Structure function calculation
The calculation of the inclusive photon-initiated cross section proceeds as described in [3]. We will briefly summarise the key elements here, but refer the reader to this work for further detailed discussion and references. For the production cross section we can write
$$\begin{aligned} \sigma _{pp}&= \frac{1}{2s} \int \mathrm{d}x_1 \mathrm{d}x_2\,\mathrm{d}^2 q_{1_\perp }\mathrm{d}^2 q_{2_\perp } \mathrm{d \Gamma } \,\alpha (Q_1^2)\alpha (Q_2^2)\nonumber \\&\qquad \frac{\rho _1^{\mu \mu '}\rho _2^{\nu \nu '} M^*_{\mu '\nu '}M_{\mu \nu }}{q_1^2q_2^2}\delta ^{(4)}\left( q_1+q_2 - k\right) , \end{aligned}$$
(1)
where \(x_i\) and \(q_{i\perp }\) are the photon momentum fractions (see [42] for precise definitions) and transverse momenta, respectively. Here the photons have momenta \(q_{1,2}\), with \(q_{1,2}^2 = -Q_{1,2}^2\), and we consider the production of a system of 4-momentum \(k = q_1 + q_2 = \sum _{j=1}^N k_j\) of N particles, where \(\mathrm{d}\Gamma = \prod _{j=1}^N \mathrm{d}^3 k_j / 2 E_j (2\pi )^3\) is the standard phase space volume. \(M^{\mu \nu }\) corresponds to the \(\gamma \gamma \rightarrow X(k)\) production amplitude, with arbitrary photon virtualities.
In the above expression, \(\rho \) is the density matrix of the virtual photon, which is given in terms of the well known proton structure functions:
$$\begin{aligned} \rho _i^{\alpha \beta }&=2\int \frac{\mathrm{d}x_{B,i}}{x_{B,i}^2} \left[ -\left( g^{\alpha \beta }+\frac{q_i^\alpha q_i^\beta }{Q_i^2}\right) F_1(x_{B,i},Q_i^2)\right. \nonumber \\&\quad \quad \left. + \frac{\left( 2p_i^\alpha -\frac{q_i^\alpha }{x_{B,i}}\right) \left( 2p_i^\beta \!-\!\frac{q_i^\beta }{x_{B,i}}\right) }{Q_i^2}\frac{ x_{B,i} }{2}F_2(x_{B,i},Q_i^2)\right] , \end{aligned}$$
(2)
where \(x_{B,i} = Q^2_i/(Q_i^2 + M_{i}^2 - m_p^2)\) for a hadronic system of mass \(M_i\). This corresponds to the general Lorentz-covariant expression that can be written down for the photon–hadron vertex, and Eq. (1), combined with a suitable input for the proton structure functions, represents the complete result we need in order to calculate the corresponding photon-initiated cross section in proton–proton collisions.
The input for the proton structure functions comes from noting that the same density matrix \(\rho \) appears in the cross section for lepton–proton scattering. One can therefore make use of the wealth of data for this process to constrain the structure functions, and hence the photon-initiated cross section, to high precision. In more detail, the structure function receives contributions from: elastic photon emission, for which we use the A1 collaboration [43] fit to the elastic proton form factors; CLAS data on inelastic structure functions in the resonance \(W^2 < 3.5\) \(\mathrm{GeV}^2\) region, primarily concentrated at lower \(Q^2\) due to the \(W^2\) kinematic requirement; the HERMES fit [44] to the inelastic low \(Q^2 < 1\) \(\mathrm{GeV}^2\) structure functions in the continuum \(W^2 > 3.5\) \(\mathrm{GeV}^2\) region; inelastic high \(Q^2 > 1\) \(\mathrm{GeV}^2\) structure functions for which the pQCD prediction in combination with PDFs determined from a global fit provide the strongest constraint (we take the ZM-VFNS at NNLO in QCD predictions for the structure functions as implemented in APFEL [45], with the MMHT2015qed_nnlo PDFs throughout, though in the MC the PDF can be set by the user). The inputs we take are as discussed in the MMHT15 photon PDF determination [46], which itself is closely based on that described in [1, 2] for the LUXqed set. However, we note that our calculation makes no explicit reference to the partonic content of the proton itself, and as discussed in detail in [3] provides by construction a more precise prediction than the result within collinear factorization that uses such a photon PDF.
