Direct measurement of the Higgs self-coupling in $e^+e^- \to ZH$

A new method to measure the trilinear Higgs self-coupling $\lambda_3^{}$ in a single Higgs production process is proposed. Time-reversal-odd (T-odd) asymmetries in the process $e^+_{}e^-_{} \to ZH$, $Z \to f\bar{f}$ are computed from the absorptive part of the electroweak one-loop amplitude. Since the T-odd asymmetries measure the tree-level $t$-channel $ZH \to ZH$ scattering, they can be direct probes of $\lambda_3^{}$. The proposed method is quite challenging; a relatively large statistics and polarized $e^+_{}e^-_{}$ beams are demanded. However, this is probably the only approach to directly measure $\lambda_3^{}$ in $e^+_{}e^-_{}$ collisions, when a beam energy above the $ZHH$ production threshold is not available.

The capabilities of the LHC and future e + e − colliders to measure the trilinear Higgs self-coupling λ 3 have been extensively studied in recent years . Unlike the couplings of the Higgs bosons with heavy fermions and gauge bosons, we do not have any meaningful information on λ 3 and its value can be very different from the one predicted in the standard model (SM). The measurements from the di-Higgs production processes, which are commonly referred as direct measurements, are challenging, because of their very small cross sections both at the LHC [1,2,4,5,9,10,15,21] and at e + e − colliders [3,6,7,17,22], at which a relatively high beam energy is required. Current LHC runs are only able to provide exclusion limits for di-Higgs production [20,24] and, even the projected sensitivity at the HL-LHC is very weak (−0.8 < λ 3 /λ SM 3 < 7.7) [19,26]. The information on λ 3 may be also obtained from measurement of the (differential) cross sections of the single Higgs production processes [8, 11-14, 16-18, 22, 23, 27]. The coupling λ 3 contributes to the electroweak one-loop correction to single Higgs processes. This approach is commonly called an indirect one due to the fact that the trilinear coupling enters into the loop. Since very precise determination of the cross sections is demanded, the indirect method in single Higgs processes is as challenging as the direct measurement from the di-Higgs productions [8,16,17,22].
In e + e − collisions, the aforementioned indirect method in the process e + e − → ZH has an obvious advantage over the direct approach with the di-Higgs production processes such as e + e − → ZHH; only a smaller beam energy is needed. However, the indirect measurements highly depend on assumptions about unknown new physics (NP) at high scale that does not modify the coupling λ 3 itself [8] 1 . Direct methods, on the other hand, are less model dependent and, therefore, can provide more reliable bounds on a possible modification of In this work, a method of measuring directly the coupling λ 3 in the single Higgs production process e + e − → ZH is proposed. The method deals with time-reversal-odd (T-odd) asymmetries in the production process with a subsequent Z boson decay into a massless fermion pair, When CP (or equally T) is conserved, T-odd quantities are generally identical to zero in the tree-level approximation and receive finite contributions only from an absorptive part of loop diagrams [32]. The T-odd asymmetries are computed at the lowest order from the absorptive part of the electroweak one-loop amplitude. The absorptive part always includes the tree level t-channel ZH → ZH re-scattering effect, a part of which is proportional to the coupling λ 3 . As a result, the T-odd asymmetries are direct probes of λ 3 . Unknown heavy NP particles, which may affect the indirect λ 3 measurement via one-loop radiative corrections, do not contribute to the T-odd asymmetries unless the beam energy is large enough to directly produce these NP particles, because the asymmetries arise only from the absorptive part 2 . frame. We neglect the initial electron mass and the final fermion mass. For given s and the electron helicity (τ ), after the summation over the final fermion helicity, the differential cross section using the narrow width approximation for the Z boson can be expressed as 3 d 3 σ(τ ) d cos Θd cos θdφ = F 1 (1 + cos 2 θ) + F 2 (1 − 3 cos 2 θ) + F 3 sin 2θ cos φ + F 4 sin 2 θ cos 2φ + F 5 cos θ + F 6 sin θ cos φ + F 7 sin θ sin φ + F 8 sin 2θ sin φ + F 9 sin 2 θ sin 2φ, 2 It should be noted that the attempt to directly measure a coupling which enters a process at the next-toleading order by using T-odd observables is not new in particle physics; see e.g. [33][34][35][36][37]. 3 s dependence is always implicit throughout the paper.
where the nine coefficients F i (i = 1 to 9) are functions of τ , s and cos Θ. After integrations over θ and φ, only F 1 remains in Eq. (2), which corresponds to the differential cross section for the ZH production process. The first six coefficients (F 1 , F 2 , F 3 , F 4 , F 5 , F 6 ) are T-even and the last three coefficients are T-odd. The leading contribution to the six T-even coefficients is calculated from the tree diagram shown in Fig. 1. The amplitude for the production process can be in general written as where u and v are the spinors for the electron and the positron, respectively, µ the Z polarization vectors, λ the Z boson helicity, and p,p and k are the four-momenta of the electron, positron and Z boson, respectively. The four-vector Γ µ in the one-loop calculation can be expanded as The six coefficients a (i)∓ (i = 1 to 3) are complex numbers, independent of τ and λ. In the tree-level calculation, only a (1)∓ are non-zero: where α = e 2 /(4π) with e being the magnitude of the electron charge, c W = cos θ W and s W = sin θ W are the weak mixing factors. We define the coordinate system of the Z rest frame as follows: the z axis is along the original Z momentum direction and the y axis is along the direction of p × (− p). The tree-level prediction for the angular coefficients in our coordinate system is with where w is the Z boson energy w = (s+m 2 , v f and a f are the vector and axial-vector couplings of the Z to the final fermion f , and τ is the final fermion helicity. At the tree-level, the T-odd coefficients are vanishing as expected. The first non-vanishing contribution comes from the interference between the tree diagram and the absorptive part of the one-loop diagrams 4 : Note that the absorptive part in this order is both ultraviolet and infrared finite. We notice that Im(a (1)∓ ) do not contribute to the T-odd coefficients. As a result, we need to calculate the absorptive part of only a limited one-loop diagrams, representatives of which are shown in Fig. 2. We divide the relevant one-loop diagrams into three categories, namely, top loop diagrams, a Higgs loop diagram that depends on the coupling λ 3 , and gauge boson loop diagrams. These are labeled as (a), (b) and (c), respectively, in Fig. 2.
They are separately gauge-independent. The electroweak one-loop diagrams and amplitude are generated with help of FeynArts [38] and FormCalc [39]. The analytic formulas for Im(a (3)∓ ) have been obtained but they are very long expressions and will be provided elsewhere [40]. The numerical values for the one-loop scalar functions are calculated with the LoopTools [39,41]. Phase space integration is performed with BASES [42]. Our calculation has been numerically checked in the following two ways. First, CP invariance of the differential cross section [43,44] has been tested. Second, the T-odd coefficients have been also calculated from the electroweak full one-loop helicity amplitudes using MadGraph5_aMC@NLO [45,46]. We have found perfect agreement for several phase space points. The leading order prediction for T-odd asymmetries is of order g 2 /(4π) and can be obtained by dividing the T-odd coefficients (F 7 , F 8 , F 9 ) in Eq. (8) by the T-even coefficient F 1 in Eq. (7) . We define the integrated T-odd asymmetries by where ξ(τ ) = (1 + τ P e − )(1 − τ P e + ) and, 1 ≤ P e − (P e + ) ≤ 1 denotes the degree of longitudinal polarization of the electron (positron) beam. The integration over cos Θ takes into account the CP invariance of the differential cross section. We have found that the coefficient F 9 receives contribution only from the gauge boson loop diagrams (c) in Fig. 2 [40], therefore an asymmetry based on it is not useful for our purpose. In Fig. (3), diagrams contributing to the numerator and denominator in the T-odd asymmetries are described. Here A f i represents the absorptive part of only the Higgs one-loop diagram 5 . It is shown that, because an absorptive part of one-loop amplitude is simply a tree amplitude times a tree amplitude, the tree diagram for e + e − → ZH drops from the ratio and only the tree diagram for the ZH → ZH scattering (t-channel) is left. This explains that the T-odd asymmetries measure the tree-level ZH → ZH scattering, and because the coupling λ 3 is no longer a part of the loop, the T-odd asymmetries are direct probes of λ 3 . The asymmetries depend also on the ZZH coupling. However, the ZZH coupling can be constrained separately via a precise measurement of the cross section. Therefore, we can use it as input to predict the asymmetries, focusing only on constraining deviation in λ 3 .
We use the following set of input parameters for the numerical results: i.e. √ s < 2m t , the top loop diagrams do not contribute to the T-odd asymmetries and we can avoid any ambiguity from a modified top Yukawa coupling due to a high scale NP, which is unavoidable in the indirect method [11]. We separate the contribution from the Higgs loop diagram and that from the gauge boson loop diagrams to the SM asymmetries as, Observation of A 7 requires the charge identification of the final fermion f . This requirement is easily met for the decay modes Z → − + . The charge of a B meson containing one b orb quark can be identified via the decay mode B → ν + X. We assume an efficiency of 20% for identifying the charges of the decaying b orb hadrons [44]. The asymmetries receive unpleasant suppressions due to the fact that the T-odd coefficients are also parity-odd in case we do not measure the spin of the initial and final states [36]. Techniques to reduce the suppressions are as follows. The asymmetry A 7 vanishes if the Z → ff decay process conserves parity. Since the coupling of the charged lepton to the Z is almost axial-vector, A 7 in the Z → e − e + and Z → µ − µ + decay modes has the suppression factor of ∼ 1/5, which is unavoidable. However, for some fraction of the Z → τ − τ + decay, we can measure the τ helicity from τ decay distributions [47,48] and reduce the suppression factor. We assume an efficiency of 40% for measuring the τ helicity [44]. Similarly, the asymmetry A 8 vanishes if parity is conserved in the production process, namely if a (i)− = a (i)+ for all i = 1, 2, 3.
Due to the coupling of the incoming electron to the Z being dominantly axial-vector, the to the asymmetries also become larger at higher c.m. energies, implying a higher sensitivity to λ 3 at a higher beam energy.

