Measurement of single-diffractive dijet production in proton-proton collisions at $\sqrt{s} =$ 8 TeV with the CMS and TOTEM experiments

Measurements are presented of the single-diffractive dijet cross section and the diffractive cross section as a function of the proton fractional momentum loss $\xi$ and the four-momentum transfer squared $t$. Both processes pp$\to$pX and pp$\to$Xp, ie with the proton scattering to either side of the interaction point, are measured, where X includes at least two jets; the results of the two processes are averaged. The analyses are based on data collected simultaneously with the CMS and TOTEM detectors at the LHC in proton-proton collisions at $\sqrt{s}$= 8 TeV during a dedicated run with $\beta^{\ast} =$ 90 m at low instantaneous luminosity and correspond to an integrated luminosity of 37.5 nb$^{-1}$. The single-diffractive dijet cross section $\sigma^\mathrm{pX}_{\mathrm{jj}}$, in the kinematic region $\xi \lt$ 0.1, 0.03 $\lt |$t$| \lt 1$ GeV$^2$, with at least two jets with transverse momentum $p_\mathrm{T}>$ 40 GeV, and pseudorapidity $|\eta| \lt$ 4.4, is 21.7$\pm$0.9 (stat)$^{+3.0}_{-3.3}$ (syst) $\pm$ 0.9 (lum) nb. The ratio of the single-diffractive to inclusive dijet yields, normalised per unit of $\xi$, is presented as a function of $x$, the longitudinal momentum fraction of the proton carried by the struck parton. The ratio in the kinematic region defined above, for $x$ values in the range $-$2.9 $\leq \log_{10} x \leq$ $-$1.6, is $R = (\sigma^\mathrm{pX}_{\mathrm{jj}}/\Delta\xi)/\sigma_{\mathrm{jj}} =$ 0.025$\pm$0.001 (stat) $\pm$ 0.003 (syst), where $\sigma^\mathrm{pX}_{\mathrm{jj}}$ and $\sigma_{\mathrm{jj}}$ are the single-diffractive and inclusive dijet cross sections, respectively. The results are compared with predictions from models of diffractive and nondiffractive interactions. Monte Carlo predictions based on the HERA diffractive parton distribution functions agree well with the data when corrected for the effect of soft rescattering between the spectator partons.

pX jj , in the kinematic region ξ < 0.1, 0.03 < |t| < 1 GeV 2 , with at least two jets with transverse momentum p T > 40 GeV, and pseudorapidity |η| < 4.4, is 21.7 ± 0.9 (stat) +3.0 −3.3 (syst) ± 0.9 (lumi) nb. The ratio of the single-diffractive to inclusive dijet yields, normalised per unit of ξ, is presented as a function of x, the longitudinal momentum fraction of the proton carried by the struck parton. The ratio in the kinematic region defined above, for x values in the range −2.9 ≤ log 10 x ≤ −1.6, is R = (σ pX jj /∆ξ)/σ jj = 0.025 ± 0.001 (stat) ± 0.003 (syst), where σ pX jj and σ jj are the single-diffractive and inclusive dijet cross sections, respectively. The results are compared with predictions from models of diffractive and nondiffractive interactions. Monte Carlo predictions based on the HERA diffractive parton distribution functions agree well with the data when corrected for the effect of soft rescattering between the spectator partons.
We dedicate this paper to the memory of our colleague and friend Sasha Proskuryakov, who started this analysis but passed away before it was completed. His contribution to the study of diffractive processes at CMS is invaluable.

The CMS and TOTEM detectors
The central feature of the CMS apparatus is a superconducting solenoid of 6 m internal diameter, providing a magnetic field of 3.8 T. Within the superconducting solenoid volume are a silicon pixel and strip tracker, a lead tungstate crystal electromagnetic calorimeter (ECAL), and a brass and scintillator hadron calorimeter (HCAL), each composed of a barrel and two endcap sections. Forward calorimeters extend the pseudorapidity coverage provided by the barrel and endcap detectors. The forward hadron (HF) calorimeter uses steel as an absorber and quartz fibers as the sensitive material. The two HFs are located 11.2 m from the interaction region, one on each end, and together they provide coverage in the range 3.0 < |η| < 5.2. Muons are measured in gas-ionisation detectors embedded in the steel flux-return yoke outside the solenoid.
When combining information from the entire detector, including that from the tracker, the jet energy resolution amounts typically to 15% at 10 GeV, 8% at 100 GeV, and 4% at 1 TeV, to be compared to about 40, 12, and 5%, respectively, obtained when ECAL and HCAL alone are used. In the region |η| < 1.74, the HCAL cells have widths of 0.087 in pseudorapidity and 0.087 in azimuth (φ). In the η-φ plane, and for |η| < 1.48, the HCAL cells map on to 5×5 arrays of ECAL crystals to form calorimeter towers projecting radially outwards from close to the nominal interaction point. For |η| > 1.74, the coverage of the towers increases progressively to a maximum of 0.174 in ∆η and ∆φ. Within each tower, the energy deposits in the ECAL and HCAL cells are summed to define the calorimeter tower energies, subsequently used to provide the energies and directions of hadronic jets.
