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Аксиальная аномалия и физика тяжелых ионов 26/11/2013, NICA-cеминар
O.В. ТеряевЛТФ
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Symmetries and conserved operators
(Global) Symmetry -> conserved current ( )
Exact: U(1) symmetry – charge
conservation - electromagnetic (vector) current
Translational symmetry – energy momentum tensor
0J
0T
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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 )
5 0J
0T
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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
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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)
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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 Also particle number violation ~EH/sinh(H/E)
–analog of Schwinger mechanism for pure electric field
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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?!
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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
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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
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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…
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Heavy quarks matrix elements
QCD at LO From anomaly cancellations (27=33-6)
“Light” terms
Dominated by s-of the order of cancellation
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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
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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
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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
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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
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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
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Magnetic fields in HIC (this and next slide by D. Kharzeev)
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Relation to anomaly
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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
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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 ?)
44 /))((/)~
)(~
( mGGFFdmGGFFcL
GGxdmGG
H
mGGFcj
~/)
~(~
~
/)~
(~
2
44
4
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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)
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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
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Baryon charge with neutrons – (Generalized) Chiral Vortical Effect
Coupling:
Current: - Uniform chemical potentials: - Rapidly (and similarly) changing
chemical potentials:
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Baryon charge with neutrons – (Generalized) Chiral Vortical Effect
Coupling:
Current: - Uniform chemical potentials: - Rapidly (and similarly) changing
chemical potentials:
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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
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Unpolarized strangeness – can it be considered as heavy? Heavy quark momentum – defined by <p|
GGG|p> 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%
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Heavy unpolarized Strangeness: possible escape Conjecture: <p|GGG|p> 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%
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Charm/Strangeness universality
Universal behaviour of\heavy quarks distributions
c(x)/s(x) = (ms /mc)2 ~ 0.01
Delta c(x)/Delta s(x)= (ms /mc)2 ~ 0.01 Delta c(x)/Delta s(x)= c(x)/s(x) Experimental tests – comparison of
strange/charmed hadrons asymmetries
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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)
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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
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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
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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)
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Perturbative PHASES IN QCD
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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)
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Twist 3 correlators
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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
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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
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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
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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)
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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
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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)
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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
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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
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Twist 3 factorization -II
Non-local operators for quark-gluon correlators
Symmetry properties (from T-invariance)
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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
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Sum rules
EOM + n-independence (GI+rotational invariance) –relation to (genuine twist 3) DIS structure functions
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Sum rules -II
To simplify – low moments
Especially simple – if only gluonic pole kept:
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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
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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
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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
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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
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Electromagnetism vs Gravity Interaction – field vs metric deviation
Static limit
Mass as charge – equivalence principle
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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
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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
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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)
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Generalization of Equivalence principle
Various arguments: AGM 0 separately for quarks and gluons – most clear from the lattice (LHPC/SESAM)
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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
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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.
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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
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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
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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?!
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Sum rules -II
To simplify – low moments
Especially simple – if only gluonic pole kept:
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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
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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
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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!