Higgs loop contribution
We parametrize the NP effect on the trilinear self-coupling in terms of a real parameter δ h as where, δ h = 0 gives the SM prediction for λ 3 . Because A Higgs i is proportional to λ 3 , the asymmetries with nonzero δ h can be described as In Fig. 5 To summarize, a direct measurement of the Higgs self-coupling λ 3 is possible even in the single Higgs production process e + e − → ZH by using the T-odd asymmetries. Due to the smallness of the asymmetries ( 1%), the method is very challenging and requires a huge statistics. Our analysis with a beam polarization (−0.80, +0.30) and an integrated luminosity of 30 ab −1 suggests that using this method we can measure λ 3 with an accuracy of ∼ 100% at √ s = 340 GeV. However, the following benefits of the proposed method should be emphasized: (1) Any ambiguity from a possible modification in the top Yukawa coupling is absent in a measured λ 3 , when the beam energy is below the tt threshold.
(2) Since the T-odd asymmetries are independent information from the ZH production cross section, the ZZH coupling which also contributes to the asymmetries can be very well constrained through the cross section measurement and, therefore, the asymmetries can be utilized to constrain λ 3 only.
(3) This is so far the only approach to directly measure λ 3 in e + e − collisions, when a beam energy above the ZHH threshold is not available.