Probing anomalous γγγγ couplings at a future muon collider

We have studied anomalous four-photon couplings in the µ + µ − → µ + γγµ − scattering at a future muon collider. The collision energies of 3 TeV, 14 TeV, and 100 TeV are addressed. Both diﬀerential and total cross sections versus invariant mass of the outgoing photons are calculated. The best 95% C.L. exclusion bounds on anomalous couplings are obtained to be g 1 = 2 . 23 × 10 − 8 TeV − 4 and g 2 = 4 . 22 × 10 − 8 TeV − 4 . They correspond to the muon collision energy of 100 TeV. The partial-wave unitary constraints on g 1 and g 2 are examined. We have demonstrated that the unitarity is not violated in a region of the anomalous couplings obtained in the present paper.


Introduction
In our previous papers, we examined the anomalous quartic gauge couplings (AQGCs) of the neutral bosons in the processes γγ → γγ [1] and γγ → γZ [2] at the CLIC.It enabled us to probe the anomalous vertices γγγγ and γγγZ.
The LHC experimental bounds on neutral AQGCs have been presented by the ATLAS [11]- [13] and CMS [14,15] collaborations.Recently new limits on AQGCs have been obtained by the CMS and TOTEM collaborations [16] (see also [17]).The LHC limits on the anomalous triple gauge couplings (ATGCs) were also given [18].The sensitivity of AQGCs at future hadronhadron colliders was investigated in [20,21].Previously a limit on neutral AQGC was obtained at the LEP using data on the Z → γγγ decay at the LEP [19].At present and future e + e − , eγ, and γγ colliders AQGCs were probed in a number of papers [1,2], [22]- [32].

Diphoton production at muon collider
Our goal is to examine the process It is shown in Fig. 1.The colliding muon beams are assumed to be unpolarized, and we sum over the photon polarization states.This process can be regarded as an exclusive process by requiring the outgoing muons to be observable in the detector coverage where θ is a scattering angle of the outgoing muons [84].
The Feynman diagrams describing diphoton production in the µ + µ − collision via vector boson fusion.
In the equivalent photon approximation (EPA) [85]- [90] the polarized distributions of photon inside unpolarized fermion beam look like [89] where f = µ − , x = E γ /E µ is the ratio of the photon energy E γ and energy of the incoming muon E µ , m µ is the muon mass.Note that in our case f = µ − .Because of C and P invariance, For collisions of massive vector bosons (W ± and Z), the effective W approximation (EWA) is usually used [91,92].It allows to treat of massive vector bosons as partons inside the colliding beams [93]- [101].In this approach the Z boson has the following distributions for its transverse (±1) and longitudinal (0) polarizations [60,98,101] where Again, it is assumed that we deal with the unpolarized fermion beam.The Z boson distributions in anti-fermions are related to those in fermions by CP relations, f The cross section of the process (1) is given by where V 1 and V 2 run over γ, Z + , Z − , Z 0 , and p ⊥ is the transverse momenta of the outgoing photons.The boson distributions inside the muon beam, are given by eqs.(3), (4).We take Q 2 = ŝ, where The differential cross section in ( 6) is a sum of squared helicity amplitudes, where λ 1 , λ 2 are helicity of the incoming bosons, and λ 3 , λ 4 are helicities of the outgoing photons.Remind that we consider the unpolarized diphoton final state.In its turn, each of the helicity amplitudes M λ 1 λ 2 λ 3 λ 4 in ( 8) is a sum of the anomaly and SM terms, In what follows, to obtain exclusion bounds for the AQGCs, we assume that there are no anomalous vertices Zγγγ and ZZγγ.It means that only the γγ → γγ subprocess can contribute to dσ(V 1 V 2 → γγ) (in other words, one has to put V 1 = V 2 = γ in eq. ( 6)).Otherwise, one may obtain only bound on some combination of anomalous couplings corresponding to the γγγγ, Zγγγ, and ZZγγ vertices.Since the distribution of the photon in the unpolarized muon does not depend on photon's helicity (3), we have to average 4-photon amplitude in (8) over helicity states λ 1 , λ 2 .
