1,359 research outputs found
A universal flow invariant in quantum field theory
A flow invariant is a quantity depending only on the UV and IR conformal
fixed points and not on the flow connecting them. Typically, its value is
related to the central charges a and c. In classically-conformal field
theories, scale invariance is broken by quantum effects and the flow invariant
a_{UV}-a_{IR} is measured by the area of the graph of the beta function between
the fixed points. There exists a theoretical explanation of this fact. On the
other hand, when scale invariance is broken at the classical level, it is
empirically known that the flow invariant equals c_{UV}-c_{IR} in massive
free-field theories, but a theoretical argument explaining why it is so is
still missing. A number of related open questions are answered here. A general
formula of the flow invariant is found, which holds also when the stress tensor
has improvement terms. The conditions under which the flow invariant equals
c_{UV}-c_{IR} are identified. Several non-unitary theories are used as a
laboratory, but the conclusions are general and an application to the Standard
Model is addressed. The analysis of the results suggests some new minimum
principles, which might point towards a better understanding of quantum field
theory.Comment: 28 pages, 3 figures; proof-corrected version for CQ
A note on the improvement ambiguity of the stress tensor and the critical limits of correlation functions
I study various properties of the critical limits of correlators containing
insertions of conserved and anomalous currents. In particular, I show that the
improvement term of the stress tensor can be fixed unambiguously, studying the
RG interpolation between the UV and IR limits. The removal of the improvement
ambiguity is encoded in a variational principle, which makes use of sum rules
for the trace anomalies a and a'. Compatible results follow from the analysis
of the RG equations. I perform a number of self-consistency checks and discuss
the issues in a large set of theories.Comment: 15 page
Renormalization of a class of non-renormalizable theories
Certain power-counting non-renormalizable theories, including the most
general self-interacting scalar fields in four and three dimensions and
fermions in two dimensions, have a simplified renormalization structure. For
example, in four-dimensional scalar theories, 2n derivatives of the fields,
n>1, do not appear before the nth loop. A new kind of expansion can be defined
to treat functions of the fields (but not of their derivatives)
non-perturbatively. I study the conditions under which these theories can be
consistently renormalized with a reduced, eventually finite, set of independent
couplings. I find that in common models the number of couplings sporadically
grows together with the order of the expansion, but the growth is slow and a
reasonably small number of couplings is sufficient to make predictions up to
very high orders. Various examples are solved explicitly at one and two loops.Comment: 38 pages, 1 figure; v2: more explanatory comments and references;
appeared in JHE
Infinite reduction of couplings in non-renormalizable quantum field theory
I study the problem of renormalizing a non-renormalizable theory with a
reduced, eventually finite, set of independent couplings. The idea is to look
for special relations that express the coefficients of the irrelevant terms as
unique functions of a reduced set of independent couplings lambda, such that
the divergences are removed by means of field redefinitions plus
renormalization constants for the lambda's. I consider non-renormalizable
theories whose renormalizable subsector R is interacting and does not contain
relevant parameters. The "infinite" reduction is determined by i) perturbative
meromorphy around the free-field limit of R, or ii) analyticity around the
interacting fixed point of R. In general, prescriptions i) and ii) mutually
exclude each other. When the reduction is formulated using i), the number of
independent couplings remains finite or slowly grows together with the order of
the expansion. The growth is slow in the sense that a reasonably small set of
parameters is sufficient to make predictions up to very high orders. Instead,
in case ii) the number of couplings generically remains finite. The infinite
reduction is a tool to classify the irrelevant interactions and address the
problem of their physical selection.Comment: 40 pages; v2: more explanatory comments; appeared in JHE
Inequalities for trace anomalies, length of the RG flow, distance between the fixed points and irreversibility
I discuss several issues about the irreversibility of the RG flow and the
trace anomalies c, a and a'. First I argue that in quantum field theory: i) the
scheme-invariant area Delta(a') of the graph of the effective beta function
between the fixed points defines the length of the RG flow; ii) the minimum of
Delta(a') in the space of flows connecting the same UV and IR fixed points
defines the (oriented) distance between the fixed points; iii) in even
dimensions, the distance between the fixed points is equal to
Delta(a)=a_UV-a_IR. In even dimensions, these statements imply the inequalities
0 =< Delta(a)=< Delta(a') and therefore the irreversibility of the RG flow.
