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Solving the strong CP problem via a ˉθ-characterized mirror symmetry

  • 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 of ˉθ to be extremely small. The significant discrepancy between theoretical expectations and experimental results in this context is widely recognized as the strong CP problem. To solve this puzzle in an appealing context of two Higgs doublets, we propose a ˉθ-characterized mirror symmetry between two Higgs singlets with respective discrete symmetries. In our scenario, the parameter ˉθ can completely disappear from the full Lagrangian after the standard model fermions take a proper phase rotation as well as the Higgs doublets and singlets. Moreover, all of new physics for solving the strong CP problem can be allowed near the TeV scale.
  • 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 [15]. In 1977, Peccei and Quinn determined that the CP violating ˉθ-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) symmetry U(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 [79], 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 [15]. 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 [1215], indicating that it must be spontaneously broken at an energy scale significantly higher than the weak scale [15].

    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 [15]. 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 parameter ˉθ 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(imqeiθq)q14GaμνGaμνθαs8πGaμν˜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:

    qeiγ5θq/2q,

    (2)

    Furthermore, their mass terms can remove phases θq from the QCD Lagrangian, i.e.,

    LQCD=qˉq(imq)q14GaμνGaμνˉθαs8πGaμν˜GaμνwithˉθθArgDet(MdMu).

    (3)

    Here, Md and Mudenote 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 of ˉθ should be extremely small rather than the theoretically expected order of unity, i.e.,

    |ˉθ|<1010.

    (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 by Z(1)3×Z(2)3 discrete symmetries as well as the two Higgs doublets ϕ1,2. Besides three generations of the SM quarks qL, dR, and uR and SM leptons lL and eR, we introduce three right-handed neutrinos NR to realize a seesaw [1822] 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 a ˉθ-characterized mirror symmetry between the two Higgs singlets ξ1,2, i.e.

    Table 1

    Table 1.  All scalars and fermions in the model. The two Higgs singlets ξ1,2 are denoted by Z(1)3×Z(2)3 discrete symmetries and the two Higgs doublets ϕ1,2. In addition to three generations of the SM quarks qL, dR , and uR and SM leptons lL and eR, we introduce three right-handed neutrinos NR 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.
    Scalars&Fermionsξ1ξ2ϕ1ϕ2qLdRuRlLeRNR
    SU(3)c1111333111
    SU(2)L1122211211
    U(1)Y00+12+12+1613+231210
    Z(1)3ei2π31ei2π311ei4π311ei4π31
    Z(2)31ei2π3ei2π311ei4π311ei4π31
    DownLoad: CSV
    Show Table

    ξ1ˉθcharacterized mirror symmetryeiˉθ/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=ydˉqLϕ1dRyuˉqL˜ϕ2uRyeˉlLϕ1eRyNˉlL˜ϕ2NR12MNˉNRNcR+H.c.with˜ϕ1,2=iτ2ϕ1,2,

    (6)

    The full scalar potential at a renormalizable level is as follows:

    V=μ21ϕ1ϕ1+μ22ϕ2ϕ2+λ1(ϕ1ϕ1)2+λ2(ϕ2ϕ2)2+λ3ϕ1ϕ1ϕ2ϕ2+λ4ϕ1ϕ2ϕ2ϕ1+μ2ξ(ξ1ξ1+ξ2ξ2)+κ1[(ξ1ξ1)2+(ξ2ξ2)2]+κ2ξ1ξ1ξ2ξ2+ρξ[(ξ31+eiˉθξ32)+H.c.]+ϵ1ϕ1ϕ1(ξ1ξ1+ξ2ξ2)+ϵ2ϕ2ϕ2(ξ1ξ1+ξ2ξ2)+ϵ3(ξ1ξ2ϕ1ϕ2+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,

    (ξ1ξ1ξ2e+iˉθ/3ξ2),(ϕ1e+iˉθ/3ϕ1ϕ2ϕ2),(qLqLdReiˉθ/3dRuRuRlLlLeReiˉθ/3eRNRNR),

    (8)

    the QCD Lagrangian (3) and the scalar potential (7) can simultaneously remove the parameter ˉθ 1 as follows:

    LQCDqˉq(imq)q14GaμνGaμν,

    (9)

    Vμ21ϕ1ϕ1+μ22ϕ2ϕ2+λ1(ϕ1ϕ1)2+λ2(ϕ2ϕ2)2+λ3ϕ1ϕ1ϕ2ϕ2+λ4ϕ1ϕ2ϕ2ϕ1+μ2ξ(ξ1ξ1+ξ2ξ2)+κ1[(ξ1ξ1)2+(ξ2ξ2)2]+κ2ξ1ξ1ξ2ξ2+ρξ[(ξ31+ξ32)+H.c.]+ϵ1ϕ1ϕ1(ξ1ξ1+ξ2ξ2)

    +ϵ2ϕ2ϕ2(ξ1ξ1+ξ2ξ2)+ϵ3(ξ1ξ2ϕ1ϕ2+H.c.),

    (10)

    The Yukawa and mass terms (6) can remain invariant with the unshown kinetic terms.

