Black disk, maximal Odderon and unitarity

We argue that the so-called maximal Odderon contribution breaks the `black disk' behaviour of the asymptotic amplitude, since the cross section of the events with Large Rapidity Gaps grows faster than the total cross section. That is the `maximal Odderon' is not consistent with the unitarity.


Introduction
Recall that the Odderon was first introduced in 1973 [11], and since then it has been the subject of intensive theoretical discussion, in particular within the context of QCD. Indeed, there have been several attempts to prove its existence experimentally (see, for example, [7,8,12] for comprehensive reviews and references). While the discovery of the long-awaited, but experimentally elusive, Odderon would be very welcome news for the theoretical community, our aim here is to try to check whether the presence of the maximal Odderon, with an amplitude A − ∝ ln 2 s, which has a real part with high energy behaviour similar to that of the imaginary part of the even-signature amplitude A + , does not violate unitarity at asymptotically large c.m.s. energy √ s → ∞. Here we use the normalization ImA = σ tot .

Multi-Reggeon processes
It was recognized already in the 1960s [13,14] that multi-Reggeon reactions, where small groups of particles (X i ), are separated from each other by Large Rapidity Gaps (LRG) (see Fig.1), 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 in the literature as the Finkelstein-Kajantie disease (FK), see [15] for a review. Let us explain the situation using the simple example of Central Exclusive Production (CEP) of only one group of particles, as shown in Fig.2. Here the double line denotes the amplitude, A, which describes the interaction across the LRG (in particular, the proton-proton elastic amplitude). Correspondingly, the CEP amplitude for Fig.2a reads where V is the vertex factor of central production and the y i are the values of the rapidity.
The full CEP cross section is given by the integral where N is the normalization constant and where we put the upper rapidity y 2 = Y = ln s. In case of the maximal Odderon the real part of amplitude A(Y ) grows as ReA = c ln 2 s = cY 2 .
On the other hand, the t-slope B ∝ R 2 , with the interaction radius limited by the Froissart [16] condition R ≤ const · Y . That is the integral leading to Thus in such a case, the CEP cross section would grow much faster than the total cross section σ tot ∼ ln 2 s = Y 2 .
The same result can be obtained in impact parameter, b, space. Now Recall that in b space the amplitude is limited 2 to |A(Y, b)| ≤ 2 by the unitarity equation where G inel denotes the total contribution of all the inelastic channels. On the other hand the area where the amplitude is large (A ∼ O(1)), that is the value of Summing up the analogous cross sections for processes with a larger number of LRGs (i.e. a larger number, n, of hadron groups X i in Fig.1) we obtain the cross section which increases faster than the power of s. Indeed, each additional gap brings a factor ln s arising from the integral over the gap size (times the 'elastic' cross section which in the Froissart limit increases as ln 2 s i,i+1 ). The sum of these ln s factors leads to the power behaviour.
Note that by 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, the 'total' cross section, σ(b) tot must be less than the corresponding CEP contribution.
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, at least as t i → 0. However this cannot be true. Indeed, as far as we have a non-vanishing high-energy elastic proton-proton cross section, we can build the diagram on the right side of Fig.3 from a lower part which is just elastic pp-scattering and an upper part which corresponds to the proton-antiproton elastic interaction. Such a diagram is generated by the t-channel two-particle unitarity equation for the amplitude, where in our case |j is the t-channel pp state. Note that the contribution of this diagram is singular at t = m 2 p (where m p is the proton mass). There are no other similar terms corresponding to the central exclusive production of a pp pair with the same pole singularity. That is, in the vertex V of Fig.2a, there exists at least one subprocess (pp CEP), which cannot be cancelled identically.
It is useful to clarify the above argument, since it is a little subtle. We have to distinguish between the momenta transfer squared, t 1 and t 2 , incoming to the vertex V in Fig.2a, and the momentum transferred squared inside the vertex V , denoted by t in Fig.3a. The value of the t is driven by the transverse momentum, p t , of the antiproton. Even if, due to a subtraction in dispersion relation in t that reconstructs the amplitude, we find at some p t point that V = 0, this will not insure that the total vertex contribution vanishes. We will have V = 0 at other p t values. Now, to calculate the total CEP cross section, we have to integrate over all available p t , so finally we obtain a non-zero contribution of this particular pp subprocess.

The 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 a LRG, tends to zero at s → ∞, and the value of σ CEP does not exceed σ tot (for a review of diffractive processes at the LHC see e.g. [17]) .
In other words, besides the contribution of Fig.2a, we have to consider the diagram of Fig.2b, where the double-dotted line denotes an additional proton-proton (incoming hadron) interaction. This diagram describes the absorptive correction to the original CEP process, and has a negative sign with respect to the amplitude A a of Fig.2a. Therefore to calculate the CEP cross section we have to square the full amplitude where Indeed, in terms of S-matrix, the elastic component S l = 1 + iA(b), and the unitarity equation (7) reflects the probability conservation condition n S * l |n n|S l = 1 (11) for the partial wave l = b √ s/2. The solution of unitarity equation (7) reads or in terms of the partial wave amplitude with orbital moment l = b √ s/2 Figure 4: The partial wave amplitude, a l = i(1 − e 2iδ l ), is constrained to lie in the 'unitarity circle' centred on (0, i). At low energy, for a resonance of orbital momentum l appearing only in the elastic channel, the energy dependence of the amplitude a l follows a counter-clockwise circle of radius η l = 1 with maximum amplitude at δ l = π/2. However, for high energy scattering, black disk asymptotics requires ReΩ(b) → ∞ (see (19)), so 2Imδ l → ∞ and η l → 0, and hence the amplitude becomes pure imaginary.
The unitarity circle bounding the partial wave amplitude is shown in Fig. 4.
The above discussion shows that −Ω(b)/2 plays the role of 2iδ l . The elastic component of S matrix S l = exp(2iδ l ). Correspondingly, the probability of inelastic interaction (with all the intermediate states n except of the incoming, elastic, state) takes the form Within the eikonal model Ω(b) is described by the sum of single Reggeon exchanges while the decomposition of the exponent generates the multi-Reggeon diagrams.
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 lines in Fig.2b. That is according to (16) Equation (17) can be rewritten as (see (12,16)) In the case of black disk asymptotics 3 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.2a), so that finally we have no problem with unitarity.
Recall that this solution of the FK problem was actually realized by Cardy in [18], where the reggeon diagrams (generated by Pomerons with intercept α P (0) > 1) were considered by assuming analyticity in the number of Pomerons; and also by Marchesini and Rabinovici in [19], where diffractive production processes were discussed for the case of α P (0) > 1.
Note that at the moment we deal with a one-channel eikonal. In other words, in Fig.2 and in the unitarity equation (7), 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.2b. The possibility of such excitations can be included by the Good-Walker [20] 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.

