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Vector Beta Function. Yu Nakayama  ( IPMU & Caltech ) arXiv:1310.0574. Vector Beta Function. Analogous to scalar beta function. Why do we care?. Poincare breaking: e.g. chemical potential Space-time dependent coupling const (localization, domain wall etc)

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vector beta function

Vector Beta Function

Yu Nakayama (IPMU & Caltech)



Vector Beta Function

  • Analogous to scalar beta function

Why do we care?

  • Poincare breaking: e.g. chemical potential
  • Space-time dependent coupling const (localization, domain wall etc)
  • Renormalization of vector operators (vector meson, non-conserved current etc)
  • Cosmology
  • Condensed matter
  • Holography

What I will show (or claim)

Vector beta functions must satisfy

  • Compensated gauge invariance
  • Orthogonality
  • Higgs-like relation with anomalous dimension
  • Gradient property
  • Non-renormalization

How am I going to show?

  • General argument based on local renormalization group flow
    • Consistency conditions
  • Direct computations
    • Conformal perturbation theories
    • Holography


  • My argument is general
  • I believe they are true in any sufficiently good relativistic field theories
  • Beta functions should make sense
  • To make the statement precise, I do assume powercounting renormalization scheme
  • It should work also in Wilsonian sense…

1. Compensated gauge invariance

Consider renormalized Schwinger functional

A priori, vector beta function is expanded as

But, I claim it must be gauge covariant


2. Orthogonality condition

Scalar beta functions and vector beta functions are orthogonal

There are 72 such relations in standard model beta functions( only depends on )


3. Anomalous dimensions

We can compute anomalous dimensions of scalar operators and vector operators

: representation matrix of symmetry group G


4. Gradient property

Vector beta functions are generated as a gradient of the local gauge invariant functional

Cf: Scalar beta functions are generated by gradient flow (strong c-theorem)


5. Non-renormalization

Vector beta functions are zero if and only if the corresponding current is conserved.


(Redundant) Conformal perturbation theory

Second order in perturbation


Checks 1

  • Compensated gauge invariance almost obvious from power-counting and current (non)-conservation
  • Orthogonality
    • Scalar beta function is gradient
    • C-function is gauge invariant

Checks 2

  • Anomalous dimensions
  • Gradient property
  • Non-renormalization
    • Essentially Higgs effect

Local Renormalization Group

  • Renormalized Schwinger functional
  • Action principle
  • Local renormalization group operator
  • Local Callan-Symanzik eq or trace identity

Gauge (scheme) ambiguity

  • Current non-conservation
  • Compensated gauge invariance
  • With this gauge (scheme) freedom, local renormalization group operator and beta functions are ambiguous

Interlude: cyclic conformal flow?

  • The choice

is very convenient because B=0  conformal

  • Alternatively, even for CFT,

is possible by gauge (scheme) choice

  • Unless you compute vector beta functions, you are uncertain…
  • You are (artificially) renormalizing the total derivative term. The flow looks cyclic…
  • But it IS CFT

Integrability condition

  • Simple observation (Osborn):
  • For this to hold
  • Consistency of Hamiltonian constraint

Anomalous dimensions

  • Start with local Callan-Symazik equation
  • Act , and integrate over x once

Anomalous dimension formula


Gradient property (conj)

  • From powercounting
  • Gradient property requires
  • Does this hold?

I don’t have a general proof, but it seems crucial in holography (S.S. Lee)


Non-renormalization (conj)

  • Non-renormalization for conserved current
  •  direction is a standard argument: conserved current is not renormalized
  •  direction is more non-trivial. If H and G is non-singular, it must be true
  • Closely related to scale vs conformal

Vector beta functions in holography

  • Non-conserved current  Spontaneously broken gauge theory in bulk
  • For simplicity I’ll consider fixed AdS
  • In a gauge
  • For sigma model with potential

Vector beta functions in holography

  • Relate 2nd order diff  1st order RG eq
    • Hamilton-Jacobi method
    • CGO singular perturbation with RG improvement method
  • Similar to (super)potential flow

Check 1

  • Gauge invariance
    • d-dim invariance is obvious
    • What is d+1-dim gauge transformation?
  • This leads to apparent cyclic flow for AdS space-time.

Check 2

  • Orthogonality
    • Gauge invariance of (super)potential

  • Anomalous dimensions  massive vector from bulk Higgs mechanism

Check 3

  • Gradient property
    • Radial Lagrangian  potential functional
    • Partly conjectured by S.S Lee

  • Non-renormalization
    • Common lore from unitarity
    • Higgs mechanism  Massive vector
    • Massive vector  Higgs mechanism
    • Can be broken at the sacrifice of NEC…

Vector Beta Function

  • To be studied more
    • 72 functions to be computed in standard model
    • What is variation of potential functional with respect to ?
    • New fixed points? Domain walls?
    • Any monotonicity?