Abstract
HTML
Reference
Related
PDF
-
In the standard model QCD Lagrangian, a term of CP violating gluon density is theoretically expected to have a physical coefficient
ˉθ , which is typically on the order of unity. However, the upper bound on the electric dipole moment of the neutron enforces the value ofˉθ to be extremely small. The huge gap between the theoretical expectation and experimental result leads to the so-called strong CP problem [1−5]. In 1977, Peccei and Quinn determined that the CP violatingˉθ -term can be effectively neutralized if the QCD Lagrangian contains a global symmetry [6]. Currently, this global symmetry is well known as the Peccei-Quinn (PQ) symmetryU(1)PQ . After the PQ global symmetry undergoes spontaneous breaking, a massless Goldstone boson typically emerges. However, in this case, the Goldstone boson gains mass due to the color anomaly [7−9], transforming into a pseudo Goldstone boson, commonly referred to as the axion [10, 11].The simplest approach to the PQ symmetry appears to consider a two-Higgs-doublet model [6]. Unfortunately, this original PQ model was quickly ruled out in experiments. However, the axion has not been observed experimentally and is still an invisible particle. This implies that the interactions between the axion and SM particles should be at an extremely weak level [1−5]. For a successful realization of the PQ symmetry with an invisible axion, Kim-Shifman-Vainstein-Zakharov (KSVZ) [12, 13] and Dine-Fischler-Srednicki-Zhitnitsky (DFSZ) [14, 15] proposed their elegant methods to effeciently decrease the couplings of the axion to the SM particles. Currently, the absence of positive outcomes in axion search experiments imposes stringent constraints on the PQ symmetry within KSVZ-type and DFSZ-type models [12−15], indicating that it must be spontaneously broken at an energy scale significantly higher than the weak scale [1−5].
In the KSVZ-type and DFSZ-type models, the new particles except the invisible axion should be too heavy to verify in experiments unless the related couplings are artificially small. This implies that all experimental attempts to test the PQ symmetry can only depend on the axion-meson mixing and hence the axion searches [1−5]. Theoretically, when there is a substantial hierarchy between the PQ and electroweak symmetry breaking scales, the inevitable Higgs portal interaction requires an exceptionally small coupling. Otherwise, there must be significant cancellation between its contribution and the rarely quadratic term of the SM Higgs scalar [16]. In some sense, the invisible axion models pay a price of additional fine tuning to solve the strong CP problem.
In this paper, we propose a new mechanism to solve the strong CP problem in the appealing context of two Higgs doublets [17]. Specifically, we introduce a
ˉθ -characterized mirror symmetry only between two Higgs singlets with respective discrete symmetries. In this scenario, the parameterˉθ can completely disappear from the full Lagrangian after the standard model fermions and the Higgs scalars take a proper phase rotation. Moreover, all of new physics for solving the strong CP problem can be allowed near the TeV scale. -
Before delving into the specifics of our proposed mechanism, let us first provide a succinct overview of the strong CP problem. The QCD Lagrangian in the SM can be characterized as follows:
LQCD=∑qˉq(iD̸−mqeiθq)q−14GaμνGaμν−θαs8πGaμν˜Gaμν,
(1) where
θq denotes the phase from the Yukawa couplings of quark fields, θ denotes the QCD vacuum angle,Gaμν denotes the gluon field strength tensor, and˜Gaμν denotes its dual. Following the application of a chiral phase transformation to the quark fields, it can be described as follows:q→e−iγ5θq/2q,
(2) Furthermore, their mass terms can remove phases
θq from the QCD Lagrangian, i.e.,LQCD=∑qˉq(iD̸−mq)q−14GaμνGaμν−ˉθαs8πGaμν˜Gaμνwithˉθ≡θ−ArgDet(MdMu).
(3) Here,
Md andMu denote the respective mass matrices of the SM down-type and up-type quarks, respectively. To satisfy the upper limits on the electric dipole moment of the neutron, the value ofˉθ should be extremely small rather than the theoretically expected order of unity, i.e.,|ˉθ|<10−10.
(4) This fine tuning of ten orders of magnitude is commonly termed as the strong CP problem.
