Transcript Forbidden Landscape from Holography

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)
• 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
Disclaimer
• 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.
Computation in conformal
perturbation theory
(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
Approach
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
A bit on Holographic computation
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…
Conclusion
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?