renormalization of gauge theories in the background-field ...eichhorn/barvinsky.pdf ·...
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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
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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
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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
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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
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Generators of gauge transformations:
Gauge theory:
DeWitt summation rule:
structure constants
BRST formalism
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gauge-breaking
term Faddeev-Popov
operator
measure
Feynman-DeWitt-Faddeev-Popov functional integral
BRST functional
integral
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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
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Assumptions on the class of theories
generators: linear
closed algebra
ireducible
Existence of gauge invariant regularization (no anomalies) – very subtle!
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Background gauge fixing
same
background field
background gauge
transformations (BGT)
Linear representation of the gauge group:
fundamental representation
adjoint representation
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Modification of BRST operator
anticommuting auxiliary field – controls
dependence on background Á
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Introduction of sources into gauge fermion
sources for sources for
background
source
Obviously
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Introduction of sources into BRST charge
Extended BRS operator
Extended gauge fermion
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Add background gauge transformation of other fields and sources:
BGT symmetry
Full set of symmetries
BRST symmetry
U(1) symmetry with ghost numbers:
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L-th order renormalization: quantum field
Apply it to a gauge theory: effective action renormalized in (L-1)-th order
-- local divergence
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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
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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
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reduced
effective action
all sources
BRST
BGT
b® multiplier
equation
Transition to effective action
subtracting gauge fixing term
Slavnov-Taylor and Ward identities
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Slavnov-Taylor identities for
antibracket
Ward identities for
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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 ¡
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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
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at any loop order L
Thus:
full action vs reduced
BRST exact term
(not yet the original
BRST charge!)
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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
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The power and beauty of the nilpotent BRST operator
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Example: 2D O(N) gauge model
Abelian gauge invariance
Gauge fixing
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One-loop divergences
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essentially nonlinear
Physical one-loop counterterm
Field redefinition
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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
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Anisotropic scaling transformations and scaling dimensions
Foliation preserving diffeomorphisms
ADM metric decomposition
space
dimensionality
extrinsic
curvature
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``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
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Divergences power counting
critical value
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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):
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Asymptotic freedom in (2+1)-dimensions
Essential coupling constants:
Off-shell extension is not unique:
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background split
background covariant
gauge-fixing term
¾, » – free parameters
localization of the
kinetic term by
auxilairy field ¼
action of ghosts:
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Diagrammatic technique in terms
of
Perturbation theory around
flat ST:
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Propagators
particular gauge parameters Prototypical loop ntegrals
Expansion in external
momenta P and
fequencies
Logarithmic divergences
and ¯-functions
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Mathematica package xAct
D. Brizuela, J. M. Martin-Garcia, and G. A. Mena
Marugan, Gen. Rel. Grav. 41, 2415 (2009),
arXiv:0807.0824
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Check in conformal gauge
Compare to regular ``relativistic’’ gauge
same
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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
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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)