arXiv subjects

Ahmadullah Zahed

Publications and source records attributed to Ahmadullah Zahed.

15 recordsLinked to original sources

arXiv2025

Analyticity and positivity of Green's functions without Lorentz

We study the properties imposed by microcausality and positivity on the retarded two-point Green's function in a theory with spontaneous breaking of Lorentz invariance. We assume invariance under time and spatial translations, so that the Green's function $G$ depends on $\omega$ and $\vec k$. We discuss that in Fourier space microcausality is equivalent to the analyticity of $G$ when $\Im (\omega,\vec k)$ lies in the forward light-cone, supplemented by bounds on the growth of $G$ as one approaches the boundaries of this domain. Microcausality also implies that the imaginary part of $G$ (its spectral density) cannot have compact support for real $(\omega,\vec k)$. Using analyticity, we write multi-variable dispersion relations and show that the spectral density must satisfy a family of integral constraints. Analogous constraints can be applied to the fluctuations of the system, via the fluctuation-dissipation theorem. A stable physical system, which can only absorb energy from external sources, satisfies $\omega \cdot \Im G(\omega,\vec k) \ge 0$ for real $(\omega,\vec k)$. We show that this positivity property can be extended to the complex domain: $\Im [\omega\, G(\omega,\vec k)] >0$ in the domain of analyticity guaranteed by microcausality. Functions with this property belong to the Herglotz-Nevanlinna class. This allows to prove the analyticity of the permittivities $\epsilon(\omega,k)$ and $\mu^{-1}(\omega,k)$ that appear in Maxwell equations in a medium. We verify the above properties in several examples where Lorentz invariance is broken by a background field, e.g. non-zero chemical potential, or non-zero temperature. We study subtracted dispersion relations when the assumption $G \to 0$ at infinity must be relaxed.

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arXiv2025

A Geometric View on Crossing-Symmetric Dispersion Relations

We introduce a general framework for constructing dispersion relations using crossing-symmetric variables, leading to infinitely many distinct representations of the 2-to-2 scattering amplitude of identical scalars. Classical formulations such as the Auberson-Khuri crossing-symmetric dispersion relations (CSDRs), the Mahoux-Roy-Wanders relations, and the local CSDR, as well as fixed-t dispersion relations emerge as special cases. Within this setting we re-derive the null constraints from a geometric perspective. Finally, we present, for the first time, an explicit extension of Roy-like equations that remain valid at arbitrarily high energies, relying only on the rigorously established analyticity domain of scattering amplitudes.

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arXiv2025

Bootstrapping leading hadronic muon anomaly

We bootstrap the leading order hadronic contribution to $a_μ$ using unitarity, analytic properties, crossing symmetry and finite energy sum rules (FESR) from quantum chromodynamics (QCD), establishing a lower bound. Combining this lower bound with the remaining precisely calculated contributions from quantum electrodynamics and electroweak interactions, we achieve a lower bound on muon anomaly $a_μ$. Since the FESRs have uncertainties, our bound depends on the choices of FESRs within these uncertainties. A conservative choice of the FESR gives a conservative lower bound, consistent with Standard Model (SM) data-driven prediction. We show that there are other valid choices of FESRs within the uncertainties that lead to lower bounds, which are inconsistent with SM data-driven prediction but consistent with the measured values of the muon anomaly. The bootstrapped spectral density shows a $ρ$-resonance peak similar to experimental hadronic cross-ratio data, providing a bootstrap prediction for $ρ$-meson mass.

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arXiv2025

Complexity for one-dimensional discrete time quantum walk circuits

We compute the complexity for the mixed state density operator derived from a one-dimensional discrete-time quantum walk (DTQW). The complexity is computed using a two-qubit quantum circuit obtained from canonically purifying the mixed state. We demonstrate that the Nielson complexity for the unitary evolution oscillates around a mean circuit depth of $k$. Further, the complexity of the step-wise evolution operator grows cumulatively and linearly with the steps. From a quantum circuit perspective, this implies a succession of circuits of (near) constant depth to be applied to reach the final state.

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arXiv2024

Bootstrapping High-Energy Observables

In this paper, we set up the numerical S-matrix bootstrap by using the crossing symmetric dispersion relation (CSDR) to write down Roy equations for the partial waves. As a motivation behind examining the local version of the CSDR, we derive a new, crossing symmetric, 3-channels-plus-contact-terms representation of the Virasoro-Shapiro amplitude in string theory that converges everywhere except at the poles. We then focus on gapped theories and give novel analytic and semi-analytic derivations of several bounds on low-energy data. We examine the high-energy behaviour of the experimentally measurable rho-parameter, introduced by Khuri and Kinoshita and defined as the ratio of the real to the imaginary part of the amplitude in the forward limit. Contrary to expectations, we find numerical evidence that there could be multiple changes in the sign of this ratio before it asymptotes at high energies. We compare our approach with other existing numerical methods and find agreement, with improvement in convergence.

