<p>The unsuppressed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> violation in QCD is a problem in the standard model. If we have some mechanism to guarantee real determinants of the quark mass matrices, the vanishing physical vacuum angle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>θ</mi><mo stretchy="false">¯</mo></mover></math> indicates the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> invariance at the fundamental level. Thus, the small <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>θ</mi><mo stretchy="false">¯</mo></mover></math> is technically natural, since we have an enhanced <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> symmetry in the limit of the vanishing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>θ</mi><mo stretchy="false">¯</mo></mover><mo>=</mo><mn>0</mn></math>. In fact, it was proved that the vacuum angle is never renormalized up to the four-loop level once it is fixed at 0 value at some high energy scale. The purpose of this paper is to construct a model that guarantees the real determinants of the quark mass matrices assuming a noninvertible symmetry. In the present model, we have only one <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> violating phase and hence all observable phases such as phases in the Cabibbo-Kobayashi-Maskawa matrix, in the neutrino oscillation and in the leptogenesis are related. It should be remarkable that the correct sign of the baryon-number asymmetry in the present Universe is obtained and a narrow region for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> violating phase <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>δ</mi></mrow><mrow><mi mathvariant="italic">CP</mi></mrow></msub><mo>≃</mo><mn>19</mn><mn>2</mn><mi>°</mi><mi>–</mi><mn>19</mn><mn>7</mn><mi>°</mi></mrow></math> is predicted.</p>

Japan

Tatsuo Kobayashi

Hajime Otsuka

Morimitsu Tanimoto

China

Tsutomu T. Yanagida

GUT-motivated noninvertible symmetry as a solution to the strong <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> problem and the neutrino <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math>-violating phase

CP invariance is a very attractive solution to the strong CP problem in quantum chromodynamics (QCD). This solution requires the vanishing <inline-formula><tex-math id='TM0001' notation='LaTeX'>${\rm arg}\, [{\rm det}\, M_d\, {\rm det} M_u]$</tex-math></inline-formula>, where <inline-formula><tex-math id='TM0002' notation='LaTeX'>$M_d$</tex-math></inline-formula> and <inline-formula><tex-math id='TM0003' notation='LaTeX'>$M_u$</tex-math></inline-formula> are the mass matrices for the down- and up-type quarks, respectively. It happens if we have several zeros in the quark mass matrices. We proceed with a systematic construction, in this paper, of two-zeros textures for the down-type quark mass matrix while the mass matrix for the up-type quarks is always diagonal. We find only three types of the mass matrices can explain the observed Cabibbo&#8211;Kobayashi&#8211;Maskawa (CKM) matrix, the masses of the quarks and the charged leptons, and the small enough vacuum angle <inline-formula><tex-math id='TM0004' notation='LaTeX'>$\theta &lt; 10^{-10}$</tex-math></inline-formula>. We extend the mass construction to the neutrino sector and derive predictions on the CP-violating parameter <inline-formula><tex-math id='TM0005' notation='LaTeX'>$\delta _{\mathrm{ CP}}$</tex-math></inline-formula> in the neutrino oscillation and the mass parameter <inline-formula><tex-math id='TM0006' notation='LaTeX'>$m_{\beta \beta }$</tex-math></inline-formula>. It is extremely remarkable that the normal (NH) and inverted (IH) hierarchies in the neutrino masses are equally possible in the case where we introduce only two right-handed neutrinos <italic>N</italic>s. Furthermore, we have a strict prediction that the <inline-formula><tex-math id='TM0008' notation='LaTeX'>$\delta _{\mathrm{ CP}} \simeq 200^\circ$</tex-math></inline-formula> or <inline-formula><tex-math id='TM0009' notation='LaTeX'>$250^\circ$</tex-math></inline-formula> in the NH case. If it is the case we can naturally explain the positive sign of the baryon asymmetry in the present universe.

Tsutomu T Yanagida

Axionless Solution to the Strong CP Problem—Two-Zeros Textures of the Quark and Lepton Mass Matrices and Neutrino CP Violation

We study the lepton mass textures, which are derived by $\textit{ℤ}$ gauging of $\textit{ℤ}$ symmetries. We can obtain various textures for the Yukawa couplings in the charged lepton sector, but the patterns of neutrino mass matrices are limited. All the obtained textures can not be realized by group-theoretical symmetries, and certain textures can lead to realistic results.

Haruki Uchida

Lepton mass textures from non-invertible multiplication rules

We study the coupling selection rules associated with non-group symmetries, i.e., ℤ gauging of ℤ symmetries. We clarify which Yukawa textures can be derived by our selection rules for $\textit{M}$ = 3$\textit{,}$ 4, and 5, and obtain various textures including the nearest neighbor interaction type and its extension. Some of them cannot be realized by a conventional group-like symmetry. They lead to interesting phenomenology such as a solution to the strong CP problem without axion.

