Analyticity and higher twists Hadron Structure’13, Tatranské Matliare , July 2, 2013

1. Analyticity and higher twists Hadron Structure’13, Tatranské Matliare, July 2, 2013 Oleg Teryaev JINR, Dubna

2. QCD without confinement problem solved – description and classification of NP inputs (~biology before Darwin)Complicated processes – complicated classification – UGDFs, TMDs, GPDs, HTs (~multiparton pdfs) Are such classifications compatible with symmetries ( -> low-energy theorems), CAUSALITY AND ANALYTICITY ?

3. Outline • HT resummation and analyticity in (spin-dependent) DIS • HT resummation and scaling variables: DIS vs SIDIS • TMDs as infinite towers of twists • Quarks in vacuum and inside the hadrons: TMDs vs non-local condensates

4. Spin dependent DIS • Two invariant tensors • Only the one proportional to contributes for transverse (appears in Born approximation of PT) • Both contribute for longitudinal • Apperance of only for longitudinal case –result of the definition for coefficients to match the helicity formalism

5. Generalized GDH sum rule • Define the integral – scales asymptotically as • At real photon limit (elastic contribution subtracted) – - Gerasimov-Drell-Hearn SR • Finite limit of infinite sum of inverse powers! • Proton- dramatic sign change at low Q2!

6. Finite limit of infinite sum of inverse powers?! • How to sum ci (- M2/Q2 )i ?! • May be compared to standard twist 2 factorization • Light cone:

7. Moments and partonic expression • Lorentz invariance: • Summed by representing

8. Summation and analyticity • Justification (in addition to nice parton picture) - analyticity! • Correct analytic properties of virtual Compton amplitude • Defines the region of x • Generalized Parton Distributions – compatibility of analyticity with factorization – non-trivial: Radon transform technique (OT’05; Anikin,OT’07,D. Muller,Kumericky’09-13)

9. Summation and analyticity-HT • Parton model with |x| < 1 – transforms poles to cuts! – justifies the representation in terms of moments • For HT series ci = <f(x) xi> - moments of HT “density”- geometric series rather than exponent: Σci (- M2/Q2 ) = < M2f(x)/(x M2+ Q2 )> • Like in parton model: pole -> cut

10. Summation and analyticity-HT • Analytic properties  proper integration region (positive x, two-pion threshold) • Finite value for Q2 =0: -< f(x)/x> - inverse moment! • Relation to matrix elements unclear (probably – Wilson lines: transverse momentum)

11. Summation and analyticity • “Chiral” expansion: - (- Q2/M2 )i <f(x)/x i+1> • “Duality” of chiral and HT expansions: analyticity allows for EITHER positive OR negative powers (no complete series!) • Analyticity – (typically)alternating series

12. Summation and analyticity • Analyticity of HT  analyticity of pQCD series – (F)APT • Finite limit -> series starts from 1/Q2 unless the density oscillates • Annihilation – (unitarity - no oscillations) extra justification of “short strings”?

13. Short strings • Confinement term in the heavy quarks potential – dimension 2 (GI OPE – 4!) scale ~ tachyonic gluon mass • Effective modification of gluon propagator

14. Back to DIS: (J. Soffer, OT ‘92) • Supported by the fact that • Linear in , quadratic term from • Natural candidate for NP, like QCD SR analysis – hope to get low energy theorem via WI (C.f. pion F.F. – Radyushkin) - smooth model • For -strong Q – dependence due to Burkhardt-Cottingham SR

15. Models for :proton • Simplest - linear extrapolation – PREDICTION (10 years prior to the data) of low (0.2 GeV) crossing point • Accurate JLAB data – require model account for PQCD/HT correction – matching of chiral and HT expansion • HT – values predicted from QCD SR (Balitsky, Braun, Kolesnichenko) • Rather close to the data For Proton

16. The model for transition to small Q (Soffer, OT ’04)

17. Access to the neutron – via the (p-n) difference – linear in -> Deuteron – refining the model eliminates the structures Models for :neutron and deuteron for neutron and deuteron

18. Duality for GDH – resonance approach • Textbook (Ioffe, Lipatov. Khoze) explanation of proton GGDH structure –contribution of dominant magnetic transition form factor • Is it compatible with explanation?! • Yes!– magnetic transition contributes entirely to and as a result to

