Smearing of causality by compositeness divides dispersive approaches into exact ones and precision-limited ones

Submission summary

 As Contributors: Felipe J. Llanes-Estrada Preprint link: scipost_202010_00009v3 Date submitted: 2021-12-28 12:32 Submitted by: Llanes-Estrada, Felipe J. Submitted to: SciPost Physics Core Academic field: Physics Specialties: High-Energy Physics - Phenomenology Nuclear Physics - Theory Approaches: Theoretical, Computational, Phenomenological

Abstract

Scattering off the edge of a composite particle or finite--range interaction can precede that off its center. An effective theory treatment with pointlike particles and contact interactions must find that the scattered experimental wave is slightly advanced, in violation of causality (the fundamental underlying theory being causal). In practice, partial--wave or other projections of multivariate amplitudes exponentially grow with ${\rm Im}(E)$, so that analyticity is not sufficient to obtain a dispersion relation for them, but only for a slightly modified function (the modified relations additionally connect different $J$). This can limit the precision of certain dispersive approaches to compositeness based on Cauchy's theorem. Awareness of this may be of interest to some dispersive tests of the Standard Model with hadrons, and to unitarization methods used to extend electroweak effective theories. Interestingly, the Inverse Amplitude Method is safe (as the inverse amplitude has the opposite, convergent behavior allowing contour closure). Generically, one-dimensional sum rules such as for the photon vacuum polarization, form factors or the Adler function are not affected by this uncertainty; nor are fixed-$t$ dispersion relations, cleverly constructed to avoid it and whose consequences are solid.

Current status:
Editor-in-charge assigned

Dear Editor,
Thank you for forwarding the last rounds of referee comments. We regret not having responded earlier, it has been quite
a hectic trimester with the overburden of teaching under pandemic conditions and there was a lot of backlog to clear.
We should like to respond to these last observations as follows.

REFEREE: The manuscript in its present form shows a clear improvement with respect the version thta was previously reviewed. The authors have taken into account the criticisms and comments in my previous reports. As it stands the paper sends a word of caution when using multi-variable dispersion relations because the convergence in the upper half plane in integrals over
a complexified energy variable is not granted.

RESPONSE: We thank the referee for the appreciative comment.

REFEREE: As a shortcoming I would mention that, having blown the whistle, they do not present a convincing calculation or example showing the actual and detailed relevance of the effect. Instead they introduce a modified partial wave (without obvious physical meaning, as they state) but having a good behaviour in the upper half plane.
In the integrals considered in Eq. 11 no complex contour is required and indeed they work with a linearized version of the exponential that yields a good behaviour. So, altogether it is not clear to the reader the relevance of the whole study. The only clear message is that there may be a problem.

RESPONSE: We can see the referee's point. Indeed, because the modified partial wave has good convergence properties,
the contour at infinity is well-behaved and the integration along the real-axis only is granted.
This allows to estimate what separation from the actual dispersion relation for the physical partial wave
can be expected in the absence of other theoretical support, which can be found for certain
kinematic configurations in fixed-t dispersion relations, but not generally for all kinematics. Auxiliary
functions such as the one introduced allow to obtain partial-wave or fixed-angle dispersion relations which
can then be related to the physical scattering functions. The linearized version is offered for clarity, but
there is no obstacle to use the full exponential (numerically) as is the case in Eq.(13).
We have attempted to make these points clearer in the manuscript. For this, section 2 has been reorganized,
with a break just before Eq.(8) to consolidate this discussion at the beginning of a subsection,
and a paragraph has been added there.

REFEREE: Two last comments: In the introduction it is said that single variable dispersion relations are unafected in any case. Yet the authors explain in some detail the dispersion relation involved in g-2. I do not see the necessity of this digression.

RESPONSE: The community is immersed in reviewing every aspect of the g-2 computation as well as the experimental setup,
in view of the disagreement with the standard model recently sharpened by the FNAL g-2 data. In view of this effort,
it appears to us of nonnegligible interest to very quickly review the foundation of the corresponding dispersion
relation, explicitly showing that it is not a problematic one, since a lot of the theory effort hangs from it.

REFEREE: Second, section 2.2 on W_L scattering is probably unnecessary as there is no evidence whatsoever that they composite.

RESPONSE: Indeed, there is no current evidence of their compositeness, but there is ample work preparing for the eventuality
that such discovery would be made. We have added reference to a recent review by Dobado and Espriu on the topic.
In addition, when/if WLWL scattering data becomes available, it can be tested against a dispersion relation to
ascertain its possible composite or pointlike nature. The discussion has been slightly revised.
We think this is an important potential application of dispersion relations in high-energy physics that should not
be discounted.

We have reread the manuscript and made minor text improvements wherever it seemed appropriate.
With these changes, we hope the editorial now finds it acceptable.