The reconstructed vertex with the largest value of summed charged-particle track p 2 T is taken to be the primary interaction vertex. Tracks are clustered based on the z coordinate of the track at the point of closest approach to the beamline. In the vertex fit, each track is assigned a weight between 0 and 1, which reflects the likelihood that it genuinely belongs to the vertex. The number of degrees of freedom in the fit is strongly correlated with the number of tracks arising from the interaction region.
The particle-flow (PF) algorithm [19] aims to reconstruct and identify each individual particle in an event with an optimised combination of information from the various elements of the CMS detector. The energy of photons is directly obtained from the ECAL measurement, corrected for zero-suppression effects. The energy of electrons is determined from a combination of the electron momentum at the primary interaction vertex as determined by the tracker, the energy of the corresponding ECAL cluster, and the energy sum of all bremsstrahlung photons spatially compatible with originating from the electron track. The energy of muons is obtained from the curvatures of the corresponding track. The energy of charged hadrons is determined from a combination of their momentum measured in the tracker and the matching ECAL and HCAL energy deposits, corrected for zero-suppression effects and for the response function of the calorimeters to hadronic showers. Finally, the energy of neutral hadrons is obtained from the corresponding corrected ECAL and HCAL energies.
features originating from isolated noise patterns in some HCAL regions [23].
A more detailed description of the CMS detector, together with a definition of the coordinate system used and the relevant kinematic variables, can be found in Ref. [24].
The TOTEM experiment [25,26] is located at the LHC interaction point (IP) 5 together with the CMS experiment. The RP system is the subdetector relevant for measuring scattered protons. The RPs are movable beam pipe insertions that approach the LHC beam very closely (few mm) to detect protons scattered at very small angles or with small ξ. The proton remains inside the beam pipe and its trajectory is measured by tracking detectors installed inside the RPs. They are organised in two stations placed symmetrically around the IP; one in LHC sector 45 (positive z), the other in sector 56 (negative z). Each station is formed by two units: near (215 m from the IP) and far (220 m from the IP). Each unit includes three RPs: one approaching the beam from the top, one from the bottom and one horizontally. Each RP hosts a stack of 10 silicon strip sensors (pitch 66 µm) with a strongly reduced insensitive region at the edge facing the beam (few tens of µm). Five of these planes are oriented with the silicon strips at a +45 • angle with respect to the bottom of the RP and the other five have the strips at a −45 • angle.
The beam optics relates the proton kinematics at the IP and at the RP location. A proton emerging from the interaction vertex (x * , y * ) at horizontal and vertical angles θ * x and θ * y , with a fractional momentum loss ξ, is transported along the outgoing beam through the LHC magnets. It arrives at the RPs at the transverse position: relative to the beam centre. This position is determined by the optical functions, characterising the transport of protons in the beamline and controlled via the LHC magnet currents. The effective length L x,y (z), magnification v x,y (z) and horizontal dispersion D x (z) quantify the sensitivity of the measured proton position to the scattering angle, vertex position, and fractional momentum loss, respectively. The dispersion in the vertical plane, D y , is nominally zero.
For the present measurement, a special beam optical setup with β * = 90 m was used, where β * is the value of the amplitude function of the beam at the IP. This optical setup features parallelto-point focussing (v y ∼ 0) and large L y , making y at RP directly proportional to θ * y , and an almost vanishing L x and v x , implying that any horizontal displacement at the RP is approximately proportional to ξ. Protons can hence be measured with large detector acceptance in the vertical RPs that approach the beam from the top and bottom.
To reduce the impact of imperfect knowledge of the optical setup, a calibration procedure [27] has been applied. This method uses elastic scattering events and various proton observables to determine fine corrections to the optical functions presented in Eq. (1). For the RP alignment, a three-step procedure [26] has been applied: beam-based alignment prior to the run (as for the LHC collimators) followed by two offline steps. First, track-based alignment for the relative positions among RPs, and second, alignment with elastic events for the absolute position with respect to the beam. The final uncertainties per unit (common for top and bottom RPs) are: 2 µm (horizontal shift), 100 µm (vertical shift), and 0.2 mrad (rotation about the beam axis).
The kinematic variables (ξ, θ * x , θ * y as well as t) are reconstructed with the use of parametrised proton transport functions [26]. The values of the optical functions vary with ξ, an effect that is taken into account by the optics parametrisation. The details of the reconstruction algorithms and optics parametrisation are discussed in Refs. [26,28]. The momentum loss reconstruction depends mostly on the horizontal dispersion, which is determined with a precision better than 10%. The scattering angle resolution depends mainly on the angular beam divergence and in the horizontal plane also on the detector resolution, whereas the momentum loss resolution depends mainly on the optics [29]. The ξ resolution is about σ(ξ) = 0.7% and the θ * y and the θ * x resolutions 2.4 µrad and 25 µrad, respectively.
3 Event kinematics Figure 1 shows a schematic diagram of the single-diffractive reaction pp → Xp with X including two high-p T jets. Single-diffractive dijet production is characterised by the presence of a high-energy proton, which escapes undetected by the CMS detector, and the system X, which contains high-p T jets, separated from the proton by an LRG.  Figure 1: Schematic diagram of single-diffractive dijet production. The exchange of a virtual object with the vacuum quantum numbers (i.e. a Pomeron) is indicated by the symbol IP. The diagram shows an example of the gg → dijet hard scattering process; the qq and gq initial states also contribute.
