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In July 2012, the
125GeV Higgs boson was discovered by ATLAS and CMS collaborations at the CERN Large Hadron Collider (LHC) [1, 2]. In addition to measuring the125GeV 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 spontaneousCP -violation, and a candidate for dark matter, as the Higgs sector of the THDM is composed of two complex scalar doublets [3-6]. In THDM, there are five Higgs bosons: two neutralCP -even Higgs bosons h and H (mh<mH ), two charged Higgs bosonsH± , and a neutralCP -odd Higgs boson A. Although bothCP -even scalars can be interpreted as being the125GeV Higgs boson in the alignment limit, we assume that the lighter scalar h is the125GeV Higgs boson in this study. As flavor changing neutral currents (FCNCs) can be induced in THDM, which have not yet been observed, an additionalZ2 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) [6]. The most investigated THDMs are Type-I and Type-II THDMs. For Type-I THDM, all fermions only couple to one of the two Higgs doublets, while for Type-II THDM, the up-type and down-type fermions couple to the two Higgs doublets respectively.Many different aspects of THDMs have been widely studied in previous works. As it is possible to introduce a spontaneous
CP -violation in THDM, it has been considered as a solution to the problem of baryogenesis in Refs. [7-9]. In Refs. [10, 11], the neutral scalar in the inert THDM is interpreted as a candidate for dark matter. The existence of charged Higgs bosonsH± is one important characteristic of new physics beyond the SM. Therefore, searching for a charged Higgs boson in various aspects is a high priority in experiments. At the LHC Run II, the charged Higgs boson has been probed in various channels, such asH±→tb,τντandWZ [12-15].To match the precise experimental data, theoretical predictions on kinematic observables should be calculated with high precision. Renormalization of THDM has been studied in detail in different renormalization schemes in Refs. [16-20]. The production mechanisms and decay modes of the charged Higgs boson have been investigated at the one-loop level in THDM. The Drell-Yan production of charged Higgs pair has been studied at NLO in Refs. [21, 22]. The full one-loop contributions for the charged Higgs production associated with a vector boson are given in Refs. [22-25]. The dominant decays of the charged Higgs boson into tb and
τντ have been studied in Refs. [26, 27], and the loop-induced decay modesH±→W±γ andH±→W±Z have also been investigated in Refs. [28-31]. In this work, we focus onH±W∓ associated production at electron-positron colliders in Type-I THDM. This production channel is a loop-induced process at the lowest order due to the absence of tree-levelH±W∓γ andH±W∓Z couplings, and has been investigated at one-loop level in THDM, as well as in the minimal supersymmetric standard model [32-35]. In order to test THDM viaH±W∓V couplings precisely, we study in detail the two-loop QCD corrections to thee+e−→H±W∓ process, provide the NLO QCD corrected integrated and differential cross sections, and discuss the dependence on the THDM parameters and thee+e− colliding energy.The rest of this paper is organized as follows. In Sec. 2, we give a brief review of 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. 3. In Sec. 4, the numerical results for both integrated and differential cross sections and some discussions are provided. Finally, a short summary is given in Sec. 5.
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The Higgs sector of THDM is composed of two complex scalar doublets,
Φ1=(ϕ+1,ϕ01)T andΦ2=(ϕ+2,ϕ02)T , which are both in the(1,2,1) representation of theSU(3)C⊗SU(2)L⊗U(1)Y gauge group. In this paper, we consider only theCP -conserving THDM with a discreteZ2 symmetry of the formΦ1→−Φ1 . Then, the renormalizable and gauge invariant scalar potential is given byVscalar=m211Φ†1Φ1+m222Φ†2Φ2−[m212Φ†1Φ2+h.c.]+12λ1(Φ†1Φ1)2+12λ2(Φ†2Φ2)2+λ3(Φ†1Φ1)(Φ†2Φ2)+λ4(Φ†1Φ2)(Φ†2Φ1)+12[λ5(Φ†1Φ2)2+h.c.].
