LHC as a photon-photon collider: bounds on ΓX→γγ

In the relatively recent CMS data, there is a hint on the existence of a resonance with the mass 28 GeV coupling to muons. Such a resonance should also couple to photons through the fermion loop, therefore it can be searched for in ultraperipheral collisions (UPC) of protons. We set an upper bound on the Xγγ coupling constant from the data on μμ pair production in UPC at the LHC. Our approach can be used for similar resonances should they appear in the future.


Introduction
LHC designed as a proton-proton collider can also be considered as a photon-photon collider in which photons are produced in ultraperipheral collisions of protons. The interest in studying γγ collisions is twofold: first, QED processes like γγ → l + l − [1][2][3], γγ → W + W − [4][5][6][7], γγ → γγ [8][9][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 particles was considered in [11]. Dark matter particles are discussed in [12][13][14]. In the paper [15] the production of exclusive γγ → µ + µ − events in proton-proton collisions at a centre-of-mass energy of 13 TeV with the AT-LAS detector was analyzed. The measurement was performed in the dimuon invariant mass interval 12 GeV < m µ + µ − < 70 GeV. If a resonance with the mass in this interval does exist and can decay to a µ + µ − pair, we can obtain an upper bound on its coupling with two photons from the data provided in [15]. Evidence of such a resonance X with the mass (28.3 ± 0.4) GeV was reported by the CMS 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 muon anomalous magnetic moment a µ from its theoretical value. Introducing the coupling Y of the scalar X to muons according to ∆L = Y µµX, it was obtained in [17] that for Y = 0.041 ± 0.006 one loop contribution δa X µ = (29 ± 8) × 10 −10 explains the deviation of the measured value of a µ from the Standard Model result. It was also shown that such couplings are consistent with other experimental bounds.
With this value of Y we get: while according to [16] the width of the peak is which is only several times bigger than the detector mass resolution for a dimuon system σ = 0.45GeV. That is why we will also consider the case of Γ X approximately equal to Γ X→µ + µ − given in (2). 2 The fiducial cross section of the pp(γγ) → ppµ + µ − reaction We are interested in the contribution of the X resonance to this cross section. In [15] the cross section of µ + µ − production was measured in four intervals of muon pair invariant mass on which the entire interval 12 GeV < m µ + µ − < 70 GeV was divided. We are interested in the interval 22 GeV < m µ + µ − < 30 GeV, for which, according to Table 3 of [15], This cross section measurement corresponds to the fiducial region p µ T >p T = 6 GeV and |η| < η T = 2.4, where p µ T is the component of the muon momentum transversal to the proton beam and η is the muon pseudorapidity: η = − ln tan(θ/2), where θ is the angle between the muon momentum and the beam. The ATLAS muon spectrometer is measuring muon momentum up to |η| = 2.7, but the trigger chambers cover the range |η| < 2.4 that corresponds to the pseudorapidity cutoff given above.
According to the equivalent photons approximation the cross section of µ + µ − pair production in ultraperipheral collisions is given by: where n(ω) is the equivalent photons spectrum. In the leading logarithmic approximation (LL) where α is the fine structure constant, γ = 6.93 × 10 3 is the Lorentz factor of the proton with the energy 6.5 TeV, andq 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 off explicitly withqγ: The value ofq is determined by the proton form factor and numericallyq ≈ 0.20 GeV [18]. It is convenient to substitute the integration over photon energies by integration over s = 4ω 1 ω 2 and x = ω 1 /ω 2 . Then Eq. (5) changes to To take the experimental cuts into account, we substitute σ(γγ → µ + µ − ) by the differential over p T cross section: Figure 1: Diagrams which contribute to the production of muon pair in ultraperipheral pp collisions It is then straightforward to implement cuts over s and p T by changing the integration limits tô . To implement the cutoff over pseudorapidity, one should integrate over x in the interval [18] 1 Let us note that in the leading logarithmic approximation from the condition ω qγ it follows that x should be always smaller than (2qγ/ √ s) 2 . For numerical values ofη,p T andŝ = {ŝ min ,ŝ max } we are interested in and for x from the interval (10) this demand is satisfied. Thus for the fiducial cross section we obtain: wherex is defined in (10). In the leading logarithmic approximation the fiducial cross section is 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, §88]: Substituting it in (12) and integrating over p T we get: where we neglected the small second term in the square brackets in (12) in order to perform integration analytically. Taking into account the omitted term and integrating numerically in (12), instead of 0.73 pb we obtain 0.68 pb.