    When the Higgs scalars ξ1, ξ2, ϕ1 , and ϕ2 develop their nonzero vacuum expectation values vξ1, vξ2, vϕ1 , and vϕ2, 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±2vϕ2ϕ±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=(hξ1hξ2)/2;

    (18)

    aϕ=(vϕ1Pϕ2vϕ2Pϕ1)/v2ϕ1+v2ϕ2,aξ=(Pξ1+Pξ2)/2,Pξ=(Pξ1Pξ2)/2.

    (19)

    With the minimum of the scalar potential, we obtain the mass-squared matrix of three scalars hϕ1, hϕ2 and hξ, i.e.

    L12[hϕ1hϕ2hξ][2λ1v2ϕ112ϵ3v2ξvϕ2vϕ1(λ3+λ4)vϕ1vϕ2+12ϵ3v2ξ2(ϵ1+12ϵ3)vϕ1vξ(λ3+λ4)vϕ1vϕ2+12ϵ3v2ξ2λ2v2ϕ212ϵ3v2ξvϕ1vϕ22(ϵ2+12ϵ3)vϕ2vξ2(ϵ1+12ϵ3)vϕ1vξ2(ϵ2+12ϵ3)vϕ2vξ2κ1v2ξ+32ρξvξ12ϵ3vϕ1vϕ2][hϕ1hϕ2hξ].

    (20)

    Hereafter, we consider the following:

    vξ1=vξ2vξ,

    (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 scalar Sξ without Yukawa couplings, it corresponds to a mass eigenstate with the following mass square,

    m2Sξ=2κ1v2ξ+32ρξvξ12ϵ3vϕ1vϕ2.

    (22)

    We then consider the pseudo scalars aϕ, aξ and Pξ. Their mass-squared matrix is given by

    L12[aϕaξPξ][12ϵ3v2ξ(vϕ1vϕ2+vϕ2vϕ1)14ϵ3vξv2ϕ1+v2ϕ2014ϵ3vξv2ϕ1+v2ϕ292ρξvξϵ3vϕ1vϕ200092ρξvξϵ3vϕ1vϕ2][aϕaξPξ].

    (23)

    Clearly, Pξ is already a mass eigenstate and its mass square is just

    m2Pξ=92ρξvξϵ3vϕ1vϕ2.

    (24)

    For aϕ and aξ, they mix with each other, and their mass eigenstates are as follows:

    a1=aϕcosαaξsinαwithm2a1=m2aϕ+m2aξ+(m2aϕm2aξ)2+4Δ42,

    (25)

    a2=aϕsinα+aξcosαwithm2a2=m2aϕ+m2aξ(m2aϕm2aξ)2+4Δ42,

    (26)

    Here, m2aϕ, m2aξ , and Δ2 are defined by

    m2aϕ=12ϵ3v2ξ(vϕ1vϕ2+vϕ2vϕ1),m2aξ=92ρξvξϵ3vϕ1vϕ2,

    Δ2=14ϵ3vξv2ϕ1+v2ϕ2,

    (27)

    while α is the mixing angle and is determined by

    tan2α=2Δ2m2Aϕm2Aξ.

    (28)

    The pseudo scalars a1,2couple 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. The ˉθ-characterized mirror symmetry is only involved in the two Higgs singlets with respective discrete symmetries. The parameter ˉθ 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.

    1The parameter \begin{document}$ \bar{\theta} $\end{document} can still appear at loop level, however, such loop corrections are much smaller than the experimental constraints [25].

    [1] H.Y. Cheng, Phys. Rep. 158, 1 (1988) doi: 10.1016/0370-1573(88)90135-4
    [2] J.E. Kim and G. Carosi, Rev. Mod. Phys. 82, 557 (2010) doi: 10.1103/RevModPhys.82.557
    [3] L. Di Luzio, M. Giannotti, E. Nardi et al., Phys. Rep. 870, 1 (2020) doi: 10.1016/j.physrep.2022.06.006
    [4] P. Sikivie, Rev. Mod. Phys. 93, 015004 (2021) doi: 10.1103/RevModPhys.93.015004
    [5] R.L. Workman et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2022, 083C01 (2022) doi: 10.1093/ptep/ptac097
    [6] R.D. Peccei and H.R. Quinn, Phys. Rev. Lett. 38, 1440 (1977) doi: 10.1103/PhysRevLett.38.1440
    [7] S.L. Adler, Phys. Rev. 177, 2426 (1969) doi: 10.1103/PhysRev.177.2426
    [8] J.S. Bell and R. Jackiw, Nuovo Cim. 60A, 47 (1969) doi: 10.1007/BF02823296
    [9] S.L. Adler and W.A. Bardeen, Phys. Rev. 182, 1517 (1969) doi: 10.1103/PhysRev.182.1517
    [10] S. Weinberg, Phys. Rev. Lett. 40, 223 (1978) doi: 10.1103/PhysRevLett.40.223
    [11] F. Wilczek, Phys. Rev. Lett. 40, 279 (1978) doi: 10.1103/PhysRevLett.40.279
    [12] J.E. Kim, Phys. Rev. Lett. 43, 103 (1979) doi: 10.1103/PhysRevLett.43.103
    [13] M.A. Shifman, A. Vainshtein, and V.I. Zakharov, Nucl. Phys. B 166, 493 (1980) doi: 10.1016/0550-3213(80)90209-6
    [14] M. Dine, W. Fischler, and M. Srednicki, Phys. Lett. B 104, 199 (1981) doi: 10.1016/0370-2693(81)90590-6
    [15] A. Zhitnitsky, Sov. J. Nucl. Phys. 31, 260 (1980)
    [16] J.D. Clarke and R.R. Volkas, Phys. Rev. D 93, 035001 (2016) doi: 10.1103/PhysRevD.93.035001
    [17] G.C. Branco, P.M. Ferreira, L. Lavoura, M.N. Rebelo, M. Sher, and J. Silva, Phys. Rept. 516, 1 (2012) doi: 10.1016/j.physrep.2012.02.002
    [18] P. Minkowski, Phys. Lett. B 67, 421 (1977) doi: 10.1016/0370-2693(77)90435-X
    [19] T. Yanagida, Proceedings of the Workshop on Unified Theory and the Baryon Number of the Universe, ed. O. Sawada and A. Sugamoto (Tsukuba 1979).
    [20] M. Gell-Mann, P. Ramond, and R. Slansky, Supergravity, ed. F. van Nieuwenhuizen and D. Freedman (North Holland 1979).
    [21] R.N. Mohapatra and G. Senjanović, Phys. Rev. Lett. 44, 912 (1980) doi: 10.1103/PhysRevLett.44.912
    [22] J. Schechter and J.W.F. Valle, Phys. Rev. D 22, 2227 (1980) doi: 10.1103/PhysRevD.22.2227
    [23] M. Fukugita and T. Yanagida, Phys. Lett. B 174, 45 (1986) doi: 10.1016/0370-2693(86)91126-3
    [24] P.H. Gu, arXiv: 2307.04236[hep-ph
    [25] J. Ellis and M.K. Gaillard, Nucl. Phys. B 150, 141 (1979) doi: 10.1016/0550-3213(79)90297-9
  • [1] H.Y. Cheng, Phys. Rep. 158, 1 (1988) doi: 10.1016/0370-1573(88)90135-4
    [2] J.E. Kim and G. Carosi, Rev. Mod. Phys. 82, 557 (2010) doi: 10.1103/RevModPhys.82.557
    [3] L. Di Luzio, M. Giannotti, E. Nardi et al., Phys. Rep. 870, 1 (2020) doi: 10.1016/j.physrep.2022.06.006
    [4] P. Sikivie, Rev. Mod. Phys. 93, 015004 (2021) doi: 10.1103/RevModPhys.93.015004
    [5] R.L. Workman et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2022, 083C01 (2022) doi: 10.1093/ptep/ptac097
    [6] R.D. Peccei and H.R. Quinn, Phys. Rev. Lett. 38, 1440 (1977) doi: 10.1103/PhysRevLett.38.1440
    [7] S.L. Adler, Phys. Rev. 177, 2426 (1969) doi: 10.1103/PhysRev.177.2426
    [8] J.S. Bell and R. Jackiw, Nuovo Cim. 60A, 47 (1969) doi: 10.1007/BF02823296
    [9] S.L. Adler and W.A. Bardeen, Phys. Rev. 182, 1517 (1969) doi: 10.1103/PhysRev.182.1517
    [10] S. Weinberg, Phys. Rev. Lett. 40, 223 (1978) doi: 10.1103/PhysRevLett.40.223
    [11] F. Wilczek, Phys. Rev. Lett. 40, 279 (1978) doi: 10.1103/PhysRevLett.40.279
    [12] J.E. Kim, Phys. Rev. Lett. 43, 103 (1979) doi: 10.1103/PhysRevLett.43.103
    [13] M.A. Shifman, A. Vainshtein, and V.I. Zakharov, Nucl. Phys. B 166, 493 (1980) doi: 10.1016/0550-3213(80)90209-6
    [14] M. Dine, W. Fischler, and M. Srednicki, Phys. Lett. B 104, 199 (1981) doi: 10.1016/0370-2693(81)90590-6
    [15] A. Zhitnitsky, Sov. J. Nucl. Phys. 31, 260 (1980)
    [16] J.D. Clarke and R.R. Volkas, Phys. Rev. D 93, 035001 (2016) doi: 10.1103/PhysRevD.93.035001
    [17] G.C. Branco, P.M. Ferreira, L. Lavoura, M.N. Rebelo, M. Sher, and J. Silva, Phys. Rept. 516, 1 (2012) doi: 10.1016/j.physrep.2012.02.002
    [18] P. Minkowski, Phys. Lett. B 67, 421 (1977) doi: 10.1016/0370-2693(77)90435-X
    [19] T. Yanagida, Proceedings of the Workshop on Unified Theory and the Baryon Number of the Universe, ed. O. Sawada and A. Sugamoto (Tsukuba 1979).
    [20] M. Gell-Mann, P. Ramond, and R. Slansky, Supergravity, ed. F. van Nieuwenhuizen and D. Freedman (North Holland 1979).
    [21] R.N. Mohapatra and G. Senjanović, Phys. Rev. Lett. 44, 912 (1980) doi: 10.1103/PhysRevLett.44.912
    [22] J. Schechter and J.W.F. Valle, Phys. Rev. D 22, 2227 (1980) doi: 10.1103/PhysRevD.22.2227
    [23] M. Fukugita and T. Yanagida, Phys. Lett. B 174, 45 (1986) doi: 10.1016/0370-2693(86)91126-3
    [24] P.H. Gu, arXiv: 2307.04236[hep-ph
    [25] J. Ellis and M.K. Gaillard, Nucl. Phys. B 150, 141 (1979) doi: 10.1016/0550-3213(79)90297-9
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Pei-Hong Gu. Solving the strong CP problem by a ˉθ-characterized mirror symmetry[J]. Chinese Physics C. doi: 10.1088/1674-1137/ad102b
Pei-Hong Gu. Solving the strong CP problem by a ˉθ-characterized mirror symmetry[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ad102b shu
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Solving the strong CP problem via a ˉθ-characterized mirror symmetry

  • School of Physics, Jiulonghu Campus, Southeast University, Nanjing 211189, China

Abstract: 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 of ˉθ to be extremely small. The significant discrepancy between theoretical expectations and experimental results in this context is widely recognized as the strong CP problem. To solve this puzzle in an appealing context of two Higgs doublets, we propose a ˉθ-characterized mirror symmetry between two Higgs singlets with respective discrete symmetries. In our scenario, the parameter ˉθ can completely disappear from the full Lagrangian after the standard model fermions take a proper phase rotation as well as the Higgs doublets and singlets. Moreover, all of new physics for solving the strong CP problem can be allowed near the TeV scale.

    HTML

    I.   INTRODUCTION
    • 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 [15]. In 1977, Peccei and Quinn determined that the CP violating ˉθ-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) symmetry U(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 [79], 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 [15]. 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 [1215], indicating that it must be spontaneously broken at an energy scale significantly higher than the weak scale [15].

      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 [15]. 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 parameter ˉθ 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.

    II.   STRONG CP PROBLEM
    • 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(imqeiθq)q14GaμνGaμνθαs8πGaμν˜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:

      qeiγ5θq/2q,

      (2)

      Furthermore, their mass terms can remove phases θq from the QCD Lagrangian, i.e.,

      LQCD=qˉq(imq)q14GaμνGaμνˉθαs8πGaμν˜GaμνwithˉθθArgDet(MdMu).

      (3)

      Here, Md and Mudenote 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 of ˉθ should be extremely small rather than the theoretically expected order of unity, i.e.,

      |ˉθ|<1010.

      (4)

      This fine tuning of ten orders of magnitude is commonly termed as the strong CP problem.

    III.   \bar{\theta} -CHARACTERIZED MIRROR SYMMETRY
    • 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 \xi_{1,2}^{} are distinguished by Z^{(1)}_{3}\times Z^{(2)}_{3} discrete symmetries as well as the two Higgs doublets \phi_{1,2}^{} . Besides three generations of the SM quarks q_L^{} , d_R^{} , and u_R^{} and SM leptons l_L^{} and e_R^{} , we introduce three right-handed neutrinos N_R^{} to realize a seesaw [1822] 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 a \bar{\theta} -characterized mirror symmetry between the two Higgs singlets \xi_{1,2}^{} , i.e.

