Variations on the SU(5) axion

Abstract

The simultaneous embeddings of an axion state and a seesaw mechanism within the SU(5) Grand Unification Theory, both minimal and flipped, are systematically studied. Generically, seesaw mechanisms prevents a manifest axion Peccei–Quinn (PQ) symmetry at the GUT scale, but this does not mean no axion could emerge at the low scale. Several viable DFSZ-like scenarios are identified, both in the minimal and flipped case, and with various seesaw mechanisms, by carefully analyzing the global and local U(1) symmetries at play and their fate after the various stages of symmetry breaking. Interestingly, though these scenarios are quite different phenomenologically, and each assign specific non-manifestly SU(5) symmetric PQ charges to the SM fermions, the resulting low energy axion phenomenology is found to be particularly resilient and universal.

Fermion masses

The fermion masses are not correctly reproduced in the minimal SU(5) model. The situation is similar in the PQ and DFSZ axion models built in that context since introducing a second Higgs fiveplets does not alter the relationship between $m_{d}$ and $m_{e}$, which both derive from the same Yukawa coupling. To cure for this, there are two solutions. The first is to introduce a new scalar multiplet. To induce non-trivial corrections to the Yukawa couplings, this multiplet cannot transform as ${\mathbf {5}}$, and by inspection, the simplest choice is as ${{\mathbf {45}}}$ [45, 46]. The second solution is to introduce higher-dimensional operators, involving both $h_{\mathbf {5}}$ and ${\mathbf {H}}_{\mathbf {24}}$, thus exploiting the fact that ${{\mathbf {45}}} \subset {\mathbf {5}}\otimes {\mathbf {24}}$. The goal of this appendix is to present, briefly and in general terms, how these two solutions can be adapted to the PQ and DFSZ models.

1.1 The $h_{\mathbf{{45}}}$ and axions

Let us first consider the introduction of the $h_{{\mathbf {45}}}$. Concentrating on the fermion masses, two new Yukawa couplings can be constructed

$$\begin{aligned} {\mathcal {L}}_{\text {Yukawa}}^{{\mathbf {45}}}=-\sqrt{3/8}\varepsilon _{ABCDE} ({\bar{\chi }}_{{{\mathbf {10}}}}^{c})^{AB}{\mathbf {Y}}_{10}^{\prime } (\chi _{{{\mathbf {10}}}})^{CF}(h_{{\mathbf {45}}})_{F}^{DE}-\sqrt{12}(\bar{\psi }_{\bar{\mathbf {{5}}}}^{c})_{C}{\mathbf {Y}}_{5}^{\prime }(\chi _{{{\mathbf {10}}}})^{AB}(h_{{\mathbf {45}}}^{\dagger })_{AB}^{C}+h.c.\;, \end{aligned}$$

(122)

and they permit to induce reasonable fermion masses once SU(5) breaks down. When plugging in the specific structure of the vacuum [45,46,47,48]

$$\begin{aligned} ({\mathbf {v}}_{{\mathbf {45}}})_{C}^{AB}\equiv \langle 0|(h_{{\mathbf {45}}})_{C} ^{AB}|0\rangle =\frac{v_{45}}{4\sqrt{3}}(\delta _{C}^{A}-4\delta _{4}^{A} \delta _{C}^{4})\delta _{5}^{B}-(A\leftrightarrow B)\;, \end{aligned}$$

(123)

one finds in the one-fiveplet case,

$$\begin{aligned} {\mathbf {Y}}_{u}&={\mathbf {Y}}_{10}\sin \alpha +{\mathbf {Y}}_{10}^{\prime } \cos \alpha ,\;\;{\mathbf {Y}}_{10}={\mathbf {Y}}_{10}^{T},\;{\mathbf {Y}}_{10}^{\prime }=-{\mathbf {Y}}_{10}^{\prime T}\;, \end{aligned}$$

(124a)

$$\begin{aligned} {\mathbf {Y}}_{d}&={\mathbf {Y}}_{5}\sin \alpha +{\mathbf {Y}}_{5}^{\prime }\cos \alpha ,\;{\mathbf {Y}}_{e}={\mathbf {Y}}_{5}^{T}\sin \alpha -3{\mathbf {Y}}_{5}^{\prime T}\cos \alpha \;, \end{aligned}$$

(124b)

where $\tan \alpha =v_{5}/v_{45}$, $v^{2}=v_{5}^{2}+v_{45}^{2}\approx 246\,$GeV, and $v{\mathbf {Y}}_{u,d,e}\rightarrow \sqrt{2}{\mathbf {m}}_{u,d,e}$ in the mass eigenstate basis.

