LHC as a photon-photon collider: bounds on Γ X→γγ
S. I. Godunov, E. K. Karkaryan, V. A. Novikov, A. N. Rozanov, M. I. Vysotsky, E. V. Zhemchugov
LLHC as a photon-photon collider: bounds on Γ X → γγ S. I. Godunov , E. K. Karkaryan , V. A. Novikov , A. N. Rozanov , M. I. Vysotsky , andE. V. Zhemchugov ∗ I.E. Tamm Department of Theoretical Physics, Lebedev Physical Institute, 119991 Moscow,Russia Institute for Theoretical and Experimental Physics, 117218 Moscow, Russia Moscow Institute of Physics and Technology (State University), 141701 Moscow, Russia Centre de Physique des Particules de Marseille, CPPM, Aix-Marseille Universite,CNRS/IN2P3, F-13288 Marseille, FranceFebruary 2, 2021
Abstract
In the relatively recent CMS data, there is a hint on the existence of a resonance with the mass28 GeV decaying to a µ + µ − pair and produced in association with a b quark jet and a secondjet. Such a resonance should also couple to photons through the fermion loop, therefore it can besearched for in ultraperipheral collisions (UPC) of protons. We set an upper bound on the Xγγ coupling constant from the data on µ + µ − pair production in UPC at the LHC. Our approach canbe used for similar resonances should they appear in the future. LHC designed as a proton-proton collider can also be considered as a photon-photon collider inwhich photons are produced in ultraperipheral collisions of protons. The interest in studying γγ collisions is twofold: first, QED processes like γγ → l + l − [1–3], γγ → W + W − [4–7], γγ → γγ [8–10]are investigated at very high energies never before accessible at particle accelerators, and second,production of new exotic particles can be looked for. The case of long-lived heavy charged particleswas considered in [11]. Dark matter particles are discussed in [12–14]. In the paper [15] the productionof exclusive γγ → µ + µ − events in proton-proton collisions at a center-of-mass energy of 13 TeV withthe ATLAS detector was analyzed. The measurement was performed in the dimuon invariant massinterval 12 GeV < m µ + µ − <
70 GeV. If a resonance with the mass in this interval does exist and candecay to a µ + µ − pair, we can obtain an upper bound on its coupling with two photons from the dataprovided in [15]. A hint of such a resonance X with the mass (28 . ± .
4) GeV was reported by theCMS Collaboration [16], and in what follows, we will obtain bounds on its coupling to two photons.However, being universal, our approach can be used for another resonance if it exists. As it was noticed in [17], X can be responsible for the deviation of the measured value of the muonanomalous magnetic moment a µ from its theoretical value. Introducing the coupling Y of the scalar X to muons according to ∆ L = Y µµX, (1) ∗ Corresponding author. E-mail: [email protected] In particular, the X production mechanism in inclusive pp collisions is not straightforward and requires introductionof other new particles [17]. This is not relevant for the X production in γγ collision. a r X i v : . [ h e p - ph ] J a n t was obtained in [17] that for Y = 0 . ± .
006 one loop contribution δa Xµ = (29 ± × − explainsthe deviation of the measured value of a µ from the Standard Model result. It was also shown thatsuch couplings are consistent with other experimental bounds.With this value of Y , we getΓ X → µ + µ − = Y π M X (cid:18) − m µ M X (cid:19) / = (1 . ± .
5) MeV , (2)while according to [16] the width of the peak isΓ exp X = (1 . ± .
8) GeV , (3)which is close to the detector mass resolution for a dimuon system σ = 0 .
