Dispersion Sum Rules Research Articles

Connection between the Dipole Polarizabilities of Charged and Neutral π-Mesons

In light of the contribution of the states with isospin I = 0in the difference of the amplitudes of the processes γγ → π+π− and 0 being very small, the dispersion sum rules for the difference betweenthe dipole polarizabilities of the charged and neutral pions are analyzed as a function of the σ meson’s parameters. Using the value of the chiral theory of perturbations for (α1 − β1)π0 = −1.9, (α1 − β1)π± =9.4−8.2 is found for the σ meson parameter within the region mσ = 400−550 MeV, Γσ = 400−600 MeV, Γσ→γγ = 0−3 keV. Estimated optimum value of decay width σ → γγ yields Γσ→γγ 0.7 keV.

The AdS Virasoro-Shapiro amplitude

We present a constructive method to compute the AdS Virasoro-Shapiro amplitude, order by order in AdS curvature corrections. At kth order the answer takes the form of a genus zero world-sheet integral involving weight 3k single-valued multiple polylogarithms. The coefficients in our ansatz are fixed, order by order, by requiring: crossing symmetry; the correct supergravity limit; the correct structure of poles, determined by dispersive sum rules; and the dimensions of the first few Konishi-like operators, available from integrability. We explicitly construct the first two curvature corrections. Our final answer then reproduces all localisation results and all CFT data available from integrability, to this order, and produces a wealth of new CFT data for planar mathcal{N} = 4 SYM at strong coupling.

Forward light-by-light scattering and electromagnetic correction to hadronic vacuum polarization

Lattice QCD calculations of the hadronic vacuum polarization (HVP) have reached a precision where the electromagnetic (e.m.) correction can no longer be neglected. This correction is both computationally challenging and hard to validate, as it leads to ultraviolet (UV) divergences and to sizeable infrared (IR) effects associated with the massless photon. While we precisely determine the UV divergence using the operator-product expansion, we propose to introduce a separation scale Λ ~ 400 MeV into the internal photon propagator, whereby the calculation splits into a short-distance part, regulated in the UV by the lattice and in the IR by the scale Λ, and a UV-finite long-distance part to be treated with coordinate-space methods, thereby avoiding power-law finite-size effects altogether. In order to predict the long-distance part, we express the UV-regulated e.m. correction to the HVP via the forward hadronic light-by-light (HLbL) scattering amplitude and relate the latter via a dispersive sum rule to γ∗γ∗ fusion cross-sections. Having tested the relation by reproducing the two-loop QED vacuum polarization (VP) from the tree-level γ∗γ∗→ e+e− cross-section, we predict the expected lattice-QCD integrand resulting from the γ∗γ∗→ π0 process.

Bootstrapping mathcal{N} = 4 sYM correlators using integrability

How much spectral information is needed to determine the correlation functions of a conformal theory? We study this question in the context of planar supersymmetric Yang-Mills theory, where integrability techniques accurately determine the single-trace spectrum at finite ’t Hooft coupling. Corresponding OPE coefficients are constrained by dispersive sum rules, which implement crossing symmetry. Focusing on correlators of four stress-tensor multiplets, we construct combinations of sum rules which determine one-loop correlators, and we study a numerical bootstrap problem that nonperturbatively bounds planar OPE coefficients. We observe interesting cusps at the location of physical operators, and we obtain a nontrivial upper bound on the OPE coefficient of the Konishi operator outside the perturbative regime.

AdS Virasoro-Shapiro from single-valued periods

We determine the full 1/ sqrt{uplambda} correction to the flat-space Wilson coefficients which enter the AdS Virasoro-Shapiro amplitude in mathcal{N} = 4 SYM theory at strong coupling. The assumption that the Wilson coefficients are in the ring of single-valued multiple zeta values, as expected for closed string amplitudes, is surprisingly powerful and leads to a unique solution to the dispersive sum rules relating Wilson coefficients and OPE data obtained in [1]. The corresponding OPE data fully agrees with and extends the results from integrability. The Wilson coefficients to order 1/ sqrt{uplambda} can be summed into an expression whose structure of poles and residues generalises that of the Virasoro-Shapiro amplitude in flat space.

Dispersive sum rules in AdS2

Dispersion relations for S-matrices and CFT correlators translate UV consistency into bounds on IR observables. In this note, we construct dispersive sum rules for 1D CFTs. We use them to prove bounds on higher-derivative couplings in weakly-coupled non-gravitational EFTs in AdS2. At the leading order in the bulk-point limit, the bounds agree with the flat-space result. We compute the leading universal effect of finite AdS radius on the bounds. Along the way, we give an explicit formula for anomalous dimensions in general higher-derivative contact Witten diagrams in AdS2.

