$\displaystyle\left(799+\frac{67043N_{f}^{2}}{12}-\frac{25N_{f}^{3}}{2}+\frac{21687N_{f}^{4}}{4}-N_{f}^{5}+\frac{2363N_{f}^{6}}{6}\right)$
(C.1)
$\displaystyle+\left(\frac{259N_{f}^{3}}{6}+\frac{5139N_{f}^{4}}{4}+\frac{35N_{f}^{5}}{6}+\frac{753N_{f}^{6}}{4}\right)\epsilon_{B}\epsilon_{L}+\left(4N_{f}+\frac{92N_{f}^{2}}{3}+2N_{f}^{3}+\frac{70N_{f}^{4}}{3}\right)\epsilon_{L}^{2}\,,$
$\displaystyle\\#\,{\cal O}_{12}^{\text{PQ}}\,=\,$
$\displaystyle\left(693+\frac{6629N_{f}^{2}}{2}-5N_{f}^{3}+\frac{3857N_{f}^{4}}{2}\right)+\left(-\frac{4N_{f}^{2}}{9}+\frac{13N_{f}^{3}}{12}+\frac{5N_{f}^{4}}{36}+\frac{71N_{f}^{5}}{12}+\frac{2135N_{f}^{6}}{36}\right)\epsilon_{B}^{2}$
$\displaystyle+\left(-2N_{f}^{2}+173N_{f}^{3}+\frac{8450N_{f}^{4}}{3}+74N_{f}^{5}+\frac{2221N_{f}^{6}}{3}\right)\epsilon_{B}\epsilon_{L}$
$\displaystyle+\left(-21N_{f}+1091N_{f}^{2}-50N_{f}^{3}+\frac{37349N_{f}^{4}}{12}+7N_{f}^{5}+\frac{8503N_{f}^{6}}{12}\right)\epsilon_{L}^{2}$
$\displaystyle+\left(-\frac{2N_{f}^{2}}{9}-\frac{N_{f}^{4}}{18}+4N_{f}^{5}+\frac{437N_{f}^{6}}{18}\right)\epsilon_{B}\epsilon_{L}^{3}\,.$

For aSMEFT without a shift symmetry, the numbers of operators at each
dimension are given by

$\displaystyle\\#\,{\cal O}_{5}^{\cancel{\text{PQ}}}\,=\,$ $\displaystyle
9+6N_{f}^{2}\,,$ $\displaystyle\\#\,{\cal O}_{6}^{\cancel{\text{PQ}}}\,=\,$
$\displaystyle\left(10+6N_{f}^{2}\right)+\left(N_{f}+N_{f}^{2}\right)\epsilon_{L}^{2}\,,$
$\displaystyle\\#\,{\cal O}_{7}^{\cancel{\text{PQ}}}\,=\,$
$\displaystyle\left(30+\frac{315N_{f}^{2}}{4}+\frac{N_{f}^{3}}{2}+\frac{107N_{f}^{4}}{4}\right)+\left(\frac{2N_{f}^{2}}{3}+N_{f}^{3}+\frac{19N_{f}^{4}}{3}\right)\epsilon_{B}\epsilon_{L}+\left(N_{f}+N_{f}^{2}\right)\epsilon_{L}^{2}\,,$
$\displaystyle\\#\,{\cal O}_{8}^{\cancel{\text{PQ}}}\,=\,$
$\displaystyle\left(43+\frac{359N_{f}^{2}}{4}+\frac{N_{f}^{3}}{2}+\frac{107N_{f}^{4}}{4}\right)+\left(3N_{f}+\frac{41N_{f}^{2}}{3}+N_{f}^{3}+\frac{37N_{f}^{4}}{3}\right)\epsilon_{L}^{2}$
$\displaystyle+\left(2N_{f}^{3}+16N_{f}^{4}\right)\epsilon_{B}\epsilon_{L}\,,$
$\displaystyle\\#\,{\cal O}_{9}^{\cancel{\text{PQ}}}\,=\,$
$\displaystyle\left(206+934N_{f}^{2}+650N_{f}^{4}\right)+\left(\frac{2N_{f}^{2}}{3}+3N_{f}^{3}+\frac{475N_{f}^{4}}{3}\right)\epsilon_{B}\epsilon_{L}+\left(4N_{f}+\frac{44N_{f}^{2}}{3}+N_{f}^{3}+\frac{37N_{f}^{4}}{3}\right)\epsilon_{L}^{2}\,,$
$\displaystyle\\#\,{\cal O}_{10}^{\cancel{\text{PQ}}}\,=\,$
$\displaystyle\left(280+\frac{2299N_{f}^{2}}{2}+\frac{1391N_{f}^{4}}{2}\right)+\left(\frac{N_{f}^{2}}{4}+\frac{61N_{f}^{3}}{24}+\frac{29N_{f}^{4}}{24}+\frac{11N_{f}^{5}}{24}+\frac{85N_{f}^{6}}{24}\right)\epsilon_{B}^{2}$
$\displaystyle+\left(-\frac{2N_{f}^{2}}{3}+\frac{62N_{f}^{3}}{3}+\frac{1256N_{f}^{4}}{3}+\frac{4N_{f}^{5}}{3}+40N_{f}^{6}\right)\epsilon_{B}\epsilon_{L}$
$\displaystyle+\left(12N_{f}+\frac{518N_{f}^{2}}{3}+\frac{73N_{f}^{3}}{12}+\frac{4187N_{f}^{4}}{12}-\frac{N_{f}^{5}}{12}+\frac{437N_{f}^{6}}{12}\right)\epsilon_{L}^{2}+\left(-N_{f}^{5}+N_{f}^{6}\right)\epsilon_{B}\epsilon_{L}^{3}\,,$
$\displaystyle\\#\,{\cal O}_{11}^{\cancel{\text{PQ}}}\,=\,$
$\displaystyle\left(1609+\frac{22461N_{f}^{2}}{2}-21N_{f}^{3}+12961N_{f}^{4}-7N_{f}^{5}+\frac{3305N_{f}^{6}}{2}\right)$
(C.2)
$\displaystyle+\left(\frac{N_{f}^{2}}{4}+\frac{61N_{f}^{3}}{24}+\frac{29N_{f}^{4}}{24}+\frac{11N_{f}^{5}}{24}+\frac{85N_{f}^{6}}{24}\right)\epsilon_{B}^{2}$
$\displaystyle+\left(-\frac{4N_{f}^{2}}{3}+\frac{695N_{f}^{3}}{6}+\frac{40555N_{f}^{4}}{12}+\frac{115N_{f}^{5}}{6}+\frac{3331N_{f}^{6}}{4}\right)\epsilon_{B}\epsilon_{L}$
$\displaystyle+\left(16N_{f}+\frac{610N_{f}^{2}}{3}+\frac{97N_{f}^{3}}{12}+\frac{1489N_{f}^{4}}{4}-\frac{N_{f}^{5}}{12}+\frac{437N_{f}^{6}}{12}\right)\epsilon_{L}^{2}$
$\displaystyle+\left(-\frac{4N_{f}^{2}}{9}-\frac{N_{f}^{3}}{3}-\frac{N_{f}^{4}}{9}-\frac{2N_{f}^{5}}{3}+\frac{14N_{f}^{6}}{9}\right)\epsilon_{B}\epsilon_{L}^{3}+\left(-\frac{N_{f}^{2}}{6}+\frac{N_{f}^{4}}{6}\right)\epsilon_{L}^{4}\,,$
$\displaystyle\\#\,{\cal O}_{12}^{\cancel{\text{PQ}}}\,=\,$
$\displaystyle\left(2302+14545N_{f}^{2}-26N_{f}^{3}+\frac{29779N_{f}^{4}}{2}-7N_{f}^{5}+\frac{3305N_{f}^{6}}{2}\right)$
$\displaystyle+\left(\frac{5N_{f}^{2}}{9}+\frac{197N_{f}^{3}}{12}+\frac{59N_{f}^{4}}{36}+\frac{175N_{f}^{5}}{12}+\frac{7661N_{f}^{6}}{36}\right)\epsilon_{B}^{2}$
$\displaystyle+\left(-\frac{16N_{f}^{2}}{3}+\frac{871N_{f}^{3}}{2}+\frac{109639N_{f}^{4}}{12}+\frac{411N_{f}^{5}}{2}+\frac{13823N_{f}^{6}}{4}\right)\epsilon_{B}\epsilon_{L}$
$\displaystyle+\left(13N_{f}+\frac{6695N_{f}^{2}}{3}-\frac{1111N_{f}^{3}}{12}+7381N_{f}^{4}-\frac{149N_{f}^{5}}{12}+\frac{7786N_{f}^{6}}{3}\right)\epsilon_{L}^{2}$
$\displaystyle+\left(-\frac{2N_{f}^{2}}{3}+\frac{2N_{f}^{3}}{3}+\frac{17N_{f}^{4}}{6}+\frac{25N_{f}^{5}}{3}+\frac{569N_{f}^{6}}{6}\right)\epsilon_{B}\epsilon_{L}^{3}+\left(-\frac{N_{f}^{2}}{6}+\frac{N_{f}^{4}}{6}\right)\epsilon_{L}^{4}\,.$

By setting $\epsilon_{B,L}\to 1$, the total number of operators can be easily
obtained.

###  C.2 aLEFT

In Section 4, the Hilbert series for aLEFT with (without) a shift symmetry is
shown up to dimension 7 (6). In this section, we provide more results for
Hilbert series at higher dimensions. For the shift-symmetric aLEFT, the
Hilbert series from dimension 5 to dimension 8 are given by

