Comparing T-odd and T-even
spin sum rules
Diffraction-2014
Primosten, Croatia,
September 13, 2014
Oleg Teryaev
JINR, Dubna, Russia
Main Topics
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Sum rules from limited region: extrapolation
with generalized cut moments
EMTSRs and coupling to gravity
Burkardt sum rule for T-odd Sivers function
and energy momentum tensor
Deuteron tensor polarization and T-even EPrelated sum rule (HERMES and JLab)
Extrapolation to 0 (case of
Bjorken sum rule)
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Interval of x’s limited from below (especially
important for JLab)
Continuation – substantial part pf SRs check
(~ 50%!)
New method (Kotlorz,Mikhailov,OT) –
generalized cut moments – bunch of function
with the same 1st moment – extra degree of
freedom to constrain continuation
Contains extra (smooth) test function; powers
– the same DGLAP evolution as original one
Continuation to zero for
Bjorken SR
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Continuation from x~0.1 seems possible
Energy-momentum tensor and
sum rules
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Related to (angular) momentum conservation
Simplest case – partons momentum
conservation (gluons discovery)
More involved Ji’s sum rules for GPDs (talks
of W.-D. Nowak, D. Mueller, O. Selyugin)
Relevant also for coupling of hadrons to
gravity (OT’99)
1-st moments - EM, 2-nd Gravitational Formfactors
Describe the partition of angular momentum between quarks and
gluons
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Conservation laws - zero Anomalous Gravitomagnetic Moment :
(g=2)
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Describe interaction with both classical and TeV gravity (and
Pomeron?? – talk of J. Soffer) – all smallness in gravity coupling
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May be extracted from high-energy experiments/NPQCD calculations
Equivalence principle
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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
(Reanalysis - Silenko,OT’07)
Anomalous gravitomagnetic moment iz ZERO
(= Ji SR)
or
Classical and QUANTUM rotators behave in
the SAME way
Generalization of Equivalence
principle
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Various arguments: AGM  0 separately
for quarks and gluons – most clear from
the lattice (LHPC/SESAM)
Recent lattice study (M. Deka
et al. arXiv:1312.4816)
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Sum of u and d for Dirac (T1) and Pauli
(T2) FFs
Extended Equivalence
Principle=Exact EquiPartition
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In pQCD – violated
Reason – in the case of ExEP- 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 generic smallness of E
(isoscalar AMM)
T-odd quantities
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P-conserving Single Spin asymmetries: Spin
pseudovector accompanied by Levi-Civita
tensor – change sign under the time
reflection
Hermiticity of density matrix: accompanied
by i (like photon circular polarization)
May appear either due to real T (CP) violation
(complex couplings) or due to non-trivial
phases (FSI)
T-odd TMDs
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Transverse momentum - like in UGDF in kT
factorization (but soft kT – sum of higher
twists) – talks of P. Zavada,O. Kouznetsov
T-odd (ST-kT correlation) Sivers TMD due to
FSI (Brodsky, Hwang, Schmidt) – relation to
AMM
Burkardt sum rule
Conjecture (OT’06) AMM ->AGM:
Ji SR <-> Burkardt SR
Direct derivation of Burkardt SR from EMT?
Burkardt SR and EnergyMomentum tensor
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Consider twist 3 (Bukhvostov, Kuraev,Lipatov;Efremov,OT’83) –
related to relevant moment of Sivers function being infinite
tower of twists
Specific vector qqG(zero momentum) correlator – gluonic pole
EMT forward matrix element: no spin-dependent structure
Naively: Valid identically due to symmetry properties implied by
T-invariance)
However: EOM – generated pole should get imaginary part due
to EMGI and related contour gauge(DY process: Anikin, OT,
PLB2010) – analog of the choice of Wilson line
Cut in Q2: cf cuts in pion dissociation to dijet (D. Ivanov et al.)
Electromagnetric Gauge
Invariance in DY process
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Extra diagram – factor 2 in transverse
(TM integrated) asymmetry
Absent in pQCD
Pole prescription and Burkardt
SR
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Pole prescription (dynamics!) provides (“T-odd”)
symmetric part!
SR:
(but relation of gluon Sivers to
twist 3 still not found – prediction!)
Can it be valid separately for each quark flavour:
nodes (related to “sign problem”)?
Valid if structures forbidden for TOTAL EMT do not
appear for each flavour
Structure contains besides S gauge vector n: If GI
separation of EMT – forbidden: SR valid separately!
Can one have T-even inclusive
spin SR?
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Spin pseudovector – calls for either
GPDs or TMDs (extra momentum
required for Levi-Civita tensor)
But! Tensor polarization (talk of
K. Slifer) is P-even – no T-oddness as
well!
Another manifestation of postNewtonian (Ex)EP for spin 1 hadrons
(SRs - Efremov,OT’82; EP- OT’09)
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Tensor polarization coupling of gravity
to spin in forward
matrix elements inclusive processes
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Second moments of
tensor distributions
should sum to zero
=0 for ExEP
HERMES – data on tensor
spin structure function
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Isoscalar target –
proportional to the
sum of u and d
quarks –
combination
required by EMT
Second moments –
compatible to zero
better than the first one
(collective glue << sea)
– for valence:
Are more accurate data
possible?
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HERMES – unlikely
JLab (talks of O.Rondon,K.Slifer) may
provide information about collective sea
and glue in deuteron and indirect new
test of Equivalence Principle
CONCLUSIONS
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Generalized cut moment may be useful for
SRs extrapolation
T-odd Burkardt SR is related to EMT
conservation and peculiar cut in soft gluon
emission
EMTSR for tensor polarization testing
deuteron collective sea/glue and Equivalence
Principle – may be tested at JLab?
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BACKUP SLIDES
Cut Mellin moments (Kotlorz,
Mikhailov
)
Electromagnetism vs Gravity
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Interaction – field vs metric deviation
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Static limit
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Mass as charge – equivalence principle
Gravitomagnetism
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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
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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, O.T.)
EEP and AdS/QCD
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Recent development – calculation of
Rho formfactors in Holographic QCD
(Grigoryan, Radyushkin)
Provides g=2 identically!
Experimental test at time –like region
possible
Preprint (in Russian) with first
derivation of sum rules for 1st
(=Close-Kumano) and 2nd moments