精确计算两希格斯双峰模型中h→WW/ZZ→4费米子的次优阶修正

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本文主要探讨了在带有Prophecy4f的两希格斯双峰模型（Two-Higgs-Doublet Model, THDM）中的轻度CP偶数希格斯玻色子h的精确电弱和量子色动力学（Quantum Chromodynamics, QCD）次优阶修正。THDM分为几种类型，包括标准型I（Type I）、标准型II（Type II）、"lepton-specific"模型和"flipped"模型。研究关注的重点是h玻色子衰变为W玻色子对(WW)或Z玻色子对(ZZ)再进一步转化为四个费米子（four fermions）的过程。作者针对四种不同的重规一化方案进行了计算，这些方案包括实验上不易直接测量的参数采用MS $$ \overline{\text{MS}} $$规范。数值结果展示了对之前文献中提出的基准情景部分衰减宽度的校正，包括对MS $$ \overline{\text{MS}} $$重新规一化等级的依赖性以及不同规一化方案间的转换效应。特别强调了解决这些问题在参数分析中的重要性，因为不同的规一化方案可能导致在接近实验排除边界或理论限制时，参数转换带来的显著修正。研究发现，对于h→4f衰变过程，超出标准模型的校正主要由轻型和重型CP粒子，以及希格斯玻色子之间的混合引起，这导致微分分布的分析并不能有效地区分不同的THDM类型。衰变宽度对于模型的具体类型并不是敏感的指标，这意味着仅依赖于宽度的测量不足以确定THDM的确切性质。此外，文章还提到了论文的发表信息，如期刊名称《Journal of High Energy Physics》(JHEP)、卷号03、年份2018以及具体的出版日期。这篇开放获取的文章在2018年3月19日发表，经过了多次修订，展示了理论物理学家在精确计算和模型分析方面的严谨工作，对于理解希格斯粒子在多希格斯模型中的行为具有重要的学术价值。

JHEP03(2018)110

Type I Type II Lepton-speciﬁc Flipped

cos α/ sin β −sin α/ cos β −sin α/ cos β cos α/ sin β

cos α/ sin β cos α/ sin β cos α/ sin β cos α/ sin β

cos α/ sin β −sin α/ cos β cos α/ sin β −sin α/ cos β

Table 2. The coupling strengths ξ

of the light, CP-even Higgs boson h to the fermions f relative

to the SM values for diﬀerent types of THDM models.

The quark loop diagrams do not contain IR singularities, so that the singular structure of

the one-loop matrix element matches the one of the corresponding SM amplitude multiplied

by s

β−α

. As the LO amplitude contains the same factor, the SM subtraction and slicing

algorithms can be applied without modiﬁcation.

2.2.4 Complex-mass scheme

To treat the vector-boson resonances in a proper way, we employ the complex-mass scheme

which is explained in detail in refs. [37–39]. This prescription consists of an analytic

continuation of the masses of unstable particles into the complex plane which preserves

gauge invariance as well as all algebraic relations between amplitudes or Green functions

that do not involve complex conjugation (such as Ward and Slavnov-Taylor identities).

The complex mass µ

of V is directly connected to the real pole mass M

and the decay

width Γ

= M

− iM

, (2.13)

with V = W, Z. For our process at NLO, it is suﬃcient to treat only the W and Z boson

in the complex-mass scheme even though the other scalar particles are not stable. We

assume that in the THDM, the light Higgs boson has properties similar to the SM Higgs

boson, i.e. its width is very small, O(Γ

) < O(10

−4

). Eﬀects of this order can be

neglected, as they are smaller than the contributions from NLO and have the same size

as the uncertainties due to the separation of h production and decay. The other unstable

Higgs bosons of the THDM enter only in loop diagrams, and the corrections from the

complex masses are negligible as Γ

 M

where Γ

and M

are the decay width and

real pole mass of the considered Higgs boson. A fully consistent replacement of the real

masses by its complex counterparts includes also a complex deﬁnition of the weak mixing

angle θ

cos

= c

= 1 − s

− iM

. (2.14)

The generalization of this prescription to the one-loop level leads to complex renormaliza-

tion constants [38], for instance to the complex mass renormalization constants

δµ

= Σ

) + (µ

− M

)Σ

0WW

) +

iα

δµ

= Σ

) + (µ

− M

)Σ

0ZZ

), (2.15)

– 10 –

JHEP03(2018)110

where Σ

V V

) denotes the transverse parts of the V -boson self-energy with momentum

transfer p and Σ

0V V

its derivative w.r.t. p

. As a consequence the renormalization constants

of the weak mixing angle are

δs

= −

δc

= −



δµ

−

δµ



. (2.16)

The ﬁeld renormalization constants of the vector bosons are given in eq. (4.30) of ref. [38].

In particular, they enter the calculation of the electric charge renormalization constant.