For axion models, the $U(1)_{1}\otimes U(1)_{2}$ symmetry forces the $h_{{\mathbf {45}}}$ to be aligned with $h_{2,{\mathbf {5}}}$, with thus only the ${\mathbf {Y}}_{5}^{\prime }$ Yukawa coupling of Eq. (122) and no coupling to ${\bar{\chi }}_{{{\mathbf {10}}}}^{c}\chi _{{{\mathbf {10}}}}$. Also, because $h_{{\mathbf {45}}}$ is aligned with $h_{2,{\mathbf {5}}}$, the symmetry is not extended and there is no additional Goldstone boson.

This may be a bit surprising because the potential can be constructed as invariant under $U(1)_{1}\otimes U(1)_{2}\otimes U(1)_{45}$. What happens in that case is that when both $h_{{\mathbf {45}}}$ and $h_{2,{\mathbf {5}}}$ couple to the same fermion pair ${\bar{\psi }}_{\bar{\mathbf {{5}}}}^{c}\chi _{{{\mathbf {10}}}}$, we can construct the dimension-four effective potential term

$$\begin{aligned} {\mathcal {L}}_{eff}=c_{eff}\langle {\mathbf {Y}}_{10}^{\dagger }{\mathbf {Y}} _{10}{\mathbf {Y}}_{5}^{\dagger }{\mathbf {Y}}_{5}^{\prime }\rangle (h_{1,{\mathbf {5}} }^{\dagger })_{C}(h_{1,{\mathbf {5}}})^{B}(h_{2,{\mathbf {5}}})^{A}(h_{{\mathbf {45}} }^{\dagger })_{AB}^{C}+h.c.\ , \end{aligned}$$

(125)

where we have used $(h_{{\mathbf {45}}})_{C}^{AB}=-(h_{{\mathbf {45}}})_{C}^{BA}$ and $(h_{{\mathbf {45}}})_{A}^{AB}=0$. Thus, the $U(1)_{1}\otimes U(1)_{2}\otimes U(1)_{45}$ symmetry cannot exist in the presence of the Yukawa couplings. Yet, this specific coupling cannot induce a mass for the pseudoscalar states because of the antisymmetry of $h_{{\mathbf {45}}}$. But with the help of $h_{1,{\mathbf {5}}}$ exchanges, the above coupling induces

$$\begin{aligned} {\mathcal {L}}_{eff}\sim c_{eff}^{2}\langle {\mathbf {Y}}_{10}^{\dagger } {\mathbf {Y}}_{10}{\mathbf {Y}}_{5}^{\dagger }{\mathbf {Y}}_{5}^{\prime }\rangle ^{2}(h_{2,{\mathbf {5}}})^{A}(h_{{\mathbf {45}}}^{\dagger })_{AB}^{C}(h_{{\mathbf {45}} }^{\dagger })_{DC}^{B}(h_{2,{\mathbf {5}}})^{D}\ , \end{aligned}$$

(126)

which does contribute to pseudoscalar masses. To check that let us plug in the polar representation of the scalar fields, with

$$\begin{aligned} h_{{\mathbf {45}}}=\frac{1}{\sqrt{2}}\exp (i\eta _{45}/v_{45} ){\mathbf {v}}_{{\mathbf {45}}}+...\ ,\ h_{i,{\mathbf {5}}}=\frac{1}{\sqrt{2}}\exp (i\eta _{i}/v_{i}){\mathbf {v}}_{i,{\mathbf {5}}}+...\ , \end{aligned}$$

