Crossing-Symmetric Polyakov Blocks in CFT

• Crossing-symmetric Polyakov blocks are basis elements in conformal bootstrap that combine s-, t-, and u-channel exchanges with contact terms to implement inherent crossing symmetry.
• They are constructed using momentum-space, Mellin-space, and dispersive methods to isolate physical exchange singularities while canceling spurious nonlocal contributions.
• The framework extends to one-dimensional cases, higher-point functions, and mixed correlators, yielding analytic sum rules and enforcing OPE consistency.

Searching arXiv for recent and foundational papers on crossing symmetric Polyakov blocks. arXiv search query: "crossing symmetric Polyakov blocks conformal bootstrap Mellin momentum space dispersion relation" Crossing-symmetric Polyakov blocks are building blocks for conformal correlators that implement crossing symmetry at the level of the basis rather than only after summing an ordinary conformal-block expansion. In the modern literature, they are constructed as exchange-like objects—typically sums of $s$-, $t$-, and $u$-channel exchange Witten diagrams together with contact-term completions—chosen so that they reproduce the physical exchange singularities of a given conformal family while avoiding spurious nonlocal or unphysical singularities. This reorganizes the conformal bootstrap in a manner close to Polyakov’s original idea: the correlator is expanded in manifestly crossing-symmetric objects, and consistency with the operator product expansion (OPE) is then imposed through cancellation conditions on spurious contributions (Isono et al., 2018).

1. Definition and conceptual role

In the crossing-symmetric reformulation of the bootstrap, a scalar four-point correlator is written schematically as

$$\langle O_1 O_2 O_3 O_4\rangle' = \sum_O\Big(W_O^{(\mathbf s)}+W_O^{(\mathbf t)}+W_O^{(\mathbf u)}\Big)+\text{(analytic terms)}.$$

Here $W_O^{(\mathbf s)}$, $W_O^{(\mathbf t)}$, and $W_O^{(\mathbf u)}$ are Polyakov blocks associated with exchange of a conformal family $O$ in the three channels. The characteristic requirement is not merely that these objects carry the right OPE poles, but that each block be engineered so that its non-analyticity matches a single physical exchange channel and does not introduce unwanted non-analyticities in the others. In momentum space this is formulated as a factorization condition on the discontinuity; in Mellin-space and dispersive formulations it is expressed through the exchange-plus-contact structure of Witten diagrams and the removal of spurious singularities (Isono et al., 2019).

This distinguishes Polyakov blocks from ordinary conformal blocks. Standard conformal blocks isolate one OPE channel and are not manifestly crossing-symmetric. By contrast, crossing-symmetric Polyakov blocks are designed so that crossing symmetry is already built into the basis. The price is that OPE consistency becomes a nontrivial constraint: the summed Polyakov expansion must cancel spurious double-trace or other unphysical contributions. In several formulations, these cancellation conditions are called Polyakov conditions or Polyakov sum rules (Mazac, 2018).

A recurrent interpretation across the literature is that Polyakov blocks are exchange Witten diagrams together with contact-term completions. This point is explicit in Mellin-space, momentum-space, and one-dimensional constructions. A plausible implication is that the notion of a Polyakov block is less a single formula than a structural prescription: one fixes the physical exchange singularities, crossing behavior, Regge or locality properties, and then determines the admissible contact completion accordingly (Gopakumar et al., 2018).

2. Momentum-space construction and factorization

The momentum-space program constructs Polyakov blocks directly from the analytic structure of momentum-space correlators. For scalar four-point functions, the central idea is that the non-analytic part of a correlator factorizes analogously to an on-shell scattering amplitude. In three dimensions, the $\mathbf s$-channel Polyakov block $W_n^{(\mathbf s)}$ is defined so that

$t$0

while remaining analytic in the variables associated with the other channels (Isono et al., 2018).

In this formulation, the correct object is not the ordinary conformal block but the Witten exchange diagram. For scalar exchange, the momentum-space Polyakov block takes the bulk form

$t$1

with $t$2 the bulk-to-bulk scalar propagator. The step-function or $t$3-ordered structure of the propagator removes the unwanted exponential growth that would otherwise generate incorrect singularities. This is also the mechanism by which shadow singularities are avoided (Isono et al., 2018).

