1. Quantum Anomalies for Heavy Quarks Heavy Quarks-08 JINR, Dubna, August 20 2008
Oleg Teryaev
JINR

2. Outline Conservation laws in classical and quantum theory
Chiral and trace anomalies for massless and massive fermions: decoupling
Trace anomaly: Heavy scalar particles couplings to nucleons
Axial anomaly: Heavy quarks polarization in nucleon
When strange quarks can be heavy: multiscale hadrons

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

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

5. 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

6. Calculation of anomalies
Many various ways
All lead to the same operator equation
UV vs IR languages-
understood in physical
picture (Gribov, Feynman, Nielsen and Ninomiya) of Landau levels flow (E||H)

7. 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 dimensional volume - e2 EH

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 breakins (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 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 – % - 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. 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

16. 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

17. Heavy Strangeness transversity
Heavy strange quarks – neglect genuine higher twist: 0 =
Strange transversity - of the same sign as helicity and enhanced by M/m!

18. 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

19. 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)

20. Perturbative PHASES IN QCD

21. 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)

22. Twist 3 correlators

23. 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

24. 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

25. 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

26. 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)

27. 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

28. 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)

29. 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

30. 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

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

32. 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

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

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

35. 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)

36. Compatibility of SSA and DIS
Extractions of and modeling of Sivers function: – “mirror” u and d
Second moment at % level
Twist 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

37. 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

38. 2nd 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

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

40. 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

41. 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

42. 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)

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

44. 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

45. 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=<E(x)>; B=<xE(x)>
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.

46. 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

47. 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

48. 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?!

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

50. Compatibility of SSA and DIS
Extractions of Sivers function: – “mirror” u and d
First moment of EGMMS = ( – )
Twist 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

51. 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/ ):
Burkardt SR for Sivers functions is now related to Ji SR for E and, in turn, to Equivalence Principle

52. 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!