What is the twist of TMDs? Como, June 12, 2013

1. What is the twist of TMDs? Como, June 12, 2013 Oleg Teryaev JINR, Dubna

2. Outline • Definitions of twist • TMDs as infinite towers of twists • Quarks in vacuum and inside the hadrons: TMDs vs non-local condensates • HT resummation and analyticity in DIS • HT resummation and scaling variables: DIS vs SIDIS

3. HT resummation in DIS • Higher Twists in spin-dependent DIS: GDH sum rule – finite sum of infinite number of divergent terms • Resummation of HT and analyticity • Comparing modified scaling variables

4. What is twist? • Power corrections ~1/Q2 • ------//------------- ~ M2 • DIS – it’s the same (~ M2/Q2) • TMD – usually ~1/Q2 - (M2/kT2 )i attributed to Leading Twist • However – tracing the powers of M is helpful for studying HT in coordinate (~impact parameter) space

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

6. Resummation in x-space (DY) • Full x/kT – dependence • DY weighted cross-section • Similarity with non-local quark condensate: quarks in vacuum ~ transverse d.o.f. of quarks in hadrons (Euclidian!) ?! –cf with Radyushkin et al • Universal hadron(type-dependent)/vacuum functions?!

7. Hadronic vs vacuum matrix elements • Hadron-> (LC) momentum; dimension-> twist; quark virtuality -> TM; (Euclidian) space separation -> impact parameter • D-term ~ Cosmological constant in vacuum; Negative D-> negative CC in space-like/positive in time-like regions: Annihilation~Inflation!

8. 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

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

10. 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: • Lorentz invariance • Summed by representing

11. Summation and analyticity? • Justification (in addition to nice parton picture) - analyticity! • Correct analytic properties of virtual Compton amlitude • Defines the region of x • Require: Analyticity of first moment in Q2 • Strictly speaking – another integration variable (Robaschik et al, Solovtsov et al)

12. Summation and analyticity! • 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 • Analytic properties  proper integration region (positive x, two-pion threshold) • Finite value for Q2 =0: -< f(x)/x> - inverse moment!

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

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

15. Decomposition of (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

16. 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

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

18. 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

19. 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

20. Bjorken Sum Rule – most clean test • Strongly differs from smooth interpolation for g1 (Ioffe,Lipatov,Khoze) • Scaling down to 1 GeV

21. New option: Analytic Perturbation Theory • Shirkov, Solovtsov: Effective coupling – analytic in Q2 • Generic processes: FAPT (BMS) • 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

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

23. PT/HT duality

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

25. 4-loop corrections included V.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!?

26. 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.

27. 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 (cf Ioffe e.a. interpolation)!

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

29. NP vs P masses • Non-monotonic! • “Phase diagram”

30. P/NP masses

31. Data at LO

32. NLO

33. Data vs NLO

34. Modification of spectral function for HT • Add const × Q2/(M22+Q2) 2 -> First of second derivative of delta-function appear – double and triple poles (single – almost cancelled) • Masses: • P= 0.68 • NP=0.76 • Expansion at low Q2 • Real scale – pion mass?!

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

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

37. 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)?!

38. Conclusions/Discussion • TMD – infinite towers of twists • Similar to non-local quark condensates – vacuum/hadrons universality?! • Infinite sums of twists – important for DIS at Q->0 • Representation for HT similar to parton model: preserves analyticity changing the poles to cuts • Modified scaling variables – models for twists towers at DIS and (TMD) SIDIS • Good description of the data at all Q2 with the single scale parameter