# Colorings of some Cayley graphs

Prajnanaswaroopa S
<EMAIL_ADDRESS>
Abstract: Cayley graphs are graphs on algebraic structures, typically groups
or group-like structures. In this paper, we have obtained a few results on
Cayley graphs on Cyclic groups, typically powers of cycles, some colorings of
powers of cycles, Cayley graphs on some non-abelian groups, and Cayley graphs
on gyrogroups.

##  1 Introduction

For a simple loopless graph $G$, we denote by $V(G)$ and $E(G)$ the vertex and
edge sets of the graph, respectively. A $k$-vertex coloring of a graph $G$ is
a map $c:V(G)\to\\{1,2,\ldots,k\\}$ such that $c(v_{i})\neq c(v_{j})$, where
$v_{i},v_{j}\in V(G)$ are adjacent vertices. The minimum $k$ required to color
vertices is called the chromatic number of $G$, denoted by $\chi(G)$.

Edge coloring of a graph $G$ is the proper Coloring of the edges of $G$ such
that no two edges incident on the same vertex receive the same color. It can
also be interpreted as the vertex coloring of its line graph, $L(G)$. In terms
of mappings, a $k$-edge coloring of $G$ is a map $c:E(G)\to\\{1,2,\ldots,k\\}$
such that $c(e)\neq c(e^{\prime})$ for any two incident edges $e,e^{\prime}\in
E(G)$. The minimum $k$ required in such a coloring is the edge chromatic
number, or the chromatic index of $G$, denoted by $\chi^{\prime}(G)$. By
Vizing’s theorem ([10]), it is known that $\chi^{\prime}(G)$ is either
$\Delta(G)$ or $\Delta(G)+1$, where $\Delta(G)$ is the maximum degree of the
graph $G$. The graphs $G$ with $\chi^{\prime}(G)=\Delta(G)$ are said to be of
class I, and those with $\chi^{\prime}(G)=\Delta(G)+1$ are said to be of class
II.

The total Coloring of a graph $G$ is the Coloring of the elements of $G$ such
that no two adjacent vertices, two adjacent edges, or an edge and its incident
vertices receive the same color. In other words, a $k$-total coloring is a map
$c:V(G)\cup E(G)\to\\{1,2,\ldots,k\\}$ such that $c(u)\neq c(v)$ for any two
adjacent vertices $u,v\in V(G)$,$c(e)\neq c(e^{\prime})$ for any two incident
edges $e,e^{\prime}\in E(G)$ and $c(v)\neq c(e)$ for any vertex $v\in V(G)$
and any edge $e\in E(G)$ incident to $v$. The minimum $k$ required in such a
coloring is called the total chromatic number of the graph, denoted by
$\chi^{\prime\prime}(G)$. A trivial bound on total Coloring is that
$\chi^{\prime\prime}(G)\geq\Delta(G)+1$, where $\Delta(G)$ is the maximum
degree of $G$. Total Coloring Conjecture(TCC) is the assertion that
$\chi^{\prime\prime}(G)\leq\Delta(G)+2$ ([1], [9]). The graphs with
$\chi^{\prime\prime}(G)=\Delta(G)+1$ are called type I, and those with
$\chi^{\prime\prime}(G)=\Delta(G)+2$ are said to be type II.

In this paper, we obtain some bounds on the total chromatic number of some
Cayley graphs on symmetric groups and powers of cycles, which are a class of
Cayley graphs on Cyclic groups. We also obtain bounds on the chromatic number,
chromatic index, and total chromatic number of some classes of Cayley graphs
on gyrogroups. The Cayley graphs on a group/ gyrogroup $\Gamma$ with symmetric
generating set $S$ (a set is called symmetric if both $s$ and $s^{-1}$ both
belong to $S$) will be denoted by $C(\Gamma,S)$. As all graphs are loopless,
$S$ does not have the identity element of the group/gyrogroup. The symmetric
group, which is the group of all bijective functions from an $n$-element set
to itself, will be denoted by $S_{n}$. $C_{n}^{k}$ will denote the $k$-th
power of $n$-cycle.

##  2 Some Results on Cayley graphs of non-abelian groups

######  Theorem 2.1.

The graph $G=C(S_{n},S)$ with $S=\\{(1,2),(1,2,\ldots,n),(1,n,\ldots,2)$ with
$3|n$ is type I.

###### Proof.

We observe that the graph $G$ is a disjoint union of a perfect matching
generated by the element $(1,2)$ and cycles generated by the elements
$(1,2,\ldots,n),(1,n,\ldots,2)$ In order to perform the total Coloring, we
first color the matching formed by the element $(1,2)$ by a unique color The
remaining graph is then a union of disjoint cycles Now, in order to color the
graph totally, we take left cosets of the cyclic group $\mathbb{Z}_{n}$ with
respect to $S_{n}$ We observe that each of the disjoint $n$-cycles formed by
taking the cosets are $3$-total colorable. In addition, the vertex coloring
induced from this total Coloring can also apply to the graph as a whole. This
is because, $g(1,2,\ldots,n)^{i}\neq g(1,2)$, where
$i\in\\{0,1,2,\ldots,n-1\\}$. Therefore, we can color the graph $G$ totally
using $4$ colors; in other words, $G$ is type I. ∎

######  Theorem 2.2.

If the graph $C(A_{n-1},S)\quad n\geq 5$ for

$S=\begin{cases}\\{(1,2,3),(1,3,2),(2,3,\ldots,n),(2,n,\ldots,3)\\}\quad,\text{if}\
n\ \text{is\ even}\\\
\\{(1,2,3),(1,3,2),(1,2,\ldots,n),(1,n,\ldots,2)\\}\quad,\text{otherwise}\end{cases}$

is colorable with $x$ colors, then $C(A_{n},S)$ is also colorable with $x$
colors.

###### Proof.

The theorem is the fact that the vertex coloring of $C(A_{n-1},S)$ can be
extended to that of $C(A_{n},S)$ by taking cosets. First, as the graph
$C(A_{n}-1,S)$ has the triangle by the generating element$(1,2,3)$, the
chromatic number should be $x\geq 3$. The procedure for taking cosets is as
follows.

Case:1-$n$ even:

The set $S$ contains the element $(2,3,\ldots,n)$, which is an $n-1$ cycle. We
take right cosets of $A_{n-1}$ with respect to $A_{n}$ and label the cosets as
$A_{n-1},A_{n-1}(2,3,\ldots,n),A_{n-1}(2,3,\ldots,n)^{2}\ldots,A_{n-1}(2,n,n-1,\ldots,3),L$,
where $L=A_{n-1}(1,3,\ldots,n)$ is the remaining set of elements. We call
$A_{n-1}$ the principal coset, and the other cosets as non-principal cosets.
For coloring the graph $C(A_{n},S)$, we arrange the cosets by shifting the
non-principal cosets alternately one down and two down. As the adjacencies of
the principal coset with respect to $(1,2,3),(1,3,2)$ are covered in the
assumption that $C(A_{n-1},S)$ is properly colored with $x$ colors. Therefore
the only thing we need to take care is that the principal coset has no
neighbors with respect to the elements $(2,3,\ldots,n),(2,n,n-1,\ldots,3)$
with any element of the non-principal cosets in the shifted arrangement. Now,
since both the cosets $A_{n-1}(2,3,\ldots,n)$ and
$A_{n-1}(2,3,\ldots,n)^{n-2}=A_{n-1}(2,n,n-1,\ldots,3)$ are shifted either one
down or two down in the above arrangement, therefore no adjacency clashes will
occur from the vertices of the principal cosets to the non-principal coset.

