Sixth order

An entirely analogous pattern of terms and coupling constants appears at sixth order. Imposing either Galilean or gauge symmetry forces 23 dependent coupling constants to be specific linear combinations of 3 independent ones:.

$$\bm{C_{00,6000}^{0000}} = -\tfrac{3}{4} \sqrt{\tfrac{3}{7}} {}{C_{00,3303}^{3303}} ,$$ (171)

$$\bm{C_{00,4000}^{2000}} = -\tfrac{3}{4} \sqrt{21} {}{C_{00,3303}^{3303}} ,$$ (172)

$$\bm{C_{00,4202}^{2202}} = -3 \sqrt{\tfrac{15}{7}} {}{C_{00,3303}^{3303}} ,$$ (173)

$${}{C_{00,5101}^{1101}} = \tfrac{9}{2} \sqrt{\tfrac{3}{7}} {}{C_{00,3303}^{3303}} ,$$ (174)

$${}{C_{00,3101}^{3101}} = \tfrac{9}{10} \sqrt{21} {}{C_{00,3303}^{3303}} ,$$ (175)

$$\bm{C_{00,5110}^{1110}} = -\tfrac{1}{2} \sqrt{\tfrac{3}{5}} {}{C_{00,4212}^{2212}}-\tfrac{9 }{\sqrt{5}}{}{C_{00,6211}^{0011}} ,$$ (176)

$$\bm{C_{00,5111}^{1111}} = -\tfrac{3 }{2 \sqrt{5}}{}{C_{00,4212}^{2212}} ,$$ (177)

$$\bm{C_{00,5112}^{1112}} = -\tfrac{1}{2} \sqrt{3} {}{C_{00,4212}^{2212}}-\tfrac{18 }{5}{}{C_{00,6211}^{0011}} ,$$ (178)

$$\bm{C_{00,5312}^{1112}} = -4 \sqrt{\tfrac{7}{15}} {}{C_{00,6211}^{0011}} ,$$ (179)

$$\bm{C_{00,3110}^{3110}} = -\tfrac{7}{10} \sqrt{\tfrac{3}{5}} {}{C_{00,4212}^{2212}}-\tfrac{63 }{5 \sqrt{5}}{}{C_{00,6211}^{0011}} ,$$ (180)

$$\bm{C_{00,3111}^{3111}} = -\tfrac{21 }{10 \sqrt{5}}{}{C_{00,4212}^{2212}} ,$$ (181)

$$\bm{C_{00,3112}^{3112}} = -\tfrac{7}{10} \sqrt{3} {}{C_{00,4212}^{2212}}-\tfrac{126 }{25}{}{C_{00,6211}^{0011}} ,$$ (182)

$$\bm{C_{00,3312}^{3112}} = -\tfrac{12}{5} \sqrt{\tfrac{21}{5}} {}{C_{00,6211}^{0011}} ,$$ (183)

$$\bm{C_{00,3312}^{3312}} = -\tfrac{1}{3 \sqrt{3}}{}{C_{00,4212}^{2212}}-\tfrac{6 }{5}{}{C_{00,6211}^{0011}} ,$$ (184)

$$\bm{C_{00,3313}^{3313}} = -\tfrac{1}{3} \sqrt{\tfrac{7}{15}} {}{C_{00,4212}^{2212}} ,$$ (185)

$$\bm{C_{00,3314}^{3314}} = -\tfrac{1}{\sqrt{15}}{}{C_{00,4212}^{2212}}-\tfrac{8 }{3 \sqrt{5}}{}{C_{00,6211}^{0011}} ,$$ (186)

$${}{C_{00,6011}^{0011}} = \tfrac{1}{4} \sqrt{\tfrac{3}{5}} {}{C_{00,4212}^{2212}}+\tfrac{3 }{2 \sqrt{5}}{}{C_{00,6211}^{0011}} ,$$ (187)

$${}{C_{00,4011}^{2011}} = \tfrac{7}{4} \sqrt{\tfrac{3}{5}} {}{C_{00,4212}^{2212}}+\tfrac{21 }{2 \sqrt{5}}{}{C_{00,6211}^{0011}} ,$$ (188)

$${}{C_{00,4211}^{2011}} = 6 {}{C_{00,6211}^{0011}} ,$$ (189)

$${}{C_{00,4011}^{2211}} = \tfrac{21 }{5}{}{C_{00,6211}^{0011}} ,$$ (190)

$${}{C_{00,4211}^{2211}} = \sqrt{\tfrac{3}{5}} {}{C_{00,4212}^{2212}}+\tfrac{12 }{\sqrt{5}}{}{C_{00,6211}^{0011}} ,$$ (191)

$${}{C_{00,4213}^{2213}} = \sqrt{\tfrac{7}{5}} {}{C_{00,4212}^{2212}}+18 \sqrt{\tfrac{3}{35}} {}{C_{00,6211}^{0011}} ,$$ (192)

$${}{C_{00,4413}^{2213}} = \tfrac{4 }{\sqrt{5}}{}{C_{00,6211}^{0011}} .$$ (193)

