Four loop anomalous dimensions of gradient operators in ${\phi^4}$ theory

The field theoretic renormalization group technique together with the operator product expansion in the second order of the perturbation theory (in the two-loop approximation) is used for the investigation of the influence of the finite time correlations of the velocity field on the anomalous dimensions of the leading set of composite operators, which drive the anomalous scaling of correlation functions of a weak magnetic field in the framework of the kinematic Kazantsev–Kraichnan model in the presence of a large scale anisotropy. The system of possible scaling regimes of the model is found and two important special limits of the model are briefly discussed. The general two-loop expressions for the anomalous and critical dimensions of the leading composite operators are found as functions of the spatial dimension d and of the renormalization group fixed point value of the parameter u, which drives the presence of the finite time correlations of the velocity field in the model. The anisotropic hierarchies among various anomalous dimensions are investigated and it is shown that, regardless of the fixed point value of the parameter u as well as regardless of the spatial dimension of the system, the leading role in the anomalous scaling properties of the model is played by the anomalous dimensions of the composite operators near the isotropic shell, in accordance with the Kolmogorov’s local isotropy restoration hypothesis. The properties of the anomalous dimensions of the leading composite operators in the Kazantsev–Kraichnan model with finite time correlations of the velocity field are compared to the properties of the corresponding anomalous dimensions of the composite operators relevant in the framework of the Kraichnan model of passively advected scalar field with finite time correlations. It is shown that, regardless of the fixed point value of the parameter u, the two-loop corrections to the anomalous dimensions are much more important in the framework of the Kazantsev–Kraichnan vector model than in the Kraichnan model of a passive scalar advection. At the same time, again regardless of the strength of time correlations, the two-loop values of the leading anomalous dimensions in the Kazantsev–Kraichnan model of the passive magnetic field are significantly more negative than the corresponding two-loop values of the relevant anomalous dimensions in the framework of the Kraichnan model. It means that the anomalous scaling of the correlation functions of the passive magnetic field, deep inside the inertial interval of the turbulent environment with finite time correlations of the velocity field, must be much more pronounced than in the case of the correlation functions of the passively advected scalar field.

The large linear superconformal algebra (generated by four spin- currents, seven spin-1 currents, four spin- currents and one spin-2 current) found by Sevrin, Troost and Van Proeyen (and other groups) was realized in the superconformal coset theory previously. The lowest 16 higher spin currents of spins , , and are obtained by starting with the operator product expansions (OPEs) between the four spin- currents from the above large linear superconformal algebra and the lowest higher spin-1 current which is the same as the one in the Wolf space coset theory. These OPEs determine the four higher spin- currents and the next six higher spin-2 currents are obtained from the OPEs between the above four spin- currents associated with the supersymmetry and these four higher spin- currents. The four higher spin- currents can be determined by calculating the OPEs between the above four spin- currents and the higher spin-2 currents. Similarly, the higher spin-3 current is obtained from the OPEs between the four spin- currents and the higher spin- currents. The explicit relations between the above 16 higher spin currents and the corresponding 16 higher spin currents which were found in the extension of large nonlinear superconformal algebra previously are given. By examining the OPEs between the 16 currents from the large linear superconformal algebra and the 16 higher spin currents, the match with the findings of Beccaria, Candu and Gaberdiel is also given. The next 16 higher spin currents of spins , , and occur from the OPEs between the above lowest 16 higher spin currents.

The soft wall model in holographic QCD has Regge trajectories but wrong oper- ator product expansion (OPE) for the two-point vectorial QCD Green function. We modify the dilaton potential to comply with the OPE. We study also the axial two-point function us- ing the same modified dilaton field and an additional scalar fie ld to address chiral symmetry breaking. OPE is recovered adding a boundary term and low energy chiral parameters, Fπ and L10, are well described analytically by the model in terms of Regge spacing and QCD condensates. The model nicely supports and extends previous theoretical analyses advocat- ing Digamma function to study QCD two-point functions in different momentum regions.

We study the operator product expansion (OPE) limit of correlation functions in field theories which possess string theory duals, from the point of view of the string worldsheet. We show how the interesting (single-trace) terms in the OPE of the field theory arise in this limit from the OPE of the worldsheet theory of the string dual, using a dominant saddle point which appears in computations of worldsheet correlation functions in the space-time OPE limit. The worldsheet OPE generically contains only non-physical operators, but all the non-physical contributions are resummed by the saddle point to a contribution similar to that of a physical operator, which exactly matches the field theory expectations. We verify that the OPE limit of the worldsheet theory does not have any other contributions to the OPE limit of space-time correlation functions. Our discussion is completely general and applies to any local field theory (conformal at high energies) that has a weakly coupled string theory dual (with arbitrary curvature). As a first application, we compare our results to a proposal of R. Gopakumar for the string theory dual of free gauge theories.

