in fact, such aQ2 dependence has been found also experimentally[93,94], although the fitted massparameter is slightly smaller thanMJ/:MV =2.7±0.5 GeV[93] andMV =2.4±0.3 GeV[94]. In Ref.[78] the sensitivity of the cross-section on this parameter has been studied. Differently from the muonproduction, the neutrino one receives substantial contribution from the highQ2 region and, therefore, issensitive to the value ofMV
(N → J/N)|MV=MJ/ ≈ 1.5× (N → J/N)|MV=2.7 GeV .
Assuming equal hadronic cross-sections forJ/ and′, and the previous relation, the′ rate can beestimated from Eq. (75)
(N → ′N)|MV=M′
(N → J/N)|MV=MJ/≈ (′ → +−)
(J/ → +−)(N → ′N)
(N → J/N)
∣∣∣∣MV=2.7 GeV
≈ 0.18 , (77)
where, the ratio(′ → +−)/(J/ → +−) 0.12 [21] has been used to take into account thedifferent vector coupling toJ/ and′.
For its nature the VDM cannot predict thec1 neutrino production rate. However, in Ref.[78], undersome assumptions, the authors showed that thec1 neutrino production rate is larger (about a factor 3)thanJ/ and′ ones.
The inclusiveJ/ neutrino production cross-section can be written as
3.6.3. Charmonium production in the non-relativistic QCD approachThe charmonium production is a sensitive probe of the gluon distribution function in the nucleon
and provides a useful laboratory to test the interplay of perturbative and non-perturbative phenomena.More precisely, the production of charmonium states can be described in the factorization formalismof NRQCD [82]. In this formalism, the non-perturbative matrix elements are predicted to scale with adefinite power of the velocity,v, of the charm quark. So, the theoretical predictions are organized as adouble expansion ins andv. The neutrino neutral-current production of charmonium production in theframework of NRQCD has been studied in Ref.[83] for diffractive process and in Ref.[84] for diffractiveand non-diffractive cases. In NRQCD we can write[83]
(A+ B → Hcc +X)=∑n
Fn
mdn−4c
〈0|OHn |0〉 , (79)
where the short-distance physics, represented by the coefficientsFn (containing the perturbatively cal-culable contributions), is separated by the long-distance physics, represented by the NRQCD matrixelements,〈0|OHn |0〉 (which can be fixed from the experiments). Here the indexn is connected to thespectral decomposition of the charmonium states in terms of the quantum numbers2S+1L
(colour)J .
A major advantage of using the neutrino beam is that, at leading order ins , the spin structure of theZ0
coupling selects a certain combination of octet operators. In particular, for the process we are discussing,the largest contribution is from the operator with3S
(8)1 quantum numbers[83,84]. The differential cross-
section is given by[83]
d(s,Q2)
dQ2 = 22s
3 sin4 2W
1
(Q2 +m2Z)
2
∑n
〈0|On|0〉m3c
∫ 1
(Q2+4m2c)/s
dx g(x,Q2)hn(y,Q2) , (80)
wheres is the total invariant mass of theN system andy = (Q2 + 4m2c)/sx. g(x,Q2) represents the
gluon distribution function in the nucleon andhn(y,Q2) are the structure functions which can be foundin [83]. The numerical results shown inTable 2 [83]are obtained formc = 1.35 GeV,Q2>(1.2 GeV)2,and adopting the parton distribution functions in Ref.[95].
3.7. The Cabibbo–Kobayashi–Maskawa matrix
In the Standard Model of particle physics the quark mass eigenstates do not take part as pure states inweak interactions. The unitary matrix connecting the mass and weak eigenstates is the so-called CKM
252 G. De Lellis et al. / Physics Reports 399 (2004) 227–320
Table 2Total cross-sections for theJ/ production inN → J/X for various incident neutrino energies
(Cabibbo–Kobayashi–Maskawa) matrix[25,26]. By convention, the charge+2/3 quarks (u, c, t) arechosen to be pure eigenstates and the flavour mixing is described in terms of a 3× 3 matrix operating onthed, s andb quark states(
d ′s′b′
)= VCKM
(d
s
b
)=(Vud Vus VubVcd Vcs VcbVtd Vts Vtb
)(d
s
b
). (81)
Thusd ′, s′ andt ′ are the partners ofu, c andt , respectively, within the weak isospin doublets. On theproperties of the CKM matrix there are several textbooks to which we refer for more details[140,141].In the following we focus on its properties that are strongly related to heavy quark production induced byneutrinos. The CKM matrix can be parametrized in terms of three angles and one phase, being the latterresponsible for the CP violation in the Standard Model. There are several parametrizations of the CKMmatrix. In the following we used the one introduced in Ref.[142] and also adopted by the Particle DataGroup[21]
wherecij = cosij , sij = sinij , ij is the mixing angle between theith and thej th generation and isthe phase angle.
A common approximation that emphasizes the experimentally observed hierarchy
|Vub|< |Vcb|< |Vus |, |Vcd |<1
sets = s12 (the sine of the Cabibbo angle) and defines the parametersA, and through the relations[139]
s23 = A2, s13e−i = A3( − i) .
By neglecting terms of orderO(4), the CKM matrix can be written as
VCKM =( 1− 2/2 A3(1− − i)
− 1− 2/2 A2
A3(1− − i) −A2 1
). (83)
The values of individual matrix elements can be, in principle, obtained from weak decays of thecorresponding quarks, or, in some cases, from deep-inelastic neutrino scattering. Tight constraints on theelements of the CKM matrix can be obtained by requiring the unitarity and assuming only three quarkgenerations. It is worth noting that the constraint of unitarity connects different elements, so choosing aspecific value for one element restricts the range of others.
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 253
Among the orthonormality relations of the row-vectors and those of column-vectors of the CKM matrix(9 independent relations), there is one which is particularly interesting
VudV∗ub + VcdV ∗
cb + VtdV ∗tb = 0 . (84)
Indeed, Eq. (84) defines on the complex plane a triangle whose sides have dimensions of the same orderof and thus subtend angles having comparable amplitude. There are many papers on measurementsof the CKM matrix elements and on the check of the unitarity[21]. Here we do not to want to enterinto the details, but stress the fact that Eq. (84) contains the elementVcd . The latter can be either tightlyconstrained by imposing unitarity or measured directly. Such a measurement can only be done by deep-inelastic scattering of neutrinos off nucleons producing a charmed hadron in the final state. The possibilityto directly assessVcd is particularly interesting since a way to probe physics beyond the Standard Modelof particle physics is to check the unitarity of the CKM matrix. Namely, measure all matrix elementsand see whether the matrix is unitary or not. The experimental methods to extract|Vcd | and|Vcs | fromneutrino experiments are discussed in Section 6.3. Finally, we want to mention the possibility to extractfrom neutrino data|Vub| and|Vcb|. These measurements are not possible with present neutrino beams,but are one of the main physics topics at a Neutrino Factory[143].
3.8. Parton distributions with neutrinos
Valence- and sea-quark distributions cannot be separated in a direct way by charged lepton scatteringexperiments: the virtual photon just couples to the charge of the quarks independent of their flavour. Thisis not the case in neutrino and anti-neutrino scattering. The charged-current weak interaction takes placebetween members of the weak isospin doublets which are left-handed for particles and right-handed forantiparticles. Only selected flavour changes are allowed because of charge conservation (e.g.d → −u,c→ −s, etc.)
Interactions between particles of the same helicity (q, q) can be distinguished from those betweenparticles of opposite helicity (q,q) due to their energy dependence. In the latter case, backward scatteringin the center-of-mass frame is forbidden by angular momentum conservation and the cross-section isproportional to(1− y)2, wherey = (E − E)/E.
In terms of the individual neutrino-quark cross-sections the double differential cross section per nucleonfor scattering off an isoscalar target can be written as
d2
dx dy= G2
FME
(M2W
M2W +Q2
)2
2x[(u+ d + 2s)+ (1− y)2(u+ d + 2c)] ,
d2
dx dy= G2
FME
(M2W
M2W +Q2
)2
2x[(u+ d + 2s)+ (1− y)2(u+ d + 2c)] .
As it can be seen for instance in the NuTeV data inFig. 11, for neutrino scattering the distribution isflat in y with a small(1− y)2 admixture arising from the sea-quarks, while for anti-neutrino scatteringthe distribution is dominated by the(1− y)2 term from the interaction with the valence quarks, a smally
dependence part arises from the sea-quarks. Valence and sea-quark distributions can directly be obtainedby extrapolation toy = 0 and 1. The same cross-sections can be written also in terms of three structurefunctions as shown in Eq. (7).
254 G. De Lellis et al. / Physics Reports 399 (2004) 227–320
0.5
1
1.5
2
x=0.275
y y
0.5
1
1.5
2
x=0.350
y y
0.5
1
x=0.450
y y
0.5
1
x=0.550
y y0 0.2 0.4 0.6 0.8
x=0.650
y yy yy0 0.2 0.4 0.6 0.8 1
NuTeV Diff. Cross Section Data (E=95 GeV)
Neutrino Anti-Neutrino
CCFR DataNuTeV Data
QCD Inspired Fit
Fig. 11. An example of they distribution from NuTeV data.
The structure functionF3 is a consequence of theV − A structure of the weak charged current. Theterm withF3 has a positive sign for neutrino scattering and a negative sign for anti-neutrino scattering.Assuming 2xF 1 =F2 (Callan–Gross relation), one obtains by comparison of the previous equations, theexpression for the structure function per nucleon for an isoscalar target in terms of quark distributions
xFN3 (x)= 12(xF
N3 (x)+ xF N
3 (x))= x(u+ d − u− d)= x(uv + dv) ,F N
2 (x)= F N2 (x)= x(u+ u+ d + d + s + s + c + c) 18
5 FN2 (x) .
The quark, antiquark and valence quark distributions are separately measurable by appropriate combina-tions of neutrino and anti-neutrino cross-section. The valence quark distribution from
d
dx− d
dx,
while the sea-quark distribution from
3d
dx− d
dx.
The universality of parton distributions has been studied by comparing neutrino and charged-leptonscattering data. Several measurements in the past have indicated thatF
2 differs fromFe/2 by 10–20% inthe low-x region. Recently the CCFR/Nu TeV Collaboration has reported the analysis of CCFR-Fe and-Fe differential cross-sections to extract the structure functions[112]. The neutrino-muon differencehas been resolved by extracting the structure functions in a physics model independent (PMI) way.
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 255
Fig. 12. The ratio ofF 2 (PMI) data divided byF
2 (NMC or BCDMS) orFe2 (SLAC). Both statistical and systematic errors areincluded. Also shown are the predictions of the TR-VFS (MRST99), ACOT-VFS (CTEQ4HQ) and FFS (GRV94) heavy flavourcalculations.
In previous analyses[108] of data, light flavour universal physics model dependent (PMD) structurefunctions were extracted by applying a slow rescaling correction to correct for the charm mass suppressionin the final state. In addition, thexF 3=xF
3−xF 3 term (used as input in the extraction) was calculated
from a leading order charm-production model. Recent calculations[33,109,110]indicate that there arelarge theoretical uncertainties in the charm-production medelling for bothxF 3 and the slow rescalingcorrections. Therefore, in the new analysis reported here, slow rescaling corrections are not applied, andxF 3 andF2 are extracted from two-parameter fits to the data. The values ofxF 3 to various charmproduction models have also been compared. The extracted PMI values forF
2 are then compared withF
2 within the framework of NLO models for charm-production.Fig. 12shows the ratio of the CCFR/NuTeVF
2 (PMI) measurements[112]divided by(18/5)F 2 (NMC
[113]or BCDMS[114]) or(18/5)F e2 (SLAC[115]) measurements[116]. The overall normalization errors
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of 2% (CCFR), and 2.5% (NMC) are not shown. Within 5%, the ratio is in agreement with the predictionsof the TR-VFS (MRST99), ACOT-VFS (CTEQ4HQ), and FFS (GRV94) calculations[117].
In conclusion, the ratio ofF2 (PMI) values measured in neutrino-iron and muon–deuterium scatteringare in agreement with the predictions of next-to-leading-order PDFs (using massive charm-productionscheme), thus resolving the long-standing discrepancy between the two sets of data.
4. Overview of the experiments
This section gives an overview of the past and present experiments that studied charm-productioninduced by (anti-)neutrinos, the collected charm samples and the analysis techniques exploited to studythe charm signal.
4.1. Electronic detector experiments
Many experiments have studied neutrino and anti-neutrino charm-production by looking at the presenceof two oppositely charged leptons in the final state: one produced at the interaction vertex (the primarymuon) and the other one at the decay vertex (the secondary lepton). A schematic view of the productionof the so-called dimuon events is given inFig. 13. This technique was firstly used by the HPWF in 1974[144] when the neutrino induced charm-production was discovered.
In the case of neutrino scattering, the underlying process is a neutrino charged-current interaction off as or d quark, producing a charm-quark that fragments into a charmed hadron. The charmed hadron maydecay semi-leptonically producing opposite sign dileptons through the process:
+ N −→ − + c + X→ s + l+ + l . (85)
Analogously an anti-neutrino can interact off as or d anti-quark, producing a charm anti-quark thatfragments into a charmed hadron, again leading to a final state with two oppositely charged leptons.
Fig. 13. Dimuon production in-nucleon DIS from scattering off a strange or down quark (LO QCD charm production).
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Table 3Available (anti-)neutrino dilepton data samples from electronic experiments. The samples are background subtracted
Since 1974, several experiments[145–156]have used this technique to study charm-production. Theycan be separated in two classes: calorimetric and bubble chamber experiments, exploiting the muonic andelectronic decay of the charmed hadron, respectively.
Calorimetric experiments are characterized by a massive iron target and a muon spectrometer to identifythe muon and measure its charge. For these experiments pion and kaon decays constitute the mainbackground. The high density of the target calorimeter minimizes this background due to the shortinteraction length of the detector. A further background reduction is obtained by requiring a minimummomentum, typicallyp>5 GeV, for the less energetic muon. The drawback of such a selection is thatthese experiments are not able to search for charm-production at relatively low neutrino energies. For atypical calorimetric experiment it is not possible to investigate energy regions below 15 GeV, where theslow-rescaling threshold effect is more important.
The main characteristic of a bubble chamber filled with a mixture of heavy liquids (Ne–H2, freon–propane) is its high efficiency in identifying electrons. Therefore, they searched for charm-productionby looking at−e+ events. In these experiments, the low threshold,pe+ >0.3 GeV, combined with highstatistics forE<30 GeV, gives good sensitivity to the slow-rescaling threshold behaviour. The mainbackground sources for these searches are0 Dalitz decays andeCC interactions.
Table 3summarizes the available (anti-)neutrino dilepton data samples.
4.1.1. The HPWF experimentThe HPWF experiment[157,158]was the first application of a calorimeter as target and detector for
neutrino interactions. This technique is particularly appropriate to detect the hadrons produced in neutrinointeractions where the small cross-section dictates the use of very thick targets which absorb the hadronshowers, but where much interesting physics does not depend upon detailed knowledge of the hadrons.
The detector consisted of four main sections in series along the beam axis, each main section had across-sectional area of 3×3 m2 and length along the beam of 1.8 m, and contained 15 ton of mineral-oil-based liquid scintillator. There were wide-gap optical spark chambers of area 3× 3 m2 after each mainsection of the ionization calorimeter. The hadronic energy resolution of the calorimeter was measured
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to be 10–12% in the energy range 10–150 GeV. Immediately downstream of the calorimeter there wasa magnetic spectrometer made up of four units of toroidal iron magnets, with narrow-gap optical sparkchambers following each magnet unit. The momentum resolution varied from 10% to 15% between 5and 150 GeV. The target for neutrino interactions had a fiducial mass of 60 ton of liquid scintillator in thecalorimeter plus another 60 ton of iron (part of the first unit of the magnetic spectrometer).
4.1.2. The CDHS experimentThe CDHS experiment[159–161]was built in the late 1970s with the aim of studying neutrino inter-
action properties by exploiting the CERN super proton synchrotron (SPS) WANF neutrino beam[162].It consisted of toroidally magnetized iron plates sandwiched with scintillator planes. The detector wasgrouped in 21 modules of three types: modules 1–10 with 2.5 cm, modules 11–15 with 5 cm and modules16–21 with 15 cm thick iron plates. The total iron thickness of each module was 50 cm for modules 1–10and 75 cm for the others. Each module had a diameter of 3.75 m. The total mass of the detector was∼ 1100 ton for a total length of∼ 21 m. In between the modules triple plane drift chambers were insertedfor muon tracking. The momentum of the muon was determined from the curvature of its track in themagnetic field of the detector, with an average resolution of 9%. The hadronic energy resolution wasmeasured to be∼ 0.58/
√Ehad for modules 1–10 and∼ 0.70/
√Ehad for modules 11–21.
The magnetized iron modules of the CDHS apparatus acted as muon spectrometer, as hadronic calorime-ter and as neutrino target. This integration of functions was a novel feature of this detector. Its mainadvantage lies in the acceptance solid angle for the muon spectrometer.
