Besides the Galilean invariance mentioned above, the standard Skyrme force has been also proved to be invariant with respect to a more general local gauge invariance, and to give rise to the energy density that is invariant under the same symmetry when specific relations between the coupling constants are set [26,27].
The gauge transformation acts on a many-body wave function by
multiplying it with a position-dependent phase factor, that is,
Apart from zero order, the terms of the pseudopotential are not trivially invariant with
respect to the transformation of the Eq. (20) and, in general,
the transformed pseudopotential is different than the
original pseudopotential
. To impose the gauge invariance on
the pseudopotential, one has to derive a list of constraints among the
parameters, which can be done using the condition
At fourth order, the gauge invariance forces seven parameters of the
pseudopotential to be specific linear combinations of four
independent ones. In Table 4, they are marked by
letters D and I, respectively. In Appendix B, we list such
relations between the dependent and independent parameters. One
should note that other choices of the four independent parameters are
also possible, that is, at fourth order, there are simply four
different gauge-invariant linear combinations of terms of the
pseudopotential (1). Moreover, at this order, there are also
two terms that alone are gauge non-invariant - those that correspond
to parameters
and
; in
Table 4, they are marked by letters N. Similarly, at
sixth order, there are six gauge-invariant linear combinations of
terms of the pseudopotential, that is, sixteen dependent parameters
are related to six independent ones, see Appendix B, and
there are also four alone gauge non-invariant terms corresponding to
parameters
,
,
, and
.
A comparison between the numbers of terms of the Galilean-invariant
pseudopotential and the gauge-invariant pseudopotential is plotted in
Fig. 1. Again we note that at each order, the numbers of
gauge-invariant parameters (2 for the zero order, 7 for the second
order, 6 for the fourth order, and 6 for the sixth order) are exactly
the same as the numbers of independent coupling constants of the EDF
in each isospin channel with the gauge invariance imposed,
cf. Table VI of Ref. [2]. Again, this observation will
be crucial when we proceed to derive relations between the isoscalar
and the isovector parts of the EDF, stemming from the gauge-invariant
pseudopotential. We also remark that whereas the second-order
spin-orbit term, corresponding to parameter
, is
gauge invariant, all higher-order spin-orbit terms, corresponding to
parameters
with
do violate the gauge symmetry.
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