19. Bjorken Sum Rule – most clean test • Strongly differs from smooth interpolation for g1 • Scaling down to 1 GeV

20. New option: Analytic Perturbation Theory • Shirkov, Solovtsov: Effective coupling – analytic in Q2 • Landau pole automatically removed • Generic processes: F(ractional)APT • Does not include full NPQCD dynamics (appears at ~ 1GeV where coupling is still small) –> Higher Twist • Depend on (A)PT • Low Q – very accurate data from JLAB

21. Bjorken Sum Rule-APT Accurate data + IR stable coupling -> low Q region

22. PT/HT duality

23. Matching in PT and APT • Duality of Q and 1/Q expansions

24. 4-loop corrections includedV.L. Khandramai, R.S. Pasechnik, D.V. Shirkov, O.P. Solovtsova, O.V. Teryaev. Jun 2011. 6 pp. e-Print: arXiv:1106.6352 [hep-ph] • HT decrease with PT order and becomes compatible to zero (V.I. Zakharov’s duality) • Analog for TMD – intrinsic/extrinsic TM duality!?

25. Asymptotic series and HT • Duality: HT can be eliminated at all (?!) • May reappear for asymptotic series - the contribution which cannot be described by series due to its asymptotic nature.

26. Another version of IR stable coupling – “gluon mass” – Cornwall,.. Simonov,.. Shirkov(NLO)arXiv:1208.2103v2 [hep-th] 23 Nov 2012 • HT – in the “VDM” form M2/(M2+ Q2 ) • Corresponds to f(x) ~ • Possible in principle to go to arbitrarily small Q • BUT NO matching with GDH achieved • Too large average slope – signal for transverse polarization role !

27. Account for transverse polarization -> description in the whole Q region (Khandramai, Shirkov, OT, in progress) • 1-st order – LO coupling with (P) gluon mass + (NP) “VDM” • GDH – relation between P and NP masses

28. P/NP

29. Data vs LO/NLO/NNLO

30. Smaller Q

31. HT – modifications of scaling variables • Various options since Nachtmann • ~ Gluon mass • -//- new (spectrality respecting) modification • JLD representation

32. Resummed twists: Q->0 (D. Kotlorz, OT)

33. Modified scaling variable for TMD • First appeared in P. Zavada model • XZ = • Suggestion – also (partial) HT resummation(M goes from denominator to numerator in cordinate/impact parameter space)?!

34. HT for TMDs - case study: Collins FF and twist 3 • x(T) –space : qq correlator ~ M - twist 3 • Cf to momentum space (kT/M) – M in denominator – “Leading Twist” • x <-> kT spaces • Moment – twist 3 (for Sivers – Boer, Mulders, Pijlman) • Higher (2D-> Bessel) moments – infinite tower of twists (for Sivers - Ratcliffe,OT)

35. Resummation in x-space (DY) • Full x/kT – dependence and its expansion • Singularities (-> power/log tail – cf Efremov, Vladimirov – causality -  arXiv:1306.3929 ) should be subtracted to get exponential falloff (required to have all moments in TM finite) • DY weighted cross-section

36. What happens in vacuum? • Suggestion : similarity with non-local quark condensates (Radyushkin et al) : quarks in vacuum ~ transverse d.o.f. of quarks in hadrons (Euclidian!) ?! • Universality in hadron( type-dependent) TMD and vacuum NLC functions?!

37. Hadronic vs vacuum matrix elements • Hadron -> (LC) momentum; dimension-> twist • Transverse dynamics – looks more similar to vacuum: quark virtuality -> TM(squared) • (Euclidian) space separation -> impact parameter

38. More complcated objects • Modification of BFKL kernel - talk of E. Levin: May be a way to modify NP part (impact factor): exponential falloff in coordinate space -> finiteness in momentum space (cf GDH) • D-term for GPDs (~ quadrupole gravitational FF) ~ Cosmological Constant in vacuum; Negative D-term -> negative CC in space-like/positive in time-like regions: Annihilation~Inflation

39. Conclusions • Representation for HT similar to parton model: preserves analyticity changing the poles to cuts • Infinite sums of twists – important for DIS at Q->0 • Good description of the data at all Q2 with the single scale parameter

40. Discussion • TMD – infinite towers of twists • Modified scaling variables – models for infinite twists towers in DIS and SIDIS • Similar to non-local quark condensates – vacuum/hadrons universality?!