Chiral-even and odd faces of transverse Sum Rule Trieste(+Dubna), November 24 2006

Oleg TeryaevJINR, Dubna

Outline Even chirality of spin and angular momentum operators

Comparing longitudinal and transverse sum rules: g_1 -> g_T Non-zero contribution of Gluon Spin to transverse SR Burkhatrdt-Cottingham sum rule: similarity of longitudinal and

transverse spin structure Chiral-odd BLT SR – test of dynamical picture of nucleon,

relations of even and odd – POSSIBLE! Belinfante invariance and equivalence principle – possible

violation in the case of transversity appearance in Spin Sun Rule

Relation to Sivers functions, Burkardt SR and Brodsky/Gardner conjecture

Free vs bounded particles – longitudinal case

Free vs bounded particles – transverse case Density matrix – 2 terms responsible for

1 2 transverse polarization - chiral odd twist

2 transversity (1) and chiral even twist –3 g_T SAME for free particles –

independent for quarks bounded in a nucleon

Field-tyheoretical origin of parton model sum rules

Momentum and Spin sum rules

How derive SR for longitudinal and transverse spin?

Different components of angular momentum tensor and Pauli-Lubanski vector do not commute – one needs yo specify projection onto space-like vector n, (nP)=0.

Different projections (T vs L)– lead to appearance of different parton distributions – but ALWAYS chiral-even

Quarks:

Various projections of axial current:

Related by EOM to quark-gluon correlations

Gluons No gluonic transversity for spin-1/2

BUT transverse twist 3 distribution analogous to quark case:

May contribute to jet double transverse asymmetries at RHIC

Transverse sum rule Similar to longitudinal

Twist 3 not suppressed in SR – no Q Spins same as L due to BC SR Orbital -?

Different L and T orbital momenta – natural from the point of view of Brodsky-

Gardner conjecture

Sivers function similar to (transverse) L and AMM

Small singlet Sivers -> Small singlet AMM -> EQUIPARTION of

momentum and TOTAL angular momentum + small gluon spin -> large (longitudinal) orbital momentum

Transversity and BLT sum rule Should imply some relation of even and odd

operators May test DYNAMICAL picture of nucleon

which ay be surprisingly simple Say, transversity may be quite well

understood kinematically(Efremov, OT, Zavada) relating even and odd terms – may justify (implicit) notion of free particles – explains larger values of transversity than helicity

Fractional sum rule for transversity (Pire, Soffer, OT)

First moment is not conserved, but

May be a candidate for models/NPQCD

Chiral-even transverse SR –supported by EQUIVALENCE principle

Belinfante invariance -> spin in (chiral-even) orbital form

Momentum+Angular momentum conservation -> JI SR

Equivalence principle

Newtonian – “Falling elevator” + Anomalous gravitomagnetic

moment iz ZERO or Classical and QUANTUM rotators

behave in the SAME way

Electromagnetism vs Gravity

Interaction – field vs metric deviation

Static limit

Mass as charge – equivalence principle

Gravitational formfactors

Conservation laws - zero Anomalous Gravitomagnetic Moment : (g=2)

Moments of GPD’s (X. Ji)- may be extracted from high-energy experiments/NPQCD calculations

Describe the partition of angular momentum between quarks and gluons

Valid for any spin projection! Appearance of chiral-odd term in angular momentum conservation may violate EP – unless it is related to chiral-even

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

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

Another arguments in favour of EEP J=1/2 -> J=1. QCD SR calculation of Rho’s AMM

gives g close to 2. Maybe because of similarity of moments. Gluons momentum fraction sizable. Direct calculation for AGM are desired!

“Valence” Parametrization of E (GPV) – remarakble relations between valence quantities - physical input – EQUIPARTITION

Relation: E -> Sivers; EP -> Burkardt SR; EEP -> Brodsky/Gardner conjecture

Conclusions

Standard derivation -> chiral-even transverse SR

Longitudinal and transverse quark and gluon spins – same if BCSR is valid

L and T Orbital momenta – related to Brodsky et al conjectures

Chiral-odd sum rules – may test dynamical picture of nucleon

Spin (L and T)sum rules – related to equivalence principle; independent chiral-odd terms may violate it