**TMD Evolution: Matching SIDIS to Drell-Yan/W/Z Production in pp collisionsFeng Yuan Lawrence Berkeley National Laboratory

Refs: Sun, Yuan, arXiv: 1304.5037; to be submitted

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Outlines General theory backgroundImplement the TMD evolution from low Q SIDIS to Drell-YanMatch to high Q Drell-Yan/W/ZCollins asymmetries**

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Collinear vs TMD factorizationTMD factorization is an extension and simplification to the collinear factorizationExtends to the region where collinear failsSimplifies the kinematicsPower 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 CSSDGLAP for integrated parton distributionsOne hard scale (Q)=H(Q/) f1()CSS for TMDsTwo scales, large double logs**

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Evolution vs resummationAny evolution is to resum large logarithmsDGLPA resum single large logarithmsCSS evolution resum double logarithms**

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Energy EvolutionCS evolution for TMD distribution/fragmentation functions, scheme-dependentCollins-Soper 81, axial gaugeJi-Ma-Yuan 04, Feynman gauge, off-lightCollins 11, y-cut-offSCET, quite a few, CSS evolution on the cross sectionsTMD factorization implicit **

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

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Semi-inclusive DISFourier transformEvolution

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Calculate at small-bSudakov **

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

This will introduce a non-perturbative form factors Generic behavior

**Collins-Soper-Sterman 85

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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 factorsFit to Drell-Yan and W/Z boson production**bmax=0.5GeV-1

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Very successful phenomenologyMost quoted comparisons at the LHC for W/Z production**ResBos: Nadolsky, et al., PRD 2003 CSS resummation built in

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BLNY form cant describe SIDISLog(Q) dependence is so strong, leading to a0.08 at HERMES energyHermes data require a0.2**

BLNY will be evenWorseAny modification willIntroduce new problem

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It could be that the functional form is not adequate to describe large-b physicsIn 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 wayParameterize at scale Q0**Ji, Ma, Yuan, 2004

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Limitations Its an approximation: both Q0 and Q are restricted to a limited range, definitely not for W/Z bosonLog(Q0 b) in the evolution kernelDo not have correct behavior at small-b (could be improved), will have uncertainties at large ptx-dependence is not integrated into the formalism**

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Advantages There is no Landau pole singularity in the integralAlmost parameter-freeNo Q-dependent non-perturbative form factorGaussian assumption at lower scale Q0**

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Almost parameter-free predictionSIDIS Drell-Yan in similar x-range**

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Fit to Sivers asymmetriesWith the evolution effects taken into account. Not so large Q difference**

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Assumptions Systematics of the SIDIS experiments are well understoodQ range is large to apply perturbative QCD and TMD factorizationSivers functions are only contributions to the observed asymmetries**

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Predictions at RHICAbout a factor of 2 reduction, as compared to previous order of magnitude difference**

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Cross checksRe-fit Rogers et als parameterization to the pt-distributions, and calculate the SSA, in similar rangeAssume 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 QExtract the transverse momentum-moment of the Sivers function, and use the b* prescription and resummation, and again obtain similar size of SSA for Drell-YanThis can be used to calculate the asymmetries up to W/Z boson production**

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Matching**b*-prescriptionwith evolutionQ=5.5GeVPT(GeV)Arbitrary unit

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High energies**Z bosonQ=5.5GeVPT(GeV)Q=7.5GeVQ=9.5GeVQ=20GeVArbitrary unitSee also D. Boer talkDGLAP evolution (1/b*) yet to be included

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Uncertainties in the Sivers functions**Up Ubar Down

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SSA for W at RHICx-range similar to HERMES/COMPASS Early calculations by Kang-Qiu, Metz-Zhou**W+W-500GeV, y=0

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Collins asymmetriesEc.m.10GeV, di-hadron azimuthal asymmetric correlation in e+e- annihilation**

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Collins asymmetries in SIDISasd**

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Test the evolution at BEPCEc.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 effectEIC will be perfect, because Q coverageAnselm 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 processThe same evolution procedure consistently describes the Collins asymmetries from HERMES/COMPASS and BELLEFurther tests are needed to nail down this issue**

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*Hard to summarize the challengesBut the whole program already answered this questionIn the following, just a personal bias to account for

Here, I want to highlight one central aspect: the energy dependence for the associated observables. All these studies are aiming at this point as well. From earlier 80, by the so-called Collins-Soper evolution. Daniel Boer was able to make some predictions based on the Collins-Soper-Sterman resummation. In 2004, we derived the Collins-Soper evolution for the kt-odd TMDs. Following latest developments, we write down the final results for the CSS resummaiton form for, in particular, the Sivers single spin asymmetries in DIS and Drell-Yan processes. Meanwhile, Collins proposed alternative definition of the tMDs (his words, more perfect one), based on this, Ted Rogers and his collaborators did numeric calculations. In particular, with Alexei. Recently, my formal collaborator Idilbi and his recent collaborators came out another version of TMD. We will hear more from his talk.*