$\displaystyle{\cal H}_{5}^{\text{aLEFT}_{\text{PQ}}}\,=\,$
$\displaystyle\partial a\,u_{L}u_{L}^{\dagger}+\partial
a\,u_{R}u_{R}^{\dagger}+\partial a\,d_{L}d_{L}^{\dagger}+\partial
a\,d_{R}d_{R}^{\dagger}+\partial a\,\nu_{L}\nu_{L}^{\dagger}+\partial
a\,e_{L}e_{L}^{\dagger}+\partial a\,e_{R}e_{R}^{\dagger}$
$\displaystyle-\partial a\,F_{L}{\cal D}-\partial a\,F_{R}{\cal D}-\partial
a\,{\cal D}^{3}\,,$ $\displaystyle{\cal
H}_{6}^{\text{aLEFT}_{\text{PQ}}}\,=\,$ $\displaystyle 0\,,$
$\displaystyle{\cal H}_{7}^{\text{aLEFT}_{\text{PQ}}}\,=\,$
$\displaystyle(\partial a)^{2}u_{L}u_{R}+(\partial
a)^{2}u_{L}^{\dagger}u_{R}^{\dagger}+(\partial a)^{2}d_{L}d_{R}+(\partial
a)^{2}d_{L}^{\dagger}d_{R}^{\dagger}+(\partial a)^{2}\nu_{L}^{2}+(\partial
a)^{2}\nu_{L}^{\dagger 2}$ $\displaystyle+(\partial a)^{2}e_{L}e_{R}+(\partial
a)^{2}e_{L}^{\dagger}e_{R}^{\dagger}+\partial
a\,u_{L}u_{L}^{\dagger}F_{L}+\partial a\,u_{L}u_{L}^{\dagger}F_{R}+\partial
a\,u_{L}u_{L}^{\dagger}G_{L}$ $\displaystyle+\partial
a\,u_{L}u_{L}^{\dagger}G_{R}+\partial a\,u_{R}u_{R}^{\dagger}F_{L}+\partial
a\,u_{R}u_{R}^{\dagger}F_{R}+\partial a\,u_{R}u_{R}^{\dagger}G_{L}+\partial
a\,u_{R}u_{R}^{\dagger}G_{R}$ $\displaystyle+\partial
a\,d_{L}d_{L}^{\dagger}F_{L}+\partial a\,d_{R}d_{R}^{\dagger}F_{L}+\partial
a\,d_{L}d_{L}^{\dagger}F_{R}+\partial a\,d_{R}d_{R}^{\dagger}F_{R}+\partial
a\,d_{L}d_{L}^{\dagger}G_{L}$ $\displaystyle+\partial
a\,d_{R}d_{R}^{\dagger}G_{L}+\partial a\,d_{L}d_{L}^{\dagger}G_{R}+\partial
a\,d_{R}d_{R}^{\dagger}G_{R}+\partial
a\,\nu_{L}\nu_{L}^{\dagger}F_{L}+\partial a\,\nu_{L}\nu_{L}^{\dagger}F_{R}$
$\displaystyle+\partial a\,e_{L}e_{L}^{\dagger}F_{L}+\partial
a\,e_{R}e_{R}^{\dagger}F_{L}+\partial a\,e_{L}e_{L}^{\dagger}F_{R}+\partial
a\,e_{R}e_{R}^{\dagger}F_{R}\,,$ $\displaystyle{\cal
H}_{8}^{\text{aLEFT}_{\text{PQ}}}\,=\,$ $\displaystyle(\partial
a)^{4}+(\partial a)^{2}u_{L}u_{L}^{\dagger}{\cal D}+(\partial
a)^{2}u_{R}u_{R}^{\dagger}{\cal D}+(\partial a)^{2}d_{L}d_{L}^{\dagger}{\cal
D}+(\partial a)^{2}d_{R}d_{R}^{\dagger}{\cal D}+(\partial
a)^{2}\nu_{L}\nu_{L}^{\dagger}{\cal D}$ $\displaystyle+(\partial
a)^{2}e_{L}e_{L}^{\dagger}{\cal D}+(\partial a)^{2}e_{R}e_{R}^{\dagger}{\cal
D}+(\partial a)^{2}F_{L}^{2}+(\partial a)^{2}F_{L}F_{R}+(\partial
a)^{2}F_{R}^{2}+(\partial a)^{2}G_{L}G_{R}$ $\displaystyle+(\partial
a)^{2}G_{L}^{2}+(\partial a)^{2}G_{R}^{2}+2\partial
a\,u_{L}^{2}u_{L}^{\dagger}u_{R}+2\partial a\,u_{L}u_{L}^{\dagger
2}u_{R}^{\dagger}+2\partial a\,u_{L}u_{R}^{2}u_{R}^{\dagger}+2\partial
a\,u_{L}^{\dagger}u_{R}u_{R}^{\dagger 2}$ $\displaystyle+4\partial
a\,u_{L}u_{L}^{\dagger}d_{L}d_{R}+4\partial
a\,u_{L}u_{L}^{\dagger}d_{L}^{\dagger}d_{R}^{\dagger}+4\partial
a\,u_{L}u_{R}d_{L}d_{L}^{\dagger}+4\partial a\,u_{L}u_{R}d_{R}d_{R}^{\dagger}$
$\displaystyle+4\partial
a\,u_{L}^{\dagger}u_{R}^{\dagger}d_{L}d_{L}^{\dagger}+4\partial
a\,u_{L}^{\dagger}u_{R}^{\dagger}d_{R}d_{R}^{\dagger}+4\partial
a\,u_{R}u_{R}^{\dagger}d_{L}d_{R}+4\partial
a\,u_{R}u_{R}^{\dagger}d_{L}^{\dagger}d_{R}^{\dagger}$
$\displaystyle+2\partial a\,d_{L}^{2}d_{L}^{\dagger}d_{R}+2\partial
a\,d_{L}d_{L}^{\dagger 2}d_{R}^{\dagger}+2\partial
a\,d_{L}d_{R}^{2}d_{R}^{\dagger}+2\partial
a\,d_{L}^{\dagger}d_{R}d_{R}^{\dagger 2}+\partial
a\,u_{L}^{2}d_{R}^{\dagger}e_{L}$ $\displaystyle+\partial
a\,u_{L}^{2}d_{L}e_{R}^{\dagger}+\partial a\,u_{L}^{\dagger
2}d_{R}e_{L}^{\dagger}+\partial a\,u_{L}^{\dagger
2}d_{L}^{\dagger}e_{R}+2\partial
a\,u_{L}^{\dagger}u_{R}d_{L}^{\dagger}e_{L}^{\dagger}+2\partial
a\,u_{L}^{\dagger}u_{R}d_{R}e_{R}$ $\displaystyle+\partial
a\,u_{R}^{2}d_{R}e_{L}^{\dagger}+\partial
a\,u_{R}^{2}d_{L}^{\dagger}e_{R}+2\partial
a\,u_{L}u_{R}^{\dagger}d_{L}e_{L}+2\partial
a\,u_{L}u_{R}^{\dagger}d_{R}^{\dagger}e_{R}^{\dagger}+\partial
a\,u_{R}^{\dagger 2}d_{R}^{\dagger}e_{L}$ $\displaystyle+\partial
a\,u_{R}^{\dagger 2}d_{L}e_{R}^{\dagger}+2\partial
a\,u_{L}d_{L}d_{R}^{\dagger}\nu_{L}+\partial
a\,u_{L}d_{L}^{2}\nu_{L}^{\dagger}+\partial a\,u_{L}d_{R}^{\dagger
2}\nu_{L}^{\dagger}+\partial a\,u_{L}^{\dagger}d_{L}^{\dagger 2}\nu_{L}$
$\displaystyle+\partial a\,u_{L}^{\dagger}d_{R}^{2}\nu_{L}+2\partial
a\,u_{L}^{\dagger}d_{L}^{\dagger}d_{R}\nu_{L}^{\dagger}+2\partial
a\,u_{R}d_{L}^{\dagger}d_{R}\nu_{L}+\partial a\,u_{R}d_{L}^{\dagger
2}\nu_{L}^{\dagger}+\partial a\,u_{R}d_{R}^{2}\nu_{L}^{\dagger}$
$\displaystyle+\partial a\,u_{R}^{\dagger}d_{L}^{2}\nu_{L}+\partial
a\,u_{R}^{\dagger}d_{R}^{\dagger 2}\nu_{L}+2\partial
a\,u_{R}^{\dagger}d_{L}d_{R}^{\dagger}\nu_{L}^{\dagger}+\partial
a\,d_{L}^{\dagger}d_{R}^{2}e_{L}+\partial a\,d_{L}d_{R}^{\dagger
2}e_{L}^{\dagger}$ (C.3) $\displaystyle+\partial
a\,d_{L}^{2}d_{R}^{\dagger}e_{R}+\partial a\,d_{L}^{\dagger
2}d_{R}e_{R}^{\dagger}+\partial a\,u_{L}u_{L}^{\dagger}\nu_{L}^{2}+\partial
a\,u_{L}u_{L}^{\dagger}\nu_{L}^{\dagger 2}+2\partial
a\,u_{L}u_{R}\nu_{L}\nu_{L}^{\dagger}$ $\displaystyle+2\partial
a\,u_{L}^{\dagger}u_{R}^{\dagger}\nu_{L}\nu_{L}^{\dagger}+\partial
a\,u_{R}u_{R}^{\dagger}\nu_{L}^{2}+\partial
a\,u_{R}u_{R}^{\dagger}\nu_{L}^{\dagger 2}+2\partial
a\,u_{L}u_{L}^{\dagger}e_{L}e_{R}+2\partial
a\,u_{L}u_{L}^{\dagger}e_{L}^{\dagger}e_{R}^{\dagger}$
$\displaystyle+2\partial a\,u_{L}u_{R}e_{L}e_{L}^{\dagger}+2\partial
a\,u_{L}u_{R}e_{R}e_{R}^{\dagger}+2\partial
a\,u_{L}^{\dagger}u_{R}^{\dagger}e_{L}e_{L}^{\dagger}+2\partial
a\,u_{L}^{\dagger}u_{R}^{\dagger}e_{R}e_{R}^{\dagger}$
$\displaystyle+2\partial a\,u_{R}u_{R}^{\dagger}e_{L}e_{R}+2\partial
a\,u_{R}u_{R}^{\dagger}e_{L}^{\dagger}e_{R}^{\dagger}+2\partial
a\,u_{L}u_{R}F_{L}{\cal D}+\partial a\,u_{L}u_{R}F_{R}{\cal D}$
$\displaystyle+2\partial a\,u_{L}u_{R}G_{L}{\cal D}+\partial
a\,u_{L}u_{R}G_{R}{\cal D}+\partial
a\,u_{L}^{\dagger}u_{R}^{\dagger}F_{L}{\cal D}+2\partial
a\,u_{L}^{\dagger}u_{R}^{\dagger}F_{R}{\cal D}$ $\displaystyle+\partial
a\,u_{L}^{\dagger}u_{R}^{\dagger}G_{L}{\cal D}+2\partial
a\,u_{L}^{\dagger}u_{R}^{\dagger}G_{R}{\cal D}+2\partial
a\,u_{L}d_{L}^{\dagger}\nu_{L}e_{L}+2\partial
a\,u_{L}d_{R}\nu_{L}e_{R}^{\dagger}$ $\displaystyle+2\partial
a\,u_{L}d_{R}\nu_{L}^{\dagger}e_{L}+2\partial
a\,u_{L}d_{L}^{\dagger}\nu_{L}^{\dagger}e_{R}^{\dagger}+2\partial
a\,u_{L}^{\dagger}d_{R}^{\dagger}\nu_{L}e_{L}^{\dagger}+2\partial
a\,u_{L}^{\dagger}d_{L}\nu_{L}e_{R}$ $\displaystyle+2\partial
a\,u_{L}^{\dagger}d_{L}\nu_{L}^{\dagger}e_{L}^{\dagger}+2\partial
a\,u_{L}^{\dagger}d_{R}^{\dagger}\nu_{L}^{\dagger}e_{R}+2\partial
a\,u_{R}d_{L}\nu_{L}e_{L}^{\dagger}+2\partial
a\,u_{R}d_{R}^{\dagger}\nu_{L}e_{R}$ $\displaystyle+2\partial
a\,u_{R}d_{R}^{\dagger}\nu_{L}^{\dagger}e_{L}^{\dagger}+2\partial
a\,u_{R}d_{L}\nu_{L}^{\dagger}e_{R}+2\partial
a\,u_{R}^{\dagger}d_{R}\nu_{L}e_{L}+2\partial
a\,u_{R}^{\dagger}d_{L}^{\dagger}\nu_{L}e_{R}^{\dagger}$
$\displaystyle+2\partial
a\,u_{R}^{\dagger}d_{L}^{\dagger}\nu_{L}^{\dagger}e_{L}+2\partial
a\,u_{R}^{\dagger}d_{R}\nu_{L}^{\dagger}e_{R}^{\dagger}+\partial
a\,d_{L}d_{L}^{\dagger}\nu_{L}^{2}+\partial
a\,d_{R}d_{R}^{\dagger}\nu_{L}^{2}+2\partial
a\,d_{L}d_{R}\nu_{L}\nu_{L}^{\dagger}$ $\displaystyle+2\partial
a\,d_{L}^{\dagger}d_{R}^{\dagger}\nu_{L}\nu_{L}^{\dagger}+\partial
a\,d_{L}d_{L}^{\dagger}\nu_{L}^{\dagger 2}+\partial
a\,d_{R}d_{R}^{\dagger}\nu_{L}^{\dagger 2}+2\partial
a\,d_{L}d_{R}e_{L}e_{L}^{\dagger}+2\partial
a\,d_{L}^{\dagger}d_{R}^{\dagger}e_{L}e_{L}^{\dagger}$
$\displaystyle+2\partial a\,d_{L}d_{L}^{\dagger}e_{L}e_{R}+2\partial
a\,d_{R}d_{R}^{\dagger}e_{L}e_{R}+2\partial
a\,d_{L}d_{L}^{\dagger}e_{L}^{\dagger}e_{R}^{\dagger}+2\partial
a\,d_{R}d_{R}^{\dagger}e_{L}^{\dagger}e_{R}^{\dagger}$
$\displaystyle+2\partial a\,d_{L}d_{R}e_{R}e_{R}^{\dagger}+2\partial
a\,d_{L}^{\dagger}d_{R}^{\dagger}e_{R}e_{R}^{\dagger}+2\partial
a\,d_{L}d_{R}F_{L}{\cal D}+\partial
a\,d_{L}^{\dagger}d_{R}^{\dagger}F_{L}{\cal D}+\partial
a\,d_{L}d_{R}F_{R}{\cal D}$ $\displaystyle+2\partial
a\,d_{L}^{\dagger}d_{R}^{\dagger}F_{R}{\cal D}+2\partial
a\,d_{L}d_{R}G_{L}{\cal D}+\partial
a\,d_{L}^{\dagger}d_{R}^{\dagger}G_{L}{\cal D}+\partial
a\,d_{L}d_{R}G_{R}{\cal D}+2\partial
a\,d_{L}^{\dagger}d_{R}^{\dagger}G_{R}{\cal D}$ $\displaystyle+\partial
a\,\nu_{L}^{2}e_{L}e_{L}^{\dagger}+\partial
a\,\nu_{L}^{2}e_{R}e_{R}^{\dagger}+2\partial
a\,\nu_{L}\nu_{L}^{\dagger}e_{L}e_{R}+2\partial
a\,\nu_{L}\nu_{L}^{\dagger}e_{L}^{\dagger}e_{R}^{\dagger}+\partial
a\,\nu_{L}^{\dagger 2}e_{L}e_{L}^{\dagger}$ $\displaystyle+\partial
a\,\nu_{L}^{\dagger 2}e_{R}e_{R}^{\dagger}+\partial
a\,e_{L}^{2}e_{L}^{\dagger}e_{R}+\partial a\,e_{L}e_{L}^{\dagger
2}e_{R}^{\dagger}+\partial a\,e_{L}e_{R}^{2}e_{R}^{\dagger}+\partial
a\,e_{L}^{\dagger}e_{R}e_{R}^{\dagger 2}+\partial a\,\nu_{L}^{2}F_{L}{\cal D}$
$\displaystyle+\partial a\,\nu_{L}^{\dagger 2}F_{R}{\cal D}+2\partial
a\,e_{L}e_{R}F_{L}{\cal D}+\partial
a\,e_{L}^{\dagger}e_{R}^{\dagger}F_{L}{\cal D}+\partial
a\,e_{L}e_{R}F_{R}{\cal D}+2\partial
a\,e_{L}^{\dagger}e_{R}^{\dagger}F_{R}{\cal D}\,.$

