Crossing-Symmetric Dispersion Relations (CSDRs)

• Crossing-Symmetric Dispersion Relations (CSDRs) are analytic representations that symmetrize scattering amplitudes across all channels using crossing-symmetric variables.
• Modern derivations employ dispersion theory, Mellin amplitudes, and geometric parametrizations to overcome spurious nonlocal terms and maintain analytic control.
• CSDRs enable rigorous sum rules and bounds in QFT, EFT, and CFT, advancing numerical S-matrix and conformal bootstrap studies through improved analytic techniques.

Crossing-Symmetric Dispersion Relations (CSDRs) are analytic representations of scattering amplitudes or correlators that are constructed to enforce full crossing symmetry at the level of the integral kernel or series structure. Crossing symmetry, together with unitarity and analyticity, is a nontrivial and powerful constraint in quantum field theory (QFT), effective field theory (EFT), and conformal field theory (CFT), with direct implications for both the determination of amplitudes and for bounds on low-energy effective couplings. The CSDR paradigm has evolved into a core tool for S-matrix and conformal bootstrap studies, with modern formulations overcoming earlier technical obstacles such as spurious “nonlocal” terms and enabling improved numerical and analytic control of sum rules and positivity constraints.

1. Foundations and Principle of Crossing-Symmetric Dispersion Relations

CSDRs encode the analytic structure of amplitudes in a representation where crossing symmetry is manifest under the full permutation of kinematic variables, typically the Mandelstam variables $(s, t, u)$ or, in CFT, the cross-ratios or Mellin variables. A key advance over conventional fixed-$t$ dispersion relations is that CSDRs treat all channels on equal footing, typically by parametrizing the kinematics in terms of crossing-symmetric variables such as

$$x = -(s_1s_2 + s_2s_3 + s_3s_1)\,, \quad y = -s_1s_2s_3\,,$$

where $s_1, s_2, s_3$ are shifted or rescaled Mandelstam variables subject to $s_1 + s_2 + s_3 = \text{constant}$ (Sinha et al., 2020, Song, 2023, Miro et al., 17 Sep 2025).

The core CSDR takes the form

$$\mathcal{M}(s_1, s_2) = \alpha_0 + \frac{1}{\pi} \int ds_1' \,\mathcal{A}(s_1'; s_2^+(s_1', a))\, H(s_1'; s_1, s_2, s_3)\,,$$

with a kernel

$$H(s_1'; s_1, s_2, s_3) = \frac{s_1}{s_1' - s_1} + \frac{s_2}{s_1' - s_2} + \frac{s_3}{s_1' - s_3}$$

and the absorptive part $\mathcal{A}(s_1'; s_2^+)$ determined via crossing (Sinha et al., 2020, Gopakumar et al., 2021, Bhat et al., 4 Jun 2025). Alternative geometric parametrizations and foliations—e.g., $y = a(x - x_0)$ or homogeneous leaves $y = \alpha \, x^{3/2}$—generate infinite families of CSDRs all related by crossing-symmetric variable choices (Miro et al., 17 Sep 2025).

CSDRs ensure that all three two-body channels enter in a symmetrized way, and by construction enforce the mutual analyticity requirements for the amplitude regarded as a function of its complexified variables.

2. Derivation and Analytical Structure: Modern Techniques

State-of-the-art CSDR derivations begin from dispersion theory and the requirements of causality, analyticity, and unitarity. For the $2 \to 2$ process, the construction may (i) introduce a parametric variable (e.g., $z$ related to $s, t, u$ via a cubic curve (Sinha et al., 2020)), (ii) foliate the crossing-symmetric $(x, y)$ plane with families of one-dimensional leaves parametrizing the physical region (Miro et al., 17 Sep 2025), or (iii) use Mellin amplitudes for CFT correlators (Gopakumar et al., 2021).

