Mixed electroweak-QCD corrections toe⁺e^-→μ⁺μ^-Hat CEPC with finite-width effect

Ever since the ground-breaking discovery of the Higgs boson at Large Hadron Collider (LHC) in 2012 [1,2], one of the highest priorities of particle physics is to nail down the properties of the Higgs boson as precise as possible. Unlike the hadron colliders, which suffer from severe contamination due to the copious background events, the electron-positron colliders provide an ideal platform to precisely measure various Higgs couplings [3]. In recent years, three next-generatione⁺e⁻colliders have been proposed for dedicated study of Higgs boson: International Linear Collider (ILC) [4], Future Circular Collider (FCC-ee) [5], and Circular Electron-Positron Collider (CEPC) [6], all of which plan to operate at the center-of-mass energy around 240~250 GeV.

There emerge several Higgs production mechanisms ate⁺e⁻colliders: Higgsstrahlung,WWfusion andZZfusion, etc.. Around $\sqrt{s}\approx 240$ GeV, which is the projected energy range of CEPC, the Higgs production is dominated by the Higgsstrahlung channele⁺e⁻→ZH, Higgs production associated with aZ⁰boson. It is anticipated that, with the aid of very high luminosity and the recoil mass technique, CEPC can measure the Higgs production cross section with an exquisite sub-per-cent accuracy. Needless to say, it is indispensable for theoretical predictions for the Higgsstrahlung channel to be commensurate with the projected experimental precision.

The leading order (LO) prediction fore⁺e⁻→ZHwas first considered in 70s [7-10]. In the early 90s, the next-to-leading order (NLO) electroweak correction for this process has also been addressed by three groups independently [11-13], which turns out to be significant. Very recently, the mixed electroweak-QCD next-to-next-to-leading (NNLO) corrections were also be independently calculated by two groups [14,15]. The ${\mathcal{O}}(\alpha {\alpha }_{s})$ correction may reach 1% of the LO prediction, thereby must be included when confronting the future measurement. Recently, the ISR effect of this process has also been carefully analyzed [16].

From the experimental angle, it is the decay products of theZ⁰boson, rather than theZ⁰itself that are tagged by detectors in the Higgsstrahlung channel, since theZ⁰is an unstable particle. Therefore, in order to get closer contact with experiment, it is advantageous to make precise predictions directly for the process ${e}^{+}{e}^{-}\to ({Z}^{* }\to )f\bar{f}+H$ , wherefrepresents leptons or quarks. Among a flurry of Higgs production channels associated with variousZdecay products, thee⁺e⁻→μ⁺μ^-Hprocess occupies a unique place for probing Higgs properties, because it is a very clean channel and possesses large cross section. The production cross section for this individual channel can be measured with 0.9% precision at CEPC [6,17]. Combining several other channels, CEPC is anticipated to measure the Higgs production rate with the accuracy of 0.51%.

The LO contribution to the ${e}^{+}{e}^{-}\to f\bar{f}+H$ process was first considered in 70s [18]. The initial-state-radiation (ISR) correction to these types of processes was addressed in 80s [19]. There exist a flurry of higher-order studies for the process ${e}^{+}{e}^{-}\to \nu \bar{\nu }H$ , where both Higgsstrahlung andWWfusion mechanisms contribute [20-27]. To our knowledge, there appears no dedicated work to investigate the NLO weak correction toe⁺e⁻→μ⁺μ^-H. Nevertheless, the NLO weak correction to a similar processe⁺e⁻→e⁺e⁻Hwere calculated by the GRACE group more than a decade ago [28,29]. One can extract the corresponding NLO weak correction toe⁺e⁻→μ⁺μ^-Hby singling out a subset of diagrams in [28,29].

