Quantum Anomalies for Heavy Quarks Heavy Quarks-08 JINR, Dubna, August 20 2008

Oleg TeryaevJINR

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

Symmetries and conserved operators

(Global) Symmetry -> conserved current ( )

Exact: U(1) symmetry – charge

conservation - electromagnetic (vector) current

Translational symmetry – energy momentum tensor

0J

0T

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 0J

0T

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

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)

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 - e2 EH

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

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

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…

Heavy quarks matrix elements

QCD at LO From anomaly cancellations (27=33-6)

“Light” terms

Dominated by s-of the order of cancellation

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

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

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

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

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

Heavy Strangeness transversity

Heavy strange quarks – neglect genuine higher twist: 0 =

Strange transversity - of the same sign as helicity and enhanced by M/m!

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

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)

Perturbative PHASES IN QCD

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)

Twist 3 correlators

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

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

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

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)

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

(1) 1( ) ( , ),

2i jSx x C T x x

Mf

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)

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

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

g2

Twist 3 factorization -II

Non-local operators for quark-gluon correlators

Symmetry properties (from T-invariance)

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

Sum rules

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

Sum rules -II

To simplify – low moments

Especially simple – if only gluonic pole kept:

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)

)(4

1),(

2

1)( xxxT

Mxx

vTf

)2)((3

4)(

1

0

2

_2

1

0

xxxdxxg fdxx T

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

g2

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

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

Electromagnetism vs Gravity Interaction – field vs metric deviation

Static limit

Mass as charge – equivalence principle

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

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

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)

Generalization of Equivalence principle

Various arguments: AGM 0 separately for quarks and gluons – most clear from the lattice (LHPC/SESAM)

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

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.

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

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

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?!

Sum rules -II

To simplify – low moments

Especially simple – if only gluonic pole kept:

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

g2

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

( ) ( )T

x x xE xf

, ,

( ) ( ) 0T

q G q G

dxx x dxxE xf

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!