Neutrino flavor conversion in a neutrino background: Single- versus multi-particle description
Alexander Friedland*Theoretical Division, T-8, MS B285, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
and School of Natural Sciences, Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540, USA
Cecilia Lunardini†
School of Natural Sciences, Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540, USA~Received 20 April 2003; published 16 July 2003!
In the early Universe, or near a supernova core, neutrino flavor evolution may be affected by coherentneutrino-neutrino scattering. We develop a microscopic picture of this phenomenon. We show that coherentscattering does not lead to the formation of entangled states in the neutrino ensemble and therefore theevolution of the system can always be described by a set of one-particle equations. We also show that thepreviously accepted formalism overcounts the neutrino interaction energy; the correct one-particle evolutionequations for both active-active and active-sterile oscillations contain additional terms. These additional termsmodify the index of refraction of the neutrino medium, but have no effect on oscillation physics.
It is well known that interaction with a medium modifieneutrino dispersion relations or, equivalently, gives the mdium a nontrivial refraction index for neutrinos. Because teffect is generically flavor dependent, it can have a profouimpact on neutrino flavor evolution. Inside the Sun andEarth, the refraction effect arises as a result of neutrinoteractions with electrons and nucleons. On the other hanthe early Universe or near a supernova core, where the nber density of neutrinos is sufficiently high, the refractiproperties of the neutrino background itself are import@1–5#. This neutrino ‘‘self-refraction’’ is the subject of thpresent work.
Early studies@2–7# of the effect treated the neutrino bacground analogously to the case of ordinary matter~electronsand nucleons!. Specifically, two assumptions were made:~1!it was assumed that for each neutrino, one can write a sinparticle evolution equation and take into account the effecall other neutrinos by adding appropriate terms to the oparticle Hamiltonian;~2! these terms are diagonal in the flvor basis. Thus, the evolution of each neutrino in the syswas described by
idn ( i )
dt5S Hvac1Hmat1(
jHnn
( i j )D n ( i ), ~1!
wherei andj label neutrinos in the system,Hvac andHmat arethe usual vacuum and ordinary matter Hamiltonian termsthe last term is the sum over the contributions of all baground neutrinos, taken to be flavor diagonal.
These assumptions were critically reexamined by Paleone@8,9# who made several important observations regaing their validity. First, he reasoned that flavor evolutionneutrinos in a neutrino background is in general a ma
body phenomenon and it isa priori not obvious that the firstassumption is justified in all cases. Second, he demonstrthat, for a system of several active neutrinos flavors, ewhen the first assumption is justified, the second one iscompatible with the symmetry of the problem. Indeed, tlow energy neutral current Hamiltonian@30#,
HNC5GF
A2S (
aj amD S (
bj bmD , ~2!
where the currentsj am[nagmna and a is a flavor index,a
51, . . . ,N, possesses aU(N) flavor symmetry, which mustbe respected by any effective description of the system. Trequirement is not satisfied by the diagonal form forHnn
( i j )
used in the earlier studies.Pantaleone proposed a modified form forHnn
( i j ) , whichcontained non-zero off-diagonal terms,
Hnn( i j )5AF une
( j )u21unm( j )u21S une
( j )u2 ne( j )nm
( j )*
ne( j )* nm
( j ) unm( j )u2 D G .
~3!
In Eq. ~3! the wave function of the background neutrinon ( j )
is normalized such that*dV(une( j )u21unm
( j )u2)51 and the co-efficient of proportionality A equals A2GF(12cosb(ij )),b ( i j ) being the angle between the two neutrino momenta.antineutrinos in the background, the form of the Hamiltoniis exactly the same, with the only difference thatA has theopposite sign. As can be easily checked, Eq.~3! is indeedconsistent with theU(2) flavor symmetry of the two-flavorsystem.
The result~3! was also later obtained by Sigl and Raffe@10# and by McKellar and Thomson@11# in the framework ofmore general analyses of the flavor evolution of a neutrsystem, which took into account both refraction effects acollisions. Equations~1!,~3! have been used as a startinpoint for extensive studies of the neutrino evolution in t
A. FRIEDLAND AND C. LUNARDINI PHYSICAL REVIEW D 68, 013007 ~2003!
early Universe@12–20# and in supernovae@21–25#. The gen-eral properties of the solutions have been investigated@26–29#.
Certain theoretical aspects of these equations, howehave not received adequate attention in the literature to dOne such aspect is the validity of a description of the prlem by a set of single-particle equations. The absencequantum correlations~entanglement! at all times during theevolution of the system isa priori not obvious, and has nobeen proven. Indeed, in the derivations@10# and @11# it isstated as an assumption. In@9#, it is suggested that for quantum correlations not to form the system must obey certphysical conditions, for example, it should contain an incherent mixture of mass eigenstates. This raises the quehow general the results in Eqs.~1!,~3! are and what physicacriteria determine their breakdown.