4.1.3. The CHARM II experimentThe principal aim of the CHARM II experiment[163,164]was a high statistics study of the reaction
()+ e− → ()+ e−for the determination of the Standard Model parameters in the leptonic sector. The detector consistedof a massive, low-density, fine-grained calorimeter (700 ton in weight and 35 m long) and of a muonspectrometer. The basic building element of the calorimeter was a module made of a 4.8 cm thick glasstarget (0.5X0 or 1/9int) followed by a plane of plastic streamer tubes. Behind each group of five modules,a plane of 3 cm thick, 15 cm wide and 3 m long plastic scintillator counters, was inserted to measure dE/dxfor e/0 discrimination.
The lowZ material and the granularity of the calorimeter ensured the required angular resolutionfor electron showers. The electromagnetic energy resolution achieved by using the digital signal of thestreamer tubes was(E)/E = 0.15/
√E(GeV) + 0.09. The spectrometer consisted of magnetized iron
toroids interleaved with scintillator counters and drift chambers. It measured the muon momentum with aresolutionp/p=13% at 20 GeV. The layout of the detector is shown inFig. 14.A dimuon event observedin the CHARM II experiment is shown inFig. 15to illustrate the fine granularity of the calorimeter.
4.1.4. The CCFR/NuTeV experimentThe CCFR Collaboration has performed a series of experiments (E616, E701, E744, E770 and E815)
at Fermilab using the Lab E neutrino detector spanning a period from 1979 through 1997[165,166].The Lab E detector, shown inFig. 16, consisted of two major parts, a target calorimeter and an irontoroid spectrometer. The target calorimeter contained 690 ton of iron sampled at 10 cm intervals byeighty-four 3 m× 3 m scintillator counters and at 20 cm interval by forty-two 3 m× 3 m drift chambers.
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 259
Fig. 14. Schematic view of the CHARM II detector.
Fig. 15. Display of a dimuon event in the CHARM II detector. The two views refer to the horizontal and vertical projections.
The measured calorimeter hadronic energy resolution was/Ehad= 0.89/√Ehad and an absolute scale
uncertainty of Ehad/Ehad= 0.5%. The toroid spectrometer consisted of five sets of drift chambers aswell as hodoscopes for triggering. The spectrometer energy resolution was limited by multiple Coulombscattering top/p = 0.11. The muon momentum scale was known to E/E = 1.0%.
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Fig. 16. Schematic view of the Lab E neutrino detector showing the target calorimeter followed by the downstream muonspectrometer.
ChambersMuon
Beam
Neutrino
V8
CalorimeterHadronic
1 meter
PreshowerModulesTRD
Dipole Magnet⊗ B = 0.4 T
Trigger Planes
ElectromagneticCalorimeterDrift Chambers
CalorimeterFront
Veto planes
x
y
z⊗
Fig. 17. A side view of the NOMAD detector.
4.1.5. The NOMAD experimentThe NOMAD experiment was designed to search for → in the CERN SPS wide-band neutrino
beam[167]. A side view of the detector is shown inFig. 17. An iron-scintillator hadronic calorimeter,denoted as front calorimeter (FCAL), was located upstream of the central part of the NOMAD detectorand formed the target for charm studies. It consisted of 23 iron plates, 4.9 cm thick, separated by 1.8 cmair gaps. The gaps were instrumented with plastic scintillators read at both ends by photomultipliers. Thetotal instrumented region had a mass of 17.7 tons and is about 5 nuclear interaction lengths thick. Theenergy resolution of the FCAL was/E = 100%/
√E(GeV).
Given the coarse granularity of FCAL, neutrino induced charm-production occurring therein wasstudied by looking at the dimuon events. A typical dimuon event reconstructed in the NOMAD FCALdetector is shown inFig. 18. FCAL was followed by a tracking detector consisting of 44 drift chambers
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Fig. 18. Example of an opposite sign dimuon event occurring in the NOMAD Front Calorimeter. The two tracks are the oppositelycharged muons in the event and the detector is shown from the side. The grey shading in FCAL indicates different level of energydeposition.
situated in a 0.4 T magnetic field. The position resolution for tracks at normal incidence was 150m (fordetails of its dependence on the incidence angle and on the drift distance we refer to[167]). For chargedhadrons and muons travelling normal to the plane of the chambers, the momentum resolution can beparametrized as:p/p ∼ 0.05/
√L⊕0.008p/
√L5. The good tracking and momentum resolutions were
exploited to reconstruct the invariant mass of charmed hadrons (see Section 5.8). A transition radiationdetector, to enhance the electron–pion separation, and an electromagnetic calorimeter, that consisted ofan array of lead-glass Cerenkov detector, followed the tracking detector. The energy resolution of theECAL was/E = 1%+ 3.2%/
√E(GeV).
A hadronic calorimeter (HCAL) with an energy resolution of/E = 100%/√E(GeV) was installed
behind the magnet coil and was followed by two muon detection stations consisting of large area driftchambers, the first after 8 and the second after 13 nuclear interaction lengths.
4.1.6. Bubble chamber experimentsBubble chambers belong to the class of visual detectors. Indeed, once an event occurs inside the
detector, images are taken allowing the reconstruction of events with high complexity with a millimeteraccuracy.
The working principle of a bubble chamber is rather simple. A liquid gas (H2, D2, Ne, C3H8, Freon,etc.) is held at pressure close to its boiling point. Before the expected event occurs, the chamber volumeis expanded leading to a pressure reduction thereby exceeding the boiling temperature of the bubblechamber liquid. If in this superheated liquid state a charged particle enters the chamber, bubbles are formedalong the particle track. The seeds for bubble production are given by the positive ions produced by theincident particle. Being the ion lifetime very short, 10−11–10−10 s, the expansion of the chamber cannot
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Fig. 19. Charm-production and subsequent decay as observed in a bubble chamber.
be triggered on the incoming charged particle. Therefore, bubble chambers are suited for experimentswhere the arrival time of particles is known and, consequently, the chamber can be expanded in advance.Depending on the size of the chamber repetition times down to 100 ms can be obtained.
In the superheated state the bubbles grow until the growth is stopped by a termination of the expansion.At this moment the bubbles are illuminated by light flashes and photographed.
In order to measure the momentum of charged particles, bubble chambers were operated in highmagnetic fields. The particle identification (mainly/ separation) was only possible by using additionalexternal detectors.
Bubble chamber experiments have made significant contributions to the study of deep-inelastic charm-production by neutrinos. The observation of charm-production by neutrinos helped to establish the exis-tence of charmed particles and provided evidence of the preferential coupling of charm to strangeness.
The majority of these studies were carried out in the decade between 1970 and 1980, primarily atCERN in Europe, Fermilab, Brookhaven, and Argonne in the USA, and several Russian laboratories byusing a variety of targets.
For a detailed review we refer to the Proceeding of the “Conference on Bubble Chamber and itsContributions to Particle Physics”[168].
An example of charm-production and subsequent decay in a bubble chamber is shown inFig. 19.
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4.2. Emulsion experiments
So far only two experiments, E531[169] and CHORUS[170], have searched for inclusive charm-production through the direct identification of charm decays in the emulsions. The main advantage ofthese experiments is that, being the charmed particle clearly identified through its decay, very loosekinematical cuts are applied. This translates into a very good sensitivity to the slow-rescaling thresholdbehaviour and consequently to the charm-quark mass.
The above experiments have a hybrid design. Electronic detectors predicted the regions in emulsionswhere the neutrino interactions occurred and contributed to the reconstruction of the event kinematics.Emulsions were used as active targets. They have the appropriate position resolution (less than 1m)and granularity to detect short-lived particles through the visual observation of their decays. The mainbackground forD0 detection in emulsion comes fromK0 anddecays, and neutron,K0 and interactionswithout any visible nuclear break-up at the interaction vertex. For charged charmed-hadrons the mainbackgrounds are andK decays in flight, and thewhite kink(hadron interaction without any visiblenuclear break-up) on any charged non-charmed hadron. These background processes can be easily ruledout by applying a cut on the transverse momentum at the decay vertex(pT >250 MeV). The contributionto the background of these processes is of the order of 10−4/CC.
4.2.1. The E531 experimentThe E531 experiment[171] was proposed to study the properties of charmed particles and their pro-
duction mechanism in neutrino interactions. For a description of the experiment we refer to[172].The neutrino beam was produced by 350 GeV protons for the first exposure (for a total of 7.2 ×
1018pot)6 and by 400 GeV protons for the second one(6.8×1018pot). The beam composition averagedover both runs was 92.3% , 7.0% , 0.5% e and 0.2% e. The neutrino target consisted of nuclearemulsions where neutrinos interacted and short lived particles were precisely measured with micrometricaccuracy.An electronic spectrometer detected the decay products. The hybrid detection technique adoptedby E531 was further developed for the CHORUS experiment described in Section 4.2.2.
The emulsion target consisted of 22.6 l in the first run and of 30 l in the second. It was made of modulescomposed of a series of sheets made up of a 300m emulsion layer coated on both sides of polystyrenefoils of 70m thickness. Immediately downstream of the emulsion modules two large lucite sheets800m thick coated on both sides with 75m emulsion layers were installed. These interface detectors(changeable emulsion sheets) ease the extrapolation of particle tracks predicted by the electronic detectorsinto the emulsion target. The changeable sheets are replaced every two or three days of data taking, inorder to limit the total number of accumulated background tracks.
Downstream of the target, a magnet was equipped with high resolution drift chambers (x ∼ 150m,1 mrad), providing track predictions into the emulsions and tracking through the magnet. The mo-
mentum resolution wasp/p =√(0.014)2 + (0.004p)2. In addition, a time-of-flight detector made of
two scintillator planes situated 2.7 m apart yields a time resolutiont1 ns. The setup was complementedby a lead glass array and a hadron calorimeter followed by a muon spectrometer. More details on theapparatus can be found in Ref.[173].
A total of 3886 neutrino interactions were located in the fiducial volume of the target.
6 pot≡proton on target.
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Fig. 20. General layout of the CHORUS detector.
4.2.2. The CHORUS experimentThe CHORUS experiment was designed to search for → oscillations in the SPS Wide Band
Neutrino Beam at CERN through the direct observation of the decay. The West Area Neutrino Facility(WANF) of the CERN SPS provides an intense beam of with an average energy of 27 GeV, well abovethe production threshold(E = 3.5 GeV). At the average beam energy, the lepton travels, on average,about 1 mm before decaying. In order to observe the, the experiment was designed hybrid, namely withelectronic detectors to complement the reconstruction of the event kinematics and with nuclear emulsionsused as active target.
Since charmed particles and the lepton have a comparable flight length, the experiment was suitableto study charm physics as well. With 770 kg and four years exposure, CHORUS accumulated a very large(∼ 104) number of charmed interactions in emulsions which makes it statistically compelling even withelectronic detector experiments. On the other hand, unlike electronic detectors, nuclear emulsions allowthe direct identification of charmed particles through the visual observation of their decay.
A schematic picture of the CHORUS apparatus is shown inFig. 20.Thehybridsetup, described in detail in Ref.[170], was made of an emulsion target, a scintillating fibre
tracker system, trigger hodoscopes, a magnetic spectrometer, a lead-scintillator calorimeter and a muonspectrometer.
The nuclear emulsions acted as the target and, simultaneously, as the detector of the interaction vertexand of the lepton decay[174]. The emulsions were subdivided in four stacks of 36 plates, orientedperpendicularly to the beam and with a surface of 1.44 × 1.44 m2. Each plate is made of a 90mtransparent plastic film with 350m emulsion sheets on both sides.
The nuclear emulsion target was equipped with a high resolution tracker made out of interface emulsionsand scintillating fibres planes. Each stack was followed by three special interface emulsion sheets: twochangeable sheets (CS), close to the fibre trackers, and a special sheet (SS), close to the emulsion stack.The sheets had a plastic base of 800m coated on both sides by 100m emulsion layers. Eight planes oftarget trackers of scintillating fibres (500m diameter)[175], interleaved between the emulsion stacks,reconstructed the trajectories of the charged particles with a precision of 150m in position and 2 mradin angle at the surface of the CS. The layout of an emulsion stack is sketched inFig. 21.
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Fig. 21. Layout of a CHORUS emulsion stack and associated fibre trackers.
Downstream of the target region, a magnetic spectrometer was deputed to the momentum and chargereconstruction for charged particles. An air-core magnet[176] of hexagonal shape produced a pulsedhomogeneous field of 0.12 T. Field lines were parallel to the sides of the hexagon and the magnetizedregion extended for a depth of 75 cm in the direction of the beam. The tracking before and after themagnet was performed by a high resolution detector made of scintillating fibres (500m diameter) andcomplemented with few planes of electronic detectors (streamer tube chambers in the 1994, 1995 andthe beginning of 1996 runs, honeycomb chambers[177] afterward). The resulting momentum resolutionp/p was 30% at 5 GeV.
In addition to the detection elements described above, the air-core hexagonal magnet region wasequipped with large area emulsion trackers for the 1996 and 1997 runs. The aim was to perform a moreprecise kinematical analysis of the decay candidates.
A 100 ton lead-scintillating fibre calorimeter[178] followed the magnetic spectrometer and mea-sured the energy and direction of electromagnetic and hadronic showers, together with a lead-scintillatorcalorimeter.
A muon spectrometer made of magnetized iron disks interleaved with plastic scintillators and trackingdevices was located downstream of the calorimeter. A momentum resolution of 19% was achieved bymagnetic deflection for muons with momenta greater than 7 GeV. At lower momenta, the measurementof the range yielded a 6% resolution.
Using nuclear emulsions, CHORUS could detect the charm through the direct observation of its decay.The observable decay topologies are classified as odd-prong decays of a charged particle (mainlyD+,D+s , +
c ) or even-prong decays of a neutral particle (mainlyD0). These are denoted as V2, V4 or V6 forneutral and C1, C3 or C5 for charged decays according to the multiplicity.
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Table 4Charm sample as observed by the CHORUS experiment[196]
Charm topology Observed events
C1 461V2 841C3 501V4 230C5 23V6 3
Events are classified according to their decay topology as explained in the text.
By applying the charm search to 95450 CC located interactions, they have observed 2059 charmcandidate events, the largest charm sample observed so far, as reported inTable 4 [196].
4.3. Analysis in electronic and bubble chamber experiments
The data selection is similar for all calorimetric experiments, although the kinematical cuts can beslightly different. In the following we describe qualitatively the typical data selection flow:
• the event must occur in coincidence with the beam and fire the penetration trigger (CC interactiontrigger);
• a fiducial cut is applied in order to ensure both the longitudinal and transverse containment of thehadronic shower;
• both muons have to be well reconstructed, with the closest approach between them being less than acertain distance (typicallyO(10 cm)). This cut allows to reject overlays of two CC events as well asobvious muons from the decay of shower hadrons.
The dilepton events are categorized as originating from an incident neutrino or anti-neutrino by assum-ing that the primary muon (the muon produced at the leptonic vertex) is the one with the largest transversemomentum with respect to the beam direction.
For each event, the reconstructed muon parameters at the vertex ( pi ,Ei , ri = pi/|pi |, i= 1,2) and theshower energyEhad, are used to compute the following kinematical variables:
vis(E1− p1· i), the visible negative four-momentum transfer squared, where p1 is the leadingmuon 3-momentum and i represents a unit vector parallel to the beam direction;
• vis = E2 + Ehad, the visible energy transferred to the hadronic system;• xvis =Q2
vis/2Mvis, the visible Bjorkenx;• yvis = vis/E
vis, the visible Bjorkeny.
In addition to the topological cuts listed above, a set of kinematical cuts is also applied. Both muons arerequired to have an energy greater than 4–10 GeV, depending of the experiment, essentially to reduce the
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E1 (GeV)
Ent
ries
/10
GeV
E2 (GeV)
Ent
ries
/ 2 G
eV
Ehad
vis (GeV)
Ent
ries
/5G
eV
E νvis (GeV)
Ent
ries
/ 5 G
eV
0
100
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400
500
600
700
0 50 100 150 2000
100
200
300
400
500
600
0 10 20 30 40 50
0
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0 25 50 75 100 1250
50
100
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300
350
0 100 200 300
Bjorken Y
Ent
ries
/ 0.0
5 un
its
Bjorken X
Ent
ries
/ 0.0
5 un
its
0
50
100
150
200
250
0 0.2 0.4 0.6 0.8 10
100
200
300
400
500
600
0 0.2 0.4 0.6 0.8 1
(a) (b)
(c) (d)
(e) (f )
Fig. 22. Comparison of the shapes of the kinematical variables of opposite sign dimuons (histogram) and data (point) as measuredin NOMAD [156]. Shown are the distributions of (a) energy of the primary muons, (b) the energy of the secondary muon candidate,(c) the visible hadronic energy, (d) the visible neutrino energy, (e) Bijorkeny and (f) Bijorkenx. The distributions are normalizedto equal area.
meson decay background. The visible hadronic energy is required to be more than 5–20 GeV to improvethe hadronic energy reconstruction. As an example, we show inFig. 22 the kinematical variables asreconstructed in the NOMAD experiment[156].