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

$${}{G_{60,0000}^{0000}} = \bm{T_{60,0000}^{0000}} ,$$ (194)

$${}{G_{60,0011}^{0011}} = {}{T_{60,0011}^{0011}} ,$$ (195)

$${}{G_{62,0011}^{0011}} = {}{T_{62,0011}^{0011}} ,$$ (196)

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

$${}{G_{00,3303}^{3303}} = \tfrac{9}{10} \sqrt{21} {}{T_{00,3101}^{3101}}+{}{T_{00,3303}^{33... ...sqrt{21} \bm{T_{00,4000}^{2000}}-3 \sqrt{\tfrac{15}{7}} \bm{T_{00,4202}^{2202}}$$

$$+\tfrac{9}{2} \sqrt{\tfrac{3}{7}} {}{T_{00,5101}^{1101}}-\tfrac{3}{4} \sqrt{\tfrac{3}{7}} \bm{T_{00,6000}^{0000}} ,$$ (197)

$${}{G_{00,6211}^{0011}} = -\tfrac{63 }{5 \sqrt{5}}\bm{T_{00,3110}^{3110}}-\tfrac{126 }{25}\... ...frac{9 }{\sqrt{5}}\bm{T_{00,5110}^{1110}}-\tfrac{18 }{5}\bm{T_{00,5112}^{1112}}$$

$$-4 \sqrt{\tfrac{7}{15}} \bm{T_{00,5312}^{1112}}-\tfrac{6 }{5}\bm{... ...}^{2011}}+6 {}{T_{00,4211}^{2011}}+\tfrac{3 }{2 \sqrt{5}}{}{T_{00,6011}^{0011}}$$

$$+{}{T_{00,6211}^{0011}}+\tfrac{21 }{5}{}{T_{00,4011}^{2211}}+\tfr... ...rac{3}{35}} {}{T_{00,4213}^{2213}}+\tfrac{4 }{\sqrt{5}}{}{T_{00,4413}^{2213}} ,$$ (198)

$${}{G_{00,4212}^{2212}} = -\tfrac{7}{10} \sqrt{\tfrac{3}{5}} \bm{T_{00,3110}^{3110}}-\tfrac... ...ac{3}{5}} \bm{T_{00,5110}^{1110}}-\tfrac{3 }{2 \sqrt{5}}\bm{T_{00,5111}^{1111}}$$

$$-\tfrac{1}{2} \sqrt{3} \bm{T_{00,5112}^{1112}}-\tfrac{1}{3 \sqrt{... ...\bm{T_{00,3314}^{3314}}+\tfrac{7}{4} \sqrt{\tfrac{3}{5}} {}{T_{00,4011}^{2011}}$$

$$+\sqrt{\tfrac{3}{5}} {}{T_{00,4211}^{2211}}+\tfrac{1}{4} \sqrt{\t... ...11}^{0011}}+{}{T_{00,4212}^{2212}}+\sqrt{\tfrac{7}{5}} {}{T_{00,4213}^{2213}} .$$ (199)

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

Apart from the 6 free and 23 dependent coupling constants, at sixth order the gauge invariance requires that all the remaining 100 coupling constants are equal to zero. These 100 constants are allowed to be non-zero if the Galilean symmetry is imposed instead of the full gauge invariance. Then, there are 80 dependent coupling constants that are forced to be linear combinations of 20 independent ones:

$$\bm{C_{40,2000}^{0000}} = -{}{C_{40,1101}^{1101}} ,$$ (200)

$$\bm{C_{42,2202}^{0000}} = -{}{C_{42,1101}^{1101}} ,$$ (201)

$$\bm{C_{20,4000}^{0000}} = \tfrac{3 }{2 \sqrt{5}}\bm{C_{20,2202}^{2202}} ,$$ (202)

$$\bm{C_{22,4202}^{0000}} = -\tfrac{1}{2} \sqrt{\tfrac{15}{7}} {}{C_{22,3303}^{1101}} ,$$ (203)

$$\bm{C_{20,2000}^{2000}} = \tfrac{1}{2} \sqrt{5} \bm{C_{20,2202}^{2202}} ,$$ (204)

$$\bm{C_{22,2202}^{2000}} = -\tfrac{1}{2} \sqrt{\tfrac{35}{3}} {}{C_{22,3303}^{1101}} ,$$ (205)

$$\bm{C_{22,2202}^{2202}} = -\tfrac{1}{2} \sqrt{\tfrac{5}{3}} {}{C_{22,3303}^{1101}} ,$$ (206)

$${}{C_{20,3101}^{1101}} = -\tfrac{6 }{\sqrt{5}}\bm{C_{20,2202}^{2202}} ,$$ (207)

