Baryon number non-conservation
as Peccei-Quinn mechanism

The $d=6$ operators of $qqq\ell$ type break both the baryon number and lepton number by one unit as $\mathbf{\Delta B}=\mathbf{\Delta L}=1$. These are given by

where SM right-handed quark singlets $u_{R}$, $d_{R}$ and left-handed quark doublet $Q$, respectively. All these operators hold global $\mathbf{B-L}$ symmetry, while the $\mathbf{B+L}$ symmetry is explicitly broken. According to the prescription discussed in the previous section, $\mathbf{B+L}$ symmetry may be restored by introducing a complex scalar $S$ as

Note that the baryon and lepton numbers of $S$ are determined as $\mathbf{B}(S)=\mathbf{L}(S)=+1$ through this operator. In general, the $\mathbf{B+L}$ conserving operator may be higher order as ${\mathcal{O}}_{8}=(S^{*})^{2}\,\widehat{\mathcal{O}}_{6},\cdots$, instead. If we commonly use $S^{*}$ or $S$ for the KSVZ Yukawa interaction as in Eq. (2) or Eq. (3), pNGB caused by $\mathbf{B+L}$ symmetry breaking is identified as an axion. At this point, how can we assign $\mathbf{B}$ and $\mathbf{L}$ for $\Psi_{L}$ and $\Psi_{R}$?

The $\mathbf{B}$ and $\mathbf{L}$ charges for $\Psi_{L}$ and $\Psi_{R}$ can be determined by introducing the one of the following mass mixing operators,

The hypercharges of $\Psi_{L}^{q}$ and $\Psi_{R}^{q}$ are chosen to be $+2/3$ (for $q=U$) or $-1/3$ (for $q=D$). These operators are obtained by replacing a SM quark singlet $q_{R}(=u_{R},d_{R})$ by $\Psi_{R}^{q}$ or $\overline{Q}\widetilde{H}(\overline{Q}H)$ by $\overline{\Psi^{U}_{L}}(\overline{\Psi^{D}_{L}})$ in the quark Yukawa interactions, i.e., $y_{U}\overline{Q}\widetilde{H}u_{R}$ and $y_{D}\overline{Q}Hd_{R}$. This is usually done in the KSVZ model in order to allow the decay of KSVZ quark into the SM particles. Through these connections, we find $\mathbf{B}(\Psi^{q}_{L})=\mathbf{B}(\Psi^{q}_{R})=+1/3$ and $\mathbf{L}(\Psi^{q}_{L})=\mathbf{L}(\Psi^{q}_{R})=0$. This is not the case for our purpose. In order to identify the pNGB as an axion, $\Psi_{L}$ and $\Psi_{R}$ must have different $\mathbf{B}$ and $\mathbf{L}$ charges. Therefore, we determine $\mathbf{B}$ and $\mathbf{L}$ only for $\Psi_{L}^{q}\,(\Psi_{R}^{q})$ by this procedure, and the other link for $\Psi_{R}^{q}\,(\Psi_{L}^{q})$ is taken from the KSVZ Yukawa interactions through Eq. (7). As a concrete example, we assume the operators of $\overline{\Psi^{D}_{L}}d_{R}$ and $S^{*}QQQL$ in our effective Lagrangian. At this point, $\mathbf{B}$ and $\mathbf{L}$ for $\Psi^{D}_{L}$ and $S$ are uniquely fixed, while that for $\Psi^{D}_{R}$ is ambiguous. Once the KSVZ Yukawa interaction $S\overline{\Psi^{D}_{L}}\Psi_{R}^{D}$ is turned on, $\mathbf{B}$ and $\mathbf{L}$ for $\Psi^{D}_{R}$ are uniquely determined.

The $\mathbf{B+L}$ conserving operator $\mathcal{O}_{7}$ can be systematically constructed from the following $\mathbf{B}$ and $\mathbf{L}$ conserving operators,

where one of a SM quark singlet $q_{R}$ is substituted by $\Psi_{R}^{q}$ in $\widehat{\mathcal{O}}_{6}$. Similarly to the method developed with ${\mathcal{O}}\,^{\prime}_{\text{mix}}$, ${\mathcal{O}}\,^{\prime}_{6}$ determines the baryon (lepton) number of KSVZ quarks as $\mathbf{B}(\Psi^{q}_{R})=-2/3$ and $\mathbf{L}(\Psi^{q}_{R})=-1$. If we take $\Psi_{L}^{q}$ from ${\mathcal{O}}\,^{\prime}_{\text{mix}}$ and $\Psi_{R}^{q}$ from ${\mathcal{O}}\,^{\prime}_{6}$, the KSVZ Yukawa interaction in Eq. (3) induces ${\mathcal{O}}_{7}$. The higher order operator ${\mathcal{O}}_{8}$ can also be derived by considering the operator ${\mathcal{O}}\,^{\prime\prime}_{6}=d_{R}\Psi_{R}^{U}\Psi_{R}^{U}e_{R}$, where $\mathbf{B}(\Psi^{U}_{R})=-1/6$ and $\mathbf{L}(\Psi^{U}_{R})=-1/2$ correspond to $\mathbf{B}(S)=\mathbf{L}(S)=+1/2$.

