Terms in the energy density

Terms in the EDF we construct here are required to be quadratic in densities, invariant with respect to time reversal (Sec. A.1), and covariant with respect to space inversion and rotations (Sec. A.2). All terms up to the N$^3$LO order in derivatives fulfilling these restrictions are constructed below.

Using notation of Eq. (24), a general term in the energy density can be written in the following form,

$$T^{m'I',n'L'v'J'}_{mI,nLvJ,Q}(\vec {r})=[\rho_{m'I',n'L'v'J',Q}(\vec {r})\rho_{mI,nLvJ,Q}(\vec {r})]_0,$$ (28)

where both densities must have the same rank $Q$ to be coupled to a scalar. Moreover, their time-reversal and space-inversion parities ($T$ and $P$) must be the same. Again, at N$^3$LO only terms with $m'+n'+m+n\leq6$ are allowed. Then, the total energy density reads

$${\cal H}(\vec {r})=\sum_{{m'I',n'L'v'J'}\atop{mI,nLvJ,Q}} C^{m'I',n'L'v'J'}_{mI,nLvJ,Q} T^{m'I',n'L'v'J'}_{mI,nLvJ,Q}(\vec {r}),$$ (29)

where $C^{m'I',n'L'v'J'}_{mI,nLvJ,Q}$ are coupling constants and the summation runs over all allowed indices.

Had we considered the case of coupling constants depending on density, all terms in Eq. (28) would have been independent of one another (up to a possible exchange of the two densities). Table 5 lists numbers of such independent terms, and they are also plotted in Fig. 1.

Table 5: Numbers of terms defined in Eq. (28) of different orders in the EDF up to N$^3$LO. Numbers of terms depending on the time-even and time-odd densities are given separately. The last two columns give numbers of terms when the Galilean or gauge invariance is assumed, respectively, see Sec. 3.2. To take into account both isospin channels, the numbers of terms should be multiplied by a factor of two.

order	T-even	T-odd	Total	Galilean	Gauge
0	1	1	2	2	2
2	8	10	18	12	12
4	53	61	114	45	29
6	250	274	524	129	54
N$^3$LO	312	346	658	188	97

Figure 1: (Color online) Numbers of terms (28) and (30) shown in Tables 5 and 6, respectively, plotted in logarithmic scale as a function of the order in derivatives.

In the present study, we concentrate on the case of density-independent coupling constants, in which case one can perform integrations by parts, so that the derivative operators $D_{m'I'}$ are transferred from one density to the other. That this can always be done is obvious by the fact that the coupled derivative operators $D_{m'I'}$ can always be expressed as sums of products of uncoupled derivatives $\nabla_{1\mu}$ or $\nabla_a$. As a result of the integration by parts, the integral (28) can now be written as a sum of terms, where each term has the form:

$$T^{n'L'v'J'}_{mI,nLvJ}(\vec {r})=[\rho_{n'L'v'J'}(\vec {r})[D_{mI}\rho_{nLvJ}(\vec {r})]_{J'}]_0,$$ (30)

where ranks $I$ and $J$ are coupled to $J'$. Here, at N$^3$LO (i) only terms with $n'+m+n\leq6$ are allowed, (ii) both densities must have the same time-reversal parity $T$, and (iii) their space-inversion parities must differ by factors $(-1)^I$. Finally, in order to avoid double-counting one takes only terms with $n'<n$, and for $n'=n$ only those with $L'<L$, and for $L'=L$ only those with $v'<v$, and for $v'=v$ only those with $J'\leq J$. Then, the total energy density reads

$${\cal H}(\vec {r})=\sum_{{n'L'v'J'}\atop{mI,nLvJ,J'}} C^{n'L'v'J'}_{mI,nLvJ} T^{n'L'v'J'}_{mI,nLvJ}(\vec {r}),$$ (31)

where $C^{n'L'v'J'}_{mI,nLvJ}$ are coupling constants and the summation again runs over all allowed indices. As we did for the local densities above, to better visualize the time-reversal characteristics of terms in the EDF, the coupling constants $C^{n'L'v'J'}_{mI,nLvJ}$ corresponding to terms that depend on time-even densities are shown in bold face.

Based on the results obtained in Secs. A.1 and A.2, and on Eqs. (25) and (26), we see that time-reversal invariance and space-inversion covariance require that

$$(-1)^{n'+v'+n+v} = 1 ,$$ (32)

$$(-1)^{n'+m+n} = 1 ,$$ (33)

respectively. This means that integers $v'+v$, $n'+n$, and $m$ must be simultaneously either even or odd. The numbers of all such allowed terms are given in Table 6 and plotted in Fig. 1. The space-inversion covariance (33) requires that for all terms, the total orders in derivatives are even numbers, which defines our classification of the EDF up to LO (0), NLO (2), NNLO (4), and N$^3$LO (6).

Table 6: Same as in Table 5, but for numbers of terms defined in Eq. (30).

order	T-even	T-odd	Total	Galilean	Gauge
0	1	1	2	2	2
2	6	6	12	7	7
4	22	23	45	15	6
6	64	65	129	26	6
N$^3$LO	93	95	188	50	21

In Appendix B, we presented terms in the EDF up to NLO, i.e., for zero and second orders, see Table 22. The EDF at NLO is exactly equivalent to the standard Skyrme functional [28,29], generalized to include all time-odd terms [25,30,24]. In both representations the functional depends, in general, on 14 coupling constants, and both sets are related by simple expressions given in Eqs. (93)-(106).

In Tables 7-18, we list all 45 and 129 terms in the EDF that are of fourth and sixth order, respectively. Together with 14 terms at NLO, listed in Table 22, this constitutes the full list of 188 terms in the EDF at N$^3$LO.

Table 7: Terms in the EDF (30) that are of fourth order, depend on time-even densities, and are built from the scalar nonlocal density $\rho(\vec {r},\vec {r}')$. Coupling constants corresponding to terms that depend on time-even densities are marked by using the bold-face font. Bullets ($\bullet$) mark coupling constants corresponding to terms that do not vanish for conserved spherical, space-inversion, and time-reversal symmetries, see Sec. 4.

