Determining Higgs boson width at electron-positron colliders

Probing Higgs width $\Gamma_h$ is critical to test the Higgs properties. In this work we propose to measure $\Gamma_h$ at the $e^+e^-$ collider with a model-independent analysis under the Standard Model Effective Field Theory framework. We demonstrate that making use of the cross section measurements from $e^+e^-\to Zh$, $e^+e^-\to \nu_e\bar{\nu}_eh$ production and Higgs decay branching ratios $h\to WW^*/ZZ^*/\gamma\gamma$, one could determine $\Gamma_h$ at a percentage level with a center of mass energy $\sqrt{s}=250$ and 350 GeV and integrated luminosity $5~{\rm ab}^{-1}$. This conclusion would not depend on the assumption of the fermion Yukawa interactions. We further apply this result to constrain the fermion Yukawa couplings and it shows that the couplings could be well constrained.


I. INTRODUCTION
After the discovery of the Higgs boson at the Large Hadron Collider (LHC), precision measurements of the properties of the Higgs at the LHC and future colliders have become a prior task of particle physics. Determining the couplings of the Higgs boson to particles in the Standard Model (SM) is one of the avenues to verify the SM and search for the possible new physics (NP) beyond the SM (BSM). Applying the narrow width approximation for the Higgs production and decay, the scattering rate of process i → h → f could be factorized as the Higgs production cross section and decay branching ratio, i.e., where g i (g f ) is the Higgs coupling from the initial (final) state and Γ h is Higgs width. It is evident that any attempts to extract the information of g i,f , one needs an assumption of Γ h . Therefore, measuring the Higgs width with a model independent method becomes crucial for us to understand the Higgs properties. However, Γ h ∼ 4 MeV in the SM for 125 GeV Higgs boson, it would be a challenge to probe the Higgs width directly with a desirable accuracy at the LHC and future colliders due to the limitation of the detector energy and momentum resolution of the final states. Alternatively, Γ h could be probed indirectly at the LHC with additional theoretical assumptions. For example, Γ h could be obtained by (1) comparing the production rates of on-shell and off-shell Higgs production [1][2][3]; (2) the invariant mass distribution of γγ and ZZ from the interference between the Higgs production and the continuum background [4,5]; (3) tth and tttt production rates [6,7]. So far the AT-LAS [8,9] and CMS [10][11][12] collaborations give the upper bound of Γ h ≤ 14.4 MeV at 95% confidence level based on the first method. Due to the clean and readily identifiable signature of the Higgs boson at the e + e − collider, we expect the accuracy of Higgs width could be much improved at the * binyan@lanl.gov lepton collider. There are three major proposals for the lepton colliders, the Circular Electron Positron Collider (CEPC) [13], the Future Circular Collider (FCC-ee) [14], and the International Linear Collider (ILC) [15]. At the e + e − collider, Γ h could be probed indirectly with a high accuracy by the measurements of Higgs production rates and the decay branching ratios [16][17][18][19][20][21][22]. For example, Γ h could be measured through the e + e − → Zh(→ ZZ * ) production channel, i.e., where Γ(h → ZZ * ) (BR(h → ZZ * )) is the partial decay width (branching ratio) of h → ZZ * , and σ(e + e − → Zh) is the production rate of e + e − → Zh. To determine Γ h via this strategy, it depends on the κ-framework assumption on the Higgs couplings; i.e. all the Higgs couplings are SM-like and the deviations are dressed by one scale factors κ i for the coupling of particle i to Higgs boson. However, an important feature of the κ framework is that the kinematics of the Higgs boson are same as the SM. Going beyond κ-framework becomes important for the NP which has the different Lorentz structures of the Higgs couplings compared to the SM, e.g. the BSM operators under the SM effective field theory (SMEFT). In that case, the presence of the new hZZ anomalous couplings will ruin the strategy in Eq. 2. To overcome this problem, we need a separate method to determine the size of each operators. In this work, we try to present the minimum number of observabels that are needed to extract Γ h at the e + e − collider under the SMEFT framework. We argue that Γ h could be determined via combining the data from the cross sections of e + e − → Zh, e + e − → ν eνe h productions and branching ratios of h → W W * /ZZ * . We emphasize that our strategy would rely on the Higgs gauge couplings alone, while not for the assumption of the Yukawa interactions. It shows that the accuracy of Γ h could be reached at percentage level at the CEPC and the result is comparable to the method in Eq. 2 [20].