45 GeV. That is why we willalso consider the case of Γ X approximately equal to Γ X → µ + µ − given in (2). pp ( γγ ) → ppµ + µ − reaction We are interested in the contribution of the X resonance to this cross section. In [15], the crosssection of µ + µ − production was measured in four intervals of the muon pair invariant mass onwhich the entire interval 12 GeV < m µ + µ − <
70 GeV was divided. We are interested in the interval22 GeV < m µ + µ − <
30 GeV, for which, according to Table 3 of [15], dσ exp dm µ + µ − = (0 . ± . , hence σ exp = (0 . ± .
04) pb . (4)This cross section measurement corresponds to the fiducial region p µT > ˆ p T = 6GeV and | η | < ˆ η = 2 . p µT is the component of the muon momentum transversal to the proton beam and η is themuon pseudorapidity: η = − ln tan( θ/ θ is the angle between the muon momentum and thebeam. The ATLAS muon spectrometer is measuring muon momentum up to | η | = 2 .
7, but the triggerchambers cover the range | η | < . µ + µ − pair production inultraperipheral collisions is given by σ ( pp ( γγ ) → ppµ + µ − ) = ∞ (cid:90) dω ∞ (cid:90) dω σ ( γγ → µ + µ − ) n ( ω ) n ( ω ) , (5)where n ( ω ) is the equivalent photons spectrum. In the leading logarithmic approximation (LL), n ( ω ) ≈ n LL ( ω ) = 2 απω ln ˆ qγω , (6)where α is the fine structure constant, γ = 6 . × is the Lorentz factor of the proton with theenergy 6.5 TeV, and ˆ q is the maximal photon momentum at which the proton does not disintegrate.In this approximation the integrals in Eq. (5) are divergent, and the integration domain is cut offexplicitly with ˆ qγ , σ LL ( pp ( γγ ) → ppµ + µ − ) = ˆ qγ (cid:90) m µ / ˆ qγ dω qγ (cid:90) m µ /ω dω σ ( γγ → µ + µ − ) n LL ( ω ) n LL ( ω ) , (7)The value of ˆ q is determined by the proton form factor and numerically ˆ q ≈ .
20 GeV [18].2 pµ + µ − pp (a) p pµ + µ − pp (b) p p µ − µ + pp X (c) Figure 1: Diagrams which contribute to the production of muon pair in ultraperipheral pp collisionsIt is convenient to substitute the integration over photon energies by integration over s = 4 ω ω and x = ω /ω . Then Eq. (5) changes to σ ( pp ( γγ ) → ppµ + µ − ) = ∞ (cid:90) (2 m µ ) σ ( γγ → µ + µ − ) ds ∞ (cid:90) dx x n (cid:18)(cid:114) sx (cid:19) n (cid:18)(cid:114) s x (cid:19) . (8)To take the experimental cuts into account, we substitute σ ( γγ → µ + µ − ) by the differential over p T cross section, σ ( pp ( γγ ) → ppµ + µ − ) = ∞ (cid:90) (2 m µ ) ds √ s/ (cid:90) dσ ( γγ → µ + µ − ) dp T dp T ∞ (cid:90) dx x n (cid:18)(cid:114) sx (cid:19) n (cid:18)(cid:114) s x (cid:19) . (9)It is then straightforward to implement cuts over s and p T by changing the integration limits toˆ s min < s < ˆ s max and ˆ p T < p T < √ s/ s min (cid:62) (2ˆ p T ) (cid:29) (2 m µ ) ]. To implement the cutoffover pseudorapidity, one should integrate over x in the interval [18],1ˆ x < x < ˆ x, where ˆ x = exp(2ˆ η ) 1 − (cid:113) − p T /s (cid:113) − p T /s . (10)Let us note that in the leading logarithmic approximation from the condition ω (cid:46) ˆ qγ , it followsthat x should be always smaller than (2ˆ qγ/ √ s ) . For numerical values of ˆ η , ˆ p T , and ˆ s = { ˆ s min , ˆ s max } ,we are interested in, and for x from the interval (10), this demand is satisfied.Thus, for the fiducial cross section we obtain σ ˆ s, ˆ p T , ˆ η fid = ˆ s max (cid:90) ˆ s min ds √ s/ (cid:90) ˆ p T dσ ( γγ → µ + µ − ) dp T dp T ˆ x (cid:90) / ˆ x dx x n (cid:18)(cid:114) sx (cid:19) n (cid:18)(cid:114) s x (cid:19) , (11)where ˆ x is defined in (10). In the leading logarithmic approximation, the fiducial cross section is σ ˆ s, ˆ p T , ˆ η fid , LL = α π s max (cid:90) ˆ s min ln (2ˆ qγ ) s dss √ s/ (cid:90) ˆ p T dσ ( γγ → µ + µ − ) dp T − (cid:32) ln ˆ x ln (2ˆ qγ ) s (cid:33) ln ˆ x dp T . (12)Let us begin with the calculation of the Standard Model contribution to the cross section of µ + µ − pair production, given by the diagrams shown in Figs. 1a, 1b.The expression for the differential cross section is [19, § dσ ( γγ → µ + µ − ) = 2 πα s (cid:18) s + tt + ts + t (cid:19) dt = 8 πα sp T − p T /s (cid:113) − p T /s dp T . (13)3ubstituting it in (12) and integrating over p T , we get σ ˆ s, ˆ p T , ˆ η fid , LL ≈ α π ˆ s max (cid:90) ˆ s min ln (2ˆ qγ ) s dss ˆ η ln (cid:113) p T /s − (cid:113) − p T /s − (cid:114) − p T s −−
14 ln (cid:113) p T /s − (cid:113) − p T /s + 12 (cid:114) − p T s ln (cid:113) p T /s − (cid:113) − p T /s = 0 .
73 pb , (14)where we neglected the small second term in the square brackets in (12) in order to perform integrationanalytically. Taking into account the omitted term and integrating numerically in (12), instead of0 .
73 pb, we obtain 0 .
68 pb.More accurate calculation depends on the internal structure of proton and the probability for theprotons to survive the collision. For the latter, we will use the expression suggested in Ref. [20], P ( b ) = (cid:18) − e − b B (cid:19) , (15)where b is the impact parameter of the collision, and B was measured to be 19 . − in the case of pp collisions with the energy 7 TeV [21]. To utilize this function, we introduce the equivalent photonspectrum at the distance b from the source particle n ( b, ω ), such that n ( ω ) = (cid:90) n ( b, ω ) d b. (16)Then the leading logarithmic spectrum [22, § n LL ( b, ω ) = αωπ γ K (cid:18) bωγ (cid:19) , (17)where K is the modified Bessel function of the second kind (the Macdonald function).In the framework of the parton model and following Ref. [23], Eq. (5) is replaced with σ ( pp ( γγ ) → ppµ + µ − ) = ∞ (cid:90) dω ∞ (cid:90) dω σ ( γγ → µ + µ − ) ×× (cid:90) d b (cid:90) d b n ( b , ω ) n ( b , ω ) P ( | b − b | ) . (18)This change is then propagated into Eq. (11), σ ˆ s, ˆ p T , ˆ η fid = ˆ s max (cid:90) ˆ s min ds √ s/ (cid:90) ˆ p T dp T dσ ( γγ → µ + µ − ) dp T ×× ˆ x (cid:90) / ˆ x dx x (cid:90) b > d b (cid:90) b > d b n (cid:18) b , (cid:114) sx (cid:19) n (cid:18) b , (cid:114) s x (cid:19) P ( | b − b | ) . (19)The internal structure of proton is characterized by the Dirac form factor [24] F ( Q ) = G D ( Q ) (cid:20) µ p − τ τ (cid:21) , G D (cid:0) Q (cid:1) = 1(1 + Q Λ ) , (20)4able 1: The measured cross section for each interval of muon pair invariant mass and the correspondingtheoretical calculations with different approximations: Eq. (12) is the calculation with the equivalentphoton spectrum taken in the leading logarithmic approximation; Eqs. (11), (21) is the calculationtaking into account the proton electromagnetic form factor; Eqs. (19), (22) also accounts for theprobability of strong interactions at small impact parameters. ”Survival ratio” is the ratio of thepreceding two columns. Note that for the interval 30–70 GeV the cutoff ˆ p T = 10 GeV as it is in [15]. m µ + µ − , GeV σ exp , pb [15] Leadinglogarithmicapprox.,Eq. (12) With the formfactor,Eqs. (11), (21) Also with thesurvival factor,Eqs. (19), (22) Survivalratio12–17 1 . ± .
