# Generation of central exclusive final states

## Abstract

We present a scheme for the generation of central exclusive final states in the program. The implementation allows for the investigation of higher-order corrections to such exclusive processes as approximated by the initial-state parton shower in . To achieve this, the spin and colour decomposition of the initial-state shower has been worked out, in order to determine the probability that a partonic state generated from an inclusive sub-process followed by a series of initial-state parton splittings can be considered as an approximation of an exclusive colour- and spin-singlet process. We use our implementation to investigate the effects of parton showers on some examples of central exclusive processes, and we find sizeable effects on di-jet production, while the effects on *e.g.* central exclusive Higgs production are minor.

## 1 Introduction

Compared to the fairly clean environment of \(\mathrm{e}^{+}\mathrm{e}^{-} \) annihilation, proton collision events are in general very messy, especially at the LHC at high luminosity. Even at lower luminosity where pile-up events are absent, the existence of multiple soft interactions and initial-state parton showers means that any hard sub-process of interest will be obscured by soft and semi-hard hadrons smearing the measurements. However, for some rare events, a central colour-singlet hard sub-process may appear in complete isolation, with rapidity gaps on both sides stretching all the way out to the (quasi-) elastically scattered protons, giving a nice and clean environment to study its properties.

Such *central exclusive processes* (CEPs) have been extensively studied in the so-called Durham formalism, first described by Khoze, Martin and Ryskin in [1] and reviewed in detail in [2]. The simplest such process is Higgs production, where two gluons in a colour- and spin-singlet state fuse together via a top-quark loop into a Higgs particle as outlined in Fig. 1. With an additional virtual exchange of a (semi-hard) gluon, the net colour exchange between the colliding protons can be zero and we may end up with a very simple and clean final state consisting only of two (quasi-) elastically scattered protons along the beam pipe and the Higgs decay products in the central rapidity region.

The formalism can be generalised to any colour-singlet hard sub-process, and the main ingredients to construct the amplitude is the matrix element for this sub-process and the so-called off-diagonal unintegrated parton densities. The latter can be interpreted as the amplitude related to the probability of finding gluons in a proton with equal but opposite transverse momentum, \(\mathbf{q}_\perp \), and carrying energy fractions *x* and \(x'\) each, one of which is being probed by a hard scale \(\mu ^2\). These densities also include a Sudakov form factor describing the probability that there is no additional initial-state radiation from the incoming gluon between the scales \(q_\perp \) and \(\mu \), which could destroy the rapidity gaps. Additional emissions below \(q_\perp \) are then suppressed since they cannot resolve the individual colours of the two gluons. A third ingredient is the so-called *soft survival probability* which gives the probability that there are no additional soft or semi-hard interactions between the colliding protons which could destroy the rapidity gap.

Our implementation is provided as an add-on^{1} to [6] and is inspired by the observation that the Sudakov form factors in the off-diagonal unintegrated parton densities used within the Durham formalism can be interpreted in terms of no-emission probabilities in the parton shower language of .

In this way, we can reformulate the cross section for producing a CEP event in terms of a probability that a standard inclusive sub-process generated by at some scale during the parton shower evolution is converted to a colour-singlet, and thereafter be considered a CEP event disallowing further initial-state shower splitting. The main advantage of this approach is that we actually are allowed to include initial-state shower splittings, and thus can approximately model higher-order corrections to the original sub-process. In addition, we have the option of using the multiple interactions machinery of to directly model soft survival probability as suggested in [7].

Consider, *e.g.*, the central exclusive production of di-jets. We would start by generating the basic \(2\rightarrow 2\) hard partonic scattering from the inclusive matrix element. We would then generate an initial-state parton emission from each of the incoming partons. This implicitly includes the probability that no emission has been made at a higher scale than the two generated splittings. If the colour and spin state of the original \(2\rightarrow 2\) is consistent with a CEP, we basically take the ratio of the corresponding exclusive cross section and the one calculated with the inclusive matrix element, using the generated scales as factorisation scale. This gives us the probability to discard the generated splittings and continue the event as a CEP, or to keep the hardest emission and continue as a normal inclusive event.

If we continue the event as inclusive, we keep the hardest splitting and again generate one initial-state splitting from each side. Now we have a three-parton final state for which it must be checked if it can be a candidate for CEP, but otherwise the procedure is repeated. It should be noted that in general does not assign spin states to particles, so the procedure here also involves a spin decomposition of the parton splitting probabilities and the matrix elements to correctly get the probability for this to be a CEP. This will become cumbersome when we go up in parton multiplicity, but it is still fairly straightforward.

The fact that we can stop the parton shower at any stage and check if we can convert the generated state exclusive, does not only mean that we can approximate higher-order contributions from initial-state radiation. If we continue the parton shower evolution to low scales we are also able to investigate the transition region between the Durham formalism and the resolved Pomeron formalism [8] which may produce similar final states.

The outline of this paper is as follows. First, we recapitulate the main features of the Durham formalism in Sect. 2. Then we describe the different parts of our implementation in the program, starting in Sect. 3 with the reinterpretation of the exclusive luminosity function in terms of the parton shower no-emission probabilities, and followed by a description (Sect. 4) of the spin and colour decomposition of a given partonic state generated by a parton shower from an inclusive hard matrix element. In Sect. 5 we then present some proof-of-concept results for some sample processes before we conclude with a summary and outlook in Sect. 6.

## 2 The Durham formalism

*pp*collision,

*i.e.*as the smaller of the virtualities of the screening gluon and the fusing gluon momenta related to the particular proton.

*x*(\(x'\)) and on two scales: the scale of the hard sub-process \(\mu ^2\); and the scale corresponding to the screening gluon transverse momentum \(Q^2\).

*z*integration

^{2}depends on the mass of the exclusive system which makes the Sudakov factor \(T_M\) also \(M_X\)-dependent as indicated by the subscript,

*M*.

*e.g.*\(A_{++}\), depend on \(M_X\), the momenta of outgoing particles, the helicities of outgoing particles as well as the helicities of incoming gluons (here both \(+1\)). The amplitudes also depend on the colours of the particles in the sub-process. Here, the amplitudes \(A\) are the result of averaging over colour indices of the incoming partons in such a way that the exclusive system is a colour-singlet.

Note that Eq. (2) does not include a soft survival probability which cannot be calculated in a perturbative way. It reduces the CEP cross section at LHC by typically two orders of magnitude and can, *e.g.*, be determined using the eikonal model [10, 11, 12].

## 3 Reinterpretation of the exclusive cross section

### 3.1 Prerequisites

*y*and

*M*denote the rapidity and mass of the exclusive system and are related to the momenta fractions \(x_1\) and \(x_2\) by the formulae

*M*(via a Sudakov form factor) as well as the arguments of \(V_{ij}^{aa}\) are written explicitly. The kinematics of all outgoing particles of the exclusive system

*X*is denoted by

*w*and \(\mathrm {d}\Sigma _w\) is the corresponding phase space element. The whole expression is integrated over phase space of these outgoing momenta

*w*, which satisfy the imposed kinematic cuts.

^{3}this differential simplifies to \(- \mathrm {d}^2 \mathbf{q} / \mathbf{q}^4\).

*i.e.*amplitudes are scaled by \(1/\sqrt{N_s}\). Letters \(\lambda \) and

*j*denote all possible helicity and colour configurations of the exclusive system.

^{4}especially because \(\mathrm {d}(1/q^2) = -\mathrm {d}q^2/q^4\).

*q*region. Whereas it dominates for

*q*of 2–\(3\, \text{ GeV }\) in the case of LHC Higgs production and for even smaller values (1–\(2\, \text{ GeV }\)) in the case of di-jet production at Tevatron. Values below the starting scale \(q_0 = 1\,\text{ GeV }\) of MMHT2014 LO PDF [14] can be extracted using a backward DGLAP evolution [15]. In reality, it was argued in [16] that for \(q \lesssim 0.85\,\text{ GeV }\) the gluon propagator would be modified by non-perturbative dynamics which effectively suppress such low

*q*contributions. Rather than a sharp cut-off, the damped gluon PDF [5] is used to calculate \(\phi _M\) below \(q_0\) in order to suppress the region of low transverse momentum:

### 3.2 Screening gluons in the interleaved parton shower

The way parton showers are included in is through an interleaved process where we normally have three competing processes, which we will denote ISR, FSR and MPI. ISR is an initial-state splitting where one of the incoming partons to the hard sub-process is evolved to lower scale and higher energy fraction, by emitting a parton into the final state; FSR is the final-state splitting of a parton in the final state; while MPI is the appearance of an additional parton–parton interaction. All of these occur at decreasing scale, where the highest scale is given by the kinematics of the hard sub-process. In each step in the shower we then pick a process which has a lower scale than the previous one, and the factorisation property of the no-emission probability means that we can generate one of each of the possible processes independently and simply pick the one which yielded the highest scale in each step.

For simplicity we will here only consider the ISR, concentrating on the initial-state \(g\rightarrow gg\) splittings, and show how we can reinterpret the formula for CEP as an extra process in the interleaved shower, which transforms an inclusive event into an exclusive one.

