Higgs boson self-coupling constraints from single Higgs, double Higgs and Electroweak measurements

We set constraints on the trilinear Higgs boson self-coupling, $\lambda_3$, by combining the information coming from the $W$ mass and leptonic effective Weinberg angle, electroweak precision observables, with the single Higgs boson analyses targeting the $\gamma \gamma,\, ZZ^*,\, WW^*, \,\tau^+ \tau^-$ and $\bar{b} b$ decay channels and the double Higgs boson analyses in the $b\bar{b}b\bar{b}, \, b\bar{b}b \tau^+ \tau^-$ and $b\bar{b}b \gamma \gamma$ decay channels, performed by the ATLAS collaboration. With the assumption that the new physics affects only the Higgs potential, values outside the interval $ -1.8\, \lambda_3^{\rm SM}<\lambda_3<9.2 \, \lambda_3^{\rm SM}$ are excluded at $95\%$ confidence level. With respect to similar analyses that do not include the information coming from the electroweak precision observables our analysis shows a stronger constraint on both positive and negative values of $\lambda_3$.


Introduction
With the discovery of the Higgs boson [1,2,3,4], the study of the Higgs boson potential and of the Higgs self-interactions [5] has become of great interest in the scientific community [6]. The shape of the Higgs boson energy potential and the value of the Higgs self-couplings have deep implications on cosmology [7,8,9] and on quantum field theory, in particular in connection with gravity [10].
In the Standard Model (SM) the coupling of the quartic term in the Higgs potential, λ, dictates the trilinear and quadrilinear Higgs self-interactions and its value is related to the Higgs field vacuum expectation value, v, and the Higgs boson mass, m H , by λ ≡ λ SM 3 = m 2 H /2v 2 . Thus, it can be expressed as a function of physically measurable quantities in terms of G F and m H as where G F is the Fermi coupling constant, linked to v via v = ( √ 2 G F ) −1/2 , whose value is obtained from the muon lifetime measurement: G F = 1.1663788×10 −5 GeV −2 [11], while m H is the Higgs boson mass measured from the Higgs boson decay products [12,13,14,15], m H = 125.14 GeV.
The Higgs self-interactions affect any observable either at the tree-level or via quantum corrections. In particular, the trilinear Higgs coupling, λ 3 , affects the double-Higgs boson production, pp → HH at the tree-level [5,16] while both single-Higgs boson production and decay processes are affected at the oneloop level [17]. Going on in the perturbative expansion, i.e. at the two-loop level, λ 3 affects observables with no Higgs bosons as external states, like the electroweak precision observables (EWPO), in particular the W boson mass, m W , and the leptonic effective Weinberg angle sin 2 θ lep eff [18]. The latter differs from the Weinberg angle θ W defined in terms of the physical W and Z boson masses through the relation cos θ W = m W /m Z , by a renormalization factor κ lep such that sin 2 θ lep eff = κ lep sin 2 θ W , where κ lep includes all higher order corrections affecting the coupling of the Z boson to leptons.
While the couplings of the Higgs boson to vector bosons and fermions have been measured at the 5% level [19,20,21], among the Higgs self-interactions only the trilinear one can be constrained experimentally, although very weakly. Then, it is worth to use all the available experimental information in order to strengthen the constraint on λ 3 , albeit under some assumptions. At present the constraint on λ 3 obtained by the ATLAS collaboration combines the information from double Higgs analyses with an integrated luminosity up to 36.1 fb −1 with single Higgs analyses up to 79.8 fb −1 reporting that values of λ 3 outside the interval −2.3 λ SM 3 < λ 3 < 10.3 λ SM 3 are excluded at 95% confidence level (CL) [22]. Instead the CMS collaboration, using the process pp → HH → bbγγ with an integrated luminosity of 137 fb −1 , is able to exclude λ 3 values outside −3.3 λ SM 3 < λ 3 < 8.5 λ SM 3 at 95 % CL [23]. In this letter we make a further step on the path of strengthening the constraint on λ 3 by combining the public information available on the double and single-Higgs processes from the ATLAS Collaboration with the information coming from the EWPO. We perform a fit to double and single-Higgs production cross sections and Higgs decay channels together with the value of W mass and sin 2 θ lep eff building a likelihood function of one parameter of interest, κ λ , that measures the deformation of the Higgs trilinear coupling with respect to its SM value, or λ 3 = κ λ λ SM 3 . The paper is organized as follows. In section 2 we discuss the theoretical framework in which our analysis is inserted. In section 3 the experimental inputs that enter in our analysis are presented and discussed. Section 4 contains the fit procedure we employed, while the next section contains the results of the various fits we perform. Finally we present our conclusions.

