XXXIII International Conference of Theoretical Physics MATTER TO THE DEEPEST: Recent Developments in...

XXXIII International Conference of Theoretical Physics

MATTER TO THE DEEPEST: Recent Developments in Physics of Fundamental Interactions, USTROŃ'09

Ustron, PolandSeptember 11-16, 2009

Hamzeh Khanpour

In collaboration with:

Ali Khorramian, Shahin Atashbar Tehrani

Semnan Universityand

School of Particles and Accelerators, IPM (Institute for Studies inTheoretical Physics and Mathematics), Tehran, IRAN

Determination of valence quark densities up to higher order of perturbative QCD

Hamzeh Khanpour (IPM & Semnan University) [email protected] USTRON 2009

Outline

• Abstract• Introduction• DIS Kinematics• Parton distributions and hard processes (1,2)• What we need for 4-loop calculations?• Splitting functions and Coefficient functions• Pade’ approximate• The method of the QCD analysis of Structure Function

• The procedure of the QCD fits of F2 data

• Results and Conclusion


Abstract

• We present the results of our QCD analysis for non-singlet unpolarized quark distributions and structure function F2(x,Q2) up to N3LO. The analysis is based on the Jacobi polynomials technique of reconstruction of the structure functions from its Mellin moments.

• The fit results for the non-singlet parton distribution function, their correlated errors and evolution are presented. Our results on and strong coupling constant up to N3LO are presented and compared with those obtained from deep inelastic scattering processes. In the N3LO analysis, the QCD scale and the strong coupling constant were determined with the help of Pade-approximation.

QCD

QCD


Introduction• Complete knowledge of the parton distributions in the nucleon is essential for calculating deep

inelastic processes at high energies.• All calculations of high energy processes with initial hadrons, whether within the standard

model or exploring new physics, require parton distribution functions (PDF's) as an essential input. The assessment of PDF's, their uncertainties and extrapolation to the kinematics relevant for future colliders such as the LHC is an important challenge to high energy physics in recent years.

• Parton distributions form indispensable ingredients for the analysis of all hard-scattering processes involving initial-state hadrons.

• Presently the next-to-leading order is the standard approximation for most important processes but the N2LO and N3LO corrections need to be included, however, in order to arrive at quantitatively reliable predictions for hard processes at present and future high-energy colliders.

• For quantitatively reliable predictions of DIS and hard hadronic scattering processes, perturbative QCD corrections beyond the next-to-leading order, N2LO and N3LO, need to be taken into account.

• For at least the next ten years, proton (anti-) proton colliders will continue to form the high-energy frontier in particle physics. At such machines, many quantitative studies of hard (high mass/scale) standard-model and new-physics processes require a precise understanding of the parton structure of the proton.


The deep-inelastic lepton-nucleon scattering is the source of important information about the nucleons structure.

DIS Kinematics

Deep inelastic scattering in the quark parton model.

The negative four momentum transfer squared at the electron vertex

The Bjorken scale variable x and the inelasticity y


Parton distributions and hard processes (1)

inclusive photon-exchange deep-inelastic scattering (DIS)

At zeroth order in the strong coupling constant the hard coefficient functions ca,i are trivial, and the momentum fraction carried by the struck quark i is equal to Bjorken-x if mass effects are neglected.

In general, the structure functions are given by


Parton distributions and hard processes (2)

Parton distributions fi : evolution equations ( = Mellin convolution)⊗

The initial conditions are not calculable in perturbative QCD.

The task of determining these distributions can be divided in two steps. The first is the determination of the non-perturbative initial distributions at some (usually rather low) scale Q0. The second is the perturbative calculation of their scale dependence (evolution) to obtain the results at the hard scales Q. The initial distributions have to be fitted using a suitable set of hard-scattering observables.


Determination of the parton density functions from experimental data is according to the following procedure:The parton density functions are parameterised by smooth analytical functions at a low starting scale Q0

2 as a function of x with a certain number of free parameters. They are evolved in Q2 using the DGLAP equations. Afterwards predictions for structure functions and cross-sections are calculated. The free parameters are determined by performing a fit to the data. Several constraints are imposed during this procedure.

For an accurate determination of the parton density functions the coefficient and splitting functions have to be precisely known as they govern the calculation of the structure functions and the evolution, respectively. Since 2005 all of them are known to next-to-next-to-leading order (N2LO).

2

Procedure


what we need for 4-loop calculations?

The splitting functions P(n) and the process-dependent hard coefficient functions ca admit expansions in powers of strong coupling constant

The first n+1 terms define the NnLO approximation.

N2LO: N3LO:

)(),( 22,2 nPnc NS

)(),( 33,2 nPnc NS


In spite of the unknown 4–loop splitting functions one may wonder whether any statement can be made on the non–singlet parton distributions and at the 4–loop level !!!!!!!

Splitting functions and Coefficient functions

The complete NNLO splitting functions and coefficient functions are known )(),( 2

)2(,2 nPnc NS

)(,3 nP NS

[J. A. M. Vermaseren, A. Vogt and S. Moch, Nucl. Phys. B 724 (2005) 3][S. Moch, J. A. M. Vermaseren and A. Vogt, Nucl. Phys. B 688 (2004) 101]

[J. A. M. Vermaseren, A. Vogt and S. Moch, Nucl. Phys. B 724 (2005) 3]

The 3-loop coefficient functions are known )()3(,2 nc NS

QCD

nP )(

,2nNSc


The answer is: Pade’- approximate


[J. Ellis, E. Gardi, M. Karliner and M.A. Samuel, Phys. Rev. D54 (1996) 6986. ]

[J. Ellis, E. Gardi, M. Karliner and M.A. Samuel, Phys. Lett. B366 (1996) 268. ]

[M.A. Samuel, J. Ellis and M. Karliner, Phys. Rev. Lett. 74 (1995) 4380. ]

Padé estimates for the large-N behaviour of the three-loop contributions to the non-singlet coefficient function C2,ns(N). The three-loop approximants are compared with exact results.

