Massless Exchange and Infrared Subtractions

Massless exchange changes both ends of a positivity proof. A $t$-channel pole makes the forward amplitude divergent, while massless loops can move branch points to $t=0$ and destroy the analytic neighborhood needed for angular derivatives. One may regulate, subtract, and take limits only after verifying that analyticity, growth, crossing, and positivity survive those operations.

Required background. Forward-limit positivity bounds supplies the gapped theorem whose assumptions are being tested. Helpful background. Soft and collinear singularities explains why exclusive amplitudes with massless quanta can require inclusive or dressed observables.

The forward point can cease to exist

Exchange of a massless spin-$J$ particle has the schematic form

$$A_{\mathrm{pole}}(s,t)\sim\frac{R_J(s)}{t}, \qquad R_J(s)\sim s^J$$

at large $s$ and fixed $t$. For a graviton, $J=2$, so the residue itself grows like the power targeted by a twice-subtracted forward relation. At loop level, massless thresholds also generate terms such as

$$A_{\mathrm{loop}}(s,t)\supset f(s)\log(-t/\mu^2).$$

Therefore $A(s,0)$ and its $t$ derivatives need not exist. More importantly, the $t=0$ point is no longer surrounded by a disk free of singularities. The analytic continuation used to prove $\partial_t^n\operatorname{Im}A(s,0)\ge0$ can fail before any inequality is written. Alberte and collaborators explain why the spin-2 pole and its associated branch point obstruct the standard argument and why simply deleting the pole can conflict with the assumed subtraction count Alberte et al. 2020, § 1, p. 2, PDF.

This is not repaired by observing that gravity is weak at low energy. A long-range pole is singular at arbitrarily small momentum transfer even when its residue is Planck suppressed.

A regulated subtraction procedure

A defensible calculation separates four questions.

1. Regulate the infrared. Work at fixed $t=-\tau<0$, introduce a small mass $m_{\mathrm{IR}}$, or use another regulator whose effect on crossing and unitarity is known.

2. Write the regulated dispersion relation. Determine its cuts, poles, and large-$|s|$ behavior before setting the regulator to zero.

3. Subtract only known singular terms. Define

   $$B_{\mathrm{reg}}(s,t;m_{\mathrm{IR}}) =A(s,t;m_{\mathrm{IR}})-A_{\mathrm{known\ pole}}(s,t;m_{\mathrm{IR}}),$$

   and recheck the asymptotic bound for $B_{\mathrm{reg}}$. Pole subtraction changes the function whose large arc is being discarded.

4. Take limits with a uniform estimate. Show that the dispersive integral, derivatives, and regulator limit may be interchanged. Otherwise retain the order of limits as part of the result.

The potentially different operations include

$$\begin{aligned} &\lim_{t\to0^-}\partial_s^2 B_{\mathrm{reg}}(s,t), &&\partial_s^2\lim_{t\to0^-} B_{\mathrm{reg}}(s,t),\\ &\lim_{m_{\mathrm{IR}}\to0}\lim_{t\to0^-} \partial_s^2B_{\mathrm{reg}}(s,t;m_{\mathrm{IR}}), &&\lim_{t\to0^-}\lim_{m_{\mathrm{IR}}\to0} \partial_s^2B_{\mathrm{reg}}(s,t;m_{\mathrm{IR}}). \end{aligned}$$

A $\log(-t)$ term is already enough to make the first two operations undefined in general. Even when a further subtraction produces finite expressions, exchanging differentiation and either limit requires a uniform estimate. A small-mass regulator restores a gap at every nonzero mass, but the constants in the analyticity and growth bounds may diverge as $m_{\mathrm{IR}}\to0$.