07 1 .
25 1 .
28 1 .
24 0 . . ± .
05 0 .
87 0 .
896 0 .
866 0 . . ± .
04 0 .
68 0 .
703 0 .
677 0 . . ± .
04 0 .
49 0 .
506 0 .
483 0 . Q = − q , q is the photon 4-momentum, τ = Q / m p , m p is the proton mass, and µ p =2 . G D ( Q ) is the dipole form factor with Λ beingstrictly fixed by the proton charge radius: Λ = 12 /r p , r p = 0 . n ( ω ) = απ ω (cid:90) (cid:126)q ⊥ F ( (cid:126)q ⊥ + ω /γ )( (cid:126)q ⊥ + ω /γ ) d q ⊥ , (21) n ( b, ω ) = απ ω (cid:20)(cid:90) d q ⊥ q ⊥ F ( q ⊥ + ω /γ ) q ⊥ + ω /γ J ( bq ⊥ ) (cid:21) , (22)where (cid:126)q ⊥ is the photon transversal momentum, J is the Bessel function of the first kind.The so-called survival factor S γγ [18,20,26] is defined as the ratio of the integrands in Eqs. (5), (18), S γγ = (cid:82) b > (cid:82) b > n ( b , ω ) n ( b , ω ) P ( | b − b | ) d b d b n ( ω ) n ( ω ) , (23)however, in this paper, it is not calculated explicitly; Eq. (19) is used instead.Calculations for each interval of muon pair invariant mass for Eq. (12), Eq. (11) with the spec-trum (21), and Eq. (19) with the spectrum (22) are presented in the Table 1. One can see thataccounting for inelastic pp scattering reduces the theoretical result by 5% approximately.The amplitude of the µ + µ − pair production through intermediate X boson in γγ collisions [seeFig. 1c] is given by the following expression: A = κF µν F µν s − M X + i Γ X M X µµY, (24)where κ is the Xγγ coupling constant so that Γ X → γγ = ( κ M X ) / (16 π ). For the cross section of the γγ → X → µ + µ − reaction, we obtain | A | = κ Y M X s − M X ) + Γ X M X , (25) Survival factor can be also defined in a more elaborate way: on the amplitude level. See [27–33] for details.Let us also note that definition of S γγ in Ref. [34] [Eq. (7)] is different: Ref. [34] requires that the new system isproduced outside of the colliding particles, while Ref. [18] imposes no such restriction. The latter is more accurate whenthe new particles do not interact strongly, so we use the Ref. [18] definition of S γγ here. In paper [33], it was specificallystressed that impact parameter cut like in Ref. [34] is unphysical. γγ → X → µ + µ − = 2 πM X Γ X → γγ Γ X → µ + µ − ( √ s − M X ) + Γ X / , (26)where the factor 2 takes into account identity of photons.In the limit m µ → m µ the interference is zero at s = M X because then the phase between the sum of the diagrams inFigs. 1a, 1b and the diagram in Fig. 1c is π/
2. For other values of s , the interference is suppressedrelatively to X contribution by the factor α Γ X √ Γ X → µ + µ − Γ X → γγ m µ M X (cid:16) − M X s (cid:17) , which is less than 10 − forthe largest allowed values of Γ X → γγ in both cases of the narrow or the wide resonance (Γ X = 1 . . dσ = | A | πs d (4 p T /s ) (cid:113) − p T /s . (27)Substituting (27) and (25) in (12) and performing integration over p T , we obtain σ ˆ s, ˆ p T , ˆ η fid , LL ( X ) = 8 α Γ X → γγ Γ X → µ + µ − πM X ˆ s max (cid:90) ˆ s min ds ( s − M X ) + Γ X M X ln (2ˆ qγ ) s ×× (cid:114) − p T s η + ln − (cid:113) − p T /s (cid:113) − p T /s − ln 4ˆ p T s . (28)In the case of a narrow resonance Γ X ≈ Γ X → µ + µ − = (1 . ± .