-
We now demonstrate our mechanism in a realistic model. All the scalars and fermions in the model are summarized in Table 1. The two Higgs singlets
ξ1,2 are distinguished byZ(1)3×Z(2)3 discrete symmetries as well as the two Higgs doubletsϕ1,2 . Besides three generations of the SM quarksqL ,dR , anduR and SM leptonslL andeR , we introduce three right-handed neutrinosNR to realize a seesaw [18−22] mechanism for the generation of tiny neutrino masses and also a leptogenesis [23] mechanism for the explanation of cosmological baryon asymmetry. Here, the family indices of the fermions are omitted for simplicity. Moreover, there is aˉθ -characterized mirror symmetry between the two Higgs singletsξ1,2 , i.e.Scalars&Fermions 
ξ1 
ξ2 
ϕ1 
ϕ2 
qL 
dR 
uR 
lL 
eR 
NR 
SU(3)c 
1 
1 
1 
1 
3 
3 
3 
1 
1 
1 
SU(2)L 
1 
1 
2 
2 
2 
1 
1 
2 
1 
1 
U(1)Y 
0 
0 
+12 
+12 
+16 
−13 
+23 
−12 
−1 
0 
Z(1)3 
ei2π3 
1 
ei2π3 
1 
1 
ei4π3 
1 
1 
ei4π3 
1 
Z(2)3 
1 
ei2π3 
ei2π3 
1 
1 
ei4π3 
1 
1 
ei4π3 
1 
Table 1. All scalars and fermions in the model. The two Higgs singlets
ξ1,2 are denoted byZ(1)3×Z(2)3 discrete symmetries and the two Higgs doubletsϕ1,2 . In addition to three generations of the SM quarksqL ,dR , anduR and SM leptonslL andeR , we introduce three right-handed neutrinosNR to realize a seesaw mechanism for the generation of tiny neutrino masses and also a leptogenesis mechanism for the explanation of cosmological baryon asymmetry. Here, the family indices of the fermions are omitted for simplicity.ξ1ˉθ−characterized mirror symmetry←−−−−−−−−−−−−−−−−−−−−−−−−−→e−iˉθ/3ξ2.
(5) This may be a minimal version of the
ˉθ -characterized mirror symmetry [24].Based on the charge assignments in Table 1, the expressions for the allowed Yukawa and mass terms involving the fermions are as follows:
LY+M=−ydˉqLϕ1dR−yuˉqL˜ϕ2uR−yeˉlLϕ1eR−yNˉlL˜ϕ2NR−12MNˉNRNcR+H.c.with˜ϕ1,2=iτ2ϕ∗1,2,
(6) The full scalar potential at a renormalizable level is as follows:
V=μ21ϕ†1ϕ1+μ22ϕ†2ϕ2+λ1(ϕ†1ϕ1)2+λ2(ϕ†2ϕ2)2+λ3ϕ†1ϕ1ϕ†2ϕ2+λ4ϕ†1ϕ2ϕ†2ϕ1+μ2ξ(ξ∗1ξ1+ξ∗2ξ2)+κ1[(ξ∗1ξ1)2+(ξ∗2ξ2)2]+κ2ξ∗1ξ1ξ∗2ξ2+ρξ[(ξ31+e−iˉθξ32)+H.c.]+ϵ1ϕ†1ϕ1(ξ∗1ξ1+ξ∗2ξ2)+ϵ2ϕ†2ϕ2(ξ∗1ξ1+ξ∗2ξ2)+ϵ3(ξ1ξ2ϕ†1ϕ2+H.c.).
(7) The Yukawa couplings and the Majorana masses involving the right-handed neutrinos are responsible for the reaization of seesaw and leptogenesis. We do not examine the details of seesaw and leptogenesis which are beyond the goal of the present work. It should be noted that the
ˉθ -characterized mirror symmetry (5) is exactly complied in the classical Lagrangian where the kinetic terms are not given for simplicity.We clarify that the fields in Table 1 with the
ˉθ -characterized mirror symmetry in Eq. (5) to aid in solving the strong CP problem. After the two Higgs singlets, the two Higgs doublets and the three generations of fermions take the phase rotations as below,(ξ1→ξ1ξ2→e+iˉθ/3ξ2),(ϕ1→e+iˉθ/3ϕ1ϕ2→ϕ2),(qL→qLdR→e−iˉθ/3dRuR→uRlL→lLeR→e−iˉθ/3eRNR→NR),
(8) the QCD Lagrangian (3) and the scalar potential (7) can simultaneously remove the parameter
ˉθ 1 as follows:LQCD⇒∑qˉq(iD̸−mq)q−14GaμνGaμν,
(9) V⇒μ21ϕ†1ϕ1+μ22ϕ†2ϕ2+λ1(ϕ†1ϕ1)2+λ2(ϕ†2ϕ2)2+λ3ϕ†1ϕ1ϕ†2ϕ2+λ4ϕ†1ϕ2ϕ†2ϕ1+μ2ξ(ξ∗1ξ1+ξ∗2ξ2)+κ1[(ξ∗1ξ1)2+(ξ∗2ξ2)2]+κ2ξ∗1ξ1ξ∗2ξ2+ρξ[(ξ31+ξ32)+H.c.]+ϵ1ϕ†1ϕ1(ξ∗1ξ1+ξ∗2ξ2)
+ϵ2ϕ†2ϕ2(ξ∗1ξ1+ξ∗2ξ2)+ϵ3(ξ1ξ2ϕ†1ϕ2+H.c.),
(10) The Yukawa and mass terms (6) can remain invariant with the unshown kinetic terms.
-
When the Higgs scalars
ξ1 ,ξ2 ,ϕ1 , andϕ2 develop their nonzero vacuum expectation valuesvξ1 ,vξ2 ,vϕ1 , andvϕ2 , respectively, they are expressed as follows:ξ1=(vξ1+hξ1+iPξ1)/√2,
(11) ξ2=(vξ2+hξ2+iPξ2)/√2,
(12) ϕ1=[ϕ+1(vϕ1+hϕ1+iPϕ1)/√2],
(13) ϕ2=[ϕ+2(vϕ2+hϕ2+iPϕ2)/√2].