Yume Nishioka

More about quark Yukawa textures from selection rules without group actions

A model of the axion-less solution to the strong CP problem has recently been constructed based on the <inline-formula><tex-math id='TM0002' notation='LaTeX'>$\bf T^2/\mathbb {Z}_3$</tex-math></inline-formula> orbifold in six-dimensional space-time. We extend, in this paper, the model to include the lepton sector with three right-handed neutrinos (RHNs). A crucial point in the extended model is that we have only one CP-violating phase and thus the CP phase in neutrino oscillation can be predicted by the CP phase in the Cabibbo&#8211;Kobayashi&#8211;Maskawa (CKM) matrix. The CP phase <inline-formula><tex-math id='TM0003' notation='LaTeX'>$\delta _{\mathrm{ CP}}$</tex-math></inline-formula> in the Pontecorvo&#8211;Maki&#8211;Nakagawa&#8211;Sakata matrix (PMNS) matrix is predicted as <inline-formula><tex-math id='TM0004' notation='LaTeX'>$\delta _{\mathrm{ CP}}\simeq 192^\circ \!-\! 195^\circ$</tex-math></inline-formula>, which can be tested in near-future experiments. We also discuss the effective mass <inline-formula><tex-math id='TM0005' notation='LaTeX'>$m_{\beta \beta }$</tex-math></inline-formula> responsible for the neutrinoless double beta decay of the nucleus, which is predicted as <inline-formula><tex-math id='TM0006' notation='LaTeX'>$m_{\beta \beta } \simeq 8 \!-\! 11$</tex-math></inline-formula>&#8201;meV. We consider leptogenesis and confirm that our model generates the right sign of the baryon asymmetry if the first family RHN is the lightest among the heavy RHNs.

Prediction of the CP Phase δ <sub>CP</sub> in Neutrino Oscillation and an Axion-less Solution to the Strong CP Problem

We have studied the leptonic EDM and the LFV decays relating with the recent data of anomalous magnetic moment of muon, $$(g-2)_{\mu }$$ in the leptonic dipole operator. We have adopted the successful $$\Gamma _2$$ modular invariant model by Meloni–Parriciatu as the flavor symmetry of leptons. Suppose the anomaly of $$(g-2)_{\mu }$$ , $$\Delta a_{\mu }$$ to be evidence of New Physics (NP), we have related it with the anomalous magnetic moment of the electron $$\Delta a_e$$ , the electron EDM $$d_e$$ and the $$\mu \rightarrow e \gamma $$ decay. We found that the NP contributions to $$\Delta a_{e(\mu )}$$ are proportional to the lepton masses squared likewise the naive scaling $$\Delta a_\ell \propto m^2_\ell $$ . The experimental constraint of $$|d_e|$$ is much tight compared with the one from the branching ratio $$\mathcal {B} (\mu \rightarrow e \gamma )$$ in our framework. Supposing the phase of our model parameter $$\delta _{\alpha }$$ for the electron to be of order one, we have estimated the upper-bound of $$\mathcal {B}(\mu \rightarrow e \gamma )$$ , which is at most $$10^{-21}-10^{-20}$$ . If some model parameters are real, leptonic EDMs vanish since the CP phase of the modular form due to modulus $$\tau $$ does not contribute to the EDM. However, we can obtain $$\mathcal {B} (\mu \rightarrow e \gamma )\simeq 10^{-13}$$ with non-vanishing $$d_e$$ in a specific case. The imaginary part of a parameter can lead to $$d_e$$ in the next-to-leading contribution. The predicted electron EDM is below $$10^{-32}$$ e cm, while $$\mathcal {B} (\mu \rightarrow e \gamma )$$ is close to the experimental upper-bound. The branching ratios of $$\tau \rightarrow e\gamma $$ and $$\tau \rightarrow \mu \gamma $$ are also discussed.

Takaaki Nomura

Xing-Yu Wang

Leptonic dipole operator with $$\Gamma _2$$ modular invariance in light of Muon $$(g-2)_\mu $$

Phenomenological aspects of non-invertible symmetries, in particular the flavor structure of quarks and leptons, are studied. We start with a ℤ discrete symmetry and gauge ℤ so as to obtain a non-invertible symmetry. We study which Yukawa textures can be derived from the non-invertible symmetries. Various textures can be realized and some of them cannot be realized by a conventional symmetry. For example, the nearest neighbor interaction texture as well as other interesting textures of quarks and leptons are obtained.

