Products Of Local Operators Research Articles

Power corrections to energy flow correlations from large spin perturbation

Dynamics of high energy scattering in Quantum Chromodynamics (QCD) are primarily probed through detector energy flow correlations. One important example is the Energy-Energy Correlator (EEC), whose back-to-back limit probes correlations of QCD on the lightcone and can be described by a transverse-momentum dependent factorization formula in the leading power approximation. In this work, we develop a systematic method to go beyond this approximation. We identify the origin of logarithmically enhanced contributions in the back-to-back limit as the exchange of operators with low twists and large spins in the local operator product expansion. Using techniques from the conformal bootstrap, the large logarithms beyond leading power can be resummed to all orders in the perturbative coupling. As an illustration of this method, we perform an all-order resummation of the leading and next-to-leading logarithms beyond the leading power in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 Super Yang-Mills theory.

Gauging quantum states with nonanomalous matrix product operator symmetries

Gauging a global symmetry of a system amounts to introducing new degrees of freedom whose transformation rule makes the overall system observe a local symmetry. In quantum systems there can be obstructions to gauging a global symmetry. When this happens the symmetry is dubbed anomalous. Such obstructions are related to the fact that the global symmetry cannot be written as a tensor product of local operators. In this paper we study nonlocal symmetries that have an additional structure: They take the form of a matrix product operator (MPO). We exploit the tensor network structure of the MPOs to construct local operators from them, satisfying the same group relations; that is, we are able to localize even anomalous MPOs. For nonanomalous MPOs, we use these local operators to explicitly gauge the MPO symmetry of a one-dimensional quantum state obtaining nontrivial gauged states. We show that our gauging procedure satisfies all the desired properties that the standard on-site case does. We also show how this procedure is naturally represented in matrix product states protected by MPO symmetries. In the case of anomalous MPOs, we shed light on the obstructions to gauging these symmetries.

Systematic parametrization of the leading B-meson light-cone distribution amplitude

We propose a parametrization of the leading B-meson light-cone distribution amplitude (LCDA) in heavy-quark effective theory (HQET). In position space, it uses a conformal transformation that yields a systematic Taylor expansion and an integral bound, which enables control of the truncation error. Our parametrization further produces compact analytical expressions for a variety of derived quantities. At a given reference scale, our momentum-space parametrization corresponds to an expansion in associated Laguerre polynomials, which turn into confluent hypergeometric functions 1F1 under renormalization-group evolution at one-loop accuracy. Our approach thus allows a straightforward and transparent implementation of a variety of phenomenological constraints, regardless of their origin. Moreover, we can include theoretical information on the Taylor coefficients by using the local operator product expansion. We showcase the versatility of the parametrization in a series of phenomenological pseudo-fits.

Theoretical predictions for inclusive B→Xuτν¯ decay

With the expected large increase in datasets, previously not measured decays will be studied at Belle II. We derive standard model predictions for the $B\ensuremath{\rightarrow}{X}_{u}\ensuremath{\tau}\overline{\ensuremath{\nu}}$ decay rate and distributions. The region in the lepton energy spectrum where higher-dimension operators in the local operator product expansion need to be resummed into the $b$-quark light-cone distribution function is a significantly greater fraction of the phase space than for massless leptons. The finite $\ensuremath{\tau}$ mass has the novel effect of shifting and squeezing how the distribution function enters the lepton energy spectrum. We also derive new predictions for the $\ensuremath{\tau}$ polarization.

Secondary Products in Supersymmetric Field Theory

The product of local operators in a topological quantum field theory in dimension greater than one is commutative, as is more generally the product of extended operators of codimension greater than one. In theories of cohomological type, these commutative products are accompanied by secondary operations, which capture linking or braiding of operators, and behave as (graded) Poisson brackets with respect to the primary product. We describe the mathematical structures involved and illustrate this general phenomenon in a range of physical examples arising from supersymmetric field theories in spacetime dimension two, three, and four. In the Rozansky–Witten twist of three-dimensional {mathcal {N}}=4 theories, this gives an intrinsic realization of the holomorphic symplectic structure of the moduli space of vacua. We further give a simple mathematical derivation of the assertion that introducing an Omega -background precisely deformation quantizes this structure. We then study the secondary product structure of extended operators, which subsumes that of local operators but is often much richer. We calculate interesting cases of secondary brackets of line operators in Rozansky–Witten theories and in four-dimensional {mathcal {N}}=4 super-Yang–Mills theories, measuring the noncommutativity of the spherical category in the geometric Langlands program.

A recipe for constructing frustration-free Hamiltonians with gauge and matter fields in one and two dimensions

State sum constructions, such as Kuperberg’s algorithm, give partition functions of physical systems, like lattice gauge theories, in various dimensions by associating local tensors or weights with different parts of a closed triangulated manifold. Here we extend this construction by including matter fields to build partition functions in both two and three space–time dimensions. The matter fields introduce new weights to the vertices and they correspond to Potts spin configurations described by an -module with an inner product. Performing this construction on a triangulated manifold with a boundary we obtain transfer matrices which are decomposed into a product of local operators acting on vertices, links and plaquettes. The vertex and plaquette operators are similar to the ones appearing in the quantum double models (QDMs) of Kitaev. The link operator couples the gauge and the matter fields, and it reduces to the usual interaction terms in known models such as gauge theory with matter fields. The transfer matrices lead to Hamiltonians that are frustration-free and are exactly solvable. According to the choice of the initial input, that of the gauge group and a matter module, we obtain interesting models which have a new kind of ground state degeneracy that depends on the number of equivalence classes in the matter module under gauge action. Some of the models have confined flux excitations in the bulk which become deconfined at the surface. These edge modes are protected by an energy gap provided by the link operator. These properties also appear in ‘confined Walker–Wang’ models which are 3D models having interesting surface states. Apart from the gauge excitations there are also excitations in the matter sector which are immobile and can be thought of as defects like in the Ising model. We only consider bosonic matter fields in this paper.

Factorization at subleading power and irreducible uncertainties in $ \bar{B} \to {X_s}\gamma $ decay

Using methods from soft-collinear and heavy-quark effective theory, a systematic factorization analysis is performed for the $\bar B\to X_s\gamma$ photon spectrum in the endpoint region $m_b-2E_\gamma={\cal O}(\Lambda_{\rm QCD})$. It is proposed that, to all orders in $1/m_b$, the spectrum obeys a novel factorization formula, which besides terms with the structure $H\,J\otimes S$ familiar from inclusive $\bar B\to X_u l\,\bar\nu$ decay distributions contains "resolved photon" contributions of the form $H\,J\otimes S\otimes\bar J$ and $H\,J\otimes S\otimes\bar J\otimes\bar J$. Here $S$ and $\bar J$ are new soft and jet functions, whose form is derived. These contributions arise whenever the photon couples to light partons instead of coupling directly to the effective weak interaction. The new contributions appear first at order $1/m_b$ and are related to operators other than $Q_{7\gamma}$ in the effective weak Hamiltonian. They give rise to non-vanishing $1/m_b$ corrections to the total decay rate, which cannot be described using a local operator product expansion. A systematic analysis of these effects is performed at tree level in hard and hard-collinear interactions. The resulting uncertainty on the decay rate defined with a cut $E_\gamma>1.6$ GeV is estimated to be approximately $\pm 5%$. It could be reduced by an improved measurement of the isospin asymmetry $\Delta_{0-}$ to the level of $\pm 4%$. We see no possibility to reduce this uncertainty further using reliable theoretical methods.

Hidden Grassmannian Structure in the XXZ Model

For the critical XXZ model, we consider the space \(\mathcal{W}_{[\alpha]}\) of operators which are products of local operators with a disorder operator. We introduce two anti-commutative families of operators \({\bf {b}}(\zeta), {\bf {c}}(\zeta)\) which act on \(\mathcal{W}_{[\alpha]}\). These operators are constructed as traces over representations of the q-oscillator algebra, in close analogy with Baxter’s Q-operators. We show that the vacuum expectation values of operators in \(\mathcal{W}_{[\alpha]}\) can be expressed in terms of an exponential of a quadratic form of \({\bf {b}}(\zeta), {\bf {c}}(\zeta)\).

Enhanced nonlocal power corrections to theB¯→Xsγdecay rate

A new class of enhanced nonperturbative corrections to the inclusive $\overline{B}\ensuremath{\rightarrow}{X}_{s}\ensuremath{\gamma}$ decay rate is identified, which contribute first at order $\ensuremath{\Lambda}/{m}_{b}$ in the heavy-quark expansion and cannot be described using a local operator product expansion. Instead, these effects are described in terms of hadronic matrix elements of nonlocal operators with component fields separated by lightlike distances. They contribute to the high-energy part of the photon-energy spectrum but do not reduce to local operators when an integral over energy is taken to obtain the total inclusive decay rate. The dominant corrections depend on the flavor of the $B$-meson spectator quark and are described by trilocal four-quark operators. Their contribution is estimated using the vacuum insertion approximation. The corresponding uncertainty in the total decay rate is found to be at the few percent level. This new effect accounts for the leading contribution to the rate difference between ${B}^{\ensuremath{-}}$ and ${\overline{B}}^{0}$ mesons.

Hidden Grassmann Structure in the XXZ Model

For the critical XXZ model, we consider the space $$\mathcal{W}_{[\alpha]}$$ of operators which are products of local operators with a disorder operator. We introduce two anti-commutative families of operators $${\bf {b}}(\zeta), {\bf {c}}(\zeta)$$ which act on $$\mathcal{W}_{[\alpha]}$$ . These operators are constructed as traces over representations of the q-oscillator algebra, in close analogy with Baxter’s Q-operators. We show that the vacuum expectation values of operators in $$\mathcal{W}_{[\alpha]}$$ can be expressed in terms of an exponential of a quadratic form of $${\bf {b}}(\zeta), {\bf {c}}(\zeta)$$ .

Factorization in deeply virtual Compton scattering: local OPE formalism and structure functions

We give a complete treatment of factorization of deeply virtual Compton scattering (DVCS) in the generalized Bjorken limit, using the local operator product expansion. The method allows a straightforward proof that, at leading twist, the DVCS amplitude factorizes into an integral over coefficient functions and skewed parton distribution functions. The integral is well defined for on-shell final state photons if the Wilson coefficients satisfy a certain factorization condition, which we derive. We also show that it enables a simple proof that soft singularities either cancel out or, in the case where the final state photon is on shell, are integrable. This confirms the argument of Collins and Freund. Further, we repeat the tree-level calculation of twist-three contributions to DVCS off a scalar target, where factorization was found to be violated. We propose a new definition of the structure functions and calculate the coefficient functions, which are such that factorization works.

Geodesic propagators and black hole holography

One of the most challenging technical aspects of the dualities between string theory on anti--de Sitter spaces and conformal field theories is understanding how location in the interior of spacetime is represented in the field theory. It has recently been argued that the interior of the spacetime can be directly probed by using intrinsically non-local quantities in the field theory. In addition, Balasubramanian and Ross [Phys. Rev. D 61, 044009 (2000)] argued that when the spacetime described the formation of an ${\mathrm{AdS}}_{3}$ black hole, the propagator in the field theory probed the whole spacetime, including the region behind the horizon. We use the same approach to study the propagator for the BTZ black hole and a black hole solution with a single exterior region, and show that it reproduces the propagator associated with the natural vacuum states on these spacetimes. We compare our result with a toy model of the CFT for the single-exterior black hole, finding remarkable agreement. The spacetimes studied in this work are analytic, which makes them quite special. We also discuss the interpretation of this propagator in more general spacetimes, shedding light on certain issues involving causality, black hole horizons, and products of local operators on the boundary.

Ghost-free spectrum of a quantum string in SL(2,R) curved spacetime.

