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Oleg Teryaev JINR

Oleg Teryaev JINR. Beta-function as Infrared ``Phenomenon” RG-2008 ( Shirkovfest ) JINR, Dubna, September 1 2008. Outline. Beta-function and trace anomaly Dispersive approach to chiral anomaly

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Oleg Teryaev JINR

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  1. Oleg Teryaev JINR Beta-function as Infrared ``Phenomenon” RG-2008 (Shirkovfest) JINR, Dubna, September 1 2008

  2. Outline • Beta-function and trace anomaly • Dispersive approach to chiral anomaly • Dispersive approach to trace anomaly: beta function as a zero mass pole • Matching UV and IR • Dispersive approach and decoupling. When strange quarks can be heavy: multiscale hadrons • “Decoupling” of light quarks at IR; approximate conformal invariance and to AdS/QCD?

  3. Dilatational anomaly • Classical and anomalous terms • Beta function – describes the appearance of scale dependence due to renormalization

  4. Dispersive (IR) approach for AXIAL anomaly (Dolgov, Zakharov) • VVA correlator • Unsubtracted dispersion relations

  5. Anomaly as a finite subtraction • Non-anomalous axial Ward identities for imaginary parts (pseudoscalar current: B -> Im G: • -> Finite subtraction for real parts • Anomaly sum rule

  6. Dispersive approach to trace anomaly (Horejsi, Schnabl; Kawka, Veretin, OT) • Scalar theory • -> Improved EMT

  7. Traiangle diagram • Transition of EMT to 4 ``mesons” • Special kinematics. C.m. ->

  8. Ward Identities • Translational and dilatational WI • Invariant formfactors

  9. Trace anomaly from dispersion relations • Anomaly-free for imaginary parts • Unsubtracted DR + translational invariance • Anomaly:

  10. Explicit calculation (Kawka, Veretin, OT) • Exact calculation of imaginary parts:

  11. IR effect • m ->0 - “Dilaton” pole • Pure dimensional reason: • Heavy mass limit: decoupling (=cancellation of classical and anomalous terms)

  12. Matching of UV and IR (axial anomaly) • Both 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)

  13. 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

  14. Beta-function in IR region • Low momentum transfer – even light fermions (quarks) may be considered heavy • Cancellation of classical and anomalous terms – approximate conformal invariance -> AdS/QCD • C.f. analytic QCD PT (D.V. Shirkov, I.L.Solovtsov; talks of N.G. Stefanis, A.P.Bakulev, A.V.Nesterenko, O.P.Solovtsova, C.Valenzuella) – amendments (e.g. Bakulev, Radyushkin, Stefanis; Nestserenko) may lead to nullifications of beta-function

  15. Heavy quarks matrix elements • QCD at LO • From anomaly cancellations (27=33-6) • “Light” terms • Dominated by s-of the order of cancellation -> “heavy”

  16. Back to axial anomaly -> 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

  17. 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

  18. 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

  19. 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

  20. Conclusions/Outlook • Trace anomaly may be calculated in dispersive approach • Approximate scale invariance may appear in IR region. Ground for AdS/QCD? Small cosmological constant? • Multiscale picture of nucleon - Strange quarks may be considered are heavy sometimes

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

  22. 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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