Institute for studies in theoretical Physics and Mathematics,IPM

Yazd University, IRAN

A Talk given by

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OutlineOutline

RS and FS dependence in QCD predictions RS and FS dependence in QCD predictions Complete RG ImprovementComplete RG Improvement Valon modelValon model Unpolarized valon distribution in a protonUnpolarized valon distribution in a proton Polarized valon distributionPolarized valon distribution Parton distribution functions in moment spaceParton distribution functions in moment space Comparison of the MomentsComparison of the Moments Moments of PPDF’s and distribution functionMoments of PPDF’s and distribution function ConclusionConclusion

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RS and FS dependence in RS and FS dependence in QCD predictionsQCD predictions

For a single case of a dimensional observable R(Q) with perterbative series

The RS can be labeled by the non-universal coefficients of the beta-function and ΛRS.

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Self-consistency of perturbation theory that is the derivative of N-th order

approximant R(n) With respect to ln(µ) is of higher order than the approximant

itself, will yields

2

where

Which is

which is a renormalization scale and scheme invariant..

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For other coefficients we will arrive first on following partial derivatives for r2

and for r3,r4,… we will have

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on integration one finds

and finally

General structure is as follows

Where Xn are Q-independent and RS-invariant and are unknown unless a complete NnLO calculation has been performed.

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Given a NLO calculation of r1, the sum of all-orders of the RG-predictable term simplifies to a geometric progression:

which is completely independent of renormalization µ scale

Complete RG ImprovementComplete RG Improvement

From pervious results for r1 , r2, …

At NLO calculation r1 is known but X2, X3,…are unknown. Thus the complete subset of known terms in above at NLO is

It is RS-invariant. Choose r1=0, c2=c3=…=cn=0 we obtain a0=a.

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At NNLO calculation X2 is unknown. Further infinite subset of terms are known and can be resummed to all orders.

Finally we will arrive at

where a0=a(0,0,0, …) is the coupling in this scheme and satisfies

The solution of this transcendental equation can be written in closed form in terms of the Lambert-W function

where Lambert-W function is defined by

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What is the valon model?What is the valon model?

The valon model describes the hadron structure relevant to multiparticle production.

The valon is defined to be a dressed valence quark in QCD with the cloud of gluons and sea quarks which can be resolved by high Q^2 probes. At sufficiently low value of Q^2 the internal structure of a valon cannot be resolved and hence it behaves as a valence quark.

For example in the proton we have 2 U type valons and one D valon.

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Valon model

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To facilitate the phenomenological analysis Hwa assumed a simple form for the exclusive valon distribution in unpolarized protonwhich is

where yi is the momentum fraction of the i-th valon.The inclusivevalon distributions are obtained by integration

Unpolarized valon distribution in a proton

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Unpolarized

U-valon distribution

PolarizedU-valon

distribution

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Polarized valon distribution

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Parton distribution functions in moment space

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Comparison of the Moments

Comparison of Non-singlet moments in standard perturbative QCD and CORGI approach.

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Comparison of singlet moments in standard perturbative QCD and CORGI approach.

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By having the moments of polarized valon distributions, the determination of the moments of parton distributions in a proton can be done strictly. The distributions that we shall calculate are δuv, δdv, δ. and δg.

• Moments of PPDF’s and distribution function

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Comparison of polarized valance u and d quark distributions and polarized gluon distribution in standard perturbative QCD and in CORGI approach.

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First we introduced the idea of complete RG- improvement and showed how we First we introduced the idea of complete RG- improvement and showed how we can reconstructed the QCD perurbatives series in this approach.can reconstructed the QCD perurbatives series in this approach.

Valon model was defined and unpolarized and polarized valon distributionsValon model was defined and unpolarized and polarized valon distributions were obtained.were obtained.

The nonsinglet and singlet moments of polarized valons were obtained and The nonsinglet and singlet moments of polarized valons were obtained and the results in standard perterbative series and the CORGI approch were the results in standard perterbative series and the CORGI approch were comparied.comparied.

Using the convelution of moments in valon and the moments of valon in proton Using the convelution of moments in valon and the moments of valon in proton and finally the inverse melleion technique, the PDF’s were obtained and and finally the inverse melleion technique, the PDF’s were obtained and compared in Standard Perturbative series and in CORGI approch.compared in Standard Perturbative series and in CORGI approch.

Since the perturbative terms in CORGI approach are scheme invariants and Since the perturbative terms in CORGI approach are scheme invariants and were obtained from a resummation of ultraviolet terms in specified order, we were obtained from a resummation of ultraviolet terms in specified order, we expect to get better consistency between polarized proton SF and available expect to get better consistency between polarized proton SF and available experimantal dataexperimantal data

. This is our new task.. This is our new task.

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ConclusionConclusion

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