The purpose of this work is to conduct a systematic investigation on the higher-order radiative corrections to the processe⁺e⁻→μ⁺μ^-H, to match the projected experimental precision at CEPC. We first compute the NLO weak correction toe⁺e⁻→μ⁺μ^-H, then proceed to include the ${\mathcal{O}}(\alpha {\alpha }_{s})$ mixed electroweak-QCD correction. Besides the integrated cross section, we also study the impact of radiative corrections to various kinematic distributions such as theμ⁺μ^-invariant mass distribution. For this purpose, the finiteZ⁰width effect must be consistently taken into account. It is also instructive to examine how our results deviate from those obtained by invoking the narrow width approximation (NWA).

The rest of the paper is structured as follows. In Section2, adopting the Breit-Wigner ansatz for the resonantZ⁰propagator, we recapitulate the LO prediction fore⁺e⁻→μ⁺μ^-Hand also show the corresponding NWA result. In Section3, we specify our strategy of implementing the finiteZ⁰-width effect in higher-order calculation. In Section4, we present the calculation for the NLO weak correction to this channel. In Section5, we describe the calculation for the mixed electroweak-QCD corrections. In Section6, we present the numerical results and phenomenological analysis. Finally we summarize in Section7.

We are considering the process

where the momenta of the incoming and outgoing particles are specified in the parentheses. For future usage, we defines≡(k₁+k₂)², ands₁₂≡(p₁+p₂)². For convenience, we also define the invariant mass of the muon pair by ${M}_{\mu \mu }\equiv \sqrt{{s}_{12}}$ , which lies in the range $0\le {M}_{\mu \mu }\le \sqrt{s}-{M}_{H}$ .

At Higgs factory, lepton masses can be safely neglected owing to their exceedingly small Yukawa couplings. Consequently at the lowest order, there is only a singles-channel diagram as depicted inFig. 1. The LO amplitude reads

Figure 1.(color online) LO diagram fore⁺e⁻→μ⁺μ^-H.

wherec_W≡ cosθ_W,s_W≡ sinθ_W, withθ_Wthe Weinberg angle,M_Zrepresents the mass of theZ⁰boson. ${\Gamma }_{V}^{\mu }={g}_{V}^{+}{\gamma }^{\mu }\frac{1+{\gamma }^{5}}{2}+{g}_{V}^{-}{\gamma }^{\mu }\frac{1-{\gamma }^{5}}{2}$ is the coupling of the gauge boson and the charged lepton. Specifically speaking, ${g}_{Z}^{+}=\frac{{s}_{W}}{{c}_{W}}$ , ${g}_{Z}^{-}=\frac{{s}_{W}}{{c}_{W}}-\frac{1}{2{s}_{W}{c}_{W}}$ . The chirality structure of the neutral current demands that,e⁺ande⁻(alsoμ⁺andμ^-) must carry opposite helicity in order to render a non-vanishing amplitude.

As can be readily seen fromFig. 1, it is possible for theμ⁺μ^-pair to be resonantly produced from the on-shellZ⁰boson, consequently the amplitude in (2) blows up at ${s}_{12}={M}_{Z}^{2}$ , which reflects that fixed-order calculation breaks down near theZpole. To tame the singularity in the limit ${s}_{12}\to {M}_{Z}^{2}$ , it is customary to replace the secondZboson propagator in (2) with the Breit-Wigner form, which amounts to include the Dyson summation for theZboson self-energy diagrams. Retaining finiteZwidth would effectively cutoff the IR singularity. One may define a new amplitude:

where $ {\mathcal F} $ is a rescaling factor, and Γ_Zsignifies the width of theZ⁰boson.

The LO cross section is then given by

where the three-body phase space in the center-of-mass (CM) frame can be conveniently parameterized as

where $(|{{\boldsymbol{p}}}_{1}^{* }|, {\Omega }_{1}^{* })$ signifies the 3-momentum of theμ^-in the rest frame of the dimuon system, |p_H|,θ_Hrepresent the magnitude of the momentum and the polar angle of the Higgs boson in the laboratory frame, respectively. Upon neglecting masses of the electron and muon, one obtains $|{{\boldsymbol{p}}}_{1}^{* }|={M}_{\mu \mu }/2$ , and $|{{\boldsymbol{p}}}_{H}|=\frac{1}{2\sqrt{s}}{\lambda }^{1/2}(s, {s}_{12}, {M}_{H}^{2})$ , where λ (a,b,c)≡a²+b²+c²−2ab−2ac−2bcis the Källén function. In deriving (5), we have utilized the axial symmetry to eliminate the trivial dependence on the azimuthal angle of the outgoing Higgs boson.