Note that the number of quark flavors $N_{u,d}$ and lepton flavors $N_{e,\nu}$
in above Hilbert series are all set to 1. In the ancillary file, we provide
the Hilbert series with all flavor dependence. Based on the full Hilbert
series, and assume that $N_{\nu}=N_{e}=N_{d}$, we can obtain the operator
counting with flavor independence, which are shown up to dimension 10 as
follows,

$\displaystyle\\#\,{\cal O}_{5}^{\text{aLEFT}_{\text{PQ}}}\,=\,$
$\displaystyle 2-2N_{d}+5N_{d}^{2}-N_{u}+2N_{u}^{2}\,,$
$\displaystyle\\#\,{\cal O}_{6}^{\text{aLEFT}_{\text{PQ}}}\,=\,$
$\displaystyle 0\,,$ $\displaystyle\\#\,{\cal
O}_{7}^{\text{aLEFT}_{\text{PQ}}}\,=\,$
$\displaystyle\left(18N_{d}^{2}+10N_{u}^{2}\right)+\left(N_{d}+N_{d}^{2}\right)\epsilon_{L}^{2}\,,$
$\displaystyle\\#\,{\cal O}_{8}^{\text{aLEFT}_{\text{PQ}}}\,=\,$
$\displaystyle\left(7+23N_{d}^{2}+36N_{d}^{4}+16N_{d}^{3}N_{u}+14N_{u}^{2}+52N_{d}^{2}N_{u}^{2}+8N_{u}^{4}\right)$
$\displaystyle+\left(-\frac{4N_{d}^{2}}{3}+\frac{16N_{d}^{4}}{3}+16N_{d}^{3}N_{u}+16N_{d}^{2}N_{u}^{2}\right)\epsilon_{B}\epsilon_{L}$
$\displaystyle+\left(-N_{d}+\frac{7N_{d}^{2}}{3}+\frac{26N_{d}^{4}}{3}+16N_{d}^{3}N_{u}+4N_{d}^{2}N_{u}^{2}\right)\epsilon_{L}^{2}\,,$
$\displaystyle\\#\,{\cal O}_{9}^{\text{aLEFT}_{\text{PQ}}}\,=\,$
$\displaystyle\left(6+\frac{429N_{d}^{2}}{4}-\frac{5N_{d}^{3}}{2}+\frac{393N_{d}^{4}}{4}+44N_{d}^{3}N_{u}+66N_{u}^{2}+142N_{d}^{2}N_{u}^{2}+22N_{u}^{4}\right)$
$\displaystyle+\left(\frac{4N_{d}^{2}}{3}+4N_{d}^{3}+\frac{44N_{d}^{4}}{3}+8N_{d}^{2}N_{u}+44N_{d}^{3}N_{u}+44N_{d}^{2}N_{u}^{2}\right)\epsilon_{B}\epsilon_{L}$
(C.4)
$\displaystyle+\left(4N_{d}^{2}-4N_{d}^{3}+24N_{d}^{4}+44N_{d}^{3}N_{u}-2N_{d}N_{u}^{2}+12N_{d}^{2}N_{u}^{2}\right)\epsilon_{L}^{2}$
$\displaystyle+\left(-\frac{N_{d}}{2}+\frac{5N_{d}^{2}}{12}-\frac{N_{d}^{3}}{2}+\frac{7N_{d}^{4}}{12}\right)\epsilon_{L}^{4}\,,$
$\displaystyle\\#\,{\cal O}_{10}^{\text{aLEFT}_{\text{PQ}}}\,=\,$
$\displaystyle\left(17+\frac{539N_{d}^{2}}{2}-2N_{d}^{3}+\frac{865N_{d}^{4}}{2}+192N_{d}^{3}N_{u}+186N_{u}^{2}+704N_{d}^{2}N_{u}^{2}+116N_{u}^{4}\right)$
$\displaystyle+\left(-\frac{4N_{d}^{2}}{3}+4N_{d}^{3}+\frac{232N_{d}^{4}}{3}+8N_{d}^{2}N_{u}+232N_{d}^{3}N_{u}+232N_{d}^{2}N_{u}^{2}\right)\epsilon_{B}\epsilon_{L}$
$\displaystyle+\left(-N_{d}+\frac{79N_{d}^{2}}{3}-7N_{d}^{3}+\frac{275N_{d}^{4}}{3}+192N_{d}^{3}N_{u}-4N_{d}N_{u}^{2}+48N_{d}^{2}N_{u}^{2}\right)\epsilon_{L}^{2}$
$\displaystyle+\left(-\frac{N_{d}^{2}}{6}+\frac{N_{d}^{4}}{6}\right)\epsilon_{L}^{4}\,.$

For aLEFT without a shift symmetry, the Hilbert series from dimension 5 to
dimension 7 are shown as follows