The second important aspect is the connection betwthe elementary neutrino-neutrino scattering processes anmacroscopic description of refraction given in Eqs.~1!,~3!.For refraction in ordinary matter, such a connection is vwell established and provides the most straightforward dvation of the effect as an interference of many elementscattering amplitudes. A similar treatment for the case ofneutrino background has not been given.
In this paper we present a description of the flavor evotion of neutrinos in a neutrino background in terms of telementary neutrino-neutrino scattering processes. Thisscription provides a transparent, physical picture of thefect, and, at the same time, allows us to prove that quancorrelations in the neutrino ensemble are negligible atherefore the description in terms of one-particle equationvalid. These equations are obtained directly from our formism and are compared to the accepted results, Eqs.~1!,~3!.
We work in the regime in which~i! neutrino-neutrino in-teractions are described by the low-energy four-fermFermi coupling,~ii ! the neutrino gas is non-degenerate, a~iii ! incoherent scattering of neutrinos with other neutrinand other particle species is negligible. The first conditimplies that the neutrino center-of-mass energies arebelow the weak scale. The second one is satisfied if a ganeutrinos has a number densitynn!E3, whereE is a typicalneutrino energy. Finally, incoherent scattering is significanthe column density of the medium exceeds the inverse ofscattering cross section:d[*n(x)dx*1/s, as happens, e.gin the early Universe at temperatures larger than few M~see, e.g.,@18#!.
The refraction effects of the neutrino background are nligible if the coupling between a neutrino and the neutrigas is significantly smaller than the vacuum oscillatiHamiltonian or than the coupling to ordinary matter. Thcondition thus depends on the neutrino energy, on the denof the neutrino background, and on the density of the onary matter. In the core of the Sun, where the number denof neutrinos isnn.106 cm23, neutrino-neutrino interactionis negligible for all relevant neutrino energies and oscillatparameters. In contrast, near a supernova core the neudensity can be as high asnn.1031 cm23 and is comparable
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with the electron density there. Therefore, neutrino-inducrefraction effects are important and must be taken intocount.
The paper is organized as follows. In Sec. II we reviethe microscopical picture of the neutrino flavor evolutionnormal matter and point out that a naive extension of tpicture to the neutrino self-refraction leads to seeminparadoxical results. In Sec. III we show how this pictushould be constructed consistently by identifying statesthe neutrino ensemble that amplify coherently. We also shthat coherent scattering does not form entangled statesSec. IV we compare the one-particle evolution equationsobtain for active-active and active-sterile oscillations in tneutrino and antineutrino backgrounds with the acceptedsults. In Sec. V we show how entanglement is effectivdestroyed by the refraction phenomenon. Finally, Sec.summarizes our conclusions.
II. FORMULATION OF THE PROBLEM
A. Conventional neutrino refraction: FCNC case
We begin by briefly reviewing the physics of the refration effect in ‘‘normal’’ matter. While there are many ways tderive the refraction properties in this case, fundamentathe relevant effect is the coherent interference of manyementary scattering events.
Let us consider the problem of neutrino oscillations inmedium which possesses flavor changing neutral cur~FCNC! interactions. Such interactions give rise to the nozero off-diagonal terms in the neutrino evolution Hamtonian,
HFCNC5A2GFn2
2 F const1S e8 e
e 2e8D G , ~4!
whereGF is the Fermi constant andn2 is the number densityof scatterers in the medium.
As a toy example, consider a beam of electron neutriincident on a thin slab of matter of thicknessL made ofFCNC interacting particles, as illustrated in Fig. 1. Assumthat the neutrino masses are sufficiently small so thateffects of vacuum oscillation can be neglected. The flaconversion rate in the slab can then be found using the
FIG. 1. Neutrino beam traversing a slab of FCNC interactmatter.
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NEUTRINO FLAVOR CONVERSION IN A NEUTRINO . . . PHYSICAL REVIEW D68, 013007 ~2003!
lowing straightforward physical argument. Letf be the am-plitude for an electron neutrino to scatter as a muon neutin a given direction on a particle in the target. If the scatting amplitudes for different target particles add up incohently, the flux of muon neutrinos in that direction is}Nsu f u2,whereNs is the number of scatterers. In the case of forwascattering, however, the scattering amplitudes add upcoher-ently and, hence, the forward flux of muon neutrinos}Ns
2u f u2. Indeed, in the smallL limit Eq. ~4! gives
Pne→nm
FCNC .e2~GFn2L !2/2, ~5!
which has the formPne→nm
FCNC }Ns2u f u2, sincee} f . Notice that
by choosing a smallL limit we were able to ignore the secondary conversion effects in the slab, i.e., to assume thaall elementary scattering events the incident neutrinos arthe ne state.
To summarize, for small enoughL, the flavor conversionrate due to coherent FC scattering in the forward directioproportional to the square of the modulus of the productthe elementary scattering amplitude and number of scaers. This quadratic dependence onNs is what makes the coherent forward scattering important even when the incohent scattering can be neglected.
