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

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 - e 2 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