Beta-function as Infrared ``Phenomenon” RG-2008

(Shirkovfest) JINR, Dubna, September 1 2008

Oleg TeryaevJINR

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?

Dilatational anomaly

Classical and anomalous terms

Beta function – describes the appearance of scale dependence due to renormalization

Dispersive (IR) approach for AXIAL anomaly (Dolgov, Zakharov)

VVA correlator

Unsubtracted dispersion relations

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

Dispersive approach to trace anomaly (Horejsi, Schnabl; Kawka, Veretin, OT)

Scalar theory

-> Improved EMT

Traiangle diagram Transition of EMT to 4 ``mesons”

Special kinematics. C.m. ->

Ward Identities

Translational and dilatational WI

Invariant formfactors

Trace anomaly from dispersion relations

Anomaly-free for imaginary parts

Unsubtracted DR + translational invariance

Anomaly:

Explicit calculation (Kawka, Veretin, OT)

Exact calculation of imaginary parts:

IR effect m ->0 - “Dilaton” pole

Pure dimensional reason:

Heavy mass limit: decoupling (=cancellation of classical and anomalous terms)

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)

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

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

Heavy quarks matrix elements QCD at LO

From anomaly cancellations (27=33-6)

“Light” terms

Dominated by s-of the order of cancellation -> “heavy”

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

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

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

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

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

Heavy Strangeness transversity

Heavy strange quarks – neglect genuine higher twist: 0 =

Strange transversity - of the same sign as helicity and enhanced by M/m!

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