Signature of Pseudo Nambu-Goldstone Higgs boson in its Decay

If the Higgs boson is a pseudo Nambu-Goldstone boson (PNGB), the $hZ\gamma$ contact interaction induced by the $\mathcal{O}(p^4)$ invariants of the non-linear sigma model is free from its nonlinearity effects. The process $h\rightarrow Z\gamma$ can be used to eliminate the universal effects of heavy particles, which can fake the nonlinearity effects of the PNGB Higgs boson in the process $h\rightarrow V^*V$ ($V=W^\pm$,\ $Z$). We demonstrate that the ratio of the signal strength of $h\rightarrow Z\gamma$ and $h\rightarrow V^*V$ is good to distinguish the signature of the PNGB Higgs boson from Higgs coupling deviations.

Introduction. Deciphering the nature of the Higgs boson is one of the major tasks of particle physics, and one can ask whether there is dynamics behind electroweak symmetry breaking (EWSB). Given no hint of new heavy particles at the Large Hadron Collider (LHC), the best strategy is to use effective theories to parametrize the ignorance of UV physics. When the Higgs boson arises from a weakly-coupled UV theory, the Standard Model (SM) Effective Field Theory (EFT) can be used. On the other hand, the Higgs boson might emerge as a pseudo Nambu-Goldstone boson (PNGB) from some strong dynamics at the TeV scale [1][2][3][4][5][6][7][8]; see Refs. [9][10][11] for recent reviews. The traditional CCWZ formalism [12,13] is often used to construct the nonlinear sigma model (NLσM) of the PNGB Higgs boson with the symmetry breaking pattern G/H. Alternatively, one can use the so-called shift symmetry [14,15] to construct NLσM even without knowing the UV group G. The nature of the PNGB Higgs boson is encoded in the NLσM, and one can characterize its signature explicitly with a parameter ξ, which is defined as the ratio of the electroweak scale v and the decay constant of the PNGB Higgs boson f . For that the parameter ξ is named as the nonlinearity of the PNGB Higgs boson.
It's crucial to tell whether the Higgs boson is a PNGB from Higgs precision measurements. In particular, the Higgs couplings to electroweak gauge bosons are of the most importance as they are directly related to the EWSB. Unfortunately, one cannot learn any useful information of the parameter ξ from the hV V (V = W ± , Z) couplings alone. For example, two effects could modify the hV V couplings and fake each other: 1. the nonlinearity of the PNGB Higgs boson ξ; 2. the shift-symmetry-breaking effects induced by heavy particles, e.g. a singlet scalar interacting with the Higgs boson [16][17][18].
To probe the nonlinearity of the PNGB Higgs boson, we propose an observable R defined as the ratio of the signal strengths of the h → Zγ and h → V * V decay channels, We demonstrate that the ratio R is sensitive only to nonlinearity of the PNGB Higgs boson in stronglycoupled models but not to the faking effects originating from unknown heavy particles. Another advantage is that R is independent of the single Higgs boson cross section and the Higgs boson width.
From the operators at the order of O(p 2 ) and O(p 4 ) of the NLσM, one can obtain the Higgs couplings to electroweak gauge bosons which respect the shift symmetry of the PNGB Higgs boson. At the order of O(p 2 ), there is only one invariant ( D µ H) † D µ H, which gives rise to both the normalized kinetic term of the physical Higgs boson and the mass term of the electroweak gauge bosons, i.e.
where θ W is the weak mixing angle and g is the gauge coupling of SU ( Note that the VEV of the physical Higgs boson, h , is not exactly equal to 246 GeV. We define the nonlinearity parameter ξ as The hV V coupling is where M V is the mass of electroweak gauge boson V . Although sensitive to ξ, the hV V coupling alone cannot provide enough information to pin down the Higgs boson nature; the hV V coupling could be modified by heavy particles which violate the shift symmetry [16][17][18]. We use SM-EFT to describe the new physics (NP) effects which violate the shift symmetry in Higgs boson physics. Only one leading operator needs to be considered at dimension-six level [19], The effect of O H is universal in all the single Higgs boson processes as it simply rescales the amplitude as h → h/ √ 1 + c H due to the renormalization of the Higgs boson field. For cancelling the universal O H effect, we further consider the hZγ coupling.
