Black disk radius constrained by unitarity

We argue that if the elastic proton-proton cross section increases with energy, the Froissart-like high energy behaviour of the elastic amplitude (which corresponds to a `black disk' of radius $R(s)=c\ln s-\beta\ln(\ln s)$) is the only possibility to satisfy the unitarity equation at each value of the impact parameter, $b$. Otherwise the cross section of events with Large Rapidity Gaps grows faster than the total cross section at the same $b$. That is, these `gap' events require {\it maximal} growth of the high-energy (asymptotic) cross section and of the interaction radius $R(s)$ in order to be consistent with unitarity.


Introduction
It was shown long ago [1] that in the so-called 'strong coupling' regime (where the cross section increases with energy) the high energy, √ s, dependence of the total and elastic cross sections take the form with the function F chosen to describe the t dependence of the elastic cross section, with the "slope" B(s) = B 0 (ln s) γ , where the parameters η and γ are limited to the intervals 0 ≤ η ≤ γ and 0 ≤ γ ≤ 2.
Note that processes with Large Rapidity Gaps (LRG) were not considered specially in [1]. In a recent paper [2] we argued that when we account for events with LRG the only possibility to satisfy unitarity is to make the disk completely black. That is, in terms of (1,2) to put η = γ.
Moreover, when we consider the contribution of LRG events at the edge of disk (where the disk is not black but 'grey', i.e. partly transparent) we find that the radius of the disk must grow as R(s) ∝ (ln s) γ/2 ∝ ln s.
That is the only solution is η = γ = 2.
In section 2 we recall the main arguments of [2] in favour of black disk asymptotics. In section 3 we study LRG events at large impact parameter b. There we show that only in the case of γ = 2 (that is R(s) ∝ ln s) does there exist a possibility of screening an increasing LRG cross section in such a way that it does not exceed the total cross section at the same partial wave, that is at the same value of b.

Finkelstein-Kajantie problem
We first explain the problem. Then we present the solution of the problem and its implications for high-energy proton-proton scattering. Further implications are discussed in Section 3 when we study the behaviour at the edge of the disk.

Growth of inelastic cross section with large rapidity gaps
It was recognized already in the 1960s [3,4] that multi-Reggeon reactions, where small groups of particles (X i ), are separated from each other by Large Rapidity Gaps (denoted by + signs), may cause a problem with unitarity. Indeed, being summed over n and integrated over the rapidities of each group, the cross section of such quasi-diffractive production increases faster than a power of s. This was termed as the Finkelstein-Kajantie disease (FK) in the literature, see [5] for a review.
Let us explain the problem using the simple example of Central Exclusive Production (CEP) of a proton-antiproton pair, as shown in Fig.1. Since the proton-proton elastic cross section does not vanish, but increases with energy as σ el ∝ (ln s) 2η−γ , the corresponding contribution to the inelastic cross section reads where the elastic amplitude A(y, t) is normalized in such a way that dt|A(y, t)| 2 = σ el (y), and the upper rapidity Y = ln s/m 2 p where m p is the proton mass. The vertex V describes the central production of a pp-pair. In other words we find In the case of a black (or grey) disk of increasing radius when η = γ this leads to The same result can be obtained in impact parameter, b, space (see [2] for details). Moreover working in b space we have a stronger constraint since for each value of b, that is for each partial wave l = b √ s/2 of the incoming proton pair, we have the unitarity equation and the 'total' cross section, σ(b) tot must be less than the corresponding CEP contribution (here G inel denotes the total contribution of all the inelastic channels). Actually one will face this FK problem in any model where the elastic cross section does not decrease with energy.
At first sight the simplest way to avoid the FK problem is to say that the production vertex (V in Fig.1) vanishes. However this cannot be fulfilled. Indeed, as far as we have a nonvanishing high-energy elastic proton-proton cross section we can build diagram Fig.1 in such a way that the lower part is just the elastic pp-scattering while the upper part corresponds to the proton-antiproton elastic interaction. Such a diagram is generated by the t-channel unitarity equation where in our case j| is the t-channel p state.
Note that this contribution is singular at t = m 2 p (where m p is the proton mass). There are no other similar terms corresponding to central exclusive production of pp pair with the same pole singularity. That is, the vertex V contains at least one subprocess (pp CEP) which cannot be cancelled identically. See [2] for more details why this establishes that V = 0.