Similarly, among the non-principal cosets, the possible adjacency clashes can
occur with respect to the elements $(1,2,3),(1,3,2)$ because of the alternate
shifting of the cosets. Let us assume that for some elements
$g_{1}(2,3,\ldots,n)^{i}$ and $g_{2}(2,3,\ldots,n)^{j}$, we have
$g_{1}(2,3,\ldots,n)^{i}(1,2,3)=g_{2}(2,3,\ldots,n)^{j}$, where $g_{2}=g_{1}s$
and $g_{1},g_{2},s\in A_{n-1}$. This would imply
$s=(2,3,ldots,n)^{i}(1,2,3)(2,3,ldots,n)^{k}$ with $k=n-1-j$. As $s\in
A_{n-1}$, and $(2,3,\ldots,n)^{i}$ sends $n$ to $i+1$ and sends $n-i-1$ to
$n$; the only way this could happen is either when $n-(i+2)=k\implies j=i+1$,
or $i=1$ and $k=n-3$ as $s$ must fix $n$. The case $i=j+1$ is impossible by
the alternate shift arrangement of the cosets; thereby the only case remaining
is $i=1$ and $k=n-3$. In this case, we have
$(2,3,\ldots,n)(1,2,3)(2,3,\ldots,n)^{n-3}=(1,2)(3,4,\ldots,n)(2,n-1,n-3,\ldots
n,n-2,n-4,\ldots,4)=(1,n-1,\ldots,2)$. Since we have taken the cosets of
$C(A_{n-1},S)$, in which $S$ had this element, again, this case is impossible.
Similarly, we see that if for some elements $g_{1}(2,3,\ldots,n)^{i}$ and
$g_{2}(2,3,\ldots,n)^{j}$, if we have
$g_{1}(2,3,\ldots,n)^{i}(1,3,2)=g_{2}(2,3,\ldots,n)^{j}$ with $g_{1}s=g_{2}$
and $g_{1},g_{2},s\in A_{n-1}$, then this would imply that
$s=(2,3,\ldots,n)^{i}(1,3,2)(2,3,\ldots,n)^{k}$ with again $k=n-2-j$. With
similar reasoning as before, this would imply either $i+1=j$ or $i=2$ and
$k=1$. For this case, computing $s=(2,3,\ldots,n)^{2}(1,3,2)(2,n,\ldots,3)$
gives us $(1,n-1,\ldots,2)$ which is, again, impossible as stated earlier. The
last coset $L$ is arranged in the same position as that of the principal
coset, because we have $(2,3,\ldots,n)(1,3,2)=(1,3,\ldots,n)$. Therefore $x$
colors suffice to color the vertices of $C(A_{n-1},S)$.

Case:2-$n$ odd:

We observe that the set $S$ contains the element $(1,2,\ldots,n)$, which is an
$n$-cycle. We take right cosets of $A_{n-1}$ with respect to $A_{n}$ and label
the cosets as
$A_{n-1},A_{n-1}(1,2,\ldots,n),A_{n-1}(1,2,\ldots,n)^{2}\ldots,A_{n-1}(1,n,n-1,\ldots,2)$.
We call $A_{n-1}$ the principal coset, and the other cosets as non-principal
cosets. For coloring the graph $C(A_{n},S)$, we arrange the non-principal
cosets by shifting the cosets alternately one down, two down, and in the same
position. Note the difference from the even case. If $3|n$, the last coset is
placed one down. As the adjacencies of the principal coset with respect to
$(1,2,3),(1,3,2)$ are covered in the assumption that $C(A_{n-1},S)$ is
properly colored with $x$ colors; therefore the only thing we need to take
care is that the principal coset has no neighbors with respect to the elements
$(1,2,\ldots,n),(1,n,n-1,\ldots,2)$ with any element of the non-principal
cosets in the shifted arrangement. Now, since both the cosets
$A_{n-1}(1,2,\ldots,n)$ and
$A_{n-1}(1,2,\ldots,n)^{n-1}=A_{n-1}(1,n,n-1,\ldots,2)$ are shifted either one
down or two down in the above arrangement, therefore no adjacency clashes will
occur from the vertices of the principal cosets to the non-principal coset.

Similarly, among the non-principal cosets, the possible adjacency clashes can
occur with respect to the elements $(1,2,3),(1,3,2)$ because of the alternate
shifting of the cosets. Let us assume that for some elements
$g_{1}(1,2,\ldots,n)^{i}$ and $g_{2}(1,2,\ldots,n)^{j}$, we have
$g_{1}(1,2,\ldots,n)^{i}(1,2,3)=g_{2}(1,2,\ldots,n)^{j}$, where $g_{2}=g_{1}s$
and $g_{1},g_{2},s\in A_{n-1}$. This would imply
$s=(1,2,ldots,n)^{i}(1,2,3)(1,2,ldots,n)^{k}$ with $k=n-j$. As $s\in A_{n-1}$,
and $(1,2,\ldots,n)^{i}$ sends $n$ to $i$ and sends $n-i$ to $n$; the only way
this could happen is either when $i=n-k\implies i=j$, or the three
cases:$i=1,k=n-2$, $i=2,k=n-3$ and $i=3,k=n-1$; as $s$ must fix $n$. The case
$i=j$ is verily impossible by our assumption that $C(A_{n-1},S)$ is properly
colored, thereby the only cases remaining are $i=1,k=n-2$, $i=2,k=n-3$ and
$i=3,k=n-1$. These cases are not possible due to the arrangement of the cosets
described; that is, we arrange the cosets one down, two down, or in the same
position as the principal coset. Similarly, we see that if for some elements
$g_{1}(1,2,\ldots,n)^{i}$ and $g_{2}(1,2,\ldots,n)^{j}$, if we have
$g_{1}(1,2,\ldots,n)^{i}(1,3,2)=g_{2}(1,2,\ldots,n)^{j}$ with $g_{1}s=g_{2}$
and $g_{1},g_{2},s\in A_{n-1}$, then this would imply that
$s=(1,2,\ldots,n)^{i}(1,3,2)(1,2,\ldots,n)^{k}$ with again $k=n-j$. With
similar reasoning as before, this would imply either $i=j$ or the three
cases:$i=1,k=n-3$, $i=2,k=n-1$ and $i=3,k=n-2$. For these cases, as before,
the alternate arrangement of cosets yields us contradictions. Therefore, again
$x$ colors suffice for vertex coloring in this case. ∎

######  Corollary 2.1.

The graph $G=C(A_{n},S)\quad,n\geq 4$ with

$S=\begin{cases}\\{(1,2,3),(1,3,2),(2,3,\ldots,n),(2,n,\ldots,3)\\}\quad,\text{if}\
n\ \text{is\ even}\\\
\\{(1,2,3),(1,3,2),(1,2,\ldots,n),(1,n,\ldots,2)\\}\quad,\text{otherwise}\end{cases}$

is type I.

###### Proof.

The above theorem, combined with the fact that $C(A_{3},S)$ has chromatic
number $3$ gives us an equitable $3$-coloring of $G$, such that any induced
bipartite graph formed by taking two independent sets of vertices is regular
of degree $2$. This can be seen by the fact that $(1,2,3)(1,2,3)=(1,3,2)$,
hence if the induced graphs formed by any two independent sets are non-
regular, then we would have had edge clashes among the non-principal cosets,
as well as by the alternating arrangement of the non-principal cosets. Thus,
every independent set of vertices can be extended to total independent sets by
taking a perfect matching from the regular graph formed by the other two
independent sets. The graph’s last two perfect matchings are given two extra
colors to give $G$ a full total coloring with $5$ colors, thereby proving that
$G$ is type I. ∎

##  3 Some results on Powers of Cycles

It is proved that all graphs $C_{n}^{k}$ with $n=s(2m+1)\pm 1\quad,\
\frac{k}{2}\leq m\leq k,\ s-\text{even}$ satisfies TCC. We show some related
results in the below discussions. Some regard conformability, and some others
focus on total Coloring of powers of cycles. The following theorems are also
though proved in [12]. We provide shorter proof.

