Vector Beta Function

Yu Nakayama　（ IPMU & Caltech）

arXiv:1310.0574 

• Analogous to scalar beta function

Vector Beta Function

• 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

Why do we care?

Vector beta functions must satisfy

• Compensated gauge invariance

• Orthogonality

• Higgs-like relation with anomalous dimension

• Gradient property

• Non-renormalization

What I will show (or claim)

• General argument based on local renormalization group flow– Consistency conditions

• Direct computations– Conformal perturbation theories– Holography

How am I going to show?

• 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…

Disclaimer

Consider renormalized Schwinger functional

A priori, vector beta function is expanded as

But, I claim it must be gauge covariant

1. Compensated gauge invariance

Scalar beta functions and vector beta functions are orthogonal

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

2. Orthogonality condition

We can compute anomalous dimensions of scalar operators and vector operators

3. Anomalous dimensions

: representation matrix of symmetry group G

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)

4. Gradient property

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

5. Non-renormalization

Computation in conformal perturbation theory

Second order in perturbation

(Redundant) Conformal perturbation theory

• Compensated gauge invariance almost obvious from power-counting and current (non)-conservation

• Orthogonality

– Scalar beta function is gradient– C-function is gauge invariant

Checks 1

• Anomalous dimensions

• Gradient property

• Non-renormalization– Essentially Higgs effect

Checks 2

Local Renormalization Group Approach

• Renormalized Schwinger functional

• Action principle

• Local renormalization group operator

• Local Callan-Symanzik eq or trace identity

Local Renormalization Group

• Current non-conservation

• Compensated gauge invariance

• With this gauge (scheme) freedom, local renormalization group operator and beta functions are ambiguous

Gauge (scheme) ambiguity

• 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

Interlude: cyclic conformal flow?

• Simple observation (Osborn):

• For this to hold

• Consistency of Hamiltonian constraint

Integrability condition

• Start with local Callan-Symazik equation

• Act , and integrate over x once

Anomalous dimension formula

Anomalous dimensions

• From powercounting

• Gradient property requires

• Does this hold?

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

Gradient property (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

Non-renormalization (conj)

A bit on Holographic computation

• 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 improvem

ent method

• Similar to (super)potential flow

Vector beta functions in holography

• 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 1

• Orthogonality– Gauge invariance of (super)potential

• Anomalous dimensions massive vector from bulk Higgs mechanism

Check 2

• 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…

Check 3

Conclusion

• To be studied more– 72 functions to be computed in standard mod

el– What is variation of potential functional with re

spect to ?– New fixed points? Domain walls?– Any monotonicity?

Vector Beta Function