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Structure of standard DGLAP inputs for initial parton densities and the role of the singular terms B.I. Ermolaev, M. Greco, S.I. Troyan Spin-05 Dubna Sept. 27- Oct. 1, 2005
Structure of standard DGLAP inputs for initial parton densities and the role of the
singular terms
B.I. Ermolaev, M. Greco, S.I. Troyan
Spin-05 Dubna Sept. 27- Oct. 1, 2005
Deep Inelastic e-p Scattering
Incoming lepton
outgoing lepton- registered
Incoming hadron
Produced hadrons - not registered
k
p
K’
X
Deeply virtual photon
q
Leptonic tensor
hadronic
tensor W
hadronic tensor consists of two terms: spindunpolarize WWW
Does not depend on spin
Spin-dependent
The spin-dependent part of Wmn is parameterized by two
structure functions:
),(),( 2
22
1 Qxgppq
SqSQxgSqi
pq
mW
spin
Structure functions
where m, p and S are the hadron mass, momentum and spin; q is the virtual photon momentum (Q2 = - q2 > 0). Both of the functions depend on Q2 and x = Q2 /2pq, 0< x < 1. At small x:
)),(),((),( 22
21
21
|| QxgQxgSQxgSqipq
mW
spin
longitudinal spin-flip transverse spin -tlip
When the total energy and Q2 are large compared to the mass scale, one can use the factorization and represent W as a convolution of the the partonic tensor and probabilities to find a polarized parton (quark or gluon) in the hadron :
spinW
Wquark
quark
Wgluon
gluon
q
q
pp
gluongluon
quarkquarkparton WWWW
Probability to find quark
Probability to find gluon
DIS off quark
DIS off gluon
In the analytic way this convolution is written as follows:
DIS off quark and gluon can be studied with perturbative QCD, with calculating involved Feynman graphs.
Probabilities, quark and gluon involve non-perturbaive QCD. There is no a regular analytic way to calculate them. Usually they are defined from experimental data at large x and small Q2 , they are called the initial quark and gluon densities and are denoted q and g . The conventional form of the hadronic tensor is:
The standard instrument for theoretical investigation of the polarized DIS is DGLAP. The DGLAP –expression for the non-singlet g1 in the Mellin space is:
gWqWW gluonquark
Dokshitzer-Gribov-Lipatov-Altarelli-Parisi
2
2
))(,(exp)()(1
2)2/(),( 2
2
222
1
Q
sDGLAPDGLAP
i
iqNSDGLAP k
k
dkqC
xi
deQxg
Coefficient function Anomalous dimension
Initial quark density
Expression for the singlet g1 is similar, though more involved. It includes more coefficient functions, the matrix of anomalousdimensions and, in addition to q , the initial gluon density g
Coefficient function CDGLAP evolves the initial quark density q : )()(),( xqyqyxC
Anomalous dimension governs the Q2 -evolution of q
Evolved quark
distribution
Pert QCD
Non-Pert QCD
... )( )2/)(()( )4/)(()(
... )( )2/)((1)( )1(22)0(2
)1(2
CQC
ssDGLAP
sDGLAP
LO
LO
NLO
NLO
In DGLAP, coefficient functions and anomalous dimensions are known with LO and NLO accuracy
One can say that DGLAP includes both Science and Art :
matrix of Gribov, Lpatov, Ahmed, Ross, Altarelli, Parisi, Dojshitzer
matrix of 1
Floratos, Ross, Sachradja, Gonzale- Arroyo, Lopes, Yandurain, Kounnas, Lacaze, Gurci, Furmanski, Peronzio, Zijlstra, Merig, van Neervan, Gluck, Reya, Vogelsang
Coefficient functions C(1)
k , C(2)k
Bardeen, Buras, Duke, Altarelli, Kodaira, Efremov, Anselmino, Leader, Zijlstra, van Neerven
SCIENCE
ART
There are different its for q and g. For example,
Altarelli-Ball-Forte-Ridolfi)]/1(ln)/1([ln
)]1()1[(
xxxNq
xxNxq
Parameters N, , , , should be fixed from experiment
This combination of science and art works well at large and small x, though strictly speaking, DGLAP is not supposed to work at the small- x region:
DGLAP
1/x
1
Q2
ln(1/x) < ln(Q2)
DGLAP accounts for ln(Q2) to all orders in s and neglects
ks
ks xx ))/1ln(( ,))/1(ln( 2 with k>2
ln(1/x)> ln(Q2)
However, these contributions become leading at small x and should be accounted for to all orders in the QCD coupling.