(1) As parameter
m12 has mass-dimension 1, terms of this type only break theZ2 symmetry softly, which can therefore be retained. The parametersm11,m22,λ1,λ2,λ3,λ4 have to be real, as the Lagrangian must be real. Though parametersm12 andλ5 can be complex, the imaginary parts of these two parameters would induce explicitCP violation that we do not consider in this paper. So, we assume all parameters in Eq. (1) are real. The minimization of the potential in Eq. (1) gives two minima,⟨Φ1⟩ and⟨Φ2⟩ , of the form⟨Φ1⟩=(0v1/√2),⟨Φ2⟩=(0v2/√2),
(2) where
v1 andv2 are the vacuum expectation values of the neutral components of the two Higgs doubletsΦ1 andΦ2 , respectively. With respect to the convention in Ref. [36], we definev1=vcosβ andv2=vsinβ , wherev=(√2GF)−1/2≈246GeV . Expanding at the minima, the two complex Higgs doubletsΦ1,2 can be expressed asΦ1=(ϕ+1(v1+ρ1+iη1)/√2),Φ2=(ϕ+2(v2+ρ2+iη2)/√2).
(3) The mass eigenstates of the Higgs fields are given by [6, 20]
(Hh)=(cosαsinα−sinαcosα)(ρ1ρ2),
(4) (G0A)=(cosβsinβ−sinβcosβ)(η1η2),
(5) (G±H±)=(cosβsinβ−sinβcosβ)(ϕ±1ϕ±2),
(6) where
α is the mixing angle of the neutralCP -even Higgs sector. After the spontaneous electroweak symmetry breaking, the charged and neutral Goldstone fieldsG± andG0 are absorbed by the weak gauge bosonsW± and Z, respectively. Thus, THDM predicts the existence of five physical Higgs bosons: two neutralCP -even Higgs bosons h and H, one neutralCP -odd Higgs boson A, and two charged Higgs bosonsH± . The masses of these physical Higgs bosons are given bym2A=m212sinβcosβ−v2λ5,m2H±=m212sinβcosβ−v22(λ4+λ5),m2H,h=12[M211+M222±√(M211−M222)2+4(M212)2],
(7) where
M2 is the mass matrix of the neutralCP -even Higgs sector. The explicit form ofM2 is expressed asM2=m2A(sin2β−sinβcosβ−sinβcosβcos2β)+v2B2,
(8) where
B2=(λ1cos2β+λ5sin2β(λ3+λ4)sinβcosβ(λ3+λ4)sinβcosβλ2sin2β+λ5cos2β).
(9) SM Higgs is the combination of the two neutral
CP -even Higgs bosons ashSM=ρ1cosβ+ρ2sinβ=hsin(β−α)+Hcos(β−α).
(10) Thus, the lighter neutral
CP -even scalar h can be identified as the SM-like Higgs boson in the so-called alignment limit ofsin(β−α)→1 . In this paper, we consider the lighterCP -even Higgs h as being the SM-like Higgs boson discovered at the LHC.The input parameters for the Higgs sector of the THDM are chosen as
{mh,mH,mA,mH±,m12,sin(β−α),tanβ},
(11) which are implemented as the “physical basis” in 2HDMC [36]. We adopt the following benchmark scenario,
mh=125.18GeV,mH=mA=mH±,m212=m2Asinβcosβ,sin(β−α)=1,mH±∈[150,400]GeV,tanβ∈[1,5],
(12) which satisfies the theoretical constraints from perturbative unitarity [37], stability of vacuum [38], and tree-level unitarity [39]. The
Z2 soft-breaking parameterm212 is chosen asm2Asinβcosβ in order to satisfy the perturbative unitarity fortanβ∈[1,5] . Considering the constraints from the experiments at the13TeV LHC in Ref. [40],cos(β−α) is very close to 0. So, we apply the alignment limit in our benchmark scenario to setsin(β−α)=1 . Regarding themH± andtanβ parameters, we concentrate on the region with low mass and smalltanβ . -
In this section, we present in detail the calculation procedure for the
e+e−→H±W∓ process at one- and two-loop levels. The Feynman diagrams and the amplitudes are generated by FeynArts-3.11 [41] using the Feynman rules of THDM in Ref. [20]. The evaluation of the Dirac trace and the contraction of the Lorentz indices are performed by the FeynCalc-9.3 package [42, 43]. In order to reduce the Feynman integrals into combinations of a small set of integrals called the master integrals (MIs), we utilize the KIRA-1.2 package [44], which adopts the integration-by-parts (IBP) method with Laporta's algorithm [45]. One can determine the numerical results of amplitudes after evaluating the MIs, which is the main difficulty encountered in multi-loop calculation.In this study, we calculate the MIs by using the ordinary differential equations (ODEs) method [46]. A L-loop Feynman integral can be expressed as