The bubble chamber experimental search for charm-production relies upon its high efficiency in iden-tifying electrons. Therefore, in order to identify−e+ dilepton candidates, the film is scanned for allevents with ane+ coming from the vertex with a momentum greater than 300 MeV. To be identified as an
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Fig. 23. Schematic drawing of the first evidence for the production and decay of short lived “X-particles”, now known as charmedparticles, in cosmic rays.
e+, the track is required to exhibit at least two characteristics of positrons in heavy liquid. The signaturesare:
• bremsstrahlung with a → e+e− conversion;• spiralisation;• production of a ray with energy comparable to the primary track;• annihilation with two → e+e− conversions.
After being scanned, measured and reconstructed, each event containing ane+ track is carefully anal-ysed, a fiducial volume cut applied and the search for at least one leaving negative (L−) track performed.The highest momentumL− track in each event is interpreted as a muon leaving the chamber. Under thisassumption, a kinematical analysis is applied in order to select genuineCC interactions.
4.4. Nuclear emulsions in hybrid experiments
Nuclear emulsions are a very old particle detector, especially suitable for the observation of short-livingparticles due to their submicron resolution. Their use notably led to the discovery of the pion[179] in 1947and to many important contributions to the development of elementary particle physics. Many cosmicray experiments were carried out after the Second World War and until the 1960s, contributing to thediscovery of strange particles in cosmic rays (see Ref.[180] and references therein).
Among cosmic ray interactions detected in ECC (emulsion cloud chamber, i.e. a sandwich of nuclearemulsions and passive material), a very peculiar event (seeFig. 23) was observed in 1971[181], three
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Fig. 24. Schematic drawing of the first hadro-producedBB pair event observed in nuclear emulsions by WA75.
years before the discovery of charm with the observation of theJ/ [182–184]. The event is due to aneutral primary in the energy range of 10 TeV. The short track segment originating from the interactionvertex was attributed to an “X-particle” of mass 2–3 GeV decaying after∼ 10−13s into a charged particleand a0 meson. The event was interpreted as the first example of the associated production of a massiveshort-lived particle with a new quantum number. After this observation, further examples ofX-particles(singly or pair produced) were then detected or dug out from previous cosmic ray exposures, supportingthe above interpretation[185]. By the time of theJ/ discovery and of the identification of theX-particlewith a charmed meson, about 20X-particle decays had been observed by using the ECC technique.
After the discovery in 1977[186]of a new heavy quark, the beauty, experiments with nuclear emulsionsaimed at the direct observation of production and decay of beauty hadrons. A successful search was firstperformed by the WA75 experiment at CERN, using a− beam of 350 GeV. In the WA75 event, shownin Fig. 24 [247], both beauty hadrons are observed to decay into a charmed particle. Nine beauty eventswere later observed by the E653 experiment[187].
The use of nuclear emulsions in neutrino experiments was firstly introduced in the E531 experimentto search for particles with lifetimes in the range 2× 10−15–3× 10−12 s and for → oscillationsearches. The similar lifetime of the lepton and charmed hadrons makes it possible indeed to searchfor both particle species in the same experiment. E531 was the first neutrino “hybrid” experiment: theemulsion analysis was fully manual and therefore it was necessary to restrict as much as possible theeye-scanning area by using other detectors: hence the idea to combine electronic devices and nuclearemulsions together.
4.5. Analysis
In hybrid experiments, the electronic detector is typically made of a tracking device and a calorimeter.The tracking device is associated to the emulsion target for the reconstruction of particle directions aimingat predicting with a suitable accuracy (typically of the order of a few mm or less) the position of one ormore particles in the event on the downstream part of the emulsion target. The required tracker precisiondepends on the event density in the emulsions. It is important to notice that the density in emulsion is alsolimited by the event physical overlap. On the other hand, the calorimeter is deputed to the measurementof kinematical global variables like the visible neutrino energy. In addition to these devices, it is usefulto have magnetic spectrometers to measure the particle momenta.
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Mean 0.3707
Position residual (µm)
0
2000
4000
6000
0 0.2 0.4 0.6 0.8 1
Fig. 25. Position residual achieved in automatic measurements in the CHORUS nuclear emulsions.
Once the particle prediction has been found in the emulsions, it is followed along its track up to thevertex. Neutrino interactions are recognized by the absence of a charged primary particle.
As an example of the impressive position resolution achievable in nuclear emulsion (about 0.4m), theresidual distribution in the position measurement of penetrating tracks in the CHORUS nuclear emulsionsis shown inFig. 25.
This hybrid technique was successfully used in the E531 experiment and about 4000 neutrino interac-tions were fully reconstructed. As already described in Section 4.2.1, a total number of 122 interactionswith charm production was observed.
Nowadays a revival of the emulsion technique is due to an impressive development in the field ofautomatic scanning microscopes, which makes the analysis of several hundred thousands events feasiblein about one year.
The first fully automated system was developed in Japan[188] in the early 1990s. After that a lot ofdevelopments were done in order to improve the speed of the system profiting of the developments inelectronics, like parallel processors working at the same frequency[189].
The impressive increase in scanning speed achieved allows the scanning of large areas to locate eventsand to study their topology. In addition, it makes it possible to perform automatically measurements ofkinematical parameters such as the particle momentum or electron identification through the measurementof the multiple Coulomb scattering[190]. In addition, electron energy measurements on the basis of theshower development can also be made.
It has to be noted that the visual inspection of the nuclear emulsions is still needed only for a limitednumber of events. A significant amount of the events (more than 95%) are fully analysed by automaticscanning procedures while only the remaining part is checked by eye. The visual inspection task is tocheck the decay topology and assess the nature of the secondary process: either a decay (signal) or aninteraction (background). Indeed the visual check allows the discrimination between interactions anddecay: the remarkable element in case of a secondary interaction is the presence of one or more heavyionizing tracks or a blob indicating either a visible nuclear recoil or Auger electrons. The remaininginteractions, “white kinks”, are indistinguishable from decays and therefore they constitute an irreducible
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 271
background source. Nevertheless, these interactions occur with a much longer interaction length andare characterized by a steeply falling transverse momentum distribution. White kinks can therefore berejected by requiring a large decaypT . A cut on the daughter momentum is also relevant to have controlover the background sources.
The white kink background source can be controlled by looking at a region where the signal is notpresent, i.e. far away from the primary interaction vertex where charm decays are negligible. A recentexperimental determination[191] has shown that, with a 250 MeV transverse momentum cut, the inter-action length for this process is about 17 m. As far as the Monte Carlo simulation of these processes isconcerned, a large effort has been put into this field recently and dedicated algorithms are now available[192] which reproduce the data with a 20% error.
5. Experimental results on charm-production rates
In the following sections we discuss all available measurements (inclusive and exclusive) on charm-production cross-sections and decay branching ratios. When several results are available, we combinethem in order to reduce the statistical error. All results obtained in the past by using out-of-date branchingratios have been recomputed by using the latest PDG results[21].
5.1. Charm-production studies with dileptons
Several experiments measured the dilepton cross-section spanning a large range of neutrino energies(0–600 GeV). The results quoted in the following have been reported in Ref.[193]. In order to combineall dilepton data (seeTables 5–7) it is necessary to make a bin by bin weighted average. Since differentexperiments have different binning, weighting criteria were defined to combine them all into an uniquere-binning of the neutrino energy range: 0–600 GeV divided into 60 bins. The data-point in thejth bin((Xj , Yj ), j ∈ 1, . . . ,60) is calculated once it was assigned a given weight to the data-point(xik, yik)
of thekth bin of theith experiments, here-after called source bin, according to the following criteria:
• the wider the source-bin(xik), the smaller the weight;• the wider the error(yik), the smaller the weight;• the larger the fractionfijk of the jth bin covered by the source bin, the larger the weight.
It is then possible to make a bin by bin weighted average of the data-points by using the followingformulas:
• data-point weight:
ijk =(
fijk
yik · xik)2
,
• data-point average:
Yj =∑i,k yik · ijk∑i,k ijk
,
272 G. De Lellis et al. / Physics Reports 399 (2004) 227–320
Table 5Dilepton neutrino cross-section, normalized to the CC cross-section, as a function of the neutrino energy, for some electronicdetector experiments[193]
Since CCFR data do not report the energy bins, it is not possible to combine them with the others atthis stage of the analysis.
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 273
Table 6Dilepton neutrino cross-section, normalized to the CC cross-section, as a function of the neutrino energy, for some electronicdetector experiments[193]
Gargamelle I 〈E〉 (GeV) l−l+/CC Gargamelle II 〈E〉 (GeV) l−l+/CC
Table 7Dilepton anti-neutrino cross-section, normalized to the CC cross-section, as a function of the neutrino energy, for some electronicdetector experiments[193]
Once the average is computed, the CCFR data are combined with the other ones in the following way:each CCFR data-point is assigned to the bin containing itsx-value, then a standard weighted averagebetween the CCFR data-point and the previous combined data is calculated. In this case, the inverse ofthey squared is used as weight.
274 G. De Lellis et al. / Physics Reports 399 (2004) 227–320
Table 8Average neutrino dilepton cross-section, normalized to CC cross-section, as a function of the neutrino energy[193]
Energy (GeV) l−l+/CC (%) Energy (GeV) l−l+/CC (%) Energy (GeV) l−l+/CC (%)
The average dilepton single-charm production cross-sections for both and are reported inTables 8and9, respectively, and shown inFig. 26.
5.2. Fully neutralD0 decay mode
The CHORUS experiment has recently reported the first measurement of the fully neutralD0 decaymode[196]. This quantity is poorly known[21] and its importance relies upon the fact that the measure-ment of fundamental quantities (i.e. the CKM matrix elementVcd , see Section 6.3) depends on it.
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 275
0
0.2
0.4
0.6
0.8
1
1.2
0 50 100 150 200 250 300 350 400
All di-lepton data
Eν(GeV)
σ -+/σ
cc(%
)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 50 100 150 200 250 300 350 400
All di-lepton data
Eν(GeV)
σ +-/
σ cc(%
)
Fig. 26. Average neutrino (left panel) and anti-neutrino (right panel) dilepton cross-sections, normalized to CC cross-section, asa function of the neutrino energy.
The CHORUS experiment has reported a confirmedD0 sample which contains 1074 (841V 2, 230V 4 and 3V 6) candidates. The background is made ofK0 and decays which affect only the two-prongsample.A background yield of 36 events in this sample has been assessed while four and six prong decaysare background free.
Given the branching ratios into two, four and six prongs as br2, br4 and br6, respectively, the fullyneutral decay mode can be written as:
br0 = 1− br4
(1+ br2
br4+ br6
br4
). (86)
br4 has been precisely measured to be(13.38± 0.58)% through the measurement of the particle widthratio,4/total [21], while the CHORUS experiment has measured the other two branching ratios needed[196]:
br2 = 0.578± 0.052 ,
br6 = (1.5± 0.9)× 10−3 .
It has to be noted that this is the first measurement of the six prong branching ratio. Given that, theyget[196]
br0 = (28.7± 5.2)% .
This result has to be compared with previous measurements that gave a value for br0 of about 5%[21].
5.3. D0 meson studies
Among the 122 observed charm events, the E531 experiment measured 57D0 [169]. After efficiencycorrections, the observed number of events is 87± 11 in 3614± 274 CC events which gives aD0
cross-section relative toCC interactions of
D0
CC× BR(D0 → V 2+ V 4)= (2.41± 0.35)× 10−2 ,
276 G. De Lellis et al. / Physics Reports 399 (2004) 227–320
whereV 2 andV 4 stand for 2 and 4 prongsD0 decays, respectively, and the error is statistical only.SinceD0’s are identified through the visual detection of either a two or a four prong decay topology, themeasured ratio includes also the branching ratio as reported in the above formula.
The available statistics has been largely increased by the CHORUS experiment which has recentlypublished the measurement ofD0 meson production inCC interactions[194,196]. We report in somedetail how the search is performed, by combining the automatic measurements and the visual inspectionat the end of the analysis chain.
A large volume scanning(1.5 × 1.5 × 6.3 mm3) is performed around the neutrino interaction vertexin order to collect all the tracks and use them to reconstruct the event topology (so-called Netscantechnique[195]). To select interesting decay topologies while preserving a goodD0 detection efficiency,the following selection is applied:
• the primary muon track and at least one of the daughter tracks are detected in more than one plate andthe direction measured in emulsions is required to match that reconstructed in the fibre tracker system;
• the impact parameter to the vertex of at least one of the daughter tracks is larger than a value which isdetermined on the basis of the resolution and depends on the track angle;
• the impact parameter must also be smaller than 400m. This cut mainly rejects spurious tracks notrelated to the neutrino interaction.
This analysis was applied to a sample of 95450 interactions, selecting 2816 events which were visuallyinspected (eye-scan) to confirm the decay topology. A secondary vertex is accepted as a decay if thenumber of prongs is consistent with charge conservation and no other activity (Auger electron orblob)is observed.
As reported in Section 5.2, the confirmedD0 sample contains 1074 (841V 2, 230V 4 and 3V 6)candidates. After background subtraction and efficiency correction, and taking into account the fullyneutral branching ratio, the CHORUS experiment published theD0 production cross-section:
D0
CC= (2.97± 0.09(stat)± 0.22(syst))× 10−2
at the average neutrino beam energy of 27 GeV.Taking into account the fully neutral decay mode, we have recomputed the E531 result which gives
D0
CC= (3.4± 0.5(stat)± 0.6(syst))× 10−2
at the average neutrino beam energy of 22 GeV. In both formulas we have used as an additional systematicuncertainty the error on the br0 measurement. This shows a good agreement between the two resultsalthough CHORUS is much more accurate.
5.4. Inclusive charm-production studies with nuclear emulsions
As discussed in the previous sections, in electronic experiments, charm-production studies have beenmainly performed by exploiting the dilepton rate. Therefore, the total charm-production rate cannot bederived, unless the semi-leptonic branching ratio of charmed particles and the charm-production fractionsare independently measured as a function of the neutrino energy. The inclusive measurement of the charm-production cross-section can only be performed by using nuclear emulsions as target (see Section 4.4
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 277
Table 10Inclusive neutrino charm-production cross-section, normalized to the CC cross-section, as a function of the neutrino energy
E (GeV) C/CC (%)
0–10 0.8+1.1−0.7
10–30 4.1± 1.030–60 5.9± 1.360–150 12.1± 3.0
for details). So far only two experiments have studied inclusive neutrino charm-production with nuclearemulsions: E531 and CHORUS. The analysis of the latter experiment is still in progress. Therefore, weonly report preliminary results obtained with a sub-sample of the whole statistics. Furthermore, veryrecently, the CHORUS Collaboration reported for the first time on the inclusive cross-section for charm-production induced by anti-neutrinos.
The average neutrino charm-production cross-section as measured by the E531 experiment[169] wasaffected by the poor knowledge on theD0 decay branching ratios. Taking into account the results reportedin Section 5.3, the total cross-section for an average neutrino energy of 22 GeV has been estimated to be(5.7+0.8
−0.7)%. In a recent CHORUS paper[197] the total number ofD++D+s and+
c is reported, although,being the paper focused on+
c production, the systematics associated with theD+ +D+s events are not
quoted. Therefore, we only give the statistical error associated with this inclusive cross-section. By usingtheD0 cross-section given in Section 5.3, the inclusive cross-section at an average neutrino energy of27 GeV is(6.4± 1.0(stat))%.
So far, the energy dependence of the cross-section has been studied only by the E531 experiment.Having applied the correction mentioned before, the inclusive neutrino charm-production cross-section,normalized to the CC cross-section is given inTable 10as a function of the neutrino energy.
The inclusive anti-neutrino induced charm-production has been studied for the first time in the CHORUSexperiment[196]by exploiting the contamination (about 5% of muonic neutrinos) of the CERN neutrinobeam. The average energy of the anti-neutrino component is about 18 GeV. This measurement has beenmade possible thanks to the good efficiency of the CHORUS muon spectrometer in measuring the chargeof the muons. Monte Carlo simulations, validated with test-beam data, have shown that the probabilityto wrongly measure the charge of the muon is of the order of few per-mill.
In doing this measurement, particular care has to be put in the determination of the normalizationsample. Indeed, given the overwhelming presence of in the beam, a poor muon charge determinationwould translate into a+ sample completely dominated by− with the charge wrongly measured. In orderto check the reliability of the reconstructed+ sample, the measured beam contamination measuredin the emulsion target has been compared with the expectations and with the measured value obtained byusing events occurring in the lead-calorimeter[198]. An anti-neutrino CC interaction rate, normalized tothe neutrino CC interaction, of(2.1± 0.1)% was obtained[196] and a good agreement is reported.
