Fixed-t and Partial-Wave Dispersion

At nonzero momentum transfer, a fixed-$t$ dispersion relation retains angular information, while partial-wave unitarity organizes the absorptive part into nonnegative components. The useful positivity statement is local in a restricted analytic neighborhood of $t=0$; it is not a claim that the amplitude is positive for arbitrary physical angle.

Required background. Subtracted dispersion relations supplies the fixed-variable contour and pole terms. Partial-wave unitarity supplies the absorptive inequalities for each angular-momentum channel.

Fixed-t dispersion in a crossing variable

For identical scalars of mass $m$, remove stable exchange poles and define

$$v=s-2m^2+\frac{t}{2}, \qquad u=4m^2-s-t, \qquad B(v,t)=B(-v,t).$$

Assume that, for every $t$ in an interval $0\le t<t_{\max}$, the pole-subtracted amplitude is analytic in $s$ away from the two unitarity cuts and obeys $B(v,t)/v^2\to0$ on the large arc. With

$$\omega(\mu,t)=\mu-2m^2+\frac{t}{2},$$

the two cuts combine into

$$B(v,t)=b_0(t)+\frac{2v^2}{\pi} \int_{4m^2}^{\infty} \frac{\operatorname{Im}A(\mu+i0,t)} {\omega(\mu,t)\,[\omega(\mu,t)^2-v^2]} \,\mathrm d\mu.$$

Crossing has removed the term linear in $v$; it has not determined $b_0(t)$. The formula is meaningful only when the $s$-plane contour and the analytic continuation in $t$ are simultaneously valid. Work on a declared compact interval $0\le t\le T<t_*$, where $t_*$ is the first $t$-channel singularity after explicit pole removal. In the simplest equal-mass benchmark $t_*$ can be $4m^2$, but the domain can be smaller in a more complicated theory. Positive $t$ is an unphysical analytic continuation from the physical range $t\le0$, not a new physical scattering angle. The assumptions and the twice-subtracted kernel are set out explicitly in Tolley, Wang, and Zhou 2021, § 2, pp. 4–6, PDF.

The crossed cut is not an optional second copy. Starting from a left-cut point $s'$, set $u'=4m^2-s'-t$ and reverse the endpoints. Only after this change of variable does the even denominator $\omega^2-v^2$ appear.

Partial waves make angular derivatives positive

For physical $s>4m^2$, introduce

$$z=\cos\theta=1+\frac{2t}{s-4m^2}.$$

Use the same four-dimensional normalization as on the prerequisite page,

$$\rho(s)=\sqrt{1-\frac{4m^2}{s}}, \qquad A(s,t)=16\pi \sum_{\ell=0}^{\infty}(2\ell+1)a_\ell(s)P_\ell(z).$$

Unitarity gives

$$\operatorname{Im}a_\ell(s) \ge \rho(s)|a_\ell(s)|^2\ge0.$$

At $t=0$, $z=1$, and

$$P_\ell^{(n)}(1) =\frac{(\ell+n)!}{2^n n!(\ell-n)!}>0 \qquad (\ell\ge n).$$

Since $\partial_tz=2/(s-4m^2)$, wherever termwise differentiation is justified,

$$\left.\partial_t^n\operatorname{Im}A(s,t)\right|_{t=0} =16\pi\left(\frac{2}{s-4m^2}\right)^n \sum_{\ell=n}^{\infty}(2\ell+1) \operatorname{Im}a_\ell(s)P_\ell^{(n)}(1) \ge0.$$

For an interacting channel the inequality is strict when some contributing partial wave has $\ell\ge n$. This argument continues $t$ from the physical range $t\le0$ into a small positive interval, where $z\ge1$. The continuation requires analyticity; partial-wave unitarity alone does not provide it. The scalar derivation and its mass-gap limitation are given in de Rham et al. 2017, pp. 1–4, PDF.

What a t derivative actually differentiates

Applying $\partial_t^M$ to the dispersive relation differentiates three objects:

$$\operatorname{Im}A(\mu,t), \qquad \omega(\mu,t), \qquad b_0(t).$$

Only the first has the immediately nonnegative partial-wave expansion. Derivatives of the kernel can carry either sign, and derivatives of $b_0$ are subtraction data. Beyond-forward positivity therefore uses carefully chosen linear combinations of $v$ and $t$ derivatives rather than declaring every mixed derivative positive.

The roles are:

Ingredient	What it controls
$\operatorname{Im}a_\ell\ge0$	sign of angular spectral components
$P_\ell^{(n)}(1)>0$	sign of forward angular derivatives
fixed-$t$ analyticity	permission to continue to and differentiate at $t=0$
crossing	conversion of the left cut and relations among low-energy coefficients
large-$	s

Domain and method checks

Check the expansion point. $v=0$ corresponds to $s=u=2m^2-t/2$, inside the Mandelstam triangle for sufficiently small positive $t$. Moving the point across a pole or cut invalidates its Taylor series.

Check the state sum. For multiple species or spin, $a_\ell$ is a matrix and crossing mixes channels. Positivity applies to appropriate elastic quadratic forms, not automatically to every component.

Check uniform convergence. Interchanging the $t$ derivative, partial-wave sum, and dispersive integral requires control stronger than pointwise positivity. Near a massless threshold this step can fail.

Check the forward limit. A $1/t$ exchange pole or a $\log(-t)$ cut destroys the analytic neighborhood used above. Pole subtraction and regulator removal then need a separate argument.

Exercises

Use the displayed formula for $P_\ell^{(n)}(1)$ to determine which partial waves can contribute to $\partial_t^2\operatorname{Im}A(s,0)$.

Check

$P_0''(1)=P_1''(1)=0$, while $P_\ell''(1)>0$ for $\ell\ge2$. Thus an amplitude supported only in $s$- and $p$-waves can have a positive forward absorptive part but a vanishing second $t$ derivative.

Continue to the derivative combinations: Beyond-Forward Positivity.

References

• de Rham, Claudia, Scott Melville, Andrew J. Tolley, and Shuang-Yong Zhou. “Positivity Bounds for Scalar Theories.” Physical Review D 96 (2017): 081702. DOI. Open PDF.
• Tolley, Andrew J., Zi-Yue Wang, and Shuang-Yong Zhou. “New Positivity Bounds from Full Crossing Symmetry.” Journal of High Energy Physics 05 (2021): 255. DOI. Open PDF.