(127)

with ${\mathbf {v}}_{{\mathbf {45}}}$ given in Eq. (123) and ${\mathbf {v}}_{i,{\mathbf {5}}}=(0,0,0,0,v_{i})^{T}$. Under the assumption that the scalar potential is invariant under $U(1)_{1}\otimes U(1)_{2}\otimes U(1)_{45}$ at leading order, only the term in Eq. (126) contributes and gives

$$\begin{aligned} V(\eta _{1},\eta _{2},\eta _{45})\sim c_{eff}^{2}\langle {\mathbf {Y}}_{10}^{\dagger }{\mathbf {Y}}_{10}{\mathbf {Y}}_{5}^{\dagger }{\mathbf {Y}}_{5}^{\prime }\rangle ^{2} \cos \left( 2\frac{\eta _{2}}{v_{2}}-2\frac{\eta _{45}}{v_{45}}\right) \ . \end{aligned}$$

(128)

The massive pseudoscalar is thus the combination $\pi ^{0}\sim \eta _{2}/v_{2}-\eta _{45}/v_{45}$. For the massless states, first note that the electrically neutral state in $h_{{\mathbf {45}}}$ belongs to a colorless $SU(2)_{L}$ doublet with hypercharge 1, as do those in $h_{1,{\mathbf {5}}}$ and $h_{2,{\mathbf {5}}}$. So, the WBG of the Z boson must be the combination $G^{0}\sim v_{1}\eta _{1}+v_{2}\eta _{2}+v_{45}\eta _{45}$. Knowing $\pi ^{0}$ and $G^{0}$, the mixing matrix is immediate to construct

$$\begin{aligned} \left( \begin{array} [c]{c} G^{0}\\ a^{0}\\ \pi ^{0} \end{array} \right) =\left( \begin{array} [c]{ccc} \sin \beta &{} \cos \beta \sin \alpha &{} \cos \beta \cos \alpha \\ \cos \beta &{} -\sin \beta \sin \alpha &{} -\sin \beta \cos \alpha \\ 0 &{} \cos \alpha &{} -\sin \alpha \end{array} \right) \left( \begin{array} [c]{c} \eta _{1}\\ \eta _{2}\\ \eta _{45} \end{array} \right) \ ,\ \end{aligned}$$

(129)

where we define

$$\begin{aligned} \tan \alpha =\frac{v_{2}}{v_{45}}\ ,\ \tan \beta =\frac{v_{1}}{\sqrt{v_{2} ^{2}+v_{45}^{2}}}\ . \end{aligned}$$

(130)

From this, the PQ charge of the weak doublets inside $h_{1,{\mathbf {5}}}$, $h_{2,{\mathbf {5}}}$, and $h_{{\mathbf {45}}}$ are found to be x, $-1/x$, and $-1/x$, respectively. Thus, fermions have the same PQ charges as without the $h_{{\mathbf {45}}}$. One can also check that their couplings to the axion are not altered. This is the simplest to see in the linear representation, in which the non-derivative couplings of the pseudoscalar states always enter in the combination $v_{i}+{\text {Re}}h_{i}^{0}+i\eta _{i}$. In particular, all dependencies on $\alpha$ cancel out of the axion couplings to fermions, leaving the same mass-dependent couplings as in the THDM,

$$\begin{aligned} {\mathcal {L}}_{a^{0}f{\bar{f}}}=\left( x{\bar{u}}_{L}{\mathbf {m}}_{u}u_{R}-\frac{1}{x}{\bar{d}}_{L}{\mathbf {m}}_{d}d_{R}-\frac{1}{x}{\bar{e}}_{L}{\mathbf {m}}_{e} e_{R}\right) \frac{a^{0}}{v}\;, \end{aligned}$$

(131)

with $x=1/\tan \beta$ and $v^{2}=v_{1}^{2}+v_{2}^{2}+v_{45}^{2}$.

1.2 Effective Yukawa couplings and axions

Alternatively to the $h_{{\mathbf {45}}}$, higher-dimensional operators can be introduced [49]. The same can be done in PQ and DFSZ axion models, provided these operators are constructed as invariant under the $U(1)_1\otimes U(1)_2$ symmetry. For instance, in the PQ model or in the singlet DFSZ model, given the charges in Eq. (21), the leading corrections to the fermion masses can arise from