Monte Carlo implementation and treatment of proton dissociation
The expression (1), in combination with the structure function inputs described above, is immediately amenable to a MC implementation of both elastic and inelastic photon-initiated production, by simply applying the elastic or inelastic structure function at the corresponding vertex. In particular, we can generate a fully differential final-state in terms of not just the centrally produced system, but the squared photon virtualities \(Q^2_i\) and the invariant masses \(M_{i}\) of the proton dissociation systems, for the case of inelastic emission, while for elastic emission the corresponding structure functions are simply \(\propto \delta (x_{B,i}-1)\), implying \(M_i=m_p\) as expected.
On the other hand, while the structure function calculation provides a precise prediction for the 4-momentum of the outgoing proton system as well as the initiating photon, to make contact with data we must also account for the decay of this system, which will as usual involve parton showering and subsequent hadronization. This is a non-trivial problem: in principle, in for example the resonance region we should take care to account for the appropriate branchings of the proton excitations, while in the high \(Q^2\) region the parton shower should match that coming from the parton-level NNLO prediction (itself an open problem) for the proton structure function.
We will take a generalised approach, which aims to capture the key physical expectations for proton dissociation via photon emission. Namely, the amount of particle production should be driven by the scale of the photon \(Q^2\) and the invariant mass \(M_i\) of the dissociation system, and should occur essentially independently of any dissociation on the other proton side, being colour disconnected from it. To achieve this, we have interfaced the appropriately formatted unweighted Les Houches events (LHE) to PYTHIA 8.2, with a suitable choice of run parameters. As this general purpose MC is set up to read in parton-level events, with collinear initiating partons, we simply map the kinematics of the \(p \rightarrow \gamma + X\) process onto the parton-level \(q\rightarrow q + \gamma \) process, where the photon 4-momenta are left unchanged, and the momentum fraction of the collinear initial-state quark is set so as to reproduce the appropriate \(x_{B,i}\), and hence invariant mass \(M_i\) of the proton final-state, provided by the structure function calculation. This may then be passed to PYTHIA for showering/hadronization. For concreteness we assign the collinear initiator to be an up quark, but the final result should not be sensitive to this choice. Indeed, a similar approach has been taken in [20], where this point was verified explicitly.
For the PYTHIA settings, we first of all set PartonLevel:MPI=off, as we only consider those events with no addition MPI, as accounted for via our calculation of the survival factor. We also use the dipole recoil scheme discussed in [47], which is specifically designed for cases where there is no colour flow between the two initiating protons (or in the parton-level LHE, quarks), as is the case here. Taking the default global recoil scheme leads to a significant overproduction of particles in the central region. A similar effect has recently been observed in [48] for the case of Higgs boson production via vector boson fusion, for which a significant enhancement of jet production at central rapidities was seen, and it was verified by comparison with the NLO calculation of Higgs boson production + 3 jets at NLO+PS accuracy that this effect was unphysical. As recommended in the PYTHIA user manual, we take SpaceShower:pTmaxMatch = 2, in order to fill the whole phase space with the parton shower, but we set SpaceShower:pTdampMatch=1 to damp emission when it is above the scale SCALUP in the LHE, which we set to the maximum of the two photon \(\sqrt{ Q_{i}^2}\); in practice, this latter option is found to have little effect on the results. We in addition set BeamRemnants:primordialKT = off, as we wish to keep the initiating quark completely collinear to fully match the kinematics from the structure function calculation.
Finally, for an elastic proton vertex, we must include the initiating photon in the event in order for Pythia to process it correctly. For the case of SD, this requires the kinematics to be modified in order to keep the elastic photon collinear and on-shell. This is achieved by setting the photon transverse momentum to zero in the event (but not in the cross section calculation), and keeping the momentum fraction fixed. We note this is only a technical necessity in order for Pythia to correctly handle the event, for the specific case of SD production, which features one elastic and one inelastic vertex. In particular we set SpaceShower:QEDshowerByQ = off, such that there is no back evolution from the photon, consistent with this being an elastic emission. Treating the initiating photon as on–shell in the event kinematics is of course an approximation to the true result, but for most purposes is a very good one.