No.		$~C_{mI,nLvJ}^{n'L'v'J'}$	$\rho_{n'L'v'J'}$	$D_{mI}$	$\rho_{nLvJ}$
1	$\bullet$	$~\bm{C^{0000}_{40,0000}}$	${[}\rho {]}_{0}$	${[}\nabla\nabla{]}_{0} ^{2}$	${[}\rho {]}_{0}$
2	$\bullet$	$~\bm{C^{0000}_{20,2000}}$	${[}\rho {]}_{0}$	${[}\nabla\nabla{]}_{0}$	${[}{[}kk {]}_{0} \rho{]}_{0}$
3	$\bullet$	$~\bm{C^{0000}_{22,2202}}$	${[}\rho {]}_{0}$	${[}\nabla\nabla{]}_{2}$	${[}{[}kk {]}_{2} \rho{]}_{2}$
4	$\bullet$	$~\bm{C^{0000}_{00,4000}}$	${[}\rho {]}_{0}$	$1$	${[}{[}kk {]}_{0} ^{2} \rho{]}_{0}$
5	$\bullet$	$~\bm{C^{2000}_{00,2000}}$	${[}{[}kk {]}_{0} \rho{]}_{0}$	$1$	${[}{[}kk {]}_{0} \rho{]}_{0}$
6	$\bullet$	$~\bm{C^{2202}_{00,2202}}$	${[}{[}kk {]}_{2} \rho{]}_{2}$	$1$	${[}{[}kk {]}_{2} \rho{]}_{2}$

Table 8: Same as in Table 7 but for terms that are built from the vector nonlocal density $\vec {s}(\vec {r},\vec {r}')$.

No.		$~C_{mI,nLvJ}^{n'L'v'J'}$	$\rho_{n'L'v'J'}$	$D_{mI}$	$\rho_{nLvJ}$
7	$$	$~\bm{C^{1110}_{20,1110}}$	${[}k s{]}_{0}$	${[}\nabla\nabla{]}_{0}$	${[}k s{]}_{0}$
8	$$	$~\bm{C^{1110}_{22,1112}}$	${[}k s{]}_{0}$	${[}\nabla\nabla{]}_{2}$	${[}k s{]}_{2}$
9	$$	$~\bm{C^{1110}_{00,3110}}$	${[}k s{]}_{0}$	$1$	${[}{[}kk{]}_{0} k s{]}_{0}$
10	$\bullet$	$~\bm{C^{1111}_{20,1111}}$	${[}k s{]}_{1}$	${[}\nabla\nabla{]}_{0}$	${[}k s{]}_{1}$
11	$\bullet$	$~\bm{C^{1111}_{22,1111}}$	${[}k s{]}_{1}$	${[}\nabla\nabla{]}_{2}$	${[}k s{]}_{1}$
12	$$	$~\bm{C^{1111}_{22,1112}}$	${[}k s{]}_{1}$	${[}\nabla\nabla{]}_{2}$	${[}k s{]}_{2}$
13	$\bullet$	$~\bm{C^{1111}_{00,3111}}$	${[}k s{]}_{1}$	$1$	${[}{[}kk{]}_{0} k s{]}_{1}$
14	$$	$~\bm{C^{1112}_{20,1112}}$	${[}k s{]}_{2}$	${[}\nabla\nabla{]}_{0}$	${[}k s{]}_{2}$
15	$$	$~\bm{C^{1112}_{22,1112}}$	${[}k s{]}_{2}$	${[}\nabla\nabla{]}_{2}$	${[}k s{]}_{2}$
16	$$	$~\bm{C^{1112}_{00,3112}}$	${[}k s{]}_{2}$	$1$	${[}{[}kk{]}_{0} k s{]}_{2}$
17	$$	$~\bm{C^{1112}_{00,3312}}$	${[}k s{]}_{2}$	$1$	${[}{[}k{[}kk{]}_{2}{]}_{3} s{]}_{2}$

Table 9: Same as in Table 7 but for terms that are built from the scalar nonlocal density $\rho(\vec {r},\vec {r}')$ and vector nonlocal density $\vec {s}(\vec {r},\vec {r}')$.

No.		$~C_{mI,nLvJ}^{n'L'v'J'}$	$\rho_{n'L'v'J'}$	$D_{mI}$	$\rho_{nLvJ}$
18	$\bullet$	$~\bm{C^{0000}_{31,1111}}$	${[}\rho {]}_{0}$	${[}\nabla\nabla{]}_{0} \nabla$	${[}k s{]}_{1}$
19	$\bullet$	$~\bm{C^{0000}_{11,3111}}$	${[}\rho {]}_{0}$	$\nabla$	${[}{[}kk{]}_{0} k s{]}_{1}$
20	$\bullet$	$~\bm{C^{2000}_{11,1111}}$	${[}{[}kk {]}_{0} \rho{]}_{0}$	$\nabla$	${[}k s{]}_{1}$
21	$\bullet$	$~\bm{C^{2202}_{11,1111}}$	${[}{[}kk {]}_{2} \rho{]}_{2}$	$\nabla$	${[}k s{]}_{1}$
22	$$	$~\bm{C^{2202}_{11,1112}}$	${[}{[}kk {]}_{2} \rho{]}_{2}$	$\nabla$	${[}k s{]}_{2}$

Table 10: Terms in the EDF (30) that are of fourth order, depend on time-odd densities, and are built from the scalar nonlocal density $\rho(\vec {r},\vec {r}')$.

No.	$~C_{mI,nLvJ}^{n'L'v'J'}$	$\rho_{n'L'v'J'}$	$D_{mI}$	$\rho_{nLvJ}$
23	$~{}{C^{1101}_{20,1101}}$	${[}k \rho {]}_{1}$	${[}\nabla\nabla{]}_{0}$	${[}k \rho {]}_{1}$
24	$~{}{C^{1101}_{22,1101}}$	${[}k \rho {]}_{1}$	${[}\nabla\nabla{]}_{2}$	${[}k \rho {]}_{1}$
25	$~{}{C^{1101}_{00,3101}}$	${[}k \rho {]}_{1}$	$1$	${[}{[}kk {]}_{0} k \rho{]}_{1}$

Table 11: Same as in Table 10 but for terms that are built from the vector nonlocal density $\vec {s}(\vec {r},\vec {r}')$.