As far as the background is concerned, the main source is the charm-production induced by neutrinoswith the charge of the primary muon wrongly measured. This background has been kept at a reasonablelevel thanks to the good muon charge reconstruction of the spectrometer.
The total number of anti-neutrino induced charm events identified in the CHORUS nuclear emulsionsis 32 with a background (K0
s and decays for neutral topologies: hadronic interactions without anyvisible recoil for charged topologies) of 2.7 events[196].
278 G. De Lellis et al. / Physics Reports 399 (2004) 227–320
Fig. 27. Flight-length distributions of charged charm candidates decaying into three particles. The histograms represent theflight-length distributions given by the Monte Carlo simulation forD+ andD+
s , normalized to the number of observed events.The plot is taken from Ref.[197].
Taking into account all the efficiencies and the background, the total charm-production rate inducedby anti-neutrinos has been estimated to be
(N → +cX)(N → +X)
= (5.1+1.4−1.0(stat)± 0.1(syst))% .
This value has to be compared with the neutrino induced charm-production cross-section measured byE531 and CHORUS,(5.7+0.8
−0.7)% and(6.4±1.0)%, respectively.Although the errors make them consistenta lower value for is observed according to the lower beam energy and the absence of quasi-elastic (QE)charm-production.
5.5. Inclusive production of+c
A dedicated analysis to search for inclusive+c production inCC interactions has been performed
by the CHORUS Collaboration[197]. The analysis exploited two event selections: one aiming at thedetection of decays occurring in the same emulsion plate as the vertex (sample, called A, enriched in+c given its shorter flight length distribution with respect to charmed mesons); the second aiming at the
detection of decays occurring in the emulsion plate downstream from the vertex one (sample, called B,enriched in charmed mesons).
Fig. 27shows the flight-length distributions for decays into three charged particles for events in thetwo regions, compared with the expected distributions for the charged charm mesonsD+ andD+
s andnormalized to the observed number of events. As expected, a difference in shape and an excess of eventsis visible in the small flight length region (below 200m) for selection A and it constitutes evidence for+c decays.Since one of the main background is the interaction of charged hadrons without any visible recoil
(“white interactions”), selection B has been also used to measure this source of background. To checkthe contamination from this background, thepT is measured using the position displacements along thetrajectory caused by multiple Coulomb scattering in the target emulsion. ThepT distributions are shownin Fig. 28separately for the two regions A and B, for events for which the measurement was possible(∼ 70% of the total). Superimposed is the expectedpT distribution from charm decay (the shape is
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 279
Decay Pt
0
2
4
6
8
10
12
14
16
0 0.2 0.4 0.6 0.8 1 1.2[GeV/c]
Num
ber
of e
vent
s / 0
.12[
GeV
/c]
Decay Pt
0
5
10
15
20
25
30
35
40
45
0 0.2 0.4 0.6 0.8 1 1.2[GeV/c]
Num
ber
of e
vent
s / 0
.12[
GeV
/c]
Fig. 28. DecaypT distribution for a subsample (about 70%) of events from selection A (left) and selection B (right) C1 events,Ref. [197]. The histograms represent the distribution of charged charm hadrons given by Monte Carlo simulation. Events withpT >250 MeV were used for normalization.
similar, regardless of whether the decaying particle is a meson or a baryon) normalized to the numberof events withpT >250 MeV. It can be seen that reasonably good agreement is obtained for selection Awhile a clear excess of data is observed at smallpT for selection B. This is consistent with a substantialcontribution fromwhite kinkinteractions, since theirpT distribution is typically peaked at small values.
In total 128 events have been selected with selectionA, with an expected background of about 20 events.For details on the background calculation and the efficiencies evaluation we refer to Ref.[197]. However,in order to extract the inclusive+
c production cross-section, there is another missing ingredient: thefraction of+
c produced through the quasi-elastic charm-production. This is an important quantity, beingthe reconstruction efficiency for DIS and QE produced+
c quite different. It has been measured that the(15± 9)% of +
c production comes from QE processes (see Section 5.7). Therefore, we can concludethat the inclusive production of+
c is
(+c )
CC= (1.50± 0.34(stat)± 0.17(syst))× 10−2 .
5.6. Inclusive production ofD+(2010)
The inclusive production of charmed vector mesons,D+(2010), in neutrino and anti-neutrino charged-current interactions has been studied by bubble chamber experiments[199,200]and, recently, by theNOMAD experiment[201].
Regardless of the experimental technique, the identification of events with aD+(2010) in the final stateis similar for all the experiments. TheD+(2010) undergoes, with a branching ratio of(67.7±0.5)%[21],the decayD+(2010) → D0+ with subsequentD0 decay. Among all possible decays, the followingones have been used by the experiments to search forD0:
D0 → K0+−, BR = (5.97± 0.35)% , (87)
280 G. De Lellis et al. / Physics Reports 399 (2004) 227–320
Fig. 29. ReconstructedD+(2010) invariant mass distribution with three normalized background estimations: MC (full circles),inverted charge combination (open circles) and with a third track taken from another event (open crosses)[201].
D0 → K−+, BR = (3.80± 0.09)% , (88)
D0 → K−++−, BR = (7.46± 0.31)% . (89)
Once aD0 candidate is found, theD+(2010) invariant mass is built by adding to low momentumhadrons theD0 candidate. Furthermore, in order to reduce the combinatorial background additional,kinematical criteria are imposed, i.e. due to the small mass difference betweenD+ andD0, the anglebetween theD0 and + momenta and the transverse momentum of the pion with respect to theD0
direction are small. An example of reconstructedD+(2010) invariant mass with the NOMAD detector isgiven inFig. 29. The parameterK is the output of a Neural Network that gives the probability for an eventto be a candidate forD+(2010) production[201]. The three different sets of points inFig. 29correspondto three different background calculations[201], where the data are reported after theK >0.6 cut.
The measuredD+(2010) rate in CC interactions (T) at different neutrino energies is given inTable 11. Notice that these results are given at the average energy of interacting neutrinos, while re-sults are usually given at the average energy of neutrino flux. We checked that the results reported in Refs.[199,200], obtained with the old branching ratios quoted in the 1994 PDG, do not change significantlyby using the new measurements. There is also a measurement with anti-neutrinos and it amounts to(1.01± 0.31)% [199].
Having measured theD+(2010) rate inCC interactions for different neutrino energies, it is possibleto predict the inclusive7 D0 andD+ cross-sections if the following assumptions hold: theD+ andD0
7 Here inclusive means the sum of charmed mesons from decays ofD and charmed mesons promptly produced.
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 281
Table 11D+(2010) rate inCC interactions for different neutrino energies
Fig. 30. Topology of the quasi-elastic charm neutrino induced events in the case of the reaction (a)n → −+c ,
(b) n→ −+c (
+c ) and (c)p → −++
c (++c ). The particles inside the box represent the+
c decay products
cross-sections are equal, as well as the direct production ofD0 andD+; the ratioR1 ≡ D+/D0 betweencharged and neutral meson inclusive production and the ratioR2 ≡ (D0 from D+)/D0 are independentof the process that generated thec-quark and of the energy. These assumptions have been experimentallychecked and the results are reported in Ref.[202]. For the purposes of this review we just recall themeasured values forR1 andR2, 0.410± 0.009 and 0.281± 0.008, respectively.
The energy ofCC interactions in the CHORUS experiment is of about 47 GeV, in between thefirst two existing measurements given inTable 11. Therefore, we considered for theD+(2010) rate inthe CHORUS nuclear emulsions the weighted average of the two measurements:T = (0.96± 0.16)%.From this rate and by assuming theR2 value given above, we infer for the inclusive neutral charm-production in CHORUS the value of(2.32± 0.39)% that is in good agreement with the measured valueof (2.97± 0.09(stat)± 0.22(syst))% given in Section 5.3.
5.7. Measurement of the quasi-elastic charm-production cross-section
In the quasi-elastic process only a muon is produced at the interaction point (primary vertex) besidesthe charmed baryon. For reaction (47) we expect the topology shown inFig. 30(a), namely, a muon trackplus the+
c then decaying. For reactions (48) and (50), since the produced++c strongly decays into a
+c and a, we expect three charged particles at the primary vertex as shown inFig. 30(c). For reactions
(49) and (51) we expect a number of charged particles produced at the primary vertex equal to the one ofreaction (47), plus a neutral pion produced in the+
c decay (seeFig. 30(b)).Indeed, all these events are characterized by a peculiar topology: two or three charged tracks produced
at the primary vertex, withc, if present, decaying strongly. In any case, the final charmed baryon canonly be a+
c . This feature is very important and will be exploited heavily in the following.A first approach was exploited by bubble chamber experiments and combines high efficiency in measur-
ing charged tracks and long-lived neutral particles (,K0s ), and good momentum resolution[203–208].
282 G. De Lellis et al. / Physics Reports 399 (2004) 227–320
Table 12Summary of the experimental measurements of quasi-elastic charm-production cross-sections
The cross-section is then extracted by using+c branching ratios, which however are known with large
uncertainties[21].Another possible approach requires an apparatus with excellent spatial resolution in order to detect the
short-lived charmed particle prior to its decay and to reconstruct the multiplicity at the primary vertex.Nuclear emulsions have such a resolution and therefore this approach is especially suitable inhybridexperiments, i.e. experiments where a nuclear emulsion target is combined with electronic detectors.E531 [169]—where three events consistent with the processn → −+
c (0) were observed—and
CHORUS are examples of such experiments.Since in the 1980s the knowledge of the+
c branching ratios was rather poor, bubble chamber experi-ments measured the quasi-elastic charm-production cross-section times the+
c branching ratios (+c →
+, +c → pK, +
c → K0s p). By using the most recent determination of the+
c branching ratios[21], we extracted from bubble chamber data the absolute quasi-elastic charm-production cross-section.The results are shown inTable 12.
In the following we focus on a recent result reported by the CHORUS experiment that observed thirteenevents consistent with quasi-elastic charm-production[209], with an expected background of 1.7 events.Thanks to the micrometric position resolution of the nuclear emulsions and their capability in recordingall passing-through charged-tracks, the event selection consisted of two steps:
• topology selection. Namely, all events showing a decay topology, reconstructed using automatic mi-croscopes, are carefully analysed by a visual inspection. Therefore, only events showing a decaytopology consistent with the ones inFig. 30, without any visible activity at the decay vertex and withflight length shorter than 200m are selected as candidates. The latter cut is applied in order to enrichthe sample in+
c , whose life-time is much shorter thanD andDs ;• kinematical selection. As discussed in Section 5.7, quasi-elastic charmed baryon production is char-
acterized by low hadronic energy. Therefore, an appreciable background reduction is obtained byapplying a cut on the visible hadronic energy. A further rejection of background from non-quasi-elastic processes is obtained by studying the distribution of the angle, the azimuthal angle betweenthe primary muon and the charmed particle trajectory in the plane transverse to the incident neutrinodirection. In this plane, if the production process is quasi-elastic, the+
c is expected to be back-to-back with respect to the muon. The experimental resolution, the Fermi motion of the nucleon, and the
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 283
transverse momentum carried by the (soft) pion when the+c is the decay product of ac would affect
causing small deviations from the ideal value of 180. The>165 cut is applied.
After the kinematical selection, 13 events survive, nine withNs = 2 (2-C1, 6-C3, 1-C5) and four withNs = 3 (1-C1, 3-C3), whereNs is the number of minimum ionizing particles measured at the primaryvertex. Sources of background are charmed hadronsD+,D+
s , and+c produced in deep-inelastic processes
as well asD+s andD+s produced diffractively.
After background subtraction and efficiency correction, the measured quasi-elastic production ofcharmed baryons relative to charged-current events is:
(QE)
CC= (0.23+0.12
−0.06(stat)+0.02−0.03(syst))× 10−2 .
In the same manner, it is possible to obtain the cross-section ratios
(C+)CC
= (0.17+0.10−0.05(stat)± 0.02(syst))× 10−2
and
(C++)CC
= (0.07+0.07−0.04(stat)± 0.01(syst))× 10−2
for the production of charmed baryons of charge+e and+2e, respectively.By using isospin properties, it is possible to show that[41–48]
(p → −++c )+ (p → −++
c )= 2((n→ −+c )+ (n→ −+c )) .
Since nuclear emulsions are approximately isoscalar, the above relation can be used with the measuredvalues of the cross-sections to conclude that in quasi-elasticN scatteringc andc baryons are producedat an almost equal rate.
By knowing the total CC neutrino-induced cross-section[1] and the average neutrino beam, it is alsopossible to determine the absolute quasi-elastic charm-production cross-sections:
that are in good agreement with the ones reported inTable 12.Among the various models, the measured value of the cross-section seems to prefer those introducing
suppressionfactors to simple quark models, like the ones reported in Refs.[44,46].Combiningtheseresultswith that given in Ref.[197]—where the total production of+
c was measured—we can conclude that 0.15± 0.09 of all c baryons are produced through quasi-elastic processes at theaverage beam energy of the CHORUS experiment.
In quasi-elastic events, conservation of transverse momentum at the primary vertex allows the momen-tum of the charmed baryon to be estimated and therefore to reconstruct completely the event kinematics.This reconstruction however is only approximate since Fermi motion and the momentum of the pion thatmight come fromc decays are neglected.
284 G. De Lellis et al. / Physics Reports 399 (2004) 227–320
Fig. 31.Q2 distribution for the observed quasi-elastic charm candidates[209]. The histogram is the result of a Monte Carlosimulation with the model of Ref.[42].
Fig. 31shows theQ2 distribution obtained from the thirteen quasi-elastic charm candidate events. Theobserved average value of this quantity is〈Q2〉obs= (2.0 ± 0.6) GeV2. The superimposed histogram isthe result of a Monte Carlo simulation with the model of Ref.[42]. Despite large discrepancies in the totalcross-section predictions, the various models[41–48]give similarQ2 behaviour and the present data donot have sufficient resolution to discriminate them.
5.8. DiffractiveDs andDs production cross-section
The theoretical aspects related to the diffractiveDs andDs production have been already discussed inSection 3.4. Here, we focus on the experimental methods developed to study this process characterized bya very low cross-section with respect to the single-charm production. This process has been observed inseveral experiments[210–212]. In particular, the CHORUS experiment has shown the evidence of aDsdiffractive production through the direct observation in nuclear emulsion of the decay chainD+s → D+
s ,D+s → +, + → +, seeFig. 32 [212]. Unfortunately, only one event has been observed in nuclear
emulsions.Therefore, we focus on the kinematical methods developed by electronic experiments to extracttheD()s signal.
From an experimental point of view, diffractively producedD()s events are characterized at the inter-actions vertex by a very simple topology:A→ −D+
s A, A→ −D+s softA, where the low energy
photon accompanies the production of theDs andA is the target nucleus for coherent production, and
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 285V
tx P
late
Vtx
Pla
te +
1
Vtx
Pla
te
Vtx
Pla
te +
1
τ+
s+
D
PrimaryVertex
µ−µ−
s+
D
τ+
+µ+µ
µ−
s+
D
τ+
+µ
30µm
400µm
60µm
60µm
400µm 30µm
νY-Projection
-beam νZ-Projection
-beam νX-Projection
-beam
Fig. 32. The double-kink event, observed in the CHORUS experiment[212], with two tracks leaving from a single grain withoutnuclear beak-up at the primary vertex. The points are the measured position of each emulsion grain with its error. Also indicatedis the borderline between two consecutive plates.
n or p for incoherent production. Depending on theDs decay mode8 two approaches are possible forelectronic detectors and are discussed in the following.
TheDs decays into a muon. The experimental signature is given by two muons in the final state plusa low hadronic energy. Furthermore, the absolute value of the smallest difference between the muonazimuthal angles() is of the order of 180, being the primary muon and the charmed hadron (theparent of the second muon) emitted back to back. This approach has been exploited by the NuTeVCollaboration that measured a rate for diffractive charm-production, normalized to the CC cross-section,equal to(3.2±0.6)×10−3 [211]. It is worth noting that in this analysis NuTeV assumed both the equalityof the diffractive cross-section for neutrino and anti-neutrino and its energy independence. We discusslater on in this section, the experimental check of the previous assumptions.
TheDs may be detected in a bubble chamber detector through one of the following decay channels:
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plus the conjugate states for theD−s production by anti-neutrinos. TheDs are selected by reconstructing
its invariant mass. The decision to restrict the analysis only to the decay channels listed above is dictatedby the good resolution of bubble chambers in measuring the 4-momentum of charged particles. Thisallows the measurement of the andK0 invariant mass giving the possibility for an additional con-straint on top of theD−
s invariant mass that reduces the combinatorial background. To further reduce thecombinatorial background, the forward–backward decay topologies are removed by retaining only eventswith | cos|<0.9, where is the angle between theDs Lorentz boost direction from the laboratory frameand the neutral decay particle (resonance) momentum, as measured in theDs rest frame. Since theDs isspinless, the acceptance of this selection is 0.9, while the background is approximately halved. However,an additional cut has to be applied to reduce the combinatorial background in the channels with aK0.Finally, the diffractive candidates are extracted out from theDs sample by applying a cut on the absolutevalue of the 4-momentum transfer squared to the nucleon,t. For the diffractive eventst is expected to besmall. For details on this search, we refer to Ref.[210].