What may survive

Infrared structure	Possible conclusion	Required extra work
isolated massless pole with a regular remainder	a pole-subtracted finite-$t$ relation	prove the remainder’s growth and the uniform forward limit
massless loop cut reaching $t=0$	a regulated or finite-$t$ relation	keep logarithms and show how inclusive/dressed quantities enter
massless charged or colored external states with uncancelled soft or collinear singularities	an infrared-safe observable may exist, but no amplitude bound follows automatically	specify the soft prescription and separately prove analyticity, crossing, growth, and a nonnegative discontinuity for any replacement object
graviton exchange	at most a conditional modified bound	state Regge, compactification, impact-parameter, or other additional assumptions
no uniform regulator limit	no standard forward positivity statement	remain at finite regulator or finite $t$

There are proposed gravitational extensions under additional high-energy assumptions. For example, a Regge ansatz can be arranged so that the $1/t$ and $\log t$ divergences cancel in a particular construction, but the Regge behavior is an extra input and fixes part of the conclusion Herrero-Valea, Santos-Garcia, and Tokareva 2021, §§ II–VI, pp. 2–7, PDF. This does not establish a universal pole-subtracted gravitational positivity theorem. The literature contains explicit cautions and different conditional schemes, so any gravitational bound must identify which one is being used.

A later string-inspired loop calculation gives a controlled example in which small-$|t|$ Regge data remain infrared-sensitive and high-energy contributions cancel inverse-mass-enhanced negative low-energy terms. It explicitly finds a sign-indefinite correction rather than a proof of $c_2(0)\ge0$, reinforcing that Regge input does not restore the elementary positive-measure argument by itself Caron-Huot and Tokuda 2024, §§ 1–2.1, pp. 1–6, PDF.

Pole subtraction is not automatically harmless

For a spin-$J<2$ exchange, subtracting $R_J(s)/t$ may be compatible with a twice-subtracted asymptotic bound, but compatibility still has to be checked. For $J=2$, the subtraction removes a term of order $s^2/t$. If the original derivation assumes $A(s,t)/s^2\to0$, the pole term does not satisfy that falloff at fixed nonzero $t$; deleting it and then imposing the same bound on the remainder is a new assertion about UV cancellations.

Three signs must also remain distinct:

• the sign of a pole residue fixed by factorization;
• the sign of a continuum discontinuity fixed by unitarity in a physical channel;
• the sign of a pole-subtracted low-energy coefficient.

Only the second is the direct positive measure in the standard proof. A subtraction can move finite pieces between the first and third without changing the physical amplitude.

Practical stopping rule

Do not quote a forward positivity inequality unless all of the following have answers:

• What is the infrared-safe object: exclusive amplitude, inclusive rate, dressed amplitude, or hard function?
• Which regulator is used, and in what order are $t\to0$, $m_{\mathrm{IR}}\to0$, and differentiation taken?
• Which poles and logarithms are subtracted, with what factorization convention?
• Does the subtracted object obey the same complex-energy growth bound?
• Is the remaining discontinuity a nonnegative physical state sum?
• Which additional assumption controls gravity or another long-range force?

If any answer is missing, the safe result is the regulated dispersion relation plus an explicit unresolved limit—not a positivity bound.

Exercises

Consider $A(s,t)=\kappa^2s^2/t+c\,s^2+\beta s^2\log(-t/\mu^2)$. After subtracting the pole, does $\partial_s^2A$ have a finite $t\to0^-$ limit?

Check

No. Pole subtraction leaves $2c+2\beta\log(-t/\mu^2)$, which diverges unless $\beta=0$ or a further, convention-explicit infrared subtraction is supplied. The existence and sign of that further-subtracted object require a new argument.

Return to finite momentum transfer: Fixed-t and Partial-Wave Dispersion. For a gapped EFT application: EFT Positivity and UV Consistency.

References

• Alberte, Lasma, Claudia de Rham, Sumer Jaitly, and Andrew J. Tolley. “Positivity Bounds and the Massless Spin-2 Pole.” Physical Review D 102 (2020): 125023. DOI. Open PDF.
• Caron-Huot, Simon, and Junsei Tokuda. “String Loops and Gravitational Positivity Bounds: Imprint of Light Particles at High Energies.” Journal of High Energy Physics 11 (2024): 055. DOI. Open PDF.
• Herrero-Valea, Mario, Raquel Santos-Garcia, and Anna Tokareva. “Massless Positivity in Graviton Exchange.” Physical Review D 104 (2021): 085022. DOI. Open PDF.