Squaring (2), summing overμ⁺μ^-helicities, and averaging upon thee⁺e⁻polarizations, one observes that the squared amplitude bears a factorized structure, thanks to the simples-channel topology. Substituting it into (4), integrating over the solid angle ${\Omega }_{1}^{* }$ , one then arrives at the following double differential cross section:

with $\alpha \equiv \frac{{e}^{2}}{4\pi }$ the electromagnetic fine structure constant.

Integrating (6) over the polar angle, one obtains the Born-order spectrum of the invariant mass ofμ⁺μ^-:

Since Γ_Z≪M_Z, one naturally expects that the NWA should be fairly reliable for the process under consideration. Inserting the limiting formula

into (6), and integrating overs₁₂, we obtain the angular distribution:

where

is the angular distribution of theZ(H) in the processe⁺e⁻→ZHat Born order, with $|{{\boldsymbol{p}}}_{H}|\equiv \frac{1}{2\sqrt{s}}{\lambda }^{1/2}(s, {M}_{Z}^{2}, {M}_{H}^{2})$ . In (9), the Born-order partial width and branching fraction ofZ→μ⁺μ^-are given by

From (9), one readily obtains the LO integrated cross section in the NWA ansatz:

where

Note that the unpolarized LO cross sectionσ₀(ZH) in (13) decreases rather mildly (∝ 1/s) in the high energy limit, reflecting the dominance of producing the longitudinally polarizedZin large $\sqrt{s}$ . However, at moderate energy such as $\sqrt{s}=250$ GeV at CEPC, the longitudinally-polarized cross section only comprises of 42% of the total unpolarized cross section.

As mentioned before, in this work we are interested in addressing the NLO weak and mixed electroweak-QCD corrections fore⁺e⁻→μ⁺μ^-H:

For simplicity, in this work we have neglected the pure QED corrections (such as ISR and FSR effect), which can instead be simulated by the package Whizard [30]. As a consequence, a simplifying feature arises that the dominant higher-order diagrams resemble thes-channel topology as depicted inFig. 1, which contains only one resonantZpropagator.

Once going beyond LO, it becomes a quite delicate issue to incorporate the finiteZwidth effect yet without spoiling gauge invariance and bringing double counting. Over the past decades, numerous practical schemes have been proposed to tackle the unstable particle, such as the pole scheme [31-33], factorization scheme [34,35], fermion-loop scheme [36,37], boson-loop scheme [38], complex mass scheme [39,40], etc.. It is worth mentioning that a systematic and model-independent approach, the unstable particle effective theory, has also emerged finally [41,42]. However, this approach is valid only near the resonance peak, and cannot be applied in the entire kinematic range.

Owing to the particularly simples-channel topology of our process, it is most convenient to employ the factorization scheme [34,35], which is particularly suitable for such resonance-dominated process. In this scheme, one rescales a gauge-invariant higher-order amplitude by a Breit-Wigner factor $ {\mathcal F} $ , and subtracting theiM_ZΓ_Zterms which potentially generates double counting. The merit of this scheme is that gauge invariance is preserved, and can be readily implemented in automated calculation. Recently this scheme has also been used by Denner et al. to analyze the NLO electroweak correction to ${e}^{+}{e}^{-}\to \nu \bar{\nu }H$ [25].

For our purpose, we specify the recipe of the factorization scheme closely following [25]:

wheren=0, 1, $\mathop{M}\limits^{\sim }$ represents the fixed-order amplitude where theZ⁰is treated as a rigorously stable particle, $ {\mathcal F} $ and ${ {\mathcal M} }_{0}$ have been defined in (3), ${\hat{\Sigma }}_{ZZ}^{T}(s)$ represents the transverse part of the renormalized one-particle irreducible self-energy diagrams forZboson.M_Zis the pole mass of theZ⁰boson, and throughout the work we take Γ_Zas the experimentally determinedZ⁰boson width¹⁾.