The general-dimensional extension replaces the three-dimensional helicity decomposition by harmonic analysis on $t$4. The relevant little group becomes $t$5, and the Fourier-mode treatment of the earlier $t$6 construction is replaced by spherical harmonics together with the Funk–Hecke formula and the addition theorem. The resulting spinning block is organized into $t$7 spin sectors, and after summing over the magnetic index the projector becomes a Gegenbauer polynomial: $t$8 A central technical claim is that the only source of non-analyticity in the relevant three-point structures comes from the triple-$t$9-type integral, while the spin-dependent prefactors are polynomial in momenta and therefore do not generate new branch cuts (Isono et al., 2019).

This momentum-space perspective also makes the separation between connected and disconnected contributions unusually sharp. For the identity operator, the corresponding Polyakov block reproduces the disconnected piece, which is especially useful in large-$u$0 CFTs. This suggests that the momentum-space basis is naturally adapted to theories where one wishes to separate exchange-like connected dynamics from generalized-free-field contributions (Isono et al., 2018).

3. Mellin-space Polyakov blocks and contact ambiguities

Mellin space provides another natural setting because Mellin amplitudes are meromorphic and closely parallel scattering amplitudes. In the Polyakov-Mellin bootstrap, a four-point Mellin amplitude is expanded as

$u$1

where the $u$2 are exchange Witten diagrams and $u$3 is a crossing-symmetric contact term. In this framework, the bootstrap condition is the cancellation of residues at the spurious double-trace poles $u$4, $u$5 (Gopakumar et al., 2018).

A central result of the Mellin-space literature is that exchange diagrams alone are not generally sufficient. Exchange Witten diagrams are only defined up to polynomial ambiguities in Mellin variables, and these ambiguities are precisely contact diagrams. For identical scalars, crossing-symmetric contact amplitudes are polynomials in the symmetric invariants built from $u$6, $u$7, and $u$8, and a convenient parametrization is

$u$9

The necessity of such contact terms is demonstrated both in the large-$$\langle O_1 O_2 O_3 O_4\rangle' = \sum_O\Big(W_O^{(\mathbf s)}+W_O^{(\mathbf t)}+W_O^{(\mathbf u)}\Big)+\text{(analytic terms)}.$$0 holographic bootstrap and in the $$\langle O_1 O_2 O_3 O_4\rangle' = \sum_O\Big(W_O^{(\mathbf s)}+W_O^{(\mathbf t)}+W_O^{(\mathbf u)}\Big)+\text{(analytic terms)}.$$1-expansion beyond leading orders (Gopakumar et al., 2018).

The problem of uniqueness was addressed by constructing “unique Polyakov blocks” in Mellin space. The key prescription is first formulated at the level of cyclic amplitudes: one demands Regge suppression in the channel limits appropriate to a chosen cyclic ordering, which removes the polynomial ambiguity. For identical external operators, the fully crossing-symmetric block is then obtained as the average of three cyclic blocks. An equivalent formulation acts directly on the fully symmetric block using the operators $$\langle O_1 O_2 O_3 O_4\rangle' = \sum_O\Big(W_O^{(\mathbf s)}+W_O^{(\mathbf t)}+W_O^{(\mathbf u)}\Big)+\text{(analytic terms)}.$$2 and $$\langle O_1 O_2 O_3 O_4\rangle' = \sum_O\Big(W_O^{(\mathbf s)}+W_O^{(\mathbf t)}+W_O^{(\mathbf u)}\Big)+\text{(analytic terms)}.$$3, with the ambiguity fixed by requiring that the remaining polynomial terms lie in $$\langle O_1 O_2 O_3 O_4\rangle' = \sum_O\Big(W_O^{(\mathbf s)}+W_O^{(\mathbf t)}+W_O^{(\mathbf u)}\Big)+\text{(analytic terms)}.$$4 (Sleight et al., 2019).

