Tauberian Theorems and Asymptotic Spectral Data

A Tauberian theorem turns a singular limit of a positive spectral transform into an asymptotic statement about an integrated spectral measure. In conformal bootstrap applications, this is the rigorous bridge from crossed-channel identity dominance to averaged high-dimension or large-spin data. It does not normally determine individual operator dimensions or coefficients.

Required background. The lightcone OPE and large-spin expansion supplies the crossed-channel singularity and the large-spin interpretation. Asymptotic scales, remainders, and uniformity distinguishes a leading equivalent from a pointwise expansion.

Helpful background. Bounded, compact, and integral operators provides language for kernels and spectral measures, although positivity rather than compactness is the decisive Tauberian input here.

Abelian reasoning is the easy direction

Let $d\mu(E)\ge0$ be a spectral measure and

$$F(t)=\int_0^\infty e^{-tE}\,d\mu(E), \qquad t>0.$$

If the cumulative measure satisfies

$$M(E)\equiv\mu([0,E]) \sim \frac{A}{\Gamma(\gamma+1)}E^\gamma,$$

then integration by parts and $y=tE$ give

$$F(t)\sim A\,t^{-\gamma} \qquad(t\to0^+).$$

This forward, or Abelian, implication follows by inserting known spectral growth into the transform. Bootstrap reasoning needs the converse: infer $M(E)$ from the singularity of $F(t)$. Oscillating signed measures can have the same leading transform while very different cumulative behavior, so the converse requires a Tauberian hypothesis such as nonnegativity or monotonicity.

A conformal-block Tauberian theorem

For a Hermitian scalar in a reflection-positive one-dimensional CFT, use

$$\langle\phi(0)\phi(z)\phi(1)\phi(\infty)\rangle =z^{-2\Delta_\phi}\mathcal G(z),$$

and the block expansion on $0<z<1$,

$$\mathcal G(z)=\int_0^\infty d\Delta\, p(\Delta)G_\Delta(z), \qquad p(\Delta)\ge0, \qquad G_\Delta(z)=z^\Delta{}_2F_1(\Delta,\Delta;2\Delta;z).$$

Assume the crossed identity is separated by a positive gap, so

$$\mathcal G(1-x)\sim x^{-2\Delta_\phi} \qquad(x\to0^+).$$

At large $\Delta$ with $\sqrt{x}\,\Delta=O(1)$,

$$G_\Delta(1-x) \sim C(\Delta)K_0(2\sqrt{x}\,\Delta), \qquad C(\Delta)=4^\Delta\sqrt{\frac{\Delta}{\pi}}.$$

Define the weighted cumulative primary density

$$Q(Y)=\int_0^Yd\Delta\,C(\Delta)p(\Delta).$$

The conformal Tauberian theorem then gives

$$Q(Y)\sim\frac{Y^\gamma}{A\gamma}, \qquad \gamma=4\Delta_\phi, \qquad A=\frac14\Gamma\!\left(\frac\gamma2\right)^2.$$

This exact prefactor follows from

$$\int_0^\infty dt\,t^{\gamma-1}K_0(2t) =\frac14\Gamma\!\left(\frac\gamma2\right)^2.$$

The proof first controls replacement of the exact block by its Bessel kernel, then uses a Wiener-type Tauberian theorem to replace that smooth kernel by a sharp cutoff. The nonvanishing Fourier transform of the logarithmically rescaled kernel is essential Qiao and Rychkov 2017, §§2–5.

What “averaged” means

For a discrete spectrum, $p(\Delta)=\sum_i a_i\delta(\Delta-\Delta_i)$, so the theorem states

$$\sum_{\Delta_i\le Y} C(\Delta_i)a_i \sim\frac{Y^{4\Delta_\phi}} {\Delta_\phi\Gamma(2\Delta_\phi)^2}.$$

It determines a cumulative weighted count. For any fixed $\eta>0$, subtraction of two cumulative equivalents also fixes a multiplicative window:

$$Q((1+\eta)Y)-Q(Y) \sim\frac{(1+\eta)^\gamma-1}{A\gamma}Y^\gamma.$$

It does not fix a window whose relative width shrinks with $Y$, and it does not imply the pointwise density

$$C(\Delta)p(\Delta)\stackrel{?}{\sim}A^{-1}\Delta^{\gamma-1}.$$

Delta functions already make that pointwise statement meaningless for a generic discrete CFT. Additional regularity, spacing, or remainder hypotheses are needed for local spectral information.