The ﬁeld renormalization constants of the stable fermions and the scalars are also aﬀected

by treating W and Z bosons in the complex-mass scheme as we do not take the real parts of

the self-energies. Due to the appearing complex parameters, the self-energies and also the

renormalization constants become complex. However, the ﬁeld renormalization constants

of internal ﬁelds drop out and those of external ﬁelds factorize from the LO, so that the

imaginary parts drop out after squaring the matrix element at NLO.

2.2.5 Implementation and checks

The implementation of our calculation is performed in two independent ways: in the ﬁrst

method, we use a FeynArts [59] model ﬁle generated with the help of FeynRules [60, 61]

as described in ref. [43] and adapt it to the speciﬁc demands of the Prophecy4f calcu-

lation, so that masses of fermions belonging to closed loops are treated with the full mass

dependence and the complex vector-boson masses are implemented. The amplitudes are

generated using FeynArts [59], processed applying FormCalc [62, 63], and implemented

into the Prophecy4f code. Additionally, the coupling factors in the Prophecy4f code

of the HV V and Hf

f couplings are adapted to the THDM so that the LO, real photonic

corrections, and the QCD corrections can be obtained by simple rescaling. In a second,

independent calculation the amplitudes are generated via a FeynArts 1 [64] model ﬁle, in

which the counterterms are inserted by hand. The amplitudes were algebraically reduced

with the same inhouse Mathematica routines as already used in the SM NLO calculation

on which the original Prophecy4f calculation was based (for details see refs. [34, 36]).

These two implementations allow us to check the results and ensure their correctness.

Apart from performing two independent loop calculations, we have veriﬁed our

one-loop matrix elements by numerically comparing our results to the ones obtained

in refs. [42, 44] for the related Wh/Zh production channels (including W/Z decays) us-

ing crossing symmetry.

3 Input parameters and scenarios for the THDM

For the SM-like input parameters we take the values recommended by the LHC Higgs Cross

section Working Group [33] which essentially follow the Particle Data Group [65]:

= 0.11663787 · 10

−4

GeV

−2

, α

= 0.118,

= 91.1876 GeV, M

= 80.385 GeV, M

= 125 GeV,

= 2.4952 GeV, Γ

= 2.085 GeV,

= 510.998928 keV, m

= 105.6583715 MeV, m

= 1.77682 GeV,

– 11 –

JHEP03(2018)110

= 100 MeV, m

= 1.51 GeV, m

= 172.5 GeV,

= 100 MeV, m

= 4.92 GeV. (3.1)

The ﬁnal-state fermions are considered massless. Although we use regulator masses in soft

and/or collinear divergent terms appearing in photon exchange or emission contributions,

those regulator masses completely drop out in the full NLO corrections. The fermion

mass values given above are employed in the evaluation of the corrections induced by

closed fermion loops.

The CKM matrix is consistently taken as the unit matrix, since all

quark-mixing eﬀects drop out in ﬂavour sums of the external fermions, since we work with

massless light quarks without mixing to the third quark generation. We employ the G

scheme where the electromagnetic coupling is derived from the Fermi constant G

√



1 −



, (3.2)

which absorbs the running of the electromagnetic coupling α

from the Thomson limit

to the electroweak scale and accounts for universal corrections to the ρ-parameter. The

strong coupling constant α

is always taken with the ﬁxed value given in eq. (3.1), which

corresponds to a QCD renormlization scale at the Z-boson mass M

. We do not consider

any QCD scale variation, since α

appears only in the QCD corrections (see section 2.2.3),

so that any QCD running eﬀects are beyond NLO.

Prophecy4f performs its calculation in the complex-mass scheme and automatically

converts the experimentally measured on-shell gauge boson masses M

to pole masses

pole

of the propagators according to

pole

= M

1 + (Γ

)

, Γ

pole

= Γ

1 + (Γ

)

. (3.3)

From these measured input values, the program recalculates the widths of the vector bosons

in O(α

) in the SM using real mass parameters everywhere. This recalculation ensures

that the branching ratios of the vector bosons are correctly normalized and add up to

one for the SM. In the THDM, the heavy Higgs bosons enter the width in the mass

counterterms, however, as we are close to the alignment limit (c

β−α

→ 0) the eﬀects are

negligible. For an easier reproducibility of our results, we keep the SM values, which are

also compatible with the measured W/Z widths.

As central renormalization scale µ

we use the average mass of all scalar degrees of

freedom,

+ M

+ 2M

). (3.4)

This scale choice might seem surprising at ﬁrst glance, since the light Higgs-boson mass is

the centre-of-mass energy of our process. However, the loop diagrams including heavy scalar

The beneﬁt in this procecure is not only to account for (small) mass corrections for bottom quarks,

charm quarks, or τ leptons, the values would also be needed in the calculation of the EW corrections in

the α(0) scheme (also an option in Prophecy4f), where all charged fermion masses are required (and are

usually adjusted to reproduce the measured photonic vacuum polarization below the EW scale).

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