CGC/saturation approach for soft interactions at high energy: survival probability of the central exclusive production

We estimate the value of the survival probability for central exclusive production, in a model, which is based on the CGC/saturation approach. Hard and soft processes are described in the same framework. At LHC energies, we obtain a small value for the survival probability. The source of the small value, is the impact parameter dependence of the hard amplitude. Our model has successfully described a large body of soft data: elastic, inelastic and diffractive cross sections,inclusive production and rapidity correlations, as well as the $t$-dependence of deep inelastic diffractive production of vector mesons


Introduction
The large body of experimental data [1][2][3][4][5][6][7][8][9] on high energy soft interactions from the LHC, calls for an approach based on QCD that allows us to comprehend this data. However, due to the embryonic stage of our understanding of the confinement of quarks and gluons, we are doomed to have to introduce phenomenological model assumptions beyond that of QCD. In our recent articles [10][11][12][13] we have proposed an approach, based on the CGC/saturation effective theory of high energy interactions in QCD (see Refs. [14][15][16][17][18][19][20] and Ref. [21] for a review) and the Good-Walker [22] approximation for the structure of hadrons. In the next section we give a brief review of our model, however, we would like to mention here, that the main ingredient of this model, is the BFKL Pomeron [23,24], which describes both hard and soft processes at high energies. In other words, in our approach, we do not separate the interactions into hard and soft, both are described in the framework of the same scheme. The second important remark concerns the description of the experimental data: we obtain a good description of the cross sections of elastic and diffractive cross sections, of inclusive productions and the rapidity correlations at high energies. Consequently, we feel that we are ready to test our model on a complicated phenomenon, the survival probability of central diffractive production.  Fig. 1-a shows the scattering amplitude of the hard process. Fig. 1-b describes the set of the eikonal diagrams in the BFKL Pomeron calculus, which suppress the di-jet quark production, due to the contamination of the LRG (large raridity gap), by gluons that can be produced by different parton showers, as shown in Fig. 1-c.
The physical meaning of "survival probability" has been clarified in the first papers on this subject (see Refs. [25][26][27]), and is illustrated by Fig. 1, using the example of the central production of a di-jet with large transverse momenta. At first sight we have to calculate the diagram of Fig. 1-a in perturbative QCD, in which only two protons and the di-jet are produced, without any other hadrons. However, this is not sufficient, since simultaneously, a number of parton showers can be produced, and gluons (quarks) from these showers will produce additional hadrons. To calculate central diffractive production, we have to exclude these processes. In other words, we multiply the cross section given by the diagram of Fig. 1-a, by the suppression factor, which reflects the probability of not having any additional parton showers. This factor is the "survival probability".
Even this brief description indicates, that we have a complex problem, since some of the produced parton showers can have perturbative QCD structures, while others can stem from the long distances, and can be non-perturbative by nature. Therefore, to attack this problem we need a model that describes both long and short distances.
As we have mentioned above, our model fulfills these requirements, and so we will proceed to discuss survival probability in this model, expecting reliable results.
The next section is a brief review of our approach. We include it in the paper, for the completeness of presentation, and to emphasize that both short and long distance phenomena are described in the same framework. Section 3 is devoted to derivation of the formulae for the survival probability using the BFKL Pomeron calculus. The numerical estimates are given in section 4, while in the section Conclusions, we summarize our results.