Within the NLσM, the leading contribution to the hZγ effective coupling arises from the order of O(p 4 ). All the relevant O(p 4 ) operators can be derived in the CCWZ formalism with the coset SO(5)/SO(4) [20,21]. Alternatively, one can derive the O(p 4 ) operators based on the shift symmetry [22,23]. The resultant operators are valid in any other cosets as long as there is an unbroken SO(4) symmetry in the IR with a Higgs boson 4-plet. The hZγ effective coupling arises from two operators, µν , which are shown as follows, where B µν and W 3 µν are the field strength tensors of external electroweak gauge bosons and g is the gauge coupling of the U (1) Y group. The hZγ coupling is with ∆κ Zγ = c HB − c HW . The hZγ coupling induced by the O(p 4 ) invariants of the NLσM does not depend on ξ, i.e. it is not sensitive to the nonlinearity of the PNGB Higgs boson at all. Both operators O HW and O HB modify the hV V and hZγ couplings. However, we only consider their effects in the hZγ coupling due to the reasons as follows. The ratio R of interest to us is where g SM hZγ (g hZγ ) and g SM hV V (g hV V ) denotes the couplings of hZγ and hV V in the SM (NP), respectively. The hV V coupling is generated at tree level while the hZγ coupling at one loop level in the SM. To match the same precision of the SM couplings, the sub-leading correction to hV V coupling from O HW and O HB are neglected due to the loop suppression [19]. Similarly all the operators are considered in the h → Zγ decay.
There is another loop-induced and shift-symmetry breaking operator O γ ∼ H † HB µν B µν contributing to the processes of h → Zγ and h → γγ. Its contribution is tightly constrained by the precision measurements of hγγ coupling, however. We thus neglect the contribution of O γ in h → Zγ in this work. The Ratio R. Next we show the observable R, the ratio of the signal strengths of h → Zγ and h → V * V , is sensitive only to the nonlinearity of the PNGB Higgs boson but not to the faking effects of O H .
The leading corrections to the signal strength of the process h → V * V from both the nonlinearity of the PNGB Higgs boson and the O H operator are where Here, σ h (σ SM h ) and Γ total (Γ SM total ) denotes the single Higgs boson production cross section and the Higgs boson total width in NP models (the SM), respectively. The substitution of ξ → −ξ describes a PNGB Higgs boson from various non-compact groups [15,24]. It is clear that one cannot distinguish the contribution from F PNGB and For the process of h → Zγ there are contributions from the top-quark loop, the W -boson loops and the hZγ effective coupling; see Fig. 1. NP effects that modify the hW W and htt effective couplings are also included. The partial width of the h → Zγ decay is where M h and M Z denotes the masses of the Higgs boson and the Z boson, respectively.
where the O H effect (F OH ) is factorized out. The ratio R follows from Eqs. (12) and (15) as The dependence of σ h and Γ total cancels out in the ratio R, and, more important, the F O H term also cancels out. Table I shows the impact of both the Higgs nonlinearity and the O H operator on the hV V and hZγ effective couplings, depending on whether the Higgs boson is a PNGB from strong dynamics at ∼ TeV scale or a SMlike scalar from weakly-coupled UV theories (ξ → 0).
The hZγ effective coupling is crucial to eliminate the universal effect of O H so as to extract out the nonlinearity (ξ) of the PNGB Higgs boson. One thus can determine F PNGB when both the ∆κ Zγ and R are known precisely from data; for example, F PNGB follows directly from Eq. 16 as For a sizable ξ one might be able to tell the PNGB Higgs boson apart from a SM-like scalar (i.e. F PNGB = 1). If the Higgs boson is a PNGB, F PNGB could be smaller than one or larger than one, depending on a specific UV group from which the PNGB Higgs boson emerges; for example, F PNGB < 1 for a compact UV group (ξ > 0) and F PNGB > 1 for a non-compact UV group (ξ < 0) [24]. It is fascinating that the ratio R distinguishes the compactness of the broken UV-group G's. Figure 2(a) displays the contour of F PNGB in the plane of R and ∆κ Zγ for F PNGB = 0.7 (black), 1.0 (red) and 1.3 (blue). Figure 2(b) shows the dependence of F PNBG on R for various ∆κ Zγ 's. We note that the discrimination power of F PNGB increases with R and is strong for a negative ∆κ Zγ but quite weak for positive ∆κ Zγ 's. For example, the blue dashed curve (∆κ Zγ = 0.01) in Fig. 2(b) is not sensitive to F P N GB . Sensitivity at the LHC and CEPC. Now consider the potential of measuring F PNGB at the High luminosity Large Hadron Collider (HL-LHC), a proton-proton collider to operate at E cm = 14 TeV with an integrated luminosity of 3 ab −1 [25] , and also at the Circular electron-positron collider (CEPC), proposed to operator at E cm = 240 GeV with an integrated luminosity of 5 ab −1 [26].