Total resummation of logs of x cannot be done because of the DGLAP-ordering – the keystone of DGLAP
2
K3
K2
K1
DGLAP –ordering:
22 3
2 2
2 1
2 k k k Q good approximation for large x when logs of x can be neglected. At x << 1 the ordering has to be lifted
q
p DGLAP small-x asymptotics of g1 is well-known:
)/(Qln n [ln(1/x)]l exp ~ 2QCD
21 g
When the DGLAP –ordering is lifted, the asymptotics is different:
2/22 1 )/(Q (1/x) ~ g Bartels- Ermolaev-
Manaenkov-Ryskin
Non- singlet intercept ,)/3(8 1/2s NS singlet intercept 1/2
s )/2(3 3.5 S
The weakest point: s is fixed at unknown scale. DGLAP : running s
)( 2Qss Arguments in favor of the DGLAP- parameterization
Bassetto-Ciafaloni-Marchesini- Veneziano, Dokshitzer-Shirkov
K
K’
K
K’
K
K’
)( 2 kss Origin: in each ladder rung
DGLAP-parameterization
However, such a parameterization is good for large x only. At x << 1 :
)/)((
))'(( 2'2
2
xkk
kk
s
ss
Ermolaev-Greco-Troyan
Obviously, this parameterization and the DGLAP oneconverge when x is large but differ a lot at small xSo, in the small-x region, it is necessary: 1. Total resummation of logs of x2. New parameterization of s
The basic idea: the formula ))/(kln /(1)( 222 bks valid when k2>>2
it is necessary to introduce an infrared cut-off for k2
It is convenient to introduce in the transverse space: k2 >> 2 Lipatov
As value of the cut-off is not fixed, one can evolve the structure functions with respect to the name of the method:
Infra-Red Evolution Equations (IREE)
IREE for the non-singlet g1 in the Mellin space looks similar to the DGLAP eq:
1221 )()2/1(
)/ln(Q gHg
new anomalous dimension H()
accounts for the total resummation of double- and single- logs of x
Contrary to DGLAP, H () and C () can be calculated with the same method.Expressions for hem are:
)]H( - /[ )C(
]))B( - ( - (1/2)[ )( 1/22
H B () is expressed through
conventional QCD parameters:
1
)(/)(ln ed)2/1(
])(
e d- (1/b)[ A
/2N)D,(C CA
22-
0
2
22
-
022
FF
bD
B
,)/122n - (33 b
)/ln(
,3/4
f
22
FC
Expression for the non-singlet g1 :
)(2221 )/(
)H( -
)q( (1/x)
2)2/(
H
i
i
qNS Q
i
deg
Expression for the singlet g1 is similar, though more involved. When x 0,
2/22gq
2q
1
2/222q
1
//1 g] Z q [Z2
e
//1 2
e
SS
NSNS
Qxg
Qxg
S
NS
The x-dependence perfectly agrees with results of several groups who fitted experimental data. The Q2 –dependence has not been checked yet
Soffer-Teryaev, Kataev-Sidorov-Parente, Kotikov-Lipatov-Parente-Peshekhonov-Krivokhijine-Zotov, Kochelev-Lipka-Vento-Novak-Vinnikov
intercepts NS = 0.42 S = 0.86.
Comparison between our and DGLAP results for g1 depends on the assumed shape of initial parton densities.