II. HIGGS ELECTROWEAK GAUGE COUPLINGS
Given the null results so far for NP searches at the LHC, the SMEFT is perfectly applicable at the future lepton colliders with center-of-mass energy √ s < 1 TeV. The NP effects under the SMEFT could be parameterized by a set of higher dimensional operators which are invariant under the Lorentz group and gauge symmetry [23][24][25][26], where L SM denotes the SM Lagrangian; c i is the Wilson coefficient of the dimension-6 operator O i and the dots denote higher dimension operators which will be ignored in this work. The operators that contribute to the Higgs gauge couplings in the SILH basis are [24], where D µ = ∂ µ − ig(τ i /2)W i µ − ig Y B µ is the gauge covariant derivative, g and g are the gauge couplings of SU (2) L and U (1) Y ; Y is the hypercharge of the field. Note H + ← → D µ H ≡ H + D µ H − (D µ H) + H and H is the SU (2) L weak doublet of the Higgs field, W i µν and B µν are the gauge boson field strength tensor of SU (2) L and U (1) Y , respectively. The operators O W,B,T are constrained strongly by the current electroweak precision measurements [24] and the bounds would be strengthened in the future lepton colliders [27], as a result, they will be neglected in this work. After the electroweak symmetry breaking H = v/ √ 2 with v = 246 GeV, above operators generate the following effective couplings of Higgs to the gauge bosons, where Wilson coefficients of the dimension-6 operators as follows, Here s W ≡ sin θ W and t W = tan θ W with θ W is the weak mixing angle. In the following, we will discuss the Higgs production and decay branching ratios under the general effective Lagrangian (see Eq. 5) at the e + e − collider. A systematic study on the sensitivities of probing the Higgs couplings at the e + e − collider under the SMEFT framework could be found in Refs. [21,[28][29][30][31][32][33]. We should note that the operators which are related to the SM fermions may also contribute to the observables of we are considering, however, it is beyond the scope of this paper and could be found in Refs. [28][29][30][31].

A. Higgs boson production cross sections
Next we discuss the cross sections of processes e + e − → Zh (σ Zh ) and e + e − → ν eνe h (σ ννh ) at the lepton collider. The generic hZZ, hZγ and hW + W − anomalous couplings in Eq. 5 could contribute to the cross sections σ Zh and σ ννh ; see Fig. 1. The excellent agreement between the SM and data indicates that deviations from the NP should be small. Hence, we restrict ourselves to the interference terms between SM and the BSM operators, i.e. the leading order of the coefficients c i . The total cross sections can be written as a linear combination of the SM contribution and NP corrections, where g i hZV and g i hW W , with i = a, b and V = γ, Z are the effective couplings between Higgs and gauge bosons; see Eq. 6. σ SM Zh and σ SM ννh are the production cross sections of e + e − → Zh and e + e − → ν eνe h in the SM , respectively. The coefficients R i ZV and R i W W describe the interference effects between the SM and the Higgs anomalous couplings and their values depend on the collider energy ( √ s). The coefficients R i ZV can be calculated with analytical method and the results are, where E Z is energy of the Z boson in the center-of-mass frame and F Zγ is the coupling ratio, Here m Z and m h are the Z boson and Higgs boson mass, respectively. e is the electron charge; g e L = g/c W (−1/2 + s 2 W ) and g e R = g/c W s 2 W with c W = cos θ W are the leftand right-handed gauge couplings of the Z boson to the electron. The analytical results for the coefficients R i W W are not available, thus we will show the numerical results only in this work. Figure 2 displays the coefficients g i hZV and g i hW W as a function of the collider energy √ s. Obviously, R a ZZ (red solid line) is much sensitive to the collider energy than R b ZZ (red dashed line), and R a Zγ (blue solid line) has a similar energy dependence as R a ZZ , but its value is highly suppressed by the coupling ratio F Zγ ; see Fig. 2(a). For the e + e − → ν eνe h production, the absolute value of the coefficient R a W W (red solid line) is much larger than R b W W (blue dashed line) and it also shows a stronger energy dependence compared to R b W W . It arises from the fact that the matrix element of R a W W is proportional to the momentum transfer t 1 = (k νe − k e − ) 2 and t 2 = (kν e − k e + ) 2 , where k i is the momentum of the par- Illustrative Feynman diagrams of h → W + W − * /ZZ * /γγ. The black dots denote the effective couplings including the new physics effects.
ticle i. As a result, the R a W W could be enhanced when the momentum k νe/νe is antiparallel to the k e − /e + .
The cross section σ Zh can be measured at the e + e − collider with the recoil mass method by tagging the decay products of the associated Z boson and the result is independently of the Higgs decay. However, the direct measurement of σ ννh is relying on the assumption of the Higgs decay branching ratios. Alternatively, we can extract σ ννh from the ratio of the cross sections of e + e − → Zh and e + e − → ν eνe h processes with one specific Higgs decay mode. The advantage of this observable is that the σ ννh could be measured without the assumption of the Higgs decay. As an example, we will focus on the bb mode since both the Zh and ν eνe h production with h → bb could be measured very well at the future lepton colliders [18,34]. The ratio is defined as, where R Zh = σ Zh /σ SM Zh , R ννh = σ ννh /σ SM ννh and the uncertainty from the unknown hbb coupling and Higgs width are cancelled.