1) By default, the pole mass of theZ⁰is determined by the condition ${\rm{Re}}\{{\hat{\Sigma }}_{ZZ}({M}_{Z}^{2})\}\equiv 0$ , whereas its width is inferred from the optical theorem, ${M}_{Z}{\Gamma }_{Z}={\rm{Im}}\{{\hat{\Sigma }}_{ZZ}({M}_{Z}^{2})\}$ . It was argued [32,33,43] that the pole mass and the corresponding width defined this way are gauge dependent. Nevertheless, the gauge-dependent terms arise at order-α³in the’t Hooft-Feynman gauge, which is beyond the accuracy targeted in this work. Thus we will pretendM_Zand Γ_Zto be gauge-invariant quantities.

The second term in the right-hand side of (15) is included to subtract the double-counting term. Fortunately, due to its orthogonal phase, the interference of this term with $\mathop{ {\mathcal M} }\limits^{\sim }$ generates a purely imaginary contribution to the cross section, thus can be safely neglected.

Once the rescaled ${\mathcal{O}}(\alpha )$ and ${\mathcal{O}}(\alpha {\alpha }_{s})$ amplitudes are obtained, we then deduce the corresponding higher-order corrections to the differential cross section through

withn=0, 1. Note even for the mixed electroweak-QCD correction, we only need consider its interference with the Born-order amplitude.

We conclude this section by stressing that, since the non-resonant diagrams are regular at ${s}_{12}={M}_{Z}^{2}$ , the rescaling procedure in (15) enforces their contributions to the amplitude to vanish on theZ⁰pole. In the vicinity of the resonance, it is intuitively appealing that the non-resonant diagrams are much more suppressed relative to the resonant diagrams. As will be seen in Section6, our numerical predictions indeed confirm this anticipation.

We now outline the calculation of the NLO weak correction toe⁺e⁻→μ⁺μ^-H, with some representative diagrams depicted inFig. 2and3. As stressed before, we will not consider the ISR and FSR types of diagrams. It is obvious that the NLO diagrams can be separated into two gauge-invariant subgroups, with either “resonant” or “non-resonant” structures. For the former subset, the diagrams are very similar to those encountered in the previous NLO weak correction fore⁺e⁻→ZH, so are the corresponding calculations; for the latter, there emerges no singularity as ${s}_{12}\to {M}_{Z}^{2}$ , so there is no need to include width effect for any particle routing around the loop.

Figure 2.(color online) Some representative higher-order diagrams fore⁺e⁻→μ⁺μ^-H, through the order-αα_s. The three solidheavy dots are explained inFig. 3. Diagrams in the first two rows correspond to the “resonant” channele⁺e⁻→(Z^*/γ^*→)μ⁺μ^-+H, while those in the last row exhibit a completely different “non-resonant” topology.

Figure 3.(color online) Representative diagrams for the radiative corrections to the renormalizedZeevertex,γ/Zself-energy, andHVVvertex, through order-αα_s. The cross represents the quark mass counterterm in QCD, cap denotes the electroweak counterterm in on-shell scheme.

The NLO amplitude is computed in Feynman gauge. Masses of all light fermions are neglected except the top quark. Dimensional regularization (DR) is employed to regularize UV divergence. The Feynman diagrams and the corresponding amplitude are generated by the package FeynArts [44]. Tensor contraction and Dirac/color matrices trace are conducted by using FeynCalc and FeynCalcFormLink [45-47]. Tensor integrals are further reduced to the Passarino-Veltman scalar functions, which are numerically evaluated by Collier [48] and LoopTools [49].

