A Pati-Salam realization of the Nelson-Barr mechanism

Abstract

We present a UV completion of the Standard Model in which quarks and leptons are unified under color SU(4). A single fermionic representation, the real antisymmetric, provides the building blocks to address the strong CP problem via the Nelson-Barr mechanism, while simultaneously correcting the charged-lepton and down-quark mass relations predicted by Pati-Salam theories for the two heaviest generations. We show that the characteristic scales of the theory are strongly constrained by its phenomenology. The interplay between the quality of the Nelson-Barr mechanism and the non-observation of baryon-number-violating processes determines the scale of spontaneous CP violation and the mass of the new vector-like down quark, while the mass of the Standard Model down quark and the upper bound on neutrino masses fix the quark-lepton unification scale. The theory predicts a distinctive baryon-number-violating decay mode of the neutron, $n \to K^+ \ell^-$ (with $\ell = e,μ$), which lies within the projected sensitivity of upcoming nucleon-decay experiments such as Hyper-Kamiokande and DUNE.

Figures (2)

• Figure 1: Allowed parameter space in the theory. The horizontal axis shows the mass of the vector-like down quark $M_D$, while the vertical axis displays the scale of spontaneous CP violation, $v_\text{CP}$. The blue-shaded region is excluded by the experimental bound on the partial lifetime of the decay mode $n \to K^+ \mu^-$, $\tau/\text{Br}(n \to K^+ \mu^-) > 3.2 \times10^{31}$ years Frejus:1991ben, corresponding to Eq. \ref{['eq:BNVbound']}. The orange-shaded region is excluded by the quality of the Nelson-Barr mechanism, obtained by requiring $\Delta \bar{\theta}_{\rm QCD} < 10^{-10}$, which leads to the bound in Eq. \ref{['eq:boundquality']}. The gray-shaded region denotes parameter values for which the Yukawa couplings $\lambda_i$ become non-perturbative. Also shown are projected sensitivities. The dashed blue line corresponds to the expected reach of $\tau/\text{Br}(n\to K^+ \mu^-) > 1.1\times10^{34}$ years from DUNE Alt:2020riiDUNE:2024wvj, while the dashed orange line indicates the anticipated sensitivity to $\bar{\theta}_{\rm QCD}$ from upcoming hadron-EDM experiments n2EDM:2021yahEuropwanEDMprojects:2025oknnEDM:2019qgkTUCAN:2022koi. As the plot shows, the currently allowed parameter space will be probed by the next generation of nucleon decay and EDM experiments in the foreseeable future.
• Figure 2: Schematic distribution of the field masses and relevant energy scales of the theory. The three main scales of the theory are shown: the electroweak scale ($M_Z$), the scale of spontaneous CP violation ($v_\text{CP} \sim \langle \phi_4^0 \rangle$), and the quark-lepton unification scale ($\Lambda_{\rm QL} \sim \langle \phi_{15}^0 \rangle$), which is of the order of the canonical seesaw scale, $\langle \phi_{10}^0 \rangle$. The matter content consists of the Standard Model fields, embedded in three copies of $F_{QL} \sim (4,2,0)_{\rm PS}$, $F_{u\nu} \sim (4,1,1/2)_{\rm PS}$, and $F_{de} \sim (4,1,-1/2)_{\rm PS}$, together with the Higgs doublet $H \sim (1,2,1/2)_{\rm PS}$. The theory also contains a vector-like down quark $D \subset F_6 \sim (6,1,0)_{\rm PS}$ responsible of (i) correcting the $\mathsf{M}_d= \mathsf{M}_e$ relation for the two heaviest generations and (ii) solving the strong CP problem à la Nelson-Barr. The scalar sector includes $\Phi_{15}$, responsible for breaking $\text{SU}(4)_C \to \text{SU}(3)_C \otimes \mathrm{U}(1)_{B-L}$, $\Phi_{10}$ which breaks $\mathrm{U}(1)_{B-L} \otimes \mathrm{U}(1)_R \to \mathrm{U}(1)_Y$, and $\Phi_4$, whose vev triggers spontaneous CP violation. The hierarchy between the quark-lepton unification scale and the electroweak scale ($\Lambda_{\rm QL} / M_Z \sim 10^{12}$) generates active active neutrino masses via the type-I seesaw mechanism, consistently with neutrino data. In turn, the gap between the quark-lepton unification scale and the Plank scale ($\Lambda_{\rm QL}/M_{\rm Pl}\sim 10^{-5}$) accounts for the observed down-quark mass through non-renormalizable operators.