Coefficient functions )(,2nNSc

)(

)()(

)1(

2)2()3(

nc

ncnc

3)1()1()2()3( )()()(2)( ncncncnc

Pade’ [0/2]

Pade’ [1/1]


The perturbative expansion of the non-singlet N-space coefficient functions

.....)()()()( )2(2)1()0(,2 ncancancnC ns

4

sa


4-loop Splitting functions )(3 nP

Pade’ [0/2]

Pade’ [1/1]

)1(

)()(

1

22

3

P

nPnP

20

310123 )(/)()(/)()(2)( nPnPnPnPnPnP


The perturbative expansion of the anomalous dimension

)()()()()( )3(4)2(3)1(2)0( nananananNS

4

sa


The method of the QCD analysis of SFOne of the simplest and fastest possibilities in the structure function reconstruction from the QCD predictions for its Mellin moments is Jacobi polynomials expansion.

[ A. N. Khorramian and S. A. Tehrani, Phys. Rev. D 78 (2008) 074019. ]


[ S. Atashbar Tehrani and A. N. Khorramian, JHEP 0707 (2007) 048 ]

Structure function in x spaceF2 Structure function as extracted from the DIS ep process can be, up to N3LO written as

The combinations of parton densities in the non-singlet regime and the valence region x>0.3 for F2

p and F2d in LO is

In the region x<0.3 for the difference of the proton and deuteron data we use


The evolution equations are solved in Mellin-N space and the Mellin transforms of the above distributions are denoted by respectively.

The non–singlet structure functions are given by

),(),,( 2,

2 QNfQNf vdp

NS

Here denotes the strong coupling constant and are the non–singlet Wilson coefficients in

4/)()( 22 QQa ss ))(( 2QNCi )( i

saO


}))(ˆ)(ˆ()(6

1)](ˆ)(

)(ˆ)(ˆ[))(ˆ)(ˆ)()((2

1

)](ˆ)2()(ˆ)()(ˆ

)(ˆ)[(3

1))(ˆ)(ˆ()(

2

1

)](ˆ)()(ˆ)(ˆ)[(2

1

)](ˆ)(ˆ)[(1

1{)(),(),(

30

0

11

303

00

0

220

21

10

120

0

11

20

2002

0

00

320

3130

31

10

220

21

20

1

330

3

0

20

0

11

202

0

00

220

21

10

12

20

2

0

00

110

0

/)(ˆ

0

20

2 00

NPNPaaNP

NPNPNPNPaaaa

NPNPNP

NPaaNPNPaa

NPNPNPaa

NPNPaaa

aQNFQNF NP

kk

The solution of the non–singlet evolution equation for the parton densities to 4–loop order reads

Here, denote the Mellin transforms of the (k + 1)–loop splitting functions.KP

[J. Blumlein, H. Bottcher and A. Guffanti, Nucl. Phys. B 774, 182 (2007)][ A. N. Khorramian and S. A. Tehrani, Phys. Rev. D 78 (2008) 074019. ]

The procedure of the QCD fits of F2 data

In the present analysis we choose the following parameterization for the valence quark densities

By QCD fits of the world data for , we can extract valence quark densities using the Jacobi polynomials method. For the non-singlet QCD analysis presented here we use the structure function data measured in charged lepton proton and deuteron deep-inelastic scattering.

dpNS FF ,22 ,

[J. Blumlein, H. Bottcher and A. Guffanti, Nucl. Phys. B 774, 182 (2007)]



Results




Table: Parameters values of the LO, NLO, NNLO and N3LO non-singlet QCD fit at Q02 = 4 GeV2

Fit results


Comparison of the parton densities xuv and xdv at the input scale Q02 = 4 GeV2 and at

different orders in QCD as resulting from the present analysis.

Parton densities at the input scale Q02 = 4 GeV2


The parton densities xuv and xdv at the input scale Q02 = 4 GeV2 (solid line)

compared to results obtained from N3LO analysis by BBG (dashed line).




The parton densities xuv(x,Q2) and xdv(x,Q2) at N3LO evolved up to Q2 = 10000 GeV2 (solid line) compared to results obtained by BBG (dashed line).

The structure functions F2p and F2

d as function of Q2 in intervals of x. Shown are the pure QCD fit in N3LO (solid line) and the contributions from target mass corrections TMC (dashed line) and higher twist HT (dashed–dotted line). The arrows indicate the regions with W2 >12.5 GeV2. The shaded areas represent the fully correlated 1σ statistical error bands.

The structure function F2NS as function of Q2 in intervals of x. Shown is the pure QCD fit in N3LO

(solid line).

Conclusion

• We performed a QCD analysis of the non–singlet world data up to N3LO and determined the valence quark densities xuv(x,Q2) and xdv(x,Q2) with correlated errors.

• Parameterizations of these parton distribution functions and their errors were derived in a wide range of x and Q2 as fit results at LO, NLO, NNLO, and N3LO.

• In the analysis the QCD scale and the strong coupling constant αs(M2Z),

were determined up to N3LO.• In spite of the unknown 4–loop splitting functions we applied Pade’-

approximate for determination of non–singlet parton distributions and QCD at the 4–loop level.

• Our calculations for non-singlet unpolarized quark distribution functions based on the Jacobi polynomials method are in good agreement with the other theoretical models.


Thanks to attention !!


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XXXIII International Conference of Theoretical Physics MATTER TO THE DEEPEST: Recent Developments in...

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