The proton is scattered at small angles, has small fractional momentum loss ξ = 1 − |p f |/|p i |, and small absolute value of the 4-momentum transfer squared t = (p f − p i ) 2 , where p i and p f are the four-momenta of the incoming and outgoing protons, respectively. The scattered proton does not leave the beam pipe and can only be detected by using the TOTEM RP detectors, which make a direct measurement of t and ξ (hereafter referred to as ξ TOTEM ).
If only CMS information is used, as in Ref.
[12], ξ can be estimated only from the energies and longitudinal momenta of the particles measured in CMS: where the sum is carried out with PF objects. The positive (negative) sign corresponds to the scattered proton moving towards the positive (negative) z direction. In this case, t cannot be measured.
The combination of the limited CMS pseudorapidity coverage (|η| < 5) and the detector inefficiency causes ξ CMS to be smaller than ξ TOTEM in general, i.e. ξ CMS − ξ TOTEM ≤ 0. However, the limited detector resolution may cause ξ CMS to be larger than ξ TOTEM .
The momentum fraction of the partons initiating the hard scattering, x + and x − , can be esti-mated from the energies and longitudinal momenta of the measured jets as: where the sum is carried out over the two highest transverse momentum jets in the event, and an additional third jet, if present. The positive (negative) sign corresponds to the incoming proton moving towards the positive (negative) z direction.
Finally, the fraction β of the Pomeron momentum carried by the interacting parton is measured from the values of x ± and ξ TOTEM as β = x ± /ξ TOTEM .

Data samples
The data were collected in July 2012 during a dedicated run with low probability (∼6-10%) of overlapping pp interactions in the same bunch crossing (pileup) and a nonstandard β * = 90 m beam optics configuration. These data correspond to an integrated luminosity of L = 37.5 nb −1 . Events were selected by trigger signals delivered simultaneously to the CMS and TOTEM detectors. The CMS orbit-counter reset signal, delivered to the TOTEM electronics at the start of the run, assures the time synchronisation of the two experiments. The CMS and the TOTEM events are combined offline based on the LHC orbit and bunch numbers.
Single-diffractive dijet events are simulated with the PYTHIA8 and POMWIG (version 2.0) [36] generators. Hard diffraction is simulated in PYTHIA8 using an inclusive diffraction model, where both low-and high-mass systems are generated [37]. High-mass diffraction is simulated using a perturbative description. Pomeron parton densities are introduced and the diffractive process is modelled as a proton-Pomeron scattering at a reduced centre-of-mass energy. The default generator settings are used, including that for the proton-Pomeron total cross section. Multiparton interactions (MPI) are included within the proton-Pomeron system to provide cross sections for parton-parton interactions. In this model, the presence of secondary interactions does not lead to a suppression of the visible diffractive cross section.
Additionally, PYTHIA8 implements a model to simulate hard-diffractive events based on a direct application of dPDFs, and a dynamical description of the rapidity gap survival probability in diffractive hadron-hadron interactions [38]. In this model an event is classified as diffractive only when no MPI are generated. We refer to this implementation as the dynamic gap (DG) model. Single-diffractive dijet events using the inclusive diffraction model are simulated with PYTHIA8, tunes 4C and CUETP8M1. The simulation of diffractive dijet events using the DG model is performed with PYTHIA8 version 8.223 [38] with the underlying event tune CUETP8M1. These PYTHIA8 tunes give a fair description of the charged-particle pseudorapidity and p T distributions in a sample with a large fraction of single-diffractive inelastic events [35,39,40].
The POMWIG generator is based on HERWIG6 and implements dPDFs to simulate harddiffractive processes. The simulation uses dPDFs from a fit to deep inelastic scattering data (H1 fit B [2]). The POMWIG generator uses a next-to-leading order dPDF fit, whereas PYTHIA8 uses a leading order dPDF fit. When using POMWIG, a constant factor S 2 = 7.4% is applied to account for the rapidity gap survival probability leading to the suppression of the diffractive cross section. This value is calculated from the ratio of the measured diffractive cross section and the prediction from POMWIG, as described in Section 8.2. Both Pomeron and Reggeon exchange contributions are generated. Reggeon exchange is not simulated in PYTHIA8.
To improve the description of the data by the MC samples, correction factors are applied eventby-event as a function of β, by a reweighting procedure. The correction modifies the event distribution as a function of β by up to 40%. The effect is considerably smaller for other distributions.
The generated events are processed through the simulation of the CMS detector, based on GEANT4 [41], and reconstructed in the same manner as the data. The acceptance and resolution of the TOTEM RP detectors are parametrised as a function of the proton kinematics, as discussed below. All samples are simulated without pileup.

Roman pot detectors acceptance and resolution
The proton path from the IP to the TOTEM RPs is calculated using a parametrisation of the LHC optics [27]. To obtain a realistic simulation of the scattered proton, the following procedure is used: • Proton transport: The simulation of the RP detectors acceptance is parametrised in terms of the vertex position, the proton scattering angles at the vertex θ * x and θ * y , and ξ. The incident beam energy spread and beam divergence are also simulated [29]. • Reconstruction of t and ξ: The detector-level distributions of t and ξ are obtained from the scattering angles θ * x and θ * y , where the correlation between the ξ and θ * x uncertainties is taken into account [26]. The generated values of θ * x and θ * y are spread by 25 µrad and 2.4 µrad, respectively. These values include the effects of detector resolution, as well as those of the beam optics and the beam divergence.
