Beta-function%20as%20Infrared%20``Phenomenon

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Beta-function as Infrared Phenomenon

RG-2008 (Shirkovfest)

JINR, Dubna, September 1 2008

• Oleg Teryaev
• JINR

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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?

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Dilatational anomaly

• Classical and anomalous terms
• Beta function describes the appearance of scale
  dependence due to renormalization

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

• VVA correlator
• Unsubtracted dispersion relations

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Anomaly as a finite subtraction

• Non-anomalous axial Ward identities for imaginary
  parts (pseudoscalar current B -gt Im G
• -gt Finite subtraction for real parts
• Anomaly sum rule

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

• Scalar theory
• -gt Improved EMT

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Traiangle diagram

• Transition of EMT to 4 mesons
• Special kinematics. C.m. -gt

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Ward Identities

• Translational and dilatational WI
• Invariant formfactors

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Trace anomaly from dispersion relations

• Anomaly-free for imaginary parts
• Unsubtracted DR translational invariance
• Anomaly

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Explicit calculation (Kawka, Veretin, OT)

• Exact calculation of imaginary parts

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IR effect

• m -gt0 - Dilaton pole
• Pure dimensional reason
• Heavy mass limit decoupling (cancellation of
  classical and anomalous terms)

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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 (EH)

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Counting the Chirality

• Degeneracy rate of Landau levels
• Transverse HS/(1/e)
  (Flux/flux quantum)
• Longitudinal Ldp eE dt L (dpeEdt)
• Anomaly coefficient in front of
  4-dimensional volume - e2 EH
  

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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 -gt 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

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Heavy quarks matrix elements

• QCD at LO
• From anomaly cancellations (2733-6)
• Light terms
• Dominated by s-of the order of cancellation -gt
  heavy

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Back to axial anomaly -gt 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

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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

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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

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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

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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

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Heavy Strangeness transversity

• Heavy strange quarks neglect genuine higher
  twist 0
• Strange transversity - of the same sign as
  helicity and enhanced by M/m!

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Other case of LT-HT relations naively leading
twists TMD functions gtinfinite 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