Higgs boson origin from a gauge symmetric theory of massive composite particles and massless W

2025-05-06 0 0 825.75KB 25 页 10玖币

侵权投诉

Higgs boson origin from a gauge

symmetric theory of massive

composite particles and massless W±

and Z0bosons at the TeV scale

She-Sheng Xue

ICRANet Piazzale della Repubblica, 10 -65122, Pescara, Italy

Physics Department, University of Rome La Sapienza, Rome, Italy

INFN, Sezione di Perugia, Perugia, Italy

ICTP-AP, University of Chinese Academy of Sciences, Beijing, China

E-mail: xue@icra.it and shesheng.xue@gmail.com

Abstract. The ultraviolet completion is the Standard Model (SM) gauge-symmetric

four-fermion couplings at the high-energy cutoﬀ. Composite particles appear in the

gauge symmetric phase in contrast with SM particles in the spontaneous symmetry-

breaking phase. The critical point between the two phases is a weak ﬁrst-order tran-

sition. It relates to an ultraviolet ﬁxed point for an SM gauge symmetric theory of

composite particles in the strong coupling regime. The low-energy SM realizes at an

infrared ﬁxed point in the weak coupling regime. Composite bosons dissolve into SM

particles at the phase transition, and in the top-quark channel, they become a compos-

ite SM Higgs boson and three Goldstone bosons. Extrapolation of SM renormalization-

group solutions to high energies implies that the gauge-symmetric theory of composite

particles has a characteristic scale of about 5.1 TeV. We discuss the phenomenological

implications of composite SM Higgs boson in the gauge symmetry-breaking phase and

massive composite bosons coupling to massless W±and Z0gauge bosons in the gauge

symmetric phase.

arXiv:2210.04825v2 [hep-ph] 29 Mar 2023

Contents

1 Theoretical ultraviolet completion 1

2 Strong-coupling gauge symmetric phase 3

3 UV scaling domain and eﬀective theory for composite bosons 5

4 First-order phase transition 7

5 Weak coupling gauge-symmetry-breaking phase 10

6 Dissociation phenomenon in ﬁrst-order phase transition 11

7 IR scaling domain and eﬀective theory for SM 13

7.1 Eﬀective SM Lagrangian and RG equations 13

7.2 Eﬀective theory of composite particles at TeV scale 16

8 Extrapolation from IR domain to UV domain 17

9 Phenomenological implications 18

9.1 Composite SM Higgs boson in IR domain at electroweak scale 18

9.2 Massive composite bosons in UV domain at TeV scale 19

10 Acknowledgment. 21

1 Theoretical ultraviolet completion

The parity-violating gauge symmetries and spontaneous/explicit breaking of these sym-

metries for the Wand Zgauge boson masses and the hierarchy pattern of fermion

masses have been at the centre of conceptual elaboration. It has played a major role

in donating to Mankind the beauty of the Standard Model (SM) and possible scenar-

ios beyond SM for fundamental particle physics. A simple description is provided on

the one hand by the composite Higgs-boson model or the Nambu-Jona-Lasinio (NJL)

model [1] with eﬀective four-fermion operators, and on the other by the phenomeno-

logical model of the elementary Higgs boson [2–4]. These two models are eﬀectively

equivalent for the SM at low energies. The ATLAS [5] and CMS [6] collaborations

have shown the ﬁrst observations of a 125 GeV scalar particle in the search for the SM

Higgs boson.

A well-deﬁned quantum ﬁeld theory for the SM Lagrangian requires a natural

regularisation at the ultraviolet energy cutoﬀ Λcut = 2π/a (the spatial lattice spacing

a), fully preserving the SM chiral gauge symmetry. The cutoﬀ Λcut could be the

Planck or the grand uniﬁed theory scale. Quantum gravity or another new physics

naturally provides such regularisation. However, the No-Go theorem [7,8] shows the

– 1 –

presence of right-handed neutrinos and the absence of consistent regularisation for

the SM bilinear fermion Lagrangian to preserve the SM chiral gauge symmetries. It

implies SM fermions’ and right-handed neutrinos’ four-fermion (quadrilinear) operators

at the cutoﬀ Λcut. As a theoretical model, we adopt the four-fermion operators of the

torsion-free Einstein-Cartan Lagrangian with all SM fermions and three right-handed

neutrinos [9–11]. Among four-fermion operators of Einstein-Cartan or NJL type, we

consider here one for the third quark family

Gcuth(¯

ψia

LtRa)(¯

RψLib)+(¯

ψia

LbRa)(¯

RψLib)i,(1.1)

where aand bare the colour indexes of the top and bottom quarks. The SUL(2)

singlets ψa

R=ta

R, ba

Rand doublet ψia

L= (ta

L, ba

L) are the eigenstates of SM electroweak

interactions. Henceforth, we suppress the colour indexes. The eﬀective four-fermion

coupling Gcut ∝ O(Λ−2

cut) and the dimensionless coupling GcutΛ2

cut depends on the run-

ning energy scale µ. Through the help of an auxiliary (static) scalar ﬁeld H0, we

rewrite the four-fermion interaction (1.1) as

gcut

H(¯

ψLtRH0+ h.c.)−M2

0H†

0H0+ (t→b).(1.2)

where the bare Yukawa coupling gcut

H=GcutΛ2

cut. Integration over the static ﬁeld

H0of mass M0= Λcut yields the four-fermion interaction (1.1) up to non-dynamical

constants.