Yukawa textures from non-invertible symmetries

We study the interplay of New Physics (NP) among the lepton magnetic moment, the lepton flavor violation (LFV) and the electron electric dipole moment (EDM) in light of recent data of the muon  $$(g-2)_\mu $$  <math> <msub> <mrow> <mo>(</mo> <mi>g</mi> <mo>-</mo> <mn>2</mn> <mo>)</mo> </mrow> <mi>μ</mi> </msub> </math>  . The NP is discussed in the leptonic dipole operator with the U(2) flavor symmetry of the charged leptons, where possible CP violating phases of the three family space are taken into account. It is remarked that the third-family contributes significantly to the LFV decay,  $$\mu \rightarrow e\gamma $$  <math> <mrow> <mi>μ</mi> <mo>→</mo> <mi>e</mi> <mi>γ</mi> </mrow> </math>  , and the electron EDM. The experimental upper-bound on  $$\mu \rightarrow e\gamma $$  <math> <mrow> <mi>μ</mi> <mo>→</mo> <mi>e</mi> <mi>γ</mi> </mrow> </math>   decay gives a severe constraint on the parameters of the flavor model. The predicted electron EDM is rather large due to the CP violating phases in the three family space. In addition, we also study  $$(g-2)_{e,\tau }$$  <math> <msub> <mrow> <mo>(</mo> <mi>g</mi> <mo>-</mo> <mn>2</mn> <mo>)</mo> </mrow> <mrow> <mi>e</mi> <mo>,</mo> <mi>τ</mi> </mrow> </msub> </math>   of the electron and tauon, and EDMs of the muon and tauon as well as the  $$\tau \rightarrow e \gamma $$  <math> <mrow> <mi>τ</mi> <mo>→</mo> <mi>e</mi> <mi>γ</mi> </mrow> </math>   and  $$\tau \rightarrow \mu \gamma $$  <math> <mrow> <mi>τ</mi> <mo>→</mo> <mi>μ</mi> <mi>γ</mi> </mrow> </math>   decays. The  $$\tau _R \rightarrow \mu _L \gamma $$  <math> <mrow> <msub> <mi>τ</mi> <mi>R</mi> </msub> <mo>→</mo> <msub> <mi>μ</mi> <mi>L</mi> </msub> <mi>γ</mi> </mrow> </math>   decay is predicted to be close to the experimental upper-bound.

Kei Yamamoto

Electron EDM and LFV decays in the light of Muon  $$(g-2)_\mu $$  <math> <msub> <mrow> <mo>(</mo> <mi>g</mi> <mo>-</mo> <mn>2</mn> <mo>)</mo> </mrow> <mi>μ</mi> </msub> </math>   with U(2) flavor symmetry

We study systematically derivation of the specific texture zeros, that is the nearest neighbor interaction (NNI) form of the quark mass matrices at the fixed point  $$\tau =\omega $$  <math> <mrow> <mi>τ</mi> <mo>=</mo> <mi>ω</mi> </mrow> </math>   in modular flavor symmetric models. We present models that the NNI forms of the quark mass matrices are simply realized at the fixed point  $$\tau =\omega $$  <math> <mrow> <mi>τ</mi> <mo>=</mo> <mi>ω</mi> </mrow> </math>   in the  $$A_4$$  <math> <msub> <mi>A</mi> <mn>4</mn> </msub> </math>   modular flavor symmetry by taking account multi-Higgs fields. Such texture zero structure originates from the ST charge of the residual symmetry  $$Z_3$$  <math> <msub> <mi>Z</mi> <mn>3</mn> </msub> </math>   of SL(2, Z). The NNI form can be realized at the fixed point  $$\tau = \omega $$  <math> <mrow> <mi>τ</mi> <mo>=</mo> <mi>ω</mi> </mrow> </math>   in  $$A_4$$  <math> <msub> <mi>A</mi> <mn>4</mn> </msub> </math>   and  $$S_4$$  <math> <msub> <mi>S</mi> <mn>4</mn> </msub> </math>   modular flavor models with two pairs of Higgs fields when we assign properly modular weights to Yukawa couplings and  $$A_4$$  <math> <msub> <mi>A</mi> <mn>4</mn> </msub> </math>   and  $$S_4$$  <math> <msub> <mi>S</mi> <mn>4</mn> </msub> </math>   representations to three generations of quarks. We need four pairs of Higgs fields to realize the NNI form in  $$A_5$$  <math> <msub> <mi>A</mi> <mn>5</mn> </msub> </math>   modular flavor models.

Shota Kikuchi

Hikaru Uchida

Texture zeros of quark mass matrices at fixed point  $$\tau =\omega $$  <math> <mrow> <mi>τ</mi> <mo>=</mo> <mi>ω</mi> </mrow> </math>   in modular flavor symmetry

Abstract We classify a supersymmetric extension of the Standard Model by discrete symmetries originating from finite modular symmetries Γ$_{N}$. Since all the couplings in supersymmetric theories of finite modular symmetries Γ$_{N}$ are described by holomorphic modular forms with even modular weights, renormalizable and non-renormalizable operators such as baryon- and/or lepton-number violating operators are severely constrained. From the modular transformation of matter multiplets with modular weight 1/M, we find  symmetries, including the generalized baryon and lepton parities, R-parity,  baryon triality and  proton hexality. Such  symmetries are enlarged to  symmetries together with the CP transformation.