Probing the muon g-2 anomaly with the Higgs boson at a Muon Collider

We point out that heavy new physics contributions in leptonic dipole moments and high-energy cross-sections of lepton pairs into Higgs bosons and photons are connected model-independently. In particular, we demonstrate that a muon collider, running at center-of-mass energies of several TeV, can provide a unique test of new physics in the muon g-2 through the study of high-energy processes such as mu+ mu- ->h gamma. This high-energy test would be of the utmost importance to shed light on the long-standing muon g-2 anomaly as it is not affected by the hadronic and experimental uncertainties entering the current low-energy determination of the muon g-2. Furthermore, we show that the current bound on the muon electric dipole moment can be improved by three orders of magnitude, down to few x 10^-22 e cm.

The forthcoming runs of the E989 experiment plan to reduce the experimental uncertainty by a factor of four. Moreover, a completely new low-energy approach to measuring the muon g-2 is being developed by the E34 collaboration at J-PARC [4]. On the theory side, there is also an ongoing effort to reduce the leading SM uncertainty stemming from hadronic corrections [5]. Given the difficulty of controlling all these effects at the required level of precision, we think it is crucial to have an independent test of NP in the muon g-2, not affected by the hadronic and experimental uncertainties entering the current low-energy determination of the muon g-2.
Incidentally, the observed muon g-2 discrepancy can be accommodated by a NP effect of the same size as the SM weak contribution ∼ 5 G F m 2 µ /24 √ 2π 2 ≈ 2×10 −9 [3]. Therefore, a very natural explanation of eq. (1) could be achieved within weakly interacting NP scenarios emerging at a scale Λ close to the electroweak scale. Remarkably, this possibility could be connected with the solution of the hierarchy problem and could provide, at the same time, a WIMP dark matter candidate. Unfortunately, the lack for new particles at LEP and LHC strongly disfavours this interpretation. As a result, two possibilities seem to emerge to solve the muon g-2 anomaly while avoiding the stringent LEP and LHC bounds. Either NP is very light (Λ 1 GeV) and feebly coupled to SM par-ticles, see e.g. [9], or NP is very heavy (Λ 1 TeV) and strongly coupled. Here, we take the second direction.
Heavy NP contributions to the muon g-2 arise from the dimension-6 dipole operator (μ L σ µν µ R ) HF µν [10] where H = v + h/ √ 2 contains both the Higgs boson field h and its vacuum expectation value v = 174 GeV and F µν is the electromagnetic field strenght tensor. After electroweak symmetry breaking H → v and we obtain the where g NP is the typical coupling of the NP sector. Therefore, the NP chiral enhancement v/m µ ∼ 10 3 with respect to the SM weak contribution, together with the assumption of a new strong dynamics with g NP ∼ 4π, bring the sensitivity of the muon g-2 to NP scales of order Λ ∼ 100 TeV [11].
Directly detecting new particles at such high scales is far beyond the capabilities of any foreseen collider. Moreover, even assuming the discovery of new particles by their direct production [12], it would be very hard, if not impossible, to unambiguously associate them to ∆a µ . In other words, it would be desirable to test the muon g-2 anomaly model-independently.
In this work, we argue that a muon collider (MC) running at energies E of several TeV would represent the only machine enabling to probe NP in the muon g-2 in a completely model-independent way. Indeed, the very same dipole operator that generates ∆a µ unavoidably induces also a NP contribution to the scattering process µ + µ − → hγ. Measuring the cross-section for this process would thus be equivalent to measuring ∆a µ . This would however be a direct determination of the NP contribution, not hampered by the hadronic uncertainties that affect the SM prediction of a µ .
At first sight, it could seem impossible to be sensitive to such a tiny value of ∆a µ ∼ 10 −9 at a collider experiment. However, analogously to the case of weak interaction cross-sections in the effective Fermi theory, the cross-section for µ + µ − → hγ as induced by the effective dipole operator grows with the square of the collider energy. As a result, a high-energy measurement with O(1) precision will be sufficient to disentangle NP effects from arXiv:2012.02769v3 [hep-ph] 30 Dec 2021 the SM background. This is the first example in highenergy particle physics of a sensitivity to a magnetic moment at this level, several orders of magnitude below all the other current and projected collider constraints. In order to reach such tiny values of ∆a µ it is however crucial to accelerate the muon pairs to the highest possible multi-TeV energies.
