Diffraction at the LHC

Abstract

We show that the diffractive pp (and $p\bar{p}$) data (on σ _tot, dσ _el/dt, proton dissociation into low-mass systems, $\sigma^{\mathrm{D}}_{\mathrm{low}M}$, and high-mass dissociation, dσ/d(Δη)) in a wide energy range from CERN-ISR to LHC energies, may be described in a two-channel eikonal model with only one ‘effective’ pomeron. By allowing the pomeron coupling to the diffractive eigenstates to depend on the collider energy (as is expected theoretically) we are able to explain the low value of $\sigma^{\mathrm{D}}_{\mathrm{low}M}$ measured at the LHC. We calculate the survival probability, S ², of a rapidity gap to survive ‘soft rescattering’. We emphasise that the values found for S ² are particularly sensitive to the detailed structure of the diffractive eigenstates.

Appendices

Appendix A: Observables in terms of GW eigenstates

In the Good–Walker approach [14], low-mass diffractive dissociation is described in terms of so-called diffractive (or GW) eigenstates, |ϕ _i 〉 with i=1,n, that diagonalise the T-matrix, and so only undergo ‘elastic’ scattering. On the other hand high-mass dissociation is described in terms of multi-pomeron diagrams. We discuss our treatment of high-mass dissociation in Appendix B.

In this appendix we recall the GW formalism. First, the incoming ‘beam’ proton wave function is written as a superposition of the diffractive eigenstates:

$$ |p\rangle= \sum a_i |\phi_i\rangle, $$

(19)

and similarly for the incoming ‘target’ proton. In this paper we use two diffractive eigenstates, i=1,2. In terms of this 2-channel eikonal model, the pp elastic cross section has the form

$$\begin{aligned} &{\frac{d\sigma_{\mathrm{el}}}{dt} = \frac{1}{4\pi} \biggl \vert \int d^2b\,e^{i \boldsymbol {q}_t \cdot \boldsymbol {b}} \sum_{i,k}|a_i|^2 |a_k|^2\bigl(1-e^{-\varOmega _{ik}(b)/2}\bigr) \biggr \vert ^2,} \\ \end{aligned}$$

(20)

where $-t=q_{t}^{2}$, and the opacity Ω _ik (b) corresponds to one-pomeron exchange between states ϕ _i and ϕ _k written in the b-representation. Also we have

$$\begin{aligned} &{\sigma_{\mathrm{el}} = \int d^2b \,\biggl \vert \sum_{i,k}|a_i|^2 |a_k|^2 \bigl(1-e^{-\varOmega_{ik}(b)/2}\bigr) \biggr \vert ^2,} \end{aligned}$$

(21)

$$\begin{aligned} &{\sigma_{\mathrm{tot}} = 2\int d^2b\,\sum _{i,k}|a_i|^2 |a_k|^2 \bigl(1-e^{-\varOmega _{ik}(b)/2}\bigr)} \end{aligned}$$

(22)

and the ‘total’ low-mass diffractive cross section

$$ \sigma_{\rm el+SD+DD} = \int d^2b ~\sum _{i,k}|a_i|^2 |a_k|^2 \bigl \vert \bigl(1-e^{-\varOmega_{ik}(b)/2}\bigr) \bigr \vert ^2, $$

(23)

where SD includes the single dissociation of both protons. So the low-mass diffractive dissociation cross section is

$$ \sigma^{\rm D}_{\mathrm{low}M} = \sigma_{\rm el+SD+DD}- \sigma_{\mathrm{el}}, $$

(24)

where $\sigma_{\rm el+SD+DD}$ corresponds to all possible low-mass dissociation caused by the dispersion of the Good–Walker eigenstate scattering amplitudes. As mentioned in footnote 2, this corresponds to M _diss∼2–3 GeV for the TOTEM data [1–4].

Appendix B: Formulae for high-mass dissociation

The process pp→X+p, where one proton dissociates into a system X of high-mass M is conventionally studied in terms of the triple-pomeron coupling, shown as the dot between the dashed lines in Fig. 6(a). In the absence of absorptive corrections, the corresponding cross section is given by

$$\begin{aligned} &{\frac{M^2 \,d\sigma}{dt\,dM^2}} \\ &{\quad = g_{3P}(t)\beta(0)\beta^2(t) \biggl( \frac {s}{M^2} \biggr)^{2\alpha(t)-2} \biggl(\frac{M^2}{s_0} \biggr)^{\alpha(0)-1},} \end{aligned}$$

(25)

where β(t) is the coupling of the pomeron to the proton and g _3P (t) is the triple-pomeron coupling. The coupling g _3P is obtained from a fit to lower energy data. Mainly it is the data on proton dissociation taken at the CERN-ISR with energies from 23.5→62.5 GeV.