No.	$~C_{mI,nLvJ}^{n'L'v'J'}$	$\rho_{n'L'v'J'}$	$D_{mI}$	$\rho_{nLvJ}$
26	$~{}{C^{0011}_{40,0011}}$	${[}s{]}_{1}$	${[}\nabla\nabla{]}_{0} ^{2}$	${[}s{]}_{1}$
27	$~{}{C^{0011}_{42,0011}}$	${[}s{]}_{1}$	${[}\nabla\nabla{]}_{0} {[}\nabla\nabla{]}_{2}$	${[}s{]}_{1}$
28	$~{}{C^{0011}_{20,2011}}$	${[}s{]}_{1}$	${[}\nabla\nabla{]}_{0}$	${[}{[}kk{]}_{0} s{]}_{1}$
29	$~{}{C^{0011}_{22,2011}}$	${[}s{]}_{1}$	${[}\nabla\nabla{]}_{2}$	${[}{[}kk{]}_{0} s{]}_{1}$
30	$~{}{C^{0011}_{20,2211}}$	${[}s{]}_{1}$	${[}\nabla\nabla{]}_{0}$	${[}{[}kk{]}_{2} s{]}_{1}$
31	$~{}{C^{0011}_{22,2211}}$	${[}s{]}_{1}$	${[}\nabla\nabla{]}_{2}$	${[}{[}kk{]}_{2} s{]}_{1}$
32	$~{}{C^{0011}_{22,2212}}$	${[}s{]}_{1}$	${[}\nabla\nabla{]}_{2}$	${[}{[}kk{]}_{2} s{]}_{2}$
33	$~{}{C^{0011}_{22,2213}}$	${[}s{]}_{1}$	${[}\nabla\nabla{]}_{2}$	${[}{[}kk{]}_{2} s{]}_{3}$
34	$~{}{C^{0011}_{00,4011}}$	${[}s{]}_{1}$	$1$	${[}{[}kk{]}_{0} ^{2} s{]}_{1}$
35	$~{}{C^{0011}_{00,4211}}$	${[}s{]}_{1}$	$1$	${[}{[}kk{]}_{0} {[}kk{]}_{2} s{]}_{1}$
36	$~{}{C^{2011}_{00,2011}}$	${[}{[}kk{]}_{0} s{]}_{1}$	$1$	${[}{[}kk{]}_{0} s{]}_{1}$
37	$~{}{C^{2011}_{00,2211}}$	${[}{[}kk{]}_{0} s{]}_{1}$	$1$	${[}{[}kk{]}_{2} s{]}_{1}$
38	$~{}{C^{2211}_{00,2211}}$	${[}{[}kk{]}_{2} s{]}_{1}$	$1$	${[}{[}kk{]}_{2} s{]}_{1}$
39	$~{}{C^{2212}_{00,2212}}$	${[}{[}kk{]}_{2} s{]}_{2}$	$1$	${[}{[}kk{]}_{2} s{]}_{2}$
40	$~{}{C^{2213}_{00,2213}}$	${[}{[}kk{]}_{2} s{]}_{3}$	$1$	${[}{[}kk{]}_{2} s{]}_{3}$

Table 12: Same as in Table 10 but for terms that are built from the scalar nonlocal density $\rho(\vec {r},\vec {r}')$ and vector nonlocal density $\vec {s}(\vec {r},\vec {r}')$.

No.	$~C_{mI,nLvJ}^{n'L'v'J'}$	$\rho_{n'L'v'J'}$	$D_{mI}$	$\rho_{nLvJ}$
41	$~{}{C^{1101}_{31,0011}}$	${[}k \rho {]}_{1}$	${[}\nabla\nabla{]}_{0} \nabla$	${[}s{]}_{1}$
42	$~{}{C^{1101}_{11,2011}}$	${[}k \rho {]}_{1}$	$\nabla$	${[}{[}kk{]}_{0} s{]}_{1}$
43	$~{}{C^{1101}_{11,2211}}$	${[}k \rho {]}_{1}$	$\nabla$	${[}{[}kk{]}_{2} s{]}_{1}$
44	$~{}{C^{1101}_{11,2212}}$	${[}k \rho {]}_{1}$	$\nabla$	${[}{[}kk{]}_{2} s{]}_{2}$
45	$~{}{C^{3101}_{11,0011}}$	${[}{[}kk {]}_{0} k \rho{]}_{1}$	$\nabla$	${[}s{]}_{1}$

Table 13: Terms in the EDF (30) that are of sixth order, depend on time-even densities, and are built from the scalar nonlocal density $\rho(\vec {r},\vec {r}')$. Coupling constants corresponding to terms that depend on time-even densities are marked by using the bold-face font. Bullets ($\bullet$) mark coupling constants corresponding to terms that do not vanish for conserved spherical, space-inversion, and time-reversal symmetries, see Sec. 4.

No.		$~C_{mI,nLvJ}^{n'L'v'J'}$	$\rho_{n'L'v'J'}$	$D_{mI}$	$\rho_{nLvJ}$
1	$\bullet$	$~\bm{C^{0000}_{60,0000}}$	${[}\rho {]}_{0}$	${[}\nabla\nabla{]}_{0} ^3$	${[}\rho {]}_{0}$
2	$\bullet$	$~\bm{C^{0000}_{40,2000}}$	${[}\rho {]}_{0}$	${[}\nabla\nabla{]}_{0} ^{2}$	${[}{[}kk {]}_{0} \rho{]}_{0}$
3	$\bullet$	$~\bm{C^{0000}_{42,2202}}$	${[}\rho {]}_{0}$	${[}\nabla\nabla{]}_{0} {[}\nabla\nabla{]}_{2}$	${[}{[}kk {]}_{2} \rho{]}_{2}$
4	$\bullet$	$~\bm{C^{0000}_{20,4000}}$	${[}\rho {]}_{0}$	${[}\nabla\nabla{]}_{0}$	${[}{[}kk {]}_{0} ^{2} \rho{]}_{0}$
5	$\bullet$	$~\bm{C^{0000}_{22,4202}}$	${[}\rho {]}_{0}$	${[}\nabla\nabla{]}_{2}$	${[}{[}kk {]}_{0} {[}kk{]}_{2} \rho{]}_{2}$
6	$\bullet$	$~\bm{C^{0000}_{00,6000}}$	${[}\rho {]}_{0}$	$1$	${[}{[}kk {]}_{0} ^3 \rho{]}_{0}$
7	$\bullet$	$~\bm{C^{2000}_{20,2000}}$	${[}{[}kk {]}_{0} \rho{]}_{0}$	${[}\nabla\nabla{]}_{0}$	${[}{[}kk {]}_{0} \rho{]}_{0}$
8	$\bullet$	$~\bm{C^{2000}_{22,2202}}$	${[}{[}kk {]}_{0} \rho{]}_{0}$	${[}\nabla\nabla{]}_{2}$	${[}{[}kk {]}_{2} \rho{]}_{2}$
9	$\bullet$	$~\bm{C^{2000}_{00,4000}}$	${[}{[}kk {]}_{0} \rho{]}_{0}$	$1$	${[}{[}kk {]}_{0} ^{2} \rho{]}_{0}$
10	$\bullet$	$~\bm{C^{2202}_{20,2202}}$	${[}{[}kk {]}_{2} \rho{]}_{2}$	${[}\nabla\nabla{]}_{0}$	${[}{[}kk {]}_{2} \rho{]}_{2}$
11	$\bullet$	$~\bm{C^{2202}_{22,2202}}$	${[}{[}kk {]}_{2} \rho{]}_{2}$	${[}\nabla\nabla{]}_{2}$	${[}{[}kk {]}_{2} \rho{]}_{2}$
12	$\bullet$	$~\bm{C^{2202}_{00,4202}}$	${[}{[}kk {]}_{2} \rho{]}_{2}$	$1$	${[}{[}kk {]}_{0} {[}kk{]}_{2} \rho{]}_{2}$