• Proton reconstruction inefficiency: The track reconstruction in the RPs may fail for several reasons: inefficiency of the silicon sensors, interaction of the proton with the RP mechanics, or the simultaneous presence of a beam halo particle or pileup. The silicon strips of the detectors in an RP are oriented in two orthogonal directions; this allows for good rejection of inclined background tracks, but makes it very difficult to reconstruct more than one track almost parallel to the beam direction [26]. These uncorrelated inefficiencies are evaluated from elastic scattering data [29], and amount to ∼6%. To correct for this, an extra normalisation factor is applied, obtained separately for protons traversing the RPs on either side of the IP.

Event selection
Dijet events are selected online by requiring at least two jets with p T > 20 GeV [42]. The efficiency of this trigger selection is estimated with a sample of minimum bias events, i.e. events collected with a loose trigger intended to select inelastic collisions with as little bias as possible, and containing a leading jet with p T , as reconstructed offline, of at least 40 GeV. The fraction of dijet events accepted by the trigger is calculated as a function of the subleading jet p T . The efficiency is above 94% for p T > 40 GeV.
The offline selection requires at least two jets with p T > 40 GeV and |η| < 4.4. Jets are recon-structed from PF objects with the anti-k T algorithm with a distance parameter R = 0.5. The reconstructed jet energy is corrected with the procedure illustrated in Ref. [22]. The parton momentum fractions x + and x − are reconstructed using Eq. (3) from the two highest transverse momentum jets and an additional third jet, if present. The latter is selected with p T > 20 GeV. In addition, the selection requires at least one reconstructed primary interaction vertex and at least one reconstructed proton track in the RP stations. The fit of the reconstructed vertex is required to have more than four degrees of freedom.
Events with protons in the RP stations on both sides are rejected if their kinematics are consistent with those of elastic scattering. Elastic scattering events, which are present in the data sample because of pileup, are identified by the presence of two proton tracks in opposite directions, in a diagonal configuration: the protons traverse the two top RPs in sector 45 and the two bottom RPs in sector 56, or vice versa. The horizontal and vertical scattering angles are required to match within the measured resolutions. These requirements are similar to those described in Ref. [29].
To avoid detector edges with rapidly varying efficiency or acceptance, as well as regions dominated by secondary particles produced by aperture limitations in the beamline upstream of the RPs, proton track candidates are selected if the corresponding hit coordinates on the RP stations satisfy the following fiducial requirements: 0 < x < 7 mm and 8.4 < |y| < 27 mm, where x and y indicate the horizontal and vertical coordinates of the hit with respect to the beam.
To suppress background from secondary particles and pileup in the RPs, the reconstructed proton track is selected if it is associated to one track element in both top or both bottom RPs on a given side. The kinematic requirements 0.03 < |t| < 1.0 GeV 2 and 0 < ξ TOTEM < 0.1 are then applied.
For signal events, one expects ξ CMS to be smaller than ξ TOTEM , i.e. ξ CMS − ξ TOTEM ≤ 0 (as discussed in Section 3). This selection is imposed to suppress the contribution of pileup and beam halo events, in which the proton is uncorrelated with the hadronic final state X measured in the CMS detector. Roughly 6% of signal events are rejected by this requirement, as estimated from a simulation of single-diffractive dijet production. Table 1 shows the number of events passing each selection. The number of events with the proton detected in the RPs in sector 45 (56) after all the selections is 368 (420).

Background
The main background is due to the overlap of a pp collision in the CMS detector and an additional track in the RP stations, originating from either a beam halo particle or an outgoing proton from a pileup interaction.
Pileup and beam halo events are not simulated, but they are present in the data. To estimate the pileup and beam halo contribution in the data, a zero bias sample consisting of events from randomly selected, nonempty LHC bunch crossings is used. Events with a proton measured in the RP stations and with any number of reconstructed vertices are selected from the zero bias data set. Such events are denoted by ZB in the following.
The RP information from events in the zero bias sample is added to the diffractive and nondiffractive samples produced with POMWIG and PYTHIA6, respectively, to describe background events with a proton.
The POMWIG sample is normalised assuming a rapidity gap survival probability factor of 7.4%, as discussed in Section 5. A mixture of MC and ZB events is then passed through the selection procedure illustrated in Section 6, except for the requirement ξ CMS − ξ TOTEM ≤ 0, which is not applied.
Such mixed events with a proton in the RPs are considered as signal if the proton originates from the MC simulated sample, or as background if it originates from the ZB sample. If an event has a proton from both the MC sample and the ZB sample, the proton with smaller ξ is chosen. However, the probability of such a combination is small and none of these events pass all the selections. Figure 2 shows the distribution of ξ CMS − ξ TOTEM for the data compared to the MC+ZB event mixture. The requirement ξ CMS − ξ TOTEM ≤ 0 selects signal events and rejects the kinematically forbidden region populated by the MC+ZB background events (filled histogram). The background distribution is normalised to the data in the ξ CMS − ξ TOTEM region from 0.048 to 0.4, which is dominated by background events.
