QCD Corrections to $e^{+}e^{-}\rightarrow H^{\pm}W^{\mp}$ in Type-I THDM at Electron Positron Colliders

In July 2012, the $125~{}{\rm GeV}$ Higgs boson has been discovered by ATLAS and CMS collaborations at CERN Large Hadron Collider (LHC) Aad et al. (2012); Chatrchyan et al. (2012). In addition to measuring the $125~{}{\rm GeV}$ Higgs boson precisely, great efforts have been made to search for exotic Higgs bosons in various scenarios beyond the standard model (SM). Among all the extensions of the SM, the two-Higgs-doublet model (THDM) is an appealing one. It provides rich phenomena such as charged Higgs bosons, explicit and spontaneous $\mathcal{CP}$-violation, and the candidate for dark matter, since the Higgs sector of the THDM is composed of two complex scalar doublets Lee (1973); Gunion et al. (2000); Gunion and Haber (2003); Branco et al. (2012). In the THDM, there are five Higgs bosons: two neutral $\mathcal{CP}$-even Higgs bosons $h$ and $H$ ($m_{h}<m_{H}$), two charged Higgs bosons $H^{\pm}$ and a neutral $\mathcal{CP}$-odd Higgs boson $A$. Although both of the two $\mathcal{CP}$-even scalars can be interpreted as the $125~{}{\rm GeV}$ Higgs boson in the alignment limit, we assume that the lighter scalar $h$ is the $125~{}{\rm GeV}$ Higgs boson in this paper. Since the flavor changing neutral currents (FCNCs) can be induced in THDM which have not been observed, an additional $Z_{2}$ symmetry is imposed to eliminate FCNCs at the tree level. Depending on the types of the Yukawa interactions between fermions and the two Higgs doublets, one can introduce several different types of THDMs (Type-I, Type-II, lepton-specific, and flipped) Branco et al. (2012). The most investigated THDMs are the Type-I and Type-II THDMs. In the Type-I THDM, all the fermions only couple to one of the two Higgs doublets, while in the Type-II THDM, the up-type and down-type fermions couple to the two Higgs doublets, respectively.

The THDMs have been widely studied in many different aspects in previous works. Since it is possible to introduce the spontaneous $\mathcal{CP}$-violation in THDM, it has been considered as a solution to the problem of baryogenesis in Refs.Turok and Zadrozny (1991); Davies et al. (1994); Fromme et al. (2006). In Refs.Deshpande and Ma (1978); Dolle and Su (2009), the neutral scalar in the inert THDM is interpreted as the candidate for dark matter. The existence of charged Higgs boson is one important characteristic of new physics beyond the SM. Therefore, searching for charged Higgs boson in various aspects is a high priority of experiments. At the LHC Run II, the charged Higgs boson has been probed in various channels, such as $H^{\pm}\rightarrow tb,\,\tau\nu_{\tau}~{}\text{and}~{}WZ$ Sirunyan et al. (2017, 2020); Aaboud et al. (2018a, b).

To match the precise experimental data, the theoretical predictions on kinematic observables should be calculated with high precision. The renormalization of the THDM has been detailedly studied in different renormalization schemes in Refs.Santos and Barroso (1997); Degrande (2015); Krause et al. (2016); Denner et al. (2016); Altenkamp et al. (2017). The production mechanisms and decay modes of the charged Higgs boson have been investigated at one-loop level in the THDM. The Drell-Yan production of charged Higgs pair has been studied at NLO in Refs.Arhrib and Moultaka (1999); Heinemeyer and Schappacher (2016). The full one-loop contributions for the charged Higgs production associated with a vector boson were given in Refs.Zhu ; Arhrib et al. (2000); Kanemura (2000a); Heinemeyer and Schappacher (2016). The dominant decays of the charged Higgs boson into $tb$ and $\tau\nu_{\tau}$ have been studied in Refs.Drees and Roy (1991); Roy (1999), and the loop-induced decay modes $H^{\pm}\to W^{\pm}\gamma$ and $H^{\pm}\to W^{\pm}Z$ have also been investigated in Refs.Capdequi Peyranere et al. (1991); Kanemura (2000b); Hernandez-Sanchez et al. (2004); Arhrib et al. (2007). In this work, we focus on the $H^{\pm}W^{\mp}$ associated production at electron-positron colliders in the Type-I THDM. This production channel is a loop-induced process at the lowest order due to the absence of tree-level $H^{\pm}W^{\mp}\gamma$ and $H^{\pm}W^{\mp}Z$ couplings, and has been investigated at one-loop level in the THDM as well as the minimal supersymmetric standard model Farris et al. (2004); Kanemura et al. (2011); Logan and Su (2002, 2003). In order to test the THDM via $H^{\pm}W^{\mp}V$ couplings precisely, we study in detail the two-loop QCD corrections to the $e^{+}e^{-}\rightarrow H^{\pm}W^{\mp}$ process, provide the NLO QCD corrected integrated and differential cross sections, and discuss the dependence on the THDM parameters and the $e^{+}e^{-}$ colliding energy.