B. Higgs decay branching ratios
The operators in Eq. 4 will also change the partial decay widths of Higgs to gauge bosons [35]. In this work, we will focus on h → ZZ * /W W * /γγ modes; see Fig. 3. The partial decay widths of Higgs to ZZ * and W W * can be expanded as follows, The hZγ anomalous couplings could contribute to ZZ * mode by h → Z + γ * (→ ff ). The Γ SM ZZ * and Γ SM W W * are the partial decay widths of h → ZZ * and h → W W * in the SM, respectively, and where Here g f LV /RV are the left-and right-handed gauge couplings of the fermion f to the gauge boson V = W, Z; N f c = 1 for the leptons and N f c = 3 for the quarks; V = m V /m h . After combining all possible final states, we obtain the effective couplings The function F ( ) in Eq. 12 is [36,37], The partial decay widths from BSM operators are The effective coupling g f γ eff is defined as where Q f is the electric charge of the fermion f in unites of e. The integration functions in Eq. 16 are,  Figure 4 shows the dependence of functions F a,b V V , F a,b Zγ and F . We note that the sign between F a(b) V V (blue solid for a and red solid for b) and F a(b) Zγ (blue dashed for a, red dashed for b) is opposite due to the off-shell . Such a behavior could be understood from the couplings in Eq. 5; i.e. there is a relative sign in the Feynman rules between the g a hV V and g b hV V terms. Compared to F a,b V V , there is an enhancement effect in F a,b Zγ due to the photon propagator. As a result, the absolute value of F a,b Zγ is much larger than F a,b V V . In the limit of → 1, Higgs boson can not decay to gauge boson pair with one gauge boson on-shell, so that all the functions tend to 0. For the Z boson, the functions are, For the W boson, The decay mode of Higgs to γγ is generated at looplevel in the SM. The contribution from dimension-6 operators could either come from the tree-level or at loop level by modifying the couplings in the SM loops. Since the contribution from hW W anomalous couplings in the loop will be highly suppressed, we only consider the SM-like hW W coupling in this decay mode. The partial decay width of h → γγ is, where F SM γγ −0.0046, induced by the W -boson and top quark loops in the SM [38,39]. Note that the W -boson loop dominates over the top quark loop, as a result, the possible impact from operator O y = −y t /v 2 H + HQ LH t R could be ignored.
The branching ratios of h → ZZ * /W W * /γγ can be measured by the cross section ratios, For a given Higgs mass m h = 125 GeV, the branching ratios could be expressed as follows,

