1. Аксиальная аномалия и физика тяжелых ионов 26/11/2013, NICA- cеминар O.В. Теряев ЛТФ

2. Symmetries and conserved operators (Global) Symmetry -> conserved current ( ) Exact: U(1) symmetry – charge conservation - electromagnetic (vector) current Translational symmetry – energy momentum tensor

3. Massless fermions (quarks) – approximate symmetries Chiral symmetry (mass flips the helicity) Dilatational invariance (mass introduce dimensional scale – c.f. energy- momentum tensor of electromagnetic radiation )

4. Quantum theory Currents -> operators Not all the classical symmetries can be preserved -> anomalies Enter in pairs (triples?…) Vector current conservation chiral invariance Translational invariance dilatational invariance

5. Calculation of anomalies Many various ways All lead to the same operator equation 2photon/gluon matrix element- triangle diagram UV vs IR languages- understood in physical picture (Gribov, Feynman, Nielsen and Ninomiya) of Landau levels flow (E||H)

6. Counting the Chirality Degeneracy rate of Landau levels “Transverse” HS/(1/e) (Flux/flux quantum) “Longitudinal” Ldp= eE dt L (dp=eEdt) Anomaly – coefficient in front of 4-dimensional volume - e 2 EH Also particle number violation ~EH/sinh(H/E) –analog of Schwinger mechanism for pure electric field

7. Anomaly from dispersion relations No anomaly for imaginary part -> Finite subtraction for real part – dispersive approach to anomaly (Dolgov, Zakharov’70) - anomaly sum rules One real and one virtual photon – Horejsi,OT’95 m->0: pole – pion (t’Hooft principle) Finite mass – anomaly is still there as a collective effect! Finite temperature does NOT work as a mass Chemical potential?!

8. Massive quarks One way of calculation – finite limit of regulator fermion contribution (to TRIANGLE diagram) in the infinite mass limit The same (up to a sign) as contribution of REAL quarks For HEAVY quarks – cancellation! Anomaly – violates classical symmetry for massless quarks but restores it for heavy quarks

9. Dilatational anomaly Classical and anomalous terms Beta function – describes the appearance of scale dependence due to renormalization For heavy quarks – cancellation of classical and quantum violation -> decoupling

10. Decoupling Happens if the symmetry is broken both explicitly and anomalously Selects the symmetry in the pair of anomalies which should be broken (the one which is broken at the classical level) For “non-standard” choice of anomalous breakings (translational anomaly) there is no decoupling Defines the Higgs coupling, neutralino scattering…

11. Heavy quarks matrix elements QCD at LO From anomaly cancellations (27=33-6) “Light” terms Dominated by s-of the order of cancellation

12. Heavy quarks polarisation Non-complete cancellation of mass and anomaly terms (97) Gluons correlation with nucleon spin – twist 4 operator NOT directly related to twist 2 gluons helicity BUT related by QCD EOM to singlet twist 4 correction (colour polarisability) f2 to g1 “Anomaly mediated” polarisation of heavy quarks

13. Numerics Small (intrinsic) charm polarisation Consider STRANGE as heavy! – CURRENT strange mass squared is 100 times larger – -5% - reasonable compatibility to the data! (But problem with DIS and SIDIS) Current data on f2 – appr 50% larger

14. Can s REALLY be heavy?! Strange quark mass close to matching scale of heavy and light quarks – relation between quark and gluon vacuum condensates (similar cancellation of classical and quantum symmetry violation – now for trace anomaly). BUT - common belief that strange quark cannot be considered heavy, In nucleon (no valence “heavy” quarks) rather than in vacuum - may be considered heavy in comparison to small genuine higher twist – multiscale nucleon picture

15. Comparison : Gluon Anomaly for massless and massive quarks Mass independent Massless – naturally (but NOT uniquely) interpreted as (on-shell) gluon circular polarization Small gluon polarizattion – no anomaly?! Massive quarks – acquire “anomaly polarization” May be interpreted as a kind of circular polarization of OFF-SHELL (CS projection -> GI) gluons Very small numerically Small strange mass – partially compensates this smallness and leads to % effect

16. Sign of polarisation Anomaly – constant and OPPOSITE to mass term Partial cancellation – OPPOSITE to mass term Naturally requires all “heavy” quarks average polarisation to be negative IF heavy quark in (perturbative) heavy hadron is polarised positively

17. Magnetic fields in HIC (this and next slide by D. Kharzeev)

18. Relation to anomaly

20. Other sources of quadratic effects Quadratic effect of induced currents – not necessary involve (C)P-violation May emerge also as C&P even quantity Complementary probes of two-current correlators desirable Natural probe – dilepton angular distributions

21. Induced current for (heavy - with respect to magnetic field strength) strange quarks Effective (Heisenberg-Euler) Lagrangian Current and charge density from c (~7/45) – term (multiscale medium!) Light quarks -> matching with D. Kharzeev et al’ -> correlation of density of electric charge with a gradient of topological one (Lattice ?)