The background is estimated separately for events with a proton traversing the two top (toptop) or the two bottom (bottom-bottom) RPs on each side. The top-top and bottom-bottom distributions are similar. Figure 2 shows the sum of the two contributions.
The background contribution for events with a proton detected in sector 56 (right panel of Fig. 2) is larger than that for events with a proton detected in sector 45 (left panel of Fig. 2). The remaining contamination of background in the signal region is estimated to be 15.7% for events in which the proton is detected in sector 45 and 16.8% for those in which the proton is detected in sector 56. Figure 3 shows the distribution of ξ TOTEM for the data and the MC+ZB sample, before and after the ξ CMS − ξ TOTEM ≤ 0 requirement, as well as the distribution of t, after the ξ CMS − ξ TOTEM ≤ 0 selection. The sum of the top-top and bottom-bottom combinations is used. The data and the MC+ZB sample are in good agreement.
An alternative method, used at HERA [4], takes two events randomly chosen from the data sample. First, ξ CMS is sampled from events that have passed the dijet selection; ξ TOTEM is then taken from events with ξ CMS > 0.12 that have passed the event selection described in Section 6, except for the ξ CMS − ξ TOTEM requirement, to select proton tracks considered to be mostly from background. These two values are used to plot the ξ CMS − ξ TOTEM distribution, which is normalised to the data in a region dominated by background. The remaining contamination in the signal region is ∼19% both for events with a proton detected in sector 45 and for those with a proton in sector 56. The ZB method is used in this analysis. Half the difference between the results of the two methods is taken as an estimate of the systematic uncertainty of the background subtraction procedure.

Results
In this section the measurements of the differential cross sections dσ/dt, dσ/dξ, and the ratio R(x) of the single-diffractive to inclusive dijet cross sections are presented. The ratio R(x), normalised per unit of ξ, is defined by: where ∆ξ = 0.1.
The cross sections are calculated in the kinematic region ξ < 0.1, 0.03 < |t| < 1 GeV 2 , with at least two jets at a stable-particle level with p T > 40 GeV and |η| < 4.4. The ratio R(x) is calculated for x values in the region −2.9 ≤ log 10 x ≤ −1.6. In the following, the estimated background is subtracted from the number of single-diffractive dijet candidates following the procedure described in the previous section.
The differential cross sections for dijet production in bins of t and ξ are evaluated as: where N i jj is the measured number of single-diffractive dijet candidates in the i-th bin after subtracting the estimated background; ∆t i and ∆ξ i are the bin widths, and L is the integrated luminosity. The factors A i include the effects of the geometrical acceptance of the apparatus. by unfolding the data using the D'Agostini method with early stopping [43]. In this method the regularisation parameter is the number of iterations used, which is optimised to obtain a relative χ 2 variation between iterations lower than 5%.
The ratio R(x) of the single-diffractive to inclusive dijet cross sections is evaluated as a function of x as: where N pX jj is the number of single-diffractive dijet candidates with ξ TOTEM < 0.1, and N jj is the total number of dijet events without the requirement of a proton detected in the RPs. This number is dominated by the nondiffractive contribution. The symbol A CMS-TOTEM indicates the acceptance of CMS and TOTEM for single-diffractive dijet events, evaluated with POMWIG, PYTHIA8 4C and PYTHIA8 CUETP8M1; A CMS is the acceptance for nondiffractive dijet production (p T > 40 GeV, |η| < 4.4), evaluated with PYTHIA6, PYTHIA8 4C, PYTHIA8 CUETP8M1, PYTHIA8 CUETP8S1, and HERWIG6. The acceptance includes unfolding corrections to the data with the D'Agostini method with early stopping, denoted by the symbol U in Eq. (6).

Systematic uncertainties
The systematic uncertainties are estimated by varying the selections and modifying the analysis procedure, as discussed in this Section. Tables 2 and 3 summarise the main systematic uncertainties of the single-diffractive cross section and the ratio of the single-diffractive and inclusive dijet cross sections, respectively, presented in Sections 8.2 and 8.3.
• Trigger efficiency: The trigger efficiency is calculated as a function of the subleading jet p T using a fit to the data. The sensitivity to the trigger efficiency determination is estimated by varying the fit parameters within their uncertainties. This variation corresponds to a trigger efficiency that increases or decreases by roughly 2% at jet p T = 40 GeV and less than 1% at p T = 50 GeV. • Calorimeter energy scale: The reconstruction of ξ CMS is affected by the uncertainty in the calorimeter energy scale and is dominated by the HF contribution. This uncertainty is estimated by changing the energy of the PF candidates by ±10% [12,44]. • Jet energy scale and resolution: The energy of the reconstructed jets is varied according to the jet energy scale uncertainty following the procedure described in Ref. [22]. The systematic uncertainty in the jet energy resolution is estimated by varying the scale factors applied to the MC, as a function of pseudorapidity. The uncertainties obtained from the jet energy scale and resolution are added in quadrature. The effect of the jet energy resolution uncertainty amounts to less than 1% of the measured cross section.
