OPE coefficients of the 3D Ising CFT

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Numbers

$\mathcal O_i$

$\mathcal O_j$

$\mathcal O_k$

$\lambda_{ijk}$

$\sigma$

$\epsilon$:

1.05185373 +/- 0.00000011

comment: The spin-spin-energy coefficient, Table 1 of [1].

$\epsilon$

$\epsilon$:

1.53244304 +/- 0.00000058

comment: The energy three-point coefficient, Table 1 of [1].

Definition

The scalar operator product expansion coefficient $\lambda_{ijk}$ of the three-dimensional Ising conformal field theory is the normalized three-point coefficient in (2) [2].

Parameters

$\mathcal O_i$

— scalar primary at the first insertion (A nonidentity scalar primary; the triple is ordered by increasing scaling dimension, $\Delta_i\leq\Delta_j\leq\Delta_k$.)

$\mathcal O_j$

— scalar primary at the second insertion (A nonidentity scalar primary; the triple is ordered by increasing scaling dimension, $\Delta_i\leq\Delta_j\leq\Delta_k$.)

$\mathcal O_k$

— scalar primary at the third insertion (A nonidentity scalar primary; the triple is ordered by increasing scaling dimension, $\Delta_i\leq\Delta_j\leq\Delta_k$.)

Formulas

(1)

$\langle\mathcal O_i(x)\mathcal O_j(0)\rangle=\delta_{ij}|x|^{-2\Delta_i}$ in an orthonormal real scalar-primary basis; $\Delta_i$ is the scaling dimension of $\mathcal O_i$ [2].

(2)

$\displaystyle \langle\mathcal O_i(x_1)\mathcal O_j(x_2)\mathcal O_k(x_3)\rangle=\frac{\lambda_{ijk}}{r_{12}^{\Delta_i+\Delta_j-\Delta_k}r_{23}^{\Delta_j+\Delta_k-\Delta_i}r_{31}^{\Delta_k+\Delta_i-\Delta_j}}$, where $r_{ab}=|x_a-x_b|$ for distinct Euclidean insertion points $x_a,x_b$, and $\Delta_i,\Delta_j,\Delta_k$ are the scaling dimensions of the three primaries [2].

(3)

$\lambda_{ijk}=\lambda_{jik}=\lambda_{ikj}$. If $p_i,p_j,p_k\in\{+1,-1\}$ are the spin-flip parities of the three scalar primaries, then $p_ip_jp_k=-1$ implies $\lambda_{ijk}=0$. In particular, $\lambda_{\sigma\sigma\sigma}=\lambda_{\sigma\epsilon\epsilon}=0$.

Comments

(4)

The setting is the unitary Euclidean continuum critical point in three dimensions. The spin primary $\sigma$ is the lowest-dimension scalar odd under spin flip, and the energy primary $\epsilon$ is the lowest-dimension nonidentity scalar even under spin flip. Their scaling dimensions specify the powers of the separations in (2). The sign of $\epsilon$ is fixed by $\lambda_{\sigma\sigma\epsilon}>0$; either sign of $\sigma$ gives the same two coefficients tabulated here.

(5)

The indices identify the three scalar operators, rather than three different quantities. Permutation symmetry identifies all orders of the same triple, so each is listed once. Among triples formed from $\sigma$ and $\epsilon$, only $\sigma\sigma\epsilon$ and $\epsilon\epsilon\epsilon$ are allowed by spin-flip symmetry. Coefficients of descendants are not independent primary data.

(6)

Each centre and radius reproduces a published numerical estimate from Table 1 of [1]. These radii describe the sampled bootstrap region in section 5.2 of that paper; they are neither statistical standard deviations nor unconditional rigorous error bounds.

References

[1]

C.-H. Chang, V. Dommes, R. S. Erramilli, A. Homrich, P. Kravchuk, A. Liu, M. S. Mitchell, D. Poland and D. Simmons-Duffin, Bootstrapping the 3d Ising stress tensor, Journal of High Energy Physics 03 (2025), 136, Table 1 and section 5.2. (arXiv) (doi)

[2]

M. Hasenbusch, Two- and three-point functions at criticality: Monte Carlo simulations of the improved three-dimensional Blume-Capel model, Physical Review E 97 (2018), 012119, equations (1), (2), (37) and (38). (arXiv) (doi)

[3]

F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi, Precision islands in the Ising and O(N) models, Journal of High Energy Physics 08 (2016), 036. (arXiv) (doi)

Links

[4]

Chang et al.: Bootstrapping the 3d Ising stress tensor, full text

Similar tables

T473 — the dimensions of the Ising primary operators whose three-point coefficients are recorded here

T472 — the bulk critical exponents of the three-dimensional universality classes, including Ising

Critical_exponents_of_the_two-dimensional_universality_classes — critical exponent data for the two-dimensional counterparts, including the two-dimensional Ising class

Critical_couplings_of_the_Ising_model_on_lattices — critical couplings of Ising lattices; the three-dimensional entries have the continuum limit described here

Data properties

Entries are of type: real number

How well the digits are known: measured

Table is complete: no (it holds the two independent symmetry-allowed coefficients made entirely from the leading spin and energy scalar primaries, which determine their nonzero three-point functions; higher scalar primaries and spinning primaries are not covered)

Sources of data: [1]

How they were obtained:

Both entries transcribe Table 1 of [1], using mixed four-point correlators of $\sigma$, $\epsilon$ and the stress tensor $T$ at derivative order $\Lambda=43$. The full text is available as [4]. The attached generator reproduces the transcription, not the bootstrap optimization. The centres and radii retain the source precision; these coefficients are not known to 100 significant digits.

The spectral assumptions in that paper's Table 6 are dimension at least $3$ for the other scalar operators in the sectors probed, dimension at least $4$ for the next spin-flip-even, spatial-parity-even spin-$2$ primary after $T$, and twist $\tau=\Delta-\ell\geq1+10^{-6}$ in the remaining sectors, where $\ell$ is spin. Section 5.2 estimates component errors by taking extrema over a cloud of allowed points. Its OPE uncertainties are not the boldface rigorous errors of Table 1. The label measured records externally reported numerical estimates, not experimental measurements or a proof that the true coefficients lie in the written intervals.

The attached checks parse the parenthesized uncertainties independently with exact rational arithmetic and compare both entries to the original source notation. Both intervals lie inside the independent Monte Carlo estimates $1.051(1)$ and $1.533(5)$ in equations (37) and (38) of [2], using the same unit two-point normalization and sign convention. They also lie inside the earlier bootstrap estimates $1.0518537(41)$ and $1.532435(19)$ in equations (3.4) and (3.5) of [3]; this is a consistency comparison within the bootstrap method. These comparisons establish agreement at the precision of the older determinations, not independent confirmation of every new digit.

The checks include known interval controls and deliberately incorrect centres and radii, finite positive widths, the complete spin-flip selection among triples of $\sigma$ and $\epsilon$, all insertion permutations, sign changes of both operators, and the dilation and inversion weights of the three-point formula. The same checks are run on values read back from the draft, in addition to Generator.verify(sample=None).