Notice that exactly the same arguments apply if one csiders the usual flavor-diagonal matter term due to the etron background in a rotated basis, for instance, in the bof vacuum mass eigenstates. In this basis, the matter Hatonian has off-diagonal terms, resulting in transitions btween the vacuum mass eigenstates.
B. Neutrino background: Physical introduction
We seek the same description for the case of neutbackground. Let us therefore modify the setup in Fig. 1 areplace the slab by a second neutrino beam, such thaneutrino momenta in the two beams are orthogonal~see Fig.2!. To keep the parallel between this case and the FCcase, we will continue to refer to the original beam as ‘‘tbeam’’ and to the second beam as ‘‘the background’’. Tneutrinos in each beam can be taken to be approxima
FIG. 2. Toy problem to illustrate neutrino flavor conversionthe neutrino background.
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monoenergetic@31#. We again assume that the neutrinmasses are sufficiently small so that, although flavor suposition states could be createdoutsidethe intersection re-gion, the effects of vacuum oscillationinsidethe intersectionregion can be neglected. Any flavor conversion that taplace in the system is therefore due to neutrino-neutrinoteractions in the intersection region.
Let us first compute the amount of flavor conversionthe beam using Eqs.~1!,~3!. The conversion is expected because of the presence of the off-diagonal terms in these etions. The result depends on the flavor composition ofbackground. If the background neutrinos are all in the saflavor state
nx5cosane1sinanm ~6!
and their density isn2, the Hamiltonian for the evolution oa beam neutrino takes the form
H5A2GFn2
2 F const1S cos 2a sin 2a
sin 2a 2cos 2a D G . ~7!
After traversing the intersection region, a neutrino in tbeam will be converted to thenm state with the probability
Pne→nm.sin22a~GFn2L !2/2, ~8!
assuming as before that the size of the regionL is small.The above descriptions of neutrino flavor conversion
the neutrino background and in the FCNC backgroundvery similar in form. We will next show, however, that despite very similar appearances of the equations, the undeing physics in the two cases is different.
First, we need to establish which elementary procesgive rise to neutrino flavor conversion in the neutrino bacground. In the absence of vacuum oscillations, the onlyteractions in the problem are neutral current interactionstween pairs of neutrinos. These are described byHamiltonian~2!, which conserves thetotal flavor of the sys-tem. The flavor composition of the beam therefore chanonly if some of the background neutrinos of different flavscatter into the beam, i.e., if a neutrino from the backgrou
and a neutrino from the beamexchangemomenta,n(kW )1n(pW )→n(pW )1n(kW ). As observed in@9#, such events canadd up coherently and give rise to the flavor off-diagonentries in the oscillation Hamiltonian.
Let us consider such an elementary event, as depicteFig. 3. Following @9#, we restrict scattering angles to thdirections in which coherent interference occurs and wthe Hamiltonian for an interacting pair of neutrinos in thform
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A. FRIEDLAND AND C. LUNARDINI PHYSICAL REVIEW D 68, 013007 ~2003!
id
dt S une~kW !ne~pW !&
une~kW !nm~pW !&
unm~kW !ne~pW !&
unm~kW !nm~pW !&
D 5A2GF
V~12cosb!S 2 0 0 0
0 1 1 0
0 1 1 0
0 0 0 2
D3S une~kW !ne~pW !&
une~kW !nm~pW !&
unm~kW !ne~pW !&
unm~kW !nm~pW !&
D , ~9!
whereV is the normalization volume. For a stateuS&, whichat t50 is unenm&, this equation formally has a solution
uS~ t !&5unenm&2unmne&
21e2 idEt
unenm&1unmne&2
,
~10!
wheredE52A2GF(12cosb)/V. In practice, however, oneonly needs a smallt expansion of Eq.~10!:
uS~ t !&.unenm&2 idEt/2~ unenm&1unmne&)
5~11 ia !unenm&1 iaunmne&, ~11!
where a52A2GF(12cosb)t/V. Indeed, since the interaction time and the normalization volume are determinedthe size of the neutrino wave packetl, t; l and V; l 3, theabsolute value ofa in any realistic physical situation is aways much less than 1. This simply means that the intetion between two neutrinos is described by the lowest orFeynman diagram and second order scattering effects masafely ignored.
We now return to the situation depicted in Fig. 2. We ainterested in the muon neutrino content of the beam aftecrosses the interaction region. Using the Hamiltonian in~9! to describe an elementary scattering eventne1nx , wefind
FIG. 3. Elementary scattering event that causes a change oflavor composition of the beam.
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uS~ t !&.unenx&1 ia~ unxne&1unenx&)
5~11 ia !unenx&1 ia cosaunene&
1 ia sinaunmne&. ~12!