C. Numerical results
Next we combine the measurements of σ Zh , σ ννh and BR Z/W/γ to determine the Higgs width. Furthermore, we have compared the result of our numerical calculations with that using the MadGraph5 [40] and found excellent agreement.
There are five variables in Eqs. (7),(10),(23), i.e. c H , c HW , c HB , c BB , Γ h /Γ 0 h . All of them can be determined by solving the linear equations and it shows that the Higgs width is depends mainly on R cs/Zh and BR W/Z . We show the energy dependence of the coefficients a, b, cBR SM W , dBR SM Z in Fig. 5(a) and various ratios of those coefficients in Fig. 5(b). We note that the size of Γ h is sensitive to the cross section R Zh and branching ratio BR W measurements, while R cs and BR Z would become important when √ s > 350 ∼ 400 GeV. The energy dependence of the coefficient a is arise from the fact that the cross section of e + e − → ν eνe h will be enhanced as the energy increase.
The Higgs width is, We plot the contours of Γ h in the plane of R cs and R Zh at √ s =250 and 350 GeV with the SM branching ratios (BR SM γ = 2.27 × 10 −3 , BR SM W = 0.214 and BR SM Z = 0.0262 [41]. ) in Fig. 6(a) and (b). The Higgs boson width in the SM prediction Γ 0 h = 4.07 MeV is used for reference. It shows that Γ h is more sensitive to R Zh than R cs at √ s = 250 GeV (see Eq. 26). However, with the increase of the collider energy, the cross section of e + e − → ν eνe h becomes larger, so that a stronger dependence on R cs is found in Fig. 6 R cs/Zh = 1. The slopes are depending on the ratio cBR SM W /(dBR SM Z ).

D. Error analysis
Now we discuss the uncertainty of Γ h from the experimental measurements. Based on the error propagation equation, we obtain the error of Γ h , which is normalized to the central value Γ 0 h , is where R 0 i and δR i are the central values and errors of those observables, respectively. The uncertainties of R i are given by, where σ 0 Zh,i (δσ Zh,i ) and σ 0 ννh,b (δσ ννh,b ) are the central values (errrors) of the processes e + e − → Zh(→ ii) and e + e − → ν eνe h(→ bb), respectively, and σ 0 Zh (δσ Zh ) is the central value (error) of the inclusive cross section of the process e + e − → Zh.
The expected uncertainties of the cross sections at the CEPC with √ s = 250 GeV and an integrated luminosity (L) of 5 ab −1 are [18,34], Therefore, we obtain the uncertainties of the R i , In the following analysis, we will assume the central values of R i to be same with SM predictions, i.e. R 0 i = 1. As a result, the error of Γ h at √ s = 250 GeV and L = 5 ab −1 is, Clearly, the cross section ratio R cs and branching ratios BR W/Z dominant the uncertainty of the Higgs width. For the general collider energy and luminosity, we could rescale the relative errors by the following method, where δσ i A,B and σ i A,B (s A,B ) are the cross section error and central value of process i with collider energy s A,B and integrated luminosity L A,B , respectively.  Figure 7 (b) shows the contours of δΓ h /Γ 0 h in the plane of the collider energy √ s (GeV) and integrated luminosity L (ab −1 ).

IV. LIMITING FERMION YUKAWA COUPLINGS
In this section, we combine the branching ratio BR(h → ff ) and Γ h measurements to constrain the fermion Yukawa couplings. The effective Lagrangian of the hff interaction could be parametrized by, where m f is the mass of fermion f = b, c, τ, µ and κ f = 1 in the SM. The κ f could by obtained through the Higgs decay branching ratio and Γ h measurements, i.e.
where Γ SM ff is the partial decay width of h → ff in the SM. Therefore, we have The uncertainty of κ f is, That yields a error on the fermion Yukawa couplings as We emphasize that the limits for the fermion Yukawa couplings are totally model-independent.

V. CONCLUSIONS
In this work we proposed a method to probe the Higgs width within the model-independent framework of the Standard Model Effective Field Theory at the future e + e − collider. The effects of the new physics are parameterized by a set of the dimension-6 operators in the SMEFT. We compute the Higgs production cross sections and decay branching ratios from the contribution of BSM operators. It shows that the Higgs width could be determined after we combine the cross sections of e + e − → Zh and e + e − → ν eνe h production processes and the branching ratio measurements h → W W * /ZZ * /γγ. We note that the size of the Higgs width is not sensitive to the BR(h → γγ) and its impact can be ignored during the numerical calculation. We further demonstrate that the Higgs width could be constrained to be percentage level at √ s = 250 and 350 GeV with integrated luminosity 5 ab −1 . As an application, we combine the Higgs width information and the decay branching ratios to constrain the fermion Yukawa couplings.