Bubble chamber experiments studied both neutrino and anti-neutrino induced diffractiveDs productionand measured the rates (including the branching ratios) relative to CC on an isoscalar target to be (1.5±0.5)×10−4 and(2.6±0.9)×10−4, respectively. This result is compatible with a 1:2 ratio, as implied bythe equality of diffractive cross-sections for neutrino and anti-neutrino, and the ratio of inclusive cross-sections[210]. The energy dependence of the diffractive cross-section has also been investigated. In theneutrino energy intervals 10–30, 30–50 and 50–200 GeV the observed rate per CC is(1.8±0.7)×10−4,(1.3± 0.6)× 10−4 and(1.6± 0.7)× 10−4, respectively, so that no variation is detected at the availablestatistical level[210]. Theoretically, no steep variation of this relative rate in the 10–200 GeV neutrinoenergy interval is expected[57].
It is worth stressing that the previous numbers have been obtained by searching for particularDs decaychannels inD()s diffractive production. Therefore, the knowledge of corresponding branching ratio isneeded in order to get the absolute production rates[210]. Since the publication of Ref.[210], newmeasurements of the branching ratios involved in the analysis came up. Therefore, we updated the resultsof Ref. [210] by using the latest branching ratios quoted in the Particle Data Group[21]. The neutrinodiffractiveD()s production rate has been evaluated to be(2.9± 1.0)× 10−3/CC [213].
The weighted average of the NuTeV and bubble chamber measurements gives a neutrino induceddiffractive production rate of(3.1 ± 0.5) × 10−3/CC that is consistent with the event observed in theCHORUS emulsion. Since the neutrino to anti-neutrino CC production rate is 2:1, the average anti-neutrino inducedD()s diffractive production rate is(6.2±1.0)×10−3/CC. The combined analysis givesan accuracy of about 15% for the diffractive production rate.
This result cannot be directly compared with the theoretical prediction reported in Section 3.4. Indeed,the latter gives only the cross-section for the production of the pseudo-scalar charmed mesonDs . The vec-torial component can be calculated by using the value ofFV , defined asV/(V +P), reported in[213]andexploiting the fact that it is independent of the process and of the energy. Therefore, the theoretical predic-tion for diffractive production ofDs plusDs relative to CC production is 0.3×10−3 and 0.6×10−3 for neu-trinos and anti-neutrinos, respectively, i.e. about one order of magnitude smaller than the measured value.
5.9. Associated charm-production cross-section in CC
As already shown in Section 4.3, the study of dimuon events has been extremely important in theunderstanding of charm-production in neutrino interactions.
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30 29 21
1010µm 6735µm
ν
12
3
µ
Fig. 33. Sketch of the topology observed in emulsion by the CHORUS experiment[191]. The emulsion layers are 350m thickand sketched in white, while the plastic support is 90m thick and sketched in dark.
In the 1970s, measurements of rates of trimuons[144] and like-sign dimuons[214,215]in high energyneutrino-nucleon scattering reported values higher than expected. For instance, in Ref.[215] a sample of47−− events in a − Fe experiment is reported, with an estimated background from andK decaysof 30± 7 events, so that 17± 7−− events had to be of direct origin. The most favoured explanationwas thought to be the associated charm-production process with subsequent charm muonic decay:
N → −ccX , (95)
wherec→ −... (same-sign dimuons) andc→ + . . . (trimuons).Nevertheless, the statistical significance of the excess was not striking and other experiments did not
confirm it in the multi-muon production searches, neither in the early 1990s[216] nor recently[217].This suggests a possible overwhelming of background in the data. On the other side, an assessment ofthe associated charm-production rate should come from the direct observation of the process that is onlypossible with nuclear emulsions.
The CHORUS experiment has searched for this process and reported the first direct observation of oneevent with the characteristics of associated charm production[191]. Since it is only one event, they did notcompute the cross-section rate. The topology of the observed event as seen in emulsion is schematicallyshown inFig. 33.
In Table 13, the list of all particles measured at the primary and secondary vertexes is shown togetherwith their emission angles. At the primary vertex there are six “black” tracks from the nuclear break-upand two minimum ionizing tracks: one is the negative muon and the other is the kink parent, denoted asparticle 1 inTable 13.
A neutral particle (C0) decays in the same emulsion plate, 340m downstream of the primary vertex.Two particles emerge from the neutral particle decay point, both matching a track in the electronic detector.The a-coplanarity of parent and daughter particles,=0.048±0.005 rad, rules out a two-body decay andthus both theK0
s and the hypotheses. Given also the short (340m) flight length, the natural assignmentis a neutral charmed meson decay: eitherD0 or D0.
The fourth track seen in the vertex plate (particle 1) has been followed-down searching for a secondaryvertex. After a path of 1010m, it shows a 417 mrad kink angle. The outgoing particle (particle 2),after travelling 7560m, shows either a kink or a secondary interaction with only one outgoing particle(particle 3) emitted with angles(−0.191,−0.164) rad, 307 mrad from the parent direction. The emulsionconditions do not allow the distinction between a secondary interaction and a decay in this case. The
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Table 13List of particles measured at primary and secondary vertexes of the first double charm event observed in the CHORUS experiment[191]
Particle ID Y (rad) Z (rad) = L〈〉− 0.009 0.104C0 −0.047 −0.055 2.8× 10−13sParticle 1 −0.102 0.020 1.4× 10−12 sC0 daughter 0.267 0.188C0 daughter −0.139 −0.054Particle 2 −0.495 −0.120
In the second and third column, the two projections of the emission angles with respect to the neutrino beam direction arereported. The fourth column shows their life-time measured according to Ref.[218].
measurement of the grain density along the track of particle 2 is compatible with minimum ionization.The ionization of particle 3 is about twice that of a minimum ionizing particle.
Thep measurement by multiple scattering[190] for particle 2 could not be performed due to its largeemission angle and short length. On the contrary, particle 3 travels across 20 emulsion plates and itsmultiple scattering could be measured. By the combined measurement of the ionization in emulsion, theyhave assessed that particle 3 is a proton of momentump3 = 780+170
−110MeV. Usingp2p3 as constraint,
a minimum transverse momentum at the first kink,pT min = 330+70−50 MeV, can be estimated.
The background to associated charm production comes predominantly from events in which a singlecharmed particle is produced and the decay or interaction of one of the non-charmed hadrons in thefinal state mimics the decay of a charmed partner. It is important to notice that the decayp⊥ cut is themost effective kinematical variable for the rejection of andK decays, which are already suppressed bytheir long flight length. Thep⊥>250 MeV cut practically eliminates this background source.All possiblebackground sources have been evaluated through simulation and an overall background of(42±8)×10−3
events is thus expected.After the first observation, CHORUS has modified the analysis strategy. As already mentioned in the
Section 5.3, the increase of automatic scanning power allowed to scan large volumes around the neutrinointeraction vertex and to select all the tracks contained. These tracks participate into a clustering algorithmwhich confirms the primary vertex and eventually find a secondary one. Afterward, if a secondary decaytopology is selected, the event is visually inspected to confirm the decay nature of the secondary vertex.
By applying this method to 95450 CC neutrino interactions, four events have been observed with atopology of double charm interaction[196]. In Fig. 34 we report one of the four candidate events. Abackground of 0.79± 0.10 has been estimated. After background subtraction and efficiency correction,the measured rate is
(cc)
CC= (3.9± 1.9± 0.6)× 10−4 .
This is the first measurement of the process through the direct observation of the double charm decaytopology, with an almost background-free experiment. The measured rate is in agreement with the theoret-ical expectation reported inFig. 8when the cross-section is convoluted with the neutrino energy spectrum.
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TT
TT
TT
TT
Plate 31 Plate 30
Fig. 34. Sketch of a double charm topology in CC interactions observed in emulsion by the CHORUS experiment[196].
Therefore we can conclude that the excess reported in some of the trimuons and like-sign-dimuons ratesin the past was of background origin.
5.10. Associated charm production cross-section in NC
The associated charm production in neutral-current interactions has been studied with nuclear emulsions[169], and electronic detectors[219]. The E531 experiment reported the observation of one event with aD0 − D0 topology. After efficiency correction (1.5+3.5
−1.2 events), the rate relative to NC interactions wasquoted to be
( → cc)
( → )= (13+31
−11)× 10−4
at the average neutrino beam energy of 22 GeV[169].CHORUS has analysed allNC interactions corresponding to 95450 CC events and has found three
candidate events with an expected background of 0.12±0.02.After background subtraction and efficiencycorrection, a relative rate of
( → cc)
( → )= (3.5± 2.1± 0.6)× 10−4
has been measured at the average neutrino beam energy of 27 GeV. We report inFig. 35the topology ofone of the three candidates found in the CHORUS emulsions[196].
The NuTeV experiment has recently reported the measurement of the associated charm rate in NCinteractions with the high energies of the Fermilab Tevatron through the detection of events with wrongsign muon (WSM) final states.
The analysis technique consists of comparing the visible inelasticity,yvis=Ehad/(Ehad+E), measuredin the and wrong sign muon (WSM) data samples to a Monte Carlo simulation containing all knownconventional WSM sources and a possible NC charm signal. The NC charm signal peaks at large values ofyvis because the decay muon from the heavy flavour hadron is usually much less energetic than the hadronshower produced in the NC interaction. The largest background, beam impurities, is concentrated at lowyvis in mode due to the characteristic(1 − y)2 behaviour of interactions of the wrong-flavouredbeam background.
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TT
TT
TT
120µm140µm210µm
TT
Plate 18 Plate 17
νµ
Fig. 35. Sketch of a double charm event topology in NC interactions observed in the CHORUS nuclear emulsions[196].
Events in the WSM sample contain a hadronic energy of at least 10 GeV (increased to 50 GeV for thefinal NC charm fit), and exactly one track (the muon) must be found. The muon is required to be well-reconstructed and to pass within the understood regions of the toroid’s magnetic field. The muon’s energymust be between 10 and 150 GeV, and its charge must be consistent with having the opposite leptonnumber as the primary beam component. Requiring that the muon energy reconstructed in differentlongitudinal sections of the toroid agree within 25% of the value measured using the full toroid reducescharge mis-identification backgrounds to the 2× 10−5 level. This latter number has been verified usingthe muon calibration beam.
Conventional WSM sources arise from beam impurities, right-flavour CC events where the charge ofthe muon is mis-reconstructed, CC and NC events where a or K meson decays in the hadron shower,and CC charm production where the primary muon is not reconstructed or the charm quark is producedvia ae interaction.
After impurities, the next largest WSM source comes from CC production of charm in which the charmquark decays semi-muonically, and its decay muon is detected in the spectrometer. The primary leptonis either an electron, which is lost in the hadron shower, or a muon which exits from or ranges out in thecalorimeter. Thee beam fraction is 1.9(1.3)% in () mode, and 22% of the CC charm events whichpass WSM cuts originate from ae.
Binned maximum likelihood fits are performed to the measured neutrino modeyvis distribution usinga model consisting of all conventional WSM sources described and a possiblecc signal. The fitter variesthe NC charm contribution in shape and level by allowing the charm mass parameter,mc, to float; it alsovaries the normalization of the beam impurities.Fig. 36shows theyvis distribution for the data with thebackground plus fitted NC charm signal superposed. The shape indicates a preference toward includingthe NC charm signal, and the fit yieldsmc = 1.42+0.77
−0.34GeV2 with a beam normalization of 1.00± 0.06,where the errors are purely statistical.
In Fig. 37theyvis distribution of WSM’s is shown with the additional requirement thatEhad be largerthan 50 GeV. This cut removes 85% of the beam impurities while keeping 75% of the NC charm events.Performing the fit again on this reduced sample with the background normalization fixed at 1.0 yields aconsistent value for the charm mass ofmc = 1.40+0.83
−0.36GeV2, with the error again purely statistical. This
value of charm mass corresponds to a production cross section(N → ccX) = (2.1+1.8−1.5) fb at an
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 291
Fig. 36. Distribution ofyvis in -mode WSM’s for data (points), background (dashed), and background plus fitted NC signal(dotted) as measured by the NuTeV experiment in searching for associated charm production in NC interactions[219].
Fig. 37. Distribution ofyvis for WSM’s for data (points), backgrounds (dashed), and background plus NC signal (dotted) withan additional requirementEhad 50 GeV as measured by the NuTeV experiment in searching for associated charm productionin NC interactions[219].
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Table 14Double charm production rate inNC interactions at different neutrino energies
Experiment E (GeV)(→cc)
(→)
E531[169] 22 (13+31−11)× 10−4
CHORUS[196] 27 (3.5± 2.1± 0.6)× 10−4
NuTeV [219] 154 (6.5+5.6−4.6)× 10−3
average neutrino energy〈E〉 = 154 GeV, i.e. to a relative ratio
( → cc)
( → )= 6.5+5.6
−4.6 × 10−3 .
In Table 14we summarize all available data on NC double charm production rate at different energies.An increasing rate as a function of the neutrino energy is observable as expected although the error barsare still large.
5.11. Measurement of the charmonium production cross-section
J/ production inNC interactions has been studied by several experiments, CDHS[220], CHARMII [154], CCFR[153,221], NuTeV[211] and CHORUS[222]. As discussed in Section 3.6, the expectedcross-section for this process is very small, although characterized by a very clean experimental signature:J/ → −+. Evidence for this rare process has been reported by the CDHS and CHORUS experiments.
As an example, in the following we describe the CHORUS analysis: details can be found in Ref.[222].In total, 4.7 million events were recorded in CHORUS lead-calorimeter. Most of these events contain
one track and an energetic hadronic shower. Only 2.3 × 105 events have two reconstructed tracks withat least 2 GeV momenta at the spectrometer entry and crossing at least two spectrometer magnets ( 1 mof iron).
To suppress the background fromCC events with+(K+)→ +X decays, both muons were requiredto have momenta above 5 GeV at the vertex and the visible energy for the event,Evis
was required to belarger than 20 GeV. 14995+− events survived the kinematical cuts.
The observed invariant-mass distribution of+− pairs is presented inFig. 38a. An approximation ofthe shape consists of 85% of the+− MC and of 15% of the−− data. The observed average visibleenergy in the dimuon events,〈E
vis〉 ≈ 85 GeV, corresponds to a total energy of about 100 GeV, i.e.twice the average energy of CC events. To reduce the background an upper cut on the shower energy wasapplied.+− invariant-mass distributions were analysed at different limits onEhad. A structure appearsin theJ/ mass region atEhad 15 GeV. The excess of events in this region is clearly seen with the cuton the shower energy varying between 14 and 10 GeV. It is gradually reduced by further tightening thecut and vanishes atEhad 5 GeV.
Fig. 38b shows the excess observed forEhad 10 GeV.9 The invariant-mass distribution of muon pairsin all +− events (Fig.38a, solid histogram) was used as a background shape to the spectrum. In total,
9 CDHS reported evidence forJ/ production by neutrinos via neutral-currents at the same upper cut on the showerenergy[220].
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 293
(a)
(b)
(c)
Fig. 38. The invariant-mass distributions for+− events in the CHORUS experiment[222]: (a) all selected events (solidhistogram is the data and dashed histogram is the sum of the Monte Carlo prediction and the−− data). The−− data aloneis also shown (dotted histogram); (b) events with cutEvis
sh 10 GeV (solid histogram is the data and dashed histogram is the
prediction for a background shape based on overall dimuon events); (c) events with cutsEvissh 10 GeV andp+p− (solid
histogram is the data and dashed histogram is the prediction for a background shape based on overall dimuon events).
1265 events survived the cut. To improve the signal-to-background ratio the selectionp+p− wasapplied. This muon momentum ‘asymmetry’ cut suppresses the background by a factor of about five as,on average,+ is much softer than−, whereas inJ/ decaysp+ ≈ p− . The resulting invariant-massdistribution is shown inFig. 38c. In total, 226 events survived the cut. In the signal region between 2.75and 3.75 GeV there are 62 events.
