Probing the electroweak symmetry breaking with Higgs production at the LHC

The electroweak symmetry breaking (EWSB) mechanism is still an undecided question in particle physics. We propose to utilize the single top quark and Higgs associated production ($th$), $Zh$ production via gluon fusion at the LHC to probe the couplings between the Higgs and the gauge bosons and further to test the EWSB. We demonstrate that the $th$ and $gg\to Zh$ productions are sensitive to the relative sign of couplings ($ht\bar{t}$, $hWW$) and ($ht\bar{t}$, $hZZ$), respectively. We find that the relative sign between $hWW$ and $hZZ$ couplings could be fully determined after combining the present measurements from $gg\to h$, $t\bar{t}h$ and the $th$, $Zh$ channels, as well as $tZj$ and $Zt\bar{t}$ production at the 13 TeV LHC, and this conclusion is not sensitive to the possible new physics contribution induced by $Zt\bar{t}$ couplings in the $gg\to Zh$ production.

The electroweak symmetry breaking (EWSB) mechanism is still an undecided question in particle physics. We propose to utilize the single top quark and Higgs associated production (th), Zh production via gluon fusion at the LHC to probe the couplings between the Higgs and the gauge bosons and further to test the EWSB. We demonstrate that the th and gg → Zh productions are sensitive to the relative sign of couplings (htt, hW W ) and (htt, hZZ), respectively. We find that the relative sign between hW W and hZZ couplings could be fully determined after combining the present measurements from gg → h, tth and the th, Zh channels, as well as tZj and Ztt production at the 13 TeV LHC, and this conclusion is not sensitive to the possible new physics contribution induced by Ztt couplings in the gg → Zh production.

Introduction:
Verifying the electroweak symmetry breaking (EWSB) mechanism is one of the major tasks of particle physics at the Large Hadron Collider (LHC) after the discovery of the Higgs-like boson [1,2]. In the Standard Model (SM), the EWSB is triggered by the Brout-Englert-Higgs mechanism, in which the couplings of the Higgs to EW gauge bosons play a crucial role. Although their coupling strengths are predicted by the SM, many new physics (NP) models could have a different prediction. Observing a deviation in the gauge couplings from the SM prediction would shed light on various NP models and also the nature of EWSB.
where g SM hV V = 2m 2 V /v with V = W , Z being the gauge couplings in the SM and v = 246 GeV. The modifier κ V could be matched to the dimension-6 SMEFT operators after the EWSB [17][18][19], and should be a leading approximation of the SMEFT to parametrize the new physics in Higgs gauge couplings [20]. A global analysis to include Higgs, diboson and top quark measurements at the LHC in the framework of SMEFT with all possible * kpxie@snu.ac.kr † Corresponding author: binyan@lanl.gov FIG. 1. Illustrative Feynman diagrams of th (a) and gg → Zh (b) production at the LHC. The red dots denote the effective couplings including both the SM and NP effects.
dimension-6 operators could be found in Ref. [21]. With higher luminosity data being accumulated, one expects the accuracy on κ V could be further improved, e.g. the uncertainty will be reduced to 2% at the high-luminosity LHC (HL-LHC) [22], which operates at the √ s = 14 TeV with an integrated luminosity of 3 ab −1 . However, the analysis based on the current Higgs signal strengths and the simulation of the future colliders can only constrain the magnitude of κ V , while not the relative sign between κ W and κ Z . It has been shown in Ref. [23] that a negative ratio λ W Z ≡ κ W /κ Z is also possible in the NP models. It is crucial to determine both the sign and the magnitude of κ V in order to further test the EWSB and search for the possible NP signals.
