Amplitudes and observables in pp elastic scattering at $\sqrt{s}=7\ \mbox{TeV}$

Abstract

A precise analysis of the pp elastic scattering data at 7 TeV in terms of its amplitudes is performed as an extension of previous studies for lower energies. Slopes B _R and B _I of the real and imaginary amplitudes are independent quantities, and a proper expression for the Coulomb phase is used. The real and imaginary amplitudes are fully disentangled, consistently with forward dispersion relations for amplitudes and for slopes. We present analytic expressions for the amplitudes that cover all t range completely, while values of total cross section σ, ratio ρ, B _I , and B _R enter consistently to describe forward scattering. It is stressed that the identification of the amplitudes is an essential step for the description of elastic scattering, and pointed out the importance of the experimental investigation of the transition range from non-perturbative to perturbative dynamics, which may confirm the three gluon exchange mechanism observed at lower energies.

Appendix: The Coulomb phase

Here we present an expression for the Coulomb interference phase appropriate for forward scattering amplitudes with B _R ≠B _I [22].

The starting point is the expression for the phase obtained by West and Yennie [23]

(A.1)

where the signs (−/+) are applied to the choices $\mathrm{pp}/\mathrm{p}\bar{\mathrm{p}}$ respectively. The quantity p is the proton momentum in center of mass system, and at high energies 4p ²≈s.

For small |t|, assuming that F ^N(s,t′) keeps the same form for large |t′| (this approximation should not have practical importance for the results), we have

(A.2)

where

$$ c\equiv\rho e^{(B_{R}-B_{I})t/2}. $$

(A.3)

The integrals that appear in the evaluation of Eq. (A.1) are reduced to the form [24, 25]

$$ I(B)=\int_{-4p^{2}}^{0}\frac{dt^{\prime}}{|t^{\prime}-t|}\bigl[1-e^{B(t^{\prime}-t)/2} \bigr], $$

(A.4)

that is solved in terms of exponential integrals [26] as

(A.5)

The real and imaginary parts of the phase are then written

$$ \varPhi_{R}(s,t) =(-/+) \biggl[\ln \biggl(- \frac{t}{s} \biggr) + \frac{1}{c^{2}+1} \bigl[ c^{2} I(B_{R})+I(B_{I}) \bigr] \biggr], $$

(A.6)

and

$$ \varPhi_{I}(s,t) =(-/+) \frac{c}{c^{2}+1} \bigl[ I(B_{I})-I(B_{R}) \bigr] . $$

(A.7)

With σ in mb and t in GeV², the practical expression for dσ/dt in terms of the parameters σ, ρ, B _I , and B _R is

(A.8)

At high energies and small |t| we simplify 4p ²+t→s and then the functional form of I(B) is written

(A.9)

For large s, the term E ₁(Bs/2) can be neglected, and the phase becomes insensitive to s.

The usual expression from West and Yennie

$$ \alpha\varPhi_{WY}=(-/+)\alpha \biggl[\gamma+\ln \biggl(- \frac{Bt}{2} \biggr) \biggr] $$

(A.10)

can be obtained from Eq. (A.6) with B _R =B _I =B and using the low t behavior

$$E_{i} \biggl[-\frac{Bt}{2} \biggr] \approx\gamma+ \log- \frac{Bt}{2} . $$

We recall that the value of the slope B usually taken from the experimental dσ/dt data is the average given by Eq. (20).

We obtain [22] that the values of αΦ _I (s,t) are very small so that the imaginary part of the phase can be safely put equal do zero.

The construction of the Coulomb phase has been studied in the eikonal formalism appropriate for the interference of Coulomb and nuclear interactions [24, 25, 27, 28] with special attention given to the influence of the proton electromagnetic form factor. These treatments keep the assumption that the real part of the nuclear amplitude has the same slope as the imaginary part, and the results are not very different from the West Yennie formula. Some of these results have been tested against the data [29].

Figure 12 shows the values of the Coulomb interference phase in the kinematical conditions of the 7 TeV LHC experiment. We show together the phase obtained with the expression [27]

(A.11)

where the proton form factor has been used with exponential form, Λ ²=0.71 GeV², and B is taken as equal to B _I .

Fig. 12

Values of the Coulomb interference phase evaluated in the kinematical conditions of pp elastic scattering at 7 TeV, with account made for the difference in the slopes of the real and imaginary amplitudes. For comparison we show the phase obtained with the basic formula [27] of Eq. (A.11)