Two-Higgs-Doublet模型的多重重整化方案：h→WW/ZZ四费米子应用

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本文主要探讨了"Two-Higgs-Doublet模型"的重归一化方案在计算高阶物理过程中的应用，特别是在h→WW/ZZ→4个费米子过程中的具体实施。作者们针对这个复杂的模型，提出了四种不同的重归一化策略，分别是基于外壳（on-shell）条件的处理和剩余场混合参数的MS/\(\overline{\text{MS}}\)规范。首先，他们着重于外壳上的重新规范化条件，这是一种确保理论在不同能量尺度下物理量保持不变的重要步骤。他们为模型中的不同参数制定了特定的外壳条件，这些条件与现有的文献方案相联系，以确保理论的一致性。在处理 tadpole 图时，他们采用了不同的方法，这涉及到模型的非线性特性，如对称性破缺和额外的费米子相互作用的处理。其次，对于那些没有明确外壳条件的场混合参数，作者们采用MS/\(\overline{\text{MS}}\)规范，这是一种常见的量子场论规范方法，用于控制无穷大项并保证理论的连续性。他们系统地解决了MS/\(\overline{\text{MS}}\)参数的重归一化群方程，使得在不同尺度下的尺度变换可以被一致地执行。为了实际应用，作者将这些重归一化规则编码到了FeynArts模型文件中，便于自动化生成振幅和平方矩阵元素。他们利用这些工具计算了模型中光子衰减（由两希格斯-双重峰模型的CP偶数希格斯玻色子引起）转化为四个费米子的过程。这种计算不仅验证了各重归一化方案的有效性，还通过对比和分析尺度依赖性，评估了由于理论截断导致的扰动不确定性和理论误差。本文的贡献在于提供了一套详尽且一致的重归一化框架，适用于"Two-Higgs-Doublet模型"，这对于理解和预测该模型在高能物理实验中的行为至关重要。通过这种方式，研究人员能够更精确地预测实验结果，同时测试理论的稳定性，为后续的模型检验和新物理搜索提供了强有力的工具。

JHEP09(2017)134

2β



2α

− M

) + 2c

2α



−

β−β

− λ

2β



β−β

−

β−β



+ c



β−β

−

β−β



2β



β−β

−

β−β



+ s



β−β

−

β−β



, (2.19c)

= λ

β−β

−

β−β

−β

α+β−β

−β

− s

β−α

−β

)

β−β

−β

α+β−β

−β

+ c

β−α

−β

)

β−β

, (2.19d)

2β

β−β



+ c



β−β



− s



β−β

(2.19e)

The tree-level relations are easily obtained by setting β

= β

= β and t

= t

= M

= 0,



+ s

− s

+ λ

)



, (2.20a)



+ c

− c

+ λ

)



, (2.20b)

2α

2β

− M

) −

− 2M

) − λ

, (2.20c)

2(M

− M

)

+ λ

, (2.20d)

2β

+ λ

). (2.20e)

Parameters of the gauge sector: mass terms of the gauge bosons arise through the

interaction of the gauge bosons with the vevs, analogous to the SM. After a rotation into

ﬁelds corresponding to mass eigenstates, one obtains relations similar to the SM ones:

= g

, M

+ g

, e =

+ g

, (2.21)

where the electric unit charge e is identiﬁed with the coupling constant of the photon ﬁeld

in the covariant derivative. Inverting these relations and introducing the weak mixing

angle θ

via cos θ

= g

+ g

, one can replace v and the gauge couplings g

and g

v =

, g

. (2.22)

Mass parameterization: the relations (2.14), (2.15), (2.19a,b,d,e), and (2.22) between

the masses, angles, and the basic parameters can be used to reparameterize the Higgs

Lagrangian and change the parameters

basic

} = {λ

, . . . , λ

, m

, v

, g

}, (2.23)

– 8 –

JHEP09(2017)134

in favour of the bare mass parameters including one parameter from scalar self-interactions

which we take as λ

mass

} = {M

, M

, e, λ

, λ

, β, t

, t

}. (2.24)

Additionally, one has to keep in mind that we keep the mixing parameters α, β

, and β

generic, and they have to be ﬁxed by additional conditions (which will be given later).

One can use eq. (2.19c) to trade λ

for α, in which case the mixing angle becomes a free

parameter of the theory. Then, one obtains the parameter set

mass

} = {M

, M

, e, λ

, α, β, t

, t

}. (2.25)

2.2 Yukawa couplings

The Higgs mechanism does not only give rise to the gauge-boson mass terms (which are

determined by the vevs), but via Yukawa couplings, it introduces masses to chiral fermions.