To calculate the anomalous helicity amplitudes we use an effective Lagrangian which contributes to anomalous quartic boson vertices.It can be found, for instance, in [102] In a broken phase this Lagrangian is expressed in terms of the physical bosons.We are interested in the part of the effective Lagrangian that describes the anomalous γγγγ interaction.It is given by where , and the couplings g 1 and g 2 have dimension TeV −4 .The couplings g 1 , g 2 are given by the following linear combinations of c i (i = 8, . . ., 16) [102] Instead of the second operator in (11) the F µν F µν F ρσ F ρσ operator are sometimes encountered, where F µν = (1/2)ǫ µναβ F αβ .The corresponding Lagrangians can be transformed into each other (up to a full derivative) using equations of motion.
As we see, two operators in (11) can arise from the SU(2)×U(1) Y effective Lagrangian which contains products of four field strengths of the hypercharge gauge field B µ and SU(2) gauge field W µ .However, we may not appeal to the effective Lagrangian (10) at all, but just to demand that our operators must be dimension-8 Lorentz invariant CP-even operators of four photon fields.Then we should come to the same operators (11) (or to a physically equivalent set of two photon operators, see comments after (12)).In other words, anomalous interactions in the broken phase (11) can be considered as independent.
In a number of papers bounds on couplings c i (often denoted as f T,i ) in EF Lagrangian (10) were searched for.While obtaining bounds on any coupling, the other couplings were set to zero (see, for instance, [31,32]). 1We prefer to examine couplings of physical operators (like couplings g 1 , g 2 in the effective Lagrangian (11)).As mentioned above, we will study four-photon anomalous interaction assuming that there are no quartic interactions of the photons with one or two Z bosons. 2  The explicit expressions for M anom λ 1 λ 2 λ 3 λ 4 are collected in Appendix.As for the SM contribution to the cross section, all the terms in ( 6) must be taken into account (V 1 , V 2 = γ, Z).For our numerical calculations we will use explicit expressions for the SM helicity amplitudes obtained for the processes γγ → γγ [103]- [106], γγ → γZ [107], and γγ → ZZ [108].The W -loop contribution to the helicity amplitudes is the leading contribution.Moreover, in the region the dominant amplitudes are M ++++ and M +−+− (for instance, see [109]), as well as helicity amplitudes related to them by Bose statistics, crossing symmetry, and parity or time-inversion invariance.Also note that in the high energy region (13) the relative size of the helicity amplitudes for γγ → γγ and γγ → ZZ collisions is given by the relation The differential cross sections for different values of the collision energy √ s are presented in Fig. 2 as functions of the diphoton invariant mass m γγ .
We have used the cut on the rapidity of the outgoing photons, |η| < 2.5.The curves for two sets of the anomalous couplings g 1 , g 2 are shown.The invariant mass m γγ of the diphoton system is expected to be close to the collision energy.But the center mass energy could be smeared by ISR/meanstrahlung ef-1 If in the unbroken phase (10) all couplings c i except one are assumed to be zero, then AQGCs are defined by one and the same quantity.For instance, if c 8 = 0, and c i = 0 for i = 9, . . ., 16, then 4g γγγγ = −(c w /s w )g Zγγγ = −(c w /s w ) 3 g ZZZγ = 4(c w /s w ) 4 g ZZZZ [102].
2 Analogously, AQGCs for the Zγγγ and ZZγγ vertices can be examined.
fects. 3 To suppress this SM background, we will use the cut m γγ < m γγ,max = 0.9 √ s.Our cut could not reduce the total cross section noticeably, since for all three values of the collision energy the differential cross section decreases rapidly as m γγ approaches √ s, see Fig.To obtain exclusion regions, we apply the following formula for the statistical significance SS [111] where S is the number of signal events and B is the number of background events.We define the regions SS 1.645 as the regions that can be excluded at the 95% C.L. To reduce the SM background, we take additional cuts on the invariant mass of the outgoing photons: i) m γγ > 0.5 TeV for √ s = 3 TeV; ii) m γγ > 3 TeV for √ s = 14 TeV; iii) m γγ > 20 TeV for √ s = 100 TeV.The 95% C.L. exclusion regions for the anomalous couplings g 1 and g 2 are shown in Figs.4-6.The expected integrated luminosities are taken from [41,56].Note that the total cross section is proportional to the coupling combination 48g 2 1 + 40g 1 g 2 + 11g 2 2 , if we take the helicity amplitudes given in Appendix [1,3].It means that the exclusion regions are ellipses in the (g 1 −g 2 ) plane, and that the cross section is approximately twice more sensitive to the coupling g 1 than to g 2 .