I({a1,…,an},D,η)=∫∏Lj=1dDlj∏nk=11(Ek+iη)ak,
(13) where
D≡4−2ϵ andEk=q2k−m2k are the denominators of Feynman propagators, in whichqk are the linear combinations of loop momenta and external momenta. The physical results of the Feynman integrals are obtained by takingη→0+ , i.e.,I({a1,…,an},D,0)=limη→0+I({a1,…,an},D,η).
(14) One can construct the ODEs with respect to
η ,∂→I(η)∂η=M(η).→I(η),
(15) where
→I is a complete set of MIs. The boundary conditions of these ODEs are chosen to be atη=∞ , which are the simple vacuum integrals. The analytical expressions for the vacuum integrals up to three-loop order can be found in Refs. [47-49]. To solve these ODEs numerically, we utilize the odeint package [50] for evaluating ODEs from an initial pointηi to a target pointηj . To perform the asymptotic expansion in the domain close toη=0 , we transform the coefficient matrix into the normalized fuchsian form with the help of the epsilon package [51]. -
For the LO calculation, we adopt the 't Hooft-Feynman gauge with the on-shell renormalization scheme at one-loop order mentioned in Ref. [20]. Some representative Feynman diagrams for the
e+e−→H±W∓ process at the LO are shown in Fig. 1, where S and V in the loops represent the Higgs/Goldstone and weak gauge bosons, respectively. Due to the tiny mass of the electron, the contributions from the diagrams involving Higgs Yukawa coupling to electron are ignored. The diagrams withV−S mixing are also not shown in Fig. 1, because these diagrams can induce a factor ofme via the Dirac equation. The last diagram in Fig. 1 is a vertex counterterm diagram induced by the renormalization constantδZG±H± at one-loop level. In the on-shell renormalization scheme, the renormalization constantδZG±H± is given byδZG±H±=−2~Re∑W±H±(m2H±)mW,
(16) where
∑W±H±(m2H±) is the transition ofW±−H± atp2=m2H± , and~Re means to take the real parts of the loop integrals in the transition. It is worth mentioning that Eq. (16) is valid at both,O(α) andO(ααs) . -
There are 24 two-loop and counterterm diagrams for
e+e−→H±W∓ at the QCD NLO. Some representative ones are depicted in Fig. 2. At the QCD NLO, all two-loop Feynman diagrams are generated from the LO quark triangle loop diagram (i.e., the first diagram in Fig. 1). The cross in the quark loop diagrams represents the renormalization constant of the quark mass atO(αs) , while the circle cross displayed in the last counterterm diagram represents the renormalization constantδZG±H± atO(ααs) . The quark mass renormalization constant used in the NLO QCD calculation is given by [52]δmq=−mqαs2πC(ϵ)(μ2m2q)ϵCF2(3−2ϵ)ϵ(1−2ϵ),
(17) where
C(ϵ)=(4π)ϵΓ(1+ϵ) andCF=43 . The lowest order fore+e−→H±W∓ is the one-loop order, therefore, renormalization should be dealt with carefully in NLO QCD calculation. As shown in Figs. 1 and 2, the wave-function renormalization constantδZG±H± is involved in both NLO and LO amplitudes. As the self-energy∑W±H±(m2H±) is non-zero atO(ααs) , i.e.δZG±H± is non-zero atO(ααs) , the contribution from the last diagram in Fig. 2 should be included in the NLO QCD calculation. Typical Feynman diagrams for theH±−W± transition atO(ααs) are shown in Fig. 3. After taking into account all contributions atO(ααs) in D-dimensional spacetime, both,1ϵ2 and1ϵ singularities are all canceled. -
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 [53]:
GF=1.1663787×10−5GeV−2,αs(mZ)=0.118,mt=173GeV,mb=4.78GeV,mW=80.379GeV,mZ=91.1876GeV,
(18) where
GF is the Fermi constant. The fine structure constantα is fixed byα=√2GFπm2W(m2Z−m2W)m2Z.