      \rm Scalars \& Fermions \xi_1^{} \xi_{2}^{} \phi_{1}^{} \phi_{2}^{} q_{L}^{} d_{R}^{} u_{R}^{} l_{L}^{} e_{R}^{} N_{R}^{}
      S U(3)_c^{} 1 1 1 1 3 3 3 1 1 1
      S U(2)_L^{} 1 1 2 2 2 1 1 2 1 1
      U(1)_Y^{} 0 0 +\dfrac{1}{2} +\dfrac{1}{2} +\dfrac{1}{6} -\dfrac{1}{3} +\dfrac{2}{3} -\dfrac{1}{2} -1 0
      Z_{3}^{(1)} {\rm e}^{{\rm i}\frac{2\pi}{3}}_{} 1 {\rm e}^{{\rm i}\frac{2\pi}{3}}_{} 1 1 {\rm e}^{{\rm i}\frac{4\pi}{3}}_{} 1 1 {\rm e}^{{\rm i}\frac{4\pi}{3}}_{} 1
      Z_{3}^{(2)} 1 {\rm e}^{{\rm i}\frac{2\pi}{3}}_{} {\rm e}^{{\rm i}\frac{2\pi}{3}}_{} 1 1 {\rm e}^{{\rm i}\frac{4\pi}{3}}_{} 1 1 {\rm e}^{{\rm i}\frac{4\pi}{3}}_{} 1

      Table 1.  All scalars and fermions in the model. The two Higgs singlets \xi_{1,2}^{} are denoted by Z^{(1)}_{3}\times Z^{(2)}_{3} discrete symmetries and the two Higgs doublets \phi_{1,2}^{} . In addition to three generations of the SM quarks q_L^{} , d_R^{} , and u_R^{} and SM leptons l_L^{} and e_R^{} , we introduce three right-handed neutrinos N_R^{} 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.

      \begin{array}{*{20}{l}} \xi_{1}^{}\stackrel{ {\rm{\; \bar{\theta}} -characterized~ mirror~ symmetry }}{\leftarrow - - - - - - - - - - - - - - - - - - - - - - - - - \rightarrow } {\rm e}^{{\rm -i}\bar{\theta}/3}_{}\xi_{2}^{} \,. \end{array}

      (5)

      This may be a minimal version of the \bar{\theta} -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:

      \begin{aligned}[b] \mathcal{L}_{Y+M}^{}=& -y_d^{}\bar{q}_L^{} \phi_1^{} d_R^{} -y_u^{}\bar{q}_L^{} \tilde{\phi}_2^{} u_R^{} -y_e^{}\bar{l}_L^{} \phi_1^{} e_R^{} \\ &-y_N^{}\bar{l}_L^{} \tilde{\phi}_2^{} N_R^{} -\frac{1}{2}M_N^{} \bar{N}_R^{} N_R^c + {\rm{H.c.}} \\ & {\rm{with}}\; \; \tilde{\phi}_{1,2}^{}= {\rm i} \tau_2^{}\phi_{1,2}^\ast\,, \end{aligned}

      (6)

      The full scalar potential at a renormalizable level is as follows:

      \begin{aligned}[b]V=& \mu_1^2 \phi_1^\dagger \phi^{}_1 + \mu_2^2 \phi^\dagger_2 \phi^{}_2 + \lambda_1^{} \left(\phi^\dagger_1 \phi^{}_1\right)^2_{}+ \lambda_2^{} \left(\phi^\dagger_2 \phi^{}_2\right)^2_{} \\ &+ \lambda_3^{} \phi^\dagger_1 \phi^{}_1 \phi^\dagger_2 \phi^{}_2 +\lambda_4^{} \phi^\dagger_1 \phi^{}_2 \phi^\dagger_2 \phi^{}_1+ \mu_\xi^2 \left(\xi_1^\ast \xi_1^{}+\xi_2^\ast\xi_2^{}\right)\\ &+\kappa_1^{}\left[\left(\xi_1^\ast \xi_1^{}\right)^2_{} + \left(\xi_2^\ast \xi_2^{}\right)^2_{}\right] +\kappa^{}_2 \xi_1^\ast \xi_1^{} \xi_2^\ast\xi_2^{}\\ &+\rho_\xi^{} \left[\left(\xi_1^3+ {\rm e}^{-{\rm i}\bar{\theta}}_{}\xi_2^3 \right)+ {\rm{H.c.}}\right]+ \epsilon_1^{}\phi^\dagger_1 \phi^{}_1 \left(\xi^\ast_1\xi^{}_1 +\xi^\ast_2 \xi^{}_2\right) \\ &+ \epsilon_2^{}\phi^\dagger_2 \phi^{}_2 \left(\xi^\ast_1\xi^{}_1 +\xi^\ast_2 \xi^{}_2\right)+ \epsilon_3^{} \left(\xi_1^{} \xi_2^{} \phi^\dagger_1 \phi_2^{} + {\rm{H.c.}}\right)\,. \end{aligned}

