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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