These Mellin constructions connect naturally to analytic bootstrap functionals. The direct-channel decomposition of cyclic Polyakov blocks generates a dual basis of functionals, and the extracted double-twist OPE data separates into analytic-in-spin and non-analytic-in-spin contributions. The literature emphasizes that some non-analytic terms are universal and cannot be shifted by lower-degree contact terms, whereas lower-spin non-analytic corrections are sensitive to the choice of contact completion (Sleight et al., 2019).

4. Dispersion relations, locality, and singularity-free blocks

A dispersive reformulation of crossing symmetry makes the same structural issues especially transparent. In crossing-symmetric variables

$$\langle O_1 O_2 O_3 O_4\rangle' = \sum_O\Big(W_O^{(\mathbf s)}+W_O^{(\mathbf t)}+W_O^{(\mathbf u)}\Big)+\text{(analytic terms)}.$$5

one introduces the symmetric combinations

$$\langle O_1 O_2 O_3 O_4\rangle' = \sum_O\Big(W_O^{(\mathbf s)}+W_O^{(\mathbf t)}+W_O^{(\mathbf u)}\Big)+\text{(analytic terms)}.$$6

Earlier crossing-symmetric dispersion relations, including the Auberson–Khuri relation, were manifestly crossing symmetric but generated negative powers of $$\langle O_1 O_2 O_3 O_4\rangle' = \sum_O\Big(W_O^{(\mathbf s)}+W_O^{(\mathbf t)}+W_O^{(\mathbf u)}\Big)+\text{(analytic terms)}.$$7, interpreted as nonlocal spurious singularities. These terms do not correspond to local contact interactions and therefore obstruct a clean Polyakov-block expansion (Chowdhury et al., 2022).

Imposing locality constraints removes these nonlocal terms and yields a local, fully crossing-symmetric expansion of amplitudes in terms of Feynman blocks: $$\langle O_1 O_2 O_3 O_4\rangle' = \sum_O\Big(W_O^{(\mathbf s)}+W_O^{(\mathbf t)}+W_O^{(\mathbf u)}\Big)+\text{(analytic terms)}.$$8 with

$$\langle O_1 O_2 O_3 O_4\rangle' = \sum_O\Big(W_O^{(\mathbf s)}+W_O^{(\mathbf t)}+W_O^{(\mathbf u)}\Big)+\text{(analytic terms)}.$$9

and the remaining channel terms obtained by permutation. The contact term $W_O^{(\mathbf s)}$0 is fixed by the requirement that the final answer contain only nonnegative powers in the local expansion

$W_O^{(\mathbf s)}$1

Within this framework, the Feynman block is explicitly identified as the analog of a crossing-symmetric Polyakov block (Chowdhury et al., 2022).

For Mellin amplitudes of scalar four-point correlators, a parallel construction shows that locality constraints $W_O^{(\mathbf s)}$2 for $W_O^{(\mathbf s)}$3 replace the role usually played by crossing symmetry in fixed-$W_O^{(\mathbf s)}$4 dispersion relations. Once locality is imposed, the contact-term ambiguities of crossing-symmetric Mellin blocks are fixed, and the resulting sum rules are identical to those derived from two-channel dispersion relations (Gopakumar et al., 2021).

A later singularity-free formulation strengthens this result by deriving a new crossing-symmetric dispersion relation in which the spurious singularities never arise in the first place. In this approach, the singularity-free block $W_O^{(\mathbf s)}$5 decomposes as

$W_O^{(\mathbf s)}$6

where the exchange term in channel $W_O^{(\mathbf s)}$7 is

$W_O^{(\mathbf s)}$8

The associated contact term becomes a manifestly crossing-symmetric polynomial. This removes the need for ad hoc locality constraints to cancel nonlocal negative powers of $W_O^{(\mathbf s)}$9, and the result is presented as a cleaner foundation for both the Polyakov bootstrap in CFT and the crossing-symmetric S-matrix bootstrap in QFT (Song, 2023).