Theoretical framework
In this section we briefly review some of the results presented in refs. [17,18,24] that were used as a basis for our analysis. We are interested in studying a where the dots represent higher orders interactions, while at the same time the effects of NP on the other SM couplings are assumed to be negligible. This scenario can be described by a Lagrangian that differs from the SM one only in the scalar potential part that is modified via an (in)finite tower of (Φ † Φ) n terms or with Φ the Higgs doublet, as shown in the Unitary gauge. The κ factors in eq.(1) can be easily related to the coefficient C 2n in eq.(2). In particular for the trilinear Higgs self-interaction one finds [18] We remark that the potential in eq.(2) is assumed to be general, i.e. the coefficients C 2n are not supposed to obey a hierarchy scaling as C 2n+2 ∼ C 2n v 2 /Λ 2 N P , with Λ N P the scale of NP, like in a well-behaved Effective Field Theory (EFT).
The effects induced on the observables by a modified trilinear Higgs selfinteraction occur at different orders in the perturbative expansion (tree or loop level) depending on the observable under consideration. When these effects appear for the first time at the loop level, i.e. in single-Higgs processes and EWPO, the modification of the observable induced by the lowest-order contribution can be parametrized as where O is a generic observable defined in the BSM scenario or in the SM respectively, and C 1 and C 2 are finite numerical coefficients, i.e. their values do not depend on Λ N P . The values of the C 1 , C 2 coefficients for single-Higgs observables are reported in ref. [17] while those for the EWPO can be found in ref. [18]. Here we use the latest SM theoretical predictions for m W [25] and sin 2 θ lep eff [26] to refine the calculation of the latter coefficients. We employ as SM predictions m W = 80.359 ± 0.06 GeV and sin 2 θ lep eff = 0.23151 ± 0.00006 where the errors reported are obtained combining in quadrature the parametric uncertainties with our estimate of the missing higher order terms [18]. In Table 1 the updated values of the C 1 , C 2 coefficients are presented.
Before concluding this section we want to comment on the total uncertainties that affect our analysis. For any measurement, besides the experimental uncertainty, we take into account a theory uncertainty that can be divided in a part related to the SM prediction and another κ λ -dependent associated to missing higher order terms. The former is usually already included in the experimental analyses while the latter is actually very difficult to estimate. In ref. [17] the κ λ -dependent uncertainty was estimated in terms of the process-dependent coefficient C 1 , however the result of that analysis showed a very mild dependence on this uncertainty. Concerning the κ λ -dependent uncertainty of the EWPO we used the same kind of estimate of ref. [17] finding also in our case a very mild dependence on this uncertainty.