Another consequence is the inequality a =< c for free scalars and fermions (but
not vectors), which can be checked explicitly. Secondly, I elaborate a more
general axiomatic set-up where irreversibility is defined as the statement that
there exist no pairs of non-trivial flows connecting interchanged UV and IR
fixed points. The axioms, based on the notions of length of the flow, oriented
distance between the fixed points and certain "oriented-triangle inequalities",
imply the irreversibility of the RG flow without a global a function. I
conjecture that the RG flow is irreversible also in odd dimensions (without a
global a function). In support of this, I check the axioms of irreversibility
in a class of d=3 theories where the RG flow is integrable at each order of the
large N expansion.Comment: 24 pages, 3 figures; expanded intro, improved presentation,
references added - CQ
Higher-spin current multiplets in operator-product expansions
Various formulas for currents with arbitrary spin are worked out in general
space-time dimension, in the free field limit and, at the bare level, in
presence of interactions. As the n-dimensional generalization of the
(conformal) vector field, the (n/2-1)-form is used. The two-point functions and
the higher-spin central charges are evaluated at one loop. As an application,
the higher-spin hierarchies generated by the stress-tensor operator-product
expansion are computed in supersymmetric theories. The results exhibit an
interesting universality.Comment: 19 pages. Introductory paragraph, misprint corrected and updated
references. CQG in pres
Theory of higher spin tensor currents and central charges
We study higher spin tensor currents in quantum field theory. Scalar, spinor
and vector fields admit unique "improved" currents of arbitrary spin, traceless
and conserved. Off-criticality as well as at interacting fixed points
conservation is violated and the dimension of the current is anomalous. In
particular, currents J^(s,I) with spin s between 0 and 5 (and a second label I)
appear in the operator product expansion of the stress tensor. The TT OPE is
worked out in detail for free fields; projectors and invariants encoding the
space-time structure are classified. The result is used to write and discuss
the most general OPE for interacting conformal field theories and
off-criticality. Higher spin central charges c_(s,I) with arbitrary s are
defined by higher spin channels of the many-point T-correlators and central
functions interpolating between the UV and IR limits are constructed. We
compute the one-loop values of all c_(s,I) and investigate the RG trajectories
of quantum field theories in the conformal window following our approach. In
particular, we discuss certain phenomena (perturbative and nonperturbative)
that appear to be of interest, like the dynamical removal of the I-degeneracy.
Finally, we address the problem of formulating an action principle for the RG
trajectory connecting pairs of CFT's as a way to go beyond perturbation theory.Comment: Latex, 46 pages, 4 figures. Final version, to appear in NPB. (v2:
added two terms in vector OPE
Large-N expansion, conformal field theory and renormalization-group flows in three dimensions
I study a class of interacting conformal field theories and conformal windows
in three dimensions, formulated using the Parisi large-N approach and a
modified dimensional-regularization technique. Bosons are associated with
composite operators and their propagators are dynamically generated by fermion
bubbles. Renormalization-group flows between pairs of interacting fixed points
satisfy a set of non-perturbative g 1/g dualities. There is an exact
relation between the beta function and the anomalous dimension of the composite
boson. Non-Abelian gauge fields have a non-renormalized and quantized gauge
coupling, although no Chern-Simons term is present. A problem of the naive
dimensional-regularization technique for these theories is uncovered and
removed with a non-local, evanescent, non-renormalized kinetic term. The models
are expected to be a fruitful arena for the study of odd-dimensional conformal
field theory.Comment: 15 pages, 3 figures; references added and some misprint correcte
Central functions and their physical implications
I define central functions c(g) and c'(g) in quantum field theory, useful to
study the flow of the numbers of vector, spinor and scalar degrees of freedom
from the UV limit to the IR limit and basic ingredients for a description of
quantum field theory as an interpolating theory between pairs of 4D conformal
field theories. The key importance of the correlator of four stress-energy
tensors is outlined in this respect. Then I focus the analysis on the
behaviours of the central functions in QCD, computing their slopes in the UV
critical point. To two-loops, c(g) and c'(g) point towards the expected IR
directions. As a possible physical application, I argue that a closer study of
the central functions might allow us to lower the upper bound on the number of
generations to the observed value. Candidate all-order expressions for the
central functions are compared with the predictions of electric-magnetic
duality.Comment: 11 pages, LaTeX. Some points stressed. Three references adde
Irreversibility and higher-spin conformal field theory
I discuss the properties of the central charges c and a for higher-derivative
and higher-spin theories (spin 2 included). Ordinary gravity does not admit a
straightforward identification of c and a in the trace anomaly, because it is
not conformal. On the other hand, higher-derivative theories can be conformal,
but have negative c and a. A third possibility is to consider higher-spin
conformal field theories. They are not unitary, but have a variety of
interesting properties. Bosonic conformal tensors have a positive-definite
action, equal to the square of a field strength, and a higher-derivative gauge
invariance. There exists a conserved spin-2 current (not the canonical stress
tensor) defining positive central charges c and a. I calculate the values of c
and a and study the operator-product structure. Higher-spin conformal spinors
have no gauge invariance, admit a standard definition of c and a and can be
coupled to Abelian and non-Abelian gauge fields in a renormalizable way. At the
quantum level, they contribute to the one-loop beta function with the same sign
as ordinary matter, admit a conformal window and non-trivial interacting fixed
points. There are composite operators of high spin and low dimension, which
violate the Ferrara-Gatto-Grillo theorem. Finally, other theories, such as
conformal antisymmetric tensors, exhibit more severe internal problems. This
research is motivated by the idea that fundamental quantum field theories
should be renormalization-group (RG) interpolations between ultraviolet and
infrared conformal fixed points, and quantum irreversibility should be a
general principle of nature.Comment: 25 pages. Presentation reorganized, with the final section moved to
the beginning. CQG in pres
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