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Energy Evolution of Sivers asymmetry in Hard Processes

Feng Yuan Lawrence Berkeley National Laboratory

Outlines

General theory background Implement the TMD evolution from low Q

SIDIS to Drell-Yan Match to high Q Drell-Yan/W/Z Collins asymmetries

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Hard processes

In the context of this talk, the hard processes means low transverse momentum hard processesSemi-inclusive DIS at low ptDrell-Yan/W/Z production Higgs production…

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Collinear vs TMD factorization

TMD factorization is an extension and simplification to the collinear factorization

Extends to the region where collinear fails Simplifies the kinematics

Power counting, correction 1/Q neglected

(PT,Q)=H(Q) f1(k1T,Q) f2(k2T, Q) S(T)There is no x- and kt-dependence in the hard

factor

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DGLAP vs CSS

DGLAP for integrated parton distributionsOne hard scale

(Q)=H(Q/) f1()… CSS for TMDs

Two scales, large double logs

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Evolution vs resummation

Any evolution is to resum large logarithms DGLPA resum single large logarithms CSS evolution resum double logarithms

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Energy Evolution CS evolution for TMD

distribution/fragmentation functions, scheme-dependentCollins-Soper 81, axial gaugeJi-Ma-Yuan 04, Feynman gauge, off-lightCollins 11, cut-offSCET, quite a few

CSS evolution on the cross sectionsTMD factorization implicit

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Energy dependence Collins-Soper Evolution, 1981 Collins-Soper-Sterman, 1985 Boer, 2001 Idilbi-Ji-Ma-Yuan, 2004 Kang-Xiao-Yuan, 2011 Collins 2010 Aybat-Collins-Rogers-Qiu, 2011 Aybat-Prokudin-Rogers,2012 Idilbi, et al., 2012

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Semi-inclusive DIS

Fourier transform Evolution

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Calculate at small-b

Sudakov

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b*-prescription and non-perturbative form factor b* always in perturbative region

This will introduce a non-perturbative form factors

Generic behavior

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Rogers et al.

Calculate the structure at two Q,

Relate high Q to low Q

Low Q parameterized as Gaussian

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BLNY form factors Fit to Drell-Yan and W/Z boson production

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BLNY form can’t describe SIDIS

Log Q dependence is so strong, leading to a≈0.08 at HERMES energy

Hermes data require a≈0.2

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BLNY will be evenWorseAny modification willIntroduce new problem

It could be that the functional form is not adequate to describe large-b physics In particular, for \ln Q term (see follows)

Or evolution has to be reconsidered in the relative (still perturbative) low Q range around HERMES/COMPASSQ>~Q0~1/b*~2GeV (for bmax=0.5GeV-1)

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One solution: back to old way

Parameterize at scale Q0

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Limitations It’s an approximation: both Q0 and Q are

restricted to a limited range, definitely not for W/Z bosonLog(Q0 b) in the evolution kernel

Do not have correct behavior at small-b (could be improved), will have uncertainties at large pt

x-dependence is not integrated into the formalism

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Advantages

There is no Landau pole singularity in the integral

Almost parameter-freeNo Q-dependent non-perturbative form factorGaussian assumption at lower scale Q0

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Almost parameter-free prediction

SIDIS Drell-Yan

in similar x-range

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Fit to Sivers asymmetries

With the evolution effects taken into account. Not so large Q difference

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Systematics of the SIDIS experiments are well understood

Q range is large to apply perturbative QCD Sivers functions are only contributions to

the observed asymmetries

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Predictions at RHIC

About a factor of 2 reduction, as compared to previous order of magnitude difference

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Cross checks

Re-fit Rogers et al’s parameterization to the pt-distributions, and calculate the SSA, in similar range

Assume a simple Gaussian for both SIDIS and Drell-Yan (Schweitzer et al.), and again obtain similar size SSA for Drell-Yan

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Match to higher Q

Extract the transverse momentum-moment of the Sivers function, and use the b* prescription and resummation, and again obtain similar size of SSA for Drell-Yan

This can be used to calculate the asymmetries up to W/Z boson production

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High energies

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w/o evolution

b*-prescription

with evolution

Z boson

Q=5.5GeV

PT(GeV)

Collins asymmetries

Ec.m.≈10GeV, di-hadron azimuthal asymmetric correlation in e+e- annihilation

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Collins asymmetries in SIDIS

asd

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Test the evolution at BEPC

Ec.m.=4.6GeV, di-hadron in e+e- annihilation BEPC-(Beijing electron-positron collider)

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It is extremely important to test this evolution effect

EIC will be perfect, because Q coverage Anselm Vossen also suggests to do it at

BELLE with ISR with various Q possible

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Conclusion We evaluate the energy dependence for

Sivers asymmetries in hard processes, from HERMES/COMPASS to typical Drell-Yan process

The same evolution procedure consistently describes the Collins asymmetries from HERMES/COMPASS and BELLE

Further tests are needed to nail down this issue

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