Table 14: Same as in Table 13 but for terms that are built from the vector nonlocal density $\vec {s}(\vec {r},\vec {r}')$.

No.		$~C_{mI,nLvJ}^{n'L'v'J'}$	$\rho_{n'L'v'J'}$	$D_{mI}$	$\rho_{nLvJ}$
13	$$	$~\bm{C^{1110}_{40,1110}}$	${[}k s{]}_{0}$	${[}\nabla\nabla{]}_{0} ^{2}$	${[}k s{]}_{0}$
14	$$	$~\bm{C^{1110}_{42,1112}}$	${[}k s{]}_{0}$	${[}\nabla\nabla{]}_{0} {[}\nabla\nabla{]}_{2}$	${[}k s{]}_{2}$
15	$$	$~\bm{C^{1110}_{20,3110}}$	${[}k s{]}_{0}$	${[}\nabla\nabla{]}_{0}$	${[}{[}kk{]}_{0} k s{]}_{0}$
16	$$	$~\bm{C^{1110}_{22,3112}}$	${[}k s{]}_{0}$	${[}\nabla\nabla{]}_{2}$	${[}{[}kk{]}_{0} k s{]}_{2}$
17	$$	$~\bm{C^{1110}_{22,3312}}$	${[}k s{]}_{0}$	${[}\nabla\nabla{]}_{2}$	${[}{[}k{[}kk{]}_{2}{]}_{3} s{]}_{2}$
18	$$	$~\bm{C^{1110}_{00,5110}}$	${[}k s{]}_{0}$	$1$	${[}{[}kk{]}_{0}^2 k s{]}_{0}$
19	$\bullet$	$~\bm{C^{1111}_{40,1111}}$	${[}k s{]}_{1}$	${[}\nabla\nabla{]}_{0} ^{2}$	${[}k s{]}_{1}$
20	$\bullet$	$~\bm{C^{1111}_{42,1111}}$	${[}k s{]}_{1}$	${[}\nabla\nabla{]}_{0} {[}\nabla\nabla{]}_{2}$	${[}k s{]}_{1}$
21	$$	$~\bm{C^{1111}_{42,1112}}$	${[}k s{]}_{1}$	${[}\nabla\nabla{]}_{0} {[}\nabla\nabla{]}_{2}$	${[}k s{]}_{2}$
22	$\bullet$	$~\bm{C^{1111}_{20,3111}}$	${[}k s{]}_{1}$	${[}\nabla\nabla{]}_{0}$	${[}{[}kk{]}_{0} k s{]}_{1}$
23	$\bullet$	$~\bm{C^{1111}_{22,3111}}$	${[}k s{]}_{1}$	${[}\nabla\nabla{]}_{2}$	${[}{[}kk{]}_{0} k s{]}_{1}$
24	$$	$~\bm{C^{1111}_{22,3112}}$	${[}k s{]}_{1}$	${[}\nabla\nabla{]}_{2}$	${[}{[}kk{]}_{0} k s{]}_{2}$
25	$$	$~\bm{C^{1111}_{22,3312}}$	${[}k s{]}_{1}$	${[}\nabla\nabla{]}_{2}$	${[}{[}k{[}kk{]}_{2}{]}_{3} s{]}_{2}$
26	$\bullet$	$~\bm{C^{1111}_{22,3313}}$	${[}k s{]}_{1}$	${[}\nabla\nabla{]}_{2}$	${[}{[}k{[}kk{]}_{2}{]}_{3} s{]}_{3}$
27	$\bullet$	$~\bm{C^{1111}_{00,5111}}$	${[}k s{]}_{1}$	$1$	${[}{[}kk{]}_{0}^2 k s{]}_{1}$

Table 14: continued.