######  Theorem 3.1.

Every graph $G=C_{n}^{k}$ with $n$ even is conformable.

###### Proof.

If we have $k+1\geq\frac{n}{4}$, then we could divide the vertices as
$[0,\frac{n}{2}],[1,\frac{n}{2}+1],\ldots,[\frac{n}{2}-1,n-1]$ to get a
conformable coloring of the vertices. On the other hand, if
$k+1<\frac{n}{2^{m}}\,,m\geq 2$, we can divide the vertices into $x+y$
classes, where $x=\lfloor\frac{n}{2^{m}}\rfloor$ and
$y=\frac{n-\lfloor\frac{n}{2^{m}}\rfloor}{2}$ with $2^{m}$ vertices in $x$
color classes and $2$ vertices in $y$ color classes. The Coloring is
conformable as all the independent sets so divided have even parity (including
the null independent sets). ∎

######  Theorem 3.2.

If $k+1<\frac{n}{3}$, then the power of cycle graphs $C_{n}^{k}$ are
conformable.

###### Proof.

The given condition implies that the independence number of the graph
$C_{n}^{k}$ is $\geq 3$. Therefore, we can put $3$ vertices in an independent
set. This condition is essential because $3$ is the minimum odd number after
$1$, and the complete graphs of odd order are the only regular and conformable
graphs having exactly $1$ vertex in all the independent sets. As the parity of
$n$ is odd, we must have no null independent sets, for $0$ has even parity.
So, let us first divide the first $2k+1$ vertices into $2k+1$ independent
sets. Now, the induced graph formed by the remaining $n-(2k+1)$ vertices can
be thought of as an induced subgraph of $k$-th power of a cycle of even order
(it is an even power of path $P_{n-(2k+1)}^{k}$). Since we have proved that
the even powers of cycles are vertex conformable, this implies that the
induced subgraph is also vertex conformable. Thus, the graph $G$, in this
case, is also vertex conformable, as we can arrange the remaining $n-(2k+1)$
vertices like the conformable Coloring of the induced graph of $k$-th power of
even cycle, which ensures odd vertices in all the independent sets of $G$. ∎

The above fact of conformability is strong evidence in favor of the Campos-de
Mello conjecture [3]. Though there are graphs that are conformable but still
not type I, the high symmetry of powers of cycles seems that conformability
implies type I for these graphs.

######  Theorem 3.3.

If $n=m(k+1)+1$ for some even integer $m$ and odd $k$, then $G=C_{n}^{k}$
satisfies TCC.

###### Proof.

We show how we can structure the total color matrix as follows. First, we
describe the method for $m=2$. The remaining cases, that is, $m>2$, follow by
repeating the copies of the case $m=2$ and the connecting edges between two
copies connected by a modification of the colors used at the top right end of
the matrix, as explained below. This is possible as $m$ is assumed even.

The induced graph formed by every set of $k$ vertices starting from $0$, that
is, the induced graph formed by the vertices $0,1,\ldots,k-1$, the induced
graph formed by the vertices $k,k+1,\ldots,2k$,$\ldots$ are given total
Coloring according to the commutative idempotent pseudo-latin square of order
$k+1$. The pseudo-latin square is not a Latin square, as it has more numbers
than its order. It is derived from the commutative idempotent Latin square of
order $k+2$ by deleting the last row and column. The last vertex is given the
color $k+2$, so the rightmost and bottom-most entry is $k+2$. Now, we fill the
remaining entries in the total color matrix using a diagonal pattern; that is,
the partial diagonals have the same number throughout.

We can say that the remaining portion of the total color matrix can be divided
into six parts, of which the first part consists of the super-diagonal
starting from the edge $0-(n-k)$ and ending the super-diagonal or entry
corresponding to edge $0-(n-1)$. The second part consists of the partial
super-diagonal starting at the entry corresponding to the edge $1-(k+1)$ and
ending at the partial super-diagonal, or entry corresponding to the edge
$k-(k+1)$. The third part corresponds to entries in the last column from the
entry corresponding to the edge $(n-k-1)-(n-1)$. The fourth, fifth, and sixth
parts are symmetric counterparts of the first, second, and third parts. The
first part of the total color matrix, that is, the partial super-diagonals
starting from the entry corresponding to the edge $0-(n-k)$ is given the color
$k+3$ (same color throughout the super-diagonal); similarly, the next partial
diagonal, that is, staring from the entry corresponding to the edge
$0-(n-k+1)$ is given the color $k+4$ (same color throughout the super-
diagonal), and so on so that, the last partial diagonal, which is just one
entry corresponding to the edge $0-(n-1)$ is given the color $2k+2$. For the
second part of the total color matrix, the partial super-diagonals starting
from the entry corresponding to the edge $1-(k+1)$ are given the color $2k+2$
(same color throughout the super-diagonal); similarly, the next partial super-
diagonal, that is, staring from the entry corresponding to the edge $2-(k+2)$
is given the color $2k+1$ (same color throughout the super-diagonal) and so
on, so that the last partial diagonal closest to the main diagonal, or the
entry corresponding to the edge $k-(k+1)$ is given the color $k+2$. The third
part of the total color matrix, which consists of the last column entries, is
just a continuation of the entries of the $k+1$ pseudo-latin square with
entries identical to that found in the Latin square of order $k+2$. The
symmetric counterparts, the fourth, fifth, and sixth parts of the total color
matrix, get the same colorings as the first, second, and third parts. In case
$m>2$, the connecting edges between the odd and even idempotent pseudo-latin
squares are given the same Coloring as for the second part of the case $m=2$.
In contrast, the connecting edges between the even and odd copies of
idempotent pseudo-latin squares are given the transpose of the colors used in
the first part of the case $m=2$. The first part of the case $m=2$ is retained
as it is for the case $m>2$.

The procedures above give us a total coloring of $G$ because there are no
clashes between the entries in the matrix. To see this, observe that for the
case $m=2$, the colors given in the pseudo-Latin squares and the super-
diagonals are entirely different. The last column entries, part of the $k+2$
Latin square, will be distinct from the entries given in the pseudo-latin
square and the third and sixth parts of the color matrix. In the case, $m>2$,
the same properties of the colors (distinctness of colors used in the pseudo-
latin squares and the connecting super-diagonals) as for $m=2$ are retained,
and the super-diagonals (connecting edges) are arranged in such a way that
clashes are averted. Thus, the graph $G$ satisfies TCC. ∎