Thus, for an elementary event, the amplitude of measuthe neutrino in the beam asnm is proportional to sina. If onemultiplies this amplitude by the number of scatterers asquares, as was previously done in the FCNC example,finds Pne→nm
}Ns2sin2a, which is in clear conflict with the
prediction of Eq.~8!, Pne→nm}Ns
2sin22a.Which of the two results is right? At first sight, the firs
possibility may appear more plausible. Consider, forstance, what happens when an electron neutrino propagthrough a background of muon neutrinos (a5p/2). Sincemuon neutrinos appear in the beam as a result of elemenexchanges between the beam and the background, onethink that for a pure muon neutrino background the convsion rate in the beam should be maximal. The first resindeed has this behavior, while the second result predictsnoconversion. One may therefore be tempted to conclude tthe second result fails to describe the system in this limit aperhaps that this failure signals the general breakdown ofvalidity of the single-particle description.
As will be shown in the next section, however, despiteseemingly paradoxical behavior, the second result is actucorrect. The explanation of the paradox lies in the procedof adding up elementary amplitudes, which in the case ofneutrino background is more subtle than in the case of onary matter.
III. MICROSCOPIC ANALYSIS OF THE EFFECT
The key observation is that a neutrino-neutrino scatterevent changes not only the state of the beam, but alsostate of the background. This implies that the refractionthe neutrino background is intrinsically different from thcase of ordinary matter, and requires a specific analysiestablish what states in the neutrino ensemble are amplcoherently.
We begin by recalling that an elementary scattering evcan only be amplified if the particles scatter ‘‘forwardmeaning that either particle momenta do not change (t chan-nel!, or that they are exchanged (u channel! as in Fig. 3. Theinteraction is described by Eq.~11!. For the sake of clarity, inthis section we consider the exchange diagrams onlyomit the effects of the non-exchange diagrams. The laproduce an overall~flavor-independent! phase shift of theneutrino states, and do not affect the neutrino flavor convsion.
We first consider a single electron neutrino in the beinteracting with several neutrinos in the background. Assuthat these background neutrinos are all in the same flastatenx , Eq. ~6!, and that the state of the background intially is a product of single-particle states,uxxx . . . xx&. Thetotal system therefore is initially in the stateue&uxxx . . . xx&and, as a result of the interaction, evolves over a time stedtaccording to
he
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NEUTRINO FLAVOR CONVERSION IN A NEUTRINO . . . PHYSICAL REVIEW D68, 013007 ~2003!
To measure thenm content of the beam, we introduce‘‘ nm number’’ operatorm[um&^mu. This operator acts onlyon the beam states and gives^xumux&5sin2a. To find theprobability that the beam neutrino after interacting withN2background neutrinos will be measured asnm , we computethe expectation value ofm in the final state of Eq.~13!:
^FumuF&5a2^xumux&~^exx . . . xu1^xex . . . xu
1 . . . !~ uexx . . . x&1uxex . . . x&1 . . . !
5a2sin2a@~N222N2!cos2a1N2#. ~15!
The last line is obtained by observing that in the sum thareN2 ‘‘diagonal’’ terms of the type
^x . . . xex . . . xux . . . xex . . . x&51
andN222N2 ‘‘off-diagonal’’ terms of the type
^x . . . xex . . . xux . . . exx . . . x&5cos2a.
The result in Eq.~15! shows how the connection betweea single scattering event and Eq.~8! is made. For a single
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scattering event,N251, one indeed finds that the conversioprobability Pne→nm
}sin2a is maximized when the back
ground consists of muon neutrinos. As the number of sctering events increases, however, the maximum of consion efficiency shifts to values ofa,p/2. In the limit oflargeN2, one findsPne→nm
}N22sin22a, precisely as predicted
by Eq. ~8!.We can now understand what happens to an electron
trino propagating in a muon neutrino background. Equat~15! shows that the conversion rate does not strictly vanas predicted by Eq.~8!, but is only proportional toN2, not toN2
2. As discussed before, the proportionality toN2 is a fea-ture of incoherent scattering. The coherent scattering paindeed strictly zero in this case: the statesuemmm . . . &,umemm . . . &, etc., aremutually orthogonal, and hence haveno overlap that could be coherently amplified.