The coefficient ∆κ Zγ can be derived from the measurement of anomalous triple gauge-boson couplings (aTGCs) [27,28] where V = γ/Z, g W W γ = −e and g W W Z = −e cot θ W . The NP contributions in the g 1,Z and ∆κ γ are It follows that ∆κ γ and ∆g 1,Z are expected to be measured with precisions as follows [29]: The uncertainty of the ratio R is where R 0 , µ 0 h→Zγ and µ 0 h→V V * denotes the central values of R, µ(h → Zγ) and µ(h → V V * ), respectively. The signal strengths µ(h → Zγ) and µ(h → V V * ) are expected to be measured at the HL-LHC [30, 31] and the CEPC [32] with errors as follows: For simplicity we assume µ 0 h→V V * = 1 (i.e. no deviation in the hV V couplings), yielding Defining F 0 PNGB and ∆κ 0 Zγ as the central values of F PNGB and ∆κ Zγ , respectively, the error of F PNGB is where we normalize the errors with the HL-LHC projections and the sign "∓" refers to Eqs. 17 and 18. A large R 0 would suppress the effects δR and we expect to reach a better measurement of F PNGB . The typical error of F PNGB is about 30% at the HL-LHC, while it reduces to ∼ 3% at the CEPC. Figure 3 displays the sensitivity to F PNGB from the R and ∆κ Zγ measurements at the HL-LHC (a, b) and the CEPC (c, d). The latest global fit results of Higgs boson couplings show that the ξ value is highly constrained [33,34]. As both the Higgs nonlinearity and heavy new resonances can contribute to Higgs coupling deviations, these two effects might accidentally cancel each other out. We choose three benchmark values of F PNGB 's for illustration; F PNGB = 0.7 (black-dashed curve) and F PNGB = 1.3 (blue-dashed) describes a PNGB Higgs boson from a compact and non-compact UV group, respectively, while F PNGB = 1.0 (red-solid) denotes a SM-like Higgs boson. The shade band along each F PNGB = 1.0 curve represents the 68% C.L. uncertainty of the F PNGB measurement, which is derived from Eq. 27. The black cross denotes a benchmark point of (∆κ Zγ , R) and the corresponding errors given by Eqs. 23  as |ξ| is too large (|ξ| ≥ 0.5). Equipped with precision measurements of aTGCs and Higgs-boson couplings, one could determine F PNGB = 1.0 ± (0.15 ∼ 0.25) at the HL-LHC in the regions of ∆κ Zγ −0.03 and R 1; see Fig. 3(a). However, it is not possible to do so for a positive ∆κ Zγ ; see Fig. 3(b). In comparison with the HL-LHC, the uncertainty of aTGCs measurements at the CEPC is reduced by a factor of 10; see Eqs. (27). As a result, the sensitivity to F PNGB at the CEPC is improved greatly; for example, given a sizable R, one can determine F PNGB = 1.0 ± (0.03 ∼ 0.1) in the negative ∆κ Zγ region and F PNGB = 1.0 ± (0.05 ∼ 0.25) in the positive ∆κ Zγ region. For a sizeable ξ, we could distinguish the Higgs boson nature since the accuracy of F PNGB is much improved at the CEPC. Conclusions. In this letter, we propose the signature of the PNGB Higgs boson can be distinguished in the ratio of µ(h → V V * ) and µ(h → Zγ) with the help of precision measurements of anomalous triple-gaugeboson couplings. The contribution of O H in the SM-EFT, which fakes the nonlinearity effect in h → V V * , is canceled out. Our result is valid in any coset G/H, as long as the unbroken group H in the IR contains the custodial SO(4), of which the Higgs boson arises from a custodial 4-plet. Depending on the magnitude of Higgs boson nonlinearity parameter ξ, at least 1σ confidence level of experimental sensitivity can be reached in general with the prospected accuracy of h → Zγ, h → V * V , aTGCs at HL-LHC and CEPC. Especially for the negative region of ∆κ Zγ , the sensitivity is good at the CEPC because aTGCs can be measured very accurately.