We also choose to use the standard on-shell renormalization scheme to sweep UV divergences, where various electroweak counterterms are tabulated in [50]. Depending on the specific recipe for the charge renormalization constantZ_e, there are three popular sub-schemes of the on-shell renormalization:α(0),α(M_Z) andG_μschemes [13]. In the first scheme, the fine structure constantαis assuming its Thomson-limit value, whereasα(0) is replaced with

in theα(M_Z) andG_μschemes, respectively. Differing from theα(0) scheme, these two schemes effectively resum either some universal large logarithms from the light fermion loop or some ${m}_{t}^{2}$ -enhanced terms from the top quark loop.

Once the ${\mathop{ {\mathcal M} }\limits^{\sim }}^{(\alpha )}$ is rendered finite after the renormalization procedure, we then employ (15) to obtain the rescaled amplitude ${ {\mathcal M} }^{(\alpha )}$ , which encapsulates the finiteZ-width effect. It is then straightforward to utilize (16) to infer the NLO weak correction to the differential cross section.

Finally we turn to the ${\mathcal{O}}(\alpha {\alpha }_{s})$ mixed electroweak-QCD correction toe⁺e⁻→μ⁺μ^-H. Since it is the quarks instead of leptons that can experience the strong color force, we only need retain those diagrams involving quark loop. Moreover, since the top quark couples the Higgs boson with the strongest strength, for simplicity we have neglected the masses of all lighter quarks, so we only retain those two-loop diagrams where only the top quark loop dressed by gluon. Some typical two-loop diagrams are shown inFig. 2and3, bearing only thes-channel “resonant” structure. As indicated inFig. 3, at this order, QCD renormalization is realized by merely inserting the one-loop top quark mass counterterm,δm_t, into the internal top-quark propagator, as well as into the $Ht\bar{t}$ vertex [15]. The calculation very much resembles our preceding work on ${\mathcal{O}}(\alpha {\alpha }_{s})$ correction toe⁺e⁻→ZH[15], and we referred the interested readers to that paper for more details.

For the actual two-loop computation, we utilize the packages Apart [51] and FIRE [52] to perform partial fraction and integration-by-parts (IBP) reduction. We then combine FIESTA [53]/CubPack [54] to perform sector decomposition and subsequent numerical integrations for master integrals with quadruple precision.

Besides the finite renormalization ofZeevertex [15], the ${\mathcal{O}}(\alpha {\alpha }_{s})$ amplitude can be expressed in terms of the Born-order amplitude supplemented with an effectiveHVVvertex:

where the sum is extended overV₁,V₂=Z⁰,γ, and $-ie{T}_{H{V}_{1}{V}_{2}}^{\mu \nu }$ is theHV₁V₂effective vertex, which depends onK=k₁+k₂andP=p₁+p₂. The gauge bosonV₁is coupled with the incominge⁺e⁻pair, whereas the gauge bosonV₂is affiliated with the outgoingμ⁺μ^-pair. ${\Gamma }_{V}^{\mu }$ represents the coupling between the gauge boson and charged leptons, whose form has already been specified in the paragraph after (2). The electromagnetic coupling of lepton is chiral symmetric, ${g}_{\gamma }^{\pm }=1$ .

By Lorentz covariance, the renormalized vertex tensor ${T}_{H{V}_{1}{V}_{2}}^{\mu \nu }$ can be decomposed as

whereT_i(i=1, ⋯ , 6) are Lorentz scalar solely depending ons,s₁₂and ${M}_{{V}_{1, 2}}^{2}$ . Furry theorem enforces thatT₆=0, technically becauseC-invariance forbids a singleγ⁵to emerge in the trace over the fermionic loop. Owing to the current conservation associated with massless leptons, it turns out that only the scalar form factorsT_{4, 5}survive in the differential cross sections.

