Galilean or gauge invariant EDF

A Galilean or gauge-invariant EDF is the one which does not change upon inserting Galilean or gauge-transformed densities (37) into the energy density of Eq. (31). Since terms quadratic in $G(\vec {r},\vec {r}')$ can be dropped, the condition for the Galilean or gauge invariance reads

$$\int{\rm d}^3\vec {r} \sum C^\beta_{mI,\gamma}\left(\left[\rho_{\... ...ec {r})\left[ D_{mI} \rho_{\gamma}^{G}(\vec {r})\right]_{J'}\right]_0\right)=0,$$ (42)

where $C^\beta_{mI,\gamma}$ is a short-hand notation for the coupling constants $C^{n'L'v'J'}_{mI,nLvJ}$, and the sum runs over all the terms in the energy density.

The task now is to group together all proportional terms in Eq. (42). In doing so, we do not aim at obtaining an invariant energy density but an invariant EDF and total energy. Therefore, after densities (40) or (41) are inserted into Eq. (42), all terms must be integrated by parts to obtain some standard form, where terms equal through integration by parts become identical.

Finally, Eq. (42) can be transformed into a sum of independent terms using recoupling. In this expression each term is multiplied by a specific linear combination of coupling constants $C^{n'L'v'J'}_{mI,nLvJ}$. The Galilean or gauge invariance of the EDF then means that these linear combinations must all vanish. This gives a set of linear equations that must be fulfilled for an invariant EDF. On the one hand, if a given coupling constant appears in none of these linear equations, the corresponding term of the EDF is invariant on its own, and the corresponding coupling constant is not restricted by the Galilean or gauge symmetry. On the other hand, for some coupling constants the only solution can be the value of zero, and then the corresponding term cannot appear in the invariant energy density.

Among all the remaining coupling constants, we may always select a subset of those that we will call the dependent ones, and express them as linear combinations of the other ones, which we will call the independent ones. This procedure is highly non-unique and can be realized in very many different ways, However, when the dependent coupling constants in function of independent ones are inserted back into the energy density (31), linear combinations of terms appearing at each independent coupling constant will all be invariant with respect to the Galilean or gauge transformations.

Then, the energy density of Eq. (31) takes the form

$${\cal H}(\vec {r})=\sum_{{n'L'v'J'}\atop{mI,nLvJ,J'}} C^{n'L'v'J'}_{mI,nLvJ} G^{n'L'v'J'}_{mI,nLvJ}(\vec {r}),$$ (43)

where the sum runs over indices that correspond to unrestricted and independent coupling constants, which we jointly call free coupling constants. For a term in Eq. (43) that corresponds to an unrestricted coupling constant, we have $G^{n'L'v'J'}_{mI,nLvJ}(\vec {r})=T^{n'L'v'J'}_{mI,nLvJ}(\vec {r})$, i.e., one term in the energy density of Eq. (31) is Galilean or gauge invariant. For a term in Eq. (43) that corresponds to an independent coupling constant, $G^{n'L'v'J'}_{mI,nLvJ}(\vec {r})$ is equal to a specific linear combination of terms $T$ from the original energy density (31). These linear combinations are listed in Appendix C.

We performed the analysis along these lines for energy densities of orders 0, 2 , 4, and 6, and the obtained results are listed in Appendix C. Derivations were performed by using symbolic programming [38] and employed the technique of forming linear equations by randomly assigning values to local densities and their derivatives, which we also used above. Numbers of linearly independent Galilean or gauge invariant terms are listed in Tables 5 and 6, and plotted in Fig. 1.

It turns out that only at orders 0 and 2, i.e., for the standard Skyrme functional, all Galilean invariant combinations of terms are also gauge invariant. At orders 4 and 6, there are only 6 gauge invariant terms available, while the numbers of Galilean invariant terms equal to 15 and 26, respectively. This is much less than the total numbers of terms at these orders, which are equal to 45 and 129, respectively. Altogether, at N$^3$LO we obtain the EDF parametrized in general by 188 coupling constants, and by 50 or 21 coupling constants if Galilean or gauge invariance is assumed. If isoscalar and isovector channels are included, all these numbers must be multiplied by a factor of two.