We stress that our results are valid for E Λ where the effective field theory (EFT) description is justified.
A high-energy MC with the luminosity needed for particle physics experiments [13] is currently not feasible. Nevertheless, several efforts to overcome the technological challenges are ongoing [14], and it is crucial to explore the broad physics potential of such a machine in order to pave the road for the forthcoming accelerator and detector studies. A MC is the ideal machine to search for NP at the highest possible energies, both directly and indirectly. Indeed, muons could in principle be accelerated to multi-TeV energies, as their larger mass greatly suppresses synchrotron radiation compared to the electronpositron case. Furthermore, the physics reach of the MC overtakes that of a proton-proton collider of the same energy since all of the beam energy is available for the hard collision, compared to the fraction of the proton energy carried by the partons: a MC in the 10 TeV range has roughly the same energy available for hard scatterings as a 100 TeV hadron collider [13].
The physics case of a high-energy determination of ∆a µ , which is unique of a MC, represents a striking example of the complementarity and interplay of the highenergy and high-intensity frontiers of particle physics. At the same time, it highlights the far reaching potential of a MC, that offers a new powerful way to probe NP which is complementary both to direct searches for new particles, and to the indirect tests conducted at low energy through high-precision experiments.
The paper is organised as follows. In section II, we introduce the SM effective field theory (SMEFT), containing operators up to dimension-6, contributing to a . After performing a one-loop calculation of a in such EFT, in section III, we study the high-energy processes at a MC which are sensitive to the same NP effects entering a . In section IV, we comment on the possibility of measuring the rare Higgs decays h → + − γ (with = µ, τ ) that are induced by the same dipole operator generating a . The huge number of Higgs bosons that could be produced at a MC [15] could in principle allow the measurement of these rare processes, and thus the extraction of a .

II. THE MUON g-2 IN THE SMEFT
New interactions emerging at a scale Λ larger than the electroweak scale can be described at energies E Λ by an effective Lagrangian containing non-renormalizable Upper row: Feynman diagrams contributing to the leptonic g-2 up to one-loop order in the Standard Model EFT. Lower row: Feynman diagrams of the corresponding highenergy scattering processes. Dimension-6 effective interaction vertices are denoted by a square.
Focusing on the leptonic g-2, the relevant effective Lagrangian contributing to them, up to one-loop order, reads [10] where it is assumed that the NP scale Λ 1 TeV. The Feynman diagrams relevant for the leptonic g-2 are displayed in figure 1. They lead to the following result where s W , c W are the sine and cosine of the weak mixing angle, C eγ = c W C eB − s W C eW and C eZ = −s W C eB − c W C eW . Additional loop contributions from the operators H † HW I µν W Iµν , H † HB µν B µν , and H † τ I HW I µν B µν are suppressed by the lepton Yukawa couplings and can be neglected. Moreover, in eq. (3), we assumed for simplicity that C eB , C eW and C T are real. Since only the first two operators of eq. (2) generate electromagnetic dipoles at tree-level, we include their one-loop renormalization effects to C eγ In order to see where we stand, let us determine the NP scale probed by ∆a . From eq. (3) we find that A few comments are in order: • The ∆a µ discrepancy can be solved for a NP scale up to Λ ≈ 250 TeV. This requires a strongly coupled NP sector where C µ eγ and/or C µt T ∼ g 2 NP /16π 2 ∼ 1 and a chiral enhancement v/m µ compared with the weak SM contribution [16]. For such large values of Λ direct NP particle production is beyond the reach of any foreseen collider. However, as we shall see, the physics responsible for ∆a µ can still be tested through high-energy processes such as µ + µ − → hγ or µ + µ − → qq (with q = c, t).
• If the underlying NP sector is weakly coupled, g NP 1, then C µ eγ and C µt T 1/16π 2 , implying Λ 20 TeV to solve the ∆a µ anomaly. In this case, a MC could still be able to directly produce NP particles [12]. Yet, the study of the processes µ + µ − → hγ and µ + µ − → qq could be crucial to reconstruct the effective dipole vertex µ + µ − γ.
• If the NP sector is weakly coupled, and further ∆a µ scales with lepton masses as the SM weak contribution, then ∆a µ ∼ m 2 µ /16π 2 Λ 2 . Here, the experimental value of ∆a µ can be accommodated only provided that Λ 1 TeV. For such a low NP scale the EFT description breaks down at the typical multi-TeV MC energies, and new resonances cannot escape from direct production.

III. HIGH-ENERGY PROBES OF THE MUON g-2
The main contribution to ∆a µ comes from the dipole operator O eγ = ¯ L σ µν e R HF µν when after electroweak symmetry breaking H → v. The same operator also induces a contribution to the process µ + µ − → hγ that grows with energy (see figure 1), and thus can become dominant over the SM cross-section at a very high-energy collider. Assuming that m h √ s, which is an excellent approximation at a MC, we find the following differential cross-section where cos θ is the photon scattering angle. Notice that there is an identical contribution also to the process µ + µ − → Zγ since H contains the longitudinal polarizations of the Z. The total µ + µ − → hγ cross-section is where in the last equation we assumed no contribution to ∆a µ other than the one from C µ eγ . Moreover, we included running effects for C µ eγ , see eq. (4), from a scale Λ ≈ 100 TeV. Given the scaling with energy of the reference 2. 95% C.L. reach on the muon anomalous magnetic moment ∆aµ, as well as on the muon EDM dµ, as a function of the collider center-of-mass energy √ s, from the processes µ + µ − → hγ (black), µ + µ − → hZ (blue), µ + µ − → tt (red), and µ + µ − → cc (orange).
one gets about 60 total hγ events at √ s = 30 TeV. The SM irreducible µ + µ − → hγ background is small. The dominant contribution arises at one-loop [18] There are two ways to isolate the hγ signal from the background: by means of the different angular distributions of the two processes -the SM Zγ peaks in the forward region, while the signal is central -and by accurately distinguishing h and Z bosons from their decay products, e.g. by precisely reconstructing their invariant mass.
To estimate the reach on ∆a µ we consider a cut-andcount experiment in the bb final state, which has the highest signal yield (with branching ratios B(h → bb) = 0.58, B(Z → bb) = 0.15). The significance of the signal -defined as N S / √ N B + N S , with N S,B the number of signal and background events -is maximized in the central region |cos θ| 0.6. At 30 TeV one gets σ cut hγ ≈ 0.53 ab ∆a µ 3 × 10 −9 2 , σ cut Zγ ≈ 82 ab .
Requiring at least one jet to be tagged as a b, and assuming a b-tagging efficiency b = 80%, we find that a value ∆a µ = 3 × 10 −9 can be tested at 95% C.L. at a 30 TeV collider if the probability of reconstructing a Z boson as a Higgs is less than 10%. The resulting number of signal events is N S = 22, and N S /N B = 0.25. In figure 2 we show as a black line the 95% C.L. reach from µ + µ − → hγ on the anomalous magnetic moment as a function of the collider energy. Note that since the number of signal events scales as the fourth power of the center-of-mass energy, only a collider with √ s 30 TeV will have the sensitivity to test the g-2 anomaly.
The Z-dipole operator O eZ = ¯ L σ µν e R HZ µν contributes to ∆a µ at one loop, and generates also the process µ + µ − → Zh (see figure 1) with the same crosssection of eq. (5) with γ ↔ Z, so that (10) Here we assume that only O eZ contributes to ∆a µ : it should be stressed that this corresponds to an unnatural scenario, where the coefficients C eB and C eW conspire to cancel out the tree-level contribution from O eγ . It is nevertheless meaningful to derive the constraint from high-energy scattering on the Z-dipole contribution to the g-2. The cross-section in eq. (10) has to be compared to the SM irreducible background given by σ SM Zh ≈ 122 ab 10 TeV √ s 2 . Considering again the h → bb channel, together with hadronic decays of the Z, one gets the 95% C.L. limit shown in figure 2 as a blue line. Next, we derive the constraints on the semi-leptonic operators. The operator O µt T that enters ∆a µ at one loop can be probed by µ + µ − → tt (see figure 1). Its contribution to the cross-section is where in the last equality we have again taken Λ ≈ 100 TeV so that |∆a µ | ≈ 3 × 10 −9 (100 TeV/Λ) 2 |C µt T |. We estimate the reach on ∆a µ simply assuming an overall 50% efficiency for reconstructing the top quarks, and requiring a statistically significant deviation from the SM µ + µ − → tt background, which has a cross-section σ SM tt ≈ 1.7 fb 10 TeV √ s 2 . Similarly, if the charm-loop contribution dominates, we can probe |∆a µ | ≈ 3 × 10 −9 (10 TeV/Λ) 2 |C µc T | through the process µ + µ − → cc. In this case, unitarity constraints on the NP coupling C µc T require a much lower NP scale Λ 10 TeV, so that our effective theory analysis will only hold for lower center-ofmass energies. Combining eq. (3) and (11), with c ↔ t, we find that The SM cross-section for µ + µ − → cc at √ s = 3 TeV is ∼ 19 fb. In figure 2 we show the 95% C.L. constraints on the top and charm contributions to ∆a µ as red and orange lines, respectively, as a function of the collider energy. Notice that the charm contribution can be probed already at √ s = 1 TeV, while the top contribution can be probed at √ s = 10 TeV. The simultaneous constraints on the NP couplings C µ eγ and C µt T are shown in figure 3 for a 30 TeV collider.