^{5}

There is, however, a problem with this approach, in that the \(p_{\mathrm{exc}}\) is very peaked at small \(\mu \), making the generation of the exclusive events very inefficient. Although a weighting procedure could be applied, it would be difficult to incorporate into the current framework of . For the purpose of this paper, we have therefore chosen to implement a simpler procedure.

### 3.3 A simpler approach

The full procedure would be to start the generation of an inclusive process in and calculate the probability in (39) of that process being exclusive. Then, after each ISR or FSR step in the interleaved shower, we would again check if the current state can be made exclusive by (39). If the event is to be considered to be exclusive we would rearrange the colour flow accordingly and insert the quasi-elastically scattered protons, but we let the shower continue without the ISR process.

Note that the soft survival probability, which we have so far left out of the exclusive luminosity function, corresponds exactly to the probability of having no additional multi-parton interactions, so any MPI in the interleaved shower (before or after the event has been made exclusive) will mean that the event will stay inclusive.

To make the generation of exclusive events more efficient we have here decided to simplify the procedure even more. A final-state emission does not modify the parton densities used in the luminosity functions, and although they may affect the exclusive cross section, we have here decided to leave them out and generate them separately. Also the calculation of the soft survival probability by vetoing any exclusive event in the case the shower gives a MPI is extremely inefficient, and we instead calculate that separately and simply multiply the inclusive cross section with that factor.^{6} It should be noted that using the MPI model in presented in [7] has not been carefully investigated. It has the interesting feature that the probability for additional scattering depends on the hardness of the primary scattering, since harder processes have larger overlap (smaller impact parameter). It also has a natural dependence on the collision energy. The downside is that it is sensitive to the soft behaviour of MPI model and may vary strongly between different tunings of the parameters. In this paper we will simply use the default tune in and postpone a proper investigation of the procedure to a future publication.

- 1.
Generate the hard sub-process of interest in , use the standard inclusive cross section.

- 2.
Use only the initial-state shower in .

- 3.
Before generating the next initial-state emission, make the event exclusive with the probability in (39).

- 4.
If the event is made exclusive, switch off the initial-state cascade, rearrange colours, remove the proton remnants, insert the scattered protons and continue with final-state radiation from the exclusive state.

- 5.
If the event stays inclusive, generate the next initial-state emission (continue with step 3).

This approach is efficient if the ratio \(\frac{\sigma '^s}{\sigma '^i}\) does not heavily depend on the number of emissions. This is normally the case for higher emission multiplicities in contrast the first few emissions where the interference effects play a role. In the program both approaches are implemented and we use each of them in such a frequency so that the event weights variation is minimal.

### 3.4 Modified luminosity for all possible helicity configurations

*j*and \(\lambda \) denote the colour state and helicity state of all final-state particles of the hard sub-process. The index

*i*denotes the colour of the fusing gluons.

## 4 Approximation of matrix elements using shower splittings

To calculate the central exclusive cross section, the amplitude for every helicity and colour combination of the studied sub-process must be known, rather than the spin- and colour-averaged sub-process cross section. For \(2\rightarrow 2\) processes the amplitude \(A^{\lambda _{l1} \lambda _{r1} \rightarrow \lambda _3 \lambda _4}_{a_{l1} a_{r1} \rightarrow x_3 x_4 }\) depends on the helicities of the incoming (\(\lambda _{l1}, \lambda _{r1}\)) and outgoing particles (\(\lambda _{3}, \lambda _{4}\)) as well as on the colours of the corresponding particles \(a_{l1}\), \(a_{r1}\), \(x_3\), \(x_4\). The additional dependency on particle momenta is not written out explicitly.

For splittings that includes quarks, the SU(3) structure constants, \(f_{abc}\), must be replaced by the Gell-Mann matrices, \(T^{a}_{ij}\), and the splitting *amplitude*, \(P^{gg}\), by the corresponding one.

### 4.1 The colour emission tensor

In appendix A we provide the complete list of linear transformations, necessary for calculating the colour emission tensor with an extra emission in the beginning of the shower. Considering the initial-state parton shower described by the tensor^{7} \(T^{em}_{p_f p_h}\), if the first parton \(p_f\) is gluon it can be evolved backward to a quark, anti-quark, or a gluon, making this parton the starting one. A quark can be evolved backward to a quark or a gluon and, finally, an anti-quark can be evolved to an anti-quark or a gluon. This gives in total seven possibilities. The parton \(p_h\) attached to the hard sub-process can be quark, anti-quark, or gluon making the overall number of possible linear transformation equal to \(7\times 3=21\). In reality, some of these transformations are independent of whether the parton is quark or anti-quark which reduces the number of non-identical linear transformations to 13.

