A.O.Barvinsky

Theory Department, Lebedev Physics Institute, Moscow

674. WE-Heraeus-Seminar “Quantum spacetime and

the Renormalization Group”

Physikzentrum Bad Honnef, June 18-22, 2018

Renormalization of gauge theories

in the background-field approach

with D. Blas

M. Herrero-Valea

S. Sibiryakov

and C. Steinwachs

Task: to prove that counterterms are covariant local functionals of the

original gauge field

Gauge-breaking and Faddeev-Popov ghost terms – counterterms to

them?

Preservation of the BRST structure and counterterms covariance by

1) The choice of background covariant gauge conditions

and

2) gauge field reparametrization

(DeWitt, Tuytin-Voronov, Batalin-Vilkovisky, Kallosh, Arefieva-Faddeev-

Slavnov, Abbot, Henneaux et al)

40+ years old topic! So what is new here?

D.Blas, M.Herrero-Valeya, S.Sibiryakov, C.Steinwachs & A.B.

Phys. Rev. D 93, 064022 (2016), arXiv:1512.02250;

arXiv:1705.03480; PRL 119,211301 (2017), arXiv:1706.06809

Background field extension of the BRST operator

Inclusion of generating functional sources into the gauge fermion

Invariance of single original gauge field counterterms via

decoupling of the background field

Quantum corrected gauge fermion is a generating functional of the

field reparameterization

No power counting or use of field dimensionalities

Extension to Lorentz symmetry violating theories

Extension to (nonrenormalizable) effective field theories

Introduction: towards local, unitary, perturbatively UV renormalizable QG

Lorentz violating extension -- Horava-Lifshitz gravity

Problems with renormalization: “regularity” of propagators

covariance of UV counterterms

BRST formalism

Background covariant gauge fixing – modification of BRST operator

Introducing sources into the gauge fermion and BRST operator

Slavnov-Taylor and Ward identities: tale of two (many) BRS operators

BRST structure of renormalization and field reparametrization

Gauge fermion as a generating functional of field reparametrization

Example: renormalization of O(N) sigma model

Asymptotic freedom in (2+1)-dimensional Horava gravity

Plan

Phys. Rev. D 93, 064022 (2016), arXiv:1512.02250;

arXiv:1705.03480; PRL 119,211301 (2017), arXiv:1706.06809

Generators of gauge transformations:

Gauge theory:

DeWitt summation rule:

structure constants

BRST formalism

gauge-breaking

term Faddeev-Popov

operator

measure

Feynman-DeWitt-Faddeev-Popov functional integral

BRST functional

integral

BRST action

nilpotent BRST operator

BRST transformations of ©

Gauge fermion (correlators of physical

observable are independent of ª ) gauge

conditions

gauge-fixing

matrix

BRST invariance

Assumptions on the class of theories

generators: linear

closed algebra

ireducible

Existence of gauge invariant regularization (no anomalies) – very subtle!

Background gauge fixing

same

background field

background gauge

transformations (BGT)

Linear representation of the gauge group:

fundamental representation

adjoint representation

Modification of BRST operator

anticommuting auxiliary field – controls

dependence on background Á

Introduction of sources into gauge fermion

sources for sources for

background

source

Obviously

Introduction of sources into BRST charge

Extended BRS operator

Extended gauge fermion

Add background gauge transformation of other fields and sources:

BGT symmetry

Full set of symmetries

BRST symmetry

U(1) symmetry with ghost numbers:

L-th order renormalization: quantum field

Apply it to a gauge theory: effective action renormalized in (L-1)-th order

-- local divergence

Main result

renormalized fields

local invariants of the

original gauge field

Renormalization of the action and gauge fermion

Renormalized generating functional:

local invariants of the

original gauge field

ghost fields

and sources

dependent

Local generally nonlinear reparametrization of quantum fields to composite operators including external sources