It can be readily seen that the part of the final stateuF&that gets amplified contains the projection of the final baground states onto theinitial background stateuxxx . . . &.Decomposing the states inuF1& according to
ux . . . xex . . . &5^xue&ux . . . xxx . . . &
1^yue&ux . . . xyx . . . &, ~16!
wherey is the state orthogonal tox, ^yux&50, we can writethe result~15! as
We now return to the problem of two intersecting beams, one containing electron neutrinos and another containing nin the flavor superposition statenx . SupposeN1 neutrinos from the first beam interact withN2 neutrinos from the secondbeam. As a result of the interaction, the system evolves over timedt according to
As in the derivation of Eq.~17!, to separate the coherent and incoherent parts, the states inuF1& need to be projected on thcorresponding initial states and orthogonal states. This is done using Eq.~16! and
We observe that the term in the last line contains a summutually orthogonal states. This term represents the partwill not amplify coherently, as can be seen by repeatingarguments given in connection with Eq.~17!. We further notethat if onedrops this term, at first order ina the final stateequals
Equations~22!,~23!,~24! represent the central result of thsection. They show that if we take the initial state to beproduct of single-particle neutrino states and evolve ittime—carefully separating the coherent effects and dropp
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the incoherent ones—the final state obtained is againa prod-uct of single particle states, each rotated according to Eq~23!,~24!. No coherent superposition of many-particle stais formed.
It is important to emphasize that to arrive at this concsion we only needed to consider elementary scattering ev~exchanges! between the beam and the background. Nosumptions, such as decoherence between mass eigenstathe background, were necessary.
IV. ONE-PARTICLE EVOLUTION EQUATIONS
A. Evolution equation: active-active oscillations
The construction in the preceding section can be useobtain a differential equation describing the time evolutiona one-particle neutrino state. We first recall that in deriviEq. ~21! the flavor blind non-exchange interactions weomitted. It is easy to see that to include their effect oshould add an additional term,N1N2ueee. . . &uxxx . . . &, tothe final stateuF1& in Eq. ~21!,
We observe that, although Eq.~26! was obtained for theinitial statesne and nx , the derivation used only particlexchanges and did not in any way rely on the particuchoice of the initial states. Therefore, at any timet, if theinitial state of the beam neutrino isc and the backgroundneutrinos are all in the statef, the evolution over a smaltime step is
r
uc~ t1dt !&2uc~ t !&
52 iA2GFN2~12cosb!/Vdt
3@1/23~11 z^fuc& z2!uc&1^c'uf&^fuc&uc'&],
~28!
wherec' is the state orthogonal toc and we restored thecoefficients, including the angular factor.
Making use of the completeness relationuc&^cu1uc'&^c'u51, we rewrite Eq.~28! as
uc~ t1dt !&2uc~ t !&
52 iA2GFN2~12cosb!/Vdt
3@1/2uc&1uf&^fuc&21/2z^fuc& z2uc&]. ~29!
The evolution of a one-particle neutrino state is therefdescribed by the following differential equation:
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NEUTRINO FLAVOR CONVERSION IN A NEUTRINO . . . PHYSICAL REVIEW D68, 013007 ~2003!
iduc ( i )&
dt5Haauc ( i )&,
Haa5(j
A2GF
V~12cosb ( i j )!
3F1
21uf ( j )&^f ( j )u2
1
2z^f ( j )uc ( i )& z2G . ~30!
Here index~i! refers to a given particle for which the eqution is written, and~j! runs over all other neutrinos in thensemble. The summation over the scattering anglesb ( i j )
was introduced to make this equation applicable to a mgeneral case of neutrinos propagating in different directio
B. Evolution equation: active-sterile oscillations
The preceding analysis dealt with flavor conversionstween active neutrino states. The method developed tcan be extended to describe the conversions between aand sterile flavor states. In doing so, one should keep in mtwo important differences between the two cases. Inactive-sterile case:~i! only active components participatethe interactions and~ii ! the interaction amplitudes are proportional to the active content of the neutrino states involvPerforming calculations similar to those in Secs. III and IV~see the Appendix! we find that the single-particle Hamiltonian for this case is given by
As before,c ( i ) denotes the state of the neutrino for which tevolution equation is written andf ( j ) represents the flavostate of thej th neutrino in the background. It is worth notinthat this Hamiltonian includes both the effects of tt-channel and theu-channel diagrams.
For comparison, the standard Hamiltonian for an actisterile neutrino system@the analogue of Eq.~3!# is
Has( i )5(
j
2A2GF
V~12cosb ( i j )!S cos2a ( j ) 0
0 0D5(
j
2A2GF
V~12cosb ( i j )!z^f ( j )ue& z2ue&^eu. ~32!
Herea ( j ) is the mixing angle of thej th neutrino in the back-ground, cos2a(j)5z^f ( j )ue& z2.
C. Background of antineutrinos
We now consider flavor transformation of the neutribeam caused by the presence of antineutrinos in the bground. For concreteness, let us envision a modificationthe thought experiment depicted in Fig. 2 in which the becontains neutrinos in a superposition of flavor states,nz5cosbne1sinbnm , and the background containsantineutri-
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nosin a flavor superpositionnx5cosane1sinanm . To fix thenotation, let nw be the flavor state orthogonal tonz ,
^nwunz&50 and ny be the flavor state orthogonal tonx ,
^nyunx&50.Consider what elementary processes are possible in
case. As can be easily seen, in addition to thet- andu-channel processes,nznx→nznx andnznx→ nxnz , it is pos-sible to have s-channel annihilation diagramsnznz
→nznz ,nwnw ,nznz ,nwnw . Notice that only thenz compo-nent of thenx state participates in thes-channel processes.