A central concern is spurious nonlocality: early CSDRs (Auberson–Khuri type) yielded expansions in crossing-symmetric variables that contained negative powers, violating locality. Modern analyses regularize the dispersion representation using operators ($\mathcal{R}$-operation), shifted Mandelstam variables, and basis changes (e.g., Feynman block expansion) to obtain singularity-free, manifestly local CSDRs:

$$M^{\rm SF}(s) = \alpha + \frac{1}{\pi} \int d\sigma \frac{(2\sigma + s)\mathcal{A}(\sigma; s) - 2\mathcal{A}(\sigma,0)}{(\sigma - s_1)(\sigma - s_2)(\sigma - s_3)}$$

with no negative powers in the expansion (Song, 2023).

Contact terms (local polynomial ambiguities) now emerge in explicit closed form, and their structure is systematically controlled so that no spurious nonlocal pieces survive (Chowdhury et al., 2022). The same program, when extended to Mellin space in CFT, fixes the so-called "Polyakov blocks" in terms of sums over Witten diagrams and contact terms (Gopakumar et al., 2021, Kaviraj, 2021).

3. Locality Constraints, Null Constraints, and Analyticity Domains

The imposition of locality—that the low-energy expansion contains only non-negative powers of crossing-invariants—leads to null constraints: linear sum rules over the partial-wave spectrum which any physical amplitude must satisfy in order to be representable via a CSDR (Sinha et al., 2020, Miro et al., 17 Sep 2025). These constraints enforce, for instance, that expansion coefficients $W_{p,q}$ vanish for $p<0$ in

$$\mathcal{M}(x,y) = \sum_{p,q \geq 0} W_{p,q}x^p y^q\,.$$

The geometric construction of CSDRs elucidates these constraints: singularities and overlapping “leaves” in the $(x, y)$-plane correspond directly to the locations where null constraints must be imposed to ensure global analyticity (Miro et al., 17 Sep 2025).

The analyticity domain of CSDRs, inherited from the Lehmann--Martin ellipse in axiomatic field theory, is expressed in the crossing-symmetric variables and determines the energy range where the corresponding dispersion relation is valid. Appropriate choices of foliation can produce Roy-like equations with validity up to arbitrarily high energies, provided the analyticity domain is not violated (Miro et al., 17 Sep 2025).

4. Applications: Sum Rules, Bounds, and Amplitude Determination

CSDRs are uniquely suited for sum rules and bootstrapping applications. By expressing the low-energy Wilson coefficients in effective actions (for instance, $g_2$ in

$$\mathcal{M}(s,t,u) = \frac{8\pi G}{stu} + g_0 + g_2(s^2 + t^2 + u^2) + g_3( stu) + \cdots$$

) as positive-definite weighted averages over the spectral density via crossing-symmetric kernels, CSDRs provide the basis for deriving rigorous upper and lower bounds on these couplings (Pasiecznik, 11 Jun 2025).

CSDRs also generate robust positivity and null constraints (sum rules), for instance,

$\sum_{r=0}^m \chi_{n}^{(r,m)} W_{n-r, r} \geq 0\,,$

for positive coefficients $\chi_{n}^{(r,m)}$ determined by the crossing-symmetric expansion (Sinha et al., 2020, Chowdhury et al., 2021). In gravitational EFTs, these sum rules function irrespective of the forward limit (where a graviton pole complicates ordinary dispersion relations), and offer improved bounds compared to improved sum rules that manually combine channel information (Pasiecznik, 11 Jun 2025). CSDRs have been successfully used to bound the couplings of higher-spin exchange, e.g., the graviton–spin-4 coupling, in tree-level amplitudes.

For CFTs, CSDRs constructed for Mellin amplitudes or direct cross-ratios impose “locality” constraints that are mathematically equivalent to the Polyakov bootstrap, producing explicit relations between OPE data and spectral integrals (Gopakumar et al., 2021, Paulos, 2020). In particular, the crossing-symmetric kernel structure allows the correlator (or amplitude) to be reconstructed directly from its single or double discontinuity, up to finite subtractions (Bissi et al., 2019, Bonomi et al., 14 Jun 2024).