Apart from what is possible new physics at the cutoﬀ Λcut explaining the origin

of these eﬀective four-fermion operators (1.1), it is essential to study their interactions

and ground state: (i) which dynamics these operators undergo in terms of their cou-

plings as functions of running energy scale µ; (ii) associating to these dynamics where

the infrared (IR) or ultraviolet (UV) stable ﬁx-point of physical couplings locates; (iii)

in the IR or UV scaling domains (scaling regions) of these stable ﬁxed points, which

physically relevant operators that become eﬀectively dimensional-4 (d= 4) renormal-

izable operators following renormalization-group (RG) equations (scaling laws); (iv)

which (d > 4) irrelevant operators, suppressed by the cutoﬀ scale, have sizable correc-

tions to the relevant operators. Here we mention the discussions on the triviality of

the four-dimensional pure m2

φφ2+λφφ4theory, namely, whether the coupling λφ→0

at its ﬁxed point for the cutoﬀ Λcut mφ.

Using the four-fermion operators (1.1), Bardeen, Hill and Lindner (BHL) [12]

investigate the dynamics of spontaneous gauge symmetry breaking (SSB), and an IR

ﬁxed point Gcat the weak coupling regime, where the β-function β=µ∂Gcut/∂µ > 0.

In the IR scaling domain, gauge symmetries break, and the eﬀective ﬁeld theory of

massive top quark, W±, Z0and h¯

tti-composite Higgs is realised for the SM model at

the electroweak scale v= (√2GF)−1/2≈246 GeV, where GFis the measured Fermi

constant. However, the compositeness conditions at the scale Λcut do not yield the

top-quark and Higgs masses consistently with experimental values.

In contrast, we investigate [11,13,14] the dynamics of SM elementary fermions

forming composite bosons Π ∝(¯

ψLψR) and fermions ΨL,R ∝(¯

ψLψR)ψL,R, and a UV

ﬁxed point at the strong coupling regime, where the β-function β=µ∂Gcut/∂µ < 0.

– 2 –

In the UV scaling domain, the SM chiral gauge symmetries preserve, and the eﬀective

ﬁeld theory of massive composite particles and massless gauge bosons W±, Z0realise at

the scale Λ (v < Λ<Λcut). The UV ﬁxed point relates to the critical point of the phase

transition from the strong-coupling symmetric phase to the weak-coupling symmetry-

breaking phase [11,15,16]. The scenario has been generalised to the strongly-coupled

Fermi liquid for the Bose-Einstein condensate [17]. Reference [18] shows the basic

phenomenology of composite bosons and fermions.

The weak and strong coupling regimes (1.1) bring us into two distinct domains:

the UV and IR domains. This is a non-perturbative issue. It is reminiscent of the QCD

dynamics: asymptotic free quark states near a UV ﬁxed point β < 0 at high energies

and bound hadron states near a possible IR ﬁxed point β > 0 at low energies. In the

present scenario, the main theoretical problems yet to be solved are the following. (i)

The quantitative features of the strong coupling domain: SM gauge couplings, and mass

spectra of composite particles. (ii) The nature of phase transition at the TeV scale:

from the exact gauge symmetric phase for the massless gauge bosons and massive

composite particles to the gauge-symmetry breaking phase for the SM elementary

gauge bosons, Higgs boson, quarks and leptons. In this article, to address these issues,

we focus on the four composite bosons H±and H0

t,b composed by tand bquarks in the

third quark family. We study the massive composite bosons Hmass terms, the eﬀective

potential V(H2) and their Yukawa coupling gHto tand bquarks at the TeV scale. We

ﬁnd how the gauge symmetry ground state evolves into the symmetry-breaking ground

state. The results show the phase transition of spontaneous symmetry breaking is the

weak ﬁrst order. Upon the transition, the composite boson H0

tbecomes the composite

SM Higgs boson h, and others become the W±and Z0gauge bosons’ longitudinal

modes. We present phenomenological implications of massless gauge bosons W±and

Z0at the TeV scale. For readers’ convenience, we brieﬂy review the results in previous

publications that are necessary gradients for discussing the issue here so that the article

is self-consistent and self-contain.