Substituting (19) into (18), utilizing the factorization scheme (15) to implement the finiteZ⁰width effect, we then obtain the rescaled amplitude ${ {\mathcal M} }^{(\alpha {\alpha }_{s})}$ . From (16), we find the ${\mathcal{O}}(\alpha {\alpha }_{s})$ mixed electroweak-QCD correction to the differential cross section to be

where

Following [15], we take $\sqrt{s}=240$ , 250 GeV as two benchmark CM energies at CEPC. We adopt the following values for the input parameters [22]:M_H= 125.09 GeV,M_Z= 91.1876(21) GeV, Γ_Z= 2.4952(23) GeV,M_W= 80.385(15) GeV,m_t= 174.2± 1.4 GeV,G_μ= 1.166 3787(6)×10⁻⁵GeV⁻²,α(0) = 1/137.035999, $\Delta {\alpha }_{{\rm{had}}}^{(5)}=0.02764(13)$ , andα(M_Z) = 1/128.943 in theα(M_Z) scheme. To analyze the mixed electroweak-QCD correction, we takeα_s(M_Z)=0.1185, and use the package RunDec [55] to evaluate the QCD running coupling constant at other scales.

The angular distribution of Higgs boson at $\sqrt{s}=240$ GeV is depicted inFig. 4, including both NLO weak and mixed electroweak-QCD corrections. We stay with theα(0) scheme, and fix $\mu =\sqrt{s}/2$ for the QCD coupling, and take ${\alpha }_{s}(\sqrt{s}/2)=0.1135$ . The impact of the ${\mathcal{O}}(\alpha )$ and ${\mathcal{O}}(\alpha {\alpha }_{s})$ corrections to the Higgs angular distribution is quite analogous to what is found in our preceding work one⁺e⁻→ZH[15]. InFig. 5, we also show theμ⁺μ^-invariant mass spectrum at $\sqrt{s}=240$ GeV, including both NLO weak and mixed electroweak-QCD corrections. As expected, the spectrum develops a sharp Breit-Wigner peak around theZresonance. The ${\mathcal{O}}(\alpha )$ and ${\mathcal{O}}(\alpha {\alpha }_{s})$ corrections play a very minor role except in the proximity of theZpole. It is interesting for the future measurement of the di-muon spectrum at CEPC to examine our predictions.

Figure 4.(color online) Angular distribution of the Higgs boson at $\sqrt{s}=240$ GeV, shown at various levels of perturbative accuracy.

Figure 5.(color online)μ⁺μ^-invariant mass spectrum at $\sqrt{s}=240$ GeV, at various levels of perturbative accuracy.

InTable 1, we supplement more details for the dimuon invariant mass spectrum. We divide the NLO weak correction into the contribution from the resonant diagrams and the one from non-resonant diagrams. As can be seen from theTable 1, the ${\mathcal{O}}(\alpha )$ correction is saturated by the resonant diagrams almost in the entire energy range, especially near theZpeak.

		50	70	80	85	90	91	92	95	100	110	σ
LO/fb		0.66	2.39	8.03	24.45	309.02	570.98	407.45	53.27	9.66	1.31	6.9828
${\mathcal{O}}(\alpha )$	resonant/fb	0.04	0.14	0.47	1.42	17.78	32.82	23.39	3.05	0.55	0.07	0.4015
	nonresonant (10−4/fb)	65	39	22	12	1	0	-0	-7	-16	-24	8.5
${\mathcal{O}}(\alpha {\alpha }_{s})$ /fb		0.01	0.04	0.13	0.35	4.54	8.37	5.97	0.79	0.15	0.02	0.103

Table 1.Differential cross section with respect to theμ⁺μ^-invariant mass at $\sqrt{s}=240$ GeV. Note the upper bound forM_μμequals $\sqrt{s}-{M}_{H}$ .

Our goal is to present to date the most comprehensive predictions for thee⁺e⁻→μ⁺μ^-Hprocess, taking into various sorts of theoretical uncertainties account. InTable 2, we present our LO, NLO, NNLO predictions for the integrated cross section at $\sqrt{s}=240(250)$ GeV. The results are provided with three renormalization sub-schemes. We also include the uncertainty inherent in the input parameters (first error) and the uncertainty due to the QCD renormalization scale (second error). To assess the parametric uncertainty, we vary the values ofM_Wandm_t, and $\Delta {\alpha }_{{\rm{had}}}^{(5)}$ around the central PDG values within the 1σbands. For the QCD scale uncertainty, we slide theμinα_sfromM_Zto $\sqrt{s}$ .

FromTable 2, we observe a very similar pattern of scheme and parametric dependence of higher-order corrections as [15]. While the parametric and scale uncertainties of the NNLO predictions in theα(0) andα(M_Z) schemes are both about 0.5% of the NNLO results, the relative errors are somewhat reduced in theG_μscheme (≈0.2%). We also find that in theG_μscheme, the mixed electroweak-QCD corrections only amount to 0.4% of LO cross section, which might be attributed to the fact that in addition to the running ofα, universal corrections to theρparameter are also absorbed into the LO cross section. As can also be seen inTable 2, though the predicted LO cross sections from three renormalization schemes differ significantly, including the NLO weak correction significantly help them converge to each other. Including mixed electroweak-QCD correction appears not to further reduce the scheme dependence. To yield a scheme-insensitive prediction, it appears to be imperative to continue to compute the NNLO electroweak correction, which is certainly an extremely daunting task.

Since Γ_Z≪M_Z, and the production rate is predominantly saturated by theZ⁰resonance. It may seem natural to anticipate that the NWA remains valid even after including higher order corrections. Under the assumption of NWA, one may approximate the LO cross section and the higher-order radiative corrections by

whereσ(ZH) represents the Higgsstrahlung cross section, withσ₀(ZH) given in (13). Br₀is defined in (11), and the radiative corrections ${{\rm{Br}}}^{(\alpha {\alpha }_{s}^{n})}$ (n= 0, 1) can be read off from

Since the width of theZ⁰is held fixed, the perturbative expansion for the branching fraction ofZ⁰→μ⁺μ^-amounts to the expansion for the corresponding partial width.

InTable 3, we compare the predictede⁺e⁻→μ⁺μ^-Hcross section from the literal full calculation with that from NWA. For the sake of concreteness, we take $\sqrt{s}=240$ GeV, and employ theα(0) scheme. At LO, the NWA prediction is about 3% higher than the full prediction, while ${\mathcal{O}}(\alpha )$ and ${\mathcal{O}}(\alpha {\alpha }_{s})$ corrections are observed to be only slightly different. As a consequence, the NWA prediction to the total cross section at NNLO accuracy turns out to be about 4% higher than the full NNLO prediction.

Higgsstrahlung is the leading Higgs production mechanism at CEPC. The mixed electroweak-QCD correction toe⁺e⁻→ZHhas recently become available [14,15]. This piece of NNLO correction appears to be surprisingly large, about 1% of the Born-order result, therefore must be considered when matching the exquisite experimental accuracy.

To make closer contact with the actual experimental measurement, in this work we have investigated both NLO weak and mixed electroweak-QCD corrections to one of the golden mode in CEPC, i.e.e⁺e⁻→μ⁺μ^-H, with the finiteZ⁰width properly accounted. At $\sqrt{s}\approx 240$ GeV, the NLO weak correction may reach 6% of the Born order cross section, while the NNLO mixed electroweak-QCD correction can reach 1.5% of the LO cross section, greater than the projected experimental accuracy of 0.9%. We also present numerical predictions to various differential cross sections at NNLO accuracy, in particular we predict theμ⁺μ^-invariant-mass spectrum of the Breit-Wigner shape. We have also compared our full predictions with those based on the NWA, and found the agreement within a few percents. It is interesting to await the future experiment to examine our predictions.

We also carefully address the issue about scheme-dependence of our predictions, at various levels of perturbative accuracy. Employing three popular renormalization sub-schemes, we find that the predicted LO cross sections substantially differ from each other. Including the NLO weak correction is crucial to stabilize the predictions from different schemes, however including mixed electroweak-QCD correction seems not to help. To yield a scheme-insensitive prediction, it appears to be compulsory to continue to include the NNLO electroweak correction.

We are grateful to Gang Li and Qing-Feng Sun for useful discussions.