Our model: generalities and the elastic amplitude
In this section we briefly review our model which successfully describes diffractive [10,11] and inclusive cross sections [12]. The main ingredient of our model is the BFKL Pomeron Green function, which we obtained using a CGC/saturation approach [10,28]. We determined this function from the solution of the non-linear Balitsky-Kovchegov (BK) equation [17,18], using the MPSI approximation [29] to sum enhanced diagrams, shown in Fig. 2-a. It has the following form: In the above formulae a = 0.65, this value was chosen so as to attain the analytical form of the solution of the BK equation. Parameters λ and φ 0 can be estimated in the leading order of QCD, but due to the large next-to-leading order corrections, we consider them as objects to be determined from a fit to the relevant experimental data. m is a non-perturbative parameter, which characterizes the large impact parameter behavior of the saturation momentum, as well, as the typical size of dipoles that take part in the interaction. The value of m = 5.25 GeV in our model, supports our main assumption, that the BFKL Pomeron calculus, based on a perturbative QCD approach, is able to describe soft physics, since m ≫ µ sof t ,where µ sof t is the natural scale for soft processes (µ sof t ∼ Λ QCD and/or pion mass).
Unfortunately, in the situation where the confinement problem is still far from being solved, we need to rely on a phenomenological approach for the structure of the colliding hadrons. We use a two channel model, which allows us also to calculate the diffractive production in the region of small masses. In this model, we replace the rich structure of the diffractively produced states, by the single state with the wave function ψ D . The observed physical hadronic and diffractive states are written in the form Functions ψ 1 and ψ 2 form a complete set of orthogonal functions The unitarity constraints take the form where G in i,k denotes the contribution of all non diffractive inelastic processes, i.e. it is the summed probability for these final states to be produced in the scattering of a state i off a state k. In Eq. (2.5) √ s = W is the energy of the colliding hadrons, and b denotes the impact parameter. A simple solution to Eq. (2.5) at high energies, has the eikonal form with an arbitrary opacity Ω ik , where the real part of the amplitude is much smaller than the imaginary part.
Eq. (2.7) implies that P S i,k = exp (−2 Ω i,k (s, b)) is the probability that the initial projectiles (i, k) will reach the final state interaction unchanged, regardless of the initial state re-scatterings.  Note, that there is no factor 1/2. Its absence stems from our definition of the dressed Pomeron. In the eikonal approximation we replace Ω i,k (s, b) by We propose a more general approach, which takes into account new small parameters that result from the fit to the experimental data (see Table 1 and Fig. 2): The second equation in Eq. (2.9) means that b ′′ in Eq. (2.8) is much smaller that b and b ′ , therefore, Eq. (2.8) can be re-written in a simpler form Selecting the diagrams using the first equation in Eq. (2.9), one can see, that the main contribution stems from the net diagrams shown in Fig. 2-b. The sum of these diagrams [11] leads to the following expression for Ω i,k (s, b) Note thatḠ dressed T does not depend on b, and is a function ofT = T (s, b = 0) = φ 0 e 0.63 λY .
In the above formulae the value of the triple BFKL Pomeron vertex is known: For further discussion we introduce with a = 0.65 . Eq. (2.13) is the analytical approximation for the numerical solution to the BK equation [28].
We recall that the BK equation sums the 'fan' diagrams shown in Fig. 3.
3 The main formulae for the survival probability

Hard amplitude in the two channel model
The expression for the hard amplitude is known, and it has been discussed in great detail (see Ref. [30]). It has the following general form (see Fig. 1-a) where Q T is the transverse momentum in the gluon loop,M is the color averaged amplitude for the process GG → X, where X denotes the final state (quark-antiquark jets in Fig. 1-a) with mass M X .
T , t i denotes the skewed unintegrated gluon densities. These functions have been discussed and we refer the reader to Ref. [30]. The t i dependence, is of great importance for the calculation of the survival probability [27,31] . We show below that the essential t i turns out to be small in our estimates, and therefore, we have to rely on some input from non-perturbative QCD. Our assumption is that at small t i we can factorize the unitegrated gluon density as Making the assumption, the hard amplitude at fixed impact parameter b, has the form The advantage of our technique is that it is based on the CGC/saturation approach, and the uninte- can be calculated in this framework. In the two channel model we have two unintegrated structure functions (see Fig. 4): We extract the b dependence of hard amplitudes i, from the experimental data for diffractive production of vector mesons in deep inelastic scattering (DIS). Presenting the t-dependance of the measure differential cross section in the form where t = −Q 2 T . The value of the slope B h can be calculated and it is equal to Using the parameters of Table 1 Therefore, the b dependence obtained from our approach, is in accord with the HERA experimental data. Generally speaking, both B h 's depend on energy. Indeed, in Regge theory [32], the scattering amplitude For hard processes we do not expect Pomeron trajectories with α ′ IP = 0. However, the effective α ′ IP is due to shadowing corrections. The hard amplitude has the following generic form Due to unitarity (see Eq. (2.5)) the amplitude cannot exceed unity. Therefore, On the other hand, the t slope of the amplitude, is equal to . Finally, the t-slope for the scattering amplitude is proportional to ln(s/s 0 ), viz.
where B h 0 is the slope for the cross section at s = s 0 . Choosing s 0 = 1 GeV 2 and ∆ = 0.2 * we obtain While the HERA experiment gives [33,34] B h el = 4.6 ± 0.06 + 4 (0.164 ± 0.41) ln (W/W 0 ). We believe that our approach provides a reasonable estimate of, and an appropriate method to understand the energy behavior of the hard amplitude. Note, that B h 0 ≈ 3.2 GeV −2 comes from the experimental formulae changing W 0 = 90 GeV to W 0 = 1 GeV . Bearing in mind this estimate, we find the following hierarchy of transverse distances, in our approach for high energy scattering (see Table 1)

ZEUS
where 4/m 2 i is the typical slope for g i (b). * Such values of ∆ come both from experiment [33,34] and from theoretical estimates.