Rapidity gaps: inclusion of survival factor
The results in the previous sections allow for a particle-level prediction of inclusive photon-initiated production, calculated with the structure function approach. However, as discussed in the Introduction, experimentally we may be interested in semi-exclusive production, that is the central production of EW objects (leptons, W pairs, sleptons, electroweakinos, ALPs...) with no additional particles produced in the final state. In this case we need to include in addition the probability of no additional particle production due to soft proton–proton interactions (i.e. underlying event activity), known as the survival factor, see [37, 38] for reviews.
We note that in all our results, and in the MC implementation, we always include this probability of no additional inelastic proton–proton interactions, however strictly speaking the result of such an interaction might still pass the experimental rapidity veto, i.e. the additional underlying event activity could lie entirely outside the rapidity veto region. However, we recall that any additional inelastic proton interactions will generate colour flow between the colliding protons, and as is well known the probability of producing such a gap from this non-diffractive interaction is exponentially suppressed by the size of the rapidity veto. Therefore for a reasonable veto region the contribution from inelastic events which pass the experimental veto should be small, with the precise amount depending on both the gap size and the \(p_\perp \) threshold, see e.g. [49] for a specific calculation. One could estimate the size of such an effect by a suitable analysis of the particle distribution generated by the model of MPI in a general purpose MC. More precisely, the value of the uncorrected survival factor should be generated using our MC, which as we discuss provides a full account of the kinematic dependence of this quantity and its variation between the different dissociative channels. From this, we could derive a contribution from the fraction \(\sim 1 - S^2\) of non-diffractive events for which additional proton–proton interactions occur, but where a rapidity gap is still present within the experimental acceptance. Roughly speaking this will be at the \(\sim \%\) level, and hence as for elastic and SD interactions (see below) we have \(S^2 \sim 0.9\), the overall correction to \(S^2\) will be extremely small, at the per mille level. For DD interactions on the other hand, where \(S^2 \sim 0.1\), we may expect the corrected survival factor to be larger by a few percent. On the other hand, this is precisely the region where the theoretical uncertainty on the original survival factor itself is larger, certainly within this percent level correction.
It should be emphasised that the impact of survival effects depends sensitively on the subprocess, through the specific proton impact parameter dependence. For example, it has been known for a long time[6, 50] that for purely elastic photon-initiated production, the low virtuality photon kinematics corresponds to a relatively large impact parameter separation between the colliding protons, and hence a smaller probability of additional proton–proton interactions, that is an average survival factor quite close to unity. Experimental evidence of this in proton–proton collisions has been seen in e.g. [32] while the same effect occurs in heavy ion collisions, for which the colliding ions are ultra-peripheral and the SM prediction for e.g. light-by-light scattering agrees well with the data [51,52,53] after including relatively mild survival effects.
To account for the above effects we follow the approach described in [18, 54]. We recap the relevant issues here, but refer the reader to these for more details. One can write the averaged survival factor as
$$\begin{aligned} \langle S_\mathrm{eik}^2\rangle = \frac{\int \mathrm{d}^2 q_{1_\perp }\,\mathrm{d}^2 q_{2_\perp }\,|T(q_{1_\perp },q_{2_\perp })+T^\mathrm{res}(q_{1_\perp },q_{2_\perp })|^2}{\int \mathrm{d}^2q_{1_\perp }\,\mathrm{d}^2q_{2_\perp }\,|T(q_{1_\perp },q_{2_\perp })|^2}\;, \end{aligned}$$
(3)
where \(q_{i_\perp }\) are as before the photon transverse momenta. Here \(T(q_{1_\perp },q_{2_\perp })\) is the amplitude corresponding to the cross section given in (1), while the so-called ‘screened’ amplitude defines the effect of potential proton–proton interactions, and is given in terms of the original amplitude T and the elastic proton–proton scattering amplitude, withFootnote 2
$$\begin{aligned} T^\mathrm{res}({q}_{1_\perp },{q}_{2_\perp }) = \frac{i}{s} \int \frac{\mathrm{d}^2 {k}_\perp }{8\pi ^2} \;T_\mathrm{el}(k_\perp ^2) \;T({q}_{1_\perp }',{q}_{2_\perp }')\;, \end{aligned}$$
(4)
where \(q_{1_\perp }=q_{1_\perp }'+k_\perp \) and \(q_{2_\perp }'=q_{1_\perp }-k_\perp \). The elastic amplitude itself can be written in impact parameter space in terms of the probability \(\exp (-\Omega (s,b_\perp ))\), that no inelastic proton–proton scattering occurs at impact parameter \(b_\perp \) between the colliding protons, where \(\Omega (s,b_\perp )\) is known as the proton opacity. As \(b_\perp \) decreases, additional short-range QCD interactions become more likely, and this probability tends to zero. Through this, the dependence of the survival factor (3) on the particular form of the amplitude T in photon \(q_{i\perp }\) and hence proton impact parameter space, enters.