The sign of λ W Z could be resolved through the Higgs golden decay channel h → ZZ * → 4 with = e, µ, due to the interference effects between the tree and loop level processes [24]. Alternatively, one can also use W + W − h [25] and vector bosons fusion production of V h processes [26] at e + e − colliders to determine the sign of λ W Z . In this work, we propose a novel method to pin down the sign of λ W Z through the measurements of a Higgs boson with a single top quark (th) and gg → Zh production at the LHC; see Fig. 1. It is well known that the interference between the diagrams containing the htt vertex and those containing the hW W vertex in th production is destructive when κ t and κ W have the same sign due to the unitarity [27,28] (see Fig. 1(a)), where κ t is the modifier of top quark Yukawa coupling, We can therefore measure the sign of the htt coupling respect to that of the hW W coupling through th production at the LHC [29][30][31][32][33][34][35]. Similarly the gluon-initiated Zh production is sensitive to the relative sign between htt and hZZ couplings due to the cancelation between the box and triangle diagrams [32,[36][37][38][39][40][41]; see Fig. 1(b). Therefore, it would be promising to probe the sign of λ W Z with the reference of htt coupling through the measurements of th and gg → Zh production at the LHC. We will demonstrate in the following that combing the information of the gg → h production, tth associated production and the two processes of we suggested, both the sign and magnitude of κ V could be well constrained. th production: The th associated production can be classified into three channels: t-channel, s-channel and tW -channel. The higher order QCD and EW corrections under the SM and SMEFT have been discussed in Refs. [32,42,43]. The three channels share the same subprocess of bW µ → th and are related to each other by crossing symmetry. At high energy limit, the amplitude of bW µ → ht scattering will be dominanted by the longitudinal polarized W boson and it could be written as, Here s, t, u are the Mandelstam variables for describing the scattering of bW → th. It clearly shows that there is a strong cancelation between htt and hW W anomalous couplings at high energy. As a result, the cross section of th production can be significantly enhanced if the relative sign between htt and hW W is reversed. In order to compare th cross section with non-standard htt and hW W couplings to the SM prediction, we define a ratio R th as, Note that we include all three channels in R th definition. SM a Κ Z 1 1Σ 3. Present constraints on the anomalous couplings κt and κW at the 13 TeV LHC. The light blue region comes from the th cross section measurement [45]. The orange and green bands correspond to the limits from tth [45,46] and gg → h → W W * [16], respectively. Figure 2 displays the contours of R th = 1, 5 and 10 in the plane of anomalous couplings κ t and κ W with CT14LO PDF [44]. The th production cross section could be enhanced up to one order of magnitude when κ t κ W < 0.
Recently, the th signal strength has been measured at the 13 TeV LHC by both the CMS (137 fb −1 ) [45] and ATLAS (139 fb −1 ) [46] collaborations, and the most stringent limit comes from the former, which is µ(th) = 5.7 ± 2.7 (stat) ± 3.0 (syst). In Fig. 3, we compare the precision on the determination of the Higgs anomalous couplings κ t and κ W via the measurements of inclusive cross section from th [45] (light blue), tth [45,46] (orange) and gg → h → W W * [16] (green), assuming κ Z = 1. We summarize the signal strengths of Higgs production at the 13 TeV LHC in table I. The higher order QCD correction for th production processes have been included by a constant k-factor. A detail analysis of QCD correction for each anomalous couplings can be found in Ref. [32] and it shows that a constant k-factor should be a good approxiamtion to parametrize the QCD effects. Furthermore, the scale and PDF uncertainties are around few percentage level at the NLO accuracy [32], and the results from 4-flavor and 5-flavor scheme provide fully consistent and similarly precise predictions for the total cross section and distributions [42]. Therefore, we expect the conclusion in this section should not strongly depend on those theoretical uncertainties. From Fig. 3, it is evident that the current measurements have favored same-sign κ t and κ W at around 2σ level, i.e. κ t κ W > 0 is required.
We remark that though we assume κ Z = 1 in the analysis, the sign of κ t κ W should not strongly dependent on this assumption since κ Z will only change the Higgs total decay width, while not for the th scattering cross section. Moreover, the magnitude of κ Z has been constrained severely at the LHC [14][15][16]. Zh production via gluon fusion: We consider the htt, hZZ and Ztt couplings to the gg → Zh production. The couplings of top quark to Z boson could be parametrized th [45] tth [45] tth [46]  generically with, where g W is the EW gauge coupling and c W is the cosine of the weak mixing angle θ W . The vector and axialvector couplings of Z boson to top quark in the SM are v t = 1/2 − 4/3s 2 W and a t = 1/2. The helicity amplitudes of g(λ 1 )g(λ 2 ) → Z(λ 3 )h with helicity λ i = ±, 0 for particle i have been calculated in Refs. [41,51,52]. It shows that the dominant amplitudes come from (±, ±, 0) helicity configurations and the results with m b = 0 are [41], where The symbols and denote the contributions from triangle and box diagrams, respectively (see Fig. 1(b)). The parameter a b = −1/2 is the axial-vector coupling of Z boson to bottom quark and parameter κ b a = 1. Note that the helicity amplitudes M , −−0 could be related to M , ++0 by Bose symmetry [52]. The definition of the scalar functions F and F 0 ++ in Eq. (6) could be found in Ref. [52]. We should note that only the axial-vector component (κ t a ) of the Ztt couplings can contribute to the gg → Zh production due to the charge conjugation invariance [41].