######  Example 3.1.

Consider the power of cycle $C_{13}^{5}$. Here, we have $m=2$ and $k+1=6$. The
total color matrix, in this case, is given in Table 1:

| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
---|---|---|---|---|---|---|---|---|---|---|---|---|---
0 | 1 | 5 | 2 | 6 | 3 | 7 |  |  | 12 | 11 | 10 | 9 | 8
1 | 5 | 2 | 6 | 3 | 7 | 4 | 8 |  |  | 12 | 11 | 10 | 9
2 | 2 | 6 | 3 | 7 | 4 | 1 | 9 | 8 |  |  | 12 | 11 | 10
3 | 6 | 3 | 7 | 4 | 1 | 5 | 10 | 9 | 8 |  |  | 12 | 11
4 | 3 | 7 | 4 | 1 | 5 | 2 | 11 | 10 | 9 | 8 |  |  | 12
5 | 7 | 4 | 1 | 5 | 2 | 6 | 12 | 11 | 10 | 9 | 8 |  |
6 |  | 8 | 9 | 10 | 11 | 12 | 1 | 5 | 2 | 6 | 3 | 7 |
7 |  |  | 8 | 9 | 10 | 11 | 5 | 2 | 6 | 3 | 7 | 4 | 1
8 | 12 |  |  | 8 | 9 | 10 | 2 | 6 | 3 | 7 | 4 | 1 | 5
9 | 11 | 12 |  |  | 8 | 9 | 6 | 3 | 7 | 4 | 1 | 5 | 2
10 | 10 | 11 | 12 |  |  | 8 | 3 | 7 | 4 | 1 | 5 | 2 | 6
11 | 9 | 10 | 11 | 12 |  |  | 7 | 4 | 1 | 5 | 2 | 6 | 3
12 | 8 | 9 | 10 | 11 | 12 |  |  | 1 | 5 | 2 | 6 | 3 | 7
Table 1: Total Color matrix of $C_{13}^{5}$

######  Example 3.2.

Consider the graph $C_{25}^{5}$. Here, $m=4$. The total color matrix, in this
case, is given in Table 2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 0 1 5 2 6 3 7
12 11 10 9 8 1 5 2 6 3 7 4 8 12 11 10 9 2 2 6 3 7 4 1 9 8 12 11 10 3 6 3 7 4 1
5 10 9 8 12 11 4 3 7 4 1 5 2 11 10 9 8 12 5 7 4 1 5 2 6 12 11 10 9 8 6 8 9 10
11 12 1 5 2 6 3 7 7 8 9 10 11 5 2 6 3 7 4 12 8 8 9 10 2 6 3 7 4 1 11 12 9 8 9
6 3 7 4 1 5 10 11 12 10 8 3 7 4 1 5 2 9 10 11 12 11 7 4 1 5 2 6 8 9 10 11 12
12 12 11 10 9 8 1 5 2 6 3 7 13 12 11 10 9 5 2 6 3 7 4 8 14 12 11 10 2 6 3 7 4
1 9 8 15 12 11 6 3 7 4 1 5 10 9 8 16 12 3 7 4 1 5 2 11 10 9 8 17 7 4 1 5 2 6
12 11 10 9 8 18 8 9 10 11 12 1 5 2 6 3 7 19 8 9 10 11 5 2 6 3 7 4 1 20 12 8 9
10 2 6 3 7 4 1 5 21 11 12 8 9 6 3 7 4 1 5 2 22 10 11 12 8 3 7 4 1 5 2 6 23 9
10 11 12 7 4 1 5 2 6 3 24 8 9 10 11 12 1 5 2 6 3 7

Table 2: Total Color matrix of $C_{25}^{5}$

##  4 Results on Cayley graphs on Gyrogroups

First, we will begin with a lemma that helps determine the isomorphism of
Cayley graphs defined on the same group. An exponent distribution of a subset
$\Sigma$ of a generating set $S$ of a group is the set of exponents of the
elements of $S$ with respect to the product of elements of $\Sigma$. That is,
if $\Sigma=\\{\sigma_{1},\sigma_{2},\ldots,\sigma_{n}\\}$ and
$S=\\{s_{1},s_{2},\ldots,s_{n}\\}$; then the exponent distribution of $\Sigma$
is the set $\\{i_{1},i_{2},\ldots,i_{n}\\}$, where
$s_{1}=(\sigma_{1}\sigma_{2}\ldots\sigma_{n})^{i_{1}},s_{2}=(\sigma_{1}\sigma_{2}\ldots\sigma_{n})^{i_{2}},\ldots$.
The following lemma gives an algorithm to determine when two Cayley Graphs on
the same group with different generating sets are isomorphic.

######  Lemma 4.1.

Two Cayley graphs $G_{1}=C(\Gamma,S_{1})$ and $G_{2}=C(\Gamma,S_{2})$ are
isomorphic if there exist two sets of generating subsets $\Sigma_{1}\subset
S_{1}$ and $\Sigma_{2}\subset S_{2}$ of $G$ of same cardinality such that
their exponent distributions with respect to some permutation of the
generating set are the same.

###### Proof.

The proof is immediate on noting that one could construct identical graphs by
starting from the identity element of $\Gamma$ and the generating subsets
$\Sigma_{1}$ and $\Sigma_{2}$. ∎

The above algorithm could be in polynomial time if the base group $\Gamma$ is
cyclic, as the following corollary shows.

######  Corollary 4.1.

If $\Gamma$ is a cyclic group of order $n$, then the graphs
$G_{1}=C(\Gamma,S_{1})$ and $G_{2}=C(\Gamma,S_{2})$ are isomorphic if there
exist generating elements of $\Gamma$ $s_{1}\in S_{1}$ and $s_{2}\in S_{2}$
such that their exponent distributions are same.

###### Proof.

The proof is immediate from the previous lemma once it is known that the
cyclic groups have a single minimal generator, and the corollary assumes that
$s_{1},s_{2}$ are those generators. ∎

A gyrogroup is a non-associative structure on a set having a left inverse and
left identity. It is a magma with a bijective function called the
gyroautomorphism that gives an associative-like structure called gyro-
associativity to the gyrogroup. Technically, a gyrogroup $\Gamma$ is a set $S$
with a binary operation $\oplus$ and a bijective function $gyr[a,b]$ which is
an automorphism for $(\Gamma,\oplus)$ satisfying:

i)Left identity: We should have an element $0\in\Gamma$ such that for all
$a\in\Gamma$, $0\oplus a=a$.

ii)Left inverse: We should have an element $\ominus a\in\Gamma$ for every
element $a\in\Gamma$, we have $(\ominus a)\oplus a=0$.

iii) Gyroassociativity: We should have, for three elements $a,b,c\in\Gamma$,
$a\oplus(b\oplus c)=(a\oplus b)\oplus gyr[a,b]c$. Thus, Gyrogroups generalize
groups, as every group is a gyrogroup (with the gyroautomorphism induced by
the identity map).

iv)Left loop property: We should have for all $a,b\in\Gamma$,
$gyr[a,b]=gyr[(a\oplus b),b]$.

In [6], the authors introduced a gyrogroup with properties resembling dihedral
groups’ properties. The authors called these 2-gyrogroups. In [7], the author
studied several graph theoretic properties of Cayley graphs associated with
these gyrogroups. Here, we give a theorem on the total Coloring of a subclass
of Cayley graphs on these gyrogroups.

######  Theorem 4.1.

Let $\Gamma$ be the $2$-gyrogroup of order $n=2^{k}$ presented in [7]. Then,
the graph $C(\Gamma,S)$ with
$S=\\{1,2,\ldots,k,m-k,m-k+1,\ldots,m-1,\frac{m}{2}+m$. Then $G$ satisfies
TCC. Further, if the power of cycle $C_{m}^{k}$ is type I, then $G$ is also
type I.

###### Proof.

For ease of reference, we state the binary operation of the $2$-gyrogroup
presented in [6] and Example $1$ of [7]. The binary operation $\oplus$ is
defined as:

$i\oplus j=\begin{cases}i+j\pmod{m}\qquad\qquad\qquad\qquad\qquad\qquad,i\in
T_{1};j\in T_{1}\\\ \\{i+j\pmod{m}\\}+m\quad\qquad\qquad\qquad\qquad,i\in
T_{1};j\in T_{2}\\\ \\{i+(\frac{m}{2}-1)j\\}\pmod{m})+m\quad\qquad\qquad,i\in
T_{2};j\in T_{2}\\\
\\{(\frac{m}{2}-1)i+(\frac{m}{2}+1)j\pmod{m}\\}+m\qquad,i\in T_{2};j\in
T_{2}\end{cases}$

. Here $T_{1}=\\{0,1,\ldots,m-1\\}$, $m=2^{n-1}$ and
$T_{2}=\\{m,m+1\ldots,n-1\\}$.