In any calculation of the survival factor we are therefore interested in extracting this vector \(q_{i\perp }\) dependence of the production amplitude as accurately as possible. Now, a potential issue is that the structure function result (1) is given only at the cross section, and not amplitude level. However, as discussed in [54], at lower photon \(q_\perp \) one can isolate the dominant contribution at the amplitude level from (2), by noting that that the first term \(\sim F_1\) is suppressed by \(\sim x_i^2/x_{B,i}^2\) with respect to the second term \(\sim F_2\). For this term we can identify the contribution at the amplitude level
$$\begin{aligned} T \propto p_{1}^\mu p_{2}^\nu M_{\mu \nu } \approx \frac{q_{1\perp }^\mu q_{2 \perp }^\nu }{x_1 x_2} M_{\mu \nu } \;, \end{aligned}$$
(5)
where the second relation comes from the high energy expansion \(q_i = x_i p_i + q_{i\perp }\) and the gauge invariance of M. In other words, we apply the usual eikonal approximation for the \(p\rightarrow X + \gamma \) vertex. We therefore make use of this relation, with the subleading terms \(\propto F_1\) included incoherently in the amplitude, as in [54]. Although such a result applies equally well for the parton-level diagrams entering a pQCD calculation of the structure functions as for elastic photon emission, nonetheless in the \(Q^2>Q_0^2\) region of inelastic production, we are applying the result within collinear factorization in terms of proton quark/gluon PDFs that are defined at the cross section and not amplitude level. This may be a cause for concern, and hence at high enough \(Q^2\) we can take an alternative approach. This comes from observing that in the \(Q^2 \gtrsim Q_0^2\) regime we have \(q_{i_\perp }' \approx q_{i_\perp }\) and the amplitude T factorizes from the integral in (4). In such a case it only ever appears at the \(|T|^2\) level in (3). More precisely, one can model the \(k_\perp \) dependence entering the integral in (4) from the inelastic cross section with reference to the dependence in the case of the ‘generalized’ PDFs [56, 57], which allowed for such a non-zero momentum transfer, in terms of the proton Dirac form factor \(F_1\). This complete factorization only applies for the case that both emitted photons are emitted inelastically with \(Q^2 \gtrsim Q_0^2\), but in the mixed case where one photon is emitted elastically or inelastically but at low \(Q^2\), a similar procedure can be performed. We emphasise again that the above discussion only serves as a summary of the key element of the calculation, and refer the reader to [18] for full details.
We note that from the discussion above we expect a transition in the appropriate calculation of the survival factors to occur in the \(Q^2 \sim Q_0^2\) region, however clearly there is some ambiguity as to the precise scale below which one can/should work at the amplitude level and above which at the cross section level. In the MC we therefore include both evaluations for each point in phase space, with a smooth interpolation performed between the two regimes around \(Q^2 = Q_0^2=1\) \(\mathrm{GeV}^2\), such that the relevant amplitude (cross section) level calculation dominates in the \(Q^2 \ll Q_0^2\) (\(Q^2 \gg Q_0^2\)) regime.
What are the consequences of the above calculation for inelastic photon-initiated production? As discussed above, the survival factor depends on the photon virtuality through its effect on the impact parameter of the colliding protons. In particular, we will expect that for inelastic photon emission, the larger photon \(Q^2\) will result in a smaller survival factor and hence a suppressed rate, with this effect becoming more significant as the photon \(Q^2\) increases. Such an effect was indeed observed in the results of [18].