The rest of this paper is organized as follows. In Sec.II, we give a brief review of the Type-I THDM and provide the benchmark scenario that we adopt. The methods and details of our LO and NLO calculations are presented in Sec.III. In Sec.IV, the numerical results for both integrated and differential cross sections and some discussions are provided. Finally, a short summary is given in Sec.V.

The Higgs sector of the THDM is composed of two complex scalar doublets $\Phi_{1}=(\phi_{1}^{+},\phi_{1}^{0})^{T}$ and $\Phi_{2}=(\phi_{2}^{+},\phi_{2}^{0})^{T}$, which are both in the $(\mathbf{1},\mathbf{2},\mathbf{1})$ representation of the $SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}$ gauge group. In this paper, we consider only the $\mathcal{CP}$-conserving THDM with a discrete $Z_{2}$ symmetry of the form $\Phi_{1}\rightarrow-\Phi_{1}$. Then the renormalizable and gauge invariant scalar potential is given by

	$\displaystyle\mathcal{V}_{\text{scalar}}$	$\displaystyle=m_{11}^{2}\Phi_{1}^{\dagger}\Phi_{1}+m_{22}^{2}\Phi_{2}^{\dagger}\Phi_{2}-\left[m_{12}^{2}\Phi_{1}^{\dagger}\Phi_{2}+\text{h.c.}\right]+\frac{1}{2}\lambda_{1}\left(\Phi_{1}^{\dagger}\Phi_{1}\right)^{2}+\frac{1}{2}\lambda_{2}\left(\Phi_{2}^{\dagger}\Phi_{2}\right)^{2}$		(1)
		$\displaystyle+\lambda_{3}\left(\Phi_{1}^{\dagger}\Phi_{1}\right)\left(\Phi_{2}^{\dagger}\Phi_{2}\right)+\lambda_{4}\left(\Phi_{1}^{\dagger}\Phi_{2}\right)\left(\Phi_{2}^{\dagger}\Phi_{1}\right)+\frac{1}{2}\left[\lambda_{5}\left(\Phi_{1}^{\dagger}\Phi_{2}\right)^{2}+\text{h.c.}\right].$

Since the parameter $m_{12}$ has the mass-dimension $1$, the terms of this kind only break the $Z_{2}$ symmetry softly which can be retained. The parameters $m_{11},\,m_{22},\,\lambda_{1},\,\lambda_{2},\,\lambda_{3},\,\lambda_{4}$ have to be real since the Lagrangian must be real. Though the parameters $m_{12}$ and $\lambda_{5}$ can be complex, the imaginary parts of these two parameters would induce explicit $\mathcal{CP}$ violation that we do not consider in this paper. So we assume all the parameters in Eq.(1) are real. The minimization of the potential in Eq.(1) gives two minima $\langle\Phi_{1}\rangle$ and $\langle\Phi_{2}\rangle$ of the form

$$\left\langle\Phi_{1}\right\rangle=\left(\begin{array}[]{c}{0}\\ {v_{1}/\sqrt{2}}\end{array}\right),\quad\left\langle\Phi_{2}\right\rangle=\left(\begin{array}[]{c}{0}\\ {v_{2}/\sqrt{2}}\end{array}\right),$$

(2)

where $v_{1}$ and $v_{2}$ are the vacuum expectation values of the neutral components of the two Higgs doublets $\Phi_{1}$ and $\Phi_{2}$, respectively. With respect to the convention in Ref.Eriksson et al. (2010), we define $v_{1}=v\cos\beta$ and $v_{2}=v\sin\beta$, where $v=(\sqrt{2}G_{F})^{-1/2}\approx 246~{}{\rm GeV}$. Expanding at the minima, the two complex Higgs doublets $\Phi_{1,2}$ can be expressed as

$$\Phi_{1}=\left(\begin{array}[]{c}{\phi_{1}^{+}}\\ {(v_{1}+\rho_{1}+i\eta_{1})/\sqrt{2}}\end{array}\right),\quad\Phi_{2}=\left(\begin{array}[]{c}{\phi_{2}^{+}}\\ {(v_{2}+\rho_{2}+i\eta_{2})/\sqrt{2}}\end{array}\right).$$

(3)

The mass eigenstates of the Higgs fields are given by Branco et al. (2012); Altenkamp et al. (2017)

$$\left(\begin{array}[]{c}{H}\\ {h}\end{array}\right)=\left(\begin{array}[]{cc}{\cos\alpha}&{\sin\alpha}\\ {-\sin\alpha}&{\cos\alpha}\end{array}\right)\left(\begin{array}[]{c}{\rho_{1}}\\ {\rho_{2}}\end{array}\right),$$

(4)

$$\left(\begin{array}[]{c}{G^{0}}\\ {A}\end{array}\right)=\left(\begin{array}[]{cc}{\cos\beta}&{\sin\beta}\\ {-\sin\beta}&{\cos\beta}\end{array}\right)\left(\begin{array}[]{c}{\eta_{1}}\\ {\eta_{2}}\end{array}\right),$$

(5)

$$\left(\begin{array}[]{c}{G^{\pm}}\\ {H^{\pm}}\end{array}\right)=\left(\begin{array}[]{cc}{\cos\beta}&{\sin\beta}\\ {-\sin\beta}&{\cos\beta}\end{array}\right)\left(\begin{array}[]{c}{\phi_{1}^{\pm}}\\ {\phi_{2}^{\pm}}\end{array}\right),$$

(6)

where $\alpha$ is the mixing angle of the neutral $\mathcal{CP}$-even Higgs sector. After the spontaneous electroweak symmetry breaking, the charged and neutral Goldstone fields $G^{\pm}$ and $G^{0}$ are absorbed by the weak gauge bosons $W^{\pm}$ and $Z$, respectively. Thus, the THDM predicts the existence of five physical Higgs bosons: two neutral $\mathcal{CP}$-even Higgs bosons $h$ and $H$, one neutral $\mathcal{CP}$-odd Higgs boson $A$, and two charged Higgs bosons $H^{\pm}$. The masses of these physical Higgs bosons are given by

	$\displaystyle m_{A}^{2}$	$\displaystyle=\frac{m_{12}^{2}}{\sin\beta\cos\beta}-v^{2}\lambda_{5},$		(7)
	$\displaystyle m_{H^{\pm}}^{2}$	$\displaystyle=\frac{m_{12}^{2}}{\sin\beta\cos\beta}-\frac{v^{2}}{2}\left(\lambda_{4}+\lambda_{5}\right),$
	$\displaystyle m_{H,h}^{2}$	$\displaystyle=\frac{1}{2}\left[\mathcal{M}_{11}^{2}+\mathcal{M}_{22}^{2}\pm\sqrt{\left(\mathcal{M}_{11}^{2}-\mathcal{M}_{22}^{2}\right)^{2}+4\left(\mathcal{M}_{12}^{2}\right)^{2}}\right],$

where $\mathcal{M}^{2}$ is the mass matrix of the neutral $\mathcal{CP}$-even Higgs sector. The explicit form of $\mathcal{M}^{2}$ is expressed as

$$\mathcal{M}^{2}=m_{A}^{2}\left(\begin{array}[]{cc}{\sin^{2}\beta}&{-\sin\beta\cos\beta}\\ {-\sin\beta\cos\beta}&{\cos^{2}\beta}\end{array}\right)+v^{2}\mathcal{B}^{2},$$

(8)

where

$$\mathcal{B}^{2}=\left(\begin{array}[]{cc}{\lambda_{1}\cos^{2}\beta+\lambda_{5}\sin^{2}\beta}&{\left(\lambda_{3}+\lambda_{4}\right)\sin\beta\cos\beta}\\ {\left(\lambda_{3}+\lambda_{4}\right)\sin\beta\cos\beta}&{\lambda_{2}\sin^{2}\beta+\lambda_{5}\cos^{2}\beta}\end{array}\right).$$

(9)

The SM Higgs is the combination of the two neutral $\mathcal{CP}$-even Higgs bosons as

	$\displaystyle h^{SM}$	$\displaystyle=\rho_{1}\cos\beta+\rho_{2}\sin\beta$		(10)
		$\displaystyle=h\sin(\beta-\alpha)+H\cos(\beta-\alpha).$

Thus, the lighter neutral $\mathcal{CP}$-even scalar $h$ can be identified as the SM-like Higgs boson in the so-called alignment limit of $\sin(\beta-\alpha)\rightarrow 1$. In this paper, we consider the lighter $\mathcal{CP}$-even Higgs $h$ as the SM-like Higgs boson discovered at the LHC.

The input parameters for the Higgs sector of the THDM are chosen as

$$\left\{m_{h},\,m_{H},\,m_{A},\,m_{H^{\pm}},\,m_{12},\,\sin(\beta-\alpha),\,\tan\beta\right\},$$

(11)

which are implemented as the “physical basis” in 2HDMC Eriksson et al. (2010). We adopt the following benchmark scenario,

		$\displaystyle m_{h}=125.18~{}{\rm GeV},$		(12)
		$\displaystyle m_{H}=m_{A}=m_{H^{\pm}},$
		$\displaystyle m_{12}^{2}=m_{A}^{2}\sin\beta\cos\beta,$
		$\displaystyle\sin(\beta-\alpha)=1,$
		$\displaystyle m_{H^{\pm}}\in[150,400]~{}{\rm GeV},$
		$\displaystyle\tan\beta\in[1,5],$

which satisfies the theoretical constraints from perturbative unitarity Grinstein et al. (2016), stability of vacuum Nie and Sher (1999), and tree-level unitarity Akeroyd et al. (2000). The $Z_{2}$ soft-breaking parameter $m_{12}^{2}$ is chosen as $m_{A}^{2}\sin\beta\cos\beta$ in order to satisfy the perturbative unitarity for $\tan\beta\in[1,5]$. Considering the constraints from experiments at the $13~{}{\rm TeV}$ LHC in Ref.Sirunyan et al. (2019), $\cos(\beta-\alpha)$ is very closed to $0$. So we apply the alignment limit in our benchmark scenario to set $\sin(\beta-\alpha)=1$. As to the $m_{H^{\pm}}$ and $\tan\beta$ parameters, we concentrate on the region with low mass and small $\tan\beta$.

In this section, we present in detail the calculation procedure for the $e^{+}e^{-}\rightarrow H^{\pm}W^{\mp}$ process at one- and two-loop levels. The Feynman diagrams and the amplitudes are generated by FeynArts-3.11 Hahn (2001), using the Feynman rules of THDM in Ref.Altenkamp et al. (2017). The evaluation of Dirac trace and the contraction of Lorentz indices are performed by the FeynCalc-9.3 package Mertig et al. (1991); Shtabovenko et al. (2016). In order to reduce the Feynman integrals into the combinations of a small set of integrals called the master integrals (MIs), we utilize the KIRA-1.2 package Maierhöfer et al. (2018), which adopts the integration-by-parts (IBP) method with Laporta’s algorithm Laporta (2000). One can get the numerical results of amplitudes after evaluating the MIs which is the main difficulty of multi-loop calculation.

In this paper, we calculate the MIs by using the ordinary differential equations (ODEs) method Liu et al. (2018). A $L$-loop Feynman integral can be expressed as

$$\mathcal{I}(\left\{a_{1},\ldots,a_{n}\right\},\,D,\,\eta)=\int\prod_{j=1}^{L}d^{D}l_{j}\prod_{k=1}^{n}\frac{1}{(E_{k}+i\eta)^{a_{k}}},$$

(13)

where $D\equiv 4-2\epsilon$ and $E_{k}=q_{k}^{2}-m_{k}^{2}$ are the denominators of Feynman propagators in which the $q_{k}$ are the linear combinations of loop momenta and external momenta. The physical results of the Feynman integrals are obtained by taking $\eta\rightarrow 0^{+}$, i.e.,

$$\mathcal{I}(\left\{a_{1},\ldots,a_{n}\right\},\,D,\,0)=\lim_{\eta\to 0^{+}}\mathcal{I}(\left\{a_{1},\ldots,a_{n}\right\},\,D,\,\eta).$$

(14)

One can construct the ODEs with respect to $\eta$,

$$\frac{\partial\vec{\mathcal{I}}(\eta)}{\partial\eta}=\mathcal{M}(\eta).\vec{\mathcal{I}}(\eta),$$

(15)

where $\vec{\mathcal{I}}$ is a complete set of MIs. The boundary conditions of these ODEs are chosen at $\eta=\infty$ which are the simple vacuum integrals. The analytical expressions for the vacuum integrals up to three-loop order can be found in Refs.Davydychev and Tausk (1993); Broadhurst (1999); Kniehl et al. (2017). To solve these ODEs numerically, we utilize the odeint package Ahnert et al. (2011) to evaluate ODEs from an initial point $\eta_{i}$ to a target point $\eta_{j}$. To perform the asymptotic expansion in the domain nearby $\eta=0$, we transform the coefficient matrix into normalized fuchsian form with the help of the epsilon package Prausa (2017).

Besides the input parameters for the Higgs sector of the THDM specified in benchmark scenario in Eq.(12), the following SM input parameters are adopted in our numerical calculation Tanabashi et al. (2018):

		$\displaystyle G_{F}=1.1663787\times 10^{-5}~{}{\rm GeV^{-2}},\quad\alpha_{s}(m_{Z})=0.118,$		(18)
		$\displaystyle m_{t}=173~{}{\rm GeV},\quad m_{b}=4.78~{}{\rm GeV},$
		$\displaystyle m_{W}=80.379~{}{\rm GeV},\quad m_{Z}=91.1876~{}{\rm GeV},$

where $G_{F}$ is the Fermi constant. The fine structure constant $\alpha$ is fixed by

$$\alpha=\frac{\sqrt{2}G_{F}}{\pi}\frac{m_{W}^{2}(m_{Z}^{2}-m_{W}^{2})}{m_{Z}^{2}}.$$

(19)

Searching for exotic Higgs boson and studying its properties are important tasks at future lepton colliders. In this work, we study in detail the $H^{\pm}W^{\mp}$ associated production at future electron-positron colliders within the framework of the Type-I THDM. We calculate the $e^{+}e^{-}\rightarrow H^{\pm}W^{\mp}$ process at the LO, and investigate the dependence of the production cross section on the THDM parameters ($m_{H^{\pm}}$ and $\tan\beta$) and the $e^{+}e^{-}$ colliding energy. The numerical results show that the cross section is very sensitive to the charge Higgs mass in the vicinity of $m_{H^{\pm}}\simeq 184~{}{\rm GeV}$ at a $500~{}{\rm GeV}$ $e^{+}e^{-}$ collider, and decreases consistently with the increment of $\tan\beta$ in the low $\tan\beta$ region. The existence of a peak in the colliding energy distribution of the cross section is explained by the resonance effect induced by loop integrals. This resonance occurs only above the threshold of $H^{+}\rightarrow t\bar{b}$, and the peak position moves towards low colliding energy as the increment of $m_{H^{\pm}}$. We also calculate the two-loop NLO QCD corrections to $e^{+}e^{-}\rightarrow H^{\pm}W^{\mp}$, and provide some numerical results for the NLO QCD corrected integrated cross section and the angular distribution of the final-state charged Higgs boson. For $\sqrt{s}=500~{}{\rm GeV}$ and $\tan\beta=2$, the QCD relative correction varies smoothly in the range of $[-10\%,\,3\%]$ as the increment of $m_{H^{\pm}}$ from $150$ to $400~{}{\rm GeV}$, except in the vicinity of $m_{H^{\pm}}\simeq 184~{}{\rm GeV}$. The QCD relative correction is sensitive to the charged Higgs mass and strongly depends on the final-state phase space. For $m_{H^{\pm}}=300~{}{\rm GeV}$ and $\tan\beta=2$, the QCD relative correction to the $H^{+}W^{-}$ production at a $500~{}{\rm GeV}$ $e^{+}e^{-}$ collider increases from about $-7.5\%$ to $-0.8\%$ as the scattering angle of $H^{+}$ increases from $0$ to $\pi$. Compared to hadron colliders, the measurement precision of Higgs associated production at future high-energy electron-positron colliders is much higher. The expected experimental error of Higgs production in association with a weak gauge boson at high-energy electron-positron colliders is less than $1\%$ through the recoil mass of the associated vector boson. For example, the measurement precision of $HZ$ production at $\sqrt{s}=240~{}{\rm GeV}$ FCC-ee and $\sqrt{s}=250~{}{\rm GeV}$ CEPC can reach about $0.4\%$ and $0.7\%$, respectively Peskin ; Bicer et al. (2014); Ruan (2016). Due to the high $W$-tagging efficiency at $e^{+}e^{-}$ colliders, the measurement precision of $H^{\pm}W^{\mp}$ production at a high-energy $e^{+}e^{-}$ collider can be also less than $1\%$. Thus, the two-loop QCD correction should be taken into consideration in precision study of the $H^{\pm}W^{\mp}$ associated production at future lepton colliders.

This work is supported in part by the National Natural Science Foundation of China (Grants No. 11775211 and No. 11535002) and the CAS Center for Excellence in Particle Physics (CCEPP).