At high energy limit, only the top quark contributes to the gg → Zh scattering and the total amplitude is, hence a strong cancellation occurs between the triangle and box diagrams in the SM where κ t = κ Z = 1. However, such relation could be violated in the NP models, so that the cancelation is spoiled and the Zh cross section would be enhanced. Similar to R th , we define a ratio R Zh to compare the Zh scattering cross section with the SM prediction,  Figure 4 displays the contours of R Zh = 1, 3 and 7 with κ t a = 1 and CT14LO PDF [44] in the plane of anomalous couplings κ t and κ Z . It shows that the cross section could be enhanced about few times compared to the SM prediction in the parameter space κ t κ Z < 0. On the other hand the gg → Zh production contributes ∼ 15% to the total cross section of the pp → Zh process in the SM at the 13 TeV LHC. Therefore, few times enhancement of gg → Zh is a large enough deviation that can be detected at the LHC.
We note that both the inclusive cross section and transverse momentum distribution of Z boson in the pp → Zh production at the 13 TeV LHC have been measured by the ATLAS and CMS collaborations with integrated luminosities 79.8 ∼ 139 fb −1 [16,[47][48][49][50]. We show the limits from the present measurements to the plane of anomalous couplings κ t and κ Z with assumption κ W = κ t a = 1 at 2σ level in Fig. 5. The light blue region denotes the constraint from the measurements of the pp → Zh production, in which both qq and gg initial states are considered. A constant k-factor has been used to mimic the higher order QCD correction effects for both qq → Zh and gg → Zh production in the analysis, i.e. k qq = 1.3 and k gg = 2.7 [53,54]. It is worthwhile discussing how much our result will be influenced by the QCD corrections. The NNLO QCD corrections to the Zh production with the anomalous couplings have been discussed in Ref. [55] and it shows a constant k-factor should be a reasonable assumption in this work [55]. Furthermore, the scale uncertainty is around 1% ∼ 2%, as a result, the high order QCD effects should not alter the conclusion in this section. The orange and green bounds show the constraints imposed by the measurements of tth [45,46] [16,[47][48][49][50]. The orange and green bands are corresponding to the limits from tth [45,46] and gg → h → ZZ * [16] production, respectively. and gg → h → ZZ * production [16]; (see tabel I for the detail of the signal strengths.) It clearly shows that the current measurements of the Zh cross sections at the LHC has resolved the ambiguity of the relative sign between κ t and κ Z , i.e. κ t κ Z > 0 is allowed. Again, we emphasize that the sign κ t κ Z should not be sensitive to the assumption of κ W = 1 due to κ W can not change the cross section of Zh scattering. Next we consider the impact of the non-standard Ztt coupling to determine the relative sign between κ t and κ Z . The Ztt couplings have been well constrained by the measurements of tZj [56,57] and Ztt [58,59] productions at the 13 TeV LHC. The limits could be potentially improved after we combining the measurement from gg → ZZ production [60]. As a conservative estimation of the impact from the Ztt coupling, we choose two benchmark points of κ t a = 0.7, 1.3 in the analysis, and show the allowed parameter space of κ t and κ Z at 2σ level with above value of κ t a in Fig. 6. Although the value of κ t a will change the allowed parameter space of the κ t and κ Z from the Zh measurements, the relative sign between them is still fixed, i.e. κ t κ Z > 0. Summary and discussion: Now equipped with the constraints for the Higgs couplings htt and hW W (see Fig. 3), htt and hZZ (see Fig. 5) at the 13 TeV LHC, we are ready to estimate the potential of pining down the sign of λ W Z through the global analysis of the gg → h, tth production and th, Zh scattering with present measurements. From the above discussion one sees that current data favors same sign for both the (htt, hW W ) and (htt, hZZ) couplings, as a result, the htt coupling could be a good reference to determine the relative sign between Higgs gauge couplings. In Fig. 7, we show the constraints on the plane of κ Z and κ W with κ t = 0.9, 1, 1.1 and κ t a = 1 from the current measurements with (blue) and without Zh data (orange) at 2σ level. Although the Zh data itself can not improve the accuracy of the κ V , the λ W Z < 0 region could be excluded almost at 2σ level by Zh measurements, and this conclusion is not sensitive to possible new physics contribution induced by Ztt coupling in the gg → Zh production (see Fig. 6). At the HL-LHC, all the experimental measurements could be much improved compared to the current data, and as a result, we expect that the gauge couplings of Higgs to W and Z bosons could be well constrained and the nature of EWSB will surface at that time. . Present constraints on the anomalous couplings κZ and κW at the 13 TeV LHC with κt = 0.9, 1, 1.1 and κ t a = 1. The blue region comes from the limits after we include all the data, while the orange band denotes the impact after we removing the Zh data (both the inclusive cross section and transverse momentum distribution of Z boson in pp → Zh production).