Edge of the disk
This subsection is not crucial for our final result, but it should be mentioned in order to demonstrate the self-consistency of the whole picture.
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 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 fact that actually the constraint on the interaction radius R is a bit stronger than just R ≤ c ln s. It was shown in [21,22] that we have to account for the 'ln ln s' correction R = c ln s − β ln ln s = cY − β ln Y .
In this case, the radius of the CEP interaction shown in Fig.2a, (that is, before we account for the screening effects of Fig.2b), turns out to be smaller than that corresponding to elastic scattering. That is the multi-Reggeon amplitude is placed inside the black disk, and its contribution is strongly suppressed by the S 2 (b) factor.
It was shown in [21,23] that the same condition (20) provides the possibility to satisfy the t-channel unitarity.

Maximal Odderon
Now let us consider the situation with the maximal Odderon, where at very high energies the real part of elastic amplitude A is comparable with its imaginary part, in the sense that the ratio 4 ReA/ImA → constant = 0.
In such a case the elastic amplitude A(b) has a non-zero real part, which violates the condition A → i at s → ∞. Now the survival factor (18) tends to some non-zero constant. That is we loose the possibility to compensate the growth of the multi-Reggeon (CEP) cross sections by the S 2 factor. Thus these cross sections, which increase faster than the total cross section, will violate unitarity.
Let us consider a dynamical model. Note that the expression for the elastic amplitude (12) is an exact solution of the s-channel two-particle unitarity equation (7), where Ω is the two-particle-irreducible amplitude which includes all possible inelastic interactions. That is, in terms of Regge theory, the Odderon contribution must be included into the opacity, and the opacity Ω(b) (i.e. the 'phase' δ l ) should be written at high energies as the sum of the even-signature (Pomeron) and the odd-signature (Odderon) terms where the Pomeron term is mainly imaginary while the Odderon contribution is mainly real. If α P (0) = 1 + ∆ > 1, then the Pomeron term increases as the s ∆ . That is the exponent exp(−Ω/2) → 0 and the second term in elastic amplitude (12) vanishes together with the Odderon contribution. In other words, in the black disk limit when the value of ReΩ increases and exp(−Ω/2) → 0 the Odderon contribution dies out. The only chance to have a sizeable Odderon as s → ∞ is to collect the contribution from the edge of black disk where the opacity ReΩ(b) ∼ O(1) is not large. From this region one may get an Odderon contribution to the forward elastic amplitude A Odd (t = 0) ∝ ln s; that is ReA Odd could grow as the area of the ring around the black disk, but certainly it cannot increase as ln 2 s.
However even the ln s asymptotic behaviour is questionable. The point is that the radius of the Odderon induced interaction is most probably smaller than the radius of black disk, generated by the even-signature bare Pomeron. Indeed, the nearest t-channel singularity of the even-signature amplitude is t = 4m 2 π , while for the Odderon the nearest singularity is at t = 9m 2 π . We cannot build the Odderon state from two pions. The most reasonable appropriate hadron state in the Odderon channel is the ω meson. Therefore the growth of the Odderon radius with energy is expected to be less than the growth of the black disk radius driven by the even-signature amplitude, and the whole Odderon contribution will be 'absorbed' (i.e. power of s suppressed) by the black disk.
Thus in this section we have demonstrated that a) the maximal Odderon violates multiparticle s-channel unitarity b) the Odderon contribution disappears in the black disk limit when ReΩ → ∞. 5

Reflective scattering
The same argument can be used to reject the so-called 'reflective scattering' asymptotics proposed in [25]. Indeed, in this regime it is assumed that the high energy interaction becomes pure elastic and the amplitude (with our normalization fixed by eq. (7)). This means that at very high energies we will have an almost pure elastic interaction with G inel → 0. In such a case S 2 = 1 (see (18)).
On the other hand, t-channel unitarity generates the inelastic CEP diagram Fig.2b with a cross section which increases faster than the elastic cross section. The contribution of such a diagram cannot be suppressed by absorptive effects since now we have S 2 = 1. That is again we face the FK problem -the cross section of multi-Reggeon processes (in particular CEP) violates the unitarity constraint.
We emphasize that black disk absorption is the only cure of the FK disease. Thus any asymptotic behaviour of a high energy cross section, increasing with energy, which does not lead to complete absorption, is not consistent with multi-particle unitarity. In particular, the amplitudes considered in [26,27,28], should be abandoned, since they do not satisfy the black disk condition.