Black disk solution of the FK problem
The only known solution of this multi-Reggeon problem comes from 'black disk' asymptotics of the high-energy cross sections. In such a case the (gap) survival probability, S 2 , of the events with LRGs tends to zero as s → ∞ and the value of σ CEP does not exceed σ tot . In other words besides the contribution of Fig.1 we have to consider the diagram of Fig.2a where the double dotted line denotes an additional (incoming) proton-proton interaction. This diagram describes the absorptive correction to the original CEP process and has a negative sign with respect to the amplitude A (1) of Fig.1. Therefore to calculate the CEP cross section we have to square the full amplitude where the survival factor and Ω(b) is the opacity of the incoming protons.
Indeed, in terms of S-matrix, the elastic component for a partial wave l = b √ s/2 has the form S l = 1 + iA(b), and the unitarity equation (8) reflects the probability conservation condition n S * l |n n|S l = 1 , where S l is the component of the S-matrix corresponding to partial wave l. The solution of unitarity equation (8) reads In terms of the partial wave amplitude a l with orbital moment l = b √ s/2 the solution is where η l = e −2Imδ l with 0 ≤ η l ≤ 1.
The above discussion shows that −Ω(b)/2 plays the role of 2iδ l . The elastic component of S matrix The gap survival factor S 2 is the probability to observe a pure CEP event where the LRG is not populated by secondaries produced in an additional inelastic interaction shown by the dotted line in Fig.2a. That is according to (15) Equation (17) can be rewritten as (see (13,15)) In the case of black disk asymptotics 1 for b < R. That is, we get S 2 (b) → 0. The decrease of the gap survival probability S 2 overcompensates the growth of the original CEP cross section (Fig.1), so that finally we have no problem with unitarity.
Recall that this solution of the FK problem was actually realized by Cardy [6], where the reggeon diagrams (generated by Pomerons with intercept α P (0) > 1) were considered by assuming analyticity in the number of Pomerons in a multi-Pomeron vertex. It was shown that the corresponding absorptive corrections (analogous to that shown in Fig.2a) suppress not only the growth of a simplest, diagram Fig.1, contribution but the growth of cross sections of processes with an arbitrary number of LRGs.
Note that at the moment we deal with a one-channel eikonal. In other words in Fig.2 and in the unitarity equation (8) we only account for the pure elastic intermediate states (that is the proton, for the case of pp collisions). In general, there may be p → N * excitations shown by the black blobs in Fig.2a. The possibility of such excitations can be included via the Good-Walker [7] formalism in terms of G-W eigenstates, |φ i , which diagonalize the high energy scattering process; that is φ k |A|φ i = A k δ ki . In this case we encounter the FK problem for each state |φ i and we then solve it for the individual eigenstates 2 .
At first sight it looks sufficient to screen not the whole CEP amplitude, as in Fig.2a, but just the central vertex V as in Fig.2b. Let us consider this so-called enhanced diagram Fig.2b in more detail. Note that we have to integrate over the rapidity-positions y 1 and y 2 of the 'effective' triple-Pomeron vertices. Since the amplitude (shown by the double dotted line) increases with energy, that is with the size of |y 2 − y 1 | interval, the main contribution comes from the configurations where y 1 → 0 and y 2 → Y . In other words the enhanced, Fig.2b, diagram acts as the non-enhanced Fig.2a graph considered above. Moreover, the physical sense of the correction Fig.2b is that simultaneously with an exclusive process some inelastic interaction occurs between the partons placed at y 1 and the partons placed at y 2 . 3 This interaction violates the 'exclusivity' of the process and in this way decreases the cross section of pure CEP events. If the central vertex is screened more or less 'locally' (i.e. within a limited |y 1 − y 2 | interval) then, by cutting the corresponding Pomerons with the help of the AGK rules [8], we get another LRG process with some more complicated central multiparticle production instead of pp production. That is we will get the same FK problem, σ CEP > σ tot but generated by another group of CEP events.
Recall that the inelastic processes generated via the AGK rules by these screening diagrams at any rapidity interval must be included in the whole G inel contribution which describes the correction shown in Fig.2a. That is, anyway, we get very large probability of inelastic interactions (η l → 0, i.e. G inel (b) → 1) and finally arrive in the black disk regime.
Note also that we cannot avoid the eikonalization which results from the iterations of twoparticle s-channel unitarity (8). The opacity Ω (i.e. the two-particle irreducible amplitude in (13)) includes different diagrams but, as far as Ω increases with energy (Ω → ∞ at s → ∞), it inevitably leads to the black disk regime.

