Why the Bethe-West-Yennie Formula for Coulomb-Nuclear Interference Is Inconsistent.

We give a new and simple proof of the inconsistency of the Bethe-West-Yennie parametrization for Coulomb-nuclear interference.

Introduction 65 years ago, in 1958, Bethe [1], when analyzing the influence of the Coulomb interaction on the pion-nucleus interaction (in a non-relativistic context), derived on the basis of the eikonal representation and a number of assumptions 1 , proposed the following parametrization for the full scattering amplitude of charged hadrons Here T N (s, t) stands for the pure strong interaction ("nuclear") scattering amplitude with usual Mandelstam variables s and t = −q 2 while T C is the Coulomb scattering amplitude (e.g., for pp scattering) with the e.m. form factor F , α is the standard notation for the fine structure constant.Since then, and especially after the modification of the phase φ by West and Yennie [2] and more detailed account for the e.m. form factor by Cahn [3], parametrization (1) has been unconditionally used in the vast majority of works, both experimental and theoretical, dealing with Coulombnuclear interference.The initial "Coulomb phase" derived by Bethe was essentially ( The West-Yennie modified this expression to and finally φ was refined by Cahn to the form with B(s) the elastic slope of the differential cross-section, C = 0.5772... the Euler constant and Λ the scale of the dipole form factor parametrization, It should be recognized that the BWY representation (1) is quite attractive due to its compactness and ease of practical use.Nonetheless, the Bethe-West-Yennie-type parameterization was later criticized in paper [4] (and subsequent similar papers of the Prague Group), as well as in paper [5].In particular, it was pointed out that the BWY parameterization is only valid if the phase of the nuclear amplitude T N (s, q2 ) , Arg(T N (s, q 2 )), does not depend on the momentum transfer 2 .
However, this criticism turned out to be not very impressive, and the use of the BWY-type parametrization continues until very recently, e.g., in recent publications of the ALFA/ATLAS group [7] and the STAR Collaboration [8].
It is not improbable that the argumentation in works [4], [5] might seem either unconvincing or too complicated.This prompts us to give a new and simple proof of the inconsistency of the BWY-type parametrization.
Specifically, below we will provide a proof that parameterization (1) as an equation relative to the value of ϕ has no solution.

The Proof
In order not to clutter up our derivation with fundamentally unimportant but cumbersome details, we will consider the case when the transfers are so small that we can neglect the value of q 2 wherever this does not lead to singularities and also we take F = 1 (as was initially assumed in BWY).Incidentally, we note that typical transfers, where Coulomb-nuclear interference is already significant, are of the order of 10 −3 GeV 2 .We will also take as the nuclear amplitude T N the "popular" expression used also in [7] T N (s, q 2 ) = (ρ(s) + i)σ tot (s)e −B(s)q 2 /2 (5) where ρ(s) = ReT N (s, 0)/ImT N (s, 0) and σ tot (s) is the pp total cross-section in GeV −2 .To proceed further we have to deal not with the very amplitudes T C+N which suffer from an IR divergency residing in the phase but with the moduli squared | T C+N | 2 which are free from these singularities 3 .So, on the one hand, the squaring of the modulus of the amplitude (1) gives On the other hand, the general expression to all orders in α reads [9] where z = Bq 2 /2 and 1 F 1 (iα, 1; z) is one of the confluent hypergeometric functions 4 (see Chapter 9.21 in [10].)Certainly, a direct extraction of φ from Eqs.( 6) and ( 7) is inconceivable.So, we follow Fermi's advise : "When in doubt, expand in a power series", do expand both Eqs( 6) and (7) in series in terms of the fine structure constant α and then equate the coefficients at the same powers of α .If the formula (1) is correct then extracting φ from the first, second etc, orders we have to obtain the same value.At small q 2 and T N as in (5) the equation r.h.s(6) = r.h.s.(7) simplifies to cos(αφ where with L = ln 2 Bq 2 .This exactly coincides with the West-Yennie expression (2) [2].However, the comparison of the second order is already disturbing: instead of just (L − C) 2 .The result of investigation of the third order looks even weirder: In other words, Eqs.( 9) - (11) show that Eq.( 8) (equivalent to Eq.( 1)) as an equation for the quantity φ independent of α has no solution.

Conclusion
Thus, we see that the use of the Bethe-West-Yennie formula (1) leads to a fatal ambiguity of the parameter φ and this conclusion still holds with a non-trivial form factor either.This means that the equation for the BWY phase has no reasonable solution at least if the nuclear amplitude T N is of the form (5).
It is worth noting that the correct general formula for taking into account the Coulomb-nuclear interference and with no ambiguities is given in papers [9] and [11].