The resulting total signal after background subtraction amounts toNJ/obs =28.1±12.3(stat)±2.7(syst)
events, with the cutsEhad 10 GeV andp+p− .The excess of events observed by CHORUS can be translated into an experimental spectrum-averaged
cross-section:〈J/〉 = (6.3 ± 3.0) × 10−41cm2/nucleon for 20E 200 GeV. The error includesboth statistical and systematical uncertainties added in quadrature. The CDHS experiment observed, forEhad<10 GeV, an excess of 45±13 events. The spectrum-averaged cross-section[220] is (5.4±1.9)×
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10−41cm2/nucleon.10 It should be mentioned that CHORUS and CDHS were using different wide-bandbeams, with slightly different energy spectra and compositions. The NuTeV experiment also searchedfor J/ production by applying a cut onEvis
sh similar to the one applied by CDHS and CHORUS, but nosignal was observed.At 90% C.L. an upper limit onJ/ production is set to 3.4×10−41cm2/nucleon and63× 10−41cm2/nucleon for〈E = 70 GeV〉 and〈E〉 = 175 GeV, respectively, which is still consistentwith the CDHS and CHORUS results. NuTeV searched for this process using also an anti-neutrinobeam. The correspondent 90% C.L. limits are 4.0 × 10−41cm2/nucleon and 73× 10−41cm2/nucleonfor 〈E = 70 GeV〉 and〈E〉 = 175 GeV, respectively.
The contribution to the cross-section from the diffractive mechanism11 is expected to be very small (seeSection 3.6). This is in agreement with CHORUS whose signal disappear when the cutEhad<5 GeV isapplied.The NuTeV experiment searched for diffractiveJ/ production by applying the cutEvis
sh <3 GeV,but no signal was observed. At 90% C.L. a limit on diffractiveJ/ production is set to 1.1×10−41cm2/
nucleon and 19× 10−41cm2/nucleon for〈E〉 = 70 GeV and〈E〉 = 175 GeV, respectively. A similarsearch with an anti-neutrino beam gave the following limits at 90%: 1.7 × 10−41cm2/nucleon and32× 10−41cm2/nucleon for〈E〉 = 70 GeV and〈E〉 = 175 GeV, respectively.
In CHORUS the theoretical expectation[78] of directJ/ production in the framework of theZ0-gluonfusion model is 8.0± 1.5 events. The main uncertainty comes from the efficiency calculation (∼ 15%).Other sources give∼ 10% contribution. The theoretical uncertainties are very difficult to calculate reliablyand are not taken into account. A higher experimental rate (28.1± 12.3± 2.7 events) suggests a sizablecontribution of excited charmonium states:c1, c2 → J/ and′ → +−J/. We recall that it maybe about as large as the directJ/ production cross-section (see Refs.[78,83]).
In conclusion, the rare process ofJ/ production via neutrino neutral-current interaction has beenobserved by CDHS and CHORUS. The measured cross-section suggests that the diffraction mechanismis unlikely to play an important rôle in the directJ/ production process. A contribution of excitedcharmonium states decaying toJ/ could be as large as the directJ/ production process, in qualitativeagreement with theoretical expectations. Therefore the neutral-current coupling to charmed quarks cannotbe extracted with sufficient precision.
6. Experimental determination of production characteristics of charmed particles in neutrinointeractions
6.1. Determination of the charm-production fractions
The problem of how a charm-quark fragments into a charmed hadron is challenging not only froma theoretical point of view, as discussed in Section 3.2, but also from an experimental point of view.Indeed, the direct identification of the charmed hadron in the final state is only possible through the directobservation of the hadron decay and the measurement of the kinematical variables. This is only feasible by
10 CDHS cross-section was rescaled from the published value of (4.2 ± 1.5) × 10−41cm2/nucleon as the measuredJ/branching ratio to muon pairs has changed from 0.07 to 0.06 andCC
0 has increased from 0.62[223] to 0.677×10−38cm2 GeV−1
[1] since 1982.11We follow the CHORUS conventional terminology, applying the term ‘diffractive’ only to the VDM process (Fig.10b).
In Ref. [220] the same term is applied for events surviving the cutEhad 10 GeV.
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Table 15Charm-production fractions as a function of the neutrino energy
using nuclear emulsions. In some papers[221,224]it is stated that charm-production fractions extractedfrom neutrino experiments are in agreement with those measured bye+e− experiments.This is not entirelytrue as pointed out in Ref.[202]. Indeed, while a charm quark once produced has probability to producea given charmed hadron that is independent of the energy and of the process, in neutrino interactions onehas also to account for diffractive and quasi-elastic charm-production. This makes a significant differencemainly at low energies where diffractive and quasi-elastic charm-production largely contribute to the totalcross-section.
In emulsion experiments decaying particles can be identified on an event by event basis by directlydetecting the decay topology. By using nuclear emulsions it is possible to identify unambiguously allD0
mesons, with a small background from neutral-hadron interactions andK0s and0 decays, while charged
charmed particles are more difficult to be separated.c can be identified either by requiring the presenceof a proton at the decay vertex (event by event classification adopted by the E531 experiment) or onstatistical basis exploiting the fact that its life-time is much smaller than the one ofD andDs (techniqueexploited by the CHORUS experiment). On the other hand,D andDs have similar masses and life-time,so that it is very difficult to separate them.
In the E531 experiment 122 events were tagged by the presence of a secondary vertex in the emulsiontarget, 119 neutrino induced and 3 induced by anti-neutrinos. Events with a candidate charmed hadron inthe final state were studied in detail in order to detect the presence of heavily ionizing particles (baryons)and fully reconstruct the kinematics at the decay vertex. However, the original analysis was biased by thefact that all events with an equal probability to be eitherD orDs were classified asD mesons.A reanalysisof E531 data that removes the previous bias can be found in Ref.[224]. Moreover, the E531 data werealso affected by the problem related to the wrong estimate, at the time of the analysis, of theD0 decayinto all neutrals, see Section 5.2. We corrected the results obtained in Ref.[224] and they are shown inTable 15, where it is possible to extract the fraction of+
c produced among charged charmed particlesexpected in the CHORUS experiment. It turned out to be(38± 9)% in agreement with the measuredvalue of(43± 8± 6)% [197].
So far, we reported the charmed fractions as measured in neutrino induced charm-production. Un-fortunately, there are no measurements for anti-neutrino interactions. It is worth noting that in principlefh $= fh, since quasi-elastic charm-production, which is dominant at low neutrino energies, cannotbe induced by anti-neutrinos. On top of that, given its quark composition (cud), the c production issuppressed by the baryon number conservation. Furthermore, as already discussed in Section 5.8, the
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Table 16The inclusive semi-muonic decay branching ratios of charmed hadrons[21]
diffractiveDs production relative to charged-current anti-neutrino interactions is twice the one inCCinteractions. All these differences contribute to makefh $= fh and very difficult the extrapolation fromthe neutrino to the anti-neutrino case.
The CHORUS experiment, given its limited anti-neutrino statistics, only measured the neutral tocharged charmed-hadron ratio
fD0
fC+= 2.4+1.5
−1.0 .
This result, although affected by a large error, shows the expected enhancement of neutral charmedhadrons in anti-neutrino with respect to the neutrino case (0.97± 0.17) [196].
6.2. Determination of the semi-muonic branching ratio
The semi-leptonic branching ratioB, defined as
B = fD0BR(D0 → )+ fD+BR(D+ → )+ fD+s
BR(D+s → )+ f+
cBR(+
c → ) (96)
is an empirical parameter, not a fundamental constant. Nevertheless, its importance is related to the factthat, by fitting dimuon rates in neutrino interactions, one extracts the quantityB|Vcd |2 that contains thefundamental constantVcd , an element of the CKM matrix (see Section 6.3). Due to its energy dependence,the value ofB measured in one experiment, with a given neutrino beam and a given acceptance, cannot bedirectly applied to extract|Vcd | from the measurement ofB|Vcd |2 performed by a different experiment,with a different neutrino beam and a different energy dependence for the given acceptance. Nevertheless,a constant value ofB for all experiments has been adopted by the PDG[21]. Here in the followingwe discuss two methods to extractB. The first one, adopted in the past by all neutrino experiments,relies upon the knowledge of the charm-production fractions and of the semi-muonic branching ratios ofcharmed hadrons. By using the energy dependence of charm-production fractions reported inTable 15andthe latest estimates of semi-muonic branching ratios given inTable 16, we recomputedB as a functionof the neutrino energy (seeTable 17). Notice that all these experiments had a kinematical analysis suchthat onlyE>30 GeV were effective. By using the charm-production fractions forE>30 GeV we getthe following average value forB:
B = 0.088± 0.006 . (97)
Without the correction for the newD0 branching ratio into all neutrals, the average semi-muonicbranching ratio would have beenB = 0.093± 0.007.
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Table 17Charmed hadron semi-muonic branching ratios at different neutrino energies
Table 18Inclusive (first row) and semi-leptonic (second row) charm selection efficiency for each charmed hadron type
D0 D+ D+s +
c
Di 0.541± 0.006 0.369± 0.010 0.440± 0.018 0.335± 0.015Di
0.50± 0.02 0.41± 0.02 0.45± 0.06 0.29± 0.07
The second method consists of a direct measurement ofB as exploited by the CHORUS Collaborationthanks to the use of nuclear emulsions[225]. In this analysis charmed hadrons are identified on an eventby event basis by directly detecting their decay topology. Therefore, it is possible to directly measure theratio of muonic charm decays normalized to the total number of produced charmed hadrons
B = Nselected2
Nselected×∑Di
DifDi∑Di
DifDi, (98)
whereNselected2 andNselectedare the number of charmed hadrons decaying into a muon and the total
number of charmed hadrons detected in the nuclear emulsions, respectively. For this particular analysisonly a subsample of 956± 35 events were observed with a charmed hadron detected in the final state,among them 88±10(stat)±8(syst) decay into a muon. The analysis of the full charm sample is in progress.The charm detection efficiencies (not including the correction for the new value of theD0 decaying intoall neutrals) for all decays and only for semi-muonic decays are given inTable 18.
Although the raw numbers (events, efficiency, systematics and so on) reported in this Section are thosequoted in Ref.[225], the main difference is that we recomputed the value ofB including the correctionfor the new value of theD0 decaying into all neutrals 5.2 and the updated charm-production fractionsreported inTable 15forE>5 GeV andE>10 GeV. The choice to study two sets of charm-productionfractions is a consequence of an intrinsic cut on the neutrino energy due to the analysis procedure. Ittranslates into an effective minimum energy of about 10 GeV once the primary muon energy has beenincluded. The value ofRhas been evaluated by letting the selection efficiencies and the charm-productionfractions variates into one sigma and it is
R = 0.89± 0.05 . (99)
Having determined the value ofR it is possible to extract the value ofB at the average neutrino energyof the CHORUS experiments (E ∼ 27 GeV)
B = 0.083± 0.010(stat)± 0.009(syst). (100)
298 G. De Lellis et al. / Physics Reports 399 (2004) 227–320
For the sake of comparison,B has been also extracted for the sub-sample of events with visible energylarger than 30 GeV. This yields a value ofB = 0.091± 0.014(stat)± 0.009(syst), that is consistent withthe one given in Eq. (97).
Finally, we would like to stress that the semi-muonic branching ratio forc-quark decays quoted inthe Particle Data Group cannot be directly compared with the one measured in neutrino induced charm-production. The reason is that, as already mentioned in Section 6.1, it is has been obtained ine+e− exper-iments where processes peculiar to neutrino interactions (diffractive and quasi-elastic charm-production)are absent.
6.3. Measuring theVcd andVcs CKM matrix elements with neutrinos
One of the “golden” measurements of neutrino experiments is the determination of the elements of theCKM matrixVcd andVcs by comparing the charm-production cross-section in neutrino and anti-neutrinosinteractions.
In the past several experiments extracted these matrix elements from neutrino and anti-neutrino dimuonevents. However, before going into details, we would like to make some remarks. Besides the CCFRCollaboration[221], all experiments performed a LO analysis of their data.
At LO, the cross-section for dimuon production induced by neutrinos can be written as
d2→2
d dy∝(
1− m2c
2ME
)×|Vcd |2u(,Q
2)+ d(,Q2)
2+ |Vcs |2s(,Q2)
× B . (101)
The corresponding cross-section for anti-neutrinos has the quarks replaced by their anti-quark partners.From Eq. (101), it is easy to understand that by a fit to neutrino and anti-neutrino dimuon cross-sections
it is possible to extract the quantities|Vcd |2 × B and/( + 2)|Vcs |2 × B, onceB and the strangecontent of the nucleon (), defined as
are independently measured.At the NLO, additional diagrams complicate the expression in Eq. (101). Indeed, as already discussed
in Section 3.1, theW+g → cs subprocess together with theW+s′ → c process should be taken intoaccount. Following the notation given in Ref.[226], at NLO one can define the CKM-rotated weakeigenstates with
s′ = |Vcs |2s + |Vcd |2dand similarly,
g′ ≡ g(|Vcs |2 + |Vcd |2)to denote its QCD evolution partner, i.e. ds′/d ln Q2 = s′ ⊗ Pqq + g′ ⊗ Pqg . Therefore, at the NLOapproach, the factorization present in Eq. (101) is completely spoiled.
As a consequence of that, the values of|Vcs | and |Vcd | depend on the approach used. The scaleuncertainty in a LO calculation is not deducible unless one performs a NLO calculation. Once the NLOcalculation is performed, the analysis of the experimental data should be performed to this order to reduce
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 299
the theoretical uncertainties (i.e. the scale dependence). Nonetheless, in all calculation available so far,LO and NLO are not treated separately. The procedure adopted by the PDG is to assign, arbitrarily, tothe LO result a scale error equal to the one of the NLO calculation[21,227]. Hence they combine the LOand NLO results in a weighted average in order to reduce the error on the parameters. Although the errorsare still too large to see any difference among the different determinations, we believe that one should becautious and average only results from different experiments obtained with the same approach.
Since the majority of the available analyses have adopted a LO approach, in the following we brieflysummarize their main features to extractVcd andVcs .
In an ideal situation one would like to measure directly the differential cross-section for charm-production in terms of Bjorken variablesx andy and at several different neutrino energies. However,charm cross-section is related to the data by model-dependent corrections for charm fragmentation anddecay and by experimental effects of resolution, acceptance and neutrino flux. Therefore, a parametricmodel is directly fitted to the data in order to extract the parameters of the model.
In order to extract the CKM matrix elements, the dimuon cross-section that has to be fitted to the datacan be expressed as
d3()→2
dx dy dz∝ |Vcd |2B ×
[dv + 1
2+ As(1− x)(
1+∣∣∣∣VcsVcd
∣∣∣∣2As(1− x))q
], (103)
where the valence quark contributiondv = 0 for anti-neutrinos. In Eq. (103)As gives the strange seanormalization and it depends on and, a parameter that relates the shape of the strange quark distributionto that of the non-strange sea.= 1 would indicate a flavourSU(3) symmetric sea;= 0 would indicatethat the strange sea has the samex dependence as the non-strange component of the quark sea.
Expression (103) shows explicitly that the fitted value ofB is correlated with|Vcd |2. Therefore,from the fit one can only extract the product|Vcd |2 × B. It is indeed important to have an independentdetermination of the charm semi-muonic branching ratio (see Section 6.2). With this parametrization thestrange quark density is multiplied by the ratio|Vcs/Vcd |2.
The adopted fitting procedure depends upon the quantities to be measured. There are two possibleapproaches (both assuming thatxs(x) = xs(x), we will discuss later results from fits investigating thepossibility thatxs(x) $= xs(x)):• Vcd andVcs elements are taken from unitarity fits to the CKM matrix and the parameters, B, andmc are extracted;
• no assumption is made on the CKM matrix elements. Therefore, the following quantities are extracted,mc, |Vcd |2 × B and/( + 2)|Vcs |2 × B.
The first approach is exploited when the analysis is focused on QCD aspects, while the second onewhen the aim is the investigation of the CKM matrix.
6.3.1. |Vcd | determinationThe measured values of|Vcd |2×B at LO for different experiments obtained by fitting simultaneously
neutrino and anti-neutrino data are:CDHS[148]: |Vcd |2 × B = (4.1± 0.7)× 10−3,CCFR[153]: |Vcd |2 × B = (5.09± 0.32(stat)+0.17
−0.16(syst))× 10−3,
CHARM II [154]: |Vcd |2 × B = (4.42+0.35−0.34(stat)± 0.34(syst))× 10−3.
300 G. De Lellis et al. / Physics Reports 399 (2004) 227–320
By combining the previous results one obtains the following weighted average:12
|Vcd |2 × B = (4.74± 0.27)× 10−3 .
Thus, by using the independent determination ofB given in Eq. (97) we obtain the value
|Vcd |LO = 0.232± 0.010 .
At the NLO there is only one measurement provided by the CCFR Collaboration[221]
|Vcd |2 × B = (5.34+0.38−0.39(stat)+0.37
−0.55(syst))× 10−3 ,
from which we can extract the following value of|Vcd | at NLO:
|Vcd |NLO = 0.246± 0.016 .
These values could be compared with the determination of|Vcd | = 0.225+0.013−0.011 obtained in Ref.[227]
by a simultaneous fit to all available data to extract|Vcd |, |Vcs |, B and. However, although this resultis not affected by the problem of the fully neutralD0 branching ratio, it is still affected by the mixing ofLO and NLO analyses.