Applies to (non-renormalizable) EFT -- nonlinear dependence on sources

For renormalizable theories:

independent of other sources

linear in and ³

Gauge fermion is a generating function of the field redefinition

reduced

effective action

all sources

BRST

BGT

b® multiplier

equation

Transition to effective action

subtracting gauge fixing term

Slavnov-Taylor and Ward identities

Slavnov-Taylor identities for

antibracket

Ward identities for

Structure of divergences: tale of two (many) BRST operators

Proceed perturbatively in ~ :

By induction ¡L-1 ¡L

minimal

sector nilpotent

-- action renormalized in the L-th loop order

ST identity for ¡

Kozhul-Tate differential has a trivial cohomology under the assumption

of local completeness and irreducibility of gauge generators for

Henneaux (1991)

Vandoren, Van Proeyen (1994)

ST+W identities:

Trivial cohomology of Batalin and

Vilkovisky (1985)

single field

counterterm

BRST exact

term (not yet the

original one!)

ghost term

decoupling of

invariant counterterm

Applies to nonrenormalizable EFT within gradient expansion: truncate in

number of derivatives + locality + Grassman statistics of

at any loop order L

Thus:

full action vs reduced

BRST exact term

(not yet the original

BRST charge!)

field redefinition

L-th order subtraction and Q+ Q transition via field

redefinition

gauge fermion

renormalization

Original gauge fermion is a generating function of the field redefinition

The power and beauty of the nilpotent BRST operator

Example: 2D O(N) gauge model

Abelian gauge invariance

Gauge fixing

One-loop divergences

essentially nonlinear

Physical one-loop counterterm

Field redefinition

Fradkin, Tseytlin (1981) Avramidy & A.B. (1985)

nonrenormalizable Einstein GR

Stelle (1977)

Higher derivative gravity

Saving unitarity in renormalizable QG

Asymptotic freedom in Horava gravity

Anisotropic scaling transformations and scaling dimensions

Foliation preserving diffeomorphisms

ADM metric decomposition

space

dimensionality

extrinsic

curvature

``Projectable’’ theory

Potential

term

Many more versions: extra

structures in non-projectable theory,

reduction of structures for

detailed balance case . . .

Horava gravity

action

kinetic term -- unitarity

Divergences power counting

critical value

Generalization of BPHZ renormalization theory (subtraction of

subdivergences) works only for Am> 0 and Bm> 0

depends on gauge fixing

Things are not so simple: power counting is not enough?

Invention of regular gauges for projectable HG

extra derivatives to

Have homogeneity

in scaling

local

Localization by auxiliary fields (cf. b-field in BRST formalism):

Asymptotic freedom in (2+1)-dimensions

Essential coupling constants:

Off-shell extension is not unique:

background split

background covariant

gauge-fixing term

¾, » – free parameters

localization of the

kinetic term by

auxilairy field ¼

action of ghosts:

Diagrammatic technique in terms

of

Perturbation theory around

flat ST:

Propagators

particular gauge parameters Prototypical loop ntegrals

Expansion in external

momenta P and

fequencies

Logarithmic divergences

and ¯-functions

Mathematica package xAct

D. Brizuela, J. M. Martin-Garcia, and G. A. Mena

Marugan, Gen. Rel. Grav. 41, 2415 (2009),

arXiv:0807.0824

Check in conformal gauge

Compare to regular ``relativistic’’ gauge

same

Renormalization flows:

AF UV fixed point

strongly coupled f. point

Cf. conformal truncation of (2+1)D HG, Benedetti and Guarneri, JHEP 03(2014)078: f. point ¸=1/2,

unitarity at G<0, ¸>1

Conclusions

BRST structure of renormalization in background field formalism:

Invariance of physical counterterms

renormalization of gauge fermion

gauge fermion is a generating functional of the field redefinition

generalization to open algebras, anomalies, composite operators?

(2+1)-dimensional Horava-Lifshitz gravity is unitary and asymptotically

free – perturbatively UV-complete theory

In IR one can have ¸ 1 – GR limit? Need for nonperturbative analysis

as G1

Towards consistent (3+1)-dimensional QG – other calculational methods

(heat kernel technique)