No.		$~C_{mI,nLvJ}^{n'L'v'J'}$	$\rho_{n'L'v'J'}$	$D_{mI}$	$\rho_{nLvJ}$
28	$$	$~\bm{C^{1112}_{40,1112}}$	${[}k s{]}_{2}$	${[}\nabla\nabla{]}_{0} ^{2}$	${[}k s{]}_{2}$
29	$$	$~\bm{C^{1112}_{42,1112}}$	${[}k s{]}_{2}$	${[}\nabla\nabla{]}_{0} {[}\nabla\nabla{]}_{2}$	${[}k s{]}_{2}$
30	$$	$~\bm{C^{1112}_{44,1112}}$	${[}k s{]}_{2}$	${[}\nabla{[}\nabla{[}\nabla\nabla{]}_{2}{]}_{3}{]}_{4}$	${[}k s{]}_{2}$
31	$$	$~\bm{C^{1112}_{22,3110}}$	${[}k s{]}_{2}$	${[}\nabla\nabla{]}_{2}$	${[}{[}kk{]}_{0} k s{]}_{0}$
32	$$	$~\bm{C^{1112}_{22,3111}}$	${[}k s{]}_{2}$	${[}\nabla\nabla{]}_{2}$	${[}{[}kk{]}_{0} k s{]}_{1}$
33	$$	$~\bm{C^{1112}_{20,3112}}$	${[}k s{]}_{2}$	${[}\nabla\nabla{]}_{0}$	${[}{[}kk{]}_{0} k s{]}_{2}$
34	$$	$~\bm{C^{1112}_{22,3112}}$	${[}k s{]}_{2}$	${[}\nabla\nabla{]}_{2}$	${[}{[}kk{]}_{0} k s{]}_{2}$
35	$$	$~\bm{C^{1112}_{20,3312}}$	${[}k s{]}_{2}$	${[}\nabla\nabla{]}_{0}$	${[}{[}k{[}kk{]}_{2}{]}_{3} s{]}_{2}$
36	$$	$~\bm{C^{1112}_{22,3312}}$	${[}k s{]}_{2}$	${[}\nabla\nabla{]}_{2}$	${[}{[}k{[}kk{]}_{2}{]}_{3} s{]}_{2}$
37	$$	$~\bm{C^{1112}_{22,3313}}$	${[}k s{]}_{2}$	${[}\nabla\nabla{]}_{2}$	${[}{[}k{[}kk{]}_{2}{]}_{3} s{]}_{3}$
38	$$	$~\bm{C^{1112}_{22,3314}}$	${[}k s{]}_{2}$	${[}\nabla\nabla{]}_{2}$	${[}{[}k{[}kk{]}_{2}{]}_{3} s{]}_{4}$
39	$$	$~\bm{C^{1112}_{00,5112}}$	${[}k s{]}_{2}$	$1$	${[}{[}kk{]}_{0}^2 k s{]}_{2}$
40	$$	$~\bm{C^{1112}_{00,5312}}$	${[}k s{]}_{2}$	$1$	${[}{[}kk{]}_{0} {[}k{[}kk{]}_{2}{]}_{3} s{]}_{2}$
41	$$	$~\bm{C^{3110}_{00,3110}}$	${[}{[}kk{]}_{0} k s{]}_{0}$	$1$	${[}{[}kk{]}_{0} k s{]}_{0}$
42	$\bullet$	$~\bm{C^{3111}_{00,3111}}$	${[}{[}kk{]}_{0} k s{]}_{1}$	$1$	${[}{[}kk{]}_{0} k s{]}_{1}$
43	$$	$~\bm{C^{3112}_{00,3112}}$	${[}{[}kk{]}_{0} k s{]}_{2}$	$1$	${[}{[}kk{]}_{0} k s{]}_{2}$
44	$$	$~\bm{C^{3112}_{00,3312}}$	${[}{[}kk{]}_{0} k s{]}_{2}$	$1$	${[}{[}k{[}kk{]}_{2}{]}_{3} s{]}_{2}$
45	$$	$~\bm{C^{3312}_{00,3312}}$	${[}{[}k{[}kk{]}_{2}{]}_{3} s{]}_{2}$	$1$	${[}{[}k{[}kk{]}_{2}{]}_{3} s{]}_{2}$
46	$\bullet$	$~\bm{C^{3313}_{00,3313}}$	${[}{[}k{[}kk{]}_{2}{]}_{3} s{]}_{3}$	$1$	${[}{[}k{[}kk{]}_{2}{]}_{3} s{]}_{3}$
47	$$	$~\bm{C^{3314}_{00,3314}}$	${[}{[}k{[}kk{]}_{2}{]}_{3} s{]}_{4}$	$1$	${[}{[}k{[}kk{]}_{2}{]}_{3} s{]}_{4}$

Table 15: Same as in Table 13 but for terms that are built from the scalar nonlocal density $\rho(\vec {r},\vec {r}')$ and vector nonlocal density $\vec {s}(\vec {r},\vec {r}')$.

No.		$~C_{mI,nLvJ}^{n'L'v'J'}$	$\rho_{n'L'v'J'}$	$D_{mI}$	$\rho_{nLvJ}$
48	$\bullet$	$~\bm{C^{0000}_{51,1111}}$	${[}\rho {]}_{0}$	${[}\nabla\nabla{]}_{0}^2 \nabla$	${[}k s{]}_{1}$
49	$\bullet$	$~\bm{C^{0000}_{31,3111}}$	${[}\rho {]}_{0}$	${[}\nabla\nabla{]}_{0} \nabla$	${[}{[}kk{]}_{0} k s{]}_{1}$
50	$\bullet$	$~\bm{C^{0000}_{33,3313}}$	${[}\rho {]}_{0}$	${[}\nabla{[}\nabla\nabla{]}_{2}{]}_{3}$	${[}{[}k{[}kk{]}_{2}{]}_{3} s{]}_{3}$
51	$\bullet$	$~\bm{C^{0000}_{11,5111}}$	${[}\rho {]}_{0}$	$\nabla$	${[}{[}kk{]}_{0}^2 k s{]}_{1}$
52	$\bullet$	$~\bm{C^{2000}_{31,1111}}$	${[}{[}kk {]}_{0} \rho{]}_{0}$	${[}\nabla\nabla{]}_{0} \nabla$	${[}k s{]}_{1}$
53	$\bullet$	$~\bm{C^{2000}_{11,3111}}$	${[}{[}kk {]}_{0} \rho{]}_{0}$	$\nabla$	${[}{[}kk{]}_{0} k s{]}_{1}$
54	$\bullet$	$~\bm{C^{2202}_{31,1111}}$	${[}{[}kk {]}_{2} \rho{]}_{2}$	${[}\nabla\nabla{]}_{0} \nabla$	${[}k s{]}_{1}$
55	$\bullet$	$~\bm{C^{2202}_{33,1111}}$	${[}{[}kk {]}_{2} \rho{]}_{2}$	${[}\nabla{[}\nabla\nabla{]}_{2}{]}_{3}$	${[}k s{]}_{1}$
56	$$	$~\bm{C^{2202}_{31,1112}}$	${[}{[}kk {]}_{2} \rho{]}_{2}$	${[}\nabla\nabla{]}_{0} \nabla$	${[}k s{]}_{2}$
57	$$	$~\bm{C^{2202}_{33,1112}}$	${[}{[}kk {]}_{2} \rho{]}_{2}$	${[}\nabla{[}\nabla\nabla{]}_{2}{]}_{3}$	${[}k s{]}_{2}$
58	$\bullet$	$~\bm{C^{2202}_{11,3111}}$	${[}{[}kk {]}_{2} \rho{]}_{2}$	$\nabla$	${[}{[}kk{]}_{0} k s{]}_{1}$
59	$$	$~\bm{C^{2202}_{11,3112}}$	${[}{[}kk {]}_{2} \rho{]}_{2}$	$\nabla$	${[}{[}kk{]}_{0} k s{]}_{2}$
60	$$	$~\bm{C^{2202}_{11,3312}}$	${[}{[}kk {]}_{2} \rho{]}_{2}$	$\nabla$	${[}{[}k{[}kk{]}_{2}{]}_{3} s{]}_{2}$
61	$\bullet$	$~\bm{C^{2202}_{11,3313}}$	${[}{[}kk {]}_{2} \rho{]}_{2}$	$\nabla$	${[}{[}k{[}kk{]}_{2}{]}_{3} s{]}_{3}$
62	$\bullet$	$~\bm{C^{4000}_{11,1111}}$	${[}{[}kk {]}_{0} ^{2} \rho{]}_{0}$	$\nabla$	${[}k s{]}_{1}$
63	$\bullet$	$~\bm{C^{4202}_{11,1111}}$	${[}{[}kk {]}_{0} {[}kk{]}_{2} \rho{]}_{2}$	$\nabla$	${[}k s{]}_{1}$
64	$$	$~\bm{C^{4202}_{11,1112}}$	${[}{[}kk {]}_{0} {[}kk{]}_{2} \rho{]}_{2}$	$\nabla$	${[}k s{]}_{2}$

