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Vector Beta Function

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

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  1. Vector Beta Function Yu Nakayama　（IPMU & Caltech） arXiv:1310.0574

  2. Vector Beta Function • Analogous to scalar beta function

  3. 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

  4. 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

  5. How am I going to show? • General argument based on local renormalization group flow • Consistency conditions • Direct computations • Conformal perturbation theories • Holography

  6. 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…

  7. 1. Compensated gauge invariance Consider renormalized Schwinger functional A priori, vector beta function is expanded as But, I claim it must be gauge covariant

  8. 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 )

  9. 3. Anomalous dimensions We can compute anomalous dimensions of scalar operators and vector operators : representation matrix of symmetry group G

  10. 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)

  11. 5. Non-renormalization Vector beta functions are zero if and only if the corresponding current is conserved.

  12. Computation in conformal perturbation theory

  13. (Redundant) Conformal perturbation theory Second order in perturbation

  14. 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

  15. Checks 2 • Anomalous dimensions • Gradient property • Non-renormalization • Essentially Higgs effect

  16. Local Renormalization Group Approach

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

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

  19. 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

  20. Integrability condition • Simple observation (Osborn): • For this to hold • Consistency of Hamiltonian constraint

  21. Anomalous dimensions • Start with local Callan-Symazik equation • Act , and integrate over x once Anomalous dimension formula

  22. 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)

  23. 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

  24. A bit on Holographic computation

  25. 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

  26. 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

  27. 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.

  28. Check 2 • Orthogonality • Gauge invariance of (super)potential  • Anomalous dimensions  massive vector from bulk Higgs mechanism

  29. 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…

  30. Conclusion

  31. 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?

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