      (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 \bar{\theta} -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 \bar{\theta} -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,

      \begin{array}{*{20}{l}} &&\left(\begin{array}{l}\xi_{1}^{}\rightarrow \xi_{1} \\ \xi_{2}^{} \rightarrow {\rm e}^{+{\rm i}\bar{\theta}/3}_{} \xi_2^{}\end{array} \right)\,,\; \; \; \; \left(\begin{array}{l} \phi_{1}^{}\rightarrow {\rm e}^{+{\rm i}\bar{\theta}/3}_{} \phi_{1}\\ \phi_{2}^{}\rightarrow \phi_{2} \end{array}\right)\,,\\ \\&&\left(\begin{array}{l}\,\,q_{L}^{}\rightarrow q_{L}\\ \,d_{R}^{}\rightarrow {\rm e}^{-{\rm i}\bar{\theta}/3}_{}d_{R}\\ \,u_{R}^{}\rightarrow u_{R}\\ \,\,\,l_{L}^{} \rightarrow l_{L}\\ \,\,e_{R}^{}\rightarrow {\rm e}^{-{\rm i}\bar{\theta}/3}_{}e_{R} \\ N_{R}^{} \rightarrow N_{R}^{}\\ \end{array}\right) \,, \end{array}

      (8)

      the QCD Lagrangian (3) and the scalar potential (7) can simultaneously remove the parameter \bar{\theta} 1 as follows:

      \begin{array}{*{20}{l}} \mathcal{L}_{ {\rm{QCD}}}^{} & \Rightarrow & \sum\limits_{q}^{} \bar{q} \left({\rm i} \not D - m_q^{} \right)q - \frac{1}{4} G^{a\mu\nu}_{} G_{\mu\nu}^{a}\,, \end{array}

      (9)

      \begin{aligned}[b] V \Rightarrow & \mu_1^2 \phi_1^\dagger \phi^{}_1 + \mu_2^2 \phi^\dagger_2 \phi^{}_2 + \lambda_1^{} \left(\phi^\dagger_1 \phi^{}_1\right)^2_{}+ \lambda_2^{} \left(\phi^\dagger_2 \phi^{}_2\right)^2_{} \\ & + \lambda_3^{} \phi^\dagger_1 \phi^{}_1 \phi^\dagger_2 \phi^{}_2 +\lambda_4^{} \phi^\dagger_1 \phi^{}_2 \phi^\dagger_2 \phi^{}_1+ \mu_\xi^2 \left(\xi_1^\ast \xi_1^{}+\xi_2^\ast\xi_2^{}\right) \\ & +\kappa_1^{}\left[\left(\xi_1^\ast \xi_1^{}\right)^2_{} + \left(\xi_2^\ast \xi_2^{}\right)^2_{}\right] +\kappa^{}_2 \xi_1^\ast \xi_1^{} \xi_2^\ast\xi_2^{}\\ & +\rho_\xi^{} \left[\left(\xi_1^3+\xi_2^3 \right)+ {\rm{H.c.}}\right] + \epsilon_1^{}\phi^\dagger_1 \phi^{}_1 \left(\xi^\ast_1\xi^{}_1 +\xi^\ast_2 \xi^{}_2\right) \\ \end{aligned}

      \begin{aligned}[b] \quad + \epsilon_2^{}\phi^\dagger_2 \phi^{}_2 \left(\xi^\ast_1\xi^{}_1 +\xi^\ast_2 \xi^{}_2\right)+ \epsilon_3^{} \left(\xi_1^{} \xi_2^{} \phi^\dagger_1 \phi_2^{} + {\rm{H.c.}}\right)\,, \end{aligned}

      (10)

      The Yukawa and mass terms (6) can remain invariant with the unshown kinetic terms.

    IV.   PHYSICAL SCALARS
    • When the Higgs scalars \xi_{1}^{} , \xi_2^{} , \phi_1^{} , and \phi_{2}^{} develop their nonzero vacuum expectation values v_{\xi_1}^{} , v_{\xi_2}^{} , v_{\phi_1}^{} , and v_{\phi_2}^{} , respectively, they are expressed as follows:

      \xi_{1}^{} =\left(v_{\xi_1}^{}+h_{\xi_1}^{}+ {\rm i} P_{\xi_1}^{}\right)/\sqrt{2}\,,

      (11)

      \xi_{2}^{} =\left(v_{\xi_2}^{}+h_{\xi_2}^{}+{\rm i}P_{\xi_2}^{}\right)/\sqrt{2}\,,

      (12)

      \phi_{1}^{}=\left[\begin{array}{c} \phi^+_{1}\\ \left(v_{\phi_1}^{}+h_{\phi_1}^{}+ {\rm i} P_{\phi_1}^{}\right)/\sqrt{2}\end{array}\right]\,,

      (13)

      \phi_{2}^{}=\left[\begin{array}{c} \phi^+_{2}\\ \left(v_{\phi_2}^{}+h_{\phi_2}^{}+ {\rm i} P_{\phi_2}^{}\right)/\sqrt{2}\end{array}\right]\,.