Table 16: Terms in the EDF (30) that are of sixth order, depend on time-odd densities, and are built from the scalar nonlocal density $\rho(\vec {r},\vec {r}')$.

No.	$~C_{mI,nLvJ}^{n'L'v'J'}$	$\rho_{n'L'v'J'}$	$D_{mI}$	$\rho_{nLvJ}$
65	$~{}{C^{1101}_{40,1101}}$	${[}k \rho {]}_{1}$	${[}\nabla\nabla{]}_{0} ^{2}$	${[}k \rho {]}_{1}$
66	$~{}{C^{1101}_{42,1101}}$	${[}k \rho {]}_{1}$	${[}\nabla\nabla{]}_{0} {[}\nabla\nabla{]}_{2}$	${[}k \rho {]}_{1}$
67	$~{}{C^{1101}_{20,3101}}$	${[}k \rho {]}_{1}$	${[}\nabla\nabla{]}_{0}$	${[}{[}kk {]}_{0} k \rho{]}_{1}$
68	$~{}{C^{1101}_{22,3101}}$	${[}k \rho {]}_{1}$	${[}\nabla\nabla{]}_{2}$	${[}{[}kk {]}_{0} k \rho{]}_{1}$
69	$~{}{C^{1101}_{22,3303}}$	${[}k \rho {]}_{1}$	${[}\nabla\nabla{]}_{2}$	${[}{[}k{[}kk{]}_{2}{]}_{3} \rho{]}_{3}$
70	$~{}{C^{1101}_{00,5101}}$	${[}k \rho {]}_{1}$	$1$	${[}{[}kk {]}_{0}^2 k \rho{]}_{1}$
71	$~{}{C^{3101}_{00,3101}}$	${[}{[}kk {]}_{0} k \rho{]}_{1}$	$1$	${[}{[}kk {]}_{0} k \rho{]}_{1}$
72	$~{}{C^{3303}_{00,3303}}$	${[}{[}k{[}kk{]}_{2}{]}_{3} \rho{]}_{3}$	$1$	${[}{[}k{[}kk{]}_{2}{]}_{3} \rho{]}_{3}$

Table 17: Same as in Table 16 but for terms that are built from the vector nonlocal density $\vec {s}(\vec {r},\vec {r}')$.

No.	$~C_{mI,nLvJ}^{n'L'v'J'}$	$\rho_{n'L'v'J'}$	$D_{mI}$	$\rho_{nLvJ}$
73	$~{}{C^{0011}_{60,0011}}$	${[}s{]}_{1}$	${[}\nabla\nabla{]}_{0} ^3$	${[}s{]}_{1}$
74	$~{}{C^{0011}_{62,0011}}$	${[}s{]}_{1}$	${[}\nabla\nabla{]}_{0}^2 {[}\nabla\nabla{]}_{2}$	${[}s{]}_{1}$
75	$~{}{C^{0011}_{40,2011}}$	${[}s{]}_{1}$	${[}\nabla\nabla{]}_{0} ^{2}$	${[}{[}kk{]}_{0} s{]}_{1}$
76	$~{}{C^{0011}_{42,2011}}$	${[}s{]}_{1}$	${[}\nabla\nabla{]}_{0} {[}\nabla\nabla{]}_{2}$	${[}{[}kk{]}_{0} s{]}_{1}$
77	$~{}{C^{0011}_{40,2211}}$	${[}s{]}_{1}$	${[}\nabla\nabla{]}_{0} ^{2}$	${[}{[}kk{]}_{2} s{]}_{1}$
78	$~{}{C^{0011}_{42,2211}}$	${[}s{]}_{1}$	${[}\nabla\nabla{]}_{0} {[}\nabla\nabla{]}_{2}$	${[}{[}kk{]}_{2} s{]}_{1}$
79	$~{}{C^{0011}_{42,2212}}$	${[}s{]}_{1}$	${[}\nabla\nabla{]}_{0} {[}\nabla\nabla{]}_{2}$	${[}{[}kk{]}_{2} s{]}_{2}$
80	$~{}{C^{0011}_{42,2213}}$	${[}s{]}_{1}$	${[}\nabla\nabla{]}_{0} {[}\nabla\nabla{]}_{2}$	${[}{[}kk{]}_{2} s{]}_{3}$
81	$~{}{C^{0011}_{44,2213}}$	${[}s{]}_{1}$	${[}\nabla{[}\nabla{[}\nabla\nabla{]}_{2}{]}_{3}{]}_{4}$	${[}{[}kk{]}_{2} s{]}_{3}$
82	$~{}{C^{0011}_{20,4011}}$	${[}s{]}_{1}$	${[}\nabla\nabla{]}_{0}$	${[}{[}kk{]}_{0} ^{2} s{]}_{1}$
83	$~{}{C^{0011}_{22,4011}}$	${[}s{]}_{1}$	${[}\nabla\nabla{]}_{2}$	${[}{[}kk{]}_{0} ^{2} s{]}_{1}$
84	$~{}{C^{0011}_{20,4211}}$	${[}s{]}_{1}$	${[}\nabla\nabla{]}_{0}$	${[}{[}kk{]}_{0} {[}kk{]}_{2} s{]}_{1}$
85	$~{}{C^{0011}_{22,4211}}$	${[}s{]}_{1}$	${[}\nabla\nabla{]}_{2}$	${[}{[}kk{]}_{0} {[}kk{]}_{2} s{]}_{1}$
86	$~{}{C^{0011}_{22,4212}}$	${[}s{]}_{1}$	${[}\nabla\nabla{]}_{2}$	${[}{[}kk{]}_{0} {[}kk{]}_{2} s{]}_{2}$
87	$~{}{C^{0011}_{22,4213}}$	${[}s{]}_{1}$	${[}\nabla\nabla{]}_{2}$	${[}{[}kk{]}_{0} {[}kk{]}_{2} s{]}_{3}$
88	$~{}{C^{0011}_{22,4413}}$	${[}s{]}_{1}$	${[}\nabla\nabla{]}_{2}$	${[}{[}k{[}k{[}kk{]}_{2}{]}_{3}{]}_{4} s{]}_{3}$
89	$~{}{C^{0011}_{00,6011}}$	${[}s{]}_{1}$	$1$	${[}{[}kk{]}_{0} ^3 s{]}_{1}$
90	$~{}{C^{0011}_{00,6211}}$	${[}s{]}_{1}$	$1$	${[}{[}kk{]}_{0}^2 {[}kk{]}_{2} s{]}_{1}$