5. One-dimensional bootstrap, inversion formulas, and mixed correlators

One dimension provides a particularly explicit realization of crossing-symmetric Polyakov blocks. For identical operators in $W_O^{(\mathbf t)}$0, a Lorentzian OPE inversion formula exists that applies directly to fully crossing-symmetric correlators: $W_O^{(\mathbf t)}$1 A key structural result is that inverting a single crossed-channel conformal block returns the coefficient function of a crossing-symmetric sum of exchange Witten diagrams in $W_O^{(\mathbf t)}$2, including the direct-channel exchange. In the bosonic case this Polyakov block takes the form

$W_O^{(\mathbf t)}$3

where $W_O^{(\mathbf t)}$4 is a contact diagram restoring the correct Regge behavior. Its direct-channel expansion contains the physical block plus an infinite tower of double-trace blocks and derivative blocks whose coefficients must cancel in the full correlator. The residues of the inversion kernel at the double-trace dimensions furnish the analytic bootstrap functionals $W_O^{(\mathbf t)}$5 and $W_O^{(\mathbf t)}$6 (Mazac, 2018).

For general systems of one-dimensional mixed correlators, Polyakov blocks become matrix-valued objects indexed by external operators. The correlator

$W_O^{(\mathbf t)}$7

admits both an OPE expansion and a Polyakov expansion,

$W_O^{(\mathbf t)}$8

where each $W_O^{(\mathbf t)}$9 packages direct-channel exchange, crossed-channel exchanges, and contact diagrams with the correct OPE orientation factors. The equality between the OPE representation and the Polyakov representation yields linear sum rules on the spectrum and OPE coefficients. In tensor products of generalized free fields these sum rules diagonalize, while in mixed systems involving elementary and composite operators the functional basis can be dressed to restore stronger orthogonality properties (Ghosh et al., 2023).

The one-dimensional literature therefore shows two complementary aspects of crossing-symmetric Polyakov blocks. First, they provide an explicit realization of Polyakov’s original crossing-symmetric exchange basis. Second, they act as generators of analytic functionals and corresponding sum rules. This suggests a deep equivalence, in one dimension, between the Polyakov-block decomposition, Lorentzian inversion, and the extremal-functional approach to the bootstrap (Mazac, 2018).

6. Extensions, variants, and scope

The concept extends beyond the standard scalar crossing-symmetric sector. In theories with global symmetries or multiple tensor structures, one encounters crossing-antisymmetric sectors. These admit a parallel basis of manifestly crossing-antisymmetric objects, called $W_O^{(\mathbf u)}$0-type Polyakov blocks, built from AdS Witten diagrams. In one dimension they encode $W_O^{(\mathbf u)}$1-type analytic functionals, while in general dimension they arise from a crossing-antisymmetric dispersion relation in Mellin space. As in the symmetric case, the construction generates locality constraints in addition to the usual Polyakov conditions (Kaviraj, 2021).

The framework also extends to higher-point correlators. In one-dimensional five-point bootstrap, the correlator can be expanded in crossing-symmetric five-point Polyakov blocks $W_O^{(\mathbf u)}$2, built from a direct double-exchange Witten diagram, its crossed permutations, a single-exchange Witten diagram, and a contact diagram. Requiring equality with the standard conformal-block expansion yields families of sum rules controlling double-twist and triple-twist OPE data. This suggests that the Polyakov-block philosophy is not confined to four-point crossing equations but can be generalized to multi-point bootstrap problems (Antunes et al., 7 Aug 2025).

Across these variants, several limitations recur. External spinning operators are not incorporated in some momentum-space constructions, where the analysis is restricted to external scalars (Isono et al., 2019). In Mellin-space and dispersive approaches, the exchange-like non-analytic part is tightly constrained, but analytic or contact terms require separate prescriptions such as Regge boundedness or locality (Sleight et al., 2019). More generally, manifest crossing symmetry does not by itself make the OPE content manifest; instead, the OPE reappears as a set of nontrivial sum rules cancelling spurious double-trace, derivative, or nonlocal contributions (Gopakumar et al., 2021).

Taken together, these developments define crossing-symmetric Polyakov blocks as a unifying language across momentum space, Mellin space, and dispersive bootstrap methods. Their common content is the replacement of a channel-by-channel conformal-block basis by exchange-plus-contact objects that are individually adapted to crossing symmetry, analyticity, and locality.