Bootstrap of Energy Correlators

Bootstrapping an energy correlator means characterizing detector distributions that simultaneously satisfy energy sum rules, positivity, permutation symmetry, local-CFT crossing, and the light-ray OPE. A closed problem must specify the source state, normalization, angular domain, contact sector, exchanged spectrum, and approximation error. Positivity alone yields useful moment bounds; reconstructing a CFT observable requires the remaining constraints as well.

Required background. Light-ray OPE and detector expansions supplies the collinear spectrum and convergence qualifications. Averaged null energy and positivity supplies the positive detector operator and its state domain.

Helpful background. Optimization from crossing explains finite positive cones, objectives, and independently checked numerical certificates.

Evidence cutoff. The primary literature and preprints discussed here were checked through 2026-08-09. The moment inequalities derived below are elementary consequences of a declared positive measure; model-specific numerical bounds, block libraries, and convergence claims after that date are not included.

A closed two-detector problem

Fix the following data before writing an equation:

• a unitary CFT in a declared dimension $d>2$;
• a normalizable scalar source state, or a regulated timelike-momentum state with its norm divided out;
• the physical detector normalization $\int d\Omega\,\mathcal E=P^0$;
• the Wightman ordering and a Regge condition sufficient for separated energy detectors to commute;
• a prescription for coincident-angle contact distributions;
• the angular variable $z=(1-\mathbf n_1\cdot\mathbf n_2)/2\in[0,1]$;
• the local and light-ray operator sectors allowed by symmetry, including parity and transverse spin.

For a four-dimensional scalar state of sharp rest energy $E$, write

$$\langle\mathcal E(\mathbf n_1)\mathcal E(\mathbf n_2)\rangle =\frac{E^2}{(4\pi)^2}F_E(z).$$

Including all contact weight, the total-energy sum rule is

$$\int_0^1dz\,F_E(z)=1.$$

If the coincident product is defined so that the separated positive, commuting detectors extend to a positive distribution, then

$$d\mu(z)=F_E(z)\,dz$$

is a probability measure on $[0,1]$. In distribution theory, a positive distribution is a measure: an uncancelled $\delta'(z)$ cannot be part of this positive completion. Algebraic intermediate expressions may contain derivatives of delta functions, but the final declared positive observable must be tested after all contact contributions are combined.

Constraints that must hold together

Constraint	Equation or test	Hidden assumption to expose
Energy normalization	$\int d\mu=1$	Sharp-energy state; otherwise divide by $\langle(P^0)^2\rangle$
Positivity	$\int \lvert p(z)\rvert^2d\mu(z)\ge0$ for every polynomial $p$	Positive coincident extension and common detector domain
Support	$0\le z\le1$	Rest-frame angular parametrization
Exchange	$G_2(\mathbf n_1,\mathbf n_2)=G_2(\mathbf n_2,\mathbf n_1)$	Regge control for separated commutativity; contacts matched
Detector OPE	Small-$z$ light-ray expansion at parent spin $J=3$	Null-limit order, transverse sectors, inversion threshold, contacts
Local crossing	Equivalent channel expansions of the underlying $\langle\phi^\dagger TT\phi\rangle$ boundary value	Correct analytic continuation and tensor basis
OPE completeness	Every allowed pole, cut, degeneracy, and contact sector included	No unbounded spectral truncation is called exact
Endpoint data	Collinear and, when used, back-to-back asymptotics	Domains do not overlap automatically

Detector exchange is not the transformation $z\leftrightarrow1-z$: exchange leaves $z$ fixed. Crossing refers instead to different operator pairings of the underlying local correlator before or after the light transforms, with their Lorentzian continuation specified.

Moment bounds from positivity

Define Hausdorff moments

$$m_k=\int_0^1z^k\,d\mu(z), \qquad m_0=1.$$

Because $0\le z\le1$,

$$0\le m_{k+1}\le m_k\le1.$$

Positivity of $(z-m_1)^2$ gives

$$m_2-m_1^2\ge0,$$

while positivity of $z(1-z)$ gives

$$m_1-m_2\ge0.$$

Thus the first nontrivial closed bound is

$$\boxed{m_1^2\le m_2\le m_1,\qquad0\le m_1\le1.}$$

It has sharp extremizers. The lower bound is saturated by a delta measure at $z=m_1$; the upper bound is saturated by a measure supported only at $z=0,1$. This conclusion uses positivity, normalization, and support—no OPE truncation.

Higher constraints are compactly expressed by moment and localizing matrices. For any finite coefficient vector $c_i$,

$$\sum_{i,j=0}^{N}c_i^*m_{i+j}c_j =\int_0^1\left|\sum_{i=0}^{N}c_i z^i\right|^2d\mu(z)\ge0,$$

so $H^{(0)}_{ij}=m_{i+j}\succeq0$. Support on $[0,1]$ also gives

$$H^{(z)}_{ij}=m_{i+j+1}\succeq0, \qquad H^{(1-z)}_{ij}=m_{i+j}-m_{i+j+1}\succeq0.$$

These are necessary conditions at every order and become sufficient for the full Hausdorff moment problem when the complete infinite sequence is imposed.