Survival probability: eikonal approach
Central exclusive production (CEP) has the typical form where LRG denotes the large rapidity gap in which no hadrons are produced. We cannot restrict ourselves to the hard amplitude (see Fig. 1-a) to describe this reaction. Indeed, Eq. (3.1) gives the amplitude of CEP, but only for one parton shower. However, the production of many parton showers , shown in Fig. 1-c, will contaminate the LRG's, and these have to be eliminated, in order to obtain the correct cross section for the reaction of Eq. (3.12). In simple eikonal models, such a suppression stems from the diagrams shown in Fig. 1-b.
For this simple case, we can derive the formula for the survival probability, using two different approaches. The first one, relies on the s-channel unitarity constraint (see Eq. (2.5)). In the eikonal approach the contribution of all inelastic states is given by Eq. (2.7) (3.13) From Eq. (3.13) we see that multiplying the hard cross section by the factor exp (− 2 Ω (s, b)), we obtain the probability that the process has no inelastic production in the entire kinematic rapidity region [25][26][27]. Therefore, the survival probability factor S 2 takes the form (3.14) The second derivation is based on summing the Pomeron diagrams of Fig. 1-b, introducing Ω = g (b) 2G dressed (s) in Eq. (3.13). The eikonal amplitude can be written as (see Eq. (2.6)) In each term with the exchange of n-Pomerons, we need to replace one of these Pomerons by the hard amplitude. Such a replacement leads to the following sum Multiplying this amplitude by its complex conjugate, and integrating over b we obtain Eq. (3.14).

Survival probability: enhanced diagrams
At first sight Eq. (3.14), provides the answer for the case of eikonal rescattering. However, this is not correct, since the dressed BFKL Pomeron Green function is the sum of enhanced diagrams of Fig. 2-a and Fig. 6-a. To find the survival probability, we need to replace one of the Pomeron lines in Fig. 6-a by the hard amplitude. As was noticed in Ref. [35] the enhanced diagrams can be reduced to a sum of diagrams which have a general form It should be noted that A hard IP is not the same as in Eq. (3.14), and its b distribution has a typical value of b ∝ 1/m. In other words A hard IP ∝ S (b). Eq. (3.18) leads to the following contribution which coincides with our hard amplitude introduced in Eq. (3.5). Using the notation in this equation, the expression for Ω hard ik takes the final form: Fig. 6-a shows the set of the diagrams for the BFKL Pomeron calculus that lead to the resulting Pomeron Green function, after integrating over rapidities of the triple Pomeron vertices, can be re-written in the form of Fig. 6-b. Fig. 6-c illustrates that to calculate the survival probability, we need to replace one of BFKL Pomerons by the hard amplitude.

Survival probability: eikonal formula for two channel model
The structure of the formula for the survival probability is shown in Fig. 4. The amplitude for the reaction of Eq. (3.12) can be written in the form A f ull (s, q 1,T , q 2,T , p 1 , The survival probability for the cross section d 2 σ/(dt 1 dt 2 ) is equal where A hard (s, q 1,T , q 2,T , p 1 , However, if we are interested in cross sections that are integrated over d 2 q 1,T and d 2 q 2,T , the expression for S 2 can be simplified, and it has the form and (3.27)

General case: Ω hard
The first problem that we need to solve, is to find a more general expression for Ω hard , than we have obtained in Eq. (3.21). Eq. (2.11) sums net diagrams, and they can be re-written in the same form as the enhanced ones [35]. Eq. (3.17) is replaced by Taking into account the enhanced diagrams forG dressed (see Eq. (3.19)) we obtain the final form for Ω hard ik : Fig. 7-a shows the set of the net diagrams in the BFKL Pomeron calculus, that lead to the resulting survival probability in the framework of high energy QCD. After integration over positions of G 3I P in rapidity, the sum of the diagrams reduces to Fig. 7-b.