We verify, both perturbatively and nonperturbatively asymptotically in the ultraviolet (UV), a special case of a low-energy theorem of the NSVZ type in QCD-like theories, recently derived in Phys. Rev. D 95 (2017) 054010, that relates the logarithmic derivative with respect to the gauge coupling, or the logarithmic derivative with respect to the renormalization-group (RG) invariant scale, of an n-point correlator of local operators in one side to an n + 1-point correlator with the insertion of TrF2 at zero momentum in the other side. Our computation involves the operator product expansion (OPE) of the scalar glueball operator, TrF2, in massless QCD, worked out perturbatively in JHEP 12 (2012) 119 — and in its RG-improved form in the present paper — by means of which we extract both the perturbative divergences and the nonperturbative UV asymptotics in both sides. We also discuss the role of the contact terms in the OPE, both finite and divergent, discovered some years ago in JHEP 12 (2012) 119, in relation to the low-energy theorem. Besides, working the other way around by assuming the low-energy theorem for any 2-point correlator of a multiplicatively renormalizable gauge-invariant operator, we compute in a massless QCD-like theory the corresponding perturbative OPE to the order of g2 and nonperturbative asymptotics. The low-energy theorem has a number of applications: to the renormalization in asymptotically free QCD-like theories, both perturbatively and nonperturbatively in the large-N ’t Hooft and Veneziano expansions, and to the way the open/closed string duality may or may not be realized in the would-be solution by canonical string theories for QCD-like theories, both perturbatively and in the ’t Hooft large-N expansion. Our computations will also enter further developments based on the low-energy theorem.

Operator product expansion in two-dimensional quantum gravity

Composite operators from the operator product expansion: what can go wrong?

We analyze a noncanonically renormalized ${\ensuremath{\varphi}}^{4}$ theory in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions with the aid of normal-product methods. By a systematic use of Callan-Symanzik equations we show how a critical point can be determined and the anomalous scale dimensions of all composite operators obtained by using Wilson's $\ensuremath{\epsilon}$ expansion. These methods are also applied to the coefficient distributions appearing in operator-product expansions.

A bstract The linearized massless wave equation in four-dimensional asymptotically flat spacetimes is known to admit two families of solutions that transform in highest-weight representations of the Lorentz group SL(2, ℂ). The two families are related to each other by a two-dimensional shadow transformation. The scattering states of one family are constructed from standard momentum eigenstates by a Mellin transformation with respect to energy. Their operator product expansion (OPE) is directly related to collinear limits of momentum space amplitudes. The scattering states of the other family are a priori non-local on the celestial sphere and lack a standard notion of OPE. Such states appear naturally in the context of asymptotic symmetries, but their properties as operators remain largely unexplored. Here we initiate a study, to be continued in a forthcoming companion paper, of a definition of an OPE for shadow operators. We present a useful technical ingredient: the transformation of the OPE coefficients associated to collinear limits under a shadow. Our results can be used to find the coefficients of all three-point functions involving any combination of celestial and shadow primaries. An OPE block is used to account for the contribution from a primary together with its global conformal descendants, all of which contribute when deriving the shadowed OPE coefficients. Applications involving U(1) currents and stress tensors as well as a chiral current algebra of soft gluons are discussed.

In the coset model based on (A_{N-1}^{(1)} \oplus A_{N-1}^{(1)}, A_{N-1}^{(1)}) at level (N, N; 2N), it is known that the N=2 superconformal algebra can be realized by the two kinds of adjoint fermions. Each Kac-Moody current of spin-1 is given by the product of fermions with structure constant (f symbols) as usual. One can construct the spin-1 current by combining the above two fermions with the structure constant and the spin-1 current by multiplying these two fermions with completely symmetric SU(N) invariant tensor of rank 3 (d symbols). The lowest higher spin-2 current with nonzero U(1) charge (corresponding to the zeromode eigenvalue of spin-1 current of N=2 superconformal algebra) can be obtained from these four spin-1 currents in quadratic form. Similarly, the other type of lowest higher spin-2 current, whose U(1) charge is opposite to the above one, can be obtained also. Four higher spin-5/2 currents can be constructed from the operator product expansions (OPEs) between the spin-3/2 currents of N=2 superconformal algebra and the above two higher spin-2 currents. The two higher spin-3 currents can be determined by the OPEs between the above spin-3/2 currents and the higher spin-5/2 currents. Finally, the ten N=2 OPEs between the four N=2 higher spin multiplets (2, 5/2, 5/2, 3), (2, 5/2, 5/2, 3), (7/2, 4, 4, 9/2) and (7/2, 4, 4, 9/2) are obtained explicitly for generic N.