As one can see from the inequalities shown above, the most stringent upper bound on g 1 looks like So, the unitarity demands that |g 1 | must be less than 7.8 × 10 −2 TeV −4 , 1.64 × 10 −4 TeV −4 , and 6.28 × 10 −8 TeV −4 , for 3 TeV, 14 TeV, and 100 TeV, respectively.The best bound on g 2 is achieved when g 1 = 0, Correspondingly, the unitary upper limits for |g 2 | are equal to 0.21 TeV −4 , 4.36 × 10 −4 TeV −4 , and 1.68 × 10 −7 TeV −4 , for 3 TeV, 14 TeV, and 100 TeV, respectively.Comparing obtained unitary bounds with our exclusion regions presented in Figs.4-6, we conclude that the unitary in not violated in our region of AQGCs.Let us underline that for numerical calculations of the unitary bounds, in all the above equations, we put s to be equal to the collision energy of the process µ + µ − → µ + γγµ − .Actually, in these equations, one has to use the squared invariant energy of the γγ → γγ subprocess ŝ = sx 1 x 2 instead of s.As a result, effective unitary bounds are even weaker than those given above.

Conclusions
In the present paper we have studied the anomalous quartic couplings of the γγγγ vertex in the unpolarized µ + µ − → µ + γγµ − scattering at the future muon collider.We require the scattering angles of the outgoing muons to be in the region 10 Using analytical expressions for the helicity amplitudes for the light-bylight scattering, we have calculated both differential and total cross sections.To reduce the SM background, we have imposed the cuts: m γγ > 0.5 TeV, 3 TeV, and 20 TeV for √ s = 3 TeV, 14 TeV, and 100 TeV, respectively.
Additionally we demand the invariant mass of the outgoing photons to satisfy the inequality m γγ < 0.9 √ s.Finally, we use the cut on the rapidity of the final photons, |η| < 2.5.The 95% C.L. exclusion regions for the anomalous four-photon couplings g 1 and g 2 are calculated.For √ s = 3 TeV the best bounds appeared to be g 1 = 6.9×10 −3 TeV −4 (for g 2 = 0) and g 2 = 1.41×10 −2 TeV −4 (for g 1 = 0).Correspondingly, for √ s = 14 TeV we have obtained The partial-wave unitary constraints on the anomalous couplings are examined in detail.We have demonstrated that the unitarity is not violated in the region of these values of couplings g 1 , g 2 .
The AQGCs can originate from one-loop contribution with a new charged particle inside the loop.Then one expect that g i ∼ α 2 /Λ 4 , where Λ is a mass scale of new physics, and our constraint g 2 = 4.2 × 10 −8 TeV −4 translates to Λ ≃ 6 TeV.However, there could be quite another origin of the AQGCs coming from s-channel diagram with a new neutral mediator.Suppose that its coupling to the SM fields is equal to 1/f .Then we expect that g i ∼ 1/(f 2 Λ 2 ).This new particle can be, for instance, a KK graviton or radion in the Randall-Sundrum (RS) model with one extra dimension and warped metric [116].Taking 1/f ∼ 1 TeV −1 (as in the RS model), we obtain from g 2 = 4.2 × 10 −8 TeV −4 that Λ ≃ 500 TeV.Thus, the use of EFT and our cuts are justified even for the 100 TeV muon collider.
The sensitivity to the γγγγ anomalous couplings in photon collisions at the HL-LHC were estimated to be g 1 (g 2 ) < 1 × 10 −2 TeV −4 [3,117].As one can see, it is comparable with the sensitivity of the 3 TeV muon collider.The sensitivity of the 100 TeV FCC-hh with integrated luminosity of 3 ab −1 is expected to be g 1 (g 2 ) < 2 × 10 −4 TeV −4 [117], while our bounds on g 1 and g 2 for the 100 TeV muon collider are four orders of magnitude stronger.It demonstrates once again an advantage of the muon collider over FCC-hh collider in searching for BSM physics.
[84]170 •[84].To derive the bounds on the four-photon couplings, we have assumed that anomalous couplings of the vertices Zγγγ and ZZγγ are zero.The collision energies of 3 TeV, 14 TeV, and 100 TeV are examined.