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 imaginary E, so that upper complex--plane 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 limits the maximum precision of certain dispersive approaches to compositeness based on Cauchy's theorem leading to partial-wave dispersion relations. 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, one-dimensional form factors or the Adler function, to name a few, are not affected by this uncertainty. Likewise, fixed-t dispersion relations were cleverly constructed to avoid it and consequences therefrom are solid.
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Reports on this Submission
Report #2 by
Anonymous
(Referee 1) on 2021-9-19
(Invited Report)
Cite as: Anonymous, Report on arXiv:scipost_202010_00009v2, delivered 2021-09-19, doi: 10.21468/SciPost.Report.3547
Report
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. 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. 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. Second, section 2.2 on W_L scattering is probably unnecessary as there is no evidence whatsoever that they composite.
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Report #1 by
Anonymous
(Referee 2) on 2021-8-18
(Invited Report)
Report
Dear Editor, the authors have responded to my questions in a satisfactory way and changed the manuscript accordingly.
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. 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. 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. Second, section 2.2 on W_L scattering is probably unnecessary as there is no evidence whatsoever that they composite.
Anonymous on 2021-09-18 [id 1765]
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.
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.
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. Second, section 2.2 on W_L scattering is probably unnecessary as there is no evidence whatsoever that they composite.