We note that the subgraph induced by the set $T_{1}$, is isomorphic to
$C_{m}^{k}$, because, we have $i\oplus j=i+j$ for
$i,j\in\\{0,1,\ldots,m-1\\}$.

Similarly, the graph $C_{m}^{k}$ is isomorphic to the subgraph induced by the
set $T_{2}$ with respect to the generating set
$\\{(\frac{m}{2}-1)+m,2m-2,\ldots,k(\frac{m}{2}-1)+m,2m-k(\frac{m}{2}-1),\ldots,2+m,(\frac{m}{2}+1)+m\\}$.

This is because of two reasons: 1) We have $i\oplus
j=i+(\frac{m}{2}-1)j\pmod{m}$ for $i\in T_{1}$ $j\in T_{2}$, and 2) We note
that the exponent distribution of $1$ for
$\\{1,2,\ldots,k,n-k,\ldots,n-2,n-1\\}$ and $z=(\frac{m}{2}-1)$ for
$\\{(\frac{m}{2}-1),m-2,\ldots,k(\frac{m}{2}-1)m-k(\frac{m}{2}-1),\ldots,2,(\frac{m}{2}+1)\\}$
are the same, in the group $\mathbb{Z}_{m}$.

Now, note that the element $\frac{m}{2}+m$ acts as a sort of reflection
equivalent for the gyrogroup $\Gamma$, in the sense that, we have
$(\frac{m}{2}+m)^{\oplus
2}=(\frac{m}{2}+m)\oplus(\frac{m}{2}+m)=(\frac{m}{2}+1)(\frac{m}{2}+m)+(\frac{m}{2}-1)(\frac{m}{2}+m)\pmod{m}$
$\equiv 2\frac{m^{2}}{4}=m(\frac{m}{2})=0\pmod{m}$. Therefore, this element
induces a perfect matching in the graph $C(\Gamma,S)$ with the end vertices of
the edges being in the two sets $T_{1}$ and $T_{2}$, respectively.

Now, if $i$ and $j$ and in the same independent set of the subgraph induced by
$T_{1}$, then $i\oplus(\frac{m}{2}+m)=(i+(\frac{m}{2}+m)\pmod{m})+m$ and
$j\oplus(\frac{m}{2}+m)=(i+(\frac{m}{2}+m)\pmod{m})+m$ are also in the same
independent set, as $i-j=i\ominus j$. Hence, we can divide the vertices of the
induced subgraph on $T_{2}$ into the same number of independent sets as that
of the induced subgraph on $T_{1}$ by shifting the translates of an
independent set with respect to the reflection element $\frac{m}{2}+m$. Thus,
the chromatic number of $G$ is the same as that of $C_{m}^{k}$. As the graphs
$C_{m}^{k}$ satisfy TCC by Theorem 16 of Campos-de Mello[3], the induced
subgraphs on $T_{1}$ and $T_{2}$ individually satisfy TCC. Since we have
arranged the induced graphs together in the same number of independent sets,
only the edge coloring of the connecting perfect matching between the induced
subgraphs formed by the element $\frac{m}{2}+m$ needs to be done in order to
complete the total Coloring of $G$. We give one extra color for this, making
$G$ satisfy TCC. This argument also shows why $G$ will be of type I if
$C_{m}^{k}$ is also of type I. ∎

The above theorem at once gives the following generalization as its corollary.

######  Corollary 4.2.

Let $\Gamma$ be the $2$-gyrogroup presented in [7]. Then, if the graph
$C(\mathbb{Z}_{m},S_{1})$ staisfies TCC, then $G=C(Gamma,S)$ with
$S_{1}\cup\\{\frac{m}{2}+m\\}$ satisfies TCC. Further, if the graph
$C(\mathbb{Z}_{m},S_{1})$ is type I, then $G$ is also type I.

###### Proof.

The proof is immediate once we replace the element $1$ in the previous theorem
with any suitable generator $s\in S_{1}$ of the group $\mathbb{Z}_{m}$. Then,
the subgraphs induced by the two sets $T_{1}$ and $T_{2}$ are isomorphic
circulant graphs. The element $\frac{m}{2}+m$ then gives us a perfect matching
of elements in $G$ with the end vertices in $T_{1}$ and $T_{2}$, respectively.
Again, by following the similar argument in the proof of the last theorem, the
chromatic number of $G$ is the same as that of the graph
$C(\mathbb{Z}_{m},S_{1})$. Then, giving the perfect matching induced by the
element $\frac{m}{2}+m$ makes $G$ satisfy TCC. ∎

The following result, an immediate consequence, gives us the chromatic index
of such graphs.

######  Theorem 4.2.

Let $\Gamma$ be the $2$-gyrogroup presented in [7]. Then, if the graph
$C(\mathbb{Z}_{m},S_{1})$ satisfies TCC, then $G=C(Gamma,S)$ with
$S_{1}\cup\\{\frac{m}{2}+m\\}$ has chromatic number equal to that of the graph
$\chi(C(\mathbb{Z}_{m},S_{1})).$ In addition, the graph is of class I.

###### Proof.

As the circulant graphs $C(\mathbb{Z}_{m},S_{1})$ are of class I by the
Corollary 2.3.1 of [8], we need to only give one extra color to the perfect
matching induced by the element $\frac{m}{2}+m$. Thus, all the edges of $G$
can be colored in precisely $\Delta(G)+1$ colors or $G$ is of class I. ∎

An immediate generalization of the above result in the context of edge
coloring is the following:

######  Theorem 4.3.

All Cayley graphs $G=Cay(\Gamma,S)$ for the $2$-gyrogroup described in the
theorems above and any generating set $S$ is of class I.

###### Proof.

The proof is immediate, once we note that all generating elements of the form
$s=j+m$, where $j\in\\{0,1,\ldots,m-1\\}$ are sort of reflections (gyro-
reflections), that is, satisfy the property $s^{\oplus 2}=s\oplus
s=(\frac{m}{2}-1)s+(\frac{m}{2}+1)s\pmod{m}\equiv ms\pmod{s}equiv0$. Hence,
all the reflections give rise to perfect matchings in the graph $G$. Since the
induced graph on the sets $\\{0,1,\ldots,m-1\\}$ and $\\{m,m+1,\ldots,n-1\\}$
are each circulant, therefore, by [8] Corollary 2.3.1 and the fact that the
perfect matchings generated by any elements of the form $s$ can be
$1$-factorized, the conclusion is immediate. ∎

## References

  * [1] Behzad, M. (1965). Graphs and their chromatic numbers. Michigan State University.
  * [2] Chen, G., Hao, Y. (2023). On the coequal values of total chromatic number and chromatic index. Journal of Combinatorial Theory, Series B, 158, 286-304.
  * [3] Campos, C. N., and de Mello, C. P. (2007). A result on the total colouring of powers of cycles. Discrete Applied Mathematics, 155(5), 585-597.
  * [4] Xie, DeZheng, and WanNian Yang. "The total chromatic number of regular graphs of high degree." Science in China Series A: Mathematics 52.8 (2009): 1743-1759.
  * [5] Heuberger, C. (2003). On planarity and colorability of circulant graphs. Discrete Mathematics, 268(1-3), 153-169.
  * [6] Mahdavi, S., Ashrafi, A. R., Salahshour, M. A., and Ungar, A. A. (2021). Construction of 2-gyrogroups in which every proper subgyrogroup is either a cyclic or a dihedral group. Symmetry, 13(2), 316.
  * [7] Maungchang, R., and Suksumran, T. (2022). On Dihedralized Gyrogroups and Their Cayley Graphs. Mathematics, 10(13), 2276.
  * [8] Stong, R. A. (1985). On 1-factorizability of Cayley graphs. Journal of Combinatorial Theory, Series B, 39(3), 298-307.
  * [9] Vizing, V. G. (1968). Some unsolved problems in graph theory. Russian Mathematical Surveys, 23(6), 125.
  * [10] Vizing, V. G. (1964). On an estimate of the chromatic class of a p-graph. Diskret analiz, 3, 25-30.
  * [11] Zmazek, B., and Zerovnik, J. (2002). Behzad-Vizing conjecture and Cartesian-product graphs. Applied mathematics letters, 15(6), 781-784.
  * [12] Zorzi, A., de Figueiredo, C., Machado, R., and Souza, U. S. (2019). Even-power of cycles with many vertices are Type 1 total colorable. Electronic Notes in Theoretical Computer Science, 346, 747-758.


