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Next: Sixth order Up: Results for the Galilean Previous: Second order


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:

$\displaystyle \bm{C_{00,4000}^{0000}}$ $\displaystyle =$ $\displaystyle \tfrac{3 }{2 \sqrt{5}}\bm{C_{00,2202}^{2202}} ,$ (126)
$\displaystyle \bm{C_{00,2000}^{2000}}$ $\displaystyle =$ $\displaystyle \tfrac{1}{2} \sqrt{5} \bm{C_{00,2202}^{2202}} ,$ (127)
$\displaystyle {}{C_{00,3101}^{1101}}$ $\displaystyle =$ $\displaystyle -\tfrac{6 }{\sqrt{5}} \bm{C_{00,2202}^{2202}} ,$ (128)
$\displaystyle \bm{C_{00,3110}^{1110}}$ $\displaystyle =$ $\displaystyle -2 \sqrt{\tfrac{3}{5}} {}{C_{00,2212}^{2212}}-\tfrac{7 }{\sqrt{5}}{}{C_{00,4211}^{0011}} ,$ (129)
$\displaystyle \bm{C_{00,3111}^{1111}}$ $\displaystyle =$ $\displaystyle -\tfrac{6 }{\sqrt{5}}{}{C_{00,2212}^{2212}} ,$ (130)
$\displaystyle \bm{C_{00,3112}^{1112}}$ $\displaystyle =$ $\displaystyle -2 \sqrt{3} {}{C_{00,2212}^{2212}}-\tfrac{14 }{5}{}{C_{00,4211}^{0011}} ,$ (131)
$\displaystyle \bm{C_{00,3312}^{1112}}$ $\displaystyle =$ $\displaystyle -2 \sqrt{\tfrac{7}{15}} {}{C_{00,4211}^{0011}} ,$ (132)
$\displaystyle {}{C_{00,4011}^{0011}}$ $\displaystyle =$ $\displaystyle \tfrac{3}{2} \sqrt{\tfrac{3}{5}} {}{C_{00,2212}^{2212}}+\tfrac{7 }{4 \sqrt{5}}{}{C_{00,4211}^{0011}} ,$ (133)
$\displaystyle {}{C_{00,2011}^{2011}}$ $\displaystyle =$ $\displaystyle \tfrac{1}{2} \sqrt{15} {}{C_{00,2212}^{2212}}+\tfrac{7}{12} \sqrt{5} {}{C_{00,4211}^{0011}} ,$ (134)
$\displaystyle {}{C_{00,2211}^{2011}}$ $\displaystyle =$ $\displaystyle \tfrac{7 }{3}{}{C_{00,4211}^{0011}} ,$ (135)
$\displaystyle {}{C_{00,2211}^{2211}}$ $\displaystyle =$ $\displaystyle \sqrt{\tfrac{3}{5}} {}{C_{00,2212}^{2212}}+\tfrac{7 }{3 \sqrt{5}}{}{C_{00,4211}^{0011}} ,$ (136)
$\displaystyle {}{C_{00,2213}^{2213}}$ $\displaystyle =$ $\displaystyle \sqrt{\tfrac{7}{5}} {}{C_{00,2212}^{2212}}+\tfrac{1}{2} \sqrt{\tfrac{21}{5}} {}{C_{00,4211}^{0011}} .$ (137)

At this order, there are 3 stand-alone Galilean and gauge invariant terms:
$\displaystyle {}{G_{40,0000}^{0000}}$ $\displaystyle =$ $\displaystyle \bm{T_{40,0000}^{0000}} ,$ (138)
$\displaystyle {}{G_{40,0011}^{0011}}$ $\displaystyle =$ $\displaystyle {}{T_{40,0011}^{0011}} ,$ (139)
$\displaystyle {}{G_{42,0011}^{0011}}$ $\displaystyle =$ $\displaystyle {}{T_{42,0011}^{0011}} ,$ (140)

and 3 Galilean and gauge invariant linear combinations of terms, corresponding to the 3 independent coupling constants:
$\displaystyle {}{G_{00,2202}^{2202}}$ $\displaystyle =$ $\displaystyle \tfrac{1}{2} \sqrt{5} \bm{T_{00,2000}^{2000}}+\bm{T_{00,2202}^{22...
...\sqrt{5}}{}{T_{00,3101}^{1101}}+\tfrac{3 }{2 \sqrt{5}}\bm{T_{00,4000}^{0000}} ,$ (141)
$\displaystyle {}{G_{00,4211}^{0011}}$ $\displaystyle =$ $\displaystyle \tfrac{7}{12} \sqrt{5} {}{T_{00,2011}^{2011}}-\tfrac{7 }{\sqrt{5}...
...12}^{1112}}+\tfrac{7 }{4 \sqrt{5}}{}{T_{00,4011}^{0011}}+{}{T_{00,4211}^{0011}}$  
    $\displaystyle +\tfrac{7 }{3}{}{T_{00,2211}^{2011}}+\tfrac{7 }{3 \sqrt{5}}{}{T_{00,2211}^{2211}}+\tfrac{1}{2} \sqrt{\tfrac{21}{5}} {}{T_{00,2213}^{2213}} ,$ (142)
$\displaystyle {}{G_{00,2212}^{2212}}$ $\displaystyle =$ $\displaystyle \tfrac{1}{2} \sqrt{15} {}{T_{00,2011}^{2011}}+\sqrt{\tfrac{3}{5}}...
...-\tfrac{6 }{\sqrt{5}}\bm{T_{00,3111}^{1111}}-2 \sqrt{3} \bm{T_{00,3112}^{1112}}$  
    $\displaystyle +\tfrac{3}{2} \sqrt{\tfrac{3}{5}} {}{T_{00,4011}^{0011}}+{}{T_{00,2212}^{2212}}+\sqrt{\tfrac{7}{5}} {}{T_{00,2213}^{2213}} ,$ (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:

$\displaystyle \bm{C_{20,2000}^{0000}}$ $\displaystyle =$ $\displaystyle -{}{C_{20,1101}^{1101}} ,$ (144)
$\displaystyle \bm{C_{22,2202}^{0000}}$ $\displaystyle =$ $\displaystyle -{}{C_{22,1101}^{1101}} ,$ (145)
$\displaystyle \bm{C_{20,1110}^{1110}}$ $\displaystyle =$ $\displaystyle -\tfrac{1}{3}{}{C_{20,2011}^{0011}}-\tfrac{1}{3} \sqrt{5} {}{C_{20,2211}^{0011}} ,$ (146)
$\displaystyle \bm{C_{22,1112}^{1110}}$ $\displaystyle =$ $\displaystyle -\bm{C_{22,1112}^{1111}}-\tfrac{2 }{\sqrt{7}}\bm{C_{22,1112}^{1112}}$  
    $\displaystyle -2 \sqrt{\tfrac{15}{7}} {}{C_{22,2213}^{0011}} ,$ (147)
$\displaystyle \bm{C_{20,1111}^{1111}}$ $\displaystyle =$ $\displaystyle \tfrac{1}{2} \sqrt{\tfrac{5}{3}} {}{C_{20,2211}^{0011}}-\tfrac{1}{\sqrt{3}}{}{C_{20,2011}^{0011}} ,$ (148)
$\displaystyle \bm{C_{22,1111}^{1111}}$ $\displaystyle =$ $\displaystyle -\tfrac{1}{\sqrt{3}}\bm{C_{22,1112}^{1111}}-3 \sqrt{\tfrac{3}{7}} \bm{C_{22,1112}^{1112}}$  
    $\displaystyle -2 \sqrt{\tfrac{5}{7}} {}{C_{22,2213}^{0011}} ,$ (149)
$\displaystyle \bm{C_{20,1112}^{1112}}$ $\displaystyle =$ $\displaystyle -\tfrac{1}{3} \sqrt{5} {}{C_{20,2011}^{0011}}-\tfrac{1}{6}{}{C_{20,2211}^{0011}} ,$ (150)
$\displaystyle {}{C_{22,2011}^{0011}}$ $\displaystyle =$ $\displaystyle \tfrac{2 }{3}\bm{C_{22,1112}^{1111}}-\tfrac{2 }{\sqrt{7}}\bm{C_{22,1112}^{1112}}$  
    $\displaystyle +\sqrt{\tfrac{5}{21}} {}{C_{22,2213}^{0011}} ,$ (151)


$\displaystyle {}{C_{22,2211}^{0011}}$ $\displaystyle =$ $\displaystyle \tfrac{1}{6} \sqrt{5} \bm{C_{22,1112}^{1111}}+\sqrt{\tfrac{5}{7}} \bm{C_{22,1112}^{1112}}$  
    $\displaystyle +\tfrac{8 }{\sqrt{21}}{}{C_{22,2213}^{0011}} ,$ (152)
$\displaystyle {}{C_{22,2212}^{0011}}$ $\displaystyle =$ $\displaystyle \tfrac{1}{2} \sqrt{3} \bm{C_{22,1112}^{1111}}+3 \sqrt{\tfrac{3}{7}} \bm{C_{22,1112}^{1112}}$  
    $\displaystyle +2 \sqrt{\tfrac{5}{7}} {}{C_{22,2213}^{0011}} ,$ (153)
$\displaystyle \bm{C_{31,1111}^{0000}}$ $\displaystyle =$ $\displaystyle {}{C_{31,0011}^{1101}} ,$ (154)
$\displaystyle \bm{C_{11,3111}^{0000}}$ $\displaystyle =$ $\displaystyle -\sqrt{\tfrac{3}{5}} {}{C_{11,2212}^{1101}} ,$ (155)
$\displaystyle \bm{C_{11,1111}^{2000}}$ $\displaystyle =$ $\displaystyle -\sqrt{\tfrac{5}{3}} {}{C_{11,2212}^{1101}} ,$ (156)
$\displaystyle \bm{C_{11,1111}^{2202}}$ $\displaystyle =$ $\displaystyle \tfrac{1}{\sqrt{3}}{}{C_{11,2212}^{1101}} ,$ (157)
$\displaystyle \bm{C_{11,1112}^{2202}}$ $\displaystyle =$ $\displaystyle {}{C_{11,2212}^{1101}} ,$ (158)
$\displaystyle {}{C_{11,2011}^{1101}}$ $\displaystyle =$ $\displaystyle -\sqrt{\tfrac{5}{3}} {}{C_{11,2212}^{1101}} ,$ (159)
$\displaystyle {}{C_{11,2211}^{1101}}$ $\displaystyle =$ $\displaystyle \tfrac{1}{\sqrt{3}}{}{C_{11,2212}^{1101}} ,$ (160)
$\displaystyle {}{C_{11,0011}^{3101}}$ $\displaystyle =$ $\displaystyle -\sqrt{\tfrac{3}{5}} {}{C_{11,2212}^{1101}} .$ (161)