22. Anomaly in medium – new external lines in VVA graph Gauge field -> velocity CME -> CVE Kharzeev,Zhitnitsky (07) – EM current Straightforward generalization: any (e.g. baryonic) current – neutron asymmeries@NICA - Rogachevsky, AS, Teryaev (10)

23. Conclusions Anomaly many derivations (some are very simple leading to the same result Anomaly cancellation for heavy (strange) quarks HIC – anomaly for external EM or velocity fields – anomalous transport (At least partial) cancellation of anomalous transport by mass effects

24. Baryon charge with neutrons – (Generalized) Chiral Vortical Effect Coupling: Current: - Uniform chemical potentials: - Rapidly (and similarly) changing chemical potentials:

25. Baryon charge with neutrons – (Generalized) Chiral Vortical Effect Coupling: Current: - Uniform chemical potentials: - Rapidly (and similarly) changing chemical potentials:

26. Heavy Strangeness transversity Heavy strange quarks – neglect higher twist: 0 = Strange transversity - of the same sign as helicity and enhanced by M/m But: only genuine HT may be be small – relation to twist 3 part of g2

27. Unpolarized strangeness – can it be considered as heavy? Heavy quark momentum – defined by matrix element (Franz,Polyakov,Goeke) IF no numerical suppression of this matrix element – charm momentum of order 0.1% IF strangeness can be also treated as heavy – too large momentum of order 10%

28. Heavy unpolarized Strangeness: possible escape Conjecture: is suppressed by an order of magnitude with respect to naïve estimate Tests in models/lattice QCD? Charm momentum of order 0.01% Strangeness momentum of order of 1%

29. Charm/Strangeness universality Universal behaviour of\heavy quarks distributions c(x)/s(x) = (m s /m c ) 2 ~ 0.01 Delta c(x)/Delta s(x)= (m s /m c ) 2 ~ 0.01 Delta c(x)/Delta s(x)= c(x)/s(x) Experimental tests – comparison of strange/charmed hadrons asymmetries

30. Higher corrections Universality may be violated by higher mass corrections Reasonable numerical accuracy for strangeness – not large for s –> negligible for c If so, each new correction brings numerically small mass scale like the first one If not, and only scale of first correction is small, reasonable validity for s may be because of HT resummation (like for non-local quark condensates or TMD parton distributions (cf talks of P. Mulders, M. Anselmino)

31. Conclusions Heavy quarks – cancellation of anomalous and explicit symmetry breaking Allows to determine some useful hadronic matrix elements Multiscale picture of nucleon - Strange quarks may be considered are heavy sometimes Possible universality of strange and charmed quarks distributions – simi;larity of spin asymetries of strange and charmed hadrons

32. Other case of LT-HT relations – naively leading twists TMD functions –>infinite sums of twists. Case study: Sivers function - Single Spin Asymmetries Main properties: – Parity: transverse polarization – Imaginary phase – can be seen T-invariance or technically - from the imaginary i in the (quark) density matrix Various mechanisms – various sources of phases

33. Phases in QCD QCD factorization – soft and hard parts- Phases form soft, hard and overlap Assume (generalized) optical theorem – phase due to on-shell intermediate states – positive kinematic variable (= their invariant mass) Hard: Perturbative (a la QED: Barut, Fronsdal (1960): Kane, Pumplin, Repko (78) Efremov (78)

34. Perturbative PHASES IN QCD

35. Short+ large overlap– twist 3 Quarks – only from hadrons Various options for factorization – shift of SH separation New option for SSA: Instead of 1-loop twist 2 – Born twist 3: Efremov, OT (85, Ferminonc poles); Qiu, Sterman (91, GLUONIC poles)