Table 17: continued.

No.	$~C_{mI,nLvJ}^{n'L'v'J'}$	$\rho_{n'L'v'J'}$	$D_{mI}$	$\rho_{nLvJ}$
91	$~{}{C^{2011}_{20,2011}}$	${[}{[}kk{]}_{0} s{]}_{1}$	${[}\nabla\nabla{]}_{0}$	${[}{[}kk{]}_{0} s{]}_{1}$
92	$~{}{C^{2011}_{22,2011}}$	${[}{[}kk{]}_{0} s{]}_{1}$	${[}\nabla\nabla{]}_{2}$	${[}{[}kk{]}_{0} s{]}_{1}$
93	$~{}{C^{2011}_{20,2211}}$	${[}{[}kk{]}_{0} s{]}_{1}$	${[}\nabla\nabla{]}_{0}$	${[}{[}kk{]}_{2} s{]}_{1}$
94	$~{}{C^{2011}_{22,2211}}$	${[}{[}kk{]}_{0} s{]}_{1}$	${[}\nabla\nabla{]}_{2}$	${[}{[}kk{]}_{2} s{]}_{1}$
95	$~{}{C^{2011}_{22,2212}}$	${[}{[}kk{]}_{0} s{]}_{1}$	${[}\nabla\nabla{]}_{2}$	${[}{[}kk{]}_{2} s{]}_{2}$
96	$~{}{C^{2011}_{22,2213}}$	${[}{[}kk{]}_{0} s{]}_{1}$	${[}\nabla\nabla{]}_{2}$	${[}{[}kk{]}_{2} s{]}_{3}$
97	$~{}{C^{2011}_{00,4011}}$	${[}{[}kk{]}_{0} s{]}_{1}$	$1$	${[}{[}kk{]}_{0} ^{2} s{]}_{1}$
98	$~{}{C^{2011}_{00,4211}}$	${[}{[}kk{]}_{0} s{]}_{1}$	$1$	${[}{[}kk{]}_{0} {[}kk{]}_{2} s{]}_{1}$
99	$~{}{C^{2211}_{20,2211}}$	${[}{[}kk{]}_{2} s{]}_{1}$	${[}\nabla\nabla{]}_{0}$	${[}{[}kk{]}_{2} s{]}_{1}$
100	$~{}{C^{2211}_{22,2211}}$	${[}{[}kk{]}_{2} s{]}_{1}$	${[}\nabla\nabla{]}_{2}$	${[}{[}kk{]}_{2} s{]}_{1}$
101	$~{}{C^{2211}_{22,2212}}$	${[}{[}kk{]}_{2} s{]}_{1}$	${[}\nabla\nabla{]}_{2}$	${[}{[}kk{]}_{2} s{]}_{2}$
102	$~{}{C^{2211}_{22,2213}}$	${[}{[}kk{]}_{2} s{]}_{1}$	${[}\nabla\nabla{]}_{2}$	${[}{[}kk{]}_{2} s{]}_{3}$
103	$~{}{C^{2211}_{00,4011}}$	${[}{[}kk{]}_{2} s{]}_{1}$	$1$	${[}{[}kk{]}_{0} ^{2} s{]}_{1}$
104	$~{}{C^{2211}_{00,4211}}$	${[}{[}kk{]}_{2} s{]}_{1}$	$1$	${[}{[}kk{]}_{0} {[}kk{]}_{2} s{]}_{1}$
105	$~{}{C^{2212}_{20,2212}}$	${[}{[}kk{]}_{2} s{]}_{2}$	${[}\nabla\nabla{]}_{0}$	${[}{[}kk{]}_{2} s{]}_{2}$
106	$~{}{C^{2212}_{22,2212}}$	${[}{[}kk{]}_{2} s{]}_{2}$	${[}\nabla\nabla{]}_{2}$	${[}{[}kk{]}_{2} s{]}_{2}$
107	$~{}{C^{2212}_{22,2213}}$	${[}{[}kk{]}_{2} s{]}_{2}$	${[}\nabla\nabla{]}_{2}$	${[}{[}kk{]}_{2} s{]}_{3}$
108	$~{}{C^{2212}_{00,4212}}$	${[}{[}kk{]}_{2} s{]}_{2}$	$1$	${[}{[}kk{]}_{0} {[}kk{]}_{2} s{]}_{2}$
109	$~{}{C^{2213}_{20,2213}}$	${[}{[}kk{]}_{2} s{]}_{3}$	${[}\nabla\nabla{]}_{0}$	${[}{[}kk{]}_{2} s{]}_{3}$
110	$~{}{C^{2213}_{22,2213}}$	${[}{[}kk{]}_{2} s{]}_{3}$	${[}\nabla\nabla{]}_{2}$	${[}{[}kk{]}_{2} s{]}_{3}$
111	$~{}{C^{2213}_{00,4213}}$	${[}{[}kk{]}_{2} s{]}_{3}$	$1$	${[}{[}kk{]}_{0} {[}kk{]}_{2} s{]}_{3}$
112	$~{}{C^{2213}_{00,4413}}$	${[}{[}kk{]}_{2} s{]}_{3}$	$1$	${[}{[}k{[}k{[}kk{]}_{2}{]}_{3}{]}_{4} s{]}_{3}$