Since both Higgs doublets can couple to fermions, the general Yukawa couplings have

the form

Yukawa

= −

k=1,2

i,j



l,k

u,k

d,k

+ h.c.



, (2.26)

with the mixing matrices ζ

f,k

, k = 1, 2, in generation space for the gauge-invariant interac-

tions with Φ

and Φ

, respectively, and the generation indices i, j = 1, 2, 3. The left-handed

SU(2) doublets of quarks and leptons are denoted Q



, d



and L



, l



while the right-handed up-type quark, down-type quark, and lepton singlets are u

, d

and l

, respectively. The primes indicate that we deal with ﬁelds in the interaction basis

here; ﬁelds without primes correspond to mass eigenstates. The ﬁeld

, k = 1, 2, is the

charge-conjugated ﬁeld of Φ

. Since, in the general THDM, there are two mass mixing

matrices for each type f of fermions, ﬂavour-changing neutral currents (FCNC) can occur

at tree level, which, however, are experimentally known to be strongly suppressed. Ac-

cording to the Paschos-Glashow-Weinberg theorem [56, 57], FCNC are absent at tree level

if each type of fermion couples only to one of the Higgs doublets. This can be achieved by

imposing an additional discrete Z

symmetry. It should be noted that the soft-Z

-breaking

term proportional to m

in the Higgs potential does not introduce FCNC. The Yukawa

Lagrangian reduces then to

Yukawa

= −

i,j



+ h.c.



, (2.27)

with n

being either 1 or 2. Depending on the exact form of the symmetry, one distin-

guishes four types of THDMs. In Type I models, all fermions couple to one Higgs doublet

(conventionally Φ

, but this is equivalent to Φ

due to possible basis changes) which can be

ensured by demanding a Φ

→ −Φ

symmetry. In Type II models, down-type fermions cou-

ple to the other doublet, which can be enforced by the symmetry Φ

→ −Φ

, d

→ −d

→ −l

. The other two possibilities are called “lepton-speciﬁc” (Type X) and “ﬂipped”

(Type Y) models. An overview over the couplings and symmetries of the diﬀerent models is

– 9 –

JHEP09(2017)134

symmetry

Type I Φ

→ −Φ

Type II Φ

(Φ

, d

, e

) → −(Φ

, d

, e

)

Lepton-speciﬁc Φ

(Φ

, e

) → −(Φ

, e

)

Flipped Φ

(Φ

, d

) → −(Φ

, d

)

Table 1. Diﬀerent types of the THDM having in common that only one of the Higgs doublet

couples to each type of fermions. This can be achieved by imposing appropriate Z

symmetry

charges to the ﬁelds.

given in table 1. For each of the fermion types, a redeﬁnition of the ﬁelds can be performed

in order to get diagonal mass matrices, analogously to the SM case. Similar to the SM,

the coupling of fermions to the Z boson is ﬂavour conserving, and a CKM matrix appears

in the coupling to the charged gauge bosons. By specifying the model type, the Higgs-

fermion interaction is determined, and one can write them, widely following the notation

of ref. [58], as

Yukawa,int

=−

f=u,d,l

f,i



h + ξ

H −2iI

W,f

−2iI

W,f



−

i,j



√

¯u

(−m

u,i

−

+ m

d,j

) d

+ h.c.



−



√

l,i

¯ν

+ h.c.



−

i,j



√

¯u

(−m

u,i

−

+ m

d,j

) d

+ h.c.



−



√

l,i

¯ν

+ h.c.



, (2.28)

where m

l,i

, m

u,i

, and m

d,i

are the lepton, the up-type, and the down-type quark masses,

respectively, and V

are the coeﬃcients of the CKM matrix. Left- and right-handed fermion

ﬁelds, f

and f

, are obtained from the corresponding Dirac spinor f by applying the

chirality projectors ω

= (1 ± γ

)/2, i.e. f = (ω

+ ω

−

)f = f

+ f

. The coupling

coeﬃcients ξ

H,h,A

are deﬁned as the couplings relative to the canonical SM value of m

and are shown in table 2. Note that we have used β

= β

= β in eq. (2.28) and table 2,

which is most relevant in applications; the generalization to independent β

, β

, β is simple.

3 The counterterm Lagrangian

The next step in calculating higher-order corrections is the renormalization of the theory.

In this section we focus on electroweak corrections of O(α

). The QCD renormalization

of the THDM is straightforward and completely analogous to the SM case, since all scalar

degrees of freedom are colour singlets and do not interact strongly. In the formulation of

– 10 –

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