Probing HZγ and Hγγ anomalous couplings in the process of e + e − → Hγ

Rare decay of the Higgs boson is a powerful tool to probe new physics beyond the standard model. The rare decays occur through quantum loops in which the standard model and new physics contributions might cancel each other out. We consider a faked-no-new-physics scenario that the new physics contributions are about minus two times the standard model contribution such that partial widths of Higgs boson rare decays are the same as the standard model predictions. We propose to measure the HZγ and Hγγ anomalous couplings in the process of e+e− → Hγ that is strongly correlated to Higgs boson rare decays. We show that the faked-no-new-physics scenario can be fully probed at a high energy electron-positron collider.

One of the major tasks of particle physics is to precisely measure the Higgs boson property. Rare decays of the Higgs boson, H → Zγ and H → γγ, offers a rich potential for new physics (NP) searches. Observing a deviation from the standard model (SM) prediction would shed light on NP models. The Hγγ and HZγ anomalous couplings are sensitive to different kind of NP and therefore are independent in principle. Measuring the HZγ and Hγγ couplings accurately is useful to test and discriminate NP models. For example, the HZγ coupling could be sizably modified in certain composite Higgs model while keeping the Hγγ coupling unchanged [1]. On the contrary, the two anomalous couplings are highly correlated in NMSSM or MSSM-like [2][3][4][5][6][7]. A large derivation in the HZγ coupling from the SM expectation, which is not correlated with a similar deviation in the Hγγ coupling, would impose strong constraints on NP models.
Our knowledge of the HZγ and Hγγ couplings are obtained from branching ratios or partial decay widths of Higgs boson rare decays. It is commonly believed that the NP effect is small if the branching ratio measurement is well consistent with the SM prediction. The rare decays of Higgs bosons occur through quantum loops in which the SM and NP contributions might cancel each other out. In this Letter, we focus on a faked-no-newphysics scenario that the NP contribution is about minus two times the SM contribution. Such a scenario yields similar partial widths of Higgs boson rare decays as the SM predictions and cannot be tested in Higgs boson decays at the Large Hadron collider (LHC). We show that the Hγ production in the electron-positron collision can verify or exclude the hidden NP effects. Measurements of the Hγ production cross section and the partial decay widths Γ(H → Zγ/γγ) are especially valuable because their correlation can be related through the anomalous couplings to the structure of NP models.
We begin with a general assumption that effects beyond the SM are described by a set of higher dimensional operators made out of the SM fields only. At the level of effective Lagrangian, effective operators after sponta-neously symmetry breaking contribute to the HZγ and Hγγ anomalous couplings as follows: where v = 246 GeV is the vacuum expectation value while Λ the NP scale. Throughout this work we choose Λ = 2 TeV. The coefficients F i are real to respect the CP parity. One can probe the anomalous couplings at the LHC from the branching ratio of H → γγ and H → Zγ rare decays. The partial decay widths are where F SM Zγ and F γγ , induced by the W -boson and topquark loops in the SM, are given by [1,8] where the functions A H 1/2 and A H 1 are given in Ref. [9] Here the bottom-quark loop contribution is ignored. The ATLAS and CMS measurements impose bounds on the signal strength relative to the SM prediction as fol- lows [10][11][12][13]: at the 95% confidence level. It yields a bound on F Zγ as −2.02 ≤ F Zγ ≤ 1.03 and two bounds on F γγ as −0.051 ≤ F γγ ≤ 0.013 and 0.55 ≤ F γγ ≤ 0.62. There are two-fold solutions of F Zγ,γγ for each fixed value of the partial decay widths. The ambiguity cannot be resolved in the branching-ratio measurements of Higgs boson rare decays. In particular, . We propose to determine both magnitude and sign of the HZγ and Hγγ couplings in the process of e + e − → Z * /γ * → Hγ at future electron-position colliders. Different from Higgs boson rare decays, the SM amplitudes of the Hγ production develop imaginary parts to enhance the production rate [14]. That helps to resolve the two-fold solutions.
The scattering process of e + e − → Hγ is absent at the tree-level in the SM when ignoring the electron mass, but it can be generated through the electroweak corrections at the loop-level [15][16][17]. The effects of the HZγ and Hγγ anomalous couplings might be comparable to those SM loop effects. One has to consider the SM loop contributions as well in the discussion of the NP effects. Figure 1 displays the representative Feynman diagrams of both NP and SM contributions.
We calculate the SM loop corrections in FormCalc [18] and LoopTools [19]. Our analytical and numerical results are consistent with those in Refs. [17]. We then incorporate the HZγ and Hγγ anomalous couplings into our calculation. In order to quantify the NP effects, we separate the total cross section of the Hγ production (σ t ) into the following three pieces: is the interference effects between the SM and NP contributions and σ (1,2,3) NP is the NP contribution. For illustration we list the total cross section (in the unit of femtobarn) for √ s = 350 GeV, 500 GeV and 1000 GeV as follows: The anomalous couplings generate strong correlations among the Hγ production and Higgs boson rare decays. Figure 2 displays the strong correlation of R σ and R Zγ/γγ (red-dashed curves) where R σ , R Zγ/γγ and the relative sign µ Zγ/γγ are defined as follows: There are two values of R σ for each