Final formula
Finally, to obtain the general formula for the survival probablity we need in Eq. (3.26), we replace ). Therefore, the survival probability is equal to . This factor takes into account the contribution from the enhanced diagrams. Fig. 8 shows that on its own, it leads to a smaller survival probability .

Survival probability in our model
Our estimates for the survival probability are shown in Fig. 9.  We predict rather small values for the survival probability. Such small values have been discussed peviously (see Ref. [31]), however, in the present model we have a different source for this small number. In Ref. [31] S 2 turns out to be small, due to contribution of the enhanced diagram, while in the present model the enhanced diagrams give a suppression factor which is moderate (see Fig. 8). The main cause for the small value of S 2 , is the b dependence of the hard amplitude. As we have mentioned our b dependence stems from the description of the soft high energy data, based on CGC/saturation approach, in which we do not introduce a special soft amplitude. In our approach we are only dealing with hard (semi-hard) amplitudes, which provide a smooth matching of the 'soft' interaction with the 'hard' one.
4.2 Importance of b-dependence of the hard amplitude.
We can illustrate the importance of the b-dependence of the hard amplitude by introducing with B h = 4 ÷ 5 GeV −2 , which follows from the experimental data, as discussed previously. At first sight Eq. (4.1) follows from the experimental observation of the vector meson production in deep inelastic scattering. As we have discussed, our hard amplitude of Eq. (3.5) leads to the slope of the differential cross section which is the same as in Eq. (4.1). Indeed, as shown in Fig. 10, the t-dependence of the differential cross sections in the region of small t (t < 0.5 GeV 2 ), are similar in both parametrizations of the hard amplitude. However, at large t there is a difference, which increases with increasing t.
The difference between the amplitudes is more pronounced when plotted as a function of the impact parameter b (see Fig. 11). Figure Fig. 11 shows that Eq. (3.5) of our model (with the old set of parameters) leads to fast decrease of the amplitude in b. As one can see from Table 1, the steepest decrease is due to the A 22 (b) amplitude, which is the smallest. This amplitude also has the smallest suppression, due to the factors exp (−2Ω ik ), since Ω 22 has the smallest value. In the numerator of Eq. (3.32), only term (22) provides the essential contribution. This term turns out to be rather large for the amplitude of Eq. (4.1), as one can see from Fig. 11. On the other hand, the same hard amplitude in Eq. (3.5) shows the steep decrease in b, resulting in the small contribution to the numerator of Eq. (3.32), as well as for the resulting S 2 .
The resulting difference for S 2 is large, values of the survival probability for the hard amplitude of Eq. (4.1), are ten or more, times larger than the results of our present model (with the old set of parameters). For example, for W = 7 T eV we obtain S 2 = 10 ÷ 15%.
We denote our fit which results in these very low values of S 2 as the "old set of parameters". Based on the our diagnosis of the problem above, we made a second fit to t he same experimental data, with the additional condition that m 2 ≤ 1.5 GeV . We will refer to this fit as "new set of parameters". The values of the parameters for both "old" and "new" sets are given in Table 1. The comparison of the results for σ tot , σ el , B el , σ sd (low and high mass) and σ dd (low and high mass) for both set of parameters are shown in Table 4. We note that the values of σ tot , σ el and B el obtained in both fits are rather close, while the diffractive cross sections, both σ sd and σ dd are smaller in the new fit.
The results for S 2 for the "new set of parameters" is shown in Fig.12, we find S 2 ≈ 3 % in the LHC energy range.

Kinematic corrections
In our approach we consider G 3IP as a point-like vertex. This assumption is a considerable simplification. As we have discussed in [13], we expect short range correlations in rapidity, with the correlation length in rapidity ∆ cor ≈ 2. Bearing in mind that the triple Pomeron vertex has a size in rapidity, we can take into consideration that in Eq. (3.17) and Eq. (3.28), the Pomerons enter not at rapidity Y but at Y − δ cor N G 3I P , where N G 3I P is the average number of triple Pomeron vertices . It is easy to see that In Fig. 12 we plot S 2 which is given by Eq. (3.32) and Eq. (3.33) but Y → Y − ∆ cor N G 3I P with N G 3I P estimated using Eq. (4.2). One can see that the effect is sizeable, and leads to larger values of the survival probability. Figure 12. S 2 versus W . S 2 is calculated using Eq. (3.32) and Eq. (3.33) with Y replaced by Y − ∆ cor N G3IP with ∆ cor = 2 ( blue curve), while the red curve is the same as in Fig. 9. The black curve is the estimates for the survival probability with new set of parameters (see Table 1).