Table 18: Same as in Table 16 but for terms that are built from the scalar nonlocal density $\rho(\vec {r},\vec {r}')$ and vector nonlocal density $\vec {s}(\vec {r},\vec {r}')$.

No.	$~C_{mI,nLvJ}^{n'L'v'J'}$	$\rho_{n'L'v'J'}$	$D_{mI}$	$\rho_{nLvJ}$
113	$~{}{C^{1101}_{51,0011}}$	${[}k \rho {]}_{1}$	${[}\nabla\nabla{]}_{0}^2 \nabla$	${[}s{]}_{1}$
114	$~{}{C^{1101}_{31,2011}}$	${[}k \rho {]}_{1}$	${[}\nabla\nabla{]}_{0} \nabla$	${[}{[}kk{]}_{0} s{]}_{1}$
115	$~{}{C^{1101}_{31,2211}}$	${[}k \rho {]}_{1}$	${[}\nabla\nabla{]}_{0} \nabla$	${[}{[}kk{]}_{2} s{]}_{1}$
116	$~{}{C^{1101}_{31,2212}}$	${[}k \rho {]}_{1}$	${[}\nabla\nabla{]}_{0} \nabla$	${[}{[}kk{]}_{2} s{]}_{2}$
117	$~{}{C^{1101}_{33,2212}}$	${[}k \rho {]}_{1}$	${[}\nabla{[}\nabla\nabla{]}_{2}{]}_{3}$	${[}{[}kk{]}_{2} s{]}_{2}$
118	$~{}{C^{1101}_{33,2213}}$	${[}k \rho {]}_{1}$	${[}\nabla{[}\nabla\nabla{]}_{2}{]}_{3}$	${[}{[}kk{]}_{2} s{]}_{3}$
119	$~{}{C^{1101}_{11,4011}}$	${[}k \rho {]}_{1}$	$\nabla$	${[}{[}kk{]}_{0} ^{2} s{]}_{1}$
120	$~{}{C^{1101}_{11,4211}}$	${[}k \rho {]}_{1}$	$\nabla$	${[}{[}kk{]}_{0} {[}kk{]}_{2} s{]}_{1}$
121	$~{}{C^{1101}_{11,4212}}$	${[}k \rho {]}_{1}$	$\nabla$	${[}{[}kk{]}_{0} {[}kk{]}_{2} s{]}_{2}$
122	$~{}{C^{3101}_{31,0011}}$	${[}{[}kk {]}_{0} k \rho{]}_{1}$	${[}\nabla\nabla{]}_{0} \nabla$	${[}s{]}_{1}$
123	$~{}{C^{3101}_{11,2011}}$	${[}{[}kk {]}_{0} k \rho{]}_{1}$	$\nabla$	${[}{[}kk{]}_{0} s{]}_{1}$
124	$~{}{C^{3101}_{11,2211}}$	${[}{[}kk {]}_{0} k \rho{]}_{1}$	$\nabla$	${[}{[}kk{]}_{2} s{]}_{1}$
125	$~{}{C^{3101}_{11,2212}}$	${[}{[}kk {]}_{0} k \rho{]}_{1}$	$\nabla$	${[}{[}kk{]}_{2} s{]}_{2}$
126	$~{}{C^{3303}_{33,0011}}$	${[}{[}k{[}kk{]}_{2}{]}_{3} \rho{]}_{3}$	${[}\nabla{[}\nabla\nabla{]}_{2}{]}_{3}$	${[}s{]}_{1}$
127	$~{}{C^{3303}_{11,2212}}$	${[}{[}k{[}kk{]}_{2}{]}_{3} \rho{]}_{3}$	$\nabla$	${[}{[}kk{]}_{2} s{]}_{2}$
128	$~{}{C^{3303}_{11,2213}}$	${[}{[}k{[}kk{]}_{2}{]}_{3} \rho{]}_{3}$	$\nabla$	${[}{[}kk{]}_{2} s{]}_{3}$
129	$~{}{C^{5101}_{11,0011}}$	${[}{[}kk {]}_{0}^2 k \rho{]}_{1}$	$\nabla$	${[}s{]}_{1}$

Table 19: Numbers of local densities $\rho_{nLvJ}$, Eq. (23), of different orders, which enter into the EDF up to N$^3$LO. Numbers of local densities constructed from the scalar $\rho(\vec {r},\vec {r}')$ or vector $\vec {s}(\vec {r},\vec {r}')$ nonlocal densities, and numbers of time-even and time-odd local densities, are given separately. In Tables 3 and 4 these densities are marked with stars ($\star$).

order	from $\rho$	from $\vec {s}$	T-even	T-odd	total
0	1	1	1	1	2
1	1	3	3	1	4
2	2	4	2	4	6
3	2	6	6	2	8
4	2	5	2	5	7
5	1	4	4	1	5
6	1	2	1	2	3
total	10	25	19	16	35

After the complete list of terms in the EDF at N$^3$LO is constructed, one can check that not all of the local densities listed in Tables 3 and 4 appear in the final EDF at N$^3$LO. This is so, because it is not possible to couple all these densities to scalars, and simultaneously fulfill conditions (32) and (33), without obtaining more than total sixth order in derivatives. It turns out that out of the 56 local densities at N$^3$LO, which are listed in Tables 3 and 4, only 35 occur in the final EDF at N$^3$LO. In Tables 3 and 4 such densities are marked with stars ($\star$). Table 19 gives their numbers determined separately at each order.