# Relativistic nucleon-nucleon potentials in a spin-dependent three-
dimensional approach

M. R. Hadizadeh  College of Engineering, Science, Technology and Agriculture,
Central State University, Wilberforce, OH 45384, USA,  Department of Physics
and Astronomy, Ohio University, Athens, OH 45701, USA,
<EMAIL_ADDRESS>M. Radin  Department of Physics, K. N. Toosi
University of Technology, Tehran, Iran<EMAIL_ADDRESS>F. Nazari
Department of Physics, K. N. Toosi University of Technology, Tehran, Iran.
<EMAIL_ADDRESS>
###### Abstract

The matrix elements of relativistic nucleon-nucleon $(NN)$ potentials are
calculated directly from the nonrelativistic potentials as a function of
relative $NN$ momentum vectors, without using a partial wave decomposition. To
this aim, the quadratic operator relation between the relativistic and
nonrelativistic $NN$ potentials is formulated in momentum-helicity basis
states. It leads to a single integral equation for the two-nucleon $(2N)$
spin-singlet state and four coupled integral equations for two-nucleon spin-
triplet states, which are solved by an iterative method. Our numerical
analysis indicates that the relativistic $NN$ potential obtained using CD-Bonn
potential reproduces the deuteron binding energy and neutron-proton elastic
scattering differential and total cross-sections with high accuracy.

###### keywords:

Relativistic nucleon-nucleon potentials, Three-dimensional scheme, Lippmann-
Schwinger equation

##  1 Introduction

Since Einstein’s theory of special relativity in the early 20th century, there
are still issues when considering systems that contain more than two nucleons,
as in such systems, pair nucleons are influenced by the presence and motion of
the other nucleons. As the three- and four-nucleon bound and scattering
problems can be numerically solved with controlled errors, they provide an
ideal theoretical laboratory for investigating the relativistic effects in the
few-nucleon systems. Several techniques are developed to study the
relativistic effects in few-body systems, and among them, the Faddeev-
Yakubovsky method provides an exact numerical treatment of three and four-
nucleon systems.

The inputs for the relativistic Faddeev-Yakubovsky equations are the fully-
off-shell (FOS) relativistic $2N$ $t-$matrices, which can be obtained with two
different approaches. In the first approach, the FOS relativistic $2N$
$t-$matrices are obtained directly from the nonrelativistic $2N$ $t-$matrices
applying a two-step process. In the first step, using an analytical relation
proposed by Coester et al., the relativistic right-half-shell (RHS)
$t-$matrices are obtained from the nonrelativistic RHS $t-$matrices [1]. In
the second step, the FOS relativistic $t-$matrices are obtained from the RHS
$t-$matrices by solving a first resolvent equation proposed by Keister et al.
[2]. This approach has been successfully implemented in few-body bound and
scattering calculations in a three-dimensional (3D) scheme [3, 4, 5, 6, 7, 8],
without using a partial wave (PW) decomposition.

In a second approach, the relativistic FOS $t-$matrices can be calculated from
the solution of the relativistic Lippmann-Schwinger (LS) equation using the
relativistic $2N$ potentials. The input relativistic $NN$ potentials can be
obtained from the nonrelativistic potentials by solving a quadratic equation,
using an iterative scheme proposed by Kamada and Glöckle [9]. We have recently
implemented this iterative technique in a 3D scheme to calculate the matrix
elements of relativistic two-body (2B) potentials for spin-independent
Malfliet-Tjon (MT) potential as a function of the magnitude of 2B relative
momenta and the angle between them. To do so, we formulated the quadratic
operator relation between the nonrelativistic and relativistic $NN$ potentials
in momentum space leading to a 3D integral equation [10]. We successfully
implemented this iterative approach to calculate the matrix elements of
boosted 2B potential from the MT potential to study the relativistic effects
in a 3B bound state [11]. Our numerical results showed that the relativistic
effects lead to a 2% reduction in 3B binding energy using MT potential. Our
exact and detailed numerical studies of relativistic effects in 3B bound
states using spin-independent MT potential demonstrates that direct
integrations in the 3D scheme can be utilized to achieve the same results
obtained using a PW method and paves the path for an extension to realistic
interactions that have a more complicated spin-isospin dependence. Considering
modern nucleon-nucleon ($NN$) potentials by including spin and isospin degrees
of freedom and calculating the relativistic $NN$ potentials from realistic
$NN$ potentials is the task we address in this paper. This is the first step
toward our goal for a fully relativistic treatment of triton and Helium-3
bound state properties and the long-term interest in studying the scattering
problems at the few-GeV energy scale in a 3D scheme. We show that the
representation of the quadratic equation in momentum helicity basis states
leads to a single and four coupled 3D integral equations for $NN$ singlet and
triplet spin state, correspondingly. The single and coupled integral equations
are solved using the mentioned iterative scheme, and the matrix elements of
relativistic $NN$ potentials are obtained from CD-Bonn potential [12]. Our
numerical analysis indicates that the calculated relativistic potential
reproduces the deuteron binding energy and differential and total cross-
sections of neutron-proton ($np$) elastic scattering with very high accuracy.

The motivation for using the 3D scheme and implementing a direct integration
method is to replace the discrete angular momentum quantum numbers of a PW
representation with continuous angle variables and consider all partial wave
components to infinite order, independent of the energy scale of the problem.
Consequently, 3D representation avoids the very involved angular momentum
algebra for permutations, transformations, and few-nucleon forces, and in
contrast to the PW approach, the number of equations in the 3D representation
is energy independent. The 3D scheme is successfully implemented in a series
of few–body bound and scattering states calculations by different few-body
groups, from Ohio-Bochum collaboration [13, 14, 15, 16, 17, 18, 19, 20, 21, 7,
5, 22, 23, 24, 25, 4] to Tehran [26, 27, 28, 29, 30, 31, 32] and Kraków [33,
34, 35, 36, 37, 38, 39, 40, 41] groups.

In Sec. 2, we present the 3D formalism for the relationship between
relativistic and nonrelativistic $NN$ potentials. By projecting the quadratic
relation between nonrelativistic and relativistic $NN$ potentials in momentum
helicity basis states, we obtain the matrix elements of relativistic $NN$
potentials as a function of the magnitude of $2N$ relative momenta, the angle
between them, and the helicity eigenvalues. We derive a single integral
equation for $NN$ total spin state $s=0$ and four coupled integral equations
for $s=1$. In Sec. 3, we present our numerical results for the matrix elements
of relativistic $NN$ potential obtained from CD-Bonn potential in different
spin and isospin channels. In Sec. 4, we test the obtained relativistic
potential by calculating and comparing deuteron binding energy and
differential and total cross-sections of $np$ elastic scattering with
corresponding nonrelativistic results. Finally, a conclusion and outlook are
provided in Sec. 5.