$\displaystyle{\cal H}_{5}^{\text{aLEFT}_{\cancel{\text{PQ}}}}\,=\,$
$\displaystyle
a^{5}+a^{2}u_{L}u_{R}+a^{2}u_{L}^{\dagger}u_{R}^{\dagger}+a^{2}d_{L}d_{R}+a^{2}d_{L}^{\dagger}d_{R}^{\dagger}+a^{2}\nu_{L}^{2}+a^{2}\nu_{L}^{\dagger
2}+a^{2}e_{L}e_{R}+a^{2}e_{L}^{\dagger}e_{R}^{\dagger}$
$\displaystyle+aF_{L}^{2}+aF_{R}^{2}+aG_{L}^{2}+aG_{R}^{2}\,,$
$\displaystyle{\cal H}_{6}^{\text{aLEFT}_{\cancel{\text{PQ}}}}\,=\,$
$\displaystyle
a^{6}+a^{3}u_{L}u_{R}+a^{3}u_{L}^{\dagger}u_{R}^{\dagger}+a^{3}d_{L}d_{R}+a^{3}d_{L}^{\dagger}d_{R}^{\dagger}+a^{3}\nu_{L}^{2}+a^{3}\nu_{L}^{\dagger
2}+a^{3}e_{L}e_{R}+a^{3}e_{L}^{\dagger}e_{R}^{\dagger}$
$\displaystyle+a^{2}F_{L}^{2}+a^{2}F_{R}^{2}+a^{2}G_{L}^{2}+a^{2}G_{R}^{2}+au_{L}u_{R}F_{L}+au_{L}u_{R}G_{L}+au_{L}^{\dagger}u_{R}^{\dagger}F_{R}+au_{L}^{\dagger}u_{R}^{\dagger}G_{R}$
$\displaystyle+ad_{L}d_{R}F_{L}+ad_{L}^{\dagger}d_{R}^{\dagger}F_{R}+ad_{L}d_{R}G_{L}+ad_{L}^{\dagger}d_{R}^{\dagger}G_{R}+ae_{L}e_{R}F_{L}+ae_{L}^{\dagger}e_{R}^{\dagger}F_{R}\,,$
$\displaystyle{\cal H}_{7}^{\text{aLEFT}_{\cancel{\text{PQ}}}}\,=\,$
$\displaystyle
a^{7}+a^{4}u_{L}u_{R}+a^{4}u_{L}^{\dagger}u_{R}^{\dagger}+a^{4}d_{L}d_{R}+a^{4}d_{L}^{\dagger}d_{R}^{\dagger}+a^{4}\nu_{L}^{2}+a^{4}\nu_{L}^{\dagger
2}+a^{4}e_{L}e_{R}+a^{4}e_{L}^{\dagger}e_{R}^{\dagger}$
$\displaystyle+a^{3}F_{L}^{2}+a^{3}F_{R}^{2}+a^{3}G_{L}^{2}+a^{3}G_{R}^{2}+a^{2}u_{L}u_{R}F_{L}+a^{2}u_{L}u_{R}G_{L}+a^{2}u_{L}^{\dagger}u_{R}^{\dagger}F_{R}$
$\displaystyle+a^{2}u_{L}^{\dagger}u_{R}^{\dagger}G_{R}+a^{2}u_{L}u_{R}{\cal
D}^{2}+a^{2}u_{L}^{\dagger}u_{R}^{\dagger}{\cal
D}^{2}+a^{2}d_{L}d_{R}F_{L}+a^{2}d_{L}^{\dagger}d_{R}^{\dagger}F_{R}+a^{2}d_{L}d_{R}G_{L}$
$\displaystyle+a^{2}d_{L}^{\dagger}d_{R}^{\dagger}G_{R}+a^{2}d_{L}d_{R}{\cal
D}^{2}+a^{2}d_{L}^{\dagger}d_{R}^{\dagger}{\cal D}^{2}+a^{2}\nu_{L}^{2}{\cal
D}^{2}+a^{2}\nu_{L}^{\dagger 2}{\cal D}^{2}+a^{2}e_{L}e_{R}F_{L}$
$\displaystyle+a^{2}e_{L}^{\dagger}e_{R}^{\dagger}F_{R}+a^{2}e_{L}e_{R}{\cal
D}^{2}+a^{2}e_{L}^{\dagger}e_{R}^{\dagger}{\cal
D}^{2}+au_{L}^{2}u_{L}^{\dagger
2}+2au_{L}^{2}u_{R}^{2}+2au_{L}u_{L}^{\dagger}u_{R}u_{R}^{\dagger}$
$\displaystyle+2au_{L}^{\dagger 2}u_{R}^{\dagger 2}+au_{R}^{2}u_{R}^{\dagger
2}+2au_{L}u_{L}^{\dagger}d_{L}d_{L}^{\dagger}+2au_{L}u_{L}^{\dagger}d_{R}d_{R}^{\dagger}+4au_{L}u_{R}d_{L}d_{R}$
$\displaystyle+2au_{L}u_{R}d_{L}^{\dagger}d_{R}^{\dagger}+2au_{L}^{\dagger}u_{R}^{\dagger}d_{L}d_{R}+4au_{L}^{\dagger}u_{R}^{\dagger}d_{L}^{\dagger}d_{R}^{\dagger}+2au_{R}u_{R}^{\dagger}d_{L}d_{L}^{\dagger}+2au_{R}u_{R}^{\dagger}d_{R}d_{R}^{\dagger}$
$\displaystyle+ad_{L}^{2}d_{L}^{\dagger
2}+2ad_{L}^{2}d_{R}^{2}+2ad_{L}d_{L}^{\dagger}d_{R}d_{R}^{\dagger}+2ad_{L}^{\dagger
2}d_{R}^{\dagger 2}+ad_{R}^{2}d_{R}^{\dagger
2}+au_{L}^{2}d_{L}e_{L}+au_{L}^{\dagger 2}d_{L}^{\dagger}e_{L}^{\dagger}$
$\displaystyle+au_{L}^{\dagger}u_{R}d_{R}e_{L}^{\dagger}+au_{L}^{\dagger}u_{R}d_{L}^{\dagger}e_{R}+au_{R}^{2}d_{R}e_{R}+au_{L}u_{R}^{\dagger}d_{R}^{\dagger}e_{L}+au_{L}u_{R}^{\dagger}d_{L}e_{R}^{\dagger}+au_{R}^{\dagger
2}d_{R}^{\dagger}e_{R}^{\dagger}$
$\displaystyle+au_{L}d_{L}^{2}\nu_{L}+au_{L}d_{L}d_{R}^{\dagger}\nu_{L}^{\dagger}+au_{L}^{\dagger}d_{L}^{\dagger}d_{R}\nu_{L}+au_{L}^{\dagger}d_{L}^{\dagger
2}\nu_{L}^{\dagger}+au_{R}d_{R}^{2}\nu_{L}+au_{R}d_{L}^{\dagger}d_{R}\nu_{L}^{\dagger}$
$\displaystyle+au_{R}^{\dagger}d_{L}d_{R}^{\dagger}\nu_{L}+au_{R}^{\dagger}d_{R}^{\dagger
2}\nu_{L}^{\dagger}+au_{L}u_{L}^{\dagger}\nu_{L}\nu_{L}^{\dagger}+au_{L}u_{R}\nu_{L}^{2}+au_{L}u_{R}\nu_{L}^{\dagger
2}+au_{L}^{\dagger}u_{R}^{\dagger}\nu_{L}^{2}$
$\displaystyle+au_{L}^{\dagger}u_{R}^{\dagger}\nu_{L}^{\dagger
2}+au_{R}u_{R}^{\dagger}\nu_{L}\nu_{L}^{\dagger}+au_{L}u_{L}^{\dagger}e_{L}e_{L}^{\dagger}+au_{L}u_{L}^{\dagger}e_{R}e_{R}^{\dagger}+2au_{L}u_{R}e_{L}e_{R}$
(C.5)
$\displaystyle+au_{L}u_{R}e_{L}^{\dagger}e_{R}^{\dagger}+au_{L}^{\dagger}u_{R}^{\dagger}e_{L}e_{R}+2au_{L}^{\dagger}u_{R}^{\dagger}e_{L}^{\dagger}e_{R}^{\dagger}+au_{R}u_{R}^{\dagger}e_{L}e_{L}^{\dagger}+au_{R}u_{R}^{\dagger}e_{R}e_{R}^{\dagger}$
$\displaystyle+au_{L}u_{L}^{\dagger}F_{L}{\cal
D}+au_{L}u_{L}^{\dagger}F_{R}{\cal D}+au_{L}u_{L}^{\dagger}G_{L}{\cal
D}+au_{L}u_{L}^{\dagger}G_{R}{\cal D}+au_{R}u_{R}^{\dagger}F_{L}{\cal D}$
$\displaystyle+au_{R}u_{R}^{\dagger}F_{R}{\cal
D}+au_{R}u_{R}^{\dagger}G_{L}{\cal D}+au_{R}u_{R}^{\dagger}G_{R}{\cal
D}+2au_{L}d_{R}\nu_{L}e_{L}+au_{L}d_{L}^{\dagger}\nu_{L}e_{R}^{\dagger}$
$\displaystyle+au_{L}d_{L}^{\dagger}\nu_{L}^{\dagger}e_{L}+au_{L}d_{R}\nu_{L}^{\dagger}e_{R}^{\dagger}+au_{L}^{\dagger}d_{L}\nu_{L}e_{L}^{\dagger}+au_{L}^{\dagger}d_{R}^{\dagger}\nu_{L}e_{R}+2au_{L}^{\dagger}d_{R}^{\dagger}\nu_{L}^{\dagger}e_{L}^{\dagger}$
$\displaystyle+au_{L}^{\dagger}d_{L}\nu_{L}^{\dagger}e_{R}+au_{R}d_{R}^{\dagger}\nu_{L}e_{L}^{\dagger}+2au_{R}d_{L}\nu_{L}e_{R}+au_{R}d_{L}\nu_{L}^{\dagger}e_{L}^{\dagger}+au_{R}d_{R}^{\dagger}\nu_{L}^{\dagger}e_{R}$
$\displaystyle+au_{R}^{\dagger}d_{L}^{\dagger}\nu_{L}e_{L}+au_{R}^{\dagger}d_{R}\nu_{L}e_{R}^{\dagger}+au_{R}^{\dagger}d_{R}\nu_{L}^{\dagger}e_{L}+2au_{R}^{\dagger}d_{L}^{\dagger}\nu_{L}^{\dagger}e_{R}^{\dagger}+ad_{L}d_{R}\nu_{L}^{2}+ad_{L}^{\dagger}d_{R}^{\dagger}\nu_{L}^{2}$
$\displaystyle+ad_{L}d_{L}^{\dagger}\nu_{L}\nu_{L}^{\dagger}+ad_{R}d_{R}^{\dagger}\nu_{L}\nu_{L}^{\dagger}+ad_{L}d_{R}\nu_{L}^{\dagger
2}+ad_{L}^{\dagger}d_{R}^{\dagger}\nu_{L}^{\dagger
2}+ad_{L}d_{L}^{\dagger}e_{L}e_{L}^{\dagger}+ad_{R}d_{R}^{\dagger}e_{L}e_{L}^{\dagger}$
$\displaystyle+2ad_{L}d_{R}e_{L}e_{R}+ad_{L}^{\dagger}d_{R}^{\dagger}e_{L}e_{R}+ad_{L}d_{R}e_{L}^{\dagger}e_{R}^{\dagger}+2ad_{L}^{\dagger}d_{R}^{\dagger}e_{L}^{\dagger}e_{R}^{\dagger}+ad_{L}d_{L}^{\dagger}e_{R}e_{R}^{\dagger}$
$\displaystyle+ad_{R}d_{R}^{\dagger}e_{R}e_{R}^{\dagger}+ad_{L}d_{L}^{\dagger}F_{L}{\cal
D}+ad_{R}d_{R}^{\dagger}F_{L}{\cal D}+ad_{L}d_{L}^{\dagger}F_{R}{\cal
D}+ad_{R}d_{R}^{\dagger}F_{R}{\cal D}+ad_{L}d_{L}^{\dagger}G_{L}{\cal D}$
$\displaystyle+ad_{R}d_{R}^{\dagger}G_{L}{\cal
D}+ad_{L}d_{L}^{\dagger}G_{R}{\cal D}+ad_{R}d_{R}^{\dagger}G_{R}{\cal
D}+a\nu_{L}^{2}\nu_{L}^{\dagger
2}+a\nu_{L}^{2}e_{L}e_{R}+a\nu_{L}^{2}e_{L}^{\dagger}e_{R}^{\dagger}$
$\displaystyle+a\nu_{L}\nu_{L}^{\dagger}e_{L}e_{L}^{\dagger}+a\nu_{L}\nu_{L}^{\dagger}e_{R}e_{R}^{\dagger}+a\nu_{L}^{\dagger
2}e_{L}e_{R}+a\nu_{L}^{\dagger
2}e_{L}^{\dagger}e_{R}^{\dagger}+ae_{L}^{2}e_{L}^{\dagger
2}+ae_{L}^{2}e_{R}^{2}+ae_{L}e_{L}^{\dagger}e_{R}e_{R}^{\dagger}$
$\displaystyle+ae_{L}^{\dagger 2}e_{R}^{\dagger 2}+ae_{R}^{2}e_{R}^{\dagger
2}+a\nu_{L}\nu_{L}^{\dagger}F_{L}{\cal D}+a\nu_{L}\nu_{L}^{\dagger}F_{R}{\cal
D}+ae_{L}e_{L}^{\dagger}F_{L}{\cal D}+ae_{R}e_{R}^{\dagger}F_{L}{\cal D}$
$\displaystyle+ae_{L}e_{L}^{\dagger}F_{R}{\cal
D}+ae_{R}e_{R}^{\dagger}F_{R}{\cal D}+aG_{L}^{3}+aG_{R}^{3}\,,$

where the flavors of fermions are set to 1, and the general Hilbert series
with flavor dependence is shown in the ancillary file. The numbers of
operators from dimension 5 to dimension 10 are presented in powers of
$\epsilon_{B,L}$, which are shown as follows,

$\displaystyle\\#\,{\cal O}_{5}^{\text{aLEFT}_{\cancel{\text{PQ}}}}\,=\,$
$\displaystyle\left(5+4N_{d}^{2}+2N_{u}^{2}\right)+\left(N_{d}+N_{d}^{2}\right)\epsilon_{L}^{2}\,,$
$\displaystyle\\#\,{\cal O}_{6}^{\text{aLEFT}_{\cancel{\text{PQ}}}}\,=\,$
$\displaystyle\left(5+10N_{d}^{2}+6N_{u}^{2}\right)+2N_{d}^{2}\epsilon_{L}^{2}\,,$
$\displaystyle\\#\,{\cal O}_{7}^{\text{aLEFT}_{\cancel{\text{PQ}}}}\,=\,$
$\displaystyle\left(7+\frac{131N_{d}^{2}}{4}+\frac{3N_{d}^{3}}{2}+\frac{87N_{d}^{4}}{4}+10N_{d}^{3}N_{u}+19N_{u}^{2}+32N_{d}^{2}N_{u}^{2}+5N_{u}^{4}\right)$
$\displaystyle+\left(-\frac{4N_{d}^{2}}{3}-2N_{d}^{3}+\frac{10N_{d}^{4}}{3}-4N_{d}^{2}N_{u}+10N_{d}^{3}N_{u}+10N_{d}^{2}N_{u}^{2}\right)\epsilon_{B}\epsilon_{L}$
$\displaystyle+\left(N_{d}+3N_{d}^{2}+2N_{d}^{3}+6N_{d}^{4}+10N_{d}^{3}N_{u}+N_{d}N_{u}^{2}+3N_{d}^{2}N_{u}^{2}\right)\epsilon_{L}^{2}$
$\displaystyle+\left(-\frac{N_{d}^{2}}{6}+\frac{N_{d}^{4}}{6}\right)\epsilon_{L}^{4}\,,$
$\displaystyle\\#\,{\cal O}_{8}^{\text{aLEFT}_{\cancel{\text{PQ}}}}\,=\,$
$\displaystyle\left(14+\frac{335N_{d}^{2}}{4}-\frac{N_{d}^{3}}{2}+\frac{303N_{d}^{4}}{4}+34N_{d}^{3}N_{u}+53N_{u}^{2}+110N_{d}^{2}N_{u}^{2}+17N_{u}^{4}\right)$
$\displaystyle+\left(-\frac{4N_{d}^{2}}{3}+2N_{d}^{3}+\frac{34N_{d}^{4}}{3}+4N_{d}^{2}N_{u}+34N_{d}^{3}N_{u}+34N_{d}^{2}N_{u}^{2}\right)\epsilon_{B}\epsilon_{L}$
$\displaystyle+\left(4N_{d}+10N_{d}^{2}-3N_{d}^{3}+19N_{d}^{4}+34N_{d}^{3}N_{u}-N_{d}N_{u}^{2}+9N_{d}^{2}N_{u}^{2}\right)\epsilon_{L}^{2}$
$\displaystyle+\left(-\frac{N_{d}^{2}}{6}+\frac{N_{d}^{4}}{6}\right)\epsilon_{L}^{4}\,,$
$\displaystyle\\#\,{\cal O}_{9}^{\text{aLEFT}_{\cancel{\text{PQ}}}}\,=\,$
$\displaystyle\left(43+220N_{d}^{2}-3N_{d}^{3}+294N_{d}^{4}+132N_{d}^{3}N_{u}+141N_{u}^{2}+462N_{d}^{2}N_{u}^{2}+75N_{u}^{4}\right)$
$\displaystyle+\left(6N_{d}^{3}+50N_{d}^{4}+12N_{d}^{2}N_{u}+150N_{d}^{3}N_{u}+150N_{d}^{2}N_{u}^{2}\right)\epsilon_{B}\epsilon_{L}$
$\displaystyle+\left(4N_{d}+14N_{d}^{2}-8N_{d}^{3}+68N_{d}^{4}+132N_{d}^{3}N_{u}-4N_{d}N_{u}^{2}+36N_{d}^{2}N_{u}^{2}\right)\epsilon_{L}^{2}$
$\displaystyle+\left(\frac{N_{d}}{2}+\frac{3N_{d}^{2}}{4}-\frac{N_{d}^{3}}{2}+\frac{5N_{d}^{4}}{4}\right)\epsilon_{L}^{4}\,,$
(C.6) $\displaystyle\\#\,{\cal
O}_{10}^{\text{aLEFT}_{\cancel{\text{PQ}}}}\,=\,$
$\displaystyle\left(60+\frac{11435N_{d}^{2}}{18}-\frac{25N_{d}^{3}}{3}+\frac{18407N_{d}^{4}}{18}+\frac{10N_{d}^{5}}{3}+\frac{865N_{d}^{6}}{9}+452N_{d}^{3}N_{u}+3N_{d}^{4}N_{u}\right.$
$\displaystyle+113N_{d}^{5}N_{u}+\frac{1309N_{u}^{2}}{3}+\frac{3409}{2}N_{d}^{2}N_{u}^{2}+2N_{d}^{3}N_{u}^{2}+\frac{623}{2}N_{d}^{4}N_{u}^{2}+70N_{d}^{3}N_{u}^{3}$
$\displaystyle\left.+286N_{u}^{4}+148N_{d}^{2}N_{u}^{4}+\frac{35N_{u}^{6}}{3}\right)+\left(\frac{N_{d}N_{u}}{2}+\frac{97}{12}N_{d}^{2}N_{u}+\frac{3}{2}N_{d}^{3}N_{u}-\frac{1}{12}N_{d}^{4}N_{u}\right.$
$\displaystyle\left.+\frac{1}{2}N_{d}N_{u}^{2}+\frac{41}{12}N_{d}^{2}N_{u}^{2}-\frac{1}{2}N_{d}^{3}N_{u}^{2}+\frac{175}{12}N_{d}^{4}N_{u}^{2}\right)\epsilon_{B}^{2}+\left(-\frac{8N_{d}^{2}}{3}+13N_{d}^{3}\right.$
$\displaystyle\left.+178N_{d}^{4}-2N_{d}^{5}+\frac{113N_{d}^{6}}{3}+\frac{82}{3}N_{d}^{2}N_{u}+\frac{3331}{6}N_{d}^{3}N_{u}-\frac{9}{2}N_{d}^{4}N_{u}+148N_{d}^{5}N_{u}\right.$
$\displaystyle\left.+\frac{1123}{2}N_{d}^{2}N_{u}^{2}-3N_{d}^{3}N_{u}^{2}+\frac{471}{2}N_{d}^{4}N_{u}^{2}-\frac{17}{6}N_{d}^{2}N_{u}^{3}+\frac{665}{6}N_{d}^{3}N_{u}^{3}+70N_{d}^{2}N_{u}^{4}\right)\epsilon_{B}\epsilon_{L}$
$\displaystyle+\left(5N_{d}+\frac{175N_{d}^{2}}{3}-\frac{46N_{d}^{3}}{3}+\frac{641N_{d}^{4}}{3}+\frac{7N_{d}^{5}}{3}+36N_{d}^{6}+9N_{d}^{2}N_{u}+\frac{1357}{3}N_{d}^{3}N_{u}\right.$
$\displaystyle\left.+2N_{d}^{4}N_{u}+\frac{350}{3}N_{d}^{5}N_{u}-\frac{15}{2}N_{d}N_{u}^{2}+\frac{697}{6}N_{d}^{2}N_{u}^{2}+4N_{d}^{3}N_{u}^{2}+\frac{280}{3}N_{d}^{4}N_{u}^{2}\right.$
$\displaystyle\left.+70N_{d}^{3}N_{u}^{3}+\frac{1}{2}N_{d}N_{u}^{4}+\frac{19}{2}N_{d}^{2}N_{u}^{4}\right)\epsilon_{L}^{2}+\left(-\frac{2N_{d}^{3}}{3}-\frac{13N_{d}^{4}}{6}-\frac{N_{d}^{5}}{3}+\frac{19N_{d}^{6}}{6}\right.$
$\displaystyle\left.+\frac{2}{3}N_{d}^{2}N_{u}-\frac{13}{2}N_{d}^{3}N_{u}-\frac{11}{6}N_{d}^{4}N_{u}+\frac{11}{3}N_{d}^{5}N_{u}-\frac{3}{2}N_{d}^{2}N_{u}^{2}+\frac{19}{2}N_{d}^{4}N_{u}^{2}\right.$
$\displaystyle\left.+\frac{1}{6}N_{d}^{2}N_{u}^{3}+\frac{35}{6}N_{d}^{3}N_{u}^{3}\right)\epsilon_{B}\epsilon_{L}^{3}+\left(\frac{N_{d}}{2}+\frac{7N_{d}^{2}}{12}+\frac{N_{d}^{3}}{2}+\frac{N_{d}^{4}}{4}-N_{d}^{5}+\frac{7N_{d}^{6}}{6}\right.$
$\displaystyle\left.-\frac{8}{3}N_{d}^{3}N_{u}-N_{d}^{4}N_{u}+\frac{11}{3}N_{d}^{5}N_{u}+\frac{1}{2}N_{d}N_{u}^{2}-\frac{7}{12}N_{d}^{2}N_{u}^{2}-\frac{1}{2}N_{d}^{3}N_{u}^{2}+\frac{7}{12}N_{d}^{4}N_{u}^{2}\right)\epsilon_{L}^{4}$
$\displaystyle+\left(-\frac{N_{d}^{2}}{36}+\frac{N_{d}^{3}}{24}+\frac{N_{d}^{4}}{72}-\frac{N_{d}^{5}}{24}+\frac{N_{d}^{6}}{72}\right)\epsilon_{L}^{6}\,.$

##  Appendix D Details on the basis change from the derivative to the Yukawa
basis

The discussion about the shift symmetry in the presence of the EOM redundancy
at dimension-5 usually does not take the effect of the field redefinition on
higher order operators into consideration. Furthermore, the effect on ALP-
independent effective operators built from SM fields are also ignored. In this
section, we will study the effect of the field redefinition on those
operators.

###  D.1 ALP-dependent operators

We will first ignore the SMEFT operators and start with the full derivatively
coupled Lagrangian up to dimension-7, i.e., all operators in Tabs. 1-3. To
keep the discussion concise, we only show the calculations for one higher
order operator, while the calculations for the other operators follow in a
straightforward way. Furthermore, we ignore the bosonic operators here, as
they are irrelevant to the discussion. The first derivatively coupled
fermionic operators beyond the leading order appears at dimension-7.
Eventually, we consider the following Lagrangian

$\begin{split}{\cal
L}=&\sum_{\psi\in\text{SM}}\bar{\psi}i\not{D}\psi-\left(\bar{Q}Y_{u}\tilde{H}u+\bar{Q}Y_{d}Hd+\bar{L}Y_{e}He+\text{h.c.}\right)+\frac{\partial_{\mu}a}{f}\sum_{\psi\in\text{SM}}\bar{\psi}\gamma^{\mu}C_{\psi}\psi\\\
&+\frac{\partial^{\mu}a}{f^{3}}\left(\bar{L}\gamma^{\nu}C_{\partial
aLB}L\right)B_{\mu\nu}+\dots\end{split}$ (D.1)

where the dots collect all the other terms in the derivatively coupled EFT
that follow the same discussion. Redefining the fermion fields by
$\psi\to\exp\left(iC_{\psi}\frac{a}{f}\right)\psi$, trades the derivatively
coupled operators at dimension-5 for the ALP-Yukawa couplings but also
generates more operators at higher dimensions. We find

$\displaystyle{\cal
L}\to\sum_{\psi\in\text{SM}}\bar{\psi}i\not{D}\psi-\cancel{\frac{\partial_{\mu}a}{f}\sum_{\psi\in\text{SM}}\bar{\psi}\gamma^{\mu}C_{\psi}\psi}-\left(\bar{Q}e^{-iC_{Q}\frac{a}{f}}Y_{u}e^{iC_{u}\frac{a}{f}}\tilde{H}u+\bar{Q}e^{-iC_{Q}\frac{a}{f}}Y_{d}e^{iC_{d}\frac{a}{f}}Hd\right.$
(D.2)
$\displaystyle\left.+\bar{L}e^{-iC_{L}\frac{a}{f}}Y_{e}e^{iC_{e}\frac{a}{f}}He+\text{h.c.}\right)+\cancel{\frac{\partial_{\mu}a}{f}\sum_{\psi\in\text{SM}}\bar{\psi}\gamma^{\mu}C_{\psi}\psi}+\frac{\partial^{\mu}a}{f^{3}}\left(\bar{L}e^{-iC_{L}\frac{a}{f}}\gamma^{\nu}C_{\partial
aLB}e^{iC_{L}\frac{a}{f}}L\right)B_{\mu\nu}+\dots$

Expanding these exponentials in the SM Yukawa couplings to leading order
yields the usual relations at dimension-5. However, we will also study how
they alter the dimension-7 operators.

Focusing only on the leptonic terms, we have after expanding the exponentials

$\displaystyle{\cal L}\to$
$\displaystyle\sum_{\psi\in\text{SM}}\bar{\psi}i\not{D}\psi-\bar{L}\left[Y_{e}+\frac{a}{f}i\left(Y_{e}C_{e}-C_{L}Y_{e}\right)+\frac{a^{2}}{f^{2}}\left(C_{L}Y_{e}C_{e}-\frac{1}{2}\left(C_{L}^{2}Y_{e}+Y_{e}C_{e}^{2}\right)\right)+\dots\right]He$
$\displaystyle+\frac{\partial^{\mu}a}{f^{3}}\left(\bar{L}\gamma^{\nu}\left[C_{\partial
aLB}+\frac{a}{f}i\left(C_{\partial aLB}C_{L}-C_{L}C_{\partial
aLB}\right)+\frac{a^{2}}{f^{2}}\left(C_{L}C_{\partial
aLB}C_{L}\vphantom{\frac{1}{2}}\right.\right.\right.$ (D.3)
$\displaystyle\left.\left.\left.\qquad\qquad\qquad-\frac{1}{2}\left(C_{L}^{2}C_{\partial
aLB}+C_{\partial
aLB}C_{L}^{2}\right)\right)+\dots\right]L\right)B_{\mu\nu}+\dots\,.$

Notice that expanding the exponential introduces more shift-breaking
interactions beyond what is usually shown in the literature. We have
constructed these terms previously with the help of the PQ-breaking isolation
condition in App. A.2. As for the dimension-5 Yukawa couplings, the Wilson
coefficients of those shift-breaking operators have to fulfill relations
dictated by the exponentiated form of the ALP interactions. We find for the
operators shown in the Lagrangian

$\begin{split}C_{ae}&=i\left(C_{L}Y_{e}-Y_{e}C_{e}\right)\,,\\\
C_{a^{2}e}&=\left(\frac{1}{2}\left(C_{L}^{2}Y_{e}+Y_{e}C_{e}^{2}\right)-C_{L}Y_{e}C_{e}\right)\,,\\\
C_{a\partial aLB}&=i\left(C_{\partial aLB}C_{L}-C_{L}C_{\partial
aLB}\right)\,,\\\ C_{a^{2}\partial aLB}&=\left(C_{L}C_{\partial
aLB}C_{L}-\frac{1}{2}\left(C_{L}^{2}C_{\partial aLB}+C_{\partial
aLB}C_{L}^{2}\right)\right)\,,\end{split}$ (D.4)

where we have used the notation introduced in App. A.2 for the shift-breaking
operators.

Note that the parameter counting in the EFT before and after the field
redefinition is still consistent after including the higher order operators as
well. The relations for the dimension-5 ALP-Yukawa operators remove exactly
the difference in physical parameters between the ALP-Yukawa and the
derivatively coupled basis as was shown in Ref. Bonnefoy _et al._ (2023). At
higher mass dimension, the relations fully saturate the freedom in the Wilson
coefficients of the shift-breaking operators and no new parameters are added,
as expected. This will always happen, since the exponential generates
interactions proportional to the Wilson coefficient of the shift-symmetric
operator that is affected by the chiral rotation. The main difference to the
dimension-5 case is that, there, the derivatively coupled operator and the
operator generated by the field redefinition are connected by an EOM
redundancy. Then, the field content of the operator is changed by the field
redefinition and a different amount of degrees of freedom is captured if the
relations are disregarded.151515To be precise, the same is true for the
dimension-5 couplings if one counts the physical parameters in the SM Yukawa
couplings as well. Then, one can also explain the CP parities of the flavor
invariants in Ref. Bonnefoy _et al._ (2023); there are 3 CP-odd relations in
the lepton sector and 9 CP-odd relations and 1 CP-even relations in the quark
sector corresponding to the numbers of physical parameters in the SM Yukawas.
The CP parities are flipped here with respect to the SM parameters because the
ALP is assigned to be odd under parity. This however does not imply that
imposing CP on the full theory (i.e. the vanishing of the CKM phase implies a
vanishing of the single CP-even relation in the ALP EFT) makes all order
parameters of shift-symmetry vanish. This is because the CP-even parameter in
the Wilson coefficient of the dimension-5 operator is independent and the
rephasing properties of the CKM matrix are unchanged by setting the CKM phase
to zero. Hence, the single CP-even invariant remains untouched.

In order to have a shift-symmetric ALP EFT in the Yukawa basis beyond
dimension-5, one needs to include these additional interactions with their
constrained Wilson coefficients. Otherwise one will run into shift-breaking
results while doing computations. Only when all the additional diagrams from
the operators generated by the field redefinitions are considered, one will
recover a shift-symmetric result.

In particular, if one allows for shift-breaking effects and wants to
understand the shift-symmetric limit it is practical to work in the Yukawa
basis instead of the basis with the derivatively coupled fermionic operators
at dimension-5. Then, to take the shift-symmetric limit, on has to impose the
relations discussed in this section on the generic Wilson coefficients of the
shift-breaking operators in order to get consistent results.

###  D.2 SMEFT operators

The ALP-dependent chiral transformation also affects SMEFT operators,
introducing a new source of ALP-dependent operators. We will give some
examples here, working with the following Lagrangian161616For simplicity we
have taken the operators with and without an ALP to be suppressed by the same
UV scale $f$. Depending on the structure of the UV theory and the details of
PQ-breaking the operators can also come with a suppression of different scales
corresponding to different UV sectors.

$\displaystyle{\cal L}_{\text{SMEFT}}=$
$\displaystyle\frac{1}{f^{2}}|H|^{2}\bar{L}C_{eH}He+\frac{1}{f^{2}}C_{lequ,ijkl}^{(1)}\left(\bar{L}_{i}e_{j}\right)\epsilon\left(\bar{Q}_{k}u_{l}\right)$
$\displaystyle\longrightarrow$
$\displaystyle\frac{|H|^{2}}{f^{2}}\bar{L}\left[C_{eH}+\frac{a}{f}i\left(C_{eH}C_{e}-C_{L}C_{eH}\right)+\frac{a^{2}}{f^{2}}\left(C_{L}C_{eH}C_{e}-\frac{1}{2}\left(C_{L}^{2}C_{eH}+C_{eH}C_{e}^{2}\right)\right)\right]He$
$\displaystyle+\frac{1}{f^{2}}\left[C_{lequ,ijkl}^{(1)}+\frac{a}{f}i\left(C_{lequ,ij^{\prime}kl}^{(1)}C_{e,j^{\prime}j}+C_{lequ,ijkl^{\prime}}^{(1)}C_{u,l^{\prime}l}-C_{lequ,i^{\prime}jkl}^{(1)}C_{L,ii^{\prime}}-C_{lequ,ijk^{\prime}l}^{(1)}C_{Q,kk^{\prime}}\right)\right.$
$\displaystyle\left.+\frac{a^{2}}{f^{2}}\left(C_{lequ,i^{\prime}j^{\prime}kl}^{(1)}C_{L,ii^{\prime}}C_{e,j^{\prime}j}-C_{lequ,i^{\prime}jk^{\prime}l}^{(1)}C_{L,ii^{\prime}}C_{Q,kk^{\prime}}+C_{lequ,i^{\prime}jkl^{\prime}}^{(1)}C_{L,ii^{\prime}}C_{u,l^{\prime}l}\right.\right.$
$\displaystyle\left.\left.+C_{lequ,ij^{\prime}k^{\prime}l}^{(1)}C_{e,j^{\prime}j}C_{Q,kk^{\prime}}-C_{lequ,ij^{\prime}kl^{\prime}}^{(1)}C_{e,j^{\prime}j}C_{u,l^{\prime}l}+C_{lequ,ijk^{\prime}l^{\prime}}^{(1)}C_{Q,kk^{\prime}}C_{u,l^{\prime}l}\right.\right.$
$\displaystyle\left.\left.-\frac{1}{2}\left(C_{lequ,i^{\prime}jkl}^{(1)}(C_{L}^{2})_{ii^{\prime}}+C_{lequ,ij^{\prime}kl}^{(1)}(C_{e}^{2})_{j^{\prime}j}+C_{lequ,ijk^{\prime}l}^{(1)}(C_{Q}^{2})_{kk^{\prime}}+C_{lequ,ijkl^{\prime}}^{(1)}(C_{u}^{2})_{l^{\prime}l}\right)\right)\vphantom{\frac{1}{f^{2}}}\right]\times$
$\displaystyle\times\left(\bar{L}_{i}e_{j}\right)\epsilon\left(\bar{Q}_{k}u_{l}\right)\,.$

Comparing to the generic shift-breaking Lagrangian, similar relations are
found as before. They become more and more complicated as more fermions appear
in the operators. One can simply read of the relations from the Lagrangian and
we will not give them explicitly again.

Instead of starting with the derivatively coupled EFT, one can also start at
the opposite end and consider the $\text{aSMEFT}_{\cancel{\text{PQ}}}$ which
is constructed without imposing a shift symmetry. Then, shifting the ALP $a\to
a+c$ and demanding that the shift $c$ vanishes in the EFT after performing
field redefinitions, while staying in the same operator basis, leads to the
same relations as we have just derived. We have checked this explicitly with
our operator basis up to dimension-8. To this end, a field redefinition should
be used that allows to remove the shift $c$ in the Lagrangian while keeping
the kinetic terms of the fermions invariant and at the same time not
generating new operators outside of the operator basis we start with. The only
such transformation is given by redefining the fermion fields with powers of
the shift $\frac{c}{f}$ as
$\psi\to\psi+i\sum_{k=1}^{\infty}c_{\psi}^{(k)}\left(\frac{c}{f}\right)^{k}\psi$,
where the $c_{\psi}^{(k)}$ are generic hermitian matrices. We keep the terms
in this expansion up to the order that is relevant for the EFT expansion in
each step of the discussion. To get consistent relations for all terms that
are proportional to the shift for operators with more than one power of an
ALP, the following choice for the coefficients in the field redefinition
$c_{\psi}^{(n)}=i^{n-1}C_{\psi}/(n!)$ has to be made, i.e.
$\psi\to\exp\left(iC_{\psi}\frac{c}{f}\right)\psi$. This is reminiscent of the
chiral transformation we have started with in the purely derivatively coupled
EFT.

###  D.3 List of additional relations in Yukawa basis

In this appendix we list all the constrained Wilson coefficients of operators
that have to be added in the Yukawa basis up to dimension 8. Here, we only
restrict to those operators which already have an ALP field before performing
the chiral rotation, i.e., we ignore the contributions from the SMEFT
operators due to their length. Those relations can be straightforwardly
constructed as shown in the previous section.

At dimension-5, we find the well-known relations

$C_{ae}=i\left(C_{L}Y_{e}-Y_{e}C_{e}\right),\quad
C_{au}=i\left(C_{Q}Y_{u}-Y_{u}C_{u}\right),\quad
C_{ad}=i\left(C_{Q}Y_{d}-Y_{d}C_{d}\right)\,.$ (D.5)

Since at dimension-6 the only existing operator is bosonic, the only relations
at this mass-dimension come again from the Yukawa-like operators. They read

$\begin{split}C_{a^{2}e}=&\left(\frac{1}{2}\left(C_{L}^{2}Y_{e}+Y_{e}C_{e}^{2}\right)-C_{L}Y_{e}C_{e}\right),\quad
C_{a^{2}u}=\left(\frac{1}{2}\left(C_{Q}^{2}Y_{u}+Y_{u}C_{u}^{2}\right)-C_{Q}Y_{u}C_{u}\right),\\\
&\qquad\qquad\qquad
C_{a^{2}d}=\left(\frac{1}{2}\left(C_{Q}^{2}Y_{d}+Y_{d}C_{d}^{2}\right)-C_{Q}Y_{d}C_{d}\right)\,.\end{split}$
(D.6)

The same is true at dimension-7

$\begin{split}C_{a^{3}e}&=\frac{i}{6}\left(Y_{e}C_{e}^{3}-C_{L}^{3}Y_{e}\right)+\frac{i}{2}\left(C_{L}^{2}Y_{e}C_{e}-C_{L}Y_{e}C_{e}^{2}\right),\\\
C_{a^{3}u}&=\frac{i}{6}\left(Y_{u}C_{u}^{3}-C_{Q}^{3}Y_{u}\right)+\frac{i}{2}\left(C_{Q}^{2}Y_{u}C_{u}-C_{Q}Y_{u}C_{u}^{2}\right),\\\
C_{a^{3}d}&=\frac{i}{6}\left(Y_{d}C_{d}^{3}-C_{Q}^{3}Y_{d}\right)+\frac{i}{2}\left(C_{Q}^{2}Y_{d}C_{d}-C_{Q}Y_{d}C_{d}^{2}\right)\,.\end{split}$
(D.7)

Only at dimension-8 there are relations introduced by new fermionic operators
at dimension-7 reading as follows

$\begin{split}C_{a^{4}e}&=\frac{1}{6}\left(C_{L}^{3}Y_{e}C_{e}+C_{L}Y_{e}C_{e}^{3}\right)-\frac{1}{4}C_{L}^{2}Y_{e}C_{e}^{2}-\frac{1}{24}\left(C_{L}^{4}Y_{e}+Y_{e}C_{e}^{4}\right)\,,\\\
C_{a^{4}u}&=\frac{1}{6}\left(C_{Q}^{3}Y_{u}C_{u}+C_{Q}Y_{u}C_{u}^{3}\right)-\frac{1}{4}C_{Q}^{2}Y_{u}C_{u}^{2}-\frac{1}{24}\left(C_{Q}^{4}Y_{u}+Y_{u}C_{u}^{4}\right)\,,\\\
C_{a^{4}d}&=\frac{1}{6}\left(C_{Q}^{3}Y_{d}C_{d}+C_{Q}Y_{d}C_{d}^{3}\right)-\frac{1}{4}C_{Q}^{2}Y_{d}C_{d}^{2}-\frac{1}{24}\left(C_{Q}^{4}Y_{d}+Y_{d}C_{d}^{4}\right)\,,\\\
C_{a\partial aeHD}^{(1,2)}&=i\left(C_{\partial
aeHD}^{(1,2)}C_{e}-C_{L}C_{\partial aeHD}^{(1,2)}\right)\,,\\\ C_{a\partial
auHD}^{(1,2)}&=i\left(C_{\partial auHD}^{(1,2)}C_{u}-C_{Q}C_{\partial
auHD}^{(1,2)}\right)\,,\\\ C_{a\partial adHD}^{(1,2)}&=i\left(C_{\partial
adHD}^{(1,2)}C_{d}-C_{L}C_{\partial adHD}^{(1,2)}\right)\,,\\\ C_{a\partial
a\psi H^{2}}&=i\left(C_{\partial a\psi H^{2}}C_{\psi}-C_{\psi}C_{\partial
a\psi H^{2}}\right)\,,\\\ C_{a\partial a\psi V}&=i\left(C_{\partial a\psi
V}C_{\psi}-C_{\psi}C_{\partial a\psi V}\right)\,.\end{split}$ (D.8)

For the last relation, the same relation holds true for the operators with the
dual field strength. Furthermore, some shift-symmetric operator at dimension-8
get shifted as follows

$\begin{split}C_{\partial a^{2}eH}&\to C_{\partial a^{2}eH}+i\left(C_{\partial
aeHD}^{(2)}C_{e}-C_{L}C_{\partial aeHD}^{(1)}\right),\\\ C_{\partial
a^{2}uH}&\to C_{\partial a^{2}uH}+i\left(C_{\partial
auHD}^{(2)}C_{u}-C_{Q}C_{\partial auHD}^{(1)}\right),\\\ C_{\partial
a^{2}dH}&\to C_{\partial a^{2}dH}+i\left(C_{\partial
adHD}^{(2)}C_{d}-C_{Q}C_{\partial adHD}^{(1)}\right),\\\ \end{split}$ (D.9)

due to the derivative acting on the fermions in the operator corresponding to
the last relation in the previous equation.

## References

  * Peccei and Quinn (1977a) R. D. Peccei and H. R. Quinn, _CP Conservation in the Presence of Instantons_ , Phys. Rev. Lett. 38 (1977a) 1440.
  * Peccei and Quinn (1977b) R. D. Peccei and H. R. Quinn, _Constraints Imposed by CP Conservation in the Presence of Instantons_ , Phys. Rev. D 16 (1977b) 1791.
  * Weinberg (1978) S. Weinberg, _A New Light Boson?_ , Phys. Rev. Lett. 40 (1978) 223.
  * Wilczek (1978) F. Wilczek, _Problem of Strong $P$ and $T$ Invariance in the Presence of Instantons_, Phys. Rev. Lett. 40 (1978) 279.
  * Kim (1979) J. E. Kim, _Weak Interaction Singlet and Strong CP Invariance_ , Phys. Rev. Lett. 43 (1979) 103.
  * Shifman _et al._ (1980) M. A. Shifman, A. Vainshtein, and V. I. Zakharov, _Can Confinement Ensure Natural CP Invariance of Strong Interactions?_ , Nucl. Phys. B 166 (1980) 493.
  * Dine _et al._ (1981) M. Dine, W. Fischler, and M. Srednicki, _A Simple Solution to the Strong CP Problem with a Harmless Axion_ , Phys. Lett. B 104 (1981) 199.
  * Zhitnitsky (1980) A. Zhitnitsky, _On Possible Suppression of the Axion Hadron Interactions. (In Russian)_ , Sov. J. Nucl. Phys. 31 (1980) 260.
  * Preskill _et al._ (1983) J. Preskill, M. B. Wise, and F. Wilczek, _Cosmology of the Invisible Axion_ , Phys. Lett. B 120 (1983) 127.
  * Abbott and Sikivie (1983) L. F. Abbott and P. Sikivie, _A Cosmological Bound on the Invisible Axion_ , Phys. Lett. B 120 (1983) 133.
  * Dine and Fischler (1983) M. Dine and W. Fischler, _The Not So Harmless Axion_ , Phys. Lett. B 120 (1983) 137.
  * Marsh (2016) D. J. E. Marsh, _Axion Cosmology_ , Phys. Rept. 643 (2016) 1, arXiv:1510.07633 [astro-ph.CO].
  * Di Luzio _et al._ (2020) L. Di Luzio, M. Giannotti, E. Nardi, and L. Visinelli, _The landscape of QCD axion models_ , Phys. Rept. 870 (2020) 1, arXiv:2003.01100 [hep-ph].
  * Hawking (1987) S. W. Hawking, _Quantum Coherence Down the Wormhole_ , Phys. Lett. B 195 (1987) 337.
  * Giddings and Strominger (1988) S. B. Giddings and A. Strominger, _Loss of Incoherence and Determination of Coupling Constants in Quantum Gravity_ , Nucl. Phys. B 307 (1988) 854.
  * Kamionkowski and March-Russell (1992) M. Kamionkowski and J. March-Russell, _Planck scale physics and the Peccei-Quinn mechanism_ , Phys. Lett. B 282 (1992) 137, arXiv:hep-th/9202003.
  * Banks and Seiberg (2011) T. Banks and N. Seiberg, _Symmetries and Strings in Field Theory and Gravity_ , Phys. Rev. D 83 (2011) 084019, arXiv:1011.5120 [hep-th].
  * Bonnefoy (2022) Q. Bonnefoy, _Heavy fields and the axion quality problem_ , (2022), arXiv:2212.00102 [hep-ph].
  * Graham _et al._ (2015) P. W. Graham, D. E. Kaplan, and S. Rajendran, _Cosmological Relaxation of the Electroweak Scale_ , Phys. Rev. Lett. 115 (2015) 221801, arXiv:1504.07551 [hep-ph].
  * Espinosa _et al._ (2015) J. R. Espinosa, C. Grojean, G. Panico, A. Pomarol, O. Pujolàs, and G. Servant, _Cosmological Higgs-Axion Interplay for a Naturally Small Electroweak Scale_ , Phys. Rev. Lett. 115 (2015) 251803, arXiv:1506.09217 [hep-ph].
  * Franceschini _et al._ (2016) R. Franceschini, G. F. Giudice, J. F. Kamenik, M. McCullough, F. Riva, A. Strumia, and R. Torre, _Digamma, what next?_ , JHEP 07 (2016) 150, arXiv:1604.06446 [hep-ph].
  * Srednicki (1985) M. Srednicki, _Axion Couplings to Matter. 1. CP Conserving Parts_ , Nucl. Phys. B 260 (1985) 689.
  * Georgi _et al._ (1986) H. Georgi, D. B. Kaplan, and L. Randall, _Manifesting the Invisible Axion at Low-energies_ , Phys. Lett. B 169 (1986) 73.
  * Bonnefoy _et al._ (2023) Q. Bonnefoy, C. Grojean, and J. Kley, _Shift-Invariant Orders of an Axionlike Particle_ , Phys. Rev. Lett. 130 (2023) 111803, arXiv:2206.04182 [hep-ph].
  * Draper and McKeen (2012) P. Draper and D. McKeen, _Diphotons from Tetraphotons in the Decay of a 125 GeV Higgs at the LHC_ , Phys. Rev. D 85 (2012) 115023, arXiv:1204.1061 [hep-ph].
  * Bauer _et al._ (2017) M. Bauer, M. Neubert, and A. Thamm, _Collider Probes of Axion-Like Particles_ , JHEP 12 (2017) 044, arXiv:1708.00443 [hep-ph].
  * Bauer _et al._ (2019) M. Bauer, M. Heiles, M. Neubert, and A. Thamm, _Axion-Like Particles at Future Colliders_ , Eur. Phys. J. C 79 (2019) 74, arXiv:1808.10323 [hep-ph].
  * Davoudiasl _et al._ (2021) H. Davoudiasl, R. Marcarelli, N. Miesch, and E. T. Neil, _Searching for flavor-violating ALPs in Higgs boson decays_ , Phys. Rev. D 104 (2021) 055022, arXiv:2105.05866 [hep-ph].
  * Brivio _et al._ (2021) I. Brivio, O. J. P. Éboli, and M. C. Gonzalez-Garcia, _Unitarity constraints on ALP interactions_ , Phys. Rev. D 104 (2021) 035027, arXiv:2106.05977 [hep-ph].
  * Bauer _et al._ (2016) M. Bauer, M. Neubert, and A. Thamm, _Analyzing the CP Nature of a New Scalar Particle via S- $>$Zh Decay_, Phys. Rev. Lett. 117 (2016) 181801, arXiv:1610.00009 [hep-ph].
  * Quevillon _et al._ (2022) J. Quevillon, C. Smith, and P. N. H. Vuong, _Axion effective action_ , JHEP 08 (2022) 137, arXiv:2112.00553 [hep-ph].
  * Björkeroth _et al._ (2018) F. Björkeroth, E. J. Chun, and S. F. King, _Flavourful Axion Phenomenology_ , JHEP 08 (2018) 117, arXiv:1806.00660 [hep-ph].
  * Bauer _et al._ (2021a) M. Bauer, M. Neubert, S. Renner, M. Schnubel, and A. Thamm, _The Low-Energy Effective Theory of Axions and ALPs_ , JHEP 04 (2021a) 063, arXiv:2012.12272 [hep-ph].
  * Martin Camalich _et al._ (2020) J. Martin Camalich, M. Pospelov, P. N. H. Vuong, R. Ziegler, and J. Zupan, _Quark Flavor Phenomenology of the QCD Axion_ , Phys. Rev. D 102 (2020) 015023, arXiv:2002.04623 [hep-ph].
  * Calibbi _et al._ (2021) L. Calibbi, D. Redigolo, R. Ziegler, and J. Zupan, _Looking forward to lepton-flavor-violating ALPs_ , JHEP 09 (2021) 173, arXiv:2006.04795 [hep-ph].
  * Bauer _et al._ (2022) M. Bauer, M. Neubert, S. Renner, M. Schnubel, and A. Thamm, _Flavor probes of axion-like particles_ , JHEP 09 (2022) 056, arXiv:2110.10698 [hep-ph].
  * Lehman and Martin (2015) L. Lehman and A. Martin, _Hilbert Series for Constructing Lagrangians: expanding the phenomenologist’s toolbox_ , Phys. Rev. D 91 (2015) 105014, arXiv:1503.07537 [hep-ph].
  * Lehman and Martin (2016) L. Lehman and A. Martin, _Low-derivative operators of the Standard Model effective field theory via Hilbert series methods_ , JHEP 02 (2016) 081, arXiv:1510.00372 [hep-ph].
  * Henning _et al._ (2016) B. Henning, X. Lu, T. Melia, and H. Murayama, _Hilbert series and operator bases with derivatives in effective field theories_ , Commun. Math. Phys. 347 (2016) 363, arXiv:1507.07240 [hep-th].
  * Henning _et al._ (2017a) B. Henning, X. Lu, T. Melia, and H. Murayama, _2, 84, 30, 993, 560, 15456, 11962, 261485, …: Higher dimension operators in the SM EFT_ , JHEP 08 (2017a) 016, [Erratum: JHEP 09, 019 (2019)], arXiv:1512.03433 [hep-ph].
  * Henning _et al._ (2017b) B. Henning, X. Lu, T. Melia, and H. Murayama, _Operator bases, $S$-matrices, and their partition functions_, JHEP 10 (2017b) 199, arXiv:1706.08520 [hep-th].
  * Kobach and Pal (2017) A. Kobach and S. Pal, _Hilbert Series and Operator Basis for NRQED and NRQCD/HQET_ , Phys. Lett. B 772 (2017) 225, arXiv:1704.00008 [hep-ph].
  * Ruhdorfer _et al._ (2020) M. Ruhdorfer, J. Serra, and A. Weiler, _Effective Field Theory of Gravity to All Orders_ , JHEP 05 (2020) 083, arXiv:1908.08050 [hep-ph].
  * Marinissen _et al._ (2020) C. B. Marinissen, R. Rahn, and W. J. Waalewijn, _…, 83106786, 114382724, 1509048322, 2343463290, 27410087742, … efficient Hilbert series for effective theories_ , Phys. Lett. B 808 (2020) 135632, arXiv:2004.09521 [hep-ph].
  * Graf _et al._ (2021) L. Graf, B. Henning, X. Lu, T. Melia, and H. Murayama, _2, 12, 117, 1959, 45171, 1170086, …: a Hilbert series for the QCD chiral Lagrangian_ , JHEP 01 (2021) 142, arXiv:2009.01239 [hep-ph].
  * Kondo _et al._ (2023) D. Kondo, H. Murayama, and R. Okabe, _23, 381, 6242, 103268, 1743183, … : Hilbert series for CP-violating operators in SMEFT_ , JHEP 03 (2023) 107, arXiv:2212.02413 [hep-ph].
  * Gráf _et al._ (2023) L. Gráf, B. Henning, X. Lu, T. Melia, and H. Murayama, _Hilbert series, the Higgs mechanism, and HEFT_ , JHEP 02 (2023) 064, arXiv:2211.06275 [hep-ph].
  * Sun _et al._ (2022) H. Sun, Y.-N. Wang, and J.-H. Yu, _Hilbert Series and Operator Counting on the Higgs Effective Field Theory_ , (2022), arXiv:2211.11598 [hep-ph].
  * Bijnens _et al._ (2022) J. Bijnens, S. B. Gudnason, J. Yu, and T. Zhang, _Hilbert series and higher-order Lagrangians for the $O(N)$ model_, (2022), arXiv:2212.07901 [hep-th].
  * Feng _et al._ (2007) B. Feng, A. Hanany, and Y.-H. He, _Counting gauge invariants: The Plethystic program_ , JHEP 03 (2007) 090, arXiv:hep-th/0701063.
  * Arzt (1995) C. Arzt, _Reduced effective Lagrangians_ , Phys. Lett. B 342 (1995) 189, arXiv:hep-ph/9304230.
  * Coleman _et al._ (1969) S. R. Coleman, J. Wess, and B. Zumino, _Structure of phenomenological Lagrangians. 1._ , Phys. Rev. 177 (1969) 2239.
  * Callan _et al._ (1969) C. G. Callan, Jr., S. R. Coleman, J. Wess, and B. Zumino, _Structure of phenomenological Lagrangians. 2._ , Phys. Rev. 177 (1969) 2247.
  * (54) C. Grojean, J. Kley, and C.-Y. Yao, _CHINCHILLA: A Mathematica package for the construction of invariants using the Hilbert series_ , To appear, 2023.
  * Bonilla _et al._ (2021) J. Bonilla, I. Brivio, M. B. Gavela, and V. Sanz, _One-loop corrections to ALP couplings_ , JHEP 11 (2021) 168, arXiv:2107.11392 [hep-ph].
  * Chala _et al._ (2022) M. Chala, A. Díaz-Carmona, and G. Guedes, _A Green’s basis for the bosonic SMEFT to dimension 8_ , JHEP 05 (2022) 138, arXiv:2112.12724 [hep-ph].
  * Weinberg (1979) S. Weinberg, _Baryon and Lepton Nonconserving Processes_ , Phys. Rev. Lett. 43 (1979) 1566.
  * Grzadkowski _et al._ (2010) B. Grzadkowski, M. Iskrzynski, M. Misiak, and J. Rosiek, _Dimension-Six Terms in the Standard Model Lagrangian_ , JHEP 10 (2010) 085, arXiv:1008.4884 [hep-ph].
  * Lehman (2014) L. Lehman, _Extending the Standard Model Effective Field Theory with the Complete Set of Dimension-7 Operators_ , Phys. Rev. D 90 (2014) 125023, arXiv:1410.4193 [hep-ph].
  * Murphy (2020) C. W. Murphy, _Dimension-8 operators in the Standard Model Eective Field Theory_ , JHEP 10 (2020) 174, arXiv:2005.00059 [hep-ph].
  * Brivio _et al._ (2017) I. Brivio, M. B. Gavela, L. Merlo, K. Mimasu, J. M. No, R. del Rey, and V. Sanz, _ALPs Effective Field Theory and Collider Signatures_ , Eur. Phys. J. C 77 (2017) 572, arXiv:1701.05379 [hep-ph].
  * Melia and Pal (2021) T. Melia and S. Pal, _EFT Asymptotics: the Growth of Operator Degeneracy_ , SciPost Phys. 10 (2021) 104, arXiv:2010.08560 [hep-th].
  * Chala _et al._ (2021) M. Chala, G. Guedes, M. Ramos, and J. Santiago, _Running in the ALPs_ , Eur. Phys. J. C 81 (2021) 181, arXiv:2012.09017 [hep-ph].
  * Gripaios and Sutherland (2016) B. Gripaios and D. Sutherland, _An operator basis for the Standard Model with an added scalar singlet_ , JHEP 08 (2016) 103, arXiv:1604.07365 [hep-ph].
  * (65) E. Bertuzzo, C. Grojean, and G. M. Salla, _ALPs, the on-shell way_ , To appear, 2023.
  * Adler (1965a) S. L. Adler, _Consistency conditions on the strong interactions implied by a partially conserved axial vector current_ , Phys. Rev. 137 (1965a) B1022.
  * Adler (1965b) S. L. Adler, _Consistency conditions on the strong interactions implied by a partially conserved axial-vector current. II_ , Phys. Rev. 139 (1965b) B1638.
  * Bonnefoy _et al._ (2021) Q. Bonnefoy, L. Di Luzio, C. Grojean, A. Paul, and A. N. Rossia, _The anomalous case of axion EFTs and massive chiral gauge fields_ , JHEP 07 (2021) 189, arXiv:2011.10025 [hep-ph].
  * Altmannshofer _et al._ (2022) W. Altmannshofer, J. A. Dror, and S. Gori, _New Insights Into Axion-Lepton Interactions_ , (2022), arXiv:2209.00665 [hep-ph].
  * Grilli di Cortona _et al._ (2016) G. Grilli di Cortona, E. Hardy, J. Pardo Vega, and G. Villadoro, _The QCD axion, precisely_ , JHEP 01 (2016) 034, arXiv:1511.02867 [hep-ph].
  * Bauer _et al._ (2021b) M. Bauer, M. Neubert, S. Renner, M. Schnubel, and A. Thamm, _Consistent Treatment of Axions in the Weak Chiral Lagrangian_ , Phys. Rev. Lett. 127 (2021b) 081803, arXiv:2102.13112 [hep-ph].
  * Liao and Liu (2012) Y. Liao and J.-Y. Liu, _Generalized Fierz Identities and Applications to Spin-3/2 Particles_ , Eur. Phys. J. Plus 127 (2012) 121, arXiv:1206.5141 [hep-ph].
  * Liao and Ma (2017) Y. Liao and X.-D. Ma, _Operators up to Dimension Seven in Standard Model Effective Field Theory Extended with Sterile Neutrinos_ , Phys. Rev. D 96 (2017) 015012, arXiv:1612.04527 [hep-ph].
  * Jenkins _et al._ (2018) E. E. Jenkins, A. V. Manohar, and P. Stoffer, _Low-Energy Effective Field Theory below the Electroweak Scale: Operators and Matching_ , JHEP 03 (2018) 016, arXiv:1709.04486 [hep-ph].
  * Liao _et al._ (2020) Y. Liao, X.-D. Ma, and Q.-Y. Wang, _Extending low energy effective field theory with a complete set of dimension-7 operators_ , JHEP 08 (2020) 162, arXiv:2005.08013 [hep-ph].
  * Murphy (2021) C. W. Murphy, _Low-Energy Effective Field Theory below the Electroweak Scale: Dimension-8 Operators_ , JHEP 04 (2021) 101, arXiv:2012.13291 [hep-ph].
  * Kim _et al._ (2023) S.-S. Kim, H. M. Lee, and K. Yamashita, _Positivity Bounds on Higgs-Portal Dark Matter_ , (2023), arXiv:2302.02879 [hep-ph].
  * Das Bakshi _et al._ (2023) S. Das Bakshi, J. Machado-Rodríguez, and M. Ramos, _Running beyond ALPs: shift-breaking and CP-violating effects_ , (2023), arXiv:2306.08036 [hep-ph].
  * Song _et al._ (2023a) H. Song, H. Sun, and J.-H. Yu, _Effective Field Theories of Axion, ALP and Dark Photon_ , (2023a), arXiv:2305.16770 [hep-ph].
  * Song _et al._ (2023b) H. Song, H. Sun, and J.-H. Yu, _Complete EFT Operator Bases for Dark Matter and Weakly-Interacting Light Particle_ , (2023b), arXiv:2306.05999 [hep-ph].