In the procedure we have developed, the two colour emission tensors are constructed, one for “left” side and one for “right”. Before starting the backward parton shower evolution these tensors describe the shower with zero emissions and the particular type (\(T_{gg}^{em}\) or \(T_{qq}^{em}\)) is chosen according to the “left” and “right” type of the parton entering to the hard sub-process. After every step in the backward evolution, one of these tensors is modified using the appropriate transformation.

### 4.2 The spin emission tensor

*z*and the azimuthal angle \(\phi \) of the emission. These can all be found in the literature [19]. Using these splitting amplitudes, the spin emission tensor can be defined, and in the particular case of only one emission, this tensor has the form

### 4.3 Amplitude definition

Within the process library the amplitude of each process is defined in the colour trace basis, in a form similar to that used for example in MadGraph [20]. For the \(gg \rightarrow gg\) process, which has the most complicated colour topology, the colour trace basis has six terms.^{8} It means that 6 amplitudes depending on Mandelstam variables *s*, *t*, *u* must be provided for every helicity combination (together \(6\times 16=96\) amplitudes). Fortunately, most of these amplitudes are equal to zero or are identical to each other, *e.g.* due to parity invariance.

## 5 Sample results

In this section we present a few sample results from our implementation of the Durham formalism. We will focus the discussion of the unique feature of our implementation, *i.e.* possibility of generation the exclusive states with higher particle multiplicities. Currently, the process library includes all hard QCD \(2\rightarrow 2\) processes; Higgs boson production via \(gg\rightarrow H\); the single \(Z^0\) production via \(q\bar{q}\rightarrow Z^0\); and two photon production via \(gg\rightarrow \gamma \gamma \) and \(q\bar{q}\rightarrow \gamma \gamma \). The program is modular, however, and new processes can easily be added.

*pp*collisions at \(\sqrt{s} = 13\,\text{ TeV }\) and are based on MMHT2014 LO PDF [14]. They incorporate hadronisation as well as the final-state radiation. The initial-state shower is evolved down to \(\mu _{\mathrm{exc}} = 1.5\,\text{ GeV }\) if not stated otherwise. All predictions incorporate the soft survival probability estimated using veto on MPI. This probability is around 0.06 with little kinematic dependency.

### 5.1 Di-jet production

We start by studying the properties of our Monte Carlo model for di-jet production at the LHC. In contrast to other implementations of the Durham formalism, our program allows for the generation the exclusive di-jet event from any \(2\rightarrow 2\) QCD hard sub-process, as long as the partons which initiate the space-like parton shower are gluons that can be in a colour-singlet state.^{9}

The variable which describes the size of the phase space available for the space-like parton shower is \(\mu _{\mathrm{exc}}\), which is the lowest allowed transverse momentum of the emission. The maximal allowed \(p_\perp \) of an emission is set to be equal to the hard scale of the sub-process, given by the transverse momentum of the leading jet. For ordinary inclusive events the cut-off scale for the initial-state radiation (ISR) in is around \(2\,\text{ GeV }\). In our discussion, we study the events with transverse momenta of the emissions starting at 1.5 GeV.

The inclusion of possible initial-state splittings in our approach will naturally increase the exclusive cross section for di-jet production and cause a smearing towards low values in the distribution of \(M_{12}/M_X\). The \(M_{12}/M_{X}\) observable can be seen as an experimental measure of the “exclusivity” of the particular event. \(M_{12}\) is here the invariant mass of the two leading jets, and the total mass, \(M_X\), of the exclusive system *X* can, in principle, be calculated from the outgoing protons relative momentum loss, \(\xi \), as \(\sqrt{\xi _1 \xi _2 s}\). Without any parton showers this ratio equals 1 on the parton level. The final-state radiation and hadronisation can smear this distribution, especially if the jet radius of the jet algorithm is small, since a final-state parton may radiate outside the jet cone, giving the smaller value of the invariant di-jet mass \(M_{12}\).

We have here used the “anti-\(k_\perp \)” jet algorithm [21] with \(R=0.7\), a minimum transverse momentum of the jets of 40 GeV, and the absolute value of the pseudorapidity of jets smaller than 2.5. As seen in Fig. 3 there is indeed a smearing from the final-state radiation (blue curve), but the smearing increases significantly if initial-state radiation is included (black curve). The distribution with initial-state radiation resembles what one would expect form the double Pomeron scattering and the final states generated by these two mechanisms overlap. However, the physical nature of both processes are different since, in DPE, where the Pomeron in the simplest approximation is a *gg* object, the colour neutralisation of the hard system comes from the Pomeron remnant gluons, while for CEP it is due to an additional gluon exchange. Despite different pictures, the final states could still be indistinguishable, as low-\(p_\perp \) initial-state emission on either side of the hard scattering in CEP could look exactly like Pomeron remnants.

The exclusive cross section is less sensitive to the space-like emissions if only the events with, for example, \(M_{12}/M_{X} > 0.8\) are accepted as is demonstrated in Fig. 4.

Here, the left plot shows that the di-jet cross section consists of events either with \(p_\perp ^{\mathrm {last}} \sim 40\,\text{ GeV }\) where there was typically no space-like emission and \(p_\perp ^{\mathrm {last}}\) was identified with the hard scale of the process, or events with \(p^{\mathrm {last}}_{\perp }\sim 3\,\text{ GeV }\). In this case, there are usually many emissions and \(p_\perp ^{\mathrm {last}}\) denotes the transverse momentum of the latest one with the smallest \(p_\perp \).

More comprehensive picture of the situation provides the two-dimensional plot (Fig. 5), where the correlation of the mass ratio and \(p_\perp ^{\mathrm {last}}\) for a particular event is shown. The depletion of the emissions for \(p_\perp ^\mathrm {last}\) slightly below \(40\,\text{ GeV }\) is partially due to the small \(gg\rightarrow ggg\) colour-singlet cross section and partially just a statistical effects given by the low probability of having no further emission below such high \(p_\perp \). The tree-level \(gg\rightarrow ggg\) spin-singlet colour-singlet cross section in the analytic form is provided in [22]. This cross section is zero in the “parton-shower” limit where one of the outgoing gluons has small \(p_T\) compared to the remaining two and \(\hat{t}=\hat{u}=-\hat{s}/2\), where the Mandelstam variables are derived from the two hardest gluons whereas the softest one is supposed to be part of the shower. Such a behaviour agrees with our calculations based on procedure introduced in Sect. 4.

*i.e.*smaller than 1 GeV, where the perturbative QCD is not justified [16].

The table demonstrates how the particular hard sub-processes in contribute to the total exclusive cross section of the di-jet production at LHC (\(\sqrt{s} = 13\,\text{ TeV }\)). The hard processes are defined using the Pythia convention and are accompanied by the Pythia process Id [23]. The letter *q* denotes any light quark flavour, therefore, *e.g.* \(gg \rightarrow q\bar{q}\) represents the sum of \(gg \rightarrow u\bar{u}\), \(gg \rightarrow d\bar{d}\) and \(gg \rightarrow s\bar{s}\) cross sections. The jets in the di-jet system are required to have \(p_\perp ^{\mathrm {jet}1,2} > 40\,\text{ GeV }\) and \(|\eta ^{\mathrm {jet}1,2}| < 2.5\). In addition the leading protons momentum loss \(\xi \) must be \(\xi _{1,2} < 0.03\). The \(\sigma ^{n_{Em}=0}_{\mathrm{exc}}\) are the exclusive cross sections with no initial-state radiation. \(\sigma _{\mathrm{exc}}\) and \(\sigma ^{M_{12}/M_X >0.8}_{\mathrm{exc}}\) are the exclusive cross section with allowed initial-state radiation down to \(1.5\,\text{ GeV }\); the last one has an additional constraint, \(M_{12}/M_X > 0.8\)

Id | Process | \(\sigma ^{n_{Em}=0}_{\mathrm{exc}}\; [\text {pb}]\) | \(\sigma _{\mathrm{exc}}\; [\text {pb}]\) | \(\sigma ^{M_{12}/M_X >0.8}_{\mathrm{exc}}\; [\text {pb}]\) |
---|---|---|---|---|

111 | \(gg \rightarrow gg\) | 23 | 173 | 57 |

112 | \(gg \rightarrow q \bar{q}\) | \(10.6\times 10^{-3}\) | 0.6 | \(56\times 10^{-3}\) |

113 | \(qg \rightarrow qg\) | − | 30 | 5.8 |

114 | \(qq' \rightarrow qq'\) | − | 1.3 | \(94\times 10^{-3}\) |

115 | \(q\bar{q} \rightarrow gg\) | − | \(10.5\times 10^{-3}\) | \(83\times 10^{-6}\) |

116 | \(q\bar{q} \rightarrow q'\bar{q}'\) | − | \(16\times 10^{-3}\) | \(0.5\times 10^{-3}\) |

121 | \(gg \rightarrow c'\bar{c}'\) | \(4.8\times 10^{-3}\) | 0.2 | \(21\times 10^{-3}\) |

122 | \(q\bar{q} \rightarrow c'\bar{c}'\) | − | \(4.5\times 10^{-3}\) | \(57\times 10^{-6}\) |

123 | \(gg \rightarrow b'\bar{b}'\) | \(20\times 10^{-3}\) | 0.3 | \(51\times 10^{-3}\) |

124 | \(q\bar{q} \rightarrow b'\bar{b}'\) | − | \(4.4\times 10^{-3}\) | \(53\times 10^{-6}\) |

All | 23 | 205 | 63 |

One can see that even with space-like parton showers enabled, the \(gg\rightarrow gg\) sub-process dominates. The second largest cross section is given by the \(qg\rightarrow qg\) process which is forbidden without ISR. Consequently, the fraction of di-jet events where at least one of them is quark-induced with respect to the total exclusive di-jet cross section is much higher than \(\sim \! 10^{-4}\) predicted in [2]. This fact makes it problematic to use the CEP as a pure source of gluonic jets.

Within the collinear approximation the \(gg\rightarrow q\bar{q}\) cross section is predicted to be suppressed as \(m_q^2/s\) with respect to the \(gg\rightarrow gg\) cross section. This is a well-known consequence of the spin-singlet selection rule. It is interesting that without using such a collinear approximation the exclusive production of light flavour \(q\bar{q}\) jets is not so heavily suppressed since the \(|J_z|=2\) contribution, absent in the collinear case, has a similar size and is quark-mass independent [2]. This effect is even stronger if the ISR is included.

Nevertheless, for higher \(M_{12}/M_X\) the fraction of the heavy flavours jets with respect to the whole CEP’s di-jet sample is still predicted to be lower compared to the DPE which makes such quantity a vital experimental variable for studying the transition region between CEP and DPE as was first done at the Tevatron [24].

Our DPE implementation uses diffractive parton densities, as measured *e.g.* by HERA. Specifically, we will here use the HERA H1 2006 Fit B DPDFs [25] with, for simplicity, the same soft survival probability as for the CEP process, although we are aware that the soft survival probability may very well be different for the DPE process as compared to the CEP one. Technically the DPE simulation is done in Pythia (with the same setting as for the CEP) by colliding two hadrons, the Pomerons, with energies \(\frac{1}{2}\xi _1 \sqrt{s}\) and \(\frac{1}{2}\xi _2 \sqrt{s}\) and with parton densities described by the HERA DPDFs.^{10}

In Fig. 6a and b we show the results of simply adding our CEP generated events for two different selection cuts (two jets above 10 and 25 GeV, respectively, and no third jet above 5 GeV). Further selection criteria, identical for both phase spaces, are given in [24]. We see that the addition of CEP severely overshoots the data in the exclusive region of high \(M_{12}/M_X\). There are, however, many uncertainties, especially when it comes to the soft survival probability, both for the CEP and DPE contribution. As a demonstration we show in Fig. 6c, d the effect of introducing a relative normalisation factor of 0.25 between the CEP and DPE contribution, which gives a quite reasonable description of the data. We note that our CEP, as expected, contributes quite noticeably also away from the purely exclusive region. A more detailed study of the differences between our new CEP procedure and the DPE one, especially in the regions of the Pomeron remnants in DPE, may result in observables that could further improve the experimental separation between the two processes.

### 5.2 Higgs production

The possibility to measure the Higgs boson in the central exclusive production was studied extensively in the last decade [27, 28]. The discussion was mainly focussed on the dominant decay channel \(gg\rightarrow H\rightarrow b \bar{b}\) with a standard model branching ratio of 59%.

The main advantage of this production mechanism is a huge suppression of the irreducible standard model background from \(gg\rightarrow b\bar{b}\) due to the \(J_z=0\) selection rule in CEP. Furthermore, the scalar nature of the Higgs boson means that the ratio of exclusive to inclusive cross sections is relatively enhanced as compared to the background,^{11} as the spin and colours have to match also in the inclusive sub-process. Both these effects improve the signal/background ratio for the Higgs boson production compared to the inclusive production.

The main background to the exclusive Higgs boson production comes from the \(gg\rightarrow gg\) sub-process, which can be substantially suppressed using b-jet tagging techniques. The other experimental challenge is the detection of the scattered protons in the forward detectors in a high pile-up environment where protons from several interactions can simultaneously hit the forward detector within one bunch crossing.^{12}

Note that the cross section of the Higgs boson production in CEP is only around \(2\,\text {fb}\), including the calculated soft survival probability of 0.06, which is about four order lower than the inclusive Higgs cross section \(\sim 20\,\text {pb}\). The signal event’s count is further reduced due to selection criteria and inefficiency of the b-jet tagging. In particular, the QCD background must be suppressed by selecting only high \(p_\perp \) b-jets (comparable to \(M_H/2\)) because the Higgs boson decay is isotropic whereas the QCD jet production is suppressed at high \(p_\perp \) at least as \(1./p_\perp ^4\).

We included the Higgs boson production in the process library of our program to study the production rates compared to the background processes. The simulation incorporates the parton showers as well as hadronisation of the resulting partons into “stable” particles, where the particles with lifetime higher than \(0.01\,\text {mm/c}\) are considered to be stable. The inclusion of initial-state showers have negligible effect on the exclusive Higgs cross section but can substantially increase the \(gg\rightarrow b\bar{b}\) background and spoil the signal significance (see Table 1).

^{13}To see the size of such a cross section we plot the b-jets cross section (both of them must still have \(p_\perp >50\,\text{ GeV }\)) in the mass window between 116 and \(127\,\text{ GeV }\) where the signal peak is expected. This cross section is shown in Fig. 8 as a function of the \(M_{12}/M_X\) ratio both for signal and background Monte Carlo sample. The ratio of these cross sections roughly matches the signal/background estimate. It is quite good for \(M_{12}/M_X>0.9\), which is the kinematic phase space shown in Fig. 7, whereas it deteriorates for lower values of \(M_{12}/M_X\).

To reach the acceptance of LHC forward detectors, the mass ratio must be lower than 0.6. Assuming \(0.5<M_{12}/M_X < 0.6\) the signal cross section of the b-jets production is around 0.05 fb^{14} and is around 200 times smaller than the QCD background. This small signal cross section and huge background contamination leads to a luminosity of \(\sim \! 60,000\,\text {fb}^{-1}\) to reach 4-sigma precision. Although the possibility of measure the Higgs production in this experimental setup is rather academic, our framework allows one to determine such cross sections as well as more realistically evaluate the contamination from the QCD background processes.

In Fig. 8 we also show the corresponding calculation from the DPE process, which becomes significant at low values of \(M_{12}/M_X\) both for the signal and background, but clearly does not give any increase in the significance.

### 5.3 \({\varvec{Z^0}}\) production

Considering the acceptance of the forward proton spectrometers of the ATLAS and CMS detectors, which is around \(0.015<\xi <0.1\), the mass of \(Z^0\) resonance is much smaller than the acceptance limit \(M_X\!\approx \!200\,\text{ GeV }\). This makes the study of direct production^{15} of the \(Z^0\) within the CEP mechanism even more impossible than the Higgs. Moreover, \(Z^0\) is produced by the \(q\bar{q}\rightarrow Z^0\) sub-process, which cannot be handled directly in the standard implementations of the Durham model.

However, our model allows for initial-state radiation from the partons entering to the hard sub-process, which can change the identity of incoming quarks to gluons, which can then be treated using the standard Durham exclusive luminosity. To do so, at least one \(g\rightarrow q\bar{q}\) emission from each side is needed.

Due to the colour-singlet nature of \(Z^0\) and the fusing quarks, there will probably be a non-negligible cross section for no space-like emissions and \(q\bar{q}\) CEP luminosity with a screening quark as discussed briefly above. Here, we will make no attempt to evaluate such cross section although our model can, in principle, be extended to cover this production mechanism as well.

The other mechanism for central (semi-)exclusive production of a \(Z^0\) is through DPE. To estimate such a cross section we use the procedure described in Sect. 5.1, where we again we assume that the DPE soft survival probability is the same as in CEP. Contrary to the CEP where the \(M_X\) mass is higher than \(M_Z\) mostly due to space-like emissions, in DPE both the space-like emissions and the Pomeron remnants contribute to the mass.

The shapes of the \(M_{\mu \mu }\) and \(M_{\mu \mu }/M_X\) distributions are compared in Fig. 9. The shapes of \(M_{\mu \mu }/M_X\) are rather similar for both processes, with the DPE curve somewhat shifted towards lower values as compared to the CEP one which prefers more “exclusive” configurations.

## 6 Conclusions

In this paper we introduced a new Monte Carlo implementation of the Durham formalism to calculate the central exclusive processes in *pp* and \(p\bar{p}\) collisions. Our model is based on generator, and naturally incorporates partonic showers and hadronisation, as well as multi-parton interactions.

The main advantage of our implementation is the possibility to study the effects of initial-state parton radiation on CEPs. This is done by allowing any inclusively produced sub-process to be converted to an exclusive at any stage in the shower. To do this we have implemented a colour and spin decomposition of the initial-state shower in which, together with a similarly decomposed (user supplied) matrix element, can be used to determine the probability that a given partonic state can be exclusive.

We have shown that this way of approximating higher jet multiplicities gives rise to new, non-trivial, physical consequences. In particular, for exclusive di-jet production, it leads to event topologies with medium values of \(M_{12}/M_X\), which naturally fill the gap between double Pomeron exchange and pure central exclusive production. Moreover, the incorporation of the parton showers enables the generation of quark-initiated processes such as \(Z^0\) production.

All predicted cross sections depend on the parameter \(\mu _{\mathrm{exc}}\), the scale relating the transition between the perturbative and non-perturbative region in the parton shower. The actual value of \(\mu _{\mathrm{exc}}\) will have to be determined from experiment. For the time being we set its value equal to \(1.5\,\text{ GeV }\).

The cross sections also depend on the soft survival probability used. Here we have used the MPI model in to simply estimate the probability of having no additional scatterings, equating this to the soft survival probability. Although this procedure was suggested long ago, it has not been properly investigated, and we intend to return with a detailed study of this model in a future publication.

Currently, the program process library includes QCD \(2\rightarrow 2\) processes, *H* production, \(Z^0\) production and \(\gamma \gamma \) production, but it can easily be extended. In particular, it would be interesting to add production of vector mesons (\(\rho \), \(\phi \), ...) and/or quarkonia \(\chi _{c,b}\). These processes have large cross sections which make them experimentally accessible even at low luminosities.

Our framework to treat colour and spin states within the partonic shower is rather general and can, in principle, be extended to simulate the central exclusive processes initiated by \(q\bar{q}\) fusion, in addition to standard *gg*-initiated processes. Here a screening quark rather than screening gluon is exchanged to cancel the colour flow. Such processes would be especially interesting for *e.g.* central exclusive \(Z^0\) production.

It should also be possible to extend our treatment of the colour and spin structure of the parton showers to treat final-state splittings. This would give an additional way of studying approximate higher-order effects in the hard sub-process matrix elements.

These and other possible improvements will be discussed in a future publication.

## Footnotes

- 1.
The code uses the UserHooks machinery and is available on request from the authors.

- 2.
Usually simply \(\epsilon (k/M_X) = k/M_X\).

- 3.
*i.e.*\(|\mathbf{p}_1| \ll |\mathbf{q}|\) and \(|\mathbf{p}_2| \ll |\mathbf{q}|\). - 4.
The integral over \(k_\perp \) in the definition of the Sudakov factor \(T_M\) (7) can be evaluated numerically by means of the Gauss–Kronrod quadrature formula [13]. The relative error of \(T_M\) is then typically \(10^{-16}\) if the function values in 15 points are used for the numerical integration.

- 5.
The function \(\tilde{g}\), resembling \(\phi _M\), depends also on mass

*M*. - 6.
In reality we calculate the exclusive cross sections with MPI switched on in several bins of the calculated variables to control possible kinematic dependence of the soft survival probability.

- 7.
\(p_f\) denotes flavour of parton which initiates the parton shower and \(p_h\) is the flavour of parton entering to the hard sub-process. Both partons are assumed to be on the “left” or on the “right” side.

- 8.
The colour basis of, for example, \(q\bar{q} \rightarrow gg\) process has two terms only.

- 9.
There is a possible extension of this approach to showers initiated by \(q\bar{q}\), where a screening quark rather than a screening gluon is exchanged in the loop to compensate the colour flow but this is not implemented in our current version.

- 10.
For single Pomeron processes this is now a standard option in [26], but we have here made our own simplified implementation of double Pomeron processes.

- 11.
Quantitatively, \(\frac{\hat{\sigma }^s}{\hat{\sigma }^i}(gg\rightarrow H) = 16\) (2 from spin \(\times \) 8 from colour), whereas \(\frac{\hat{\sigma }^s}{\hat{\sigma }^i}(gg\rightarrow b\bar{b}) = \frac{128}{7} \frac{m_b^2}{\hat{s}}\approx 0.02\).

- 12.
The background protons typically originate from single diffractive excitation. Two such soft single diffractive events together with one inclusive can fake the CEP topology. Fortunately, this kind of experimental background is suppressed for higher masses of the exclusive system (higher \(\xi \)).

- 13.
Due to the colour-singlet nature of Higgs production, at least two emissions are needed.

- 14.
Compare to 0.4 fb for \(M_{12}/M_X>0.9\).

- 15.
Without additional hadronic activity.

## Notes

### Acknowledgements

We are very grateful for many useful discussions with Tobjörn Sjöstrand. This work was supported in part by the MCnetITN FP7 Marie Curie Initial Training Network, Contract PITN-GA-2012-315877, the Swedish Research Council (Contracts 621-2012-2283 and 621-2013-4287)

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