$${}{C_{22,3101}^{1101}} = \sqrt{\tfrac{21}{5}} {}{C_{22,3303}^{1101}} ,$$ (208)

$$\bm{C_{40,1110}^{1110}} = \tfrac{1}{\sqrt{5}}\bm{C_{40,1112}^{1112}}-\tfrac{3 }{2 \sqrt{5}}{}{C_{40,2211}^{0011}} ,$$ (209)

$$\bm{C_{20,3110}^{1110}} = -\tfrac{4 }{5}{}{C_{20,2011}^{2011}}-\tfrac{2 }{\sqrt{5}}{}{C_{20,2211}^{2011}} ,$$ (210)

$$\bm{C_{22,3312}^{1110}} = \tfrac{6 }{\sqrt{35}}{}{C_{22,2212}^{2011}}+\sqrt{\tfrac{5}{21}} \bm{C_{22,3112}^{1110}}$$

$$+\tfrac{2}{3} \sqrt{\tfrac{5}{21}} \bm{C_{22,3112}^{1111}}-2 \sqrt{2} \bm{C_{22,3313}^{1111}} ,$$ (211)

$$\bm{C_{40,1111}^{1111}} = \sqrt{\tfrac{3}{5}} \bm{C_{40,1112}^{1112}}+\sqrt{\tfrac{3}{5}} {}{C_{40,2211}^{0011}} ,$$ (212)

$$\bm{C_{42,1112}^{1111}} = 2 \sqrt{3} \bm{C_{42,1111}^{1111}}+2 \sqrt{3} {}{C_{42,2212}^{0011}} ,$$ (213)

$$\bm{C_{20,3111}^{1111}} = \sqrt{\tfrac{3}{5}} {}{C_{20,2211}^{2011}}-\tfrac{4}{5} \sqrt{3} {}{C_{20,2011}^{2011}} ,$$ (214)

$$\bm{C_{22,3111}^{1111}} = \tfrac{1}{\sqrt{3}}\bm{C_{22,3112}^{1111}}-\tfrac{6 }{5}{}{C_{22,2212}^{2011}} ,$$ (215)

$$\bm{C_{22,3312}^{1111}} = \tfrac{6 }{\sqrt{35}}{}{C_{22,2212}^{2011}}-2 \sqrt{2} \bm{C_{22,3313}^{1111}} ,$$ (216)

$$\bm{C_{42,1112}^{1112}} = \tfrac{1}{\sqrt{7}}\bm{C_{42,1112}^{1110}}-\sqrt{\tfrac{3}{7}} \bm{C_{42,1111}^{1111}} ,$$ (217)

$$\bm{C_{44,1112}^{1112}} = -{}{C_{44,2213}^{0011}} ,$$ (218)

$$\bm{C_{22,3110}^{1112}} = \bm{C_{22,3112}^{1110}} ,$$ (219)

$$\bm{C_{22,3111}^{1112}} = -\bm{C_{22,3112}^{1111}} ,$$ (220)

$$\bm{C_{20,3112}^{1112}} = -\tfrac{4 }{\sqrt{5}}{}{C_{20,2011}^{2011}}-\tfrac{1}{5}{}{C_{20,2211}^{2011}} ,$$ (221)

$$\bm{C_{22,3112}^{1112}} = \tfrac{6}{5} \sqrt{\tfrac{3}{7}} {}{C_{22,2212}^{2011}}+\tfrac{2 }{\sqrt{7}}\bm{C_{22,3112}^{1110}}$$

$$-\tfrac{1}{\sqrt{7}}\bm{C_{22,3112}^{1111}} ,$$ (222)

$$\bm{C_{20,3312}^{1112}} = -2 \sqrt{\tfrac{3}{35}} {}{C_{20,2211}^{2011}} ,$$ (223)

$$\bm{C_{22,3312}^{1112}} = \tfrac{2 }{7 \sqrt{5}}{}{C_{22,2212}^{2011}}+\tfrac{2}{7} \sqrt{\tfrac{5}{3}} \bm{C_{22,3112}^{1110}}$$

$$+\tfrac{4}{21} \sqrt{\tfrac{5}{3}} \bm{C_{22,3112}^{1111}} ,$$ (224)

$$\bm{C_{22,3313}^{1112}} = \tfrac{1}{\sqrt{2}}\bm{C_{22,3313}^{1111}}-\tfrac{4 }{\sqrt{35}}{}{C_{22,2212}^{2011}}$$

$$-2 \sqrt{\tfrac{2}{7}} \bm{C_{22,3313}^{1111}} ,$$ (225)

$$\bm{C_{22,3314}^{1112}} = \tfrac{4 }{7 \sqrt{15}}{}{C_{22,2212}^{2011}}+\tfrac{4}{21} \sqrt{5} \bm{C_{22,3112}^{1110}}$$

$$+\tfrac{8}{63} \sqrt{5} \bm{C_{22,3112}^{1111}} ,$$ (226)

$${}{C_{40,2011}^{0011}} = -\tfrac{3 }{\sqrt{5}}\bm{C_{40,1112}^{1112}}-\tfrac{1}{2 \sqrt{5}}{}{C_{40,2211}^{0011}}$$

$$-\tfrac{1}{\sqrt{42}}\bm{C_{22,3313}^{1111}} ,$$ (227)

$${}{C_{42,2011}^{0011}} = \tfrac{4 }{\sqrt{3}}\bm{C_{42,1111}^{1111}}-\tfrac{1}{2}\bm{C_{42,1112}^{1110}}$$

$$+\sqrt{3} {}{C_{42,2212}^{0011}} ,$$ (228)

$${}{C_{42,2211}^{0011}} = -\tfrac{4 }{\sqrt{15}}\bm{C_{42,1111}^{1111}}-\tfrac{1}{\sqrt{5}}\bm{C_{42,1112}^{1110}}$$

$$-\sqrt{\tfrac{3}{5}} {}{C_{42,2212}^{0011}} ,$$ (229)

$${}{C_{42,2213}^{0011}} = -\tfrac{6 }{\sqrt{35}}\bm{C_{42,1111}^{1111}}-\tfrac{3}{2} \sqrt{\tfrac{3}{35}} \bm{C_{42,1112}^{1110}}$$

$$-\sqrt{\tfrac{7}{5}} {}{C_{42,2212}^{0011}} ,$$ (230)

$${}{C_{20,4011}^{0011}} = \tfrac{3 }{5}{}{C_{20,2011}^{2011}} ,$$ (231)

$${}{C_{22,4011}^{0011}} = -\tfrac{1}{10} \sqrt{3} {}{C_{22,2212}^{2011}}-\tfrac{1}{4}\bm{C_{22,3112}^{1110}}$$

$$+\tfrac{1}{3}\bm{C_{22,3112}^{1111}} ,$$ (232)

$${}{C_{20,4211}^{0011}} = \tfrac{3 }{7}{}{C_{20,2211}^{2011}} ,$$ (233)

$${}{C_{22,4211}^{0011}} = \tfrac{1}{7} \sqrt{\tfrac{3}{5}} {}{C_{22,2212}^{2011}}-\tfrac{1}{7} \sqrt{5} \bm{C_{22,3112}^{1110}}$$

$$-\tfrac{2}{21} \sqrt{5} \bm{C_{22,3112}^{1111}} ,$$ (234)

$${}{C_{22,4212}^{0011}} = \tfrac{3 }{7}{}{C_{22,2212}^{2011}} ,$$ (235)

$${}{C_{22,4213}^{0011}} = -\tfrac{3 }{7 \sqrt{35}}{}{C_{22,2212}^{2011}}-\tfrac{3}{14} \sqrt{\tfrac{15}{7}} \bm{C_{22,3112}^{1110}} ,$$ (236)

$${}{C_{22,4413}^{0011}} = -\tfrac{2}{7} \sqrt{\tfrac{5}{3}} {}{C_{22,2212}^{2011}}-\tfrac{1}{21} \sqrt{5} \bm{C_{22,3112}^{1110}}$$

$$-\tfrac{1}{7} \sqrt{\tfrac{15}{7}} \bm{C_{22,3112}^{1111}}$$

$$-\tfrac{2}{63} \sqrt{5} \bm{C_{22,3112}^{1111}}+\sqrt{\tfrac{7}{6}} \bm{C_{22,3313}^{1111}} ,$$ (237)

$${}{C_{22,2011}^{2011}} = -\tfrac{1}{2 \sqrt{3}}{}{C_{22,2212}^{2011}}-\tfrac{5 }{12}\bm{C_{22,3112}^{1110}}$$

$$+\tfrac{5 }{9}\bm{C_{22,3112}^{1111}} ,$$ (238)

$${}{C_{22,2211}^{2011}} = \tfrac{1}{\sqrt{15}}{}{C_{22,2212}^{2011}}-\tfrac{1}{3} \sqrt{5} \bm{C_{22,3112}^{1110}}$$

$$-\tfrac{2}{9} \sqrt{5} \bm{C_{22,3112}^{1111}} ,$$ (239)

$${}{C_{22,2213}^{2011}} = -\tfrac{1}{\sqrt{35}}{}{C_{22,2212}^{2011}}-\tfrac{1}{2} \sqrt{\tfrac{15}{7}} \bm{C_{22,3112}^{1110}}$$

$$-\sqrt{\tfrac{5}{21}} \bm{C_{22,3112}^{1111}} ,$$ (240)

$${}{C_{20,2211}^{2211}} = \tfrac{2 }{5}{}{C_{20,2011}^{2011}}+\tfrac{1}{2 \sqrt{5}}{}{C_{20,2211}^{2011}} ,$$ (241)

$${}{C_{22,2211}^{2211}} = -\tfrac{43 }{25 \sqrt{3}}{}{C_{22,2212}^{2011}}-\tfrac{1}{3}\bm{C_{22,3112}^{1110}}$$

$$-\tfrac{17 }{90}\bm{C_{22,3112}^{1111}} ,$$ (242)

$${}{C_{22,2212}^{2211}} = \tfrac{4 }{5 \sqrt{5}}{}{C_{22,2212}^{2011}}+\tfrac{1}{\sqrt{15}}\bm{C_{22,3112}^{1111}}$$

$$-\tfrac{3}{5} \sqrt{14} \bm{C_{22,3313}^{1111}}+\tfrac{9}{5} \sqrt{\tfrac{21}{10}} \bm{C_{22,3313}^{1111}} ,$$ (243)

$${}{C_{22,2213}^{2211}} = -\tfrac{4 }{25 \sqrt{7}}{}{C_{22,2212}^{2011}}-\sqrt{\tfrac{3}{7}} \bm{C_{22,3112}^{1110}}$$

$$+\tfrac{4 }{5 \sqrt{21}}\bm{C_{22,3112}^{1111}} ,$$ (244)

$${}{C_{20,2212}^{2212}} = \tfrac{2 }{\sqrt{15}}{}{C_{20,2011}^{2011}}-\tfrac{1}{2 \sqrt{3}}{}{C_{20,2211}^{2011}}$$

$$+\tfrac{3}{5} \sqrt{\tfrac{2}{5}} \bm{C_{22,3313}^{1111}} ,$$ (245)

$${}{C_{22,2212}^{2212}} = \tfrac{1}{5} \sqrt{7} {}{C_{22,2212}^{2011}}-\tfrac{1}{6} \sqrt{\tfrac{7}{3}} \bm{C_{22,3112}^{1111}}$$

$$-\tfrac{3 }{\sqrt{10}}\bm{C_{22,3313}^{1111}} ,$$ (246)

$${}{C_{22,2213}^{2212}} = -\tfrac{2}{5} \sqrt{\tfrac{2}{35}} {}{C_{22,2212}^{2011}}-\tfrac{2}{3} \sqrt{\tfrac{14}{15}} \bm{C_{22,3112}^{1111}}$$

$$+\tfrac{3 }{5}\bm{C_{22,3313}^{1111}} ,$$ (247)

$${}{C_{20,2213}^{2213}} = \tfrac{2}{5} \sqrt{\tfrac{7}{3}} {}{C_{20,2011}^{2011}}+\tfrac{1}{\sqrt{105}}{}{C_{20,2211}^{2011}} ,$$ (248)

$${}{C_{22,2213}^{2213}} = -\tfrac{9}{25} \sqrt{\tfrac{6}{7}} {}{C_{22,2212}^{2011}}-\tfrac{1}{\sqrt{14}}\bm{C_{22,3112}^{1110}}$$

$$+\tfrac{2}{15} \sqrt{\tfrac{2}{7}} \bm{C_{22,3112}^{1111}}+\tfrac{1}{5} \sqrt{\tfrac{3}{5}} \bm{C_{22,3313}^{1111}} ,$$ (249)

$$\bm{C_{51,1111}^{0000}} = {}{C_{51,0011}^{1101}} ,$$ (250)

$$\bm{C_{31,3111}^{0000}} = -\sqrt{\tfrac{3}{5}} {}{C_{31,2212}^{1101}} ,$$ (251)

$$\bm{C_{33,3313}^{0000}} = {}{C_{33,2213}^{1101}} ,$$ (252)

$$\bm{C_{11,5111}^{0000}} = -\tfrac{1}{14} \sqrt{15} {}{C_{11,2212}^{3101}} ,$$ (253)

$$\bm{C_{31,1111}^{2000}} = -\sqrt{\tfrac{5}{3}} {}{C_{31,2212}^{1101}} ,$$ (254)

$$\bm{C_{11,3111}^{2000}} = -\sqrt{\tfrac{5}{3}} {}{C_{11,2212}^{3101}} ,$$ (255)

$$\bm{C_{31,1111}^{2202}} = \tfrac{1}{\sqrt{3}}{}{C_{31,2212}^{1101}} ,$$ (256)

$$\bm{C_{33,1111}^{2202}} = \sqrt{6} {}{C_{33,2213}^{1101}} ,$$ (257)

$$\bm{C_{31,1112}^{2202}} = {}{C_{31,2212}^{1101}} ,$$ (258)

$$\bm{C_{33,1112}^{2202}} = -\sqrt{3} {}{C_{33,2213}^{1101}} ,$$ (259)

$$\bm{C_{11,3111}^{2202}} = \tfrac{1}{\sqrt{3}}{}{C_{11,2212}^{3101}} ,$$ (260)

$$\bm{C_{11,3112}^{2202}} = {}{C_{11,2212}^{3101}} ,$$ (261)

$$\bm{C_{11,3312}^{2202}} = -\tfrac{2}{3} \sqrt{\tfrac{5}{21}} {}{C_{11,2212}^{3101}} ,$$ (262)

$$\bm{C_{11,3313}^{2202}} = -\tfrac{2}{3} \sqrt{\tfrac{10}{21}} {}{C_{11,2212}^{3101}} ,$$ (263)

$$\bm{C_{11,1111}^{4000}} = -\tfrac{1}{2} \sqrt{\tfrac{5}{3}} {}{C_{11,2212}^{3101}} ,$$ (264)

$$\bm{C_{11,1111}^{4202}} = \tfrac{5 }{7 \sqrt{3}}{}{C_{11,2212}^{3101}} ,$$ (265)

$$\bm{C_{11,1112}^{4202}} = \tfrac{5 }{7}{}{C_{11,2212}^{3101}} ,$$ (266)

$${}{C_{31,2011}^{1101}} = -\sqrt{\tfrac{5}{3}} {}{C_{31,2212}^{1101}} ,$$ (267)

$${}{C_{31,2211}^{1101}} = \tfrac{1}{\sqrt{3}}{}{C_{31,2212}^{1101}} ,$$ (268)

$${}{C_{33,2212}^{1101}} = -2 \sqrt{2} {}{C_{33,2213}^{1101}} ,$$ (269)

$${}{C_{11,4011}^{1101}} = -\tfrac{1}{2} \sqrt{\tfrac{5}{3}} {}{C_{11,2212}^{3101}} ,$$ (270)

$${}{C_{11,4211}^{1101}} = \tfrac{5 }{7 \sqrt{3}}{}{C_{11,2212}^{3101}} ,$$ (271)

$${}{C_{11,4212}^{1101}} = \tfrac{5 }{7}{}{C_{11,2212}^{3101}} ,$$ (272)

$${}{C_{31,0011}^{3101}} = -\sqrt{\tfrac{3}{5}} {}{C_{31,2212}^{1101}} ,$$ (273)

$${}{C_{11,2011}^{3101}} = -\sqrt{\tfrac{5}{3}} {}{C_{11,2212}^{3101}} ,$$ (274)

$${}{C_{11,2211}^{3101}} = \tfrac{1}{\sqrt{3}}{}{C_{11,2212}^{3101}} ,$$ (275)

$${}{C_{33,0011}^{3303}} = {}{C_{33,2213}^{1101}} ,$$ (276)

$${}{C_{11,2212}^{3303}} = -\tfrac{2}{3} \sqrt{\tfrac{5}{21}} {}{C_{11,2212}^{3101}} ,$$ (277)

$${}{C_{11,2213}^{3303}} = -\tfrac{2}{3} \sqrt{\tfrac{10}{21}} {}{C_{11,2212}^{3101}} ,$$ (278)

$${}{C_{11,0011}^{5101}} = -\tfrac{1}{14} \sqrt{15} {}{C_{11,2212}^{3101}} .$$ (279)

Finally, we list 20 combinations of terms that are invariant with respect to the Galilean symmetry and correspond to the independent coupling constants:

$${}{G_{20,2202}^{2202}} = \tfrac{1}{2} \sqrt{5} \bm{T_{20,2000}^{2000}}+\bm{T_{20,2202}^{22... ...\sqrt{5}}{}{T_{20,3101}^{1101}}+\tfrac{3 }{2 \sqrt{5}}\bm{T_{20,4000}^{0000}} ,$$ (280)

$${}{G_{42,1112}^{1110}} = \bm{T_{42,1112}^{1110}}+\tfrac{1}{\sqrt{7}}\bm{T_{42,1112}^{1112}... ...{T_{42,2211}^{0011}}-\tfrac{3}{2} \sqrt{\tfrac{3}{35}} {}{T_{42,2213}^{0011}} ,$$ (281)

$${}{G_{22,3112}^{1110}} = \bm{T_{22,3110}^{1112}}+\bm{T_{22,3112}^{1110}}+\tfrac{2 }{\sqrt{... ...c{5}{3}} \bm{T_{22,3312}^{1112}}+\tfrac{4}{21} \sqrt{5} \bm{T_{22,3314}^{1112}}$$

$$-\tfrac{5 }{12}{}{T_{22,2011}^{2011}}-\tfrac{1}{3} \sqrt{5} {}{T_... ...}{T_{22,4211}^{0011}}-\tfrac{3}{14} \sqrt{\tfrac{15}{7}} {}{T_{22,4213}^{0011}}$$

$$-\tfrac{1}{21} \sqrt{5} {}{T_{22,4413}^{0011}}-\tfrac{1}{3}{}{T_{... ...frac{3}{7}} {}{T_{22,2213}^{2211}}-\tfrac{1}{\sqrt{14}}{}{T_{22,2213}^{2213}} ,$$ (282)

$${}{G_{42,1111}^{1111}} = \bm{T_{42,1111}^{1111}}+2 \sqrt{3} \bm{T_{42,1112}^{1111}}-\sqrt{... ...{\sqrt{15}}{}{T_{42,2211}^{0011}}-\tfrac{6 }{\sqrt{35}}{}{T_{42,2213}^{0011}} ,$$ (283)

$${}{G_{22,3112}^{1111}} = \tfrac{1}{\sqrt{3}}\bm{T_{22,3111}^{1111}}-\bm{T_{22,3111}^{1112}... ...m{T_{22,3312}^{1110}}+\tfrac{4}{21} \sqrt{\tfrac{5}{3}} \bm{T_{22,3312}^{1112}}$$

$$+\tfrac{5 }{9}{}{T_{22,2011}^{2011}}+\tfrac{8}{63} \sqrt{5} \bm{T... ...{}{T_{22,4211}^{0011}}-\tfrac{1}{7} \sqrt{\tfrac{15}{7}} {}{T_{22,4213}^{0011}}$$

$$-\tfrac{2}{63} \sqrt{5} {}{T_{22,4413}^{0011}}-\tfrac{2}{9} \sqrt... ...}{}{T_{22,2212}^{2211}}-\tfrac{1}{6} \sqrt{\tfrac{7}{3}} {}{T_{22,2212}^{2212}}$$

$$-\sqrt{\tfrac{5}{21}} {}{T_{22,2213}^{2011}}+\tfrac{4 }{5 \sqrt{2... ...T_{22,2213}^{2213}}-\tfrac{2}{3} \sqrt{\tfrac{14}{15}} {}{T_{22,2213}^{2212}} ,$$ (284)

$${}{G_{22,3313}^{1111}} = -2 \sqrt{2} \bm{T_{22,3312}^{1110}}-2 \sqrt{2} \bm{T_{22,3312}^{1... ...312}^{1112}}+\bm{T_{22,3313}^{1111}}+\tfrac{1}{\sqrt{2}}\bm{T_{22,3313}^{1112}}$$

$$-\tfrac{1}{\sqrt{42}}\bm{T_{22,3314}^{1112}}+\tfrac{9}{5} \sqrt{\... ...}{}{T_{22,2212}^{2212}}+\tfrac{3}{5} \sqrt{\tfrac{2}{5}} {}{T_{22,2213}^{2211}}$$

$$+\tfrac{3 }{5}{}{T_{22,2213}^{2212}}+\sqrt{\tfrac{7}{6}} {}{T_{22,4413}^{0011}}+\tfrac{1}{5} \sqrt{\tfrac{3}{5}} {}{T_{22,2213}^{2213}} ,$$ (285)

$${}{G_{40,1112}^{1112}} = \tfrac{1}{\sqrt{5}}\bm{T_{40,1110}^{1110}}+\sqrt{\tfrac{3}{5}} \b... ...1}^{1111}}+\bm{T_{40,1112}^{1112}}-\tfrac{3 }{\sqrt{5}}{}{T_{40,2011}^{0011}} ,$$ (286)

$${}{G_{40,1101}^{1101}} = {}{T_{40,1101}^{1101}}-\bm{T_{40,2000}^{0000}} ,$$ (287)

$${}{G_{42,1101}^{1101}} = {}{T_{42,1101}^{1101}}-\bm{T_{42,2202}^{0000}} ,$$ (288)

$${}{G_{22,3303}^{1101}} = -\tfrac{1}{2} \sqrt{\tfrac{35}{3}} \bm{T_{22,2202}^{2000}}-\tfrac... ...T_{22,3303}^{1101}}-\tfrac{1}{2} \sqrt{\tfrac{15}{7}} \bm{T_{22,4202}^{0000}} ,$$ (289)

$${}{G_{51,0011}^{1101}} = {}{T_{51,0011}^{1101}}+\bm{T_{51,1111}^{0000}} ,$$ (290)

$${}{G_{31,1112}^{2202}} = -\sqrt{\tfrac{5}{3}} \bm{T_{31,1111}^{2000}}+\tfrac{1}{\sqrt{3}}\... ...{1}{\sqrt{3}}{}{T_{31,2211}^{1101}}-\sqrt{\tfrac{3}{5}} \bm{T_{31,3111}^{0000}}$$

$$+{}{T_{31,2212}^{1101}}-\sqrt{\tfrac{3}{5}} {}{T_{31,0011}^{3101}} ,$$ (291)

$${}{G_{33,2213}^{1101}} = {}{T_{33,0011}^{3303}}+\sqrt{6} \bm{T_{33,1111}^{2202}}-\sqrt{3} ... ...sqrt{2} {}{T_{33,2212}^{1101}}+{}{T_{33,2213}^{1101}}+\bm{T_{33,3313}^{0000}} ,$$ (292)

$${}{G_{11,2212}^{3101}} = -\sqrt{\tfrac{5}{3}} \bm{T_{11,3111}^{2000}}+\tfrac{1}{\sqrt{3}}\... ...{T_{11,3312}^{2202}}-\tfrac{2}{3} \sqrt{\tfrac{10}{21}} \bm{T_{11,3313}^{2202}}$$

$$-\tfrac{1}{14} \sqrt{15} \bm{T_{11,5111}^{0000}}-\tfrac{1}{2} \sq... ...\bm{T_{11,1112}^{4202}}-\tfrac{1}{2} \sqrt{\tfrac{5}{3}} {}{T_{11,4011}^{1101}}$$

$$+\tfrac{5 }{7 \sqrt{3}}{}{T_{11,4211}^{1101}}+\tfrac{5 }{7}{}{T_{... ...{\tfrac{5}{3}} {}{T_{11,2011}^{3101}}+\tfrac{1}{\sqrt{3}}{}{T_{11,2211}^{3101}}$$

$$+{}{T_{11,2212}^{3101}}-\tfrac{2}{3} \sqrt{\tfrac{5}{21}} {}{T_{11,2212}^{3303}}-\tfrac{2}{3} \sqrt{\tfrac{10}{21}} {}{T_{11,2213}^{3303}} ,$$ (293)

$${}{G_{40,2211}^{0011}} = -\tfrac{3 }{2 \sqrt{5}}\bm{T_{40,1110}^{1110}}+\sqrt{\tfrac{3}{5}... ...1}^{1111}}-\tfrac{1}{2 \sqrt{5}}{}{T_{40,2011}^{0011}}+{}{T_{40,2211}^{0011}} ,$$ (294)

$${}{G_{42,2212}^{0011}} = 2 \sqrt{3} \bm{T_{42,1112}^{1111}}+\sqrt{3} {}{T_{42,2011}^{0011}... ...11}^{0011}}+{}{T_{42,2212}^{0011}}-\sqrt{\tfrac{7}{5}} {}{T_{42,2213}^{0011}} ,$$ (295)

$${}{G_{44,2213}^{0011}} = {}{T_{44,2213}^{0011}}-\bm{T_{44,1112}^{1112}} ,$$ (296)

$${}{G_{20,2011}^{2011}} = {}{T_{20,2011}^{2011}}+\tfrac{2 }{5}{}{T_{20,2211}^{2211}}-\tfrac... ...\tfrac{4 }{\sqrt{5}}\bm{T_{20,3112}^{1112}}+\tfrac{3 }{5}{}{T_{20,4011}^{0011}}$$

$$+\tfrac{2 }{\sqrt{15}}{}{T_{20,2212}^{2212}}+\tfrac{2}{5} \sqrt{\tfrac{7}{3}} {}{T_{20,2213}^{2213}} ,$$ (297)

$${}{G_{20,2211}^{2011}} = {}{T_{20,2211}^{2011}}-\tfrac{2 }{\sqrt{5}}\bm{T_{20,3110}^{1110}... ...sqrt{\tfrac{3}{35}} \bm{T_{20,3312}^{1112}}+\tfrac{3 }{7}{}{T_{20,4211}^{0011}}$$

$$+\tfrac{1}{2 \sqrt{5}}{}{T_{20,2211}^{2211}}-\tfrac{1}{2 \sqrt{3}}{}{T_{20,2212}^{2212}}+\tfrac{1}{\sqrt{105}}{}{T_{20,2213}^{2213}} ,$$ (298)

$${}{G_{22,2212}^{2011}} = -\tfrac{6 }{5}\bm{T_{22,3111}^{1111}}+\tfrac{6}{5} \sqrt{\tfrac{3... ...\sqrt{35}}\bm{T_{22,3312}^{1111}}+\tfrac{2 }{7 \sqrt{5}}\bm{T_{22,3312}^{1112}}$$

$$-\tfrac{4 }{\sqrt{35}}\bm{T_{22,3313}^{1112}}+\tfrac{4 }{7 \sqrt{... ... \sqrt{\tfrac{3}{5}} {}{T_{22,4211}^{0011}}+\tfrac{3 }{7}{}{T_{22,4212}^{0011}}$$

$$-\tfrac{3 }{7 \sqrt{35}}{}{T_{22,4213}^{0011}}-\tfrac{2}{7} \sqrt... ...\sqrt{15}}{}{T_{22,2211}^{2011}}-\tfrac{43 }{25 \sqrt{3}}{}{T_{22,2211}^{2211}}$$

$$+{}{T_{22,2212}^{2011}}+\tfrac{4 }{5 \sqrt{5}}{}{T_{22,2212}^{221... ...} \sqrt{7} {}{T_{22,2212}^{2212}}-\tfrac{4 }{25 \sqrt{7}}{}{T_{22,2213}^{2211}}$$

$$-\tfrac{2}{5} \sqrt{\tfrac{2}{35}} {}{T_{22,2213}^{2212}}-\tfrac{9}{25} \sqrt{\tfrac{6}{7}} {}{T_{22,2213}^{2213}} .$$ (299)