Comparison with other estimates
Nearly ten years ago we summarized the situation regarding the evaluation of the S 2 by different models [37]. Unfortunately, as the results of this paper illustrate, the values obtained for S 2 , are highly dependent on the characteristics of the models used to parametrize the soft and hard amplitudes. For details of the parametrizations used by the three groups quoted in Table 2, we refer the reader to [37][38][39]. Their values for the S 2 are given in Table 2.
In the summary of our previous approach for constructing a model based on N =4 SYM for strong coupling, and matching with the perturbative QCD approach [40], we discuss the results for S 2 obtained from this approach. We compared our results with those obtained by the Durham group [41]. The Durham W (GeV) KKMR (CD) [ Table 3. Values of S 2 at different energies from the different groups [41] and [40] and from this model: I -the hard amplitude is given by Eq. (3.5) with B = 4.5GeV −2 (see Fig. 10 and Fig. 11 ),II-the hard amplitude is given by Eq. (3.6) with old set of parameters; IIn-is the same as II but with the new set of parameters; and III-is the same as II but the kinematic corrections are include.The S 2 results are in percentages.
model is a two-channel eikonal model where the Pomeron coupling to the diffractive eigenstates are energy dependent. They presented four different versions, in Table 3, we quote their results for model 4, their "favoured version".
As can be seen from Table 3 , our results for S 2 obtained from the N =4 SYM approach are slightly larger than those given by the KMR approach, and larger than the results of our present model. The reasons for this have been discussed in subsection 4.2.
Comparing the results of different calculations given in Tables 2 and 3 with our present calculation, we see that the estimates using the same b-dependence of the hard amplitude A hard ∝ exp −b 2 /4B ( see Table 3 'this model(I)') , leads to results that are similar to the estimates obtained by the other groups. The b-dependence of the hard amplitude that follows from our present approach, produces small values for the survival probabilities with the old set of parameters, and reasonable values with the new set of parameters. We wish to emphasize that our b -dependence of our present model, has two advantages: it leads to the correct Froissart limit at large b, A hard ∝ exp (−µ b); and at large momentum transfer (Q T ) it decreases as a power of Q T , as one expects in perturbative QCD.
We wish to emphasis, that we made a second (new) fit to the experimental data, and obtained a new set of parameters, which leads to an increase in the values of the survival probabilities as shown in Table 3.

Conclusions
In this paper we calculated the survival probability for central diffractive production, and found at LHC energies, that its value is small. The small value obtained does not stem from the sum of enhanced diagrams, as in our previous models [42], but is due to the impact parameter dependence of the hard amplitude.   The distinguishing feature of our model based on CGC/saturation approach, is that we use a framework where soft and hard processes are treated on the same footing. Hence, there is no need to introduce a special hard amplitude, as has been done in all previous attempts, to estimate the survival probability. It should be stressed that the main source for our small values of S 2 , is the impact parameter dependence of the hard amplitude, for which we do not have any theoretical estimate. This is usually assumed to have a Gaussian form A hard ∝ exp −b 2 / (2B) . The value of B was taken from the experimental data on the deep inelastic diffractive production of vector mesons. We demonstrated in this paper, that in spite of the fact, that our hard amplitude leads to experimental values of B, at small t, it yields a different behaviour than the Gaussian input, leading to small values of S 2 at high energies. We note, that the impact parameter dependence of our hard amplitude satisfies two theoretical features that are violated in the Gaussian b-dependence: at large b , A hard ∝ exp (−µb), and at large Q T it decreases as a power of Q T , as required by perturbative QCD.
We wish to stress that the values obtained for the survival probability depend mostly on the bdependence of the hard amplitude. The most interesting result is that we can describe on the same footing both the soft and the hard amplitude. At first sight, the small values of S 2 contradict this the most basic idea of our approach. To show that this is not an inherent problem of our approach, we made a new fit to the all available soft data to show that we can obtain substantially larger values of the survival probability. It demonstrates that experimental measurements of this observable is a sensitive tool to determine the values of the phenomenological parameters of our model. We present in this paper, the result of the first consistent approach to obtain both the soft and the hard amplitude from the same model. We hope that the data from the LHC on the survival probability, will be instrumental in determining the impact parameter dependence of the scattering amplitude.