So far, we assumed CP conservation. If however the coefficients C eγ , C eZ or C T are complex, the muon electric dipole moment (EDM) d µ is unavoidably generated. Since the cross-sections in eq. (5) and (11) are proportional to the absolute values of the same coefficients, a MC offers a unique opportunity to test also d µ . The current experimental limit d µ < 1.9 × 10 −19 e cm was set by the BNL E821 experiment [19] and the new E989 experiment at Fermilab aims to decrease this by two orders of magnitude [20]. Similar sensitivities could be reached also by the J-PARC g-2 experiment [21].
From the model-independent relation [17] d µ tan φ µ = ∆a µ 2m µ e 3 × 10 −22 ∆a µ 3×10 −9 e cm , (13) where φ µ is the argument of the dipole amplitude, the bounds on ∆a µ in figure 2 can be translated into a modelindependent constraint on d µ . We find that already a 10 TeV MC can reach a sensitivity comparable to the ones expected at Fermilab [20] and J-PARC [21], while at a 30 TeV collider one gets the bound d µ 3×10 −22 e cm.

IV. RARE HIGGS DECAYS
We finally discuss the connection between the lepton g-2 and the radiative Higgs decays h → + − γ. Due to the large luminosity, and the growth with energy of the vector-boson-fusion cross-section, a huge number of Higgs bosons is expected to be produced at a high-energy lepton collider [15]. In particular, a MC running at √ s = 30 TeV with an integrated luminosity of 90 ab −1 will produce O(10 8 ) Higgs bosons. With the precision of Higgs couplings measurements most likely limited by systematic errors, the main advantage of having such a large number of events is the possibility to look for very rare decays of the Higgs.
The operator O eZ affects the h → + − Z decay in a way analogous to eq. (14). While the contribution in the h → µ + µ − Z channel is still too small to be observed, a measurement of B(h → τ + τ − Z) at the percent level could be sensitive to values of ∆a τ 10 −4 . It is worth pointing out that at a high-energy lepton collider ∆a τ can also be efficiently probed through the processes µ + µ − → τ + τ − , and especially µ + µ − → µ + µ − τ + τ − (νν τ + τ − ) which enjoys a very large crosssection driven by vector-boson-fusion [22].

V. CONCLUSIONS
The muon g-2 discrepancy is one of most intriguing hints of new physics emerged so far in particle physics, which has recently been reinforced with the confirmation of the BNL result [1] by the E989 experiment at Fermilab [2]. However, these low-energy determinations of ∆a µ rely on the assumption that systematic and hadronic uncertainties are under control at the outstanding level of ∆a µ ∼ 10 −9 . Therefore, an independent test of ∆a µ , not contaminated by the above sources of uncertainty, is very desirable.
In this work, we have demonstrated that a muon collider running at center-of-mass energies of several TeV can achieve this goal, providing a unique, modelindependent test of new physics in the muon g-2 through the study of the high-energy processes µ + µ − → hγ, hZ, qq. In particular, a 30 TeV collider with the baseline integrated luminosity of 90 ab −1 would be able to reach a sensitivity to the electromagnetic dipole operator of few ×10 −9 , comparable to the present value of ∆a µ . If on the other hand the g-2 anomaly arises at loop-level from quark-lepton interactions, this could already be tested at a few TeV collider. Furthermore, we have shown that the current bound on the muon electric dipole moment can be improved by three orders of magnitude, down to few × 10 −22 e cm.
These results rely on measurements with O(1) accuracy, and thus do not require a precise control of systematic or theoretical uncertainties. We stress that our findings are completely model-independent, being formulated in terms of the very same effective operators that control the lepton dipole moments. Should the muon g-