Solving the strong CP problem via a ${ \bar{\boldsymbol\theta} }$-characterized mirror symmetry

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Pei-Hong Gu. Solving the strong CP problem by a $ \bar{\theta} $-characterized mirror symmetry[J]. Chinese Physics C. doi: 10.1088/1674-1137/ad102b
Pei-Hong Gu. Solving the strong CP problem by a $ \bar{\theta} $-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 ${ \bar{\boldsymbol\theta} }$-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 $ \bar{\theta} $, which is typically on the order of unity. However, the upper bound on the electric dipole moment of the neutron enforces the value of $ \bar{\theta} $ 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 $ \bar{\theta} $-characterized mirror symmetry between two Higgs singlets with respective discrete symmetries. In our scenario, the parameter $ \bar{\theta} $ 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.

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    I.   INTRODUCTION
    • In the standard model QCD Lagrangian, a term of CP violating gluon density is theoretically expected to have a physical coefficient $ \bar{\theta} $, which is typically on the order of unity. However, the upper bound on the electric dipole moment of the neutron enforces the value of$ \bar{\theta} $ 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 $ \bar{\theta} $-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)_{ {\rm{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 $ \bar{\theta} $-characterized mirror symmetry only between two Higgs singlets with respective discrete symmetries. In this scenario, the parameter $ \bar{\theta} $ 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:

      $ \begin{aligned}[b] \mathcal{L}_{ {\rm{QCD}}}^{}=&\sum\limits_{q}^{} \bar{q} \left({\rm i} \not D - m_q^{} {\rm e}^{{\rm i}\theta_q}_{} \right)q - \frac{1}{4} G^{a\mu\nu}_{} G_{\mu\nu}^{a} \\ & - \theta \frac{\alpha_s^{}}{8\pi} G^{a\mu\nu}_{}\tilde{G}_{\mu\nu}^{a}\,, \end{aligned} $

      (1)

      where $ \theta_q^{} $ denotes the phase from the Yukawa couplings of quark fields, θ denotes the QCD vacuum angle, $ G^{a}_{\mu\nu} $ denotes the gluon field strength tensor, and $ \tilde{G}^{a}_{\mu\nu} $ denotes its dual. Following the application of a chiral phase transformation to the quark fields, it can be described as follows:

      $ \begin{array}{*{20}{l}} q\rightarrow {\rm e}^{-{\rm i} \gamma_5^{} \theta_q/2}_{} q \,, \end{array} $

      (2)

      Furthermore, their mass terms can remove phases $ \theta_q^{} $ from the QCD Lagrangian, i.e.,

      $ \begin{aligned}[b] \mathcal{L}_{ {\rm{QCD}}}^{}=&\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} - \bar{\theta} \frac{\alpha_s^{}}{8\pi} G^{a\mu\nu}_{}\tilde{G}_{\mu\nu}^{a}\\ & {\rm{with}}\; \; \bar{\theta}\equiv\theta- {\rm{Arg}} {\rm{ Det}} \left(M_d^{} M_u^{} \right)\,. \end{aligned} $

      (3)

      Here, $ M_{d}^{} $ and $ M_{u}^{} $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 of $ \bar{\theta} $ should be extremely small rather than the theoretically expected order of unity, i.e.,

      $ \begin{array}{*{20}{l}} \left|\bar{\theta}\right|<10^{-10}_{}\,. \end{array} $

      (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.

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