5) MeV, the integration can beperformed analytically, and we obtain σ ˆ s, ˆ p T , ˆ η fid , LL ( X ) = 8 α Γ X → γγ Γ X → µ + µ − Γ X M X ln (2ˆ qγ ) M X ×× (cid:115) − p T M X η + ln − (cid:113) − p T /M X (cid:113) − p T /M X − ln 4ˆ p T M X ≈≈ . × Γ X → µ + µ − M X Γ X → γγ Γ X pb . (29)However, if Γ X = (1 . ± .
8) GeV, then the width of the resonance almost equals √ ˆ s max − M X =2 GeV, so the integration should be done numerically, and we obtain σ ˆ s, ˆ p T , ˆ η fid , LL ( X ) ≈
49 Γ X → γγ M X pb . (30) From the third line of the second and the fifth columns of the Table 1, we see that the contribution ofthe resonance X into the fiducial cross section of muon pair production is bounded in the followingway: σ fid ( X ) (cid:46) .
10 pb at 99.5% confidence level. (31) When calculating the upper limit, in the case of negative signal, Ref. [35] suggests using zero instead of the negativevalue. This makes the upper limit a little less strong. This approach is widely used in the LHC experimental community,so we follow it here. γγ Figure 2: Coupling of X to two photons through a fermion loopComparing this number with the expression (29), we get that if Γ X ≈ Γ X → µ + µ − then the upperbound on Γ X → γγ isBr( X → γγ ) < . × − , Γ X → γγ <
46 keV ≈ . × − M X at 99.5% confidence level (Γ X = 1 . . (32)If the width of X is given by (3) then the bound extracted from Eq.(30) isBr( X → γγ ) < . × − , Γ X → γγ <
58 MeV ≈ × − M X at 99.5% confidence level (Γ X = 1 . . (33)Resonance X couples with photons through a triangle diagram with fermion running in the loop (seeFig. 2). Let us check that the corresponding decay probability does not violate bounds just obtained.The amplitude generated by the triangle diagram with a fermion f running in the loop equals [36] A = αF π Y Xff m f XF µν F µν . (34)The width equals Γ X → γγ = α F π Y Xff (cid:16) M X m f (cid:17) M X , (35)where F = − β [(1 − β ) κ + 1] , β = 4 m f M X , (36) κ = arctan (cid:16) √ β − (cid:17) , β > (cid:104) i ln (cid:16) √ − β −√ − β (cid:17) + π (cid:105) , β < . (37)For m f (cid:28) M X , we obtain F ∼ ( m f /M X ) , and for m f (cid:29) M X , we obtain F → − / X → γγ ≈ − M X , which is much smaller thanbounds (32), (33). For a hypothetical fermion with a mass much larger than M X , the width is also verysmall. However, for m f ∼ M X and Y Xff ∼
1, it approaches keV: Γ X → γγ ( m f = M X / ≈ Y Xff keV.
A scalar resonance with the mass 28 GeV coupling to muons in the way consistent with the recentCMS data [16] is also consistent with the measurements of the cross section for muon pair productionin ultraperipheral collisions at the LHC [15] provided that the width of its decay to a pair of photonsΓ X → γγ <
46 keV or 58 MeV depending on whether the width Γ X = 1 . X (28 GeV) to our attention. We are grateful to V.A. Khoze for drawing our attention to papers [27–33].We are supported by the Russian Science Foundation Grant No. 19-12-00123. References [1] CMS Collaboration,
Exclusive photon-photon production of muon pairs in proton-proton collisionsat √ s = 7 T eV , JHEP (2012) 052; arXiv:1111.5536 [hep-ex].[2] ATLAS Collaboration, Measurement of exclusive γγ → l + l − production in proton-proton collisionsat √ s = 7 T eV with the ATLAS detector , Phys. Lett. B (2015) 242; arXiv:1506.07098 [hep-ex].[3] CMS Collaboration,
Search for exclusive or semi-exclusive photon pair production and observationof exclusive and semi-exclusive electron pair production in pp collisions at √ s = 7 T eV , JHEP (2012) 080; arXiv:1209.1666 [hep-ex].[4] CMS Collaboration, Study of exclusive two-photon production of W + W − in pp collisions at √ s =7 T eV and constraints on anomalous quartic gauge couplings , JHEP (2013) 116; arXiv:1305.5596[hep-ex].[5] CMS Collaboration, Evidence for exclusive γγ → W + W − production and constraints on anomalousquartic gauge couplings in pp collisions at √ s = 7 and T eV , JHEP (2016) 119; arXiv:1604.04464[hep-ex].[6] ATLAS Collaboration, Measurement of exclusive γγ → W + W − production and search for exclusiveHiggs boson production in pp collisions at √ s = 8 T eV using the ATLAS detector , Phys. Rev. D (2016) 032011; arXiv:1607.03745 [hep-ex].[7] ATLAS Collaboration, Observation of photon-induced W + W − production in pp collisions at √ s = 13 T eV using the ATLAS detector , CERN-EP-2020-165 (2020); arXiv:2010.04019 [hep-ex].[8] ATLAS Collaboration,
Evidence for light-by-light scattering in heavy-ion collisions with the ATLASdetector at the LHC , Nat. Phys. (2017) 852; arXiv:1702.01625 [hep-ex].[9] CMS Collaboration, Evidence for light-by-light scattering in ultraperipheral PbPb collisions at √ s NN = 5 . TeV , Nucl. Phys. A (2019) 791 ; arXiv:1808.03524 [hep-ex].[10] ATLAS Collaboration,
Observation of light-by-light scattering in ultraperipheral Pb+Pb collisionswith the ATLAS detector , Phys. Rev. Lett. (2019) 052001; arXiv:1904.03536 [hep-ex].[11] Godunov S I, Novikov V A, Rozanov A N, Vysotsky M I, Zhemchugov E V,
Quasistable charginosin ultraperipheral proton-proton collisions at the LHC , JHEP (2020) 143; arXiv: 1906.08568[hep-ph].[12] Harland-Lang L A, Khoze V A, Ryskin M G, Tasevsky M, LHC Searches for Dark Matter inCompressed Mass Scenarios: Challenges in the Forward Proton Mode , JHEP (2019) 010;arXiv:1812.04886 [hep-ph].[13] Khoze V A, Martin A D, Ryskin M G, Can invisible objects be ‘seen’ via forward proton detectorsat the LHC? , J. Phys. G: Nucl. Part. Phys. (2017) 055002; arXiv:1702.05023 [hep-ph].[14] Tasevsky M, Harland-Lang L A, Khoze V A, Ryskin M G, Searches for Dark Matter at the LHCin forward proton mode , EPS-HEP2019 (2019); arXiv:1910.01703 [hep-ph].815] ATLAS Collaboration,
Measurement of the exclusive γγ → µ + µ − process in proton–protoncollisions at √ s = 13 TeV with the ATLAS detector , Phys. Lett. B (2018) 303; arXiv:1708.04053[hep-ex].[16] CMS Collaboration,
