Conformal Partial Wave Research Articles

On instanton effects in the operator product expansion

We revisit the computation of instanton effects to various correlation functions in mathcal{N}=4 SYM and clarify a controversy existing in the literature regarding their consistency with the OPE and conformal symmetry. To check these properties, we examine the conformal partial wave decomposition of four-point correlators involving combinations of half-BPS and Konishi operators and isolate the contribution from the conformal primary scalar operators of twist four. We demonstrate that the leading instanton correction to this contribution is indeed consistent with conformal symmetry and compute the corresponding corrections to the OPE coefficients and the scaling dimensions of such twist-four operators. Our analysis justifies the regularization procedure used to compute ultraviolet divergent instanton contribution to correlation functions involving unprotected operators.

Holographic conformal partial waves as gravitational open Wilson networks

We propose a method to holographically compute the conformal partial waves in any decomposition of correlation functions of primary operators in conformal field theories using open Wilson network operators in the holographic gravitational dual. The Wilson operators are the gravitational ones where gravity is written as a gauge theory in the first order Hilbert-Palatini formalism. We apply this method to compute the global conformal blocks and partial waves in 2d CFTs reproducing many of the known results.

On the equivalence of GPD representations

Phenomenological representations of generalized parton distributions (GPDs) implementing the non-trivial field theoretical requirements are employed in the present day strategies for extracting of hadron structure information encoded in GPDs from the observables of hard exclusive reactions. Showing out the equivalence of various GPD representations can help to get more insight into GPD properties and allow to build up flexible GPD models capable of satisfactory description of the whole set of available experimental data. We review the mathematical aspects of establishing equivalence between the the double partial wave expansion of GPDs in the conformal partial waves and in the $t$-channel ${\rm SO}(3)$ partial waves and the double distribution representation of GPDs.

Large spin systematics in CFT

Using conformal field theory (CFT) arguments we derive an infinite number of constraints on the large spin expansion of the anomalous dimensions and structure constants of higher spin operators. These arguments rely only on analiticity, unitarity, crossing-symmetry and the structure of the conformal partial wave expansion. We obtain results for both, perturbative CFT to all order in the perturbation parameter, as well as non-perturbatively. For the case of conformal gauge theories this provides a proof of the reciprocity principle to all orders in perturbation theory and provides a new "reciprocity" principle for structure constants. We argue that these results extend also to non-conformal theories.

Deconstructing conformal blocks in 4D CFT

We show how conformal partial waves (or conformal blocks) of spinor/tensor correlators can be related to each other by means of differential operators in four dimensional conformal field theories. We explicitly construct such differential operators for all possible conformal partial waves associated to four-point functions of arbitrary traceless symmetric operators. Our method allows any conformal partial wave to be extracted from a few “seed” correlators, simplifying dramatically the computation needed to bootstrap tensor correlators.

Dual parametrization of generalized parton distributions in two equivalent representations

The dual parametrization and the Mellin-Barnes integral approach represent two frameworks for handling the double partial wave expansion of generalized parton distributions (GPDs) in the conformal partial waves and in the $t$-channel ${\rm SO}(3)$ partial waves. Within the dual parametrization framework, GPDs are represented as integral convolutions of forward-like functions whose Mellin moments generate the conformal moments of GPDs. The Mellin-Barnes integral approach is based on the analytic continuation of the GPD conformal moments to the complex values of the conformal spin. GPDs are then represented as the Mellin-Barnes-type integrals in the complex conformal spin plane. In this paper we explicitly show the equivalence of these two independently developed GPD representations. Furthermore, we clarify the notions of the $J=0$ fixed pole and the $D$-form factor. We also provide some insight into GPD modeling and map the phenomenologically successful Kumeri\v{c}ki-M\"uller GPD model to the dual parametrization framework by presenting the set of the corresponding forward-like functions. We also build up the reparametrization procedure allowing to recast the double distribution representation of GPDs in the Mellin-Barnes integral framework and present the explicit formula for mapping double distributions into the space of double partial wave amplitudes with complex conformal spin.

Towards holographic higher-spin interactions: four-point functions and higher-spin exchange

Within holography, we calculate the contribution of an arbitrary spin-s gauge boson exchange in AdS d+1 to the four-point function with scalar operators on the boundary. As an important ingredient, we first compute the complete bulk-to-bulk propagators for massless bosonic higher-spin fields in the metric-like formulation, in any dimension and in various gauges. The split representation of the bulk-to-bulk propagators in terms of bulk-to-boundary propagators allows to present the higher-spin exchange diagram in the form of a conformal partial wave expansion. Our results provide a step towards the larger goal of the holographic reconstruction of bulk interactions, and of clarifying bulk locality.

Tailoring non-compact spin chains

We study three-point correlation functions of local operators in planar $\mathcal{N}=4$ SYM at weak coupling using integrability. We consider correlation functions involving two scalar BPS operators and an operator with spin, in the so called SL(2) sector. At tree level we derive the corresponding structure constant for any such operator. We also conjecture its one loop correction. To check our proposals we analyze the conformal partial wave decomposition of known four-point correlation functions of BPS operators. In perturbation theory, we extract from this decomposition sums of structure constants involving all primaries of a given spin and twist. On the other hand, in our integrable setup these sum rules are computed by summing over all solutions to the Bethe equations. A perfect match is found between the two approaches.

Spinning AdS propagators

We develop the embedding formalism to describe symmetric traceless tensors in Anti-de Sitter space. We use this formalism to construct the bulk-to-bulk propagator of massive spin J fields and check that it has the expected short distance and massless limits. We also find a split representation for the bulk-to-bulk propagator, by writing it as an integral over the boundary of the product of two bulk-to-boundary propagators. We exemplify the use of this representation with the computation of the conformal partial wave decomposition of Witten diagrams. In particular, we determine the Mellin amplitude associated to AdS graviton exchange between minimally coupled scalars of general dimension, including the regular part of the amplitude.

Higher order light-cone distribution amplitudes of the Lambda baryon

The improved light-cone distribution amplitudes (LCDAs) of the $\Lambda$ baryon are examined on the basis of the QCD conformal partial wave expansion approach. The calculations are carried out to the next-to-leading order of conformal spin accuracy with consideration of twist 6. The next leading order conformal expansion coefficients are related to the nonperturbative parameters defined by the local three quark operator matrix elements with different Lorentz structures with a covariant derivative. The nonperturbative parameters are determined with the QCD sum rule method. The explicit expressions of the LCDAs are provided as the main results.

Higher order corrections to the light-cone distribution amplitudes of the sigma baryon

The light-cone distribution amplitudes (LCDAs) of the ${\mathrm{\ensuremath{\Sigma}}}^{\ifmmode\pm\else\textpm\fi{}}$ baryons up to twist-6 are investigated on the basis of the QCD conformal partial wave expansion approach. The calculations are carried out to the next-to-leading order of conformal spin accuracy. The nonperturbative parameters relevant to the LCDAs are determined in the framework of the QCD sum rule method. The explicit expressions of the LCDAs are given as the main results.

Conformal Regge theory

We generalize Regge theory to correlation functions in conformal field theories. This is done by exploring the analogy between Mellin amplitudes in AdS/CFT and S-matrix elements. In the process, we develop the conformal partial wave expansion in Mellin space, elucidating the analytic structure of the partial amplitudes. We apply the new formalism to the case of four point correlation functions between protected scalar operators in N=4 Super Yang Mills, in cases where the Regge limit is controlled by the leading twist operators associated to the pomeron-graviton Regge trajectory. At weak coupling, we are able to predict to arbitrary high order in the 't Hooft coupling the behaviour near J=1 of the OPE coefficients C_{OOJ} between the external scalars and the spin J leading twist operators. At strong coupling, we use recent results for the anomalous dimension of the leading twist operators to improve current knowledge of the AdS graviton Regge trajectory - in particular, determining the next and next to next leading order corrections to the intercept. Finally, by taking the flat space limit and considering the Virasoro-Shapiro S-matrix element, we compute the strong coupling limit of the OPE coefficient C_{LLJ} between two Lagrangians and the leading twist operators of spin J.

USES OF Q2 EVOLUTION IN GPD PHENOMENOLOGY

We shortly review the status of perturbative corrections to hard exclusive processes, which were partially obtained by utilizing conformal symmetry. We outline the conformal partial wave expansion of generalized parton distributions (GPDs) and hard exclusive amplitudes, which can be used for a model dependent extraction of the two-dimensional momentum fraction shape of GPDs by means of evolution. We exemplify this method and point out it importance for the description of collider data in the small x Bj region.

Deep inelastic scattering in conformal QCD

We consider the Regge limit of a CFT correlation function of two vector and two scalar operators, as appropriate to study small-x deep inelastic scattering in N=4 SYM or in QCD assuming approximate conformal symmetry. After clarifying the nature of the Regge limit for a CFT correlator, we use its conformal partial wave expansion to obtain an impact parameter representation encoding the exchange of a spin j Reggeon for any value of the coupling constant. The CFT impact parameter space is the three-dimensional hyperbolic space H3, which is the impact parameter space for high energy scattering in the dual AdS space. We determine the small-x structure functions associated to the exchange of a Reggeon. We discuss unitarization from the point of view of scattering in AdS and comment on the validity of the eikonal approximation. We then focus on the weak coupling limit of the theory where the amplitude is dominated by the exchange of the BFKL pomeron. Conformal invariance fixes the form of the vector impact factor and its decomposition in transverse spin 0 and spin 2 components. Our formalism reproduces exactly the general results predict by the Regge theory, both for a scalar target and for gamma*-gamma* scattering. We compute current impact factors for the specific examples of N=4 SYM and QCD, obtaining very simple results. In the case of the R-current of N=4 SYM, we show that the transverse spin 2 component vanishes. We conjecture that the impact factors of all chiral primary operators of N=4 SYM only have components with 0 transverse spin.

Unparticle self-interactions

We develop techniques for studying the effects of self-interactions in the conformal sector of an unparticle model. Their physics is encoded in the higher n-point functions of the conformal theory. We study inclusive processes and argue that the inclusive production of unparticle stuff in standard model processes due to the unparticle self-interactions can be decomposed using the conformal partial wave expansion and its generalizations into a sum over contributions from the production of various kinds of unparticle stuff, corresponding to different primary conformal operators. Such processes typically involve the production of unparticle stuff associated with operators other than those to which the standard model couples directly. Thus just as interactions between particles allow scattering processes to produce new particles in the final state, so unparticle self-interactions cause the production of various kinds of unparticle stuff. We discuss both inclusive and exclusive methods for computing these processes. The resulting picture, we believe, is a step towards understanding what unparticle stuff "looks like" because it is quite analogous to way we describe the production and scattering of ordinary particles in quantum field theory, with the primary conformal operators playing the role of particles and the coefficients in the conformal partial wave expansion (and its generalization to include more fields) playing the role of amplitudes. We exemplify our methods in the 2D toy model that we discussed previously in which the Banks-Zaks theory is exactly solvable.

Light-cone distribution amplitudes ofΞand their applications

We present the light-cone distribution amplitudes of the $\ensuremath{\Xi}$ baryons up to twist six on the basis of QCD conformal partial wave expansion to the leading order conformal spin accuracy. The nonperturbative parameters relevant to the DAs are determined in the framework of the QCD sum rule. The light-cone QCD sum rule approach is used to investigate both the electromagnetic form factors of $\ensuremath{\Xi}$ and the exclusive semileptonic decay of ${\ensuremath{\Xi}}_{c}$ as applications. Our estimations on the magnetic moments are ${\ensuremath{\mu}}_{{\ensuremath{\Xi}}^{0}}=\ensuremath{-}(1.92\ifmmode\pm\else\textpm\fi{}0.34){\ensuremath{\mu}}_{N}$ and ${\ensuremath{\mu}}_{{\ensuremath{\Xi}}^{\ensuremath{-}}}=\ensuremath{-}(1.19\ifmmode\pm\else\textpm\fi{}0.03){\ensuremath{\mu}}_{N}$. The decay width of the process ${\ensuremath{\Xi}}_{c}\ensuremath{\rightarrow}\ensuremath{\Xi}{e}^{+}{\ensuremath{\nu}}_{e}$ is evaluated to be $\ensuremath{\Gamma}=8.73\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}\text{ }\text{ }\mathrm{GeV}$, which is in accordance with the experimental measurements and other theoretical approaches.

Distribution amplitudes of Σ and Λ and their electromagnetic form factors

Based on QCD conformal partial wave expansion to leading order conformal spin accuracy, we present the light-cone distribution amplitudes (DAs) of Σ and Λ baryons up to twist 6. It is concluded that fourteen independent DAs are needed to describe the valence three-quark states of the baryons at small transverse separations. The nonperturbative parameters relevant to the DAs are determined within the framework of QCD sum rule method. With the obtained DAs, a simple investigation on the electromagnetic form factors of these baryons are given. The magnetic moments of the baryons are estimated by fitting the magnetic form factor with the dipole formula.

Towards a fitting procedure for deeply virtual Compton scattering at next-to-leading order and beyond

Combining dispersion and operator product expansion techniques, we derive the conformal partial wave decomposition of the virtual Compton scattering amplitude in terms of complex conformal spin to twist-two accuracy. The perturbation theory predictions for the deeply virtual Compton scattering (DVCS) amplitude are presented in next-to-leading order for both conformal and modified minimal subtraction scheme. Within a conformal subtraction scheme, where we exploit predictive power of conformal symmetry, the radiative corrections are presented up to next-to-next-to-leading order accuracy. Here, because of the trace anomaly, the mixing of conformal moments of generalized parton distributions (GPD) at the three-loop level remains unknown. Within a new proposed parameterization for GPDs, we then study the convergence of perturbation theory and demonstrate that our formalism is suitable for a fitting procedure of DVCS observables.

Light-cone distribution amplitudes of axial-vector mesons

We have presented a detailed study of twist-2 and twist-3 light-cone distribution amplitudes of 1 P 1 3 and 1 P 1 1 axial-vector mesons, based on QCD conformal partial wave expansion. Applying equations of motion, the twist-three two-parton light-cone distribution amplitudes can be expressed in terms of leading-twist and twist-three three-parton light-cone distribution amplitudes. The relevant G-parity invariant and violating parameters, containing the corrections due to the SU ( 3 ) breaking effects, are evaluated from the QCD sum rule method. The results for axial-vector decay constants of 1 P 1 3 states are presented. The values of tensor decay constants and Gegenbauer moments of the leading twist distribution amplitudes for 1 P 1 1 states are updated. Using Gell-Mann–Okubo mass formula, the mixing angle for the f 8 and f 1 of 1 P 1 3 states is θ P 1 3 ∼ 38 ° , and that for h 8 and h 1 of 1 P 1 1 states is θ P 1 1 ∼ 10 ° . The detailed properties for physical states f 1 ( 1285 ) , f 1 ( 1420 ) , h 1 ( 1170 ) , and h 1 ( 1380 ) are given. Assuming the mixing angle between K 1 A and K 1 B to be θ K = 45 ° or −45°, we also give the detailed study for K 1 ( 1270 ) and K 1 ( 1400 ) . Using the conformal partial wave expansion, we obtain the models for light-cone distribution amplitudes, containing contributions up to conformal spin 9/2. It is interesting to note that some distribution amplitudes have significant asymmetric behaviors, which should be phenomenologically attractive.

Eikonal approximation in AdS/CFT: Conformal partial waves and finite N four-point functions

We introduce the impact parameter representation for conformal field theory correlators of the form A ∼ 〈 O 1 O 2 O 1 O 2 〉 . This representation is appropriate in the eikonal kinematical regime, and approximates the conformal partial wave decomposition in the limit of large spin and dimension of the exchanged primary. Using recent results on the two-point function 〈 O 1 O 1 〉 shock in the presence of a shock wave in anti-de Sitter, and its relation to the discontinuity of the four-point amplitude A across a kinematical branch cut, we find the high spin and dimension conformal partial wave decomposition of all tree-level anti-de Sitter Witten diagrams. We show that, as in flat space, the eikonal kinematical regime is dominated by the T-channel exchange of the massless particle with highest spin ( graviton dominance). We also compute the anomalous dimensions of the high spin O 1 O 2 composites. Finally, we conjecture a formula re-summing crossed-ladder Witten diagrams to all orders in the gravitational coupling.