36. Twist 3 correlators

37. Twist 3 vs Sivers function(correlation of quark pT and hadron spin) Twist 3 – Final State Interaction -qualitatively similar to Brodsky-Hwang-Schmidt model Path order exponentials (talk of N. Stefanis) – Sivers function (Collins; Belitsky, Ji, Yuan) Non-suppressed by 1/Q-leading twist? How it can be related to twist 3? Really – infinite sum of twists – twist 3 selected by the lowest transverse moment Non-suppression by 1/Q – due to gluonic pole = quarks correlations with SOFT gluons

38. Sivers and gluonic poles at large PT Sivers factorized (general!) expression M – in denominator formally leading twist (but all twists in reality) Expand in kT = twist 3 part of Sivers

39. From Sivers to twist 3 - II Angular average : As a result M in numerator - sign of twist 3. Higher moments – higher twists. KT dependent function – resummation of higher twist. Difference with BFKL IF – subtracted UV; Taylor expansion in coordinate soace – similar to vacuum non-local condensates

40. From Sivers to gluonic poles - III Final step – kinematical identity Two terms are combined to one Key observation – exactly the form of Master Formula for gluonic poles (Koike et al, 2007)

41. Effective Sivers function Expressed in terms of twist 3 Up to Colour Factors ! Defined by colour correlation between partons in hadron participating in (I)FSI SIDIS = +1; DY= -1: Collins sign rule Generally more complicated Factorization in terms of twist 3 but NOT SF

42. Colour correlations SIDIS at large pT : -1/6 for mesons from quark, 3/2 from gluon fragmentation (kaons?) DY at large pT: 1/6 in quark antiquark annihilation, - 3/2 in gluon Compton subprocess – Collins sign rule more elaborate – involve crossing of distributions and fragmentations - special role of PION DY (COMPASS). Direct inclusive photons in pp = – 3/2 Hadronic pion production – more complicated – studied for P- exponentials by Amsterdam group + VW IF cancellation – small EFFECTIVE SF (cf talk of F. Murgia) Vary for different diagrams – modification of hard part FSI for pions from quark fragmentation -1/6 x (non-Abelian Compton) +1/8 x (Abelian Compton)

43. Colour flow Quark at large PT:-1/6 Gluon at large PT : 3/2 Low PT – combination of quark and gluon: 4/3 (absorbed to definition of Sivers function) Similarity to colour transparency phenomenon

44. Twist 3 factorization (Bukhvostov, Kuraev, Lipatov;Efremov, OT; Ratcliffe;Qiu,Sterman;Balitsky,Braun) Convolution of soft (S) and hard (T) parts Vector and axial correlators: define hard process for both double ( ) and single asymmetries

45. Twist 3 factorization -II Non-local operators for quark-gluon correlators Symmetry properties (from T- invariance)

46. Twist-3 factorization -III Singularities Very different: for axial – Wandzura-Wilczek term due to intrinsic transverse momentum For vector-GLUONIC POLE (Qiu, Sterman ’91) – large distance background

47. Sum rules EOM + n-independence (GI+rotational invariance) –relation to (genuine twist 3) DIS structure functions

48. Sum rules -II To simplify – low moments Especially simple – if only gluonic pole kept:

49. Gluonic poles and Sivers function Gluonic poles – effective Sivers functions-Hard and Soft parts talk, but SOFTLY Implies the sum rule for effective Sivers function (soft=gluonic pole dominance assumed in the whole allowed x’s region of quark-gluon correlator)

50. Compatibility of SSA and DIS Extractions of and modeling of Sivers function: – “mirror” u and d Second moment at % level Twist -3 - similar for neutron and proton and of the same sign – no mirror picture seen –but supported by colour ordering! Scale of Sivers function reasonable, but flavor dependence differs qualitatively. Inclusion of pp data, global analysis including gluonic (=Sivers) and fermionic poles HERMES, RHIC, E704 –like phonons and rotons in liquid helium; small moment and large E704 SSA imply oscillations JLAB –measure SF and g2 in the same run

51. CONCLUSIONS (At least) 2 reasons for relations between various twists: exact operator equations naively leading twist object contains in reality an infinite tower Strange quark (treated as heavy) polarization – due to (twist 4, anomaly mediated) gluon polarization Sivers function is related to twist 4 gluonic poles – relations of SSA’s to DIS