Angular partial waves and CFT data

In general $d$, a scalar rest-state shape can be expanded in $SO(d-1)$ harmonics,

$$F_E(z)=\sum_{\ell=0}^{\infty}a_\ell\, \widehat C_\ell^{(d-3)/2}(1-2z),$$

with a declared normalization for the Gegenbauer polynomial. The energy Ward identity fixes the zeroth coefficient; momentum conservation constrains the first. Positivity of the angular measure bounds harmonic moments, but positivity of finitely many $a_\ell$ is not equivalent to pointwise or measure positivity of a truncated series.

The same observable descends from the local Wightman correlator $\langle\phi^\dagger TT\phi\rangle$. A source–detector channel organizes $\phi\times T$ primaries, while the detector–detector channel is the parent-spin-three light-ray OPE. Schematically, a partial-wave coefficient has a quadratic-form expansion

$$a_\ell=\sum_{\mathcal O,A,B} \lambda_{\phi T\mathcal O}^{A} K_{\mathcal O,AB}^{(\ell)} \lambda_{\phi T\mathcal O}^{B},$$

where $A,B$ label tensor structures. Reflection positivity can make the appropriate kernels positive in a fixed convention, but that property must be proved rather than inferred from a scalar-block analogy.

A 2025 primary preprint constructs source–detector conformal blocks for symmetric-traceless exchange, treats parity-even and parity-odd structures, studies convergence in a tensor-product example, and derives positivity bounds on OPE data Meçaj et al. 2025, §§2–5. Its model applications and proposed moment optimization are current research, not a theorem that every truncation converges. A separate 2025 preprint combines local $\mathcal N=4$ superconformal-bootstrap input with EEC dispersive and numerical constraints Dempsey et al. 2025, §§2, 6, and 7; those theory-specific numerical results are not generalized here.

A defensible numerical formulation

A finite computation can target, for example, the minimum or maximum of $m_2$ at fixed $m_1$. A reproducible formulation records:

1. the source theory or allowed local spectrum and every tensor-sector assumption;
2. the number of retained moments or harmonics and the angular basis normalization;
3. exact normalization, momentum, permutation, and contact equations;
4. positive moment and localizing matrices;
5. the light-ray and source–detector blocks, their valid angular domains, and versioned numerical evaluation;
6. a rigorous or explicitly estimated tail for omitted operators and harmonics;
7. solver precision, residual tolerances, primal and dual data, and an independent certificate check.

Without a tail bound, the result is a statement about the finite relaxation, not the full CFT. Without explicit contacts, the normalization row may be wrong. Without a proven positive cone, semidefinite optimization has no exclusion meaning even if the solver converges.

The analytic bound $m_1^2\le m_2\le m_1$ is an exact baseline for such a computation. Any purported stronger result must approach this allowed region when all CFT-specific input is removed.

Convergence and reconstruction

Three convergence questions must be kept separate:

• Local OPE convergence: the source–detector expansion converges in its declared Lorentzian or analytically continued domain after smearing.
• Light-ray OPE convergence: the detector–detector expansion is defined on the common null plane and its contour deformation is justified.
• Harmonic reconstruction: the partial-wave series converges as a function or distribution on the angular interval, including endpoints.

Agreement of a finite number of moments does not uniquely reconstruct a measure. Positivity narrows the possibilities; endpoint asymptotics, OPE data, and more moments narrow them further. A unique reconstruction claim needs a determinate moment problem or an independently proved completeness result.

A planned exact continuation

A reproducible calculation should supply exact four-dimensional free-field fixtures, one normalized two-detector distribution, and a declared light-ray limit. Such a calculation can test normalization, exchange, endpoint behavior, and finite-moment code; it cannot establish completeness or a truncation tail for an interacting CFT.

Common pitfalls

Calling $F_E(z)$ a function before contacts are fixed. Treat it as a distribution or measure until endpoint terms are known.

Using crossing as $z\leftrightarrow1-z$. Detector exchange leaves $z$ unchanged. Channel crossing belongs to the underlying local correlator and includes Lorentzian continuation.

Assuming finite harmonic positivity is enough. A truncated series can become negative between sample points. Impose a positive-measure or certified polynomial condition.

Reporting a bound without a tail. Solver feasibility at fixed truncation does not control omitted operators, high harmonics, or contact sectors.

Exercises

1. Test a proposed moment pair. Can a positive normalized measure on $[0,1]$ have $(m_1,m_2)=(0.4,0.1)$?

Solution

No. Positivity of the variance requires $m_2\ge m_1^2=0.16$, but $0.1<0.16$.

2. Identify the extremizer. Find a measure with $m_1=0.4$ that saturates the upper bound $m_2=m_1$.

Solution

Take $d\mu=0.6\,\delta_0+0.4\,\delta_1$, where $\delta_a$ is unit point mass at $z=a$. Then $m_k=0.4$ for every $k\ge1$, so $m_2=m_1=0.4$.

References

• Dempsey, Ross, Robin Karlsson, Silviu S. Pufu, Zahra Zahraee, and Alexander Zhiboedov. “Conformal Collider Bootstrap in $\mathcal N=4$ SYM.” arXiv preprint (2025). arXiv:2512.10796.
• Meçaj, Bianka, Ian Moult, Matthew T. Walters, and Yuan Xin. “Energy Correlator Conformal Blocks and Positivity.” arXiv preprint (2025). arXiv:2512.09986.