Our result is also in agreement with the PDG value. However, besides the LO/NLO problem, there arestill differences:
• in the determination of the average value of|Vcd |2 × B, the PDG does not include the CHARM IIand the LO CCFR results. Therefore, their average value is(4.9 ± 0.5) × 10−3, instead of(4.74±0.27)× 10−3;
• the PDG uses an old determination ofB affected by the problem of the fully neutralD0 branchingratio.
Our best value at LO (|Vcd | = 0.232± 0.010) and at NLO (|Vcd | = 0.246± 0.016) are within the 90%C.L. allowed range for determined by imposing the unitarity to the CKM matrix: 0.209< |Vcd < |0.227.
6.3.2. |Vcs | determinationFor historical reasons13 we report here below the|Vcs |determination with neutrinos and anti-neutrinos.
Note that, on top of the poor precision achievable with neutrinos, there is also the problem of dis-entangling LO from NLO measurements. In the following we do not treat separately LO and NLOmeasurements.
12 In doing the average we symmetrize the errors taking the average of positive and negative errors, and statistical andsystematic errors are added in quadrature.
13Before the recent LEP2 data onW± decay studies, this was the only way to directly measure|Vcs |.
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 301
As mentioned before, from a fit to dimuon rates CDHS, CCFR and CHARM II obtained the followingresults:
CDHS :∣∣∣∣VcsVcd
∣∣∣∣2 × = 9.3± 1.6 ,
CCFR:
+ 2× |Vcs |2 × B = (2.00± 0.10(stat)+0.09
−0.15(syst))× 10−2 ,
CHARM II : 1
+ 2×(
1+∣∣∣∣VcsVcd
∣∣∣∣2 ×
)= 3.58+0.49
−0.41(stat)± 0.44(syst).
All the results have been obtained assumingxs(x)= xs(x). However, CCFR verified that its removaldoes not lead to a significant variation of the results[221]. The extraction ofVcs from neutrino inducedcharm-production data has been made possible by a new analysis of CCFR[228] that determined thevalue of with an independent measurement to be = 0.453± 0.106+0.028
−0.096.In Ref. [227] in order to compare results from different experiments forVcs , for each experiment the
quantity × |Vcs |2B has been computed . The average value has been estimated to be
× |Vcs |2B = (4.53± 0.37)× 10−2 ,
from which one can derive the following value of|Vcs |:|Vcs | = 1.07± 0.16 .
This measurement is rather poor, if compared with other determinations obtained by studying semi-leptonic charmed meson andW+ decays. For a review of the available determinations of|Vcs | and on theadopted experimental methods we refer to Refs.[21,227,229]. The best direct measurement of|Vcs | hasbeen obtained in Ref.[227] (|Vcs | = 0.996± 0.024) and it can be compared with the value quoted in thePDG obtained by imposing the unitarity constraint:|Vcs | = 0.996± 0.013.
Given the much poorer precision achievable with neutrinos with respect to other approaches, in order toimprove the determination of the other variables, it is much better to fix the value of|Vcs |when performingthe fit to dilepton data.
6.4. Kinematical variables describing the charm hadronization:mc, z andp2T
6.4.1. Determination of themc parameterThe determination of the quark masses is a very challenging theoretical problem. For a discussion of the
quark masses at a fundamental level, we refer to Ref.[21] and references therein. From an experimentalpoint of view it is very difficult to talk about quark masses. Indeed, unlike leptons, quarks are confinedinside hadrons and are not observed as physical particles. Therefore, they cannot be directly measured,but must be determined through their influence on hadronic properties. Given these features, one shouldbe careful in speaking about quark masses. Any quantitative statement about the value of quark massesshould be referred to the particular theoretical framework that is used to define it. It is important to keepin mind this when quark mass values are compared.
In the following we discuss experimental results on the determination of the charm-quark mass (mc)through the study of charm-production in charged-current neutrino-induced interactions. Once more we
302 G. De Lellis et al. / Physics Reports 399 (2004) 227–320
Table 19Leading order determination of the parametermc for various experiments
The experimental error includes both statistical and systematic errors. The theoretical error does not include the dependenceonmc, which is shown explicitly.
want to stress the fact that the value ofmc does not have any absolute meaning, but it is just a parameterthat allows to describe charm-production and its threshold behaviour.
Several experiments performed the analysis of dimuon data at LO and extracted, among other param-eters, the value ofmc. The results are shown inTable 19together with their average value. As far as theNLO analysis is concerned, there is only one determination available[221] for mc (=1.70± 0.19 GeV).As expected, this value is different from the LO determination. The reason is the following: while atLO mc describes the observed threshold effect, through the slow-rescaling mechanism, at NLO it alsoincludes the kinematic effects associated with heavy quark production and provides a better comparisonwith measurements derived from other processes involving high-order perturbative QCD calculations.For example, the photon–gluon–fusion analysis of photoproduction data findsmc=1.74+0.13
−0.18GeV[237],which is in agreement with the neutrino measurement. Very recently the NOMAD Collaboration pub-lished preliminary NLO results by using a dilepton sample of 13659± 169 induced by neutrinos[238].Themc value extracted at NLO ismc = 1.58± 0.09+0.04
−0.09GeV, that is in agreement with the NuTeVdetermination.
Following the procedure described in Ref.[154], the average value ofmc at LO can be used to extract amore accurate value of sin2 W from the ratio of neutral-current to charged-current deep-inelastic cross-sections. InTable 20the results of sin2 W measurement by CDHS[231,232], CHARM [233,234]andCCFR [235] parametrized as a function ofmc are shown.14 Using the average value ofmc given in
14 In Ref. [154] the original values from CHARM and CDHS have been corrected by using the latest reevaluations of theradiative corrections formtop = 175 GeV andmHiggs= 150 GeV.
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 303
Table 19, we obtain a new estimate of the electroweak parameter
sin2 W = 0.2267± 0.0026 ,
where all the errors are summed in quadrature. The quoted value of sin2 W is defined in the Sirlin(on-shell) renormalization scheme, and therefore can be translated into theW-boson mass using the highprecision measurement of theZ-boson massMZ = (91.1876± 0.0021)GeV [21]
MW = (80.19± 0.14)GeV .
This result is compatible within 1.3 with the direct measurementMW = (80.425± 0.038)GeV.
6.4.2. Charm quark fragmentation studies: thez variableIn Section 3.1, the semi-inclusive cross-section has been written as a convolution of a perturbative
part through the coefficient functionHq,gi , and a non-perturbative part,Dhc , that incorporates the long-distance, non-perturbative physics of the hadronization process. Parameterizations (43), (44) and (45)contain non-perturbative parameters that have to be fitted from experimental data. In the following werefer to this approach as the “old approach”.
Although it is behind the purpose of this Review, we briefly mention here a new approach[239] (andreferences therein) to describe the fragmentation of heavy quarks. The fragmentation functionDhc can bewritten as
Dhc =Dc(z, )⊗Dhnp(z, ) , (104)
whereDc(z, ) is the probability for a massless parton to fragment, via the perturbative QCD, into amassive charm quarkc. Dhnp(z, ) is instead a non-perturbative fragmentation function, describing thetransition from the heavy quark to a hadron. ForDhnp(z, ) the functional form given by the parameteriza-tions (43)–(45) can be employed, although the meaning of the parameters is different with respect to theold approach. In general, these parameters do not have an absolute meaning, being them dependent on theparton type, on the heavy hadron considered and on the energy scale. Furthermore, they are fitted togethera model of hard radiation and their numerical value depends also on it. For an exhaustive discussion ofthese topics we refer to[240,241]and references therein.
The available results in neutrino interactions have been obtained with the old approach, i.e. the functionDhc is directly fitted to the data. No re-analysis with the new approach has been performed, although itwould be welcome.
It has to be noted that thezvariable is usually assumed to range between 0 and 1. This assumption comesfrom collider experiments where this is obviously true. This is not the case in fixed target experimentswhere it is necessary to distinguish two cases: once the charm quark is produced, it may hadronize eitherwith a valence or with a sea quark. If it takes a quark from the sea, it has to “spend” some energy to extractit from the sea and therefore the momentum of the final hadron will be less than the one of the charmquark, and hencez 1. On the contrary, if it hadronizes with a valence quark, the hadron momentum maybe well above the one of the charm quark, and hencez>1 is possible. In a neutrino experiment, bothsituations are possible and therefore there is no strict limit on this variable. In particular, since charmedmesons are bound states of a charm quark and sea-quarks,z 1 for these events, whilez may be largerthan one for thec baryon which is a bound state of thec quark withu andd quarks, and they may comefrom the valence too. InFig. 39, thez variable as predicted by the JETSET[242] simulation program isreported forc baryons and charmed mesons.
304 G. De Lellis et al. / Physics Reports 399 (2004) 227–320
Fig. 39.z distribution as obtained with JETSET for charmed mesons (left panel) and charmed baryons (right panel).
Given that, in order to see the two differentzdistribution, one would need to separate between charmedbaryons and mesons. This is possible in nuclear emulsion experiments.
This fact makes the comparison between collider and fixed target experiments somehow meaninglessand we should by no means expect the shape of the distributions, e.g. the value, to be the same.Nevertheless, in the following we will also report parameters frome+e− at center of mass energy of10.6 GeV.
The parameter that characterizes the fragmentation functions of heavy quarks can be determined byusing two different experimental methods:
• Direct measurements: thez distribution is reconstructed and fitted in order to extract the parameter. Such analysis has been performed by E531, NOMAD (seeFig. 40) and is currently in progress inCHORUS, although preliminary results are available at[236].
• Indirect measurements: is left as one of the free parameters of the fit to the dimuon data. Note thatin dimuon data the main contributors to the sample areD0 andD+.
The available results from both methods are summarized inTable 21together withe+e− results at√s = 10.6 GeV. Being the Peterson parametrization the only one used by all the experiments, we only
give in the table the results forP .All neutrino experiments, but CCFR, give a similar value for the parameterP , ∼ 0.075, while CCFR
measured a larger value,∼ 0.2. This “discrepancy” is not surprising given the large difference in thecentre of mass energy,∼ 10 and∼ 5 GeV for CCFR and other experiments, respectively. Therefore, thetwo determinations cannot even be compared as well as theP at
√s ∼ 5 GeV cannot be compared with
thee+e− results at√s ∼ 10 GeV. Conversely,e+e− results inTable 21can be compared to some extend
with neutrino data at√s ∼ 10 GeV. If one assumes the charmed fractions forE>30 GeV given in
Table 15, the semi-leptonic branching ratios inTable 16and theP measured by CLEO+ARGUS givenin Table 21, it is possible to predictP for the CCFR experiment. By doing so, one gets
expP = 0.22 that
is in good agreement with the measured value 0.22± 0.05.
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 305
0
2
4
6
8
10
12
14
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ε = 0.13 ± 0.08c
ε = 0.075 ± 0.028P
z (D*+)
Fig. 40. NOMAD results forD mesons:z distribution. The results of fits with the functions (44) and (45) are also shown.
Table 21Summary of all available determinations of. Results from a reanalysis ofe+e− (CLEO+ARGUS) experiments are also shown.C stands for all charmed hadrons;D for all charmed mesons.
The mean values ofz in the range 0.54–0.68 clearly show that charmed hadrons are produced forwardwith respect to the jet and carry a large fraction of the jet energy.
The variablepT is defined as the transverse momentum of the charmed hadron with respect to theW-boson direction. The charmed quark is initially generated along theW-boson direction. However, because
306 G. De Lellis et al. / Physics Reports 399 (2004) 227–320
Fig. 41. The calculatedpT single-inclusive heavy-quark distribution and the corresponding fits illustrated in the text[243].
of hadronization effects, thepT distribution of the hadron cannot be calculated, but has to be assignedby using an empirically measured transverse momentum spectrum. The transverse momentum of thecharmed particles with respect to theW-boson direction is parametrized by
dn
dp2T
=N × e−B×p2T , (105)
whereN ensures the normalization andB is the parameter to be extracted from a fit to the data. Althoughit is behind the purpose of this Review, we report some comments about the parametrization used for thep2T distribution, while for details we refer to Ref.[243] and the paper quoting it. It has been shown in
Ref. [243] that the parametrization given in Eq. (105) holds only for small values ofp2T (smaller than
4 GeV2) and the value of the parameterb is very sensitive to the upper bound of thep2T . This is due to
the fact that the fall-off of the cross-section at largep2T is not exponential, but rather follows a power law,
seeFig. 41. In Ref.[243] the authors found that the formula
dn
dp2T
=(
C
bm2c + p2
T
)
(106)
provides an excellent fit to the theoretical distributions. Nevertheless, given the fact that in neutrinointeractions the maximump2
T is well below 4 GeV2, we consider the parametrization given in Eq. (105)to describe neutrino data. Before, discussing the results obtained with neutrinos, we remind that in hadro-production experiments a fit with Eq. (105) to all available data in the center of mass energy range20–40 GeV gives a value ofb ranging in the interval 0.8–1.4 GeV−2. The fitted value ofb is rather
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 307
0
2
4
6
8
10
12
14
16
18
20
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
PT2 22[GeV /c ]
B = (8.26 ± 0.19) GeV , m = (1.14 ± 0.03) GeV/c
-1
2
B = (3.38 ± 0.40) GeV , m = 0 GeV/c
-1
2
Fig. 42. NOMAD results forD mesons:p2T
distribution.
constant versus the energy in the centre of mass, while perturbative QCD predicts a decrease with theenergy. However, the large errors on the fitted values ofb do not allow any firm conclusion.
The measurements of thep2T distribution in neutrino interactions are rather scarce. So far only two
experiments performed such a measurement: E531[244] and NOMAD [201]. Thep2T distribution as
measured in NOMAD is shown inFig. 42. The measured values ofbare(3.25±0.37)GeV−2 and(3.38±0.40)GeV−2 for E531 and NOMAD, respectively. The two results, obtained with two similar neutrinobeams, are in good agreement and can be combined giving the average value of(3.31± 0.27)GeV−2.Having in mind that the average energy center of mass energy in neutrino interactions is less than 10 GeV,the neutrino results are in good agreement with the theoretical predictions that foresee an increase ofbat low energies. Furthermore, the good agreement between neutrino and hadro-production data has beendemonstrated in Ref.[245], where a fit using the E531 data sample withW2>30 GeV2 (correspondingto an average energy in the center of mass of about 15 GeV) has been performed. The extracted valueof b from this sub-sample is(1.21± 0.34)GeV−2, which is in good agreement with hadro-productionmeasurements.
6.5. Strange parton distributions
Charm production in neutrino interactions is a powerful tool to investigate the strange quark contentof the nucleon. Indeed, given also the CKM couplings, about 50% of the charmed hadrons in neutrinointeractions come from as to c transition, while in anti-neutrino interactions the fraction is even larger,about 90%, given also the need for anti-quarks, hence sea quarks, in the transition.
308 G. De Lellis et al. / Physics Reports 399 (2004) 227–320
It is then clear that the combined analysis of neutrino and anti-neutrino data allows the extractionof the strange and anti-strange nucleon content. In the past, several experiments have performed theanalysis of the strange quark content of the nucleon, with different detectors and techniques which weare going to summarize. The analysis is always a data fit using a theoretical model which presents severalparameters: one of these is the strange nucleon content. The conceptual difference in these analyses isanyway two-fold: from one side the theoretical model used could be either based on a leading order or ona next-to-leading order calculation; from the other side the assumption on the shape of thesands quarkdistributions could be either conservative (same shape) or not.
In this respect, we present in detail the analysis of the CCFR Collaboration which has done botha leading order[153] and the first next-to-leading order[221] analysis of their dimuon data samples.Moreover, they have also done the analysis of the strange quark content by assuming both the same anda different shape for these partons. We shall focus on their NLO analysis with both the conservative andnon-conservative approach[221].
In the CCFR experiment, charged-current single muon events are required to haveEvis>30 GeV,Evis
had>10 GeV,Q2vis>1 GeV2 andp1>9 GeV. Dimuon events are selected by making the further
requirement that the second muon hasp2>5 GeV and that both muons have an emission angle withrespect to the beam direction<0.250 rad. The second muon’s momentum is measured in the magneticspectrometer whenever possible, otherwise it is determined from the muon’s range in the target. In order toreduce non-prompt sources of second muons, events in which the second muon does not reach the toroidmust also satisfyEvis
had<130 GeV. The final dimuon sample contains 6090 events and is characterized by〈Evis〉 = 192 GeV,〈W2
The charm-initiated dimuon signal is contaminated by non-prompt pion and kaon decays. The high-density calorimeter minimizes this contamination due to the short interaction length of the detector.A combination of hadronic test beam muonproduction data and Monte Carlo simulations predicts a small/K decay background of 797± 118 and 118± 25 events[96].
To calculate the probability of producing charm, the CCFR collaboration employed the NLO QCDcharm production differential cross-section calculation of Aivazis et al.[97], including the Born andgluon–fusion diagrams. Electromagnetic radiative corrections to the cross-section were calculated usingthe method of Bardin et al.[98].