Search for resonances in the mass spectrum of muon pairs produced inassociation with b quark jets in proton-proton collisions at √ s = 8 and TeV , JHEP (2018)161; arXiv:1808.01890 [hep-ex].[17] Godunov S I, Novikov V A, Vysotsky M I, Zhemchugov E V, Dimuon resonance near 28 GeVand muon anomaly , JETP Lett. (2019) 358; arXiv:1808.02431 [hep-ph].[18] Vysotsky M I, Zhemchugov E V,
Equivalent photons in proton-proton and ion-ion collisionsat the LHC , Phys. Usp. (2019) 910–919; Translated from Russian: UFN, (2019) 975;arXiv:1806.07238 [hep-ph].[19] Berestetskii V B, Lifshitz E M, Pitaevskii L P Quantum Electrodynamics (Oxford: PergamonPress, 1982); Translated from Russian:
Kvantovaya Elektrodinamika (Moscow: Nauka, 1989).[20] Frankfurt L, Hyde–Wright C E, Strikman M, Weiss C,
Generalized parton distributions and rapiditygap survival in exclusive diffractive pp scattering , Phys. Rev. D (2007) 054009; arXiv:0608271[hep-ph].[21] ATLAS Collaboration, Measurement of the total cross section from elastic scattering in pp collisionsat √ s = 7 TeV with the ATLAS detector , Nucl. Phys. B (2014) 486; arXiv:1408.5778 [hep-ex].[22] Jackson J D,
Classical Electrodynamics (New York City: John Wiley & Sons, 1962).[23] Cahn R N, Jackson J D,
Realistic equivalent-photon yields in heavy-ion collisions , Phys. Rev. D (1990) 3690.[24] Pacetti S, Ferroli R B, Tomasi-Gustafsson E Proton electromagnetic form factors: Basic notions,present achievements and future perspectives , Phys. Rep
550 – 551 (2015) 1.[25] Mohr P J, Newell D B and Taylor B N,
CODATA Recommended Values of the FundamentalPhysical Constants: 2014 , Rev. Mod. Phys. (2016) 035009; arXiv:1507.07956 [physics.atom-ph].[26] Godunov S I, Novikov V A, Rozanov A N, Vysotsky M I, Zhemchugov E V, paper in preparation.[27] Khoze V A, Martin A D, Orava R and Ryskin M G, Luminosity measuring processes at the LHC ,Eur. Phys. J. C (2001), 313-322; arXiv:hep-ph/0010163 [hep-ph].[28] Khoze V A, Martin A D and Ryskin M G, Photon exchange processes at hadron colliders as aprobe of the dynamics of diffraction , Eur. Phys. J. C (2002), 459-468; arXiv:hep-ph/0201301[hep-ph].[29] Khoze V A, Martin A D and Ryskin M G, Elastic scattering and Diffractive dissociation in thelight of LHC data , Int. J. Mod. Phys. A (2015) no.08, 1542004; arXiv:1402.2778 [hep-ph].[30] Harland-Lang L A, Khoze V A, Ryskin M G and Stirling W J, Central exclusive production withinthe Durham model: a review , Int. J. Mod. Phys. A (2014), 1430031; arXiv:1405.0018 [hep-ph].[31] Harland-Lang L A, Khoze V A and Ryskin M G, Exclusive physics at the LHC with SuperChic 2 ,Eur. Phys. J. C (2016) no.1, 9; arXiv:1508.02718 [hep-ph].[32] Khoze V A, Martin A D and Ryskin M G, Multiple interactions and rapidity gap survival , J. Phys.G (2018) no.5, 053002; arXiv:1710.11505 [hep-ph].933] Harland-Lang L A, Tasevsky M, Khoze V A and Ryskin M G, A new approach to modelling elasticand inelastic photon-initiated production at the LHC: SuperChic 4 , Eur. Phys. J. C (2020) no.10,925; arXiv:2007.12704 [hep-ph].[34] Dyndal M, Schoeffel L, The role of finite-size effects on the spectrum of equivalent photons inproton-proton collisions at the LHC , Phys. Lett. B (2015) 66; arXiv:1410.2983 [hep-ph].[35] Cowan G, Cranmer K, Gross E and Vitells O,
Asymptotic formulae for likelihood-based tests of newphysics , Eur. Phys. J. C (2011) 1554 [erratum: Eur. Phys. J. C (2013) 2501]; arXiv:1007.1727[physics.data-an].[36] Shifman M A, Vainshtein A I, Voloshin M B, Zakharov V I, Low-energy theorems for Higgs bosoncouplings to photons , Sov. J. Nucl. Phys.30