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arXiv2023

A Celestial Route to AdS Bulk Locality

We prove a precise form of AdS bulk locality by deriving analytical two-sided bounds on bulk Wilson coefficients. Our bounds are on the Wilson coefficients themselves, rather than their ratios, as is typically found in the literature. Inspired by the Celestial amplitudes program in flat space, we perform a Celestial transform of the CFT Mellin amplitude of four identical scalars. Using the crossing symmetric dispersion relation (CSDR), we express the resulting amplitude in terms of crossing symmetric conformal partial waves. The partial waves satisy remarkable positivity properties which along with the unitarity of the CFT prove sufficient to derive the bounds. We then employ our methods in the limit of large AdS radius and recover known bounds on flat space Wilson coefficients and new bounds on their large radius corrections. We check that the planar Mellin amplitude of four stress-tensor multiplets in $\mathcal{N} = 4$ SYM satisfies our bounds. Finally, using null constraints, we derive a form of low-spin dominance in AdS EFTs. We find that the low-spin dominance is strongest in flat space and weakens as we move away from flat space towards higher AdS curvature.

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arXiv2023

Bell inequalities in 2-2 scattering

We consider Bell inequalities in 2-2 scattering of photons, gravitons, fermions and pions. We choose measurement settings that give maximum Bell violation for maximally entangled states and calculate the relevant Bell inequalities for these processes. For photon scattering at low energies, QED exhibits Bell violation for all scattering angles except for a small transverse region. This leads to a fine-tuning problem. Incorporating a light axion/axion-like particle (ALP) removes the fine-tuning problem and constrains the axion-coupling--axion-mass parameters. Allowing for graviton exchange and demanding Bell violation in photon scattering, we find that the Weak Gravity Conjecture is satisfied. Quantum gravity effect on axion coupling is discussed. For 2-2 graviton scattering, we find that CEMZ bounds allow for at most small Bell violations. Restriction on the Weinberg angle is found by demanding Bell violation in Bhabha scattering. We use recent S-matrix bootstrap data for pions and photons to study the Bell parameter in the space of allowed S-matrices. In the photon case, we study the Bell parameters as a function of energy and find support for the EFT observations. We discuss Bell parameter for pion S-matrices, which are qutrits. For pions, we find that there is a minimization of a suitable Bell parameter for S-matrices which exhibit Regge behaviour.

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arXiv2023

A simple coin for a $2d$ entangled walk

We analyze the effect of a simple coin operator, built out of Bell pairs, in a $2d$ Discrete Quantum Random Walk (DQRW) problem. The specific form of the coin enables us to find analytical and closed form solutions to the recursion relations of the DQRW. The coin induces entanglement between the spin and position degrees of freedom, which oscillates with time and reaches a constant value asymptotically. We probe the entangling properties of the coin operator further, by two different measures. First, by integrating over the space of initial tensor product states, we determine the {\it Entangling Power} of the coin operator. Secondly, we compute the {\it Generalized Relative Rényi Entropy} between the corresponding density matrices for the entangled state and the initial pure unentangled state. Both the {\it Entangling Power} and {\it Generalized Relative Rényi Entropy} behaves similar to the entanglement with time. Finally, in the continuum limit, the specific coin operator reduces the $2d$ DQRW into two $1d$ massive fermions coupled to synthetic gauge fields, where both the mass term and the gauge fields are built out of the coin parameters.

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arXiv2022

Locality and Analyticity of the Crossing Symmetric Dispersion Relation

This paper discusses the locality and analyticity of the crossing symmetric dispersion relation (CSDR). Imposing locality constraints on the CSDR gives rise to a local and fully crossing symmetric expansion of scattering amplitudes, dubbed as Feynman block expansion. A general formula is provided for the contact terms that emerge from the expansion. The analyticity domain of the expansion is also derived analogously to the Lehmann-Martin ellipse. Our observation of type-II super-string tree amplitude suggests that the Feynman block expansion has a bigger analyticity domain and better convergence.

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arXiv2021

Positivity and Geometric Function Theory Constraints on Pion Scattering

This paper presents the fascinating correspondence between the geometric function theory and the scattering amplitudes with $O(N)$ global symmetry. A crucial ingredient to show such correspondence is a fully crossing symmetric dispersion relation in the $z$-variable, rather than the fixed channel dispersion relation. We have written down fully crossing symmetric dispersion relation for $O(N)$ model in $z$-variable for three independent combinations of isospin amplitudes. We have presented three independent sum rules or locality constraints for the $O(N)$ model arising from the fully crossing symmetric dispersion relations. We have derived three sets of positivity conditions. We have obtained two-sided bounds on Taylor coefficients of physical Pion amplitudes around the crossing symmetric point (for example, $π^+π^-\to π^0π^0$) applying the positivity conditions and the Bieberbach-Rogosinski inequalities from geometric function theory.