More accurate calculation depends on the internal structure of proton and the probability for the protons to survive the collision. The latter is [20] where b is the impact parameter of the collision, and B was measured to be 19.7 GeV −2 in the case of pp collisions with the energy 7 TeV [21]. To utilize this function we introduce the equivalent photon spectrum at the distance b from the source particle n(b, ω) such that Then the leading logarithmic spectrum [22, §15.5] where K 1 is the modified Bessel function of the second kind (the Macdonald function).
In this framework Eq. (5) is replaced with This change is then propagated into Eq. (11): The internal structure of proton is characterized by the Dirac form factor [23] where Q 2 = −q 2 , q is the photon 4-momentum, τ = Q 2 /4m 2 p , m p is the proton mass and µ p = 2.7928473508(85) is the proton magnetic moment [24], G D (Q 2 ) is the dipole form factor with Λ being strictly fixed by the proton charge radius: Λ 2 = 12/r 2 p , r p = 0.8751(61) fm [24]. The form factor enters Eqs. (5), (19) through the equivalent photon spectrum [18]: where J 1 is the Bessel function of the first kind.  (12) is the calculation with the equivalent photon spectrum taken in the leading logarithmic approximation; Eqs. (11), (21) is the calculation taking into account the proton electromagnetic form factor; Eq. (19), (22) also accounts for the probability of strong interactions at small impact parameters. "Survival ratio" is the ratio of the preceding two columns. Note that for the interval 30 − 70 GeV the cutoffp T = 10 GeV as it is in [15].  The amplitude of the µ + µ − pair production through intermediate X boson in γγ collisions (see Fig. 1c) is given by the following expression where κ is the Xγγ coupling constant so that Γ X→γγ = (κ 2 M 3 X )/(16π). For the cross section of the γγ → X → µ + µ − reaction we obtain: where the factor 2 takes into account identity of photons. In the limit m µ → 0 chiralities of the muons produced through the diagrams in Figs. 1a, 1b are not the same as in Fig. 1c. Consequently, these diagrams do not interfere in this limit. Even with nonzero m µ the interference is zero at s = M 2 X because then the phase between the sum of the diagrams in Figs. 1a, 1b and the diagram in Fig. 1c is π/2. For other values of s the interference is suppressed relatively to X contribution by the factor which is less than 10 −2 for the largest allowed values of Γ X→γγ in both cases of the narrow or the wide resonance (Γ X = 1.8 MeV or 1.8 GeV respectively).
In order to impose the cut on the transverse momentum of muons with the help of expression (11) the following differential cross section is used: 1 Definitions of S 2 γγ in Ref. [18] (Eq. (C.15)) and in Ref. [26] (Eq. (7)) are different: Ref. [26] requires that the new system is produced outside of the colliding particles, while Ref. [18] imposes no such restriction. The latter is more accurate when the new particles do not interact strongly, so we use the Ref. [18] definition of S 2 γγ here.
Substituting (27) and (25) in (12) and performing integration over p T we obtain: In the case of narrow resonance Γ X ≈ Γ X→µ + µ − = (1.8±0.5)MeV the integration can be performed analytically and we obtain: X γ γ Figure 2: Coupling of X to two photons through a fermion loop The width equals: where For m f ≪ M X we obtain F ∼ (m f /M X ) 2 and for m f ≫ M X we obtain F → −4/3.
In the case of muon running in the loop we get Γ X→γγ ≈ 10 −11 M X which is much smaller than bounds (32), (33). For a hypothetical fermion with mass much larger than M X the width is also very small. However for m f ∼ M X and Y Xf f ∼ 1 it approaches keV: Xf f keV.

Conclusions
A scalar resonance with the mass 28 GeV coupling to muons in the way consistent with the recent CMS data [16] is also consistent with the measurements of the cross section for muon pair production in ultraperipheral collisions at the LHC [15] provided that the width of its decay to a pair of photons Γ X→γγ < 46 keV or 58 MeV depending on whether the width Γ X = 1.8 GeV reported in Ref. [16] is the real width of the resonance or an artifact of the detector mass resolution. The difference between the leading logarithmic approximation and the calculation that takes into account both the proton form factor and the survival factor for the protons colliding with the energy 13 TeV is at the level of few percent. Integration of the logarithmic approximation can often be performed analytically while the form factor and especially the survival factor require computationally expensive numerical calculations. Therefore cross sections for ultraperipheral collisions of protons in the lower region of invariant masses of the produced system can be estimated in the logarithmic approximation with the form factor and the survival factor taken into account as needed.
Our study demonstrates that we can look for New Physics in ultraperipheral collisions at the LHC. We are grateful to V.B. Gavrilov and A.N. Nikitenko, who have brought the CMS observation of X(28 GeV) to our attention. We are supported by the Russian Science Foundation grant No 19-12-00123.