Measurements of theF2 andxF 3 structure functions by CCFR[99,100] are used to determine thesinglet and the non-singlet quark distributions,xqSI(x,
2)= xq(x, 2)+ xq(x, 2) andxqNS(x, 2)=
xq(x, 2)−xq(x, 2), respectively, and the gluon distribution,xg(x, 2) (see[221]). These distributionsare obtained from next-to-leading-order QCD fits to the structure function data[101] using the QCDevolution programs of Duke and Owens[102].
To resolve the strange component of the quark sea, the singlet and non-singlet quark distributions areseparated by flavour. Insofar as isospin is a good symmetry, the experiment is insensitive to the exact formof the up and down valence and sea quark distributions, because the neutrino target is nearly isoscalar.An isoscalar correction accounts for the 5.67% neutron excess in the CCFR target.
The non-strange quark and antiquark components of the sea are assumed to be symmetric,so thatxu(x, 2) = xuS(x, 2), xd(x, 2) = xdS(x, 2). The isoscalar correction is applied assum-ing xu(x, 2) = xd(x, 2). As already mentioned, the strange component of the quark sea is allowedto have a different magnitude and shape from the non-strange component. The value of = 1, where is defined in Eq. (102), would indicate a flavourSU(3) symmetric sea. The shape of the strangequark distribution relates to that of the non-strange sea by a shape parameter, where
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 309
Table 22The strange quark nucleon content and the shape parameter for all the experiments. Errors are statistical and systematic
=0 would indicate that the strange sea has the samex dependence as the non-strange component of thequark sea.
In the first CCFR data fit, the strange quark and antiquark distributions are assumed to be the same.The sea quark distributions are parameterized by
xq(x, 2)= 2
[xu(x, 2)+ xd(x, 2)
2
]+ xs(x, 2) ,
xs(x, 2)= As(1− x)[xu(x, 2)+ xd(x, 2)
2
], (107)
whereAs is defined in terms of and.CCFR performed a2 minimization to find the strange sea parameters and, the values ofBc and
mc, and the fragmentation parameterε, by fitting to thexvis, Evis andzvis distributions of the dimuondata. Taking|Vcd | = 0.221± 0.003 and|Vcs | = 0.9743± 0.0008 [21] as input values and using theCollins–Spiller fragmentation function, the extracted NLO parameters with their statistical and systematicerrors are presented in the second line ofTable 22. An excellent agreement between the data and the NLOtheoretical model is reported.
CCFR previous LO results[153], which were found by fitting to thexvis andEvis distributions of thesame data sample and using the Peterson fragmentation function[38],D(z)=N z[1− (1/z)− εP /(1−z)]2 −1 with εP = 0.20, are listed in the first line ofTable 22. For comparison with these results, theyhave also made the NLO fit using the same Peterson fragmentation function used in the LO analysis. Thevalue found for was 0.468+0.061 +0.024
−0.046 −0.025 while for they found−0.05+0.46 +0.28−0.47 −0.26. It is important to stress
310 G. De Lellis et al. / Physics Reports 399 (2004) 227–320
Fig. 43. Strange quark distribution in CCFR data for LO and NLO analyses[221].
once again that the comparison between results obtained with two different orders of calculations is notproperly done. We report it just for sake of completeness.
As reported inTable 22, at NLO, the nucleon strange quark content is found by CCFR to be =0.477+0.051
−0.050, indicating that the sea is notSU(3) symmetric, qualitatively the same result as from the LOanalyses carried out also by many other experiments and reported in the same table. The strange quarkcontent may alternatively be given by
so that by comparing to the total non-strange quark content, is less sensitive to changes in the determi-nation of the sea quark content alone. At2 = 22.2 GeV2 the ratio of antiquarks to quarks in the nucleonat NLO is found to be
∫dxxq(x, 2)/
∫dxxq(x, 2) =Q/Q = 0.245± 0.005 and thereby the strange
quark content with respect to the non-strange quarks is[221]
= 0.099± 0.008± 0.004−0.003+0.006 .
Since a non-zero value of would indicate a shape difference betweenxq(x) andxs(x), the value=−0.02+0.66
−0.60 indicates no shape difference at NLO. At leading order, they find the strange quarks softerthan the overall quark sea by a factor(1− x) with = 2.5± 0.7.
The strange quark distribution from thexs(x, 2) = xs(x, 2) fit is plotted at2 = 4 GeV2 inFig. 43. The distribution can be parameterized by a function of the forma (1 − x)bx−c. The valuesof the coefficients are given in Ref.[221]. Very recently the NOMAD Collaboration published prelimi-nary NLO results by using a dilepton sample of 13659± 169 induced by neutrinos[238]. The valueextracted at NLO is = 0.458+0.187+0.057
−0.131−0.013GeV, that is in agreement with the CCFR determination.
6.6. Possible asymmetry of the strange sector
Theoretical work has explored the possibility that the nucleon contains a sizable heavy quark componentat moderatex, i.e. the possibility of the so-called intrinsic heavy quark states within the nucleon[103,104].
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 311
Postulating intrinsic strange quark states leads to the prediction that thes quark momentum distributionwill be harder than thes quark distribution[105,106]. This possibility was explored by CCFR[221] byperforming a fit in which the momentum distributions of thes ands quarks are allowed to be different.For this study the sea quark distributions were parameterized by
xq(x, 2)= 2
(xu(x, 2)+ xd(x, 2)
2
)+ xs(x, 2)+ xs(x, 2)
2,
xs(x, 2)= As(1− x)[xu(x, 2)+ xd(x, 2)
2
],
xs(x, 2)= A′s(1− x)′
[xu(x, 2)+ xd(x, 2)
2
]. (109)
Thes ands are constrained to have the same number∫ 1
0s(x, 2)dx =
∫ 1
0s(x, 2)dx . (110)
As andA′s are defined in terms of, and′.
In order to reduce the number of free parameters, the fit constrained the average charmed hadronbranching ratio to the value obtained from other measurements,B I
c = 0.099± 0.012. The fit was donefor four parameters: the strange quark parameters, , and = − ′ and the charm quark massmc.The result was:
where the first error is statistical, the second is systematic, the third is due to the uncertainty inB Ic, and
the fourth is the error due to2 scale uncertainty.The value of = −0.46± 0.85± 0.17 indicates that the momentum distributions ofs and s are
consistent and their difference is limited to−1.9<<1.0 at the 90% confidence level. This was thefirst quantitative comparison of the components of thes ands quark sea.
It was also checked the assumption that the same average semileptonic branching ratio applies to the-and-induced samples. A two parameter fit finds the branching ratio of-induced eventsBc = 0.1147±0.0056, andBc=Bc−B ′
c=0.011±0.011, whereB ′c is the branching ratio for-induced events and the
errors are statistical only. The result indicates that there is no significant difference in the semileptonicdecays of charmed particles and antiparticles at these energies.
Recently, the NuTeV collaboration has published a precise measurement of dimuon production inneutrino and anti-neutrino scattering[230] by analysing 5102-induced and 1458-induced eventscollected with their detector. They have made in the same paper a re-analysis of 5030-induced and1060 -induced events collected from the exposure of the same detector to a quad-triplet beam by the
312 G. De Lellis et al. / Physics Reports 399 (2004) 227–320
CCFR experiment. Both analyses are done at LO. The NuTeV results and the combined analysis aboutthe strange nucleon content are reported in the third and fourth line ofTable 22, respectively.
The data from NuTeV and CCFR have also been used to perform a combined analysis on the possiblestrange quark asymmetry[130]. The strangeness momentum densitiesS±(x) and their integrals[S±] areusually defined as
[S±] ≡∫ 1
0S±(x)dx ≡
∫ 1
0x[s(x)± s(x)]dx . (112)
The global analysis performed in Ref.[130]presents several new elements when compared to the latestCTEQ6M[22] and CTEQ6HQ[134] analyses. On the experimental side, it has been added the CDHSinclusiveF2 andF3 data sets[136], the CCFR-NuTeV dimuon data sets[230] and the new E866ppDrell-Yan data set[137]. On the theoretical side, it has been expanded the parameter space to include thestrangeness sector.
As it is usual in this field, the experimental measurement is presented as a series of forward differentialcross-section with kinematic cuts, whereas the theoretical quantities that are most directly related to theparton distribution analysis are the underlying charm quark production cross-sections. Therefore in theanalysis reported in Ref.[130], it was used a Monte Carlo program that incorporates kinematic cutsas well as fragmentation and decay models. The parameters of the model were tuned to reproduce, asclosely as possible, the detailed differential dimuon cross-sections published in Ref.[230]. LO theoreticalformulas for the charm cross-section were used in this analysis. The results show that the uncertaintiesresulting from existing experimental constraints are quite large, so the LO treatment is adequate for thisfirst study of the dimuon data in a global QCD analysis context. All fully inclusive cross-sections used inthis global analysis are treated in NLO QCD instead; the precision of these data sets demands that levelof accuracy. As it is also discussed in the paper[130], the usage of a LO approximation for the dimuondata in this NLO analysis is not correct; however, within the current experimental uncertainties, the NLOcorrections to the charm production cross-section are not large enough to affect the qualitative featuresof this analysis.
In Fig. 44it is shown the strangeness asymmetry together with the momentum asymmetry as obtainedin the global analysis. The classes of solutions with the best fit results are also reported.
The global analysis shows several classes of solutions in the strangeness sector that are consistentwith all relevant world data used in the global analysis. The constraints provided by other inclusivemeasurements, labeled as “inclusive I” in the text, are consistent with those provided by dimuon data,although much weaker. The allowed solutions generally prefer the momentum integral[S−] to be positive.This conclusion is quite robust. However, the size of this strangeness momentum asymmetry is still quiteuncertain; the only conclusion is that[S−] lies in the range from−0.001 to+0.004. The Lagrangemultiplier method explicitly demonstrates that both the dimuon data and the “inclusive I” data sets stronglydisfavour a large negative value of[S−], although they may still be consistent with zero asymmetry.
6.7. Determination of the Weinberg angle in neutrino scattering
A rather long standing puzzle in particle physics in recent years is the 3 deviation of the NuTeVmeasurement of sin2 W (0.2277± 0.0013± 0.0009) reported in Ref.[120], from the world average ofother measurements[121] (0.2227± 0.0004).
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 313
10-5
0.001 0.01 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
x
-0.04
-0.02
0
0.02
0.04s- (
x,Q
) dx
/dz
class Aclass Bclass C
Strangeness Asymmetry Q2 = 10 GeV
10-5 0.001 0.01 0.05 0.1 0.2 0.3 0.4 0.5 0.60.7
x
-2×10-3
0
2×10-3
4×10-3
6×10-3
8×10-3
S- (
= x
s- (x,Q
)) d
x/dz
Momentum Asymmetry
(scale: linear in z = x1/3)
(scale: linear in z = x1/3)
Fig. 44. Strangeness asymmetrys−(x) and the associated momentum asymmetry as obtained in the global analysis[130]. Thehorizontal axis is linear inz=x1/3 so that both large and smallx regions are adequately represented; the functions are multipliedby a Jacobian factor dx/dz so that the area under the curve is the corresponding integral overx.
Possible sources of the NuTeV anomaly, both within and beyond the standard model, have beenexamined in Ref.[122]. No consistent picture has yet emerged in spite of the extensive literature[123–127]on this subject. The measurement in Ref.[120] was based on a correlated fit to the ratios of charged andneutral current (CC and NC) interactions in sign-selected neutrino and anti-neutrino scattering events ona iron target at Fermilab. This procedure is closely related to measuring the Paschos–Wolfenstein ratio[128], which provides the theoretical underpinning of the analysis. Specifically, the Paschos–WolfensteinratioR− is related to the Weinberg angleW by
R− ≡ NC −
NC
CC −
CC
1
2− sin2 W + R−
A + R−QCD + R−
EW , (113)
314 G. De Lellis et al. / Physics Reports 399 (2004) 227–320
where the three correction terms are due to the non-isoscalarity of the target ( R−A ), next-to-leading-order
(NLO) and non-perturbative QCD effects ( R−QCD), and higher-order electroweak effects ( R−
EW). SinceR− is a ratio of differences of cross-sections, the correction terms are expected to be rather small.
In a recent paper[133], it has been reported on QCD corrections to the process, which are generallyrecognized[123–127]to be the least well known. The analysis reported in Ref.[133] is based on therecent NLO calculation reported in Ref.[129], together with new parton analyses that explicitly allowstrangeness asymmetry[130] and isospin violation[131]. The results provide more realistic estimates ofthe sizes and uncertainties of the QCD corrections, and a new look at the significance of the “anomaly”(see also Ref.[132]).
As already discussed in the previous sections, except for the strangeness number sum rule,∫[s(x)− s(x)]dx = 0 , (114)
there is no fundamental symmetry that relates the strange quark PDFs(x) to the antiquark PDFs(x). Therecently published CCFR-NuTeV data on dimuon cross-sections inN andN scattering yield a directhandle ons(x) ands(x), and hence ons− [130], as seen in the previous Section. It is important to noticethat an asymmetric strange sea in the nucleon (s−(x) $= 0) would contribute to a correction term toR−at LO [122].
By including the dimuon data, and by exploring the full allowed parameter space in a global QCDanalysis, Ref.[130] presents a general picture of the strangeness sector of nucleon structure. The stronginterplay between the existing experimental constraints and the global theoretical constraints, especiallythe sum rule (114), places useful limits on acceptable values of the strangeness asymmetry momentumintegral[S−]. The limit quoted in Ref.[130] is −0.001< [S−]<+ 0.004.
In Ref. [133] it has been quantified the impact of the PDFs of Ref.[130] on the Paschos–Wolfensteinrelation in Eq. (113) by employing the NLO neutrino cross-section calculations of Ref.[129]. Moreover,for a given measurement ofR−, a shift of the theoretical prediction, such as R−
s , would lead to a shiftin the extracted sin2 W value according to (see Eq. (113)):
(sin2 W)= R−s . (115)
The results of the calculations presented in Ref.[133], along with the range−0.001< [S−]< + 0.004of Ref. [130], lead to estimate the range of (sin2 W) to be−0.005< (sin2 W)<+ 0.001.
The shift in sin2 W corresponding to a given data fit bridges only part of the original 3 discrepancybetween the NuTeV result and the world average of other measurements of sin2 W . For PDF sets with ashift toward the negative end, such as−0.004, the discrepancy is reduced to less than 1. On the otherhand, for PDF sets with a shift toward the positive end, such as+0.001, the discrepancy remains.
A new reanalysis of the NuTeV data has been recently presented[135]. It is the first analysis of dimuonevents with a Monte Carlo simulation using the DISCO[226] NLO cross-section code, differential inall variable required. In order to accomodate the sin2 W discrepancy, the value ofS− would need tobe positive and as large as 0.007[125]. Several fits have been performed, but they always find a valuewithin +0.002 which is even lower than the upper bound found in the combined analysis of CCFR andNuTeV with a different cross-section modelling[130]. Therefore it is generally agreed that the sin2 Wdiscrepancy is unlikely to be the product of an asymmetry in the strange and anti-strange PDFs.
G. De Lellis et al. / Physics Reports 399 (2004) 227–320 315
7. Summary
The study of neutrino induced charm-production is an important physics topic having a strong impactnot only on charm physics, but also on other fields of research likeb-physics and neutrino oscillations.Indeed, an improved knowledge of charmed hadron properties will help in the understanding of bottomhadrons through their decays in charmed hadrons. Neutrino oscillation physics will also profit of charmphysics being the charm-production one of the main background of experiments exploiting high-energyneutrino beams.
Among the several theoretical and experimental topics discussed in this paper, the highlights of thisreview are:
• a combined analysis of all available data on inclusive and exclusive charm-production cross-sections;• the impact of the new determination of theD0 fully neutral decay mode both on the semi-muonic
branching ratio and, consequently on the determination of the CKM matrix elementVcd , on thecharm-production fractions and on the cross-sections;
• the study of associated charm-production both in neutral- and charged-current interactions;• the status of the measurements of the kinematical variables describing the charm hadronization:mc,
z andp2T . In particular we thoroughly discuss the main features of neutrino-induced processes and
compare them withe+e− data;• the extraction of strange-quark and strange-antiquark parton distribution functions and the impact of
a possible strange/anti-strange asymmetry on the determination of the Weinberg angle in neutrinoscattering.
In the near future final results from CHORUS, NOMAD and NuTeV experiments should allow:
• detailed studies of threshold behaviour of the charm-production cross-section;• a better understanding and determination of charm-quark hadronization;• model independent extraction of strange-sea PDF at the NLO;• better direct determination ofVcd .
In the long term future, neutrino factories have the potentiality to provide an enormous amount ofneutrino-induced events with charmed particles in the final state. Moreover, at a neutrino factory it willbe possible to also produce bottom-quark. For details on the potentiality of a neutrino factory to studyheavy-quarks we refer to[143,246].
Acknowledgements
We would like to warmly thank Paolo Strolin. His encouragement to start this Review has been afundamental motivation for us. We are also indebted to him for his careful reading of the manuscript.
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