##  2 Relativistic $NN$ potentials in a momentum helicity representation

In this section, we show how to obtain the matrix elements of relativistic
$NN$ interactions in a 3D scheme from the nonrelativistic interactions by
solving a nonlinear equation derived by Kamada and Glöckle. The relativistic
interactions are designed to accurately reproduce the $NN$ bound and
scattering observables. To this aim and to check the accuracy of obtained
relativistic interactions, as we show in Sec. 4, one needs to solve the
homogeneous and inhomogeneous LS integral equations (28), (31), and (35) in a
momentum helicity representation to calculate relativistic deuteron binding
energy and the scattering amplitudes to obtain the differential and total
cross-sections.

The relativistic and nonrelativistic $NN$ potentials, i.e., ${\cal V}_{r}$ and
$V_{nr}$, are related together by a quadratic operator equation as [9]

$V_{nr}=\frac{1}{4m}\biggl{(}\omega(\hat{p}){\cal V}_{r}+{\cal
V}_{r}\omega(\hat{p})+{\cal V}_{r}^{2}\biggr{)},$ (1)

where $m$ is the mass of nucleons, $p$ is the relative momentum of two
nucleons ($\hat{p}$ is the operator), and
$\omega(p)=2E(p)=2\sqrt{m^{2}+p^{2}}$. To calculate the relativistic $NN$
potential ${\cal V}_{r}$ from a nonrelativistic potential $V_{nr}$ in a 3D
representation, we present Eq. (1) in momentum helicity basis states. The
antisymmetrized momentum helicity basis states for a $2N$ system with total
spin and isospin $s$ and $t$, and the relative momentum ${\bf p}$ are
introduced as [16]

$|{\bf p};\hat{\bf p}s\lambda;t\rangle^{\pi
a}\equiv\frac{1}{\sqrt{2}}\biggl{(}1-\eta_{\pi}(-)^{s+t}\biggr{)}\ |{\bf
p};\hat{\bf p}\ s\lambda\rangle_{\pi}\ |t\rangle,$ (2)

where $\hat{\bf p}$ is the unit momentum operator, $\lambda$ is the eigenvalue
of the helicity operator ${\bf s\cdot\hat{\bf p}}$, with the parity
eigenvalues $\eta_{\pi}=\pm 1$ and eigenstates $|{\bf p};\hat{\bf p}\
s\lambda\rangle_{\pi}=\frac{1}{\sqrt{2}}(1+\eta_{\pi}P_{\pi})|{\bf p};\hat{\bf
p}\ s\lambda\rangle$. The $2N$ helicity basis states are orthogonal and
normalized as

$\ {}^{\pi^{\prime}a}\langle{\bf p}^{\prime};\hat{\bf
p}^{\prime}s^{\prime}\lambda^{\prime};t^{\prime}|{\bf p};\hat{\bf
p}s\lambda;t\rangle^{\pi a}=\biggl{(}1-\eta_{\pi}(-)^{s+t}\biggr{)}\
\delta_{\eta_{\pi^{\prime}}\eta_{\pi}}\delta_{s^{\prime}s}\delta_{t^{\prime}t}\
\biggl{\\{}\delta({\bf p}^{\prime}-{\bf
p})\delta_{\lambda\lambda^{\prime}}+\eta_{\pi}(-)^{s}\delta({\bf
p}^{\prime}+{\bf p})\delta_{\lambda^{\prime}-\lambda}\biggr{\\}},$ (3)
$\displaystyle\sum_{s\lambda\pi t}\int d{\bf p}\ |{\bf p};\hat{\bf
p}s\lambda;t\rangle^{\pi a}\ \frac{1}{4}\ \ ^{\pi a}\langle{\bf p};\hat{\bf
p}s\lambda;t|=1.$ (4)

The matrix elements of $NN$ nonrelativistic and relativistic potentials in
$2N$ helicity basis states, introduced in Eq. (2), are given as

$\displaystyle V^{\pi st}_{nr,\lambda\lambda^{\prime}}({\bf p},{\bf
p}^{\prime})$ $\displaystyle\equiv$ $\ {}^{\pi a}\langle{\bf p};\hat{\bf
p}s\lambda;t|V_{nr}|{\bf p}^{\prime};\hat{\bf
p}^{\prime}s\lambda^{\prime};t\rangle^{\pi a},$ (5) $\displaystyle{\cal
V}^{\pi st}_{r,\lambda\lambda^{\prime}}({\bf p},{\bf p}^{\prime})$
$\displaystyle\equiv$ $\ {}^{\pi a}\langle{\bf p};\hat{\bf p}s\lambda;t|{\cal
V}_{r}|{\bf p}^{\prime};\hat{\bf p}^{\prime}s\lambda^{\prime};t\rangle^{\pi
a}.$ (6)

Representation of the quadratic relation between nonrelativistic and
relativistic $NN$ potentials, given in Eq. (1), in $2N$ helicity basis states
is as

$\ {}^{\pi a}\langle{\bf p};\hat{\bf p}s\lambda;t|4mV_{nr}|{\bf
p}^{\prime};\hat{\bf p}^{\prime}s\lambda^{\prime};t\rangle^{\pi a}$
$\displaystyle=$ $\ {}^{\pi a}\langle{\bf p};\hat{\bf
p}s\lambda;t|\omega(\hat{p}){\cal V}_{r}|{\bf p}^{\prime};\hat{\bf
p}^{\prime}s\lambda^{\prime};t\rangle^{\pi a}$ (7) $\displaystyle+$ $\ {}^{\pi
a}\langle{\bf p};\hat{\bf p}s\lambda;t|{\cal V}_{r}\omega(\hat{p})|{\bf
p}^{\prime};\hat{\bf p}^{\prime}s\lambda^{\prime};t\rangle^{\pi a}$
$\displaystyle+$ $\ {}^{\pi a}\langle{\bf p};\hat{\bf p}s\lambda;t|{\cal
V}_{r}\cdot{\cal V}_{r}|{\bf p}^{\prime};\hat{\bf
p}^{\prime}s\lambda^{\prime};t\rangle^{\pi a}.$

The first and the second terms on the right-hand side of Eq. (7) can be
evaluated straightforwardly. For evaluation of the third term, the
completeness relation of Eq. (3) should be inserted. By these considerations,
Eq. (7) reads as

$\displaystyle\frac{4mV^{\pi st}_{nr,\lambda\lambda^{\prime}}({\bf p},{\bf
p}^{\prime})}{\omega(p)+\omega(p^{\prime})}$ $\displaystyle=$
$\displaystyle{\cal V}^{\pi st}_{r,\lambda\lambda^{\prime}}({\bf p},{\bf
p}^{\prime})+\frac{1}{\omega(p)+\omega(p^{\prime})}\frac{1}{4}\sum_{\lambda^{\prime\prime}=-1}^{+1}\int
d{\bf p}^{\prime\prime}\ {\cal V}^{\pi
st}_{r,\lambda\lambda^{\prime\prime}}({\bf p},{\bf p}^{\prime\prime}){\cal
V}^{\pi st}_{r,\lambda^{\prime\prime}\lambda^{\prime}}({\bf
p}^{\prime\prime},{\bf p}^{\prime}).$ (8)

By considering the following properties of $NN$ potentials, one can obtain the
negative helicity eigenvalue components of the potential from the positive
ones as