Edge of the disk
While the survival factor S 2 solves the FK problem for the central part of the black disk, we still have to address the question of what happens at the edge of the disk where the optical density is not large? That is, when ReΩ(b) ∼ O(1). For the large partial waves which occur in this domain we still may have CEP (and other diffractive LRG) cross sections larger than the total cross section corresponding to such l-waves.
The solution is provided by the condition that actually the interaction radius corresponding to a screening amplitude must be larger than the sum of the radii of amplitudes which describe the interactions across the gaps (i.e. -the large rapidity intervals). In particular, in the case of Fig.1 the energy/rapidity dependence of interaction radius must satisfy the inequality Using the parametrization given in (2), that is R = R 0 (ln s) γ/2 , we see that in order to satisfy (20) we have to choose γ ≥ 2. On the other hand we must satisfy the Froissart limit γ ≤ 2 [10].
That is the only solution is γ = 2; which gives R ∝ ln s.
To be more precise and to provide the inequality (20) we have to add the ln ln s correction to R R = c ln s − β ln ln s .
Such a correction was obtained for example in [11] and [12]. In the latter paper the factor ln s was replaced by ln(s/σ el ) which in the case of σ el ∝ ln 2 s generates the ln ln s correction in (21).
Thanks to the fact that (for a large y 1 ∼ O(Y )) the value of ln Y = ln ln s < ln y 1 +ln(Y −y 1 ) we get now R(Y ) > R(y 1 ) + R(Y − y 1 ) .
That is even taking the intermediate amplitudes A(y 1 ) and A(Y − y 1 ) at the largest possible impact parameters (at the edge of their disks) we get the total CEP amplitude, like Fig.1, inside the completely black disk of the screening amplitude. Thus the bare LRG multi-Reggeon contribution will be totally suppressed by the absorptive corrections.
The mechanism which generates the ln ln s correction during the development of the parton cascade (after accounting for processes of diffraction dissociation) was considered in [13]. It was shown in [13,14] that the same condition (20,21) provides the possibility to satisfy the t-channel unitarity.

Summary
We emphasize that when high-energy pp cross sections grow with energy, black disk absorption is the only cure of the FK disease. Thus any asymptotic behaviour of a high energy increasing cross section which does not lead to complete absorption is not consistent with multi-particle unitarity. Moreover, in order not to violate the unitarity equation at the edge of disk, where the opacity is not large (ReΩ(b) ∼ O(1)), the interaction radius should increase linearly with ln s R = c ln s − β ln ln s , with a small correction of the order of ln ln s.
The R ∝ (ln s) δ behaviour with δ < 1 is rejected since in such a case the cross section of central exclusive events, σ CEP (b), at large impact parameters b (that is for large partial waves l = b √ s/2 occurring at the edge of black disk) grows faster than the total cross section, σ tot (b), in the same partial wave.
The remarkable conclusion is that LRG events require maximal growth of the high-energy (asymptotic) cross section and an interaction radius R(s) of the form of (23) in order to be consistent with unitarity.
Finally recall that when we say 'black disk' asymptotics we actually refer to the area covered by an 'almost' black disk; that is the area where the amplitude A(b) i is close to black disk limit. Clearly this area should be larger than the area covered by the disk-edge. At present in pp scattering at the LHC we are close to black disk saturation only for b < b 0 = 0.2 − 0.3 fm while the width of disk-edge is about 1 fm [15]. That is the black disk asymptopia that we refer to should start when b 0 becomes much larger than 1 fm; say, at b 0 > 2 − 3 fm. This will correspond to σ tot (pp) = 2πb 2 0 > 300 − 1000 mb.