The sin 2 θ lep eff measurements are slightly inconsistent due to a discrepancy at the level of 3σ between the LEP and the SLD most accurate measurements, namely the measurement obtained from the forward-backward asymmetry in the e + e − → Z → bb at LEP and the one obtained from the left-right asymmetry A LR in e + e − → Z → l + l − at SLD. The χ 2 of the fit is 11.5 with 5 degrees of freedom. In order to not underestimate the error on the average, from combining discrepant measurements, and to be conservative, we assume that the discrepancy is due to an underestimated systematic error that affects all measurements. Therefore all measurement uncertainties are multiplied by a scaling factor κ = 11.5/5 such that the χ 2 of the fit of the combined measurements equals its expectation value, this in turn consists in multiplying by 1.52 the error of the average computed in ref. [25]: sin 2 θ lep eff = 0.23151 ± 0.00014. The single-Higgs boson measurements were taken from the ATLAS collaboration results, nevertheless few measurements were excluded from the fit to extract κ λ . In particular the ttH production mode with the H → γγ decay mode was excluded to avoid double counting between this channel and the pp → HH → bbγγ channel, as discussed in ref. [22].  For the single-Higgs boson production and decay modes, it is conventional to fit data using the production cross section and decay branching fraction signal strengths (µ i ,µ f ), defined as the ratio between the observed values and their SM expectations: In this work the signal strengths are taken from ref. [19] where the product µ i × µ f is tabulated for each production and decay mode; the values used in this fit are summarised in Table 3. The fit performed in ref. [19] combines the ZH and the W H channel assuming that the ratio of their cross section equals its SM expectation. Such assumption cannot be made in our case because κ λ = 1 affects differently the W H and ZH cross sections. Nevertheless the sensitivity of the V H result is dominated by the H → bb channel where ZH provides the most accurate measurement of the V H signal strength value [50]. In this work the V H signal strength is therefore assigned to ZH, the impact of this assumption has been tested assigning the V H signal strength of the γγ channel (the second most relevant channel after H → bb) to W H and decoupling the W H and ZH signal strengths in the H → bb channel using inputs from ref. [50]. The impact on the result has been found to be negligible. In addition, the signal strength  Table 3: Values of the product µ i × µ f used in the fit procedure. The values corresponds to the product σ × Br normalised to its Standard Model expectation. Rows span over the decay mode f while colums over the production mode i. relative to ttH + tH of the original paper has been assigned to ttH being the tH contribution negligible, and the V V channel indicated in ref. [19] has been assigned to W + W − that dominates the sensitivity. Finally the uncertainties on the signal strengths have been symmetrised by averaging the squares of the positive and negative uncertainties. The measurements shown in Table 3 are correlated due to the cross contamination of signal events among different production and decay channels and the correlation matrix is provided in Figure 6 of ref. [19]; in the present work only correlations larger than 0.05 are taken into account for simplicity; this choice doesn't have impact on the final results. The correlation coefficients are shown in Table 4.
The determination of constraints on κ λ has been performed by the ATLAS collaboration in ref. [27] using data from the pp → HH production measurements, with the HH pair decaying to the final states bbγγ, bbτ + τ − and bbbb, finding −5.0 < κ λ < 12.0 (−5.8 < κ λ < 12.0) at 95% CL in observation (expectation). In that paper only the HH production cross section was parameterised as a function of κ λ while the decay branching fractions were assumed to be independent from κ λ and equal to their SM expectation. This assumption was removed in a conference note of the same collaboration [22] where the HH constraints were combined with single-Higgs differential measurements. In the pp → HH process the κ λ value has a big impact on the dynamic of the decay products affecting strongly the Higgs-pair invariant mass distribution, therefore it is not possible to extract sensible information from the final result without the information on the likelihood of each channel expressed as a function of κ λ . The likelihood shapes have been taken from ref. [51,52]. The shape of the expected and observed likelihood of the combination HH → bbbb, HH → bbγγ and HH → bbτ + τ − has been first scanned with 200 points and then interpolated using a third degree polynomial. Continuity of the first and the second derivative has been imposed at each point of the scan. The resulting likelihood function used in the fit procedure is shown in Figure 1.

Fit procedure
The fit procedure is performed by building up a likelihood function as a product of the likelihood function associated to each experimental measurement: The likelihood L is a function of one parameter of interest κ λ , the ratio: Λ(κ λ ) = L(κ λ )/L(κ λ ) is used to extract the best fit values and confidence intervals on κ λ , whereκ λ is the value that maximises L.