Let us show a UV complete example by introducing a singlet scalar $\xi$ with $Y=-1/3$. The leptoquark $\xi$ transforms as a fundamental representation of QCD, and the baryon and lepton numbers are assigned to $\mathbf{B}(\xi)=-1/3$ and $\mathbf{L}(\xi)=+1$. The baryon and lepton number conservation are assumed as PQ symmetry. Then, the dimension-six interaction of $\Psi_{R}^{U}d_{R}QL$ is decomposed to the renormalizable interactions. The relevant Lagrangian is given by

We here show the color indices ($a,b,c=1,2,3$) and the flavor indices $(i,j=1,2,3)$ explicitly, and $\epsilon_{abc}$ is the Levi-Civita tensor. The Lagrangian is assumed to hold global $\mathbf{B}+\mathbf{L}$ symmetry as PQ symmetry. The operators of $QQ\,\xi$ and $u_{R}d_{R}\,\xi$, which lead dangerous dimension-six nucleon decays, are forbidden by $\mathbf{B}+\mathbf{L}$ symmetry unlike ordinary leptoquark models. Note that $\xi$ has definite $\mathbf{B}$ and $\mathbf{L}$ by construction, no diquark interaction is allowed. On the other hand, the model predicts nucleon decays via ${\mathcal{O}}_{7}$. The corresponding Feynman diagram is depicted in the left of Fig. 3. The low energy effective Lagrangian for the nucleon decay is derived as

Here and hereafter, we omit the flavor indices. The Wilson coefficients are given by

where mass of leptoquark is $M_{\xi}$, and that of KSVZ quark is $M_{\Psi}=y_{\Psi}\langle S\rangle$.³³3 The typical mass scale of KSVZ quark can be as large as $\langle S\rangle$, while a TeV scale mass is possible by assuming the small Yukawa coupling constant $y_{\Psi}\ll 1$.

Using these effective interactions, the proton decay rates are

where the nucleon matrix elements are taken from the lattice simulation result[35]. For simplicity, we assume $y_{QL}^{ij}=y_{ue}^{ij}=y_{q\ell}$, $y_{\Psi D}^{i}=y_{\Psi D}$ and $\mu_{U}^{i}=\mu_{U}$ in numerical estimates. The proton (partial) lifetime is evaluated as functions of model parameters as

where $\tau_{p\to M\ell_{i}}=\Gamma_{p\to M\ell_{i}}^{-1}$, $\mu_{U}/M_{\Psi}$ characterizes the mixing between the SM and KSVZ quark. Among the current experimental bounds[36, 37, 38, 39, 40, 41, 42] on the proton decay modes, we find that $p\to\pi^{0}e^{+}$ mode gives the strongest constraint on this model. In the above formula, we normalize the partial lifetime of the proton by their current lower bound, e.g. the lower bound for $p\to\pi^{0}e^{+}$ mode is $\tau_{p\to\pi^{0}e^{+}}<2.4\times 10^{34}\,\text{yrs}$[36]. Among all the proton decay constraints, $p\to\pi^{0}e^{+}$ gives a strongest one. If we assume the order one Yukawa couplings $|y_{\Psi D}|\sim|y_{q\ell}|\sim 1$ and the order one mixing $\mu_{U}/M_{\Psi}\sim 0.2$, the proton decay has already given a very stringent constraint on the new physics scale $M_{\xi}$ as $M_{\xi}\gtrsim 10^{15}\,$GeV, which is much larger than typical PQ scale of $10^{12-14}\,$GeV. We note that the scale of direct search of new colored particles at hadron colliders is only a few TeV, and those of indirect searches through the flavor changing observables are about $100$–$1000$ TeV by assuming the order one coupling constants. Therefore, the proton decay search is the most promising way to explore this model.

Note that the other type of UV completion is also possible with more particles. For instance, if we introduce a QCD anti-fundamental scalar singlet diquark $\zeta$ with $Y=+1/3$ together with $\xi$, then the new Yukawa interaction $u_{R}d_{R}\zeta^{*}$ and the source of the mass mixing between the diquark and the leptoquark, $\zeta^{*}\xi^{*}S$, are allowed. The corresponding Feynman diagram for this UV completion is shown in the right of Fig. 3 as an example.

We may add right-handed neutrinos without imposing the lepton number symmetry in these setup. Since the Majorana mass term for right-handed neutrinos breaks the continuous lepton number symmetry explicitly, the baryon number symmetry is solely identified as PQ symmetry in this case. Thus, the ‘Sakharaxion’ scenario is realized in a minimal way, where the Sakharon[43], pNGB from the spontaneous baryon number violation, is identified as an axion.