In Secs. 2and 3, we discuss the strong-coupling symmetric phase and the critical

point of phase transition as a UV ﬁxed point to realise an eﬀective theory for composite

particles. In Secs. 4and 5, we show the weak ﬁrst-order phase transition to the

symmetry-breaking phase in the weak coupling regime. In Sec. 6, we discuss how

composite particles dissolve into their constituents at the phase transition, leading to

a composite SM Higgs boson. In Secs. 7and 8, we present the investigations on the IR

ﬁxed point for the low-energy SM model in connection with the UV ﬁxed point for the

theory of composite particles. In the last section 9, we discuss the phenomenological

implications of composite SM Higgs boson in the IR domain, massive composite bosons

and massless W±and Z0gauge bosons in the UV domain.

2 Strong-coupling gauge symmetric phase

In the strong-coupling limit Gcuta−21, the theory (1.1) is in the strong-coupling

symmetric phase, composite particles’ spectra preserve the SM gauge symmetries

SUc(3) ×SUL(2) ×UY(1) [13,15]. We calculate the propagators of composite fermion

– 3 –

and boson ﬁelds by the strong coupling expansion in powers of kinetic term 1/g1/2

Skinetic =1

2ag1/2X

x,µ

ψ(x)γµ∂µψ(x),

Sint =X

x(¯

ψia

LtRa)(¯

RψLib)+(¯

ψia

LbRa)(¯

RψLib),(2.1)

where g≡Gcut/a4(ga21) and scaling ψ(x)→ψ(x) = a2g1/4ψ(x). In the lowest

non-trivial order we obtain (see Sections 4 and 5 in Ref. [13]) the composite bosons

(SUL(2)-doublet)

Hi=Z−1/2

H(¯

ψia

LtRa), M2

H=4

Ncg−2Nc

a2,(2.2)

and tRa →bRa, where Z1/2

Hand M2

Hare the composite boson form-factor (wave-function

renormalization) and mass term. They relate to each other Z1/2

H∝M2

Hby the Ward

identities due to exact symmetries. It is analogous to the relation between the QED

wave function and mass renormalizations (Z2=Z1) due to the Uem(1) gauge symmetry.

The spectra of composite bosons show in Table 1. They are complex ﬁelds, forming

two SUL(2) doublets

H−1

2=H0

H−and H1

2=H+

b(2.3)

of UY(1) charge −1/2 and 1/2. The doublet H−1

2(H1

2) comes from the top-quark

(bottom-quark) channel, namely the ﬁrst (second) term of four-fermion interaction

(2.1). The composite bosons’ masses and form factors are the same at the lowest non-

trivial order. The mass degeneracy breaks if we consider the SM family mixing and

gauge interactions. They are physical particles, diﬀerently from auxiliary static ﬁelds

H0(1.2). After the wave function renormalization, the composite bosons behave as

elementary particles 1.

The one-particle-irreducible (1PI) vertex of the self-interacting term λ(H†H)2is

positive and suppressed by powers (1/g)2. The composite bosons mass term M2

HHH†

(2.2) implies the second order phase transition of Landau type from M2

H>0 to M2

0. M2

H= 0 gives rise to the critical coupling Gc

cut,

cut = 2Nca2= 2Nc(2π)2Λ−2

cut,(2.5)

and Gc

cutΛ2

cut = 2Nc(2π)21. It is the transition from the strong-coupling SM

symmetric phase to the weak-coupling SM symmetry breaking phase.

1In this article, we ignore the composite Dirac fermions: SUL(2)-doublet Ψib

D= (ψib

L,Ψib

R) and

SUL(2)-singlet Ψb

D= (Ψb

L, tb

R), where the renormalized composite three-fermion states are:

Ψib

R= (ZR)−1(¯

ψia

LtRa)tb

R;Ψb

L= (ZL)−1(¯

ψia

LtRa)ψb

iL,(2.4)

with mass MF= 2ga and form-factor ZR,L =MFa, where ZL,R ≈Z1/2

HZ1/2

ψ, and Z1/2

ψis the wave

renormalization of elementary fermion ﬁelds. Their mass terms MF(¯

ΨLψR+¯

ψLΨR) exactly preserve

the SM chiral (parity-violating) gauge symmetries. The case for SU(5) chiral gauge symmetry theory

is discussed [19,20].

– 4 –

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

HiggsbosonoriginfromagaugesymmetrictheoryofmassivecompositeparticlesandmasslessWandZ0bosonsattheTeVscaleShe-ShengXueICRANetPiazzaledellaRepubblica,10-65122,Pescara,ItalyPhysicsDepartment,UniversityofRomeLaSapienza,Rome,ItalyINFN,SezionediPerugia,Perugia,ItalyICTP-AP,UniversityofChineseAcademyofScie...

展开>> 收起<<

Higgs boson origin from a gauge symmetric theory of massive composite particles and massless W.pdf

共25页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Higgs boson origin from a gauge symmetric theory of massive composite particles and massless W

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: