Fourth order

At fourth order, imposing either Galilean or gauge symmetry forces 12 dependent coupling constants to be specific linear combinations of 3 independent ones:

$$\bm{C_{00,4000}^{0000}} = \tfrac{3 }{2 \sqrt{5}}\bm{C_{00,2202}^{2202}} ,$$ (126)

$$\bm{C_{00,2000}^{2000}} = \tfrac{1}{2} \sqrt{5} \bm{C_{00,2202}^{2202}} ,$$ (127)

$${}{C_{00,3101}^{1101}} = -\tfrac{6 }{\sqrt{5}} \bm{C_{00,2202}^{2202}} ,$$ (128)

$$\bm{C_{00,3110}^{1110}} = -2 \sqrt{\tfrac{3}{5}} {}{C_{00,2212}^{2212}}-\tfrac{7 }{\sqrt{5}}{}{C_{00,4211}^{0011}} ,$$ (129)

$$\bm{C_{00,3111}^{1111}} = -\tfrac{6 }{\sqrt{5}}{}{C_{00,2212}^{2212}} ,$$ (130)

$$\bm{C_{00,3112}^{1112}} = -2 \sqrt{3} {}{C_{00,2212}^{2212}}-\tfrac{14 }{5}{}{C_{00,4211}^{0011}} ,$$ (131)

$$\bm{C_{00,3312}^{1112}} = -2 \sqrt{\tfrac{7}{15}} {}{C_{00,4211}^{0011}} ,$$ (132)

$${}{C_{00,4011}^{0011}} = \tfrac{3}{2} \sqrt{\tfrac{3}{5}} {}{C_{00,2212}^{2212}}+\tfrac{7 }{4 \sqrt{5}}{}{C_{00,4211}^{0011}} ,$$ (133)

$${}{C_{00,2011}^{2011}} = \tfrac{1}{2} \sqrt{15} {}{C_{00,2212}^{2212}}+\tfrac{7}{12} \sqrt{5} {}{C_{00,4211}^{0011}} ,$$ (134)

$${}{C_{00,2211}^{2011}} = \tfrac{7 }{3}{}{C_{00,4211}^{0011}} ,$$ (135)

$${}{C_{00,2211}^{2211}} = \sqrt{\tfrac{3}{5}} {}{C_{00,2212}^{2212}}+\tfrac{7 }{3 \sqrt{5}}{}{C_{00,4211}^{0011}} ,$$ (136)

$${}{C_{00,2213}^{2213}} = \sqrt{\tfrac{7}{5}} {}{C_{00,2212}^{2212}}+\tfrac{1}{2} \sqrt{\tfrac{21}{5}} {}{C_{00,4211}^{0011}} .$$ (137)

At this order, there are 3 stand-alone Galilean and gauge invariant terms:

$${}{G_{40,0000}^{0000}} = \bm{T_{40,0000}^{0000}} ,$$ (138)

$${}{G_{40,0011}^{0011}} = {}{T_{40,0011}^{0011}} ,$$ (139)

$${}{G_{42,0011}^{0011}} = {}{T_{42,0011}^{0011}} ,$$ (140)

and 3 Galilean and gauge invariant linear combinations of terms, corresponding to the 3 independent coupling constants:

$${}{G_{00,2202}^{2202}} = \tfrac{1}{2} \sqrt{5} \bm{T_{00,2000}^{2000}}+\bm{T_{00,2202}^{22... ...\sqrt{5}}{}{T_{00,3101}^{1101}}+\tfrac{3 }{2 \sqrt{5}}\bm{T_{00,4000}^{0000}} ,$$ (141)

$${}{G_{00,4211}^{0011}} = \tfrac{7}{12} \sqrt{5} {}{T_{00,2011}^{2011}}-\tfrac{7 }{\sqrt{5}... ...12}^{1112}}+\tfrac{7 }{4 \sqrt{5}}{}{T_{00,4011}^{0011}}+{}{T_{00,4211}^{0011}}$$

$$+\tfrac{7 }{3}{}{T_{00,2211}^{2011}}+\tfrac{7 }{3 \sqrt{5}}{}{T_{00,2211}^{2211}}+\tfrac{1}{2} \sqrt{\tfrac{21}{5}} {}{T_{00,2213}^{2213}} ,$$ (142)

$${}{G_{00,2212}^{2212}} = \tfrac{1}{2} \sqrt{15} {}{T_{00,2011}^{2011}}+\sqrt{\tfrac{3}{5}}... ...-\tfrac{6 }{\sqrt{5}}\bm{T_{00,3111}^{1111}}-2 \sqrt{3} \bm{T_{00,3112}^{1112}}$$

$$+\tfrac{3}{2} \sqrt{\tfrac{3}{5}} {}{T_{00,4011}^{0011}}+{}{T_{00,2212}^{2212}}+\sqrt{\tfrac{7}{5}} {}{T_{00,2213}^{2213}} ,$$ (143)

Altogether, these 6 free coupling constants (3 unrestricted and 3 independent) occur in both the Galilean and gauge invariant EDF at fourth order, cf. Table 6.

Apart from these 6 free and 12 dependent coupling constants, the gauge invariance requires that all the remaining 27 coupling constants are equal to zero. These 27 constants are allowed to be non-zero if the Galilean symmetry is imposed instead of the full gauge invariance. Then, there are 18 dependent coupling constants that are forced to be linear combinations of 9 independent ones:

$$\bm{C_{20,2000}^{0000}} = -{}{C_{20,1101}^{1101}} ,$$ (144)

$$\bm{C_{22,2202}^{0000}} = -{}{C_{22,1101}^{1101}} ,$$ (145)

$$\bm{C_{20,1110}^{1110}} = -\tfrac{1}{3}{}{C_{20,2011}^{0011}}-\tfrac{1}{3} \sqrt{5} {}{C_{20,2211}^{0011}} ,$$ (146)

$$\bm{C_{22,1112}^{1110}} = -\bm{C_{22,1112}^{1111}}-\tfrac{2 }{\sqrt{7}}\bm{C_{22,1112}^{1112}}$$

$$-2 \sqrt{\tfrac{15}{7}} {}{C_{22,2213}^{0011}} ,$$ (147)

$$\bm{C_{20,1111}^{1111}} = \tfrac{1}{2} \sqrt{\tfrac{5}{3}} {}{C_{20,2211}^{0011}}-\tfrac{1}{\sqrt{3}}{}{C_{20,2011}^{0011}} ,$$ (148)

$$\bm{C_{22,1111}^{1111}} = -\tfrac{1}{\sqrt{3}}\bm{C_{22,1112}^{1111}}-3 \sqrt{\tfrac{3}{7}} \bm{C_{22,1112}^{1112}}$$

$$-2 \sqrt{\tfrac{5}{7}} {}{C_{22,2213}^{0011}} ,$$ (149)

$$\bm{C_{20,1112}^{1112}} = -\tfrac{1}{3} \sqrt{5} {}{C_{20,2011}^{0011}}-\tfrac{1}{6}{}{C_{20,2211}^{0011}} ,$$ (150)

$${}{C_{22,2011}^{0011}} = \tfrac{2 }{3}\bm{C_{22,1112}^{1111}}-\tfrac{2 }{\sqrt{7}}\bm{C_{22,1112}^{1112}}$$

$$+\sqrt{\tfrac{5}{21}} {}{C_{22,2213}^{0011}} ,$$ (151)

$${}{C_{22,2211}^{0011}} = \tfrac{1}{6} \sqrt{5} \bm{C_{22,1112}^{1111}}+\sqrt{\tfrac{5}{7}} \bm{C_{22,1112}^{1112}}$$

$$+\tfrac{8 }{\sqrt{21}}{}{C_{22,2213}^{0011}} ,$$ (152)

$${}{C_{22,2212}^{0011}} = \tfrac{1}{2} \sqrt{3} \bm{C_{22,1112}^{1111}}+3 \sqrt{\tfrac{3}{7}} \bm{C_{22,1112}^{1112}}$$

$$+2 \sqrt{\tfrac{5}{7}} {}{C_{22,2213}^{0011}} ,$$ (153)

$$\bm{C_{31,1111}^{0000}} = {}{C_{31,0011}^{1101}} ,$$ (154)

$$\bm{C_{11,3111}^{0000}} = -\sqrt{\tfrac{3}{5}} {}{C_{11,2212}^{1101}} ,$$ (155)

$$\bm{C_{11,1111}^{2000}} = -\sqrt{\tfrac{5}{3}} {}{C_{11,2212}^{1101}} ,$$ (156)

$$\bm{C_{11,1111}^{2202}} = \tfrac{1}{\sqrt{3}}{}{C_{11,2212}^{1101}} ,$$ (157)

$$\bm{C_{11,1112}^{2202}} = {}{C_{11,2212}^{1101}} ,$$ (158)

$${}{C_{11,2011}^{1101}} = -\sqrt{\tfrac{5}{3}} {}{C_{11,2212}^{1101}} ,$$ (159)

$${}{C_{11,2211}^{1101}} = \tfrac{1}{\sqrt{3}}{}{C_{11,2212}^{1101}} ,$$ (160)

$${}{C_{11,0011}^{3101}} = -\sqrt{\tfrac{3}{5}} {}{C_{11,2212}^{1101}} .$$ (161)

Finally, we list 9 combinations of terms that are invariant with respect to the Galilean symmetry and correspond to the independent coupling constants:

$${}{G_{22,1112}^{1111}} = -\tfrac{1}{\sqrt{3}}\bm{T_{22,1111}^{1111}}-\bm{T_{22,1112}^{1110... ... \sqrt{5} {}{T_{22,2211}^{0011}}+\tfrac{1}{2} \sqrt{3} {}{T_{22,2212}^{0011}} ,$$ (162)

$${}{G_{22,1112}^{1112}} = -3 \sqrt{\tfrac{3}{7}} \bm{T_{22,1111}^{1111}}-\tfrac{2 }{\sqrt{7... ...112}^{1110}}+\bm{T_{22,1112}^{1112}}-\tfrac{2 }{\sqrt{7}}{}{T_{22,2011}^{0011}}$$

$$+\sqrt{\tfrac{5}{7}} {}{T_{22,2211}^{0011}}+3 \sqrt{\tfrac{3}{7}} {}{T_{22,2212}^{0011}} ,$$ (163)

$${}{G_{20,1101}^{1101}} = {}{T_{20,1101}^{1101}}-\bm{T_{20,2000}^{0000}} ,$$ (164)

$${}{G_{22,1101}^{1101}} = {}{T_{22,1101}^{1101}}-\bm{T_{22,2202}^{0000}} ,$$ (165)

$${}{G_{31,0011}^{1101}} = {}{T_{31,0011}^{1101}}+\bm{T_{31,1111}^{0000}} ,$$ (166)

$${}{G_{11,2212}^{1101}} = -\sqrt{\tfrac{5}{3}} \bm{T_{11,1111}^{2000}}+\tfrac{1}{\sqrt{3}}\... ...{\tfrac{5}{3}} {}{T_{11,2011}^{1101}}+\tfrac{1}{\sqrt{3}}{}{T_{11,2211}^{1101}}$$

$$-\sqrt{\tfrac{3}{5}} \bm{T_{11,3111}^{0000}}+{}{T_{11,2212}^{1101}}-\sqrt{\tfrac{3}{5}} {}{T_{11,0011}^{3101}} ,$$ (167)

$${}{G_{20,2011}^{0011}} = -\tfrac{1}{3}\bm{T_{20,1110}^{1110}}-\tfrac{1}{\sqrt{3}}\bm{T_{20... ...^{1111}}-\tfrac{1}{3} \sqrt{5} \bm{T_{20,1112}^{1112}}+{}{T_{20,2011}^{0011}} ,$$ (168)

$${}{G_{20,2211}^{0011}} = -\tfrac{1}{3} \sqrt{5} \bm{T_{20,1110}^{1110}}+\tfrac{1}{2} \sqrt... ..._{20,1111}^{1111}}-\tfrac{1}{6}\bm{T_{20,1112}^{1112}}+{}{T_{20,2211}^{0011}} ,$$ (169)

$${}{G_{22,2213}^{0011}} = -2 \sqrt{\tfrac{5}{7}} \bm{T_{22,1111}^{1111}}-2 \sqrt{\tfrac{15}... ...frac{5}{21}} {}{T_{22,2011}^{0011}}+\tfrac{8 }{\sqrt{21}}{}{T_{22,2211}^{0011}}$$

$$+2 \sqrt{\tfrac{5}{7}} {}{T_{22,2212}^{0011}}+{}{T_{22,2213}^{0011}} .$$ (170)