The t-channel process is flavor diagonal and hence dnot cause flavor conversion, just like the corresponding pcess in the neutrino background. Theu-channel process putan antineutrino in the beam and therefore cannot intercoherently with the incident neutrino wave. Any cohereflavor changes, therefore, can only be due to thes-channelprocess.
Since only thenz component of thenx state participates inthe s-channel process, the beam neutrinos will not chanflavor if the background antineutrinos are in the orthogoflavor state,nw . While this is similar to what was found foa background of neutrinos in the statenw , on the micro-scopical level the two cases are quite different. For thenwbackground, the amplitude of flavor conversion for a sinelementary event is nonzero, but the conversion rate is oproportional toN, because the amplitudes add up incohently. By comparison, for thenw background already theelementary amplitude vanishes and thenw appearance rate inthe beam is strictly zero.~Instead,nw antineutrinoswill ap-pear in the beam at the rate proportional toN, due to theu-channel process.!
The elementary scattering event can be written as
unznx&→unznx&2 ia^nzunx&~ unznz&1unwnw&), ~33!
where thet-channel process as well as the processes thatan antineutrino in the beam have been omitted. The misign appears because the amplitudes for neutrino-neuand neutrino-antineutrino scattering processes have oppsigns.
As before, we project the final state on the initial staand orthogonal states,
unznx&→unznx&2 ia^nzunx&
3~^nxunz&unznx&1^nyunz&unzny&
1^nxunw&unwnx&1^nyunw&unwny&). ~34!
The rest of the argument proceeds in complete analogy tocase of the neutrino background. The first three termsparentheses in Eq.~34! will amplify coherently and the lasterm will not. Summing over many elementary scatterievents we obtain an expression similar to Eq.~21!, whichgives a one-particle evolution equation
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iduc ( i )&
dt5Haauc ( i )&,
Haa52(j
A2GF
V~12cosb ( i j )!
3F1
21uf ( j )&^f ( j )u2
1
2z^f ( j )uc ( i )& z2G , ~35!
wheref ( j ) denotes the flavor state of thej th antineutrino inthe background and the contributions of thet-channel pro-cesses have been included. Thus, the effects of the neuand antineutrino backgrounds on the coherent neutrino flaevolution have exactly the same form~but opposite signs!,even though at the microscopical level the two casesdifferent.
When both neutrinos and antineutrinos are present inbackground, their refractive effects add up linearly. Thefore, the Hamiltonian describing the flavor evolution ofneutrino beam equals the sum of the two contributions, E~30! and~35!. Further generalization to include the effectsother matter~electrons, nucleons! and vacuum oscillations isstraightforward. Just like in the case of the usual MSWfect, one should add the Hamiltonian termsHvac andHmat tothe neutrino induced Hamiltonian@see Eq.~1!#.
D. Comparison to the standard results
We now compare the one-particle Hamiltonians we haobtained, Eqs.~30!, ~31!, and~35!, to the corresponding accepted results, Eqs.~3! and ~32!. We see that, while the accepted results are similar to ours, there are important difences: in all three cases, our Hamiltonians contain additioterms. It is important to understand both the origin of thdifference and whether it leads to any physical conquences.
First, we would like to establish whether the presencethe additional terms in our results affects the flavor evolutof the neutrino system. As can be readily seen, in all thcases the additional terms are proportional to the idenmatrix in the flavor space. The evolution equations thus hthe form
idc
dt5@H01C~c,f!I#c, ~36!
whereH0 is the ‘‘standard’’ Hamiltonian given in Eq.~3! or~32! andI is the identity matrix. We observe that, ifc0(t) isthe solution of the equationidc/dt5H0c, thenc1(t) givenby
c1~ t !5expF2 i E t
C„c0~ t !,f0~ t !…d t Gc0~ t ! ~37!
is a solution of Eq.~36!. This means that the extra terms givan overall shift to both energy levels, without affecting netrino flavor evolution.
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Indeed, the physical origin of the difference in all threcases can be traced to the part of the interaction thatdoes notchange flavor. For concreteness, we for a moment specialto the first of the three cases, the active-active conversionthe neutrino background. It proves instructive to return toevolution of the beam neutrino over an infinitesimal timstep. In our case, the result is given by Eq.~26!, while theaccepted result, Eq.~3!, gives
ue8&5ue&1 iN2a@~11cos2a!ue&1sina cosaum&].~38!
The difference between the two is the factor of 1/2, whimultiplies the stateue& ~the flavor-preserving part! in thesquare brackets in Eq.~26!. This factor, in turn, can be traceto Eq. ~25!: it comes from the first term inuF1&, which mustbe splitbetween the beam and the background to avoid ovcounting. This is the origin of the factors of 1/2 in Eqs.~26!and ~27!.