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

$\displaystyle {}{G_{22,1112}^{1111}}$ $\displaystyle =$ $\displaystyle -\tfrac{1}{\sqrt{3}}\bm{T_{22,1111}^{1111}}-\bm{T_{22,1112}^{1110...
... \sqrt{5} {}{T_{22,2211}^{0011}}+\tfrac{1}{2} \sqrt{3} {}{T_{22,2212}^{0011}} ,$ (162)
$\displaystyle {}{G_{22,1112}^{1112}}$ $\displaystyle =$ $\displaystyle -3 \sqrt{\tfrac{3}{7}} \bm{T_{22,1111}^{1111}}-\tfrac{2 }{\sqrt{7...
...112}^{1110}}+\bm{T_{22,1112}^{1112}}-\tfrac{2 }{\sqrt{7}}{}{T_{22,2011}^{0011}}$  
    $\displaystyle +\sqrt{\tfrac{5}{7}} {}{T_{22,2211}^{0011}}+3 \sqrt{\tfrac{3}{7}} {}{T_{22,2212}^{0011}} ,$ (163)
$\displaystyle {}{G_{20,1101}^{1101}}$ $\displaystyle =$ $\displaystyle {}{T_{20,1101}^{1101}}-\bm{T_{20,2000}^{0000}} ,$ (164)
$\displaystyle {}{G_{22,1101}^{1101}}$ $\displaystyle =$ $\displaystyle {}{T_{22,1101}^{1101}}-\bm{T_{22,2202}^{0000}} ,$ (165)
$\displaystyle {}{G_{31,0011}^{1101}}$ $\displaystyle =$ $\displaystyle {}{T_{31,0011}^{1101}}+\bm{T_{31,1111}^{0000}} ,$ (166)
$\displaystyle {}{G_{11,2212}^{1101}}$ $\displaystyle =$ $\displaystyle -\sqrt{\tfrac{5}{3}} \bm{T_{11,1111}^{2000}}+\tfrac{1}{\sqrt{3}}\...
...{\tfrac{5}{3}} {}{T_{11,2011}^{1101}}+\tfrac{1}{\sqrt{3}}{}{T_{11,2211}^{1101}}$  
    $\displaystyle -\sqrt{\tfrac{3}{5}} \bm{T_{11,3111}^{0000}}+{}{T_{11,2212}^{1101}}-\sqrt{\tfrac{3}{5}} {}{T_{11,0011}^{3101}} ,$ (167)
$\displaystyle {}{G_{20,2011}^{0011}}$ $\displaystyle =$ $\displaystyle -\tfrac{1}{3}\bm{T_{20,1110}^{1110}}-\tfrac{1}{\sqrt{3}}\bm{T_{20...
...^{1111}}-\tfrac{1}{3} \sqrt{5} \bm{T_{20,1112}^{1112}}+{}{T_{20,2011}^{0011}} ,$ (168)
$\displaystyle {}{G_{20,2211}^{0011}}$ $\displaystyle =$ $\displaystyle -\tfrac{1}{3} \sqrt{5} \bm{T_{20,1110}^{1110}}+\tfrac{1}{2} \sqrt...
..._{20,1111}^{1111}}-\tfrac{1}{6}\bm{T_{20,1112}^{1112}}+{}{T_{20,2211}^{0011}} ,$ (169)
$\displaystyle {}{G_{22,2213}^{0011}}$ $\displaystyle =$ $\displaystyle -2 \sqrt{\tfrac{5}{7}} \bm{T_{22,1111}^{1111}}-2 \sqrt{\tfrac{15}...
...frac{5}{21}} {}{T_{22,2011}^{0011}}+\tfrac{8 }{\sqrt{21}}{}{T_{22,2211}^{0011}}$  
    $\displaystyle +2 \sqrt{\tfrac{5}{7}} {}{T_{22,2212}^{0011}}+{}{T_{22,2213}^{0011}} .$ (170)


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Next: Sixth order Up: Results for the Galilean Previous: Second order
Jacek Dobaczewski 2008-10-06