Rapidity gap survival factors caused by remnant fragmentation for $W^+ W^-$ pair production via $\gamma^*\gamma^* \to W^+ W^-$ subprocess with photon transverse momenta

We calculate the cross section for $p p \to W^{+} W^-$ in the recently developed $k_{\rm T}$-factorisation approach, including transverse momenta of the virtual photons. We focus on processes with single and double proton dissociation. First we discuss the gap survival on the parton level as due to the emission of extra jet. Both the role of valence and sea contributions is discussed. The hadronisation of proton remnants is performed with {PYTHIA 8} string fragmentation model, assuming a simple quark-diquark model for proton. Highly excited remnant systems hadronise producing particles that can be vetoed in the calorimeter. We calculate associated effective gap survival factors. The gap survival factors depend on the process, mass of the remnant system and collision energy. The rapidity gap survival factor due to remnant fragmentation for double dissociative (DD) collisions ($S_{R,DD}$) is smaller than that for single dissociative (SD) process ($S_{R,SD}$). We observe the approximate factorisation $S_{R,DD} \approx (S_{R,SD})^2$, however it is expected that this property will be violated by soft rescattering effects not accounted for in this letter.


I. INTRODUCTION
The processes with partonic γγ → O 1 O 2 (O 1 and O 2 being electroweak states) subprocesses become recently very topical. Experimentally they can be separated from other competing processes by imposing rapidity gaps around the electroweak vertex. Both charged lepton pairs l + l − [1][2][3][4][5] and electroweak gauge bosons W + W − [6,7] were recently studied experimentally at the Large Hadron Collider. In particular processes with W + W − are of special interest as here one can study e.g. anomalous quartic gauge boson coupling [8,9]. Precise data may therefore provide a useful information allowing to test the Standard Model in a sector, which is so far not accessible otherwise.
There are, in general, different categories of such processes depending on whether the proton stays intact or undergoes an electromagnetic dissociation (see e.g. [10,11]).
The W + W − production in proton-proton processes via the γγ → W + W − subprocess was recently studied in collinear [12] and transverse momentum dependent factorisation [13] approaches.
Without additional requirements it is impossible to separate the γγ → W + W − mechanism from qq → W + W − , gg → W + W − or higher-order QCD processes. To enhance the sample for the wanted mechanism one may impose a rapidity gap condition around the e + µ − or e − µ + vertex 1 .
In Fig. 1 we show a schematic picture of the single and double dissociative two-photon processes. In our recent paper [13] we have shown that rather large photon virtualities and large mass proton excitation are characteristic for the γγ → W + W − induced processes. The highly excited hadronic systems hadronise producing (charged) particles that may destroy the rapidity gap around the e + µ − or e − µ + vertex. The minimal requirement is to impose a condition of no charged particles in the main ATLAS or CMS trackers.
We will focus on such effects in the present letter. The hadronisation of the proton remnants will be performed and conditions on charged particles will be imposed. Our main 1 The e + e − or µ + µ − final states seems less useful due to contamination by the Drell-Yan processes.
aim is to estimate gap survival factor associated with the remnant hadronisation which destroys the rapidity gap. Dependence on kinematic variables will be studied. As has been stressed in [14], the ordinary collinear photon parton distribution functions (PDFs) -which imply a fully inclusive sum over remnant final states -cannot be used if additional gap requirements are imposed on the final states. For this purpose, in Ref. [14] a concept of "photon PDF in events with rapidity gaps" was introduced. There a requirement, that the parton emissions related to evolution do not contaminate the central rapidity region is implemented. The authors tried to answer the question how to approximately modify the collinear photon PDF to include rapidity gap requirement(s) used in modern experiments. The calculations in [14] are kept at the parton level, and no explicit remnant hadronisation effects were discussed there.
Remnant fragmentation is not the only effect that can destroy the rapidity gap. There are also possible interactions between remnant systems X and Y which can produce additional particles, for a recent review, see [15]. The gap survival factors for these processes are beyond the scope of the present letter. For recent estimates in photon induced processes, see e.g. [14,16,17].

II. SKETCH OF OUR CALCULATIONAL SCHEME
We calculate cross section for the pp → W + W − reaction with double proton dissociation as: Here y ± are the rapidities and p ± ⊥ the transverse momenta of W ± bosons. The M X -dependent photon fluxes can be decomposed into fluxes corresponding to the relevant proton staying intact or dissociating (see Fig. 1): and similarly for ( , so that the cross section for single dissociative process is less differential, as one of the two integrations over the remnant masses is unnecessary. These photon fluxes can be understood as a type of fully unintegrated parton distributions [18]. They allow us to generate events containing remnants of mass M X , M Y . The details of the relation of the photon fluxes to proton structure functions and the used matrix element M (λ W + , λ W − ; k 1⊥ , k 2⊥ ) can be found in [13] and references therein.
We use an implementation of the above process in CepGen [19] for the Monte-Carlo generation of unweighted events.
The hadronisation of remnant states X and/or Y systems is performed using the Lund fragmentation algorithm implemented in PYTHIA 8 [20], and interfaced to CepGen. We model the incoming photon as emitted from a valence (up) quark collinear to the incoming proton direction. Other flavour combinations are also expected to contribute to the process, but we observe the kinematics of the outgoing X and Y systems is not sensitive to this choice. The fractional quark momentum x Bj is determined event-by-event from the photon virtuality Q 2 and the relevant remnant mass M X through: We check the condition for each "stable" (pions, kaons, protons, . . . ) charged particle produced in the hadronisation of the X and Y remnants: Each event for which at least one charged particle fulfils condition (2.3) is discarded. We introduce the ratio: where ω denotes a set of kinematic variables describing details of the reaction. S R (ω) can be considered a phase-space-point-dependent rapidity gap survival factor associated with remnant(s) fragmentation. For example we will show such number for different ranges of masses of the produced system both for double and single dissociation.