• Background: Half the difference between the results of the ZB and HERA methods used to estimate the background, described in Section 7, is an estimate of the effect of the systematic uncertainty of the background.
• RP acceptance: The sensitivity to the size of the fiducial region for the impact position of the proton in the RPs is estimated by modifying its vertical boundaries by 200 µm and by reducing the horizontal requirement by 1 mm, to 0 < x < 6 mm. Half the difference of the results thus obtained and the nominal ones is used as a systematic uncertainty. The uncertainties obtained when modifying the vertical and horizontal boundaries are added in quadrature.
• Resolution: The reconstructed variables t and ξ are calculated by applying two methods: either directly, with a resolution function depending on each of these variables, or indirectly from the scattering angles θ * x and θ * y . Half the difference between the results using the two methods is taken as a systematic uncertainty.
• Horizontal dispersion: The reconstructed ξ value depends on the optical functions describing the transport of the protons from the interaction vertex to the RP stations, and specifically the horizontal dispersion. This uncertainty is calculated by scaling the value of ξ by ±10%. This value corresponds to a conservative limit of the possible horizontal dispersion variation with respect to the nominal optics.
• t-slope: The sensitivity to the modelling of the exponential t-slope is quantified by replacing its value in POMWIG by that measured in the data. Half the difference between the results thus found and the nominal results is used as an estimate of the uncertainty.
• β-reweighting: Half the difference of the results with and without the reweighting as a function of β in POMWIG (as discussed in Section 5) is included in the systematic uncertainty. The effect amounts to less than 1% of the single-diffractive cross section and less than about 6% of the single-diffractive to inclusive dijet cross section ratio versus x.
• Acceptance and unfolding: Half the maximum difference between the single-diffractive cross section results found by unfolding with POMWIG, PYTHIA8 4C, and PYTHIA8 CUETP8M1 is taken as a further component of the systematic uncertainty. Likewise for the results obtained with PYTHIA6 Z2*, PYTHIA8 4C, PYTHIA8 CUETP8M1 and PYTHIA8 CUETP8S1 for the inclusive dijet cross section. • Unfolding regularisation: The regularisation parameter used in the unfolding, given by the number of iterations in the D'Agostini method used in this analysis, is optimised by calculating the relative χ 2 variation between iterations. The value is chosen such that the χ 2 variation is below 5%. The number of iterations when the relative variation of χ 2 is below 2% is also used and half the difference with respect to the nominal is taken as a systematic uncertainty.
• Unfolding bias: A simulated sample, including all detector effects, is unfolded with a different model. The difference between the corrected results thus obtained and those at the particle level is an estimate of the bias introduced by the unfolding procedure. Half the maximum difference obtained when repeating the procedure with all generator combinations is a measure of the systematic uncertainty related to the unfolding.
• Integrated luminosity: The uncertainty in the integrated luminosity is 4%, measured using a dedicated sample collected by TOTEM during the same data taking period [29].
The total systematic uncertainty is calculated as the quadratic sum of the individual contributions. The uncertainties in the jet energy scale and horizontal dispersion are the dominant contributions overall. Figure 4 shows the differential cross section as a function of t and ξ, integrated over the conjugate variable. The results from events in which the proton is detected on either side of the IP are averaged.  Figure 4: Differential cross section as a function of t (left) and as a function of ξ (right) for single-diffractive dijet production, compared to the predictions from POMWIG, PYTHIA8 4C, PYTHIA8 CUETP8M1, and PYTHIA8 DG. The POMWIG prediction is shown with no correction for the rapidity gap survival probability ( S 2 = 1) and with a correction of S 2 = 7.4%. The vertical bars indicate the statistical uncertainties and the yellow band indicates the total systematic uncertainty. The average of the results for events in which the proton is detected on either side of the interaction point is shown. The ratio between the data and the POMWIG prediction, when no correction for the rapidity gap survival probability is applied, is shown in the bottom.

Extraction of the cross section as a function of t and ξ
The data are compared to POMWIG, PYTHIA8 4C, PYTHIA8 CUETP8M1, and PYTHIA8 DG. The POMWIG prediction is shown for two values of the suppression of the diffractive cross section, i.e. the rapidity gap survival probability, represented by S 2 . When S 2 = 1, no correction is applied. The resulting cross sections are higher than the data by roughly an order of magnitude, in agreement with the Tevatron results [9][10][11]. The POMWIG prediction is also shown with the correction S 2 = 7.4%, calculated from the ratio of the measured diffractive cross section and the MC prediction, as discussed below. After this correction, POMWIG gives a good description of the data. The POMWIG prediction is shown in Fig. 4 as the sum of the Pomeron (pIP), Reggeon (pIR) and Pomeron-Pomeron (IPIP) exchange contributions, while PYTHIA8 includes only the Pomeron (pIP) contribution. The PYTHIA8 4C and PYTHIA8 CUETP8M1 setups predict cross sections higher than the data by up to a factor of two. The PYTHIA8 DG model shows overall a good agreement with the data. No correction is applied to the normalisation of the PYTHIA8 samples. The PYTHIA8 DG model is the only calculation that predicts the cross section normalisation without an additional correction.
The ratio between the data and the POMWIG predictions is shown in the bottom of the left and right panels of Fig. 4. No correction is applied for the rapidity gap survival probability ( S 2 = 1). Within the uncertainties, no significant dependence on t and ξ is observed.