5. Numerical Implementation, Series Representations, and Geometric Function Theory

Practical utilization of CSDRs for bootstrapping S-matrices and CFT correlators relies on their favorable analytic and numerical properties:

• Modern CSDRs provide series expansions and partial-wave representations (e.g., for the Veneziano and Virasoro–Shapiro amplitudes), in which the pole structure in all channels is manifest and the convergence domain is extended relative to fixed-$t$ expressions (Bhat et al., 4 Jun 2025).
• For amplitudes with Regge growth, or gravitational EFTs with massless exchanges (e.g., the graviton pole), the introduction of an auxiliary parameter (e.g., $\lambda$ or $a$) allows moving the subtraction point to ensure convergence of the expansion in the physical region, including in the forward limit (Bhat et al., 4 Jun 2025).
• Crossing symmetry enables the direct connection of CSDRs to geometric function theory, resulting in rigorous coefficient bounds (e.g., Bieberbach–Rogosinski bounds) and revealing phenomena such as low-spin dominance (LSD), where only low-spin partial waves control the bounds for absorptive parts and Wilson coefficients (Chowdhury et al., 2021, Bissi et al., 2022).
• The analytic structure, including the size and mapping of the analyticity domain in the complexified $(x, y)$-plane, can be mapped and optimized for improved convergence and data fitting (Chowdhury et al., 2022).

CSDRs have been deployed to analyze experimental data (e.g., $\pi\pi$ and $\pi\pi \to K\bar K$ scattering), leading to improved consistency with crossing symmetry and a more accurate determination of phase shifts, pole positions (notably the f₀(500) resonance), and inelasticities (Kaminski, 2011, Bydzovsky et al., 2012, Pelaez et al., 2018).

6. Extensions and Generalizations

Extensions of the CSDR paradigm encompass:

• Construction of Roy–like equations valid to arbitrarily high energy, via homogeneous foliations in crossing-symmetric variables, ensuring coverage of the full physical region consistent with the Martin–Lehmann analyticity domain (Miro et al., 17 Sep 2025).
• Crossing-symmetric partial-wave dispersion relations for amplitudes with spin, permitting two-sided analytic bounds on Wilson coefficients in photon, graviton, and Majorana EFTs (Chowdhury et al., 2021).
• Multivariable generalizations: fully symmetric CSDRs for $n$-particle amplitudes are being developed, with roots in generalized series representations for the multi-variable Veneziano/Virasoro–Shapiro amplitudes. The approach relies on constructing CSDR kernels and spectral densities with total $S_n$ symmetry, crucial for future $n$-body S-matrix bootstrap programs (Bhat et al., 4 Jun 2025).
• In CFT, direct connections have been established between CSDRs and analytic functionals, the Polyakov bootstrap, and extremal functional bases, leading to both analytic reconstructions of correlators and rigorous upper/lower bounds within the allowed space of Euclidean four-point functions (Paulos, 2020, Bonomi et al., 14 Jun 2024).

A notable cross-disciplinary connection is the discovery that the CSDR dispersive kernel encodes knot invariants (Alexander polynomials for $(2,2n+1)$ torus knots) and $q$-deformed oscillator structures, so that S-matrix bootstrap data for low-energy coefficients is bounded by topological invariants (Sinha, 2022).

7. Impact and Ongoing Developments

The evolution of crossing-symmetric dispersion relations has fundamentally changed both how amplitude constraints are derived in analytic S-matrix theory and how numerical S-matrix and CFT bootstrap programs are structured. CSDRs now enable a unification of analyticity, unitarity, and crossing in a form where (i) null constraints and positivity emerge naturally, (ii) the domain of analytic validity is maximized, (iii) sum rules and bounds are transparent and accessible to both analytic and SDP-based numerical study, and (iv) novel mathematical connections (to geometry, topology, and algebra) are foregrounded in physical predictions.

CSDRs are thus a foundational element in contemporary studies of quantum scattering, effective field theory constraints, conformal data, amplitude geometry, and string theory, with a continuing trajectory toward higher-point, higher-spin, and more general analytic and bootstrap applications.