.
A Talk given by
Institute for studies in theoretical Physics and Mathematics,IPM
Yazd University, IRAN
Outline
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RS and FS dependence in QCD predictions
Complete RG Improvement
Valon model
Unpolarized valon distribution in a proton
Polarized valon distribution
Parton distribution functions in moment space
Comparison of the Moments
Moments of PPDF’s and distribution function
Conclusion
Spin04 Symposium-Italy
RS and FS dependence in
QCD 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
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 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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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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Unpolarized valon distribution in a proton
To facilitate the phenomenological analysis Hwa assumed a
simple form for the exclusive valon distribution in unpolarized proton
which is
where yi is the momentum fraction of the i-th valon.The inclusive
valon distributions are obtained by integration
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Polarized valon distribution
Polarized
U-valon
distribution
Unpolarize
d
U-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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• Moments of PPDF’s and distribution function
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.
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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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Conclusion
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First we introduced the idea of complete RG- improvement and showed how we
can reconstructed the QCD perurbatives series in this approach.
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Valon model was defined and unpolarized and polarized valon distributions
were obtained.
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The nonsinglet and singlet moments of polarized valons were obtained and
the results in standard perterbative series and the CORGI approch were
comparied.
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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
compared in Standard Perturbative series and in CORGI approch.
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Since the perturbative terms in CORGI approach are scheme invariants and
were obtained from a resummation of ultraviolet terms in specified order, we
expect to get better consistency between polarized proton SF and available
experimantal data
. This is our new task.
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