The value of the cross section for single-diffractive dijet production, measured in the kinematic region p T > 40 GeV, |η| < 4.4, ξ < 0.1 and 0.03 < |t| < 1 GeV 2 , is: Table 2 summarises the main systematic uncertainties of the measured cross section. The cross section is calculated independently for events in which the proton scatters towards the positive and negative z directions, namely the processes pp → pX and pp → Xp, and the results are averaged. They are compatible within the uncertainties. The PYTHIA8 DG model predicts in the same kinematic region a cross section of 23.7 nb, consistent with the measurement. Table 2: Individual contributions to the systematic uncertainties in the measurement of the single-diffractive dijet production cross section in the kinematic region p T > 40 GeV, |η| < 4.4, ξ < 0.1, and 0.03 < |t| < 1 GeV 2 . The total uncertainty is the quadratic sum of the individual contributions. The uncertainty of the integrated luminosity is not shown. The minimum relative uncertainty is not shown when it is below 1%. +14/−15 % The differential cross section as a function of t is well described by an exponential function for |t| values up to about 0.4 GeV 2 . A fit is performed with the function dσ/dt ∝ exp (−b|t|) for t values in the range 0.03 < |t| < 0.45 GeV 2 .
The resulting exponential slope is: where the systematic uncertainties include the contributions discussed in Section 8.1. The results for the exponential slope of the cross section calculated independently for events in which the proton scatters towards the positive and negative z directions are compatible within the uncertainties.
The parametrisation obtained from the fit is shown in Fig. 4. In the fit range (0.03 < |t| < 0.45 GeV 2 ), the horizontal position of the data points is calculated as the value for which the parametrised function equals its average over the bin width. The data points in the larger-|t| region outside the fit range (|t| > 0.45 GeV 2 ) are shown at the centre of the bins.
The slope measured by CDF is b ≈ 5-6 GeV −2 for |t| 0.5 GeV 2 [10]. In the larger-|t| region, the CDF data exhibit a smaller slope that becomes approximately independent of t for |t| 2 GeV 2 .
The present measurement of the slope is consistent with that by CDF at small-|t|. The data do not conclusively indicate a flattening of the t distribution at larger-|t|.
An estimate of the rapidity gap survival probability can be obtained from the ratio of the measured cross section in Eq. (7) and that predicted by POMWIG with S 2 = 1. Alternatively, the PYTHIA8 hard-diffraction model can be used if the DG suppression framework is not applied. The two results are consistent.
The overall suppression factor obtained with respect to the POMWIG cross section is S 2 = 7.4 +1.0 −1.1 %, where the statistical and systematic uncertainties are added in quadrature. A similar result is obtained when the PYTHIA8 unsuppressed cross section is used as reference value.
The H1 fit B dPDFs used in this analysis include the contribution from proton dissociation in ep collisions. They are extracted from the process ep → eXY, where Y can be a proton or a low-mass excitation with M Y < 1.6 GeV [2]. The results found when the proton is detected are consistent, apart from a different overall normalisation. The ratio of the cross sections is σ(M Y < 1.6 GeV)/σ(M Y = M p ) = 1.23 ± 0.03 (stat) ± 0.16 (syst) [2,3]. No dependence on β, Q 2 , or ξ is observed. To account for the different normalisation, the ratio is used to correct S 2 ; this yields S 2 = (9 ± 2) % when the POMWIG cross section is taken as the reference value. A similar result is obtained with PYTHIA8. Figure 5 shows the ratio R(x) in the kinematic region p T > 40 GeV, |η| < 4.4, ξ < 0.1, 0.03 < |t| < 1 GeV 2 and −2.9 ≤ log 10 x ≤ −1.6. The average of the results for events in which the proton is detected on either side of the IP is shown. The yellow band represents the total systematic uncertainty (cf. Section 8.1). The data are compared to the ratio of the singlediffractive and nondiffractive dijet cross sections from different models. The single-diffractive contribution is simulated with POMWIG, PYTHIA8 4C, PYTHIA8 CUETP8M1, and PYTHIA8 DG. The nondiffractive contribution is simulated with PYTHIA6 and HERWIG6 if POMWIG is used for the diffractive contribution. When using PYTHIA8 the diffractive and nondiffractive contributions are simulated with the same underlying event tune. When no correction for the rapidity gap survival probability is applied ( S 2 = 1), POMWIG gives a ratio higher by roughly an order of magnitude, consistent with the results discussed in Section 8.2. The suppression seen in the data with respect to the simulation is not substantially different when using PYTHIA6 or HER-WIG6 for the nondiffractive contribution. POMWIG with a correction of S 2 = 7.4% gives, as expected, a good description of the data. When HERWIG6 is used for the nondiffractive contribution the agreement is worse, especially in the lower-and higher-x regions. The agreement for PYTHIA8 4C is fair in the intermediate x region, but worse at low-and high-x. The agreement is worse for PYTHIA8 CUETP8M1, with values of the ratio higher than those in the data by up to a factor of two. The PYTHIA8 DG predictions agree well with the data, though the agreement is worse in the low-x region. No correction is applied to the PYTHIA8 normalisation.