(19) -
In Fig. 4, we display the LO cross section for
e+e−→H±W∓ as a function ofmH± andtanβ in the benchmark scenario in Eq. (12) at√s=500GeV (left) and1000GeV (right). From the left plot, we can see clearly that the LO cross section forH±W∓ production at a500GeV e+e− collider peaks atmH±≃184GeV due to the resonance effect of loop integrals. The cross section is sensitive to the mass of the charged Higgs boson, it can exceed3fb in the vicinity ofmH±≃184GeV at smalltanβ . In the region ofmH±<180GeV , the cross section increases slowly with the increase ofmH± , while it drops rapidly whenmH±>184GeV . Whentanβ increases from 1 to 5, the cross section decreases consistently due to the decline of theH+ˉtb Yukawa coupling strength in the lowtanβ region. Comparing the two plots in Fig. 4, we can see that the cross section at√s=1000GeV is much smaller than that at√s=500GeV because of s-channel suppression. As thee+e− colliding energy increases from 500 to1000GeV , the peak position of the cross section as a function ofmH± moves towards highmH± and themH± dependence of the cross section is reduced significantly. Moreover, the production cross section at√s=1000GeV also decreases quickly with the increase oftanβ in the plottedtanβ region.Figure 4. (color online) Contours of LO cross section for
e+e−→H±W∓ at√s=500GeV (left) and1000GeV (right) on themH±−tanβ plane.In Fig. 5, we present the dependence of the LO cross section for
e+e−→H±W∓ on thee+e− colliding energy for some typical values ofmH± andtanβ . As shown in this figure, the behaviors of the production cross section as a function of the colliding energy at different values oftanβ are quite similar. FormH±=160GeV , the cross section increases sharply in the range of√s<360GeV , reaches its maximum at√s≃360GeV , and then decreases slowly with the increase of√s . The existence of the peak at√s≃360GeV can be attributed to the competition between the phase-space enlargement and the s-channel suppression with the increase of√s . The maximum value of the cross section can exceed1fb fortanβ=1 and decreases to about0.3fb and0.1fb fortanβ=2 andtanβ=3 , respectively. Comparing the upper two plots of Fig. 5, we can see that the√s dependence of the cross section formH±=180GeV is very close to that formH±=160GeV , but there is a small peak at√s≃630GeV formH±=180GeV . Such resonance induced by loop integrals only occurs above the threshold ofH+→tˉb , i.e.,mH±>mt+mb . With the increase ofmH± , this resonance effect becomes more considerable and the peak position moves towards low√s . As shown in the bottom-left plot of Fig. 5, the resonance peak formH±=185GeV is located at√s≃490GeV , and is more distinct compared with that formH±=180GeV . Regarding the√s dependence of theH±W∓ production cross section formH±=200GeV shown in the bottom-right plot of Fig. 5, it looks quite different from those formH±=160,180 and185GeV . There is a sharp peak at√s≃390GeV for each value oftanβ∈{1,2,3} , which was also mentioned in previous works [22, 23, 25]. This peak is a consequence of competition among the phase-space enlargement, s-channel suppression and the resonance induced by loop integrals. We can see that the peak cross section can reach about4fb fortanβ=1 , and will decrease to around1fb and0.4fb whentanβ increases to 2 and 3, respectively. In the region of√s<350GeV , the cross section increases quickly with the increase of√s due to the enlargement of phase space, while in the region√s>450GeV , it is close to the result of s-channel suppression, especially at high√s . -
In this subsection, we calculate the
e+e−→H±W∓ process at the QCD NLO, and discuss the dependence of the integrated cross section on thee+e− colliding energy and the charged Higgs mass, as well as the angular distribution of the final-state charged Higgs boson.The LO, NLO QCD corrected integrated cross sections and the corresponding QCD relative correction for
