36 research outputs found
Exact Correlators from Conformal Ward Identities in Momentum Space and the Perturbative Vertex
We present a general study of 3-point functions of conformal field theory in
momentum space, following a reconstruction method for tensor correlators, based
on the solution of the conformal Ward identities (CWI' s), introduced in recent
works by Bzowski, McFadden and Skenderis (BMS). We investigate and detail the
structure of the CWI's, their non-perturbative solutions and the transition to
momentum space, comparing them to perturbation theory by taking QED as an
example. We then proceed with an analysis of the correlator, presenting
independent and detailed re-derivations of the conformal equations in the
reconstruction method of BMS, originally formulated using a minimal tensor
basis in the transverse traceless sector. A careful comparison with a second
basis introduced in previous studies shows that this correlator is affected by
one anomaly pole in the graviton (T) line, induced by renormalization. The
result shows that the origin of the anomaly, in this correlator, should be
necessarily attributed to the exchange of a massless effective degree of
freedom. Our results are then exemplified in massless QED at one-loop in
-dimensions, expressed in terms of perturbative master integrals. An
independent analysis of the Fuchsian character of the solutions, which bypasses
the 3K integrals, is also presented. We show that the combination of field
theories at one-loop - with a specific field content of degenerate massless
scalar and fermions - is sufficient to generate the complete non-perturbative
solution, in agreement with a previous study in coordinate space. The result
shows that free conformal field theories, in specific dimensions, arrested at
one-loop, reproduce the general result for the . Analytical checks of this
correspondence are presented in and spacetime dimensions[..].Comment: 79 pages, 3 Figures. Final version, with changes in section 8.
Accepted for publication in Nuclear Physics
The Generalized Hypergeometric Structure of the Ward Identities of CFT's in Momentum Space in
We review the emergence of hypergeometric structures (of Appell
functions) from the conformal Ward identities (CWIs) in conformal field
theories (CFTs) in dimensions . We illustrate the case of scalar 3- and
4-point functions. 3-point functions are associated to hypergeometric systems
with 4 independent solutions. For symmetric correlators they can be expressed
in terms of a single 3K integral - functions of quadratic ratios of momenta -
which is a parametric integral of three modified Bessel functions. In the
case of scalar 4-point functions, by requiring the correlator to be conformal
invariant in coordinate space as well as in some dual variables (i.e. dual
conformal invariant), its explicit expression is also given by a 3K integral,
or as a linear combination of Appell functions which are now quartic ratios of
momenta. Similar expressions have been obtained in the past in the computation
of an infinite class of planar ladder (Feynman) diagrams in perturbation
theory, which, however, do not share the same (dual conformal/conformal)
symmetry of our solutions. We then discuss some hypergeometric functions of 3
variables, which define 8 particular solutions of the CWIs and correspond to
Lauricella functions. They can also be combined in terms of 4K integral and
appear in an asymptotic description of the scalar 4-point function, in special
kinematical limits.Comment: 31 pages, 1 figure. Invited contribution to appear in: Axioms (MDPI)
"Geometric Analysis and Mathematical Physics" Ed. Sorin Dragomir, revised
final version, typos correcte
Conformal Ward Identities and the Coupling of QED and QCD to Gravity
We present a general study of 3-point functions of conformal field theory
(CFT) in momentum space, following a reconstruction method for tensor
correlators, based on the solution of the conformal Ward identities (CWIs),
introduced in recent works. We investigate and detail the structure of the CWIs
and their non-perturbative solutions, and compare them to perturbation theory,
taking QED and QCD as examples. Exact solutions of CFT's in the flat background
limit in momentum space are matched by the perturbative realizations in free
field theories, showing that the origin the conformal anomaly is related to
efffective scalar interactions, generated by the renormalization of the
longitudinal components of the corresponding operators.Comment: 5 pages. Proceedings of the Workshop QCD@work 2018, 25-28 June 2018,
Matera, Ital
Exact Correlators from Conformal Ward Identities in Momentum Space and Perturbative Realizations
The general solution of the conformal Ward identities (CWI's) in momentum
space, and their matching to perturbation theory, allows to uncover some
specific characteristics of the breaking of conformal symmetry, induced by the
anomaly. It allows to compare perturbative features of the 1-particle
irreducible (1PI, nonlocal) anomaly action with the prediction of a similar
(but exact) nonlocal action identified by the CWI's. The two predictions can be
exactly matched at the level of 3-point functions. The analysis of the
and shows that both approaches - based either on 1PI or on the exact
solutions of the CWI's - predict massless (dynamical) scalar exchanges in
3-point functions as the signature of the conformal anomaly. In a local
formulation such 1PI actions exhibit a ghost in the spectrum which may induce
ghost condensation. We also discuss alternative approaches, which take to
Wess-Zumino forms of the action with an asymptotic dilaton, which should be
considered phenomenological alternatives to the exact nonlocal action. If
derived by a Weyl gauging, they also include a ghost in the spectrum. The two
formulations, nonlocal and of WZ type, can be unified under the assumption that
they describe the same anomaly phenomenon at two separate (UV/IR) ends of the
renormalization group flow, possibly separated by a vacuum rearrangement at an
intermediate scale. A similar analysis is presented for an
supersymmetric Yang-Mills theory. We comment on the possibile cosmological
implications of such quasi Nambu-Goldstone modes as ultralight dilatons and
axions.Comment: 48 pages, 12 figures, (typos corrected) Proceedings of the Corfu
Summer Institute 2018 "School and Workshops on Elementary Particle Physics
and Gravity" 1-27 September 2018 Corfu, Greec
Gravitational coupling of QED and QCD: 3- and 4- point functions in momentum space
Conformal symmetry has important consequences for strong interactions at
short distances and provides powerful tools for practical calculations. Even if
the Lagrangians of Quantum Chromodynamics (QCD) and Electrodynamics (QED) are
invariant under conformal transformations, this symmetry is broken by quantum
corrections. The signature of the symmetry breaking is encoded in the presence
of massless poles in correlators involving stress-energy tensors. We present a
general study of the correlation functions and of conformal field theory (CFT) in the flat background limit in
momentum space, following a reconstruction method for tensor correlators.
Furthermore, our analysis also focuses on studying the dimensional degeneracies
of the tensor structures related to these correlators.Comment: 7 pages, 1 figure, QCD@work 2022 conference proceedin
Renormalization, Conformal Ward Identities and the Origin of a Conformal Anomaly Pole
We investigate the emergence of a conformal anomaly pole in conformal field
theories in the case of the correlator. We show how it comes to be
generated in dimensional renormalization, using a basis of 13 form factors (the
-basis), where only one of them requires renormalization ,
extending previous studies. We then combine recent results on the structure of
the non-perturbative solutions of the conformal Ward identities (CWI's) for the
in momentum space, expressed in terms of a minimal set of 4 form factors
( basis), with the properties of the -basis, and show how the singular
behaviour of the corresponding form factors in both basis can be related. The
result proves the centrality of such massless effective interactions induced by
the anomaly, which have recently found realization in solid state, in the
theory of topological insulators and of Weyl semimetals. This pattern is
confirmed in massless abelian and nonabelian theories (QED and QCD)
investigated at one-loop.Comment: 15 pages, 1 figure, few typos corrections, final version accepted for
publication in Physics Letters
The General 3-Graviton Vertex () of Conformal Field Theories in Momentum Space in
We present a study of the correlation function of three stress-energy tensors
in dimensions using free field theory realizations, and compare them to the
exact solutions of their conformal Ward identities (CWI's) obtained by a
general approach in momentum space. The identification of the corresponding
form factors is performed within a reconstruction method, based on the
identification of the transverse traceless components of the same
correlator. The solutions of the primary CWI' s are found by exploiting the
universality of the Fuchsian indices of the conformal operators and a
re-arrangement of the corresponding inhomogenous hypergeometric systems. We
confirm the number of constants in the solution of the primary CWI's of
previous analysis. In our comparison with perturbation theory, we discuss
scalar, fermion and spin 1 exchanges at 1-loop in dimensional regularization.
Explicit checks in and prove the consistency of this
correspondence. By matching the 3 constants of the CFT solution with the 3 free
field theory sectors available in d=4, the general solutions of the conformal
constraints is expressed just in terms of ordinary scalar 2- and 3-point
functions . We show how the renormalized TTT vertex separates
naturally into the sum of a traceless and an anomaly part, the latter
determined by the anomaly functional and generated by the renormalization of
the correlator in dimensional regularization. The result confirms the emergence
of anomaly poles and effective massless exchanges as a specific signature of
conformal anomalies in momentum space, directly connected to the
renormalization of the corresponding gravitational vertices, generalizing the
behaviour found for the vertex in previous works.Comment: 71 pages, 5 figures. Final version with corrected typos. Accepted for
publication on Nucl. Phys.
Einstein Gauss-Bonnet Theories as Ordinary, Wess-Zumino Conformal Anomaly Actions
Recently, the possibility of evading Lovelock's theorem at , via a
singular redefinition of the dimensionless coupling of the Gauss-Bonnet term,
has been very extensively discussed in the cosmological context. The term is
added as a quadratic contribution of the curvature tensor to the
Einstein-Hilbert action, originating theories of "Einstein Gauss-Bonnet" (EGB)
type. We point out that the action obtained by the dimensional regularization
procedure, implemented with the extraction of a single conformal factor,
correspond just to an ordinary Wess-Zumino anomaly action, even though it is
deprived of the contribution from the Weyl tensor. We also show that a purely
gravitational version of the EGB theory can be generated by allowing a finite
renormalization of the Gauss-Bonnet topological contribution at , as pointed out by Mazur and Mottola. The result is an effective
action which is quadratic, rather then quartic, in the dilaton field, and scale
free, compared to the previous derivations. The dilaton, in this case can be
removed from the spectrum, leaving a pure gravitational theory, which is
nonlocal. We comment on the physical meaning of the two types of actions, which
may be used to describe such topological terms both below and above the
conformal breaking scale
Electroweak Corrections to Photon Scattering, Polarization and Lensing in a Gravitational Background and the Near Horizon Limit
We investigate the semiclassical approach to the lensing of photons in a
spherically symmetric gravitational background, starting from Born level and
include in our analysis the radiative corrections obtained from the electroweak
theory for the graviton/photon/photon vertex. In this approach, the cross
section is related to the angular variation of the impact parameter (),
which is then solved for as a function of the angle of deflection, and
measured in horizon units (). Exact numerical solutions
for the angular deflection are presented. The numerical analysis shows that
perturbation theory in a weak background agrees with the classical Einstein
formula for the deflection already at distances of the order of horizon
units () and it is optimal in the description both of very
strong and weak lensings. We show that the electroweak corrections to the cross
section are sizeable, becoming very significant for high energy gamma rays. Our
analysis covers in energy most of the photon spectrum, from the cosmic
microwave background up to very high energy gamma rays, and scatterings with
any value of the photon impact parameter. We also study the helicity-flip
photon amplitude, which is of in the weak coupling , and
its massless fermion limit, which involves the exchange of a conformal anomaly
pole. The corresponding cross section is proportional to the Born level result
and brings to a simple renormalization of Einsten's formula.Comment: 38 pages, 18 figures. Revised final version, accepted on JHE