Results for the Galilean or gauge invariant energy density functional

As discussed in Sec. 3.2.3, when the Galilean or gauge invariance is imposed on the EDF, this induces specific constraints on the coupling constants and terms of the functional. We pointed out that there can be three disconnected classes of terms in the EDF with related properties of the coupling constants:

1. Terms that are invariant with respect to the Galilean or gauge transformation, and, therefore, the corresponding coupling constants are not restricted by the imposed symmetries.

2. Terms that cannot appear in the energy density when the Galilean or gauge symmetry is imposed, and, therefore, the corresponding coupling constants must be equal to zero.

3. Terms that can appear in the energy density only in certain specific linear combinations with other terms. This means that the coupling constants corresponding to these terms must obey specific linear conditions. We then distinguish:
   1. independent coupling constants, which multiply invariant combinations of terms and, therefore, their values are not restricted by the imposed symmetries and

   2. dependent coupling constants, which are equal to specific linear combinations of independent coupling constants, and, therefore, their values are in this way restricted by the imposed symmetries.
   Division into the sets of independent and dependent coupling constants is not unique, and below, in each case, we present only one specific choice thereof.

In Table 23 we show numbers of unrestricted, vanishing, independent, and dependent coupling constants that appear at a given order when either Galilean or gauge symmetry is imposed. In what follows, we use the name of a free coupling constant to denote either the unrestricted or independent one. Indeed, in the Galilean or gauge invariant energy density (43), these two groups of coupling constants become free parameters.

Table 23: Numbers of unrestricted, vanishing, independent, and dependent coupling constants in the EDF at zero, second, fourth, and sixth orders. Left and right columns correspond to the Galilean and gauge symmetries imposed, respectively.

	Galilean				Gauge
order	0	2	4	6	0	2	4	6
unrestricted	2	3	3	3	2	3	3	3
vanishing	0	0	0	0	0	0	27	100
independent	0	4	12	23	0	4	3	3
dependent	0	5	30	103	0	5	12	23

Below we simultaneously discuss the Galilean and gauge symmetries. In doing so, we use the fact that the Galilean symmetry is a special case of the gauge symmetry, and, therefore, the latter may impose more restrictions on the EDF than the former. At NLO, this is not the case, and the Galilean and gauge symmetries impose, in fact, identical restrictions on the EDF [25,36]. However, at higher orders, restrictions imposed by the Galilean and gauge symmetries are very different.

Both zero-order terms in the EDF, which correspond to the contact interaction, are Galilean and gauge invariant, i.e., these symmetries do not restrict the form of the EDF at LO. In the three following sections we give results for second, fourth, and sixth orders, respectively.