The likelihoods L H , L HH , L m W and L sin 2 θ lep eff are relative to the single Higgs production and decay measurements, the HH production, the m W and the sin 2 θ lep eff measurements respectively. The minimisation is performed on the quantity −2 lnΛ whose expression is: The −2 lnL HH is obtained directly from Figure 1, −2 lnL m W and −2 lnL sin 2 θ lep eff are χ 2 function built as: In this expression the labels "exp" and "theo" refer to the experimental and theoretical quantities respectively, while σ is the uncertainty on the observable denoted at the subscript. The experimental uncertainties are listed in Table 2 while theoretical uncertainties are discussed in section 2. The likelihood function L H contains information from single Higgs boson production and decay measurements, the quantity −2 lnL H is a χ 2 function defined as: where µ exp is a fifteen dimensional vector containing the measurements µ i × µ f listed in Table 3 and their theoretical expectation as a function of κ λ described in section 2. The matrix C(κ λ ) −1 is the inverse of the covariance matrix C(κ λ ) = C theo (κ λ ) + C exp where C exp is built from the uncertainties shown in Table 3 and the correlation matrix ρ shown in Table 4, while C theo (κ λ ) is a diagonal matrix containing the square of the theoretical uncertainties on µ i × µ f due to missing higher order terms as discussed in section 2.

Results
The value of −2 lnΛ as a function of the κ λ parameter is shown in the left panel of Figure 2. For positive κ λ values, i.e. when the interference between the box and the self-coupling diagram in the pp → HH process is destructive and brings to a sensitivity loss of the double-Higgs channel, all the three measurements (HH, single-H and EWPO) show a comparable constraining power, with a stronger impact of the EWPO for low values of κ λ . On the other hand, for negative κ λ values, the higher statistics of the single-Higgs analyses allows   to reach a better constraint on κ λ while the EWPO have a smaller impact on the result. The fit results are summarised in Table 5.
In order to compare the impact on the fit of the two EWPO we disentangle the likelihood functions of m W and sin 2 θ lep eff from the m W +sin 2 θ lep eff combination and from the combination of single-Higgs plus double-Higgs results, as shown on the right panel of Figure 2. The sensitivity of the EWPO is dominated by the m W measurement that represents an important addition to the single-Higgs and double-Higgs combination. In order to investigate if this result is due to the intrinsic sensitivity of the EWPO, we have performed the likelihood scan setting all the fit parameters to their SM expectations. For the HH analyses the expected likelihood function has been taken directly from ref. [52], while for the single-Higgs analyses and the EWPO, it has been assumed that the correlation matrix and the fractional error on the fitted parameters don't change when the parameters move from their observed values to their expected ones. The resulting -2 lnΛ functions are shown in Figure 3. The functional shapes of -2 lnΛ show that the constraining power of the EWPO is expected to be lower than what observed in data, in fact the full combined -2 lnΛ doesn't show large differences with respect to the combination of only the single-Higgs and double Higgs -2 lnΛ. From Figure 2 is possible to see that the combined H and HH -2 lnΛ has a minimum far from its SM expectation of κ λ = 1, while the minimum of the EWPO -2 lnΛ is closer to its SM expectation. Therefore the EWPO have an higher impact on the final observed result, in particular at the upper edge of the confidence interval.

Conclusion
In this paper we combine the ATLAS data analyses of the single-Higgs and double-Higgs processes with the information coming from the EWPO in order to constraint the Higgs boson trilinear self-coupling modifier κ λ = λ 3 /λ SM 3 . Under the assumption that NP affects only the Higgs potential we find as the best fit value of the trilinear self-coupling modifier κ λ = 4.0 +2.9 −3.3 excluding values outside the interval −1.8 < κ λ < 9.2 at 95% CL. With respect to analyses where single-Higgs data [22] or double-Higgs data [27] or a combination of both [22] are taken into account, our study shows that the inclusion in the fit of the information coming from the EWPO m W and sin 2 θ lep eff gives rise to a stronger constraint on κ λ , in particular on the positive side of the CL interval.
At the moment the information coming from EWPO gives an indication for λ 3 values closer to λ SM