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arXiv2021

Quantum field theory and the Bieberbach conjecture

An intriguing correspondence between ingredients in geometric function theory related to the famous Bieberbach conjecture (de Branges' theorem) and the non-perturbative crossing symmetric representation of 2-2 scattering amplitudes of identical scalars is pointed out. Using the dispersion relation and unitarity, we are able to derive several inequalities, analogous to those which arise in the discussions of the Bieberbach conjecture. We derive new and strong bounds on the ratio of certain Wilson coefficients and demonstrate that these are obeyed in one-loop $ϕ^4$ theory, tree level string theory as well as in the S-matrix bootstrap. Further, we find two sided bounds on the magnitude of the scattering amplitude, which are shown to be respected in all the contexts mentioned above. Translated to the usual Mandelstam variables, for large $|s|$, fixed $t$, the upper bound reads $|\mathcal{M}(s,t)|\lesssim |s^2|$. We discuss how Szegö's theorem corresponds to a check of univalence in an EFT expansion, while how the Grunsky inequalities translate into nontrivial, nonlinear inequalities on the Wilson coefficients.

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arXiv2021

Crossing Symmetric Dispersion Relations for Mellin Amplitudes

We consider manifestly crossing symmetric dispersion relations for Mellin amplitudes of scalar four point correlators in conformal field theories (CFTs). This allows us to set up the non-perturbative Polyakov bootstrap for CFTs in Mellin space on a firm foundation, thereby fixing the contact term ambiguities in the crossing symmetric blocks. Our new approach employs certain "locality" constraints replacing the requirement of crossing symmetry in the usual fixed-$t$ dispersion relation. Using these constraints we show that the sum rules based on the two channel dispersion relations and the present dispersion relations are identical. Our framework allows us to connect with the conceptually rich picture of the Polyakov blocks being Witten diagrams in anti-de Sitter (AdS) space. We also give two sided bounds for Wilson coefficients for effective field theories in AdS space.

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arXiv2021

Crossing Symmetric Dispersion Relations in QFTs

For 2-2 scattering in quantum field theories, the usual fixed $t$ dispersion relation exhibits only two-channel symmetry. This paper considers a crossing symmetric dispersion relation, reviving certain old ideas in the 1970s. Rather than the fixed $t$ dispersion relation, this needs a dispersion relation in a different variable $z$, which is related to the Mandelstam invariants $s,t,u$ via a parametric cubic relation making the crossing symmetry in the complex $z$ plane a geometric rotation. The resulting dispersion is manifestly three-channel crossing symmetric. We give simple derivations of certain known positivity conditions for effective field theories, including the null constraints, which lead to two sided bounds and derive a general set of new non-perturbative inequalities. We show how these inequalities enable us to locate the first massive string state from a low energy expansion of the four dilaton amplitude in type II string theory. We also show how a generalized (numerical) Froissart bound, valid for all energies, is obtained from this approach.

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arXiv2020

Crossing symmetry, transcendentality and the Regge behaviour of 1d CFTs

We develop the technology for Polyakov-Mellin (PM) bootstrap in one-dimensional conformal field theories (CFT$_1$). By adding appropriate contact terms, we bootstrap various effective field theories in AdS$_2$ and analytically compute the CFT data to one loop. The computation can be extended to higher orders in perturbation theory, if we ignore mixing, for any external dimension. We develop PM bootstrap for $O(N)$ theories and derive the necessary contact terms for such theories (which also involves a new higher gradient contact term absent for $N=1$). We perform cross-checks which include considering the diagonal limit of the $2d$ Ising model in terms of the $1d$ PM blocks. As an independent check of the validity of the results obtained with PM bootstrap, we propose a suitable basis of transcendental functions, which allows to fix the four-point correlators of identical scalar primaries completely, up to a finite number of ambiguities related to the number of contact terms in the PM basis. We perform this analysis both at tree level (with and without exchanges) and at one loop. We also derive expressions for the corresponding CFT data in terms of harmonic sums. Finally, we consider the Regge limit of one-dimensional correlators and derive a precise connection between the latter and the large-twist limit of CFT data. Exploiting this result, we study the crossing equation in the three OPE limits and derive some universal constraints for the large-twist limit of CFT data in Regge-bounded theories with a finite number of exchanges.

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arXiv2019

Positive geometry in the diagonal limit of the conformal bootstrap

We consider the diagonal limit of the conformal bootstrap in arbitrary dimensions and investigate the question if physical theories are given in terms of cyclic polytopes. Recently, it has been pointed out that in $d=1$, the geometric understanding of the bootstrap equations for unitary theories leads to cyclic polytopes for which the faces can all be written down and, in principle, the intersection between the unitarity polytope and the crossing plane can be systematically explored. We find that in higher dimensions, the natural structure that emerges, due to the inclusion of spin, is the weighted Minkowski sum of cyclic polytopes. While it can be explicitly shown that for physical theories, the weighted Minkowski sum of cyclic polytopes is not a cyclic polytope, it also turns out that in the large conformal dimension limit it is indeed a cyclic polytope. We write down several analytic formulae in this limit and show that remarkably, in many cases, this works out to be very good approximation even for $O(1)$ conformal dimensions. Furthermore, we initiate a comparison between usual numerics obtained using linear programming and what arises from positive geometry considerations.

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