This thesis is dedicated to certain nonlocal aspects of conformal quantum field theories (CFTs). Two main directions are the study of CFTs on a particular globally-nontrivial spacetime corresponding to finite temperature, and the study of particular nonlocal CFT observables localized on light-rays. Specifically, we introduce bootstrap techniques for determining finite-temperature data of CFTs, and make novel predictions for the 2+1-dimensional Ising model. We propose the “stringy equivalence principle”, stating that coincident gravitational shocks commute, as a generalization of the strong equivalence principle of Einstein’s General Relativity that must hold in all consistent theories of gravity. We prove it in Anti-de Sitter (AdS) spacetimes by studying light-ray operators in the holographically dual CFT. We also derive an operator product expansion (OPE) for light-ray operators in CFT, by which two light-ray operators on the same light-sheet can be expanded as a sum of single light-ray operators. Light-ray operators model detectors — such as calorimeters. We use the light-ray OPE to compute energy event shape observables suitable for conformal collider physics. An additional part of this thesis determines the low-energy vacua of two-dimensional maximal super-Yang-Mills theory, which describes the dynamics of stacks of D-strings in Type IIB string theory. By computing an invariant of the renormalization group (RG) flow from high to low energy — a modified thermal partition function named the refined elliptic genus — we prove the existence of multiple vacua, and identify the superconformal field theories capturing their dynamics. The vacua correspond to bound states of (p,q)-strings in Type IIB string theory. Our computation serves as a check of the strong-weak S-duality of the Type IIB string.

Several different analysis methods have been developed to determine the strong coupling via finite-energy sum-rule analyses of hadronic $\tau$ decay data. While most methods agree on the existence of the well-known ambiguity in the choice of a resummation scheme due to the slow convergence of QCD perturbation theory at the $\tau$ mass, there is an ongoing controversy over how to deal properly with non-perturbative effects. These are small, but not negligible, and include quark-hadron "duality violations" (i.e., resonance effects) which are not described by the operator product expansion (OPE). In one approach, an attempt is made to suppress duality violations enough that they might become negligible. The number of OPE parameters to be fit, however, then exceeds the number of available sum rules, necessitating an uncontrolled OPE truncation, in which a number of higher-dimension OPE contributions in general present in QCD are set to zero by hand. In the second approach, truncation of the OPE is avoided by construction, and duality violations are taken into account explicitly, using a physically motivated model. In this article, we provide a critical appraisal of a recent analysis employing the first approach and demonstrate that it fails to properly account for non-perturbative effects, making the resulting determination of the strong coupling unreliable. The second approach, in contrast, passes all self-consistency tests, and provides a competitive determination of the strong coupling from $\tau$ decays.

We derive a nonperturbative, convergent operator product expansion (OPE) for null-integrated operators on the same null plane in a CFT. The objects appearing in the expansion are light-ray operators, whose matrix elements can be computed by the generalized Lorentzian inversion formula. For example, a product of average null energy (ANEC) operators has an expansion in the light-ray operators that appear in the stress-tensor OPE. An important application is to collider event shapes. The light-ray OPE gives a nonperturbative expansion for event shapes in special functions that we call celestial blocks. As an example, we apply the celestial block expansion to energy-energy correlators in mathcal{N} = 4 Super Yang-Mills theory. Using known OPE data, we find perfect agreement with previous results both at weak and strong coupling, and make new predictions at weak coupling through 4 loops (NNNLO).

We show, within the framework of the massive Euclidean $${\varphi^4}$$ -quantum field theory in four dimensions, that the Wilson operator product expansion (OPE) is not only an asymptotic expansion at short distances as previously believed, but even converges at arbitrary finite distances. Our proof rests on a detailed estimation of the remainder term in the OPE, of an arbitrary product of composite fields, inserted as usual into a correlation function with further “spectator fields”. The estimates are obtained using a suitably adapted version of the method of renormalization group flow equations. Convergence follows because the remainder is seen to become arbitrarily small as the OPE is carried out to sufficiently high order, i.e. to operators of sufficiently high dimension. Our results hold for arbitrary, but finite, loop orders. As an interesting side-result of our estimates, we can also prove that the “gradient expansion” of the effective action is convergent.

We study the operator product expansion (OPE) and quark-hadron duality for two-and three-point correlators of the axial (A) and pseudoscalar (P) currents of the light quarks. In the chiral limit, these correlators are often dominated by nonperturbative power corrections leading to subtleties of quark-hadron duality relations and of the extraction of properties of light pseudoscalars. For the two-point correlators, we show the sum rule for 〈PP〉 to be sensitive to the excited light pseudoscalar. For the three-point correlators, we derive the Ward identities which provide the normalization of the pion electromagnetic form factor at zero momentum transfer. For large momentum transfer, we demonstrate the way the correct behavior of the pion form factor in agreement with perturbative QCD emerges from condensate terms in the OPE for the 〈PV P〉 and 〈AV P〉 correlators. The local-duality sum rule for 〈AV A〉 is shown to lead to the pion form factor with the required properties for all values of the momentum transfer.