OPE convergence is what permits the positive block expansion to be used away from coincident points; its exponential tail bound is a separate result from the Tauberian asymptotic Pappadopulo et al. 2012, §§3–5.

Hypotheses and claim limits

Method	Positivity or analyticity input	Transform and limiting variable	Arc, subtraction, or smoothing	Supported domain	Uncontrolled data
Hardy–Littlewood for powers	Nonnegative primary-plus-descendant coefficients	Laplace variable $t=\lvert\log z\rvert\to0$	Sharp cumulative count obtained from a positive exponential kernel	Integrated descendant-weighted density	Individual degeneracies and shrinking windows
Conformal-block Tauberian theorem	$p(\Delta)\ge0$ and crossed identity dominance with a gap	Bessel kernel with $x\to0$, $x\Delta^2=O(1)$	Smooth Bessel average replaced by a sharp cutoff	$Q(Y)$ with the exact leading prefactor	Pointwise primary density and generic subleading terms
Lightcone application in $d>2$	Positive collinear spectral measure in the chosen reflection-positive configuration	$1-\bar z\to0$ and large spin at bounded twist	Fixed-twist or smeared spectral sector must be isolated	Averaged large-spin OPE weight	An individual trajectory without extra analytic input
Lorentzian inversion	Declared double discontinuity and Regge bound, not a Tauberian positivity theorem	Complex spin and spectral dimension	Arc vanishes only above a spin threshold	Meromorphic data above that threshold	Low spins and zero-discontinuity terms
Subtracted dispersion	Cut-plane analyticity plus polynomial growth	Correlator cross-ratios	Explicit subtraction/contact basis	Correlator modulo supplied constants	Contact coefficients not fixed by cuts

This comparison matters: inversion can establish individual analytic trajectories under stronger Lorentzian assumptions, while a Tauberian theorem gives a robust averaged result from positivity. One conclusion cannot be silently promoted into the other.

Generalized-free check

Generalized-free bosonic data have

$$\Delta_n=2\Delta_\phi+2n,$$

and known positive OPE coefficients. At large $\Delta$ their weighted density is a comb with spacing two. Summing the comb up to $Y$ reproduces the power and prefactor of $Q(Y)$ above, whereas the unsmeared density continues to oscillate between delta-function support and zero. This simultaneously checks the theorem and displays why the conclusion must be cumulative Qiao and Rychkov 2017, app. B.

Remainders and subleading singularities

If

$$\mathcal G(1-x)=x^{-2\Delta_\phi} \bigl(1+O(x^\alpha)\bigr),$$

it is tempting to append an $O(Y^{-2\alpha})$ relative error to $Q(Y)$. That conclusion does not follow from the leading Tauberian theorem. Subleading terms can mix with discreteness and oscillatory spectral structure; a quantitative remainder theorem needs explicit hypotheses on the transform, kernel, and smoothing window. Slowly varying factors can sometimes be carried through by an appropriate generalized theorem, but they must be stated rather than inferred by pattern matching.

Exercises

Take $\Delta_\phi=1/2$. Compute $\gamma$, $A$, and the leading weighted cumulative density $Q(Y)$. Then state what the theorem predicts for the fixed multiplicative window $[Y,2Y]$.

Solution

Here $\gamma=2$ and $A=\Gamma(1)^2/4=1/4$. Therefore $Q(Y)\sim Y^2/(A\gamma)=2Y^2$. The window contains weighted mass $Q(2Y)-Q(Y)\sim2(4-1)Y^2=6Y^2$. The theorem says nothing pointwise about an individual coefficient inside the window.

References

• Pappadopulo, Duccio, Slava Rychkov, Johnny Espin, and Riccardo Rattazzi. “OPE Convergence in Conformal Field Theory.” Physical Review D 86, no. 10 (2012): 105043. doi:10.1103/PhysRevD.86.105043.
• Qiao, Jiaxin, and Slava Rychkov. “A Tauberian Theorem for the Conformal Bootstrap.” Journal of High Energy Physics 2017, no. 12 (2017): 119. doi:10.1007/JHEP12(2017)119.