fixed R Zγ/γγ ; the larger R σ corresponds to µ Zγ/γγ < 0 while the smaller to µ Zγ/γγ > 0. Now we discuss how to detect the Higgs anomalous coupling at the e + e − collider. We focus on the bb mode of Higgs boson decay. The collider signature of interest to us is one hard photon and two b-jets. We choose F Zγ = 1 to model the kinetics of the anomalous couplings. The cut efficiencies obtained also apply to other values of F Zγ/γγ . The dominant SM background, e + e − → γγ * /γZ → γbb, is generated in MadGraph [20]. At the analysis level, all signal and background events are required to pass the selection cuts: where p i T and η i denotes the transverse momentum and pseudo-rapidity of the particle i, respectively. The separation ∆R in the azimuthal angle (φ)-pseudo-rapidity (η) plane between the objects k and l is ∆R kl ≡ (η k − η l ) 2 + (φ k − φ l ) 2 . For simplicity we ignore the effects due to the finite resolution of the detector and assume a perfect b-tagging efficiency. Table I shows the rates of the signals (S) and backgrounds (B) before and after cuts, with F Zγ = 1, for four values of the c.m. energy. We assume an integrated luminosity of 1 ab −1 . The numbers of the signal and background events after imposing the above selection cuts are summarized in the second and fifth rows of Table I. The signal consisting of both the SM and NP contributions is shown in the fourth to sixth rows. Obviously, the backgrounds are larger than the signals by three or four order of magnitudes. One has to impose other cuts to extract the small signal out of the huge background.
The photon exhibits a recoil energy to balance the Higgs boson production, E γ = (s − m 2 H )/2 √ s, which could be used to trigger the signal events. The two b-jets in the signal originate from the Higgs boson while those in the background are mainly from the on-shell Z-boson. A mass-window cut on m bb , ∆M ≡ |m bb − m H | ≤ 5 GeV, suppresses the background dramatically. For instance, less than 1% of the background remains after the ∆M cut. A large anomalous coupling, e.g. F Zγ = +1, could lead to a few hundreds of the signal events after all the cuts and is testable experimentally. The significance (S/ √ B) increases with √ s owing both to the nonrenormalizable feature of the high-dimensional operators and also to the decreasing SM backgrounds.
Demanding the 5σ significance, S = 5 √ B, yields the discovery potential of the HZγ and Hγγ couplings in the scattering of e + e − → Hγ. Figure 2 displays the discovery region of R σ (bold-gray band) when one anomalous coupling is considered at a time. The two-fold ambiguity of F Zγ in the measurement of Γ(H → Zγ) can be fully resolved by precise knowledge of R σ if |F Zγ | is large enough to reach a discovery at the e + e − collider. The discrimination power of the two-fold R σ for a fixed R Zγ/γγ increases dramatically with √ s; for example, for R Zγ = 9, R σ is equal to 24 and 45 at a √ s = 350 GeV collider while it is equal to 40 and 110 at a √ s = 1000 GeV collider. We note that, the two solutions of F Zγ of R Zγ 1, the no-new-physics solution F Zγ 0 and the faked-no-new-physics solution v 2 /Λ 2 F Zγ −2F SM Zγ , can be fully resolved in the Hγ production when √ s ≥ 500 GeV. The R γγ is highly limited by the current LHC data. It yields two solutions of F γγ : one is v 2 /Λ 2 F γγ ∼ −2F SM γγ which could be detected in the Hγ production for √ s ≥ 500 GeV, the other is F γγ ∼ 0 which cannot be probed. It turns to be slightly complicated when the HZγ and Hγγ couplings both contribute to the Hγ production. Assuming an universal NP scale Λ, one is still able to determine the value of F Zγ and F γγ from the measurements of R σ and R Zγ/γγ and their correlations; see Eqs. 5 and 6. Now we turn to the effective Lagrangian discussion. The new physics effects are described by an effective La-grangian of the form [21,22] where the coefficients c i 's are numerical constants parametrizing the strength of the nonstandard interactions.
The relevant dimension-6 operators for our study are where φ denotes the SM scalar Higgs doublet, W I µν and B µν are the field-strength tensors of the SU (2) L and U (1) Y gauge bosons, respectively, and τ I = σ I /2 is the usual SU (2) L generator in the fundamental representation.
The O φφ and O BW are constrained strongly by electroweak precision measurements [23,24] and are neglected in our study. After spontaneously symmetry breaking the other two operators are related to F Zγ/γγ a CMS Projection 300fb 1 F γγ = c W W sin 2 θ W + c BB cos 2 θ W , F Zγ = (c W W − c BB ) sin(2θ W ). (10) The measurements of the Hγ production and Higgs rare decay width could probe both size and sign of c W W and c BB . For example, (i) c W W −3c BB < 0 for F γγ = 0 and F Zγ < 0; (ii) c W W = c BB for F Zγ = 0, c W W/BB > 0 for F γγ > 0 while c W W/BB < 0 for F γγ < 0; (iii) c W W < c BB −c W W /3 and c W W < 0 for F Zγ < 0 and F γγ < 0; (iv) c BB > c W W > 0 for F Zγ < 0 and F γγ > 0, etc. Those relations would shed light on new physics searches.

CMS Current Exclusion
When F Zγ ∼ F γγ 0, no excess of R σ would be observed. One can constrain tightly on F Zγ and F γγ , however. Figure 3 displays the exclusion region of F Zγ/γγ at 95% confidence level (gray region) as a function of √ s. For comparison we also plot the CMS current limits (red) and the CMS projections (blue) at a high luminosity LHC [25]. The faked-no-new-physics scenario, R Zγ/γγ ∼ 1 but v 2 /Λ 2 F Zγ/γγ ∼ −2F SM Zγ/γγ , can be excluded at a high energy electron-positron collider. For example, the blue region of F Zγ ∼ −1 in Fig. 3(a) can be completely excluded by the Hγ production; while the red and blue regions of F γγ ∼ 0.6 in Fig. 3(b) can also be excluded by the Hγ production.