Fig. 6

(a) A schematic diagram showing the notation of the impact parameters arising in the calculation of the screening corrections to the triple-pomeron contributions to the cross section; (b) a symbolic diagram of multi-pomeron effects

The problem, in the above determination of g _3P , is that this is an effective vertex with coupling

$$ g_{\mathrm{eff}} = g_{3P}*S^2 $$

(26)

which already includes the suppression S ²—the probability that no other secondaries, simultaneously produced in the same pp interaction, populate the rapidity gap region. Recall that the survival factor S ² depends on the energy of the collider. Since the opacity Ω increases with energy, the number of multiple interactions, N∝Ω, grows,⁶ leading to a smaller S ². Thus, we have to expect that the naive triple-pomeron formula with the coupling [20, 55], measured at relatively low collider energies will appreciably overestimate the cross section for high-mass dissociation at the LHC. A more precise analysis [34] accounts for the survival effect $S^{2}_{\mathrm{eik}}$ caused by the eikonal rescattering of the fast ‘beam’ and ‘target’ partons. In this way, a coupling g _3P about a factor of 3 larger than g _eff is obtained, namely g _3P ≃0.2g _N , where g _N is the coupling of the pomeron to the proton. The analysis of Ref. [34] enables us to better take account of the energy dependence of $S^{2}_{\mathrm{eik}}$.

To account for the absorptive effect, it is easier to work in the impact parameter, b, representation. To do this we follow the procedure of Ref. [34]. We first take Fourier transforms with respect to the impact parameters specified in Fig. 6(a). Then (25) becomes

$$\begin{aligned} \frac{M^2 \,d\sigma_{ik}}{dt\,dM^2} =& A\int\frac{d^2b_2}{2\pi}e^{i\boldsymbol {q}_t \cdot \boldsymbol {b}_2} F_i(b_2)\int\frac{d^2b_3}{2\pi}e^{i\boldsymbol {q}_t \cdot \boldsymbol {b}_3} F_i(b_3) \\ &{}\times \int\frac{d^2b_1}{2\pi} F_k(b_1), \end{aligned}$$

(27)

where F _i (b) is described by the opacity corresponding to the interaction of eigenstate ϕ _i with an intermediate parton placed at the position of the triple-pomeron vertex, while F _k (b) describes the opacity of eigenstate ϕ _k from the proton which dissociates and interacts with the same intermediate parton. After integrating (27) over t, the cross section becomes

$$\begin{aligned} &{\frac{M^2 \,d\sigma_{ik}}{dM^2}} \\ &{\quad = A\int\frac{d^2b_2}{\pi}\int\frac {d^2b_1}{2\pi} \big|F_i(b_2)\big|^2 F_k(b_1) \cdot S_{ik}^2(\boldsymbol {b}_2-\boldsymbol {b}_1),} \end{aligned}$$

(28)

where here we have included the screening correction $S_{ik}^{2}$, which depends on the separation in impact parameter space, (b ₂−b ₁), of states ϕ _i ,ϕ _k coming from the incoming protons

$$ S_{ik}^2(\boldsymbol {b}_2-\boldsymbol {b}_1)~ \equiv~\exp\bigl(-\varOmega_{ik}(\boldsymbol {b}_2- \boldsymbol {b}_1)\bigr). $$

(29)

If we now account for more complicated multi-pomeron vertices, coupling m to n pomerons, and assume an eikonal form of the vertex with coupling

$$ g^m_n=(g_N\lambda)^{m+n-2}, $$

(30)

then we have to replace F _i by the eikonal elastic amplitude and F _k by the inelastic interaction probability. That is, instead of F _i =Ω _i (b ₂) and F _k =Ω _k (b ₁), we put

$$ F_i \to2\bigl(1-e^{-\varOmega_i(b_2)/2}\bigr),\qquad F_k \to\bigl(1-e^{-\varOmega_k(b_1)}\bigr). $$

(31)

Figure 6(b) symbolically indicates multi-pomeron couplings. In (30), g _N is the proton-pomeron coupling and λ determines the strength of the triple-pomeron coupling.