The simplest case: the bare quark input
(x) )( xq 1 )( q
in x- space in Mellin space
Numerical comparison shows hat impact of the total resummation of logs of x becomes quite sizable at x = 0.05 approx. Hence, DGLAP should fail at x < 0.05. However, it does not take place. In order to understand what could be the reason to it, let us give more attention to structure of Standard DGLAP fits for initial parton densities. For example,
])1)(x 1[( x)( - xNxq Altarelli-Ball-Forte-Ridolfi
normalization singularfactor
75.0,5.17,5.2,53.0
In the Mellin space this fit is
)])1()(()[()( 11
1
1
kkcNqk
k
Leading pole
Non-leading poles <
the small-x DGLAP asymptotics of g1 is (inessential factors dropped )
(1/x) ~1DGLAPg
Comparison it to our asymptotics
NSx /1~g 1
shows that the singular factor x- in the DGLAP fit mimics the total resummation of ln(1/x) . However, the value = 0.53 differs from our intercept
phenomenology
calculations
Comparison between our and DGLAP results for g1 depends on the assumed shape of initial parton densities.
The simplest case: the bare quark input
Numerical comparison shows hat impact of the total resummation of logs of x becomes quite sizable at x = 0.05 approx.
(x) )( xq 1 )( q
in x- space in Mellin space
Hence, DGLAP should fail at x < 0.05. However, it does not take place. In order to understand what could be the reason to it, let us give more attention to structure of Standard DGLAP fits for initial parton densities.
For example,
])1)(x 1[( x)( - xNxq Altarelli-Ball-Forte-Ridolfi
normalization singularfactor
75.0,5.17,5.2,53.0
Although both our and DGLAP formulae lead to x- asymptotisc ofRegge type, they predict different Q2 -asymptotics: our predictionIs the scaling
2/222 1 /~g
xQ
whereas DGLAP predicts the steeper x-behavior and the flatter Q2 -behavior:
)(21 )(ln(1/x) ~ Qg DGLAP
x-asymptotics is checked with extrapolating available exp data to x 0.
Agrees with our values of Contradicts DGLAP
Q2 –asymptotics has not been checked yet.
our calculations
DGLAP fit
Structure of DGLAP fit
])1)(x 1[( x)( - xNxq
Can be dropped when ln(x) are resummed
x-dependence is weak at x<<1 and can be dropped
Common opinion: fits for q are singular but convoluting them with coefficient functions weakens the singularity
)()(),( xqyqyxC Obviously, it is not true,q and q are equally singular
Common opinion: DGLAP fits mimic structure of hadrons, they describe effects of Non-Perturbative QCD, using many phenomenological parameters fixed from experiment.
Actually, singular factors in the fits mimic effects of Perturbative QCD and can be dropped when logarithms of x are resummed
Non-Perturbative QCD effects are accumulated in the regular parts of DGLAP fits. Obviously, impact of Non-Pert QCD is not strong in the region of small x. In this region, the fits approximately = overall factor N
WAY OUT – synthesis of our approach and DGLAP
1. Expand our formulae for coeff functions and anom dimensions into
series in s
2. Replace the first- and second- loop terms of the expansion by
corresponding DGLAP –expressions
New, “synthetic” formulae accumulate all advantages of the both
approaches and are equally good at large and small x
DGLAP
Good at large x because
includes
exact two-loop calculations for
C and but lacks the total
resummaion of ln(x)
our approach
Good at small x , includes
the total resummaion of ln(x) for C
and but bad at large x because
Neglects some contributions
essential in this region
Conclusion
Total resummation of the double- and single- logarithmic contributions
New anomalous dimensions and coefficient functions
At x 0, asymptotics of g1 is power-like in x and Q2
New scaling:
g1 ~ (Q2/x2)-
With fits regular in x, DGLAP would become unreliable at x=0.05 approx
Singular terms in the DGLAP fits ensure a steep rise of g1 and mimic the resummation of logs of x. With the resummation accounted for,they can be dropped.
Regular factors can be dropped at x<<1, so the fits can be reduced down to constants
DGLAP fits are expected to correspond to Non-Pert QCD. Instead, they basically correspond to Pert QCD Non-Pert effects are surprisingly small at x<<1
