Small t elastic scattering and the rho parameter

A simple application of Regge theory, with 9 free parameters, provides a good fit to elastic scattering data at small t from 13.76 GeV to 13 TeV. It yields a value for rho, the ratio of the real part of the forward amplitude to its imaginary part, close to 0.14 at 13 TeV. Although the exact value obtained for rho is sensitive to what functional form is chosen for the fit, there is no strong case for the presence of an odderon contribution to forward scattering.

For the reggeon exchanges Regge theory introduces a trajectory α(t) associated with Chew-Frautschi plots of the squares of masses of particles with the same quantum numbers but different spins. According to figure 2.13 of reference 2 the trajectories are found to be exchange-degenerate, α + (t) for f 2 , a 2 and α − (t) for ρ, ω, with to a good approximation α ± (t) = 1 + ǫ ± + α ± (t) (1a) where ǫ + = −0.3 α ′ + = 0.8 GeV −2 ǫ − = −0.56 α ′ − = 0.92 GeV −2 (1b) The trajectory for pomeron exchange is assumed similarly to be linear in t, with intercept 1 + ǫ I P and slope α ′ I P determined by the fit. Each of these exchanges contributes to the elastic amplitude, where and the signature factor according to whether the C-parity of the exchange is even or odd. The signature factor determines the complex phase of the contribution. For each exchange there is a real factor X i F (t) which is not determined by the theory. For simplicity we assume the same function F (t) for each Regge theory has had many successes over more than half a century, with the above exchanges giving the main contributions to a wide variety of reactions. But it is necessary also to take account of the double-exchange contributions IP IP, IP ρ, IP ω, . . . to the amplitude, and perhaps triple or more. For simplicity we include only the first of these A I P I P (s, t). The trajectory for this exchange is known [2] : α I P I P (t) = 1 + 2ǫ I P + 1 2 α ′ I P t (2a) However, while this determines the power of s at large s, and the complex phase of the term, that is all that is known about it. To construct a simple model, we introduce the eikonal function (see for example equation (2.49) of reference 2) A model that is sometimes used to calculate A I P I P (s, t) is to take χ(s, b) to include only the singleexchange contribution, which would give, if we omitted the term 1 2 t in (1d) (which is justified because we are working only at small t) Then the χ 2 term in (2c) would be the double-exchange contribution to the amplitude, However, if instead we were to choose χ(s, b) to include also a double-exchange contribution, when it is inserted into (2c) this would multiply (2f) by some constant C. Our fit finds that C should be close to 1 2 . The final contribution to the amplitude is the photon-exchange term Here G(t) is a squared form factor, equal to 1 at t = 0. Choosing the square of either the Dirac or the Pauli form factor, or a combination of them, gives almost the same result, because the term is negligbly small except at extremely small t. Similarly, including a West-Yennie phase factor [4] makes neglible difference.
So we have 8 free parameters in addition to C. The fits shown in figures 1 to 5 are with the choices As is seen in figure 5, the value 0.14 obtained for ρ at 13 TeV is rather different from that of at most 0.1 concluded by TOTEM from their data [5] . The reason why the curves rise to a maximum and then fall again as the energy increases is that the IP IP term becomes progressively more important.

Comments
1 With just 9 adjustable parameters, Regge theory provides a fit to data over a range of energies differing by a factor of 1000. The fit is extremely good, though less than perfect in some cases. There are also some anomalies in the data. An example is shown in figure 6: the data at 7, 8 and 13 TeV agree well with a single exponential in t, but the slope for the 7 TeV data lies between that for 8 and 13 TeV, which is surely anomalous.
2 In their extraction of ρ from their 13 TeV data, TOTEM assume [5] that the ratio of the real to the imaginary part of the hadronic amplitude is independent of t from 0 to −0.1 GeV 2 or more. Figure 7 shows how it varies with t for the fit described here.
3 TOTEM also extract ρ by using the data only at the one energy. This ignores information linking the phase of the amplitude to its variation with s: see the signature factors (1e) for example.

4
The fit yields ρ = 0.14 at 13 TeV, a value very different from that extracted by TOTEM. It should be recognised that its value extracted from data inevitably depends on just what functional form is used to fit the data. This is illustrated in figure 8, which shows that over a wide range of values of √ s the real part of log(−s) agrees very well with a power of s, but the corresponding imginary parts are somewhat different.
5 Those who fit total cross section and elastic scattering data often replace powers of s by log factors. In Regge theory this is unnatural: it would correspond to more than just a simple pole in the complex angular momentum plane. The excuse for including a log, or the square of a log, in the amplitude is often said to be to to saturate the Froissart-Lukaszuk-Martin bound [6] . However, the bound is about 20 barns at LHC energies and so is irrelevant.

Concluding remarks
The value 0.14 obtained for ρ does not encourage the belief [3] that there is an odderon contribution at t = 0. However, there is good reason to believe that there is an odderon contribution at large t and that it is identified with triple-gluon exchange. Indeed, this led us to predict [7] that pp andpp scattering would be different, as was confirmed [8] at the CERN ISR. We included such a term in a previous fit [9] . Note, however, that to lowest order in the strong coupling triple-gluon exchange's contribution to the pp amplitude is real positive, while of the TOTEM odderon would have negative real part at t = 0.
We are grateful to Professor J R Cudell for his comments in early stages of this work.