III. NUMERICAL RESULTS
Here we wish to present some results of our Monte Carlo simulations. We consider separately the case of double dissociation as well as the case of single dissociation. The most important ingredient of our calculation is realistic hadronisation of the proton remnants, which allows to estimate gap survival factor associated with spoiling the rapidity gap in the central pseudorapidity region, assumed here to be −2.5 < η < 2.5, by individual (charged!) particles. This corresponds to a realistic situation relevant for recent CMS [6] and ATLAS [7] measurements.
As mentioned in the Introduction, in the present letter we will not discuss effects related to soft interactions between remnants. They are rather small and similar for different categories of processes (elastic-elastic, elastic-inelastic, inelastic-inelastic). In general, they are expected to be dependent on the W + W − invariant mass M W W [16,17] as well as collisionenergy dependent [17]. It was shown e.g. in [13] that: Can this ordering be changed when the rapidity gap requirement is taken into account? As will be shown below, absorption effects are the biggest for inelastic-inelastic processes, so that in principle the ordering in (3.1) can be changed when a rapidity veto is imposed.

A. Double dissociation
We start from the largest contribution, in the inclusive case, the inelastic-inelastic (double dissociative) [13] processes. In this case both remnants fragment and we have to include their fragmentation simultaneously. In Fig. 2 we show two-dimensional distributions in pseudorapidity of particles from X (η ch X ) and Y (η ch Y ) for different ranges of masses of the centrally produced system. For illustration we show by the thin square the region relevant for ATLAS and CMS pseudorapidity coverage. One can see that the gap survival weakly depends on the invariant mass of the centrally produced system.
We quantify this effect, see Table I, by showing average remnant rapidity gap factors for different ranges of M W W masses. The remnant rapidity gap survival factor at fixed η cut becomes larger at higher collision energies. We hope this can be verified by the new data for √ s = 13 TeV. We will leave it for future studies.
In Fig. 3 we show the distribution in η cut for the double dissociation process. We predict a strong dependence on η cut . It would be valuable to perform experimental measurements   with different η cut .

B. Single dissociation
We repeat a similar analysis for the single dissociative process. In Fig. 4 we show the rapidity distribution of charged particles produced in the fragmentation of the X system. The contamination of the detector is only weakly correlated with the mass of the centrally produced system.
Again we quantify the effect by showing the average remnant rapidity gap survival factor for the same windows of M W W . The conclusions here are similar as for the double dissociation, except that the effect of destroying the rapidity gap is smaller.
In Table II we show the rapidity gap survival factor for single dissociation processes and those values squared. By comparing to the results collected in Table I we observe that with good precision:   Table I.
Such an effect is naively expected when the two fragmentations are independent, which is the case by the model construction.
In Fig. 5 we show distribution in η cut for single dissociative process. The numbers here are somewhat larger than those shown in Fig. 3, consistently with factorisation. Detailed inspection shows (3.2) holds for each M W W .
Very useful can be the dependence of the rapidity gap survival factor on the mass of the dissociated protonic system. The corresponding result is shown in Fig. 6. We observe that for an η cut value of 2.5 the rapidity gap survival factor S R stays very close to 1 for M max X < 100 GeV. Increasing the mass of the dissociative system leads to graduate destroying of the (pseudo)rapidity gap, arbitrarily fixed here to be −2.5 < η < 2.5 (ATLAS, CMS).
From Fig. 6 one may infer which masses can be allowed in the dissociation still ensur-  ing the gap and avoiding the more complicated Monte Carlo simulations of the remnant hadronisation.
The hadronisation part is independent of the system centrally produced. Hence, this method can be used to perform calculations for processes for which there are no direct procedures to perform full Monte Carlo simulations.

IV. CONCLUSIONS
In the present letter we discuss the quantity called "remnant gap survival factor" for the pp → W + W − reaction initiated via photon-photon fusion. We use a recent formalism developed for the inclusive case [13] which includes transverse momenta of incoming photons. This formalism has been supplemented here by including remnant fragmentation that can spoil the rapidity gap usually used to select the wanted subprocess of interest. We quantify this effect by defining the remnant gap survival factor which in general depends on the reaction, kinematic variables and details of the experimental set-ups. We discuss dependence on invariant mass of the produced W + W − central system. We find different values for double and single dissociative processes. In general, S R,DD < S R,SD and S R,DD ≈ (S R,SD ) 2 . Furthermore the larger η cut (uppper limit on charged particles pseudorapidity) the smaller rapidity gap survival factor S R , both for double and single dissociation. Finally the effect becomes smaller for larger collision energies. We have found that the crucial variable for S R is (are) masses of the final protonic systems.
The present approach is a step towards realistic modelling of gap survival in photon induced interactions and definitely requires further detailed studies and comparisons to the existing and future experimental data. In the present analyses we have neglected other effects such as soft interactions or multiple-parton interactions (see e.g. [15,21]). More detailed studies including such effects in a consistent manner will be given elsewhere.