The unitarity problem in curved spacetime is solved for the string described by the SL(2,R) WZW model. The spectrum is computed exactly and demonstratedto be ghost-free. The new features include (i) SL(2,R) left/right symmetrycurrents that have logarithmic cuts on the world sheet but that satisfy theusual local operator products or commutation rules, (ii) physical statesconsistent with the monodromy condition of closed strings despite thelogarithmic singularity in the currents, and (iii) a new free boson realization for these currents which render the SL(2,R) WZW model completely solvable.

TWO-DIMENSIONAL (HALF-) INTEGER SPIN CONFORMAL THEORIES WITH CENTRAL CHARGE c<1

A generalized integral representation involving two types of charges is explored to construct correlation functions on the plane for c=1–6/(m(m+1))<1 discrete unitary Virasoro series. The various local operator product algebras emerging contain integer, or half-integer, spin fields along with scalar fields. The examples also include a generalization for arbitrary m of the ℤ2-statistics of the Ising model order-disorder fields.

Local operator products for some nonpolynomial interactions

Short distance behaviour of Bose quantum field models which are described by the self-interaction of the form ∫ R 1 e iαϕε ( x ⇀ ) dv(α) is found to be the same as that of the free field.

Scale-covariant quantization

It is argued that conventional operator field equations derived from a stationary action principle are improper when local operator products involve infinite multiplicative renormalizations. An alternative, scale-covariant operator field equation is derived by making scale variations of the field, variations which are everywhere proportional to the field itself. The resultant equation of motion is the same as that previously found on the basis of augmented field theory.

Local operator products in gauge theories, II

We study the operator product expansion of two gauge-invariant currents. We discuss the operators that may appear in the expansion. We study the hadronic matrix elements of gauge-variant operators that may appear in the operator product expansion and show that in the physical applications contemplated so far their hadronic physical matrix elements either vanish or cannot be isolated from strong interactions. We also study the gauge dependence of the Green's functions of the product of two (weak or e.m.) currents and show that is corresponds to the gauge dependence of the additional renormalization counterterms due to weak and electromagnetic interactions.

Local operator products in gauge theories. I

In this and a subsequent paper, we discuss the renormalization and gauge invariance of products of local gauge-invariant operators and their operator product expansions in gauge theories in detail. In this paper, we study in detail the following problem which when solved leads to the complete understanding of the operator product expansion of a product of two gauge-invariant operators. The problem is to consider the complete set of color singlet local functionals of gauge, matter, and ghost fields (F[A,c,c¯,η0]) in a non-Abelian unbroken gauge theory of strong interactions containing fermions and find what subset of these local operators have the property that all of their “physical matrix elements” are η-independent to all orders in the perturbation theory. (Note: All the phrases in quotes and the above statement are to be defined in the text by certain limiting procedures where-upon they become precise mathematical statements.) Here, η is a gauge parameter. We believe that the discussion given here can be extended to spontaneously broken gauge theories in a straightforward manner, where of course, the physical matrix elements exist in perturbation theory. The conclusion is that the only nontrivial operators that have this property to all orders are all the gauge-invariant operators and certain gauge variant operators which are a subset of the renormalization counterterms a gauge-invariant operator at zero momentum needs. However, these gauge-variant operators are not new physical entities in the sense that their “renormalized physical matrix elements” are not independent to the “S-matrix elements.” These results will be useful in understanding which operators appear in the operator product expansion of, say two currents, letting one use these methods with full understanding in gauge theories. As a by-product, certain technical discussions in this paper can be used to considerably shorten the proof of a theorem in the renormalization of gauge-invariant operators previously given.

Local operator products in gauge theories: I. Satish D. Joglekar, The Institute for Advanced Study, Princeton, New Jersey 08540, and Stanford Linear Accelerator Center, Stanford University, Stanford, California 94305

Local operator products in gauge theories: I. Satish D. Joglekar, The Institute for Advanced Study, Princeton, New Jersey 08540, and Stanford Linear Accelerator Center, Stanford University, Stanford, California 94305

Spectral properties of bilocal operators in the expansion of a product of local operators on the light cone

Spectral properties of bilocal operators in the expansion of a product of local operators on the light cone