The situation is not unlike what happens in electrostatThe interaction energy in a system of charges is given1/2( iqif i and the factor of 1/2 ensures that the interactenergy between pairs of charges is not counted twice. Incase, the extra terms serve the same purpose, to precounting the interaction energy twice. This can be seenfollows. Both active-active and active-sterile evolution equtions are particular cases of a general case when theflavor states have different couplings to theZ boson. Asshown in the Appendix, the evolution equation in this geeral case can be written in a form~A12!:
iduc ( i )&
dt5FH0
( i )21
2^c ( i )uH0
( i )uc ( i )&G uc ( i )&, ~39!
whereH0( i ) is the generalization of the standard Hamiltoni
@10#,
H0( i )5(
j
A2GF
V~12cosb ( i j )!@G~h!uf ( j )&^f ( j )uG~h!
1G~h!^f ( j )uG~h!uf ( j )&#, ~40!
with G being the matrix of couplings,
G~h!5S 1 0
0 h D . ~41!
The second term in the evolution equation~39! has theform of the expectation value 1/2^c ( i )uH0
( i )uc ( i )&. This formmakes explicit the physical meaning of this term as a corrtion to avoid double counting of the energy of the systemis worth emphasizing that this term isalwaysproportional tothe identity in the flavor space, even when the two flavstates have different couplings. Physically, this is becausepart of the interaction that needs to be split is always the pthat conserves flavor~see the Appendix!.
The extra term, 1/2c ( i )uH0( i )uc ( i )&, depends not only on
the state of the background but also on the state of the bitself. We therefore caution the reader that the superposiprinciple, which is always used for the MSW effect in no
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NEUTRINO FLAVOR CONVERSION IN A NEUTRINO . . . PHYSICAL REVIEW D68, 013007 ~2003!
mal matter, does not apply in this case. For instance, supthat a beam neutrino which isne at t50 as a result of theevolution becomes a staten8 and, similarly, a beam neutrinwhich isnm at t50 becomes a staten9. Then, it is in generalnot true that the statenx5cosane1sinanm will becomecosan81sinan9.
It would be incorrect to conclude that the extra terms hano physical effect whatsoever. While they indeed dochange neutrino flavor evolution, they do modify the ablute value of the refraction index of a neutrino medium anhence, at least in principle, change the bending of a neutbeam in a dense neutrino medium with a density gradiThis effect is present even if there is only one neutrino flain the system.
V. MORE ON THE ENTANGLED SYSTEM
We have shown that if the neutrino system initially donot contain entangled states, such states are not formedresult of coherent evolution in the system. It can be argumoreover, that such evolution can lead to an effective loscoherence between entangled states. To illustrate this, leconsider a beam in an entangled flavor state,
uent&[~ uxxx. . . .&1uyyy. . . .&)/A2, ~42!
~herex andy are not necessarily orthogonal! propagating inthe ~unentangled! backgrounduzzz. . . &.
At time t50 the expectation value of some operator,example, thenm number operator,m5(um&^mu, where thesum runs over all particles in the beam, is given by
The first two terms on the right-hand side~rhs! simply countthe muon neutrino content in the statesuxxx . . . & anduyyy . . . & and the last two terms represent the effect of etanglement.
Let us consider the effects of time evolution on the exp
tation value ofm. Each of the two terms inuent& is a productof single-particle states and according to our earlier findinover time will remain a product of single-particle states. Lus write the state at timet5t1 as
~ ux8x8x8 . . . .&uz8z8z8 . . . &
1uy9y9y9 . . . .&uz9z9z9 . . . &)/A2, ~44!
wherez8, z9, x8, andy9 are the results of solving a systeof equations given in Eq.~30!. ~For example,z8 andx8 arefound by solving the equations for the initial stauxxx . . . &uzzz. . . &.!
The expectation value ofm in the state~44! is given by
Since the statesz8 and z9 will generically be different, theabsolute value of the inner product^z9uz8& will be ,1. Thelast two terms in Eq.~45! therefore contain suppression fators z^z9uz8& zN2 and vanish as the number of neutrinos in tbackgroundN2 is taken to infinity. As already mentionedthese terms represent the entanglement between thestates; the system therefore behaves as if the beam waincoherent mixture ofux8x8x8 . . . & and uy9y9y9 . . . &.
Of course, rigorously speaking, the entanglement infmation is not completely lost in the system. It may happthat at some timet the statesz8 and z9 will be such thatz^z9uz8& z51. In this case, the entanglement effect will reapear. We, however, regard this as an artificial arrangemand therefore maintain that for practical purposes the coence is destroyed.
It is curious to note as an aside that, as the entanglemeffect reappears, the phases of the sta
woan
r-n
-ntr-
nts
ux8x8x8 . . . .&uz8z8z8 . . . & and uy9y9y9 . . . .&uz9z9z9 . . . &)will have an effect on the expectation value ofm and hencethe phases due to the additional term, introduced in Eq.~30!to avoid overcounting of energies, will have a physicalfect.