(14) Three would-be-Goldstone bosons:
G±W=(vϕ1ϕ±1+vϕ2ϕ±2)/√v2ϕ1+v2ϕ2,
(15) GZ=(vϕ1Pϕ1+vϕ2Pϕ2)/√v2ϕ1+v2ϕ2,
(16) eaten by the longitudinal components of the SM gauge bosons
W± and Z. Therefore, besides a pair of massive charged scalars,H±=(vϕ1ϕ±2−vϕ2ϕ±1)/√v2ϕ1+v2ϕ2withm2H±=−[λ4+ϵ3v2ξ/(2v1v2)](v21+v22),
(17) we eventually obtain seven massive neutral scalars including four scalars and three pseudo scalars, i.e.,
hϕ1,hϕ2,hξ=(hξ1+hξ2)/√2,Sξ=(hξ1−hξ2)/√2;
(18) aϕ=(vϕ1Pϕ2−vϕ2Pϕ1)/√v2ϕ1+v2ϕ2,aξ=(Pξ1+Pξ2)/√2,Pξ=(Pξ1−Pξ2)/√2.
(19) With the minimum of the scalar potential, we obtain the mass-squared matrix of three scalars
hϕ1 ,hϕ2 andhξ , i.e.L⊃−12[hϕ1hϕ2hξ][2λ1v2ϕ1−12ϵ3v2ξvϕ2vϕ1(λ3+λ4)vϕ1vϕ2+12ϵ3v2ξ√2(ϵ1+12ϵ3)vϕ1vξ(λ3+λ4)vϕ1vϕ2+12ϵ3v2ξ2λ2v2ϕ2−12ϵ3v2ξvϕ1vϕ2√2(ϵ2+12ϵ3)vϕ2vξ√2(ϵ1+12ϵ3)vϕ1vξ√2(ϵ2+12ϵ3)vϕ2vξ2κ1v2ξ+3√2ρξvξ−12ϵ3vϕ1vϕ2][hϕ1hϕ2hξ].
(20) Hereafter, we consider the following:
vξ1=vξ2≡vξ,
(21) which can be easily deduced from the minimization of the scalar potential. By diagonalizing the mass-squared matrix (20), we obtain three mass eigenstates
H1,2,3 with Yukawa couplings. For simplicity, we do not perform this diagonalization in the present work. For the forth scalarSξ without Yukawa couplings, it corresponds to a mass eigenstate with the following mass square,m2Sξ=2κ1v2ξ+3√2ρξvξ−12ϵ3vϕ1vϕ2.
(22) We then consider the pseudo scalars
aϕ ,aξ andPξ . Their mass-squared matrix is given byL⊃−12[aϕaξPξ][−12ϵ3v2ξ(vϕ1vϕ2+vϕ2vϕ1)−14ϵ3vξ√v2ϕ1+v2ϕ20−14ϵ3vξ√v2ϕ1+v2ϕ2−9√2ρξvξ−ϵ3vϕ1vϕ2000−9√2ρξvξ−ϵ3vϕ1vϕ2][aϕaξPξ]. 
(23) Clearly,
Pξ is already a mass eigenstate and its mass square is justm2Pξ=−9√2ρξvξ−ϵ3vϕ1vϕ2.
(24) For
aϕ andaξ , they mix with each other, and their mass eigenstates are as follows:a1=aϕcosα−aξsinαwithm2a1=m2aϕ+m2aξ+√(m2aϕ−m2aξ)2+4Δ42,
(25) a2=aϕsinα+aξcosαwithm2a2=m2aϕ+m2aξ−√(m2aϕ−m2aξ)2+4Δ42,
(26) Here,
m2aϕ ,m2aξ , andΔ2 are defined bym2aϕ=−12ϵ3v2ξ(vϕ1vϕ2+vϕ2vϕ1),m2aξ=−9√2ρξvξ−ϵ3vϕ1vϕ2,
Δ2=−14ϵ3vξ√v2ϕ1+v2ϕ2,
(27) while α is the mixing angle and is determined by
tan2α=2Δ2m2Aϕ−m2Aξ.
(28) The pseudo scalars
a1,2 couple to the axial currents of the SM quarks, and thus, they act as heavy axions [24]. -
In this paper, we propose a novel
ˉθ -characterized mirror symmetry to naturally solve the strong CP problem. In our scenario, the scalars include two Higgs singlets and two Higgs doublets, while the fermions include three generations of the SM fermions and the right-handed neutrinos. Theˉθ -characterized mirror symmetry is only involved in the two Higgs singlets with respective discrete symmetries. The parameterˉθ can completely disappear from the full Lagrangian after the fermions and the Higgs scalars take a proper phase rotation. Our mechanism ensures that the new physics for solving the strong CP problem is near the TeV scale.
Solving the strong CP problem via a ˉθ -characterized mirror symmetry
- Received Date: 2023-10-18
- Available Online: 2024-03-15
Abstract: In the standard model QCD Lagrangian, a term of CP violating gluon density is theoretically expected to have a physical coefficient
DownLoad: