Large-N Crossing, Double-Trace Data, and Contact Ambiguities

Large-N crossing is an inhomogeneous problem. Declared single-trace exchanges generate logarithms and force corrections to double-trace data, but the exchange singularities do not determine the entire solution. Crossing-symmetric homogeneous terms remain, appearing in Mellin space as contact polynomials subject to spin and Regge restrictions.

Required background. Large-N and Sparse-Spectrum CFT Data fixes the expansion and operator taxonomy. Mellin-Space CFT Correlators fixes the Gamma measure, pole, and contour conventions. Helpful background. Analytic Functionals and Polyakov Blocks gives a complementary reconstruction of exchange and contact contributions.

Crossing order by order

For identical scalars, use the reduced-correlator convention

$$V^{\Delta_\phi}\mathcal G(U,V) =U^{\Delta_\phi}\mathcal G(V,U).$$

Insert

$$\mathcal G=\mathcal G^{(0)}+g^2\mathcal G^{(1)}+O(g^4).$$

Because the external dimension is held fixed in this equation, each order obeys the same linear crossing relation. If $\Delta_\phi$ itself varies with $g$, its expansion produces additional terms proportional to $\log U$ and $\log V$ and must be included explicitly.

At leading order, generalized-free double-trace operators have dimensions $\Delta^{(0)}_{n,J}=2\Delta_\phi+2n+J$. Expanding one conformal-block contribution gives

$$\begin{aligned} a_{n,J}g_{\Delta_{n,J},J} ={}&a^{(0)}_{n,J}g_{\Delta^{(0)}_{n,J},J}\\ &+g^2\left[ a^{(1)}_{n,J}g_{\Delta^{(0)}_{n,J},J} +a^{(0)}_{n,J}\gamma_{n,J} \partial_\Delta g_{\Delta,J}\big|_{\Delta=\Delta^{(0)}_{n,J}} \right]+O(g^4). \end{aligned}$$

Since $g_{\Delta,J}(U,V)\sim U^{(\Delta-J)/2}$ in the direct OPE limit, the derivative term contains

$$\frac12 a^{(0)}_{n,J}\gamma_{n,J} U^{\Delta_\phi+n}\log U.$$

Thus crossed-channel exchange data determine weighted anomalous dimensions through logarithmic terms. Nonlogarithmic terms also involve $a^{(1)}_{n,J}$ and derivatives of the regular block coefficients. Degenerate leading operators yield matrices; one correlator usually fixes OPE-weighted averages rather than individual eigenvectors. This order-by-order logic underlies the sparse-spectrum solutions of Heemskerk et al. 2009, §§3–5.

Exchange completion and homogeneous solutions

Let $\mathcal G_{\mathcal X}$ denote the contribution forced by a declared single-trace primary $\mathcal X$. A complete order-$g^2$ solution has the schematic form

$$\mathcal G^{(1)} =\sum_{\mathcal X} c_{\mathcal X}^2 \mathcal G_{\mathcal X}^{\rm completed} +\sum_A \alpha_A\mathcal H_A,$$

where each $\mathcal G_{\mathcal X}^{\rm completed}$ includes the crossed-channel pieces needed for crossing, and $\mathcal H_A$ solves the homogeneous crossing equation. The split is basis-dependent: a polynomial can be shifted between an “exchange completion” and the homogeneous sector. Only the full correlator and a declared basis make $\alpha_A$ meaningful.

Lorentzian discontinuities efficiently reconstruct sufficiently high-spin data, but contributions with vanishing double discontinuity can survive at low spin Caron-Huot 2017, §§3–4. Regge growth controls how many subtractions or contact terms must be retained. The Lorentzian inversion formula therefore determines data only in its stated spin domain; it does not erase the low-spin completion.

Contact ambiguity in Mellin space

With the Mellin convention fixed on the preceding page, a single-trace exchange has pole ladders and polynomial residues Penedones 2011, §§2–3. The most general solution with those specified poles is

$$M(s,t)=M_{\rm poles}(s,t)+P(s,t),$$

where $P$ is an entire crossing-symmetric polynomial if polynomial boundedness is assumed. For identical scalars,

$$P(s,t)=P(t,s)=P(s,u), \qquad s+t+u=4\Delta_\phi.$$

A convenient symmetric basis begins with

$$1, \qquad \sigma_2=s^2+t^2+u^2, \qquad \sigma_3=stu, \qquad \sigma_2^2,\ldots.$$

The linear symmetric invariant $s+t+u$ is constant, so it is not independent. A degree cutoff follows only after a Regge or large-variable growth condition has been chosen. Without that condition, “enumerate all contact terms” has no finite answer.

For the pole toy

$$M_{\rm poles} =g_\chi^2\left[ \frac{1}{s-\tau_\chi} +\frac{1}{t-\tau_\chi} +\frac{1}{u-\tau_\chi} \right],$$

both $M_{\rm poles}+\lambda_0$ and $M_{\rm poles}+\lambda_2\sigma_2$ preserve the declared poles and residues. They do not preserve the double-trace CFT data. This is precisely why pole reconstruction and correlator reconstruction are different tasks.

Spin support and a finite reconstruction procedure

At fixed polynomial degree, the contact contribution has restricted spin support in the corresponding partial-wave decomposition, with details depending on dimension and convention. A practical reconstruction proceeds as follows:

1. freeze external dimensions, two-point normalization, and the order in $g$;
2. list every single-trace pole ladder and residue polynomial;
3. build a crossing-symmetric exchange completion;
4. choose a Regge bound and derive the allowed polynomial degree;
5. expand a complete symmetric polynomial basis through that degree;
6. solve crossing for double-trace averages, retaining mixing matrices;
7. verify the direct OPE, crossed OPE, Mellin inverse transform, and high-spin asymptotics;
8. report the coefficients $\alpha_A$ as undetermined unless independent CFT data fix them.

A reproducible calculation should compare constant and degree-two contact shifts against a frozen exchange fixture.

Evidence carried across protected and large-N interfaces

This comparison is shared with the protected-data chapter because both interfaces require a versioned source, an invertible convention map, and a strict limit on the resulting claim.

Imported datum	Canonical source and version	Convention transform	Evidence carried	Allowed CFT conclusion	Optional later interpretation	Does not prove
Shortening-fixed datum	Versioned protected-data export; version required	algebra label, charges, two-point convention, recombination round trip	exact only if the source says exact	fix the stated protected representation datum	none required	its OPE coefficient or an unprotected gap
Localized sphere derivative	Sphere partition functions and matrix models; version required	source map, mixing subtraction, local normalization	integrated observable with contacts	impose a derived integral constraint	none required	an unsmeared local correlator
Index coefficient	Index inversion and protected-spectrum limits; version required	character decomposition and recombination quotient	graded protected count	constrain a proven multiplicity combination	none required	a positive raw degeneracy or OPE coefficient
Large-N factorization	Large-N CFT Data and Vector Models; cited assumptions frozen in this edition	unit-normalized operators and explicit $g$ power counting	asymptotic CFT correlators	organize connected pieces and double-trace corrections	approximate multiparticle language	a bulk dictionary or Lagrangian
Mellin pole set plus contact basis	Mellin-Space CFT Correlators; convention on that page	Gamma measure, contours, residue normalization, symmetric-polynomial basis	analytic CFT representation	reconstruct exchange data modulo displayed contact coefficients	exchange/contact terminology after a separate dictionary	unique low-spin data or a local vertex
Large higher-spin gap	Large-Gap Constraints and CFT-Side Locality Tests; dated evidence boundary there	gap definition, $g$–gap order, Regge and finite-gap normalization	conditional analytic evidence	state a gap-suppressed CFT hierarchy	Volume 15 may assess approximate locality	that the diagnostics are sufficient

The table deliberately separates “optional interpretation” from “allowed conclusion.” An interpretation may be useful, but it is not evidence supplied by the input row.

Failure modes

Solving only the logarithms. Logarithms determine anomalous-dimension combinations, not the complete nonlogarithmic OPE correction. Crossing must be checked at the full order.

Declaring the pole part unique. Exchange diagrams or Polyakov blocks have convention-dependent polynomial completions. Specify the completion basis before comparing coefficients.

Using a discontinuity to eliminate contact terms. Vanishing discontinuity means the term is invisible to that reconstruction, not absent from the correlator.

Ignoring mixing. A degenerate double-trace family requires an anomalous-dimension matrix. Single-correlator averages cannot be promoted to its spectrum.

Exercises

Show that every symmetric polynomial of total degree at most two in $(s,t,u)$, subject to $s+t+u=4\Delta_\phi$, is a linear combination of $1$ and $\sigma_2=s^2+t^2+u^2$.

Solution

Symmetric polynomials are generated by the elementary invariants $e_1=s+t+u$, $e_2=st+tu+us$, and $e_3=stu$. Through degree two only $1$, $e_1$, $e_1^2$, and $e_2$ occur. Because $e_1=4\Delta_\phi$ is constant, only one nonconstant invariant remains. Using $\sigma_2=e_1^2-2e_2$, the basis may be chosen as $1$ and $\sigma_2$.

References

• Caron-Huot, S. “Analyticity in Spin in Conformal Theories.” Journal of High Energy Physics 2017, 078 (2017), §§3–4. arXiv. DOI.
• Heemskerk, I., Penedones, J., Polchinski, J., and Sully, J. “Holography from Conformal Field Theory.” Journal of High Energy Physics 2009, 079 (2009). arXiv. DOI.
• Penedones, J. “Writing CFT Correlation Functions as AdS Scattering Amplitudes.” Journal of High Energy Physics 2011, 025 (2011). arXiv. DOI.