VI. CONCLUSIONS
In summary, we have developed a conceptually simpphysical picture of coherent neutrino evolution in a mediuof neutrinos and have shown how coherent interferencemany elementary scattering events gives rise to the refracphenomenon. Unlike the case of ordinary matter, in whcoherent scattering leaves the background unchanged, inneutrino background the scattering events change the stathe background, thus requiring a different type of analys
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A. FRIEDLAND AND C. LUNARDINI PHYSICAL REVIEW D 68, 013007 ~2003!
We have found that only part of the elementary scatteramplitude is amplified coherently. This explains certaseemingly paradoxical results, such as why a neutrino teling through the medium of neutrinos of opposite flavdoes not undergo coherent flavor conversion.
We have shown that refraction does not lead to the cation of entangled states in a neutrino system, i.e., if the sof the system is initially described by a product of singparticle states, the state remains a product of single-parstates as the system is evolved in time. Furthermore,evolution effectively destroys initial entanglement in the stem. It follows that for each neutrino the result of the cohent evolution can be described by a one-particle Schro¨dingerequation, as was assumeda priori in the literature.
We have derived the one-particle equation for actiactive and active-sterile flavor transformation scenarios foneutrino in a neutrino background. We also derived the eqtion for a neutrino in an antineutrino background. In all thecases, we found that in order to avoid overcounting ofinteraction energy one has to introduce an extra term inevolution equations that is not present in the standard anses. We have proven that this extra term does not affectflavor evolution under normal conditions. It does, howevaffect the value of the refraction index and hence the bendof a neutrino beam in a dense neutrino medium.
ACKNOWLEDGMENTS
We thank Georg Raffelt, James Pantaleone, and Johncall for stimulating discussions and for valuable commeon the manuscript. We also acknowledge valuable commfrom John Cornwall, Wick Haxton, Maxim Perelstein, Ramond Sawyer, and Raymond Volkas. A.F. was supporteLANL by the Department of Energy, under contract W-740ENG-36, and at IAS by the Keck Foundation. C.L. was suported by the Keck Foundation and by the National ScieFoundation grant PHY-0070928. We acknowledge the hotality of the Kavli Institute for Theoretical Physics, whethis research was supported in part by the National ScieFoundation under Grant No. PHY99-07949.
APPENDIX: THE CASES OF ACTIVE-STERILENEUTRINOS AND NEUTRINOS WITH DIFFERENT
COUPLINGS
Let us derive the result~31!, which refers to a system oone active and one sterile neutrino,ne and ns . We start byconsidering the elementary interaction between two neu
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nos, as discussed in Sec. II B. The analogue of Eq.~9! in thiscase is
id
dt S une~kW !ne~pW !&
une~kW !ns~pW !&
uns~kW !ne~pW !&
uns~kW !ns~pW !&
D 5A2GF
V~12cosb!S 2 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
D3S une~kW !ne~pW !&
une~kW !ns~pW !&
uns~kW !ne~pW !&
uns~kW !ns~pW !&
D , ~A1!
which shows that, given an initial stateunz(kW )nx(pW )&, onlyits active-active component,une(kW )ne(pW )&, is affected by theevolution:
unz~kW !nx~pW !&⇒unz~kW !nx~pW !&2 idt2A2GF
V~12cosb!^euz&
3^eux&une~kW !ne~pW !&. ~A2!
Next, we apply this result to the case of two orthogonneutrino beams, in the spirit of what was done in Sec. III. Wtake neutrinos in the first and second beams to be instatesuz& and ux&, respectively, and omit neutrino momenfor simplicity. Similarly to Eq.~18!, we get
NEUTRINO FLAVOR CONVERSION IN A NEUTRINO . . . PHYSICAL REVIEW D68, 013007 ~2003!
where uy& and uw& are the orthogonal states toux& and uz&,respectively (xuy&5^zuw&50). The last term in Eq.~A5! isnot coherently enhanced and therefore can be dropped.allows us to obtain, at first order ina, a factorized form:
3 z^f ( j )ue& z2@ ue&^eu21/2z^c ( i )ue& z2#uc ( i )&,
~A8!
which proves the result~31!.We now discuss the generalization of our findings to t
active neutrinos with different couplings to theZ boson. Thestudy of this case provides a unified description of the resfor active-active and active-sterile cases we have discusso far. Furthermore, it allows us to compare our results wthe corresponding discussion given in Ref.@10#.
Let us consider two neutrino states,ne andnr , and takene as having the ordinary standard model coupling,ge , withthe Z boson. The coupling ofnr with the Z is defined asgr
[hge . In terms ofh, the two-neutrinos equation,~A1!, isgeneralized by the replacementsns→nr and
S 2 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
D →B~h