$\displaystyle V^{\pi st}_{nr,-\lambda\lambda^{\prime}}({\bf p},{\bf
p}^{\prime})$ $\displaystyle=$ $\displaystyle\eta_{\pi}(-)^{s}\ V^{\pi
st}_{nr,\lambda\lambda^{\prime}}(-{\bf p},{\bf p}^{\prime}),$ (9)
$\displaystyle V^{\pi st}_{nr,\lambda-\lambda^{\prime}}({\bf p},{\bf
p}^{\prime})$ $\displaystyle=$ $\displaystyle\eta_{\pi}(-)^{s}\ V^{\pi
st}_{nr,\lambda\lambda^{\prime}}({\bf p},-{\bf p}^{\prime}).$ (10)

The above relations are also valid for the relativistic potential ${\cal
V}^{\pi st}_{r,\lambda\lambda^{\prime\prime}}({\bf p},{\bf
p}^{\prime\prime})$. By using the properties of Eq. (9), one can show that the
second term of Eq. (8), for $\lambda^{\prime\prime}=-1$ and
$\lambda^{\prime\prime}=+1$ are equal together

$\displaystyle\int d{\bf p}^{\prime\prime}\ {\cal V}^{\pi
st}_{r,\lambda-1}({\bf p},{\bf p}^{\prime\prime}){\cal V}^{\pi
st}_{r,-1\lambda^{\prime}}({\bf p}^{\prime\prime},{\bf p}^{\prime})$
$\displaystyle=$ $\displaystyle\int d{\bf p}^{\prime\prime}\
\eta_{\pi}(-)^{s}\ {\cal V}^{\pi st}_{r,\lambda 1}({\bf p},-{\bf
p}^{\prime\prime})\eta_{\pi}(-)^{s}\ {\cal V}^{\pi
st}_{r,1\lambda^{\prime}}(-{\bf p}^{\prime\prime},{\bf p}^{\prime})$ (11)
$\displaystyle=$ $\displaystyle\underbrace{\eta_{\pi}^{2}(-)^{2s}}_{1}\int
d{\bf p}^{\prime\prime}\ {\cal V}^{\pi st}_{r,\lambda 1}({\bf p},-{\bf
p}^{\prime\prime}){\cal V}^{\pi st}_{r,1\lambda^{\prime}}(-{\bf
p}^{\prime\prime},{\bf p}^{\prime})$ $\displaystyle=$ $\displaystyle\int d{\bf
p}^{\prime\prime}\ {\cal V}^{\pi st}_{r,\lambda 1}({\bf p},{\bf
p}^{\prime\prime}){\cal V}^{\pi st}_{r,1\lambda^{\prime}}({\bf
p}^{\prime\prime},{\bf p}^{\prime}).$

For $2N$ singlet spin state, $s=0$, Eq. (8) leads to a single integral
equation to obtain the matrix elements of relativistic potential ${\cal
V}^{\pi 0t}_{r,00}({\bf p},{\bf p}^{\prime})$

$\displaystyle\frac{4mV^{\pi 0t}_{nr,00}({\bf p},{\bf
p}^{\prime})}{\omega(p)+\omega(p^{\prime})}$ $\displaystyle=$
$\displaystyle{\cal V}^{\pi 0t}_{r,00}({\bf p},{\bf
p}^{\prime})+\frac{1}{\omega(p)+\omega(p^{\prime})}\frac{1}{4}\int d{\bf
p}^{\prime\prime}\ {\cal V}^{\pi 0t}_{r,00}({\bf p},{\bf
p}^{\prime\prime}){\cal V}^{\pi 0t}_{r,00}({\bf p}^{\prime\prime},{\bf
p}^{\prime}).$ (12)

For triplet spin states, $s=1$, Eq. (8) leads to four coupled integral
equations, corresponding to helicity eigenvalues
$\lambda,\lambda^{\prime}=0,+1$, to calculate the matrix elements of
relativistic potentials ${\cal V}^{\pi 1t}_{r,00}({\bf p},{\bf p}^{\prime})$,
${\cal V}^{\pi 1t}_{r,01}({\bf p},{\bf p}^{\prime})$, ${\cal V}^{\pi
1t}_{r,10}({\bf p},{\bf p}^{\prime})$, and ${\cal V}^{\pi 1t}_{r,11}({\bf
p},{\bf p}^{\prime})$

$\displaystyle\frac{4mV^{\pi 1t}_{nr,\lambda\lambda^{\prime}}({\bf p},{\bf
p}^{\prime})}{\omega(p)+\omega(p^{\prime})}$ $\displaystyle=$
$\displaystyle{\cal V}^{\pi 1t}_{r,\lambda\lambda^{\prime}}({\bf p},{\bf
p}^{\prime})$ (13) $\displaystyle+$
$\displaystyle\frac{1}{\omega(p)+\omega(p^{\prime})}\biggl{(}\frac{1}{2}\int
d{\bf p}^{\prime\prime}\ {\cal V}^{\pi 1t}_{r,\lambda 1}({\bf p},{\bf
p}^{\prime\prime}){\cal V}^{\pi 1t}_{r,1\lambda^{\prime}}({\bf
p}^{\prime\prime},{\bf p}^{\prime})+\frac{1}{4}\int d{\bf p}^{\prime\prime}\
{\cal V}^{\pi 1t}_{r,\lambda 0}({\bf p},{\bf p}^{\prime\prime}){\cal V}^{\pi
1t}_{r,0\lambda^{\prime}}({\bf p}^{\prime\prime},{\bf p}^{\prime})\biggr{)}.$
(14)

Eqs. (12) and (13) should be solved for each value of $2N$ total isospin
$t=0,1$. For the numerical solution of single and coupled integral equations,
i.e., Eqs. (12) and (13), by choosing momentum vector ${\bf p}^{\prime}$
parallel to the $z-$axis, the azimuthal angular dependence of the matrix
elements of nonrelativistic and relativistic potentials can be factored out as
an exponential phase as

$\displaystyle V^{\pi st}_{nr,\lambda\lambda^{\prime}}({\bf
p},p^{\prime}\hat{z})$ $\displaystyle=$ $\displaystyle
e^{{i\lambda^{\prime}\phi}}V^{\pi
st}_{nr,\lambda\lambda^{\prime}}(p,p^{\prime},x),$ (15) $\displaystyle{\cal
V}^{\pi st}_{r,\lambda\lambda^{\prime}}({\bf p},p^{\prime}\hat{z})$
$\displaystyle=$ $\displaystyle e^{{i\lambda^{\prime}\phi}}{\cal V}^{\pi
st}_{r,\lambda\lambda^{\prime}}(p,p^{\prime},x),$ (16) $\displaystyle{\cal
V}^{\pi st}_{r,\lambda^{\prime\prime}\lambda^{\prime}}({\bf
p}^{\prime\prime},p^{\prime}\hat{z})$ $\displaystyle=$ $\displaystyle
e^{{i\lambda^{\prime}\phi^{\prime\prime}}}{\cal V}^{\pi
st}_{r,\lambda^{\prime\prime}\lambda^{\prime}}(p^{\prime\prime},p^{\prime},x^{\prime\prime}),$
(17)

where $x\equiv\hat{\bf p}\cdot\hat{\bf p}^{\prime}$,
$x^{\prime\prime}\equiv\hat{\bf p}^{\prime\prime}\cdot\hat{\bf p}^{\prime}$.
One can show that the matrix elements of ${\cal V}^{\pi
st}_{r,\lambda\lambda^{\prime\prime}}({\bf p},{\bf p}^{\prime\prime})$ can be
obtained as

${\cal V}^{\pi st}_{r,\lambda\lambda^{\prime\prime}}({\bf p},{\bf
p}^{\prime\prime})=\frac{{\cal V}^{\pi
st}_{r,\