Finally, we comment on the axion physics of the models. The model predictions for the axions are not changed from the minimal KSVZ model. This feature is common to the Majoraxion model. The constraints on the axion decay constant, e.g. from the duration time of neutrinos from the supernova SN 1987A[44], are applicable without modification. The possible axion dark matter scenario[45] can also be combined. Therefore, the model is examined only through the baryon number violation or the effect of new particles.

The $d=7$ operators of $qqq\overline{\ell}\varphi$ violate the baryon and lepton number as $\Delta\mathbf{B}=-\Delta\mathbf{L}=1$. which are given by

The global $\mathbf{B+L}$ symmetry is kept, while the $\mathbf{B-L}$ symmetry is not. The $\mathbf{B-L}$ symmetry may be restored with a complex scalar $S$ as

The baryon and lepton numbers of $S$ are fixed to be $\mathbf{B}(S)=-\mathbf{L}(S)=+1$.

As discussed in the previous subsection, the $\mathbf{B}$ and $\mathbf{L}$ charges for $\Psi_{L}\,(\Psi_{R})$ can be fixed by ${\mathcal{O}}\,^{\prime}_{\text{mix}}$. On the other hand, those for $\Psi_{R}\,(\Psi_{L})$ is determined by the following $\mathbf{B}$ and $\mathbf{L}$ conserving operators,

where the SM singlet quarks are substituted by $\Psi_{R}$, and

where $QH^{*}$ is replaced by $\Psi^{D}_{L}$ in Eq.(26). Thus, we obtain $\mathbf{B}(\Psi^{q}_{L/R})=-2/3$ and $\mathbf{L}(\Psi^{q}_{L/R})=+1$ from these connections. The extensions to higher order operators are straightforward.

As an example, we give a UV completion of this type of models by introducing a doublet scalar $\Xi$ with $Y=-2/3$. The $\Xi$ transforms as a fundamental representation of QCD, and the baryon and lepton numbers are assigned to $\mathbf{B}(\Xi)=+2/3$ and $\mathbf{L}(\Xi)=-1$. The operator of $d_{R}\Psi^{D}_{L}Q\overline{L}$ is decomposed by the following renormalizable interactions

The global $\mathbf{B}-\mathbf{L}$ symmetry is imposed as PQ symmetry. This model predicts $\mathbf{B}+\mathbf{L}$ conserving nucleon decays via ${\mathcal{O}}_{8}$ as shown in Fig. 4. The low energy effective Lagrangian for the nucleon decay is derived as

with

where $M_{\Xi}$ denotes the mass of $\Xi$, and $v_{\text{EW}}$ is the Higgs VEV of electroweak symmetry breaking.

In this model, no charged lepton mode of proton decay is induced. Only neutrino modes are available

where the nucleon matrix elements are determined by the lattice calculation[35]. We assume $y_{\overline{L}D}^{ij}=y_{\overline{L}D}$, ${y_{D}^{i}}^{\prime}=y_{D}^{\prime}$, $y_{\Psi Q}^{i}=y_{\Psi Q}$ in the following numerical estimation for simplicity. The partial lifetimes calculated from inverse partial widths are

where $(\tfrac{y_{D}^{\prime}v_{\text{EW}}}{\sqrt{2}}/M_{\Psi})$ characterizes the mixing between the SM and KSVZ quark. Comparing the experimental bound of these two proton decay mode, $p\to K^{+}\bar{\nu}$ mode gives about an order of magnitude stronger bound than that on $p\to\pi^{+}\bar{\nu}$ mode. Thus, $p\to K^{+}\bar{\nu}$ mode is the most promising mode to explore this model.

The impact of the Higgs field insertion of the operators appears in the mixing between the SM and KSVZ quark, which is strongly suppressed by $v_{\text{EW}}/M_{\Psi}$ Note that $v_{\text{EW}}$ cannot be taken to be very large unlike the model discussed in the previous subsection. Even though the proton decay search has potential to probe high energy scale far beyond the collider reach if we assume the order one coupling constants.

At $d=7$, there is a dressed operator $LLHH|H|^{2}$ with $\Delta\mathbf{L}=2$. At $d=8$, we have $\Delta\mathbf{B}=\Delta\mathbf{L}=1$ operators[15]

where $\widehat{\mathcal{O}}_{6}$ expresses dimension-six $\mathbf{B+L}$ violating operators. These operators can also be used to construct an axion model. Since the model is controlled by the same symmetry, ${\mathcal{O}}_{7}=S\widehat{\mathcal{O}}_{6}$ is allowed simultaneously with ${\mathcal{O}}_{9}=S\widehat{\mathcal{O}}_{8}$. The effects of $\mathcal{O}_{7}$ are dominated in the low energy phenomena such as nucleon decays.

We can continue the same discussions with $d>8$ operators which contain lepton and baryon number violations. There are many variations of operators, which are characterized by $\Delta\mathbf{B}$ and $\Delta\mathbf{L}$ (see FIG.1 of [15]). These operators can also be used to construct a model of axions.