In Fig. 1 we show results for the soft survival factor as function of various kinematic variables for muon pair production at \(\sqrt{s}=13\) TeV with \(|p_\perp ^l|> 10\) GeV (1 GeV for the dilepton invariant mass distribution), \(|\eta _l|<2.5\), though the results are largely insensitive to this precise choice of cuts. Considering first the dependence on the dilepton invariant mass, we can see that broadly there is a large difference in the magnitude of the survival factor between the DD and elastic/SD cases, with the former being significantly smaller. This in line with the expectations above, being in particular driven by the fact that in the DD case the photon \(Q^2\) is generally much higher, and so the collision is less peripheral; the most peripheral elastic interaction has the highest survival factor. We can also see that as the invariant mass increases, the survival factor decreases, due to effect of the kinematic requirement for producing an on-shell proton at the elastic vertex for larger photon momentum fractions, which implies a larger photon \(Q^2\), see [54]. For the DD case the survival instead increases somewhat, due to the smaller phase space in photon \(Q^2\) at the highest \(M_{ll}\) values.
The photon \(Q^2\) distribution, which while not individually an observable (with the exception of the elastic case with proton tagging) is nonetheless an illustrative demonstration of the underlying physics and is plotted as well. We show the inclusive binned photon \(Q^2\) from both vertices in the elastic and DD cases, while for SD we distinguish between the elastic and inelastic photon vertices. For the elastic and SD (inelastic) cases we observe a mild reduction with increasing \(Q^2\), due to the fact that the average photon \(Q^2\) from the other, elastic, vertex is always low, leading to a peripheral interaction and higher survival factor. In contrast, for the SD (elastic) case we observe a steep fall with \(Q^2\), as now the other inelastic, vertex has a relatively large \(Q^2\) and hence the larger \(Q^2\) region on the figure corresponds to a less peripheral interaction. Note that at lower \(Q^2\), below the limit of the plot, the survival factor in the SD (elastic) case becomes larger than in the SD (inelastic) case, as required by the fact that these should integrate to the same average value. For the DD case a similar fall off is evident, with the result being constant at large enough \(Q^2\). We note that this effect occurs by construction in our approach: in particular, as discussed below (5) at larger \(Q^2\) we assume that the substructure of the squared \(p\rightarrow \gamma ^* X\) vertex can be factorized entirely from the survival factor calculation, with the \(k_\perp \) dependence (corresponding to the impact parameter profile of the interaction) given in terms of the proton Dirac form factor. Thus we see a fall off as we interpolate between the lower \(Q^2\) region, where we treat the kinematics differentially and work at the amplitude level when calculating the survival factor, and the constant tail, where we apply this factorized approach. Strictly speaking, this factorized approach is derived by considering the cross section integrated over the photon \(Q^2>Q_0^2\) and over the mass of the dissociation system. A more detailed, though necessarily rather model-dependent approach would account for this region differentially in these variables. However, we can see that in this region the survival factor is already rather low, and we consider such an effect to be within the uncertainties of the overall approach, though potentially worth further investigation in the future. We in addition note a slight kink at the transition region \(Q^2 \approx Q_0^2\) in both the DD and SD (inel.) cases, which is a feature of the fact that the calculation of the survival factor in the higher \(Q^2\) region corresponds to a slightly larger value of the \(S^2\) in this region. A more complete, and smoother treatment of this region again may warrant further investigation, but is certainly within the uncertainty of the overall approach.
We also show the dependence of the survival factor on the invariant mass of the dissociation system in the SD and DD cases and the dilepton rapidity in all cases. While the former variable is again not an observable on its own it highlights some of the underlying physics. We can see that for larger masses, where the interaction tends to be less peripheral, the survival factor becomes smaller. For the rapidity distribution, we can see in the elastic and SD cases a clear trend for the survival factor to decrease at larger rapidities. This effect is driven by the same kinematic requirement for the on-shell elastic proton as in the case of the invariant distribution above. For DD, where neither proton remains intact and hence this requirement is absent, the opposite trend is observed and the survival factor is found to increase at forward rapidities, which is driven by the smaller phase space for dissociation and hence lower average photon \(Q^2\).