      (14)

      Three would-be-Goldstone bosons:

      G_W^{\pm}=\left(v_{\phi_1}^{}\phi^{\pm}_{1} +v_{\phi_2}^{} \phi^{\pm}_{2} \right)/\sqrt{v_{\phi_1}^2 + v_{\phi_2}^2}\,,

      (15)

      G_Z^{}=\left(v_{\phi_1}^{}P^{}_{\phi_1} +v_{\phi_2}^{} P^{}_{\phi_2} \right)/\sqrt{v_{\phi_1}^2 + v_{\phi_2}^2}\,,

      (16)

      eaten by the longitudinal components of the SM gauge bosons W^{\pm}_{} and Z. Therefore, besides a pair of massive charged scalars,

      \begin{aligned}[b] H^{\pm}_{}=&\left(v_{\phi_1}^{}\phi^{\pm}_{2} - v_{\phi_2}^{} \phi^{\pm}_{1} \right)/\sqrt{v_{\phi_1}^2 + v_{\phi_2}^2}\; \; {\rm{with}}\\ m_{H^{\pm}}^2 =& -\left[\lambda_4^{}+\epsilon_3^{} v_\xi^2 /\left(2 v_1^{} v_2^{}\right)\right]\left(v_1^2+v_2^2\right)\,,\; \; \end{aligned}

      (17)

      we eventually obtain seven massive neutral scalars including four scalars and three pseudo scalars, i.e.,

      \begin{aligned}[b] &h_{\phi_1}^{}\,,\; \; h_{\phi_2}^{}\,,\; \; h_\xi^{}=\left(h^{}_{\xi_1} +h^{}_{\xi_2} \right)/\sqrt{2}\,,\\ &S_\xi^{} =\left(h^{}_{\xi_1} -h^{}_{\xi_2}\right)/\sqrt{2}\,; \end{aligned}

      (18)

      \begin{aligned}[b] &a_\phi^{}=\left(v_{\phi_1}^{}P^{}_{\phi_2} -v_{\phi_2}^{} P^{}_{\phi_1} \right)/\sqrt{v_{\phi_1}^2 + v_{\phi_2}^2}\,,\\ &a_\xi^{}=\left(P^{}_{\xi_1} +P^{}_{\xi_2} \right)/\sqrt{2}\,,\; \; P_\xi^{}=\left(P^{}_{\xi_1} -P^{}_{\xi_2} \right)/\sqrt{2}\,.\; \; \; \; \end{aligned}

      (19)

      With the minimum of the scalar potential, we obtain the mass-squared matrix of three scalars h_{\phi_1}^{} , h_{\phi_2}^{} and h_\xi^{} , i.e.

      \begin{array}{*{20}{l}}\\ \mathcal{L}\supset-\dfrac{1}{2}\left[h_{\phi_1}^{}\; h_{\phi_2}^{}\; h_\xi^{}\right] \left[\begin{array}{ccc} 2\lambda_1^{}v_{\phi_1}^2 -\dfrac{1}{2}\epsilon_3^{} v_\xi^2 \dfrac{v_{\phi_2}^{}}{v_{\phi_1}^{}} & \left(\lambda_3^{}+\lambda_4^{}\right) v_{\phi_1}^{} v_{\phi_2}^{} +\dfrac{1}{2}\epsilon_3^{} v_\xi^2 & \sqrt{2} \left(\epsilon_1^{} +\dfrac{1}{2}\epsilon_3^{}\right) v_{\phi_1}^{} v_\xi^{}\\ \left(\lambda_3^{}+\lambda_4^{}\right) v_{\phi_1}^{} v_{\phi_2}^{} +\dfrac{1}{2}\epsilon_3^{} v_\xi^2 & 2\lambda_2^{}v_{\phi_2}^2 -\dfrac{1}{2}\epsilon_3^{} v_\xi^2 \dfrac{v_{\phi_1}^{}}{v_{\phi_2}^{}} & \sqrt{2} \left(\epsilon_2^{} +\dfrac{1}{2}\epsilon_3^{}\right) v_{\phi_2}^{} v_\xi^{} \\ \sqrt{2} \left(\epsilon_1^{} +\dfrac{1}{2}\epsilon_3^{}\right) v_{\phi_1}^{} v_\xi^{} & \sqrt{2} \left(\epsilon_2^{} +\dfrac{1}{2}\epsilon_3^{}\right) v_{\phi_2}^{} v_\xi^{} & 2\kappa_1^{} v_\xi^2 + \dfrac{3}{\sqrt{2}}\rho_\xi^{} v_\xi^{} - \dfrac{1}{2} \epsilon_3^{} v_{\phi_1}^{} v_{\phi_2}^{} \end{array}\right] \left[\begin{array}{c}h_{\phi_1}^{}\\ h_{\phi_2}^{}\\ h_\xi^{}\end{array}\right] .\;\\ \end{array}

      (20)

      Hereafter, we consider the following:

      \begin{array}{*{20}{l}} v_{\xi_1}^{}= v_{\xi_2}^{}\equiv v_{\xi}^{}\,, \end{array}

      (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 H_{1,2,3}^{} with Yukawa couplings. For simplicity, we do not perform this diagonalization in the present work. For the forth scalar S_\xi^{} without Yukawa couplings, it corresponds to a mass eigenstate with the following mass square,

      m_{S_\xi}^2= 2\kappa_1^{} v_\xi^2 + \frac{3}{\sqrt{2}}\rho_\xi^{} v_\xi^{} - \frac{1}{2} \epsilon_3^{} v_{\phi_1}^{} v_{\phi_2}^{} \,.