e+e−→H±W∓ as functions of thee+e− colliding energy√s are depicted in Fig. 6, wheremH±=200GeV andtanβ=2 . As shown in the upper panel of this figure, the NLO QCD corrected integrated cross section peaks at√s≃375GeV . It increases sharply for√s<375GeV and decreases approximately linearly in the region of√s>500GeV with the increase of√s . From the lower panel of Fig. 6, we can see that the QCD relative correction increases rapidly from about 9% to above 60% with the increment of√s from 300 to345GeV and then decreases back to about 3% as√s increases to385GeV . The variation of the QCD relative correction with√s in the region of√s>385GeV is also plotted in the inset in the upper panel of Fig. 6 for clarity. It clearly shows that the QCD relative correction decreases approximately linearly from about 3% to around -4% with the increment of√s from 385 to1000GeV . In Table 1, we list the LO, NLO QCD corrected cross sections and the corresponding QCD relative corrections at some specific colliding energies. At√s=340GeV , which can be reached by both the International Linear Collider (ILC) [54] and the Future Circular Electron-Positron Collider (FCC-ee) [55], the cross section is about0.235fb at the QCD NLO. At the ILC with√s=500 and1000GeV , the NLO QCD corrected cross sections reach about 0.193 and0.0778fb , respectively, and the corresponding relative corrections are 0.26% and −4.27%.√s /GeV300 320 340 400 500 600 700 800 900 1000 σLO /fb0.04592 0.07868 0.1712 0.3870 0.1920 0.1481 0.1230 0.1054 0.09196 0.08126 σNLO /fb0.05004 0.09163 0.2353 0.3963 0.1925 0.1466 0.1206 0.1024 0.08865 0.07779 δ (%)8.97 16.4 37.4 2.40 0.260 −1.01 −1.95 −2.85 −3.60 −4.27 Table 1. LO, NLO QCD corrected cross sections and the corresponding QCD relative corrections for
e+e−→H±W∓ at some specific colliding energies. (mH±=200GeV andtanβ=2 ).Figure 6. (color online) LO, NLO QCD corrected integrated cross sections and the corresponding QCD relative correction for
e+e−→H±W∓ as functions ofe+e− colliding energy formH±=200GeV andtanβ=2 .In order to study the
mH± dependence of the QCD correction, we plot the NLO QCD corrected cross section, as well as the LO cross section, and the QCD relative correction as functions ofmH± in Fig. 7 withtanβ=2 and√s=500GeV . The numerical results for some typical values ofmH± are also given in Table 2. From Fig. 7, we can see that the LO and NLO QCD corrected cross sections increase from about0.18fb to around 0.29 and0.32fb , respectively, asmH± increases from 150 to184GeV , and drop to less than0.01fb formH±=400GeV . Similarly, there is also a notable spike atmH±≃184GeV in the QCD relative correction, as shown in the lower panel of Fig. 7. The QCD relative correction is less than 0.5% and thus, can be neglected formH±=150GeV , while it is expected to increase to about 11% asmH± increases to184GeV . WhenmH±>185GeV , the QCD relative correction decreases slowly to about -10% asmH± increases to400GeV .mH± /GeV150 160 170 180 190 200 250 300 350 400 σLO /fb0.1828 0.1876 0.1951 0.2140 0.2230 0.1920 0.1090 0.05699 0.02553 0.007595 σNLO /fb0.1836 0.1896 0.1990 0.2249 0.2240 0.1925 0.1062 0.05376 0.02343 0.006817 δ (%)0.438 1.07 2.00 5.09 0.448 0.260 −2.57 −5.67 −8.22 −10.24 Table 2. LO, NLO QCD corrected cross sections and the corresponding QCD relative corrections for
H±W∓ production at a√s=500GeV e+e− collider for some typical values ofmH± . (tanβ=2 ).Figure 7. (color online) LO, NLO QCD corrected cross sections and the corresponding QCD relative correction for
H±W∓ associated production at a√s=500GeV e+e− collider as functions of charged Higgs mass fortanβ=2 .The LO, NLO QCD corrected angular distributions of the final-state charged Higgs boson and the corresponding QCD relative corrections for
H+W− associated production at a500GeV e+e− collider fortanβ=2 andmH±=200 and300GeV are depicted in Fig. 8, whereθ denotes the scattering angle ofH+ with respect to the electron beam direction. Due toCP conservation, the distribution of the scattering angle ofH− with respect to the positron beam direction fore+e−→H−W+ is the same as the angular distribution ofH+ fore+e−→H+W− . From this figure, we can see that the charged Higgs boson is mostly produced in the transverse direction formH±=200 and300GeV . FormH+=200GeV , the QCD relative correction decreases rapidly from 10% to nearly 0% with the increment ofcosθ from -1 to -0.5, and is steady at around 0% in the region of−0.5<cosθ<1 . It implies that the NLO QCD correction can be neglected in most of the phase space region, except whenθ→π . FormH±=300GeV , the LO differential cross section is suppressed by the NLO QCD correction in the whole phase space region. The corresponding QCD relative correction decreases from about -0.8% to -7.5% ascosθ increases from -1 to 1. This QCD correction should be taken into consideration in a precision study of theH±W∓ production at lepton colliders. -
Searching for exotic Higgs bosons and studying its properties are important tasks for future lepton colliders. In this work, we study in detail the
H±W∓ associated production at future electron-positron colliders within the framework of Type-I THDM. We calculate thee+e−→H±W∓ process at the LO and investigate the dependence of the production cross section on the THDM parameters (mH± andtanβ ) and thee+e− colliding energy. The numerical results show that the cross section is very sensitive to the charged Higgs mass in the vicinity ofmH±≃184GeV at a500GeV e+e− collider, and it decreases consistently with the increase oftanβ in the lowtanβ 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 ofH+→tˉb , and the peak position moves towards the low colliding energy with the increase ofmH± . We also calculate two-loop NLO QCD corrections toe+e−→H±W∓ 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√s=500GeV andtanβ=2 , the QCD relative correction varies smoothly in the range of [-10%, 3%] with the increase ofmH± from150 to400GeV , except in the vicinity ofmH±≃184GeV . The QCD relative correction is sensitive to the charged Higgs mass and strongly depends on the final-state phase space. FormH±=300GeV andtanβ=2 , the QCD relative correction to theH+W− production at a500GeV e+e− collider increases from about -7.5% to -0.8% as the scattering angle ofH+ increases from 0 toπ . 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 ofHZ production at√s=240GeV FCC-ee and√s=250GeV CEPC can reach about 0.4% and 0.7%, respectively [56-58]. Due to the highW -tagging efficiency ate+e− colliders, the measurement precision ofH±W∓ production at a high-energye+e− collider can be also less than 1%. Thus, two-loop QCD correction should be taken into consideration in precision studies ofH±W∓ associated production at future lepton colliders.
QCD corrections to e+e−→H±W∓ in Type-I THDM at electron positron colliders
- Received Date: 2020-04-23
- Available Online: 2020-09-01
Abstract: We investigate in detail the charged Higgs production associated with a W boson at electron-positron colliders within the framework of the Type-I two-Higgs-doublet model (THDM). We calculate the integrated cross section at the LO and analyze the dependence of the cross section on the THDM parameters and the colliding energy in a benchmark scenario of the input parameters of the Higgs sector. The numerical results show that the integrated cross section is sensitive to the charged Higgs mass, especially in the vicinity of