Borel Singularities, Lateral Sums, and Stokes Data

A singularity on the Borel integration ray prevents a unique ordinary Laplace integral. Passing just above or below it defines two lateral sums whose difference is exponentially small in the coupling. That difference is a Stokes jump of the asymptotic representation; whether either lateral value, their median, or a larger transseries combination represents the physical observable is fixed by boundary conditions or an integration cycle, not by Borel analysis alone.

Required background. Large-order growth and the Borel transform fixes the coefficient, Borel, and Laplace conventions. Stokes jumps, saddle dominance, and contour dependence explains how a fixed integration cycle can acquire a discontinuous saddle decomposition.

Helpful background. Contour deformation, pinches, and causal prescriptions distinguishes a mathematical boundary value from a physically selected prescription.

Lateral sums around one Borel pole

Use the chapter convention

$$\widehat\Phi(\zeta)=\sum_{n\ge0}\frac{a_n}{n!}\zeta^n, \qquad \mathcal S_\theta\Phi(g) =\frac1g\int_0^{e^{i\theta}\infty} \mathrm d\zeta\,e^{-\zeta/g}\widehat\Phi(\zeta).$$

The minimal nonsummable model is

$$\widehat\Phi(\zeta)=\frac1{1-\zeta/A}, \qquad A>0.$$

Its Taylor coefficients give $a_n=n!A^{-n}$, so the perturbative series is nonalternating. For real $g>0$, define the upper and lower boundary values

$$\mathcal S_{0^\pm}\Phi(g) =\frac1g\int_0^{\infty e^{\pm i0}} \frac{e^{-\zeta/g}}{1-\zeta/A}\,\mathrm d\zeta.$$

On the real axis,

$$\frac1{1-(x\pm i0)/A} =A\,\operatorname{PV}\frac1{A-x} \pm i\pi A\,\delta(A-x).$$

Therefore

$$\begin{aligned} \mathcal S_{0^\pm}\Phi(g) &=\frac{A}{g}\operatorname{PV} \int_0^\infty\frac{e^{-x/g}}{A-x}\,\mathrm dx \pm i\pi\frac{A}{g}e^{-A/g},\\ \operatorname{Disc}_0\Phi(g) &\equiv \mathcal S_{0^+}\Phi(g)-\mathcal S_{0^-}\Phi(g) =2\pi i\frac{A}{g}e^{-A/g}. \end{aligned}$$

This fixes both the sign and the $A/g$ prefactor in our convention. A contour check gives the same result: the closed path formed by the upper ray and the reversed lower ray is clockwise, while the pole residue of $(1-\zeta/A)^{-1}$ is $-A$. For the numerical fixture $A=1$ and $g=0.1$, the jump magnitude is

$$\left|\operatorname{Disc}_0\Phi(0.1)\right| =\frac{2\pi}{0.1}e^{-10} =0.002852561716306300\ldots.$$

For a branch singularity, the same argument integrates the discontinuity of $\widehat\Phi$ along its cut. If

$$\operatorname{Disc}_A\widehat\Phi(A+s) \sim D\,s^{\alpha-1}, \qquad s>0,$$

then

$$\operatorname{Disc}_0\Phi(g) \sim D\,\Gamma(\alpha)g^{\alpha-1}e^{-A/g}.$$

The exponential weight records the singularity position; the power of $g$ records its local type. Mariño 2015, §3.2, pp. 85–89 develops this discontinuity calculation for poles and branch cuts.

Stokes data describe a change of representation

A Stokes ray is a direction in the small parameter for which a Borel singularity lies on the Laplace ray. Equivalently in a saddle problem, two exponential weights can have aligned phases. The Stokes data quantify how a resummed asymptotic basis changes across that direction. Their numerical value depends on how sectors are normalized; only a complete, convention-matched relation is meaningful.

Suppose a one-parameter transseries begins

$$\mathcal O(g;\sigma) =\Phi_0(g)+\sigma e^{-A/g}\Phi_1(g)+\cdots.$$

If the perturbative sector jumps by

$$\mathcal S_{0^+}\Phi_0 =\mathcal S_{0^-}\Phi_0 +S_{01}e^{-A/g}\mathcal S_{0^-}\Phi_1,$$

then the same resummed function can be represented on the other side by shifting the parameter,

$$\sigma_+=\sigma_- - S_{01}.$$

This is a basis change: the left-hand observable need not jump. The sign of $S_{01}$ changes if one reverses the definition of the discontinuity or rescales $\Phi_1$, so the convention must accompany the number.

The Borel and transseries map places this lateral step between Borel singularities and ambiguity-canceling sectors. The exact and rigorous status comparison prevents a solvable one-pole model from being mislabeled as a theorem about a general QFT.

Representation jump versus physical discontinuity

Three cases must be separated.

1. Stable real observable. If the exact quantity is real and continuous across the chosen parameter ray, lateral imaginary parts must cancel against other sectors or be removed by a justified median prescription.
2. Metastable state. An outgoing-wave or false-vacuum boundary condition may select one analytic continuation. Its imaginary part can encode a decay width, but the sign is fixed by that physical condition, not by calling one contour “upper.”
3. Actual phase boundary or branch cut. The exact observable itself may have different boundary values. Then a physical discontinuity remains after the representation has been assembled.

The thimble analogue is exact: a fixed cycle may have different integer decompositions into saddle cycles on two sides of a Stokes wall. The cycle is the physical datum; the decompositions are representations. A pole-skipping mnemonic that omits the cycle or boundary condition cannot settle the physics.

Several singularities and directional limitations

For singularities $A_k$ on different rays, each direction has its own lateral operation. The nearest singularity usually controls leading large order, but a vanishing Stokes coefficient can remove its contribution to a particular observable, and equal-modulus singularities must be summed with their phases. Singularities may also accumulate or form natural boundaries, defeating simple analytic continuation.

The one-pole derivation does not supply full alien calculus, identify every singularity, or decide field-theory renormalons. It provides a normalized local test: compute two boundary values, verify the jump, and then ask which independently derived sector can carry the opposite ambiguity.

Common pitfalls

The upper sign is not a universal physical sign. It follows from the declared contour and discontinuity convention. Decay, causality, or reality conditions must still select the physical combination.

A Stokes jump is not necessarily a jump in the observable. It may be a discontinuous change of asymptotic coordinates that leaves the exact function unchanged.

Principal value is not automatically the answer. It is real in the one-pole model, but a physical median or principal-value prescription requires the rest of the sector structure and boundary data.

Exercises

1. Repeat the one-pole calculation for $\widehat\Phi(\zeta)=C/(1-\zeta/A)$ and state how the result changes if $\operatorname{Disc}_0$ is defined in the reverse order.

Solution

Linearity gives

$$\operatorname{Disc}_0\Phi =2\pi i C\frac{A}{g}e^{-A/g}.$$

Defining the discontinuity as lower minus upper multiplies this expression and the associated Stokes constant by $-1$. No physical conclusion changes if every sector uses the same convention.

2. Let $A=|A|e^{i\varphi}$. For what coupling directions does the exponential $e^{-A/g}$ become maximally small and nonoscillatory, and when does the Borel singularity obstruct the Laplace ray?

Solution

Writing $g=|g|e^{i\theta}$ gives $A/g=(|A|/|g|)e^{i(\varphi-\theta)}$. It is positive real when $\theta=\varphi$, so the exponential is $e^{-|A|/|g|}$ and the singularity lies on the Laplace ray used for that direction. This is the Stokes direction in the present convention.

References

• Mariño, Marcos. Instantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory. Cambridge University Press, 2015. doi:10.1017/CBO9781107705968.