52. 2 nd Spin structure - TOTAL Angular Momenta - Gravitational Formfactors Conservation laws - zero Anomalous Gravitomagnetic Moment : (g=2) May be extracted from high-energy experiments/NPQCD calculations Describe the partition of angular momentum between quarks and gluons Describe interaction with both classical and TeV gravity

53. Electromagnetism vs Gravity Interaction – field vs metric deviation Static limit Mass as charge – equivalence principle

54. Equivalence principle Newtonian – “Falling elevator” – well known and checked Post-Newtonian – gravity action on SPIN – known since 1962 (Kobzarev and Okun’) – not checked on purpose but in fact checked in atomic spins experiments at % level Anomalous gravitomagnetic moment iz ZERO or Classical and QUANTUM rotators behave in the SAME way

55. Gravitomagnetism Gravitomagnetic field – action on spin – ½ from spin dragging twice smaller than EM Lorentz force – similar to EM case: factor ½ cancelled with 2 from Larmor frequency same as EM Orbital and Spin momenta dragging – the same - Equivalence principle

56. Equivalence principle for moving particles Compare gravity and acceleration: gravity provides EXTRA space components of metrics Matrix elements DIFFER Ratio of accelerations: - confirmed by explicit solution of Dirac equation (Silenko,OT)

57. Generalization of Equivalence principle Various arguments: AGM 0 separately for quarks and gluons – most clear from the lattice (LHPC/SESAM)

58. Extended Equivalence Principle=Exact EquiPartition In pQCD – violated Reason – in the case of EEP - no smooth transition for zero fermion mass limit (Milton, 73) Conjecture (O.T., 2001 – prior to lattice data) – valid in NP QCD – zero quark mass limit is safe due to chiral symmetry breaking Supported by smallness of E (isoscalar AMM) Polyakov Vanderhaeghen: dual model with E=0

59. Vector mesons and EEP J=1/2 -> J=1. QCD SR calculation of Rho’s AMM gives g close to 2. Maybe because of similarity of moments g-2= ; B= Directly for charged Rho (combinations like p+n for nucleons unnecessary!). Not reduced to non-extended EP: Gluons momentum fraction sizable. Direct calculation of AGM are in progress.

60. EEP and AdS/QCD Recent development – calculation of Rho formfactors in Holographic QCD (Grigoryan, Radyushkin) Provides g=2 identically! Experimental test at time –like region possible

61. Another relation of Gravitational FF and NP QCD (first reported at 1992: hep-ph/9303228 ) BELINFANTE (relocalization) invariance : decreasing in coordinate – smoothness in momentum space Leads to absence of massless pole in singlet channel – U_A(1) Delicate effect of NP QCD Equipartition – deeply related to relocalization invariance by QCD evolution

62. Conclusions 2 spin structures of nucleon – related to fundamental properties of NPQCD Axial anomaly – may reside not only in gluon, but in strangeness polarization (connection of laeding and higher twists) Multiscale (“fluctonic”) nucleon: M>>m(strange)>>HT Total angular momenta – connection to (extended) equivalence principle and AdS/QCD Relation of 2 spin structures?!

63. Sum rules -II To simplify – low moments Especially simple – if only gluonic pole kept:

64. Compatibility of SSA and DIS Extractions of Sivers function: – “mirror” u and d First moment of EGMMS = 0.0072 (0.0042 – 0.014) Twist -3 - similar for neutron and proton (0.005) and of the same sign – nothing like mirror picture seen –but supported by colour ordering! Current status: Scale of Sivers function – seems to be reasonable, but flavor dependence differs qualitatively. Inclusion of pp data, global analysis including gluonic (=Sivers) and fermionic poles

65. Relation of Sivers function to GPDs Qualitatively similar to Anomalous Magnetic Moment (Brodsky et al) Quantification : weighted TM moment of Sivers PROPORTIONAL to GPD E (hep-ph/0612205 ): Burkardt SR for Sivers functions is now related to Ji SR for E and, in turn, to Equivalence Principle

66. Sivers function and Extended Equivalence principle Second moment of E – zero SEPARATELY for quarks and gluons –only in QCD beyond PT (OT, 2001) - supported by lattice simulations etc.. -> Gluon Sivers function is small! (COMPASS, STAR, Brodsky&Gardner) BUT: gluon orbital momentum is NOT small: total – about 1/2, if small spin – large (longitudinal) orbital momentum Gluon Sivers function should result from twist 3 correlator of 3 gluons: remains to be proved!