Extraction of the ratio of the single-diffractive to inclusive dijet yields
The measured value of the ratio, normalised per unit of ξ, in the full kinematic region defined above is: R = σ pX jj /∆ξ /σ jj = 0.025 ± 0.001 (stat) ± 0.003 (syst). Table 3 summarises the main contributions to the systematic uncertainty of the ratio. The uncertainty of the jet energy scale is considerably smaller than in the case of the single-diffractive cross section. Figure 6 shows the comparison between the results of Fig. 5 and those from CDF [10]. The CDF results are shown for jets with Q 2 of roughly 100 GeV 2 and pseudorapidity |η| < 2.5, with Table 3: Individual contributions to the systematic uncertainty in the measurement of the single-diffractive to inclusive dijet yields ratio in the kinematic region p T > 40 GeV, |η| < 4.4, ξ < 0.1, 0.03 < |t| < 1 GeV 2 , and −2.9 ≤ log 10 x ≤ −1.6. The total uncertainty is the quadratic sum of the individual contributions. The minimum relative uncertainty is not shown when it is below 1%.   Figure 5: Ratio per unit of ξ of the single-diffractive and inclusive dijet cross sections in the region given by ξ < 0.1 and 0.03 < |t| < 1 GeV 2 , compared to the predictions from the different models for the ratio between the single-diffractive and nondiffractive cross sections. The POMWIG prediction is shown with no correction for the rapidity gap survival probability ( S 2 = 1) (left) and with a correction of S 2 = 7.4% (right). The vertical bars indicate the statistical uncertainties and the yellow band indicates the total systematic uncertainty. The average of the results for events in which the proton is detected on either side of the interaction point is shown. The ratio between the data and the POMWIG prediction using PYTHIA6 or HERWIG6 as the nondiffractive contribution, when no correction for the rapidity gap survival probability is applied, is shown in the bottom of the left panel. 0.03 < ξ < 0.09. In this case Q 2 is defined, per event, as the mean transverse energy of the two leading jets squared. CDF measures the ratio for Q 2 values up to 10 4 GeV 2 . A relatively small dependence on Q 2 is observed. The present data are lower than the CDF results. A decrease of the ratio of diffractive to inclusive cross sections with centre-of-mass energy has also been observed by CDF by comparing data at 630 and 1800 GeV [11].  Figure 6: Ratio per unit of ξ of the single-diffractive and inclusive dijet cross sections in the kinematic region given by ξ < 0.1 and 0.03 < |t| < 1 GeV 2 . The vertical bars indicate the statistical uncertainties and the yellow band indicates the total systematic uncertainty. The red squares represent the results obtained by CDF at √ s = 1.96 TeV for jets with Q 2 ≈ 100 GeV 2 and |η| < 2.5, with 0.03 < ξ < 0.09.

Summary
The differential cross section for single-diffractive dijet production in proton-proton (pp) collisions at √ s = 8 TeV has been measured as a function of the proton fractional momentum loss ξ and the squared four momentum transfer t, using the CMS and TOTEM detectors. The data, corresponding to an integrated luminosity of 37.5 nb −1 , were collected using a nonstandard optics configuration with β * = 90 m. The processes considered are pp → pX or pp → Xp, with X including a system of two jets, in the kinematic region ξ < 0.1 and 0.03 < |t| < 1.0 GeV 2 . The two jets have transverse momentum p T > 40 GeV and pseudorapidity |η| < 4.4. The integrated cross section in this kinematic region is σ pX jj = 21.7 ± 0.9 (stat) +3.0 −3.3 (syst) ± 0.9 (lumi) nb; it is the average of the cross sections when the proton scatters to either side of the interaction point. The exponential slope of the cross section as a function of t is b = 6.5 ± 0.6 (stat) +1.0 −0.8 (syst) GeV −2 . This is the first measurement of hard diffraction with a measured proton at the LHC.
The data are compared with the predictions of different models. After applying a normalisation shift ascribed to the rapidity gap survival probability, POMWIG agrees well with the data. The PYTHIA8 dynamic gap model describes the data well, both in shape and normalisation. In this model the effects of the rapidity gap survival probability are simulated within the framework of multiparton interactions. The PYTHIA8 dynamic gap model is the only calculation that predicts the cross section normalisation without an additional correction.
The ratios of the measured single-diffractive cross section to those predicted by POMWIG and PYTHIA8 give estimates of the rapidity gap survival probability. After accounting for the correction of the dPDF normalisation due to proton dissociation, the value of S 2 is (9 ± 2) % when using POMWIG as the reference cross section value, with a similar result when PYTHIA8 is used.

Acknowledgments
We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC and thank the technical and administrative staffs at CERN and at other CMS and TOTEM institutes for their contributions to the success of the common CMS-TOTEM effort. We gratefully acknowledge work of the beam optics development team at CERN for the design and the successful commissioning of the high β * optics and thank the LHC machine coordinators for scheduling dedicated fills. In addition, we gratefully acknowledge the computing centres and personnel of the Worldwide LHC Computing Grid for delivering so effectively the computing infrastructure essential to our analyses. Finally, we acknowledge the enduring support for the construction and operation of the LHC and the CMS and TOTEM detectors provided by the following funding agencies: BMBWF and FWF (Austria); FNRS and FWO