$$\begin{aligned} {\mathcal {L}}_{\text {Yukawa}}^{\dim -5}&=-\frac{\sqrt{2}}{\Lambda }(h_{2,{\mathbf {5}}}^{\dagger })_{A}({\mathbf {H}}_{{\mathbf {24}}})_{C}^{B}(\bar{\psi }_{\bar{\mathbf {{5}}}}^{c})_{B}{\mathbf {Y}}_{5}^{\prime }(\chi _{{{\mathbf {10}}}})^{AC}+\frac{2}{\Lambda }\varepsilon _{ABCDE}(\bar{\chi }_{{{\mathbf {10}}}}^{c})^{AB}{\mathbf {Y}}_{10}^{\prime }(\chi _{{\mathbf {10}} })^{CF}(h_{1,{\mathbf {5}}})^{D}({\mathbf {H}}_{{\mathbf {24}}})_{F}^{E}\nonumber \\&\;\;\;\;+\frac{1}{\Lambda }(h_{2,{\mathbf {5}}}^{\dagger })_{C}({\mathbf {H}} _{{\mathbf {24}}})_{A}^{C}({\bar{\psi }}_{\bar{\mathbf {{5}}}}^{c} )_{B}{\mathbf {Y}}_{5}^{\prime \prime }(\chi _{{{\mathbf {10}}}})^{AB}+\frac{1}{\Lambda }\varepsilon _{ABCDE}({\bar{\chi }}_{{{\mathbf {10}}}}^{c})^{AB} {\mathbf {Y}}_{10}^{\prime \prime }(\chi _{{{\mathbf {10}}}})^{CD}(h_{1,{\mathbf {5}}} )^{F}({\mathbf {H}}_{{\mathbf {24}}})_{F}^{E}\;. \end{aligned}$$

(132)

The ${\mathbf {Y}}_{5}^{\prime \prime }$ and ${\mathbf {Y}}_{10}^{\prime \prime }$ corrections can be absorbed into the leading Yukawa couplings since $(h_{1,{\mathbf {5}}})^{A}({\mathbf {H}}_{{\mathbf {24}}})_{A}^{B}$ transforms as ${\mathbf {5}}$. The ${\mathbf {Y}}_{5}^{\prime }$ and ${\mathbf {Y}}_{10}^{\prime }$ represent genuine corrections, and give

$$\begin{aligned} {\mathcal {L}}_{\text {Yukawa}}^{\dim -5}\overset{\text {SSB}}{\rightarrow } -\frac{v_{5}v_{24}}{\sqrt{2}\Lambda }\cos \beta ({\bar{d}}_{R}^{i}{\mathbf {Y}} _{5}^{\prime }d_{L}^{i}-\frac{3}{2}{\bar{e}}_{R}{\mathbf {Y}}_{5}^{\prime T} e_{L})-\frac{v_{5}v_{24}}{\sqrt{2}\Lambda }\sin \beta {\bar{u}}_{R}^{i} (4{\mathbf {Y}}_{10}^{\prime T}-{\mathbf {Y}}_{10}^{\prime })u_{L}^{i}+h.c.\;. \end{aligned}$$

(133)

Note that compared to adding the $h_{{\mathbf {45}}}$, effective operators in general alter both the down and up-type Yukawa couplings.

The situation is a bit different in the adjoint DFSZ model, because the ${\mathbf {H}}_{{\mathbf {24}}}$ has a $U(1)_{1}\otimes U(1)_{2}$ charge. As a result, no operator can be constructed at the dimension-five level, and the leading corrections arise rather at the dimension-six level, with for example

$$\begin{aligned} {\mathcal {L}}_{\text {Yukawa}}^{\dim -6}=\frac{\sqrt{2}}{\Lambda ^{2}}\bar{\psi }_{\bar{\mathbf {{5}}}}^{c}{\mathbf {Y}}_{5}^{a}\chi _{{\mathbf {10}} }{\mathbf {H}}_{{\mathbf {24}}}^{2}h_{1,{\mathbf {5}}}^{\dagger }+\frac{\sqrt{2} }{\Lambda ^{2}}{\bar{\psi }}_{\bar{\mathbf {{5}}}}^{c}{\mathbf {Y}}_{5}^{b} \chi _{{{\mathbf {10}}}}{\mathbf {H}}_{{\mathbf {24}}}^{\dagger }{\mathbf {H}}_{{{\mathbf {24}}} }h_{2,{\mathbf {5}}}^{\dagger }+... \ , \end{aligned}$$

(134)

where for both terms, the SU(5) indices can be contracted in four different ways. Provided $v_{24}$ is not too small compared to $\Lambda$, these corrections are as effective as those of dimension-five to correct the fermion mass relations.