      (22)

      We then consider the pseudo scalars a_{\phi}^{} , a_{\xi}^{} and P_\xi^{} . Their mass-squared matrix is given by

      \begin{array}{*{20}{l}} \mathcal{L}\supset-\dfrac{1}{2}\left[a_{\phi}^{}\; \; a_\xi^{}\; \; P_\xi^{}\right] \left[\begin{array}{ccc} -\dfrac{1}{2}\epsilon_3^{}v_\xi^2\left(\dfrac{v_{\phi_1}^{}}{v_{\phi_2}^{}} + \dfrac{v_{\phi_2}^{}}{v_{\phi_1}^{}}\right) & \; \; -\dfrac{1}{4} \epsilon_3^{} v_\xi^{}\sqrt{v_{\phi_1}^2+v_{\phi_2}^2} &\; \; 0\\ -\dfrac{1}{4} \epsilon_3^{} v_\xi^{}\sqrt{v_{\phi_1}^2+v_{\phi_2}^2} &\; \; -9\sqrt{2} \rho_\xi^{} v_\xi^{} -\epsilon_3^{} v_{\phi_1}^{} v_{\phi_2}^{} &\; \; 0 \\ \; \; 0 &\; \; 0 & \; \; -9\sqrt{2} \rho_\xi^{} v_\xi^{} -\epsilon_3^{} v_{\phi_1}^{} v_{\phi_2}^{} \end{array}\right] \left[\begin{array}{c}a_{\phi}^{}\\ a_\xi^{}\\ P_\xi^{} \end{array}\right]. \\\end{array}

      (23)

      Clearly, P_\xi^{} is already a mass eigenstate and its mass square is just

      \begin{array}{*{20}{l}} m_{P_\xi}^2= -9\sqrt{2} \rho_\xi^{} v_\xi^{} -\epsilon_3^{} v_{\phi_1}^{} v_{\phi_2}^{} \,. \end{array}

      (24)

      For a_\phi^{} and a_\xi^{} , they mix with each other, and their mass eigenstates are as follows:

      \begin{aligned}[b] a_1^{}=& a_\phi^{} \cos\alpha - a_\xi^{} \sin\alpha\; \; {\rm{with}}\\ m_{a_1}^2=&\frac{m_{a_\phi}^2 + m_{a_\xi}^2 + \sqrt{\left(m_{a_\phi}^2 - m_{a_\xi}^2 \right)^2_{}+4\Delta^4 }}{2}\,, \end{aligned}

      (25)

      \begin{aligned}[b] a_2^{}=& a_\phi^{} \sin\alpha + a_\xi^{} \cos\alpha\; \; {\rm{with}}\\ m_{a_2}^2=&\frac{m_{a_\phi}^2 + m_{a_\xi}^2 - \sqrt{\left(m_{a_\phi}^2 - m_{a_\xi}^2 \right)^2_{}+4\Delta^4 }}{2}\,, \end{aligned}

      (26)

      Here, m_{a_\phi}^2 , m_{a_\xi}^2 , and \Delta^2_{} are defined by

      \begin{aligned}[b] m_{a_\phi}^2 =& -\frac{1}{2}\epsilon_3^{}v_\xi^2\left(\frac{v_{\phi_1}^{}}{v_{\phi_2}^{}} + \frac{v_{\phi_2}^{}}{v_{\phi_1}^{}}\right) \,,\\ m_{a_\xi}^2 =&-9\sqrt{2} \rho_\xi^{} v_\xi^{} -\epsilon_3^{} v_{\phi_1}^{} v_{\phi_2}^{}\,, \end{aligned}

      \begin{aligned}[b]\Delta^2=& -\frac{1}{4} \epsilon_3^{} v_\xi^{}\sqrt{v_{\phi_1}^2+v_{\phi_2}^2}\,, \end{aligned}

      (27)

      while α is the mixing angle and is determined by

      \tan 2\alpha = \frac{2\Delta^2_{}}{m_{A_\phi}^2 - m_{A_\xi}^2} \,.

      (28)

      The pseudo scalars a_{1,2}^{} couple to the axial currents of the SM quarks, and thus, they act as heavy axions [24].

    V.   CONCLUSION
    • In this paper, we propose a novel \bar{\theta} -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. The \bar{\theta} -characterized mirror symmetry is only involved in the two Higgs singlets with respective discrete symmetries. The parameter \bar{\theta} 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.

Reference (25)

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