AdS Virasoro-Shapiro from dispersive sum rules

We consider the four-point correlator of the stress-energy tensor in mathcal{N} = 4 SYM, to leading order in inverse powers of the central charge, but including all order corrections in 1/λ. This corresponds to the AdS version of the Virasoro-Shapiro amplitude to all orders in the small α′/low energy expansion. Using dispersion relations in Mellin space, we derive an infinite set of sum rules. These sum rules strongly constrain the form of the amplitude, and determine all coefficients in the low energy expansion in terms of the CFT data for heavy string operators, in principle available from integrability. For the first set of corrections to the flat space amplitude we find a unique solution consistent with the results from integrability and localisation.

Spinning sum rules for the dimension-six SMEFT

We construct new dispersive sum rules for the effective field theory of the standard model at mass dimension six. These spinning sum rules encode information about the spin of UV states: the sign of the IR Wilson coefficients carries a memory of the dominant spin in the UV completion. The sum rules are constructed for operators containing scalars and fermions, although we consider the dimension-six SMEFT exhaustively, outlining why equivalent relations do not hold for the remaining operators. As with any dimension-six dispersive argument, our conclusions are contingent on the absence of potential poles at infinity — so-called boundary terms — and we discuss in detail where these are expected to appear. There are a number of phenomenological applications of spinning sum rules, and as an example we explore the connection to the Peskin-Takeuchi parameters and, more generally, the set of oblique parameters in universal theories.

Perturbative QCD Core of Hadrons and Color Transparency Phenomena

In the current paper, we argue that the ground state of a hadron contains a significant perturbative quantum chromodynamics (pQCD) core as the result of color gauge invariance and the values of chiral and gluon vacuum condensates. The evaluation within the method of dispersion sum rules (DSR) of the vacuum matrix elements of the correlator of local currents with the proper quantum numbers leads to the value of the radius of the pQCD core of a nucleon of about 0.4–0.5 fm. The selection of the initial and final states allows to select processes in which the pQCD core of the projectile gives the dominant contribution to the process. It is explained that the transparency of nuclear matter for the propagation of a spatially small and color-neutral wave packet of quarks and gluons—a color transparency (CT) phenomenon—for a group of hard processes off nuclear targets can be derived in the form of the QCD factorization theorem accounting for the color screening phenomenon. Based on the success of the method of DSR, we argue that a pQCD core in a hadron wave function is surrounded by the layer consisting of quarks interacting with quark and gluon condensates. As a result, in the quasi-elastic processes e+A→e′+N+(A−1)∗, the quasi-Feynman mechanism could be dominating in a wide range of the momentum transfer squared, Q2. In this scenario, a virtual photon is absorbed by a single quark, which carries a large fraction of the momentum of the nucleon and dominates in a wide range of Q2. CT should reveal itself in these processes at extremely large Q2 as the consequence of the presence of the Sudakov form factors, which squeeze a nucleon.

Mixed correlator dispersive CFT sum rules

Conformal field theory (CFT) dispersion relations reconstruct correlators in terms of their double discontinuity. When applied to the crossing equation, such dispersive transforms lead to sum rules that suppress the double-twist sector of the spectrum and enjoy positivity properties at large twist. In this paper, we construct dispersive CFT functionals for correlators of unequal scalar operators in position- and Mellin-space. We then evaluate these functionals in the Regge limit to construct mixed correlator holographic CFT functionals which probe scalar particle scattering in Anti-de Sitter spacetime. Finally, we test properties of these dispersive sum rules when applied to the 3D Ising model, and we use truncated sum rules to find approximate solutions to the crossing equation.

On the light-ray algebra in conformal field theories

We analyze the commutation relations of light-ray operators in conformal field theories. We first establish the algebra of light-ray operators built out of higher spin currents in free CFTs and find explicit expressions for the corresponding structure constants. The resulting algebras are remarkably similar to the generalized Zamolodchikov’s W∞ algebra in a two-dimensional conformal field theory. We then compute the commutator of generalized energy flow operators in a generic, interacting CFTs in d > 2. We show that it receives contribution from the energy flow operator itself, as well as from the light-ray operators built out of scalar primary operators of dimension ∆ ≤ d − 2, that are present in the OPE of two stress-energy tensors. Commutators of light-ray operators considered in the present paper lead to CFT sum rules which generalize the superconvergence relations and naturally connect to the dispersive sum rules, both of which have been studied recently.

AdS bulk locality from sharp CFT bounds

It is a long-standing conjecture that any CFT with a large central charge and a large gap ∆gap in the spectrum of higher-spin single-trace operators must be dual to a local effective field theory in AdS. We prove a sharp form of this conjecture by deriving numerical bounds on bulk Wilson coefficients in terms of ∆gap using the conformal bootstrap. Our bounds exhibit the scaling in ∆gap expected from dimensional analysis in the bulk. Our main tools are dispersive sum rules that provide a dictionary between CFT dispersion relations and S-matrix dispersion relations in appropriate limits. This dictionary allows us to apply recently-developed flat-space methods to construct positive CFT functionals. We show how AdS4 naturally resolves the infrared divergences present in 4D flat-space bounds. Our results imply the validity of twice-subtracted dispersion relations for any S-matrix arising from the flat-space limit of AdS/CFT.

Generalized Kramers\u2013Kronig relations and sum rules for moments and powers of degenerate four wave mixing susceptibility

Generalized Kramers–Kronig (K–K) type dispersion relations and sum rules are derived in the static limit for the moments of the degenerate four wave mixing susceptibility. The degenerate nonlinear susceptibility is different from a typical use of the conventional K–K dispersion relations, which assume absence of complex poles of a function in the upper half of complex frequency plane, whereas degenerate susceptibility has simultaneous poles in both half planes. In the derivation of the generalized K–K relations the poles and their order are taken into account by utilization of the theorem of residues. The conventional K–K relations can be used to estimate the real and imaginary parts of the second and higher powers of the susceptibility as the effect of the poles is reduced due to a faster convergence of the dispersion relations. The present theory is directly applicable to higher order susceptibilities and can be used in testing of theoretical models describing the degenerate four wave mixing susceptibility in nonlinear optical and terahertz spectroscopy.

Sharp boundaries for the swampland

We reconsider the problem of bounding higher derivative couplings in consistent weakly coupled gravitational theories, starting from general assumptions about analyticity and Regge growth of the S-matrix. Higher derivative couplings are expected to be of order one in the units of the UV cutoff. Our approach justifies this expectation and allows to prove precise bounds on the order one coefficients. Our main tool are dispersive sum rules for the S-matrix. We overcome the difficulties presented by the graviton pole by measuring couplings at small impact parameter, rather than in the forward limit. We illustrate the method in theories containing a massless scalar coupled to gravity, and in theories with maximal supersymmetry.

Applications of dispersive sum rules: $ε$-expansion and holography

We use Mellin space dispersion relations together with Polyakov conditions to derive a family of sum rules for Conformal Field Theories (CFTs). The defining property of these sum rules is suppression of the contribution of the double twist operators. Firstly, we apply these sum rules to the Wilson-Fisher model in d=4-\epsilond=4−ϵ dimensions. We re-derive many of the known results to order \epsilon^4ϵ4 and we make new predictions. No assumption of analyticity down to spin 00 was made. Secondly, we study holographic CFTs. We use dispersive sum rules to obtain tree-level and one-loop anomalous dimensions. Finally, we briefly discuss the contribution of heavy operators to the sum rules in UV complete holographic theories.

Dispersive CFT sum rules

We give a unified treatment of dispersive sum rules for four-point correlators in conformal field theory. We call a sum rule “dispersive” if it has double zeros at all double-twist operators above a fixed twist gap. Dispersive sum rules have their conceptual origin in Lorentzian kinematics and absorptive physics (the notion of double discontinuity). They have been discussed using three seemingly different methods: analytic functionals dual to double-twist operators, dispersion relations in position space, and dispersion relations in Mellin space. We show that these three approaches can be mapped into one another and lead to completely equivalent sum rules. A central idea of our discussion is a fully nonperturbative expansion of the correlator as a sum over Polyakov-Regge blocks. Unlike the usual OPE sum, the Polyakov-Regge expansion utilizes the data of two separate channels, while having (term by term) good Regge behavior in the third channel. We construct sum rules which are non-negative above the double-twist gap; they have the physical interpretation of a subtracted version of “superconvergence” sum rules. We expect dispersive sum rules to be a very useful tool to study expansions around mean-field theory, and to constrain the low-energy description of holographic CFTs with a large gap. We give examples of the first kind of applications, notably we exhibit a candidate extremal functional for the spin-two gap problem.

Hadronic light-by-light scattering amplitudes from lattice QCD versus dispersive sum rules

The hadronic contribution to the eight forward amplitudes of light-by-light scattering ($\gamma^*\gamma^*\to \gamma^*\gamma^*$) is computed in lattice QCD. Via dispersive sum rules, the amplitudes are compared to a model of the $\gamma^*\gamma^*\to {\rm hadrons}$ cross sections in which the fusion process is described by hadronic resonances. Our results thus provide an important test for the model estimates of hadronic light-by-light scattering in the anomalous magnetic moment of the muon, $a_\mu^{\rm HLbL}$. Using simple parametrizations of the resonance $M\to \gamma^*\gamma^*$ transition form factors, we determine the corresponding monopole and dipole masses by performing a global fit to all eight amplitudes. Together with a previous dedicated calculation of the $\pi^0\to \gamma^*\gamma^*$ transition form factor, our calculation provides valuable information for phenomenological estimates of $a_\mu^{\rm HLbL}$. The presented calculations are performed in two-flavor QCD with pion masses extending down to 190\,MeV at two different lattice spacings. In addition to the fully connected Wick contractions, on two lattice ensembles we also compute the (2+2) disconnected class of diagrams, and find that their overall size is compatible with a parameter-free, large-$N$ inspired prediction, where $N$ is the number of colors. Motivated by this observation, we estimate in the same way the disconnected contribution to $a_\mu^{\rm HLbL}$.

Evaluation of the forward Compton scattering off protons. II. Spin-dependent amplitude and observables

The forward Compton scattering off the proton is determined by substituting the empirical total photoabsorption cross sections into dispersive sum rules. In addition to the spin-independent amplitude evaluated previously [Phys. Rev. D 92, 074031 (2015)], we obtain the spin-dependent amplitude over a broad energy range. The two amplitudes contain all the information about this process, and we, hence, can reconstruct the nonvanishing observables of the proton Compton scattering in the forward kinematics. The results are compared with predictions of chiral perturbation theory where available. The low-energy expansion of the spin-dependent Compton scattering amplitude yields the Gerasimov-Drell-Hearn (GDH) sum rule and relations for the forward spin polarizabilities (FSPs) of the proton. Our evaluation provides an empirical verification of the GDH sum rule for the proton, and yields empirical values of the proton FSPs. For the GDH integral, we obtain $204.5(21.4)$ $\mu$b, in agreement with the sum rule prediction: $204.784481(4)$ $\mu$b. For the FSPs, we obtain: $\gamma_0=-92.9(10.5) \times 10^{-6}$ fm$^4$, and $\bar{\gamma_0}=48.4(8.2) \times 10^{-6}$ fm$^6$, improving on the accuracy of previous evaluations.

Lattice QCD Calculation of Hadronic Light-by-Light Scattering.

We perform a lattice QCD calculation of the hadronic light-by-light scattering amplitude in a broad kinematical range. At forward kinematics, the results are compared to a phenomenological analysis based on dispersive sum rules for light-by-light scattering. The size of the pion pole contribution is investigated for momenta of typical hadronic size. The presented numerical methods can be used to compute the hadronic light-by-light contribution to the anomalous magnetic moment of the muon. Our calculations are carried out in two-flavor QCD with the pion mass in the range of 270-450MeV and contain so far only the diagrams with fully connected quark lines.

Accurate decay-constant ratiosfB*/fBandfBs*/fBsfrom Borel QCD sum rules

We present our analysis of the decay constants of the beauty vector mesons $B^*$ and $B^*_s$ within the framework of dispersive sum rules for the two-point correlator of vector currents in QCD. While the decay constants of the vector mesons $f_{B^*}$ and $f_{B_s^*}$ $-$ similar to the decay constants of the pseudoscalar mesons $f_B$ and $f_{B_s}$ $-$ individually have large uncertainties induced by theory parameters not known with a satisfactory precision, these uncertainties almost entirely cancel out in the ratios of vector over pseudoscalar decay constants. These ratios may be thus predicted with very high accuracy due to the good control over the systematic uncertainties of the decay constants gained upon application of our hadron-parameter extraction algorithm. Our final results read $f_{B^*}/f_B=0.944\pm0.011_{\rm OPE}\pm0.018_{\rm syst}$ and $f_{B_s^*}/f_{B_s}=0.947\pm0.023_{\rm OPE}\pm0.020_{\rm syst}$. Thus, both $f_{B^*}/f_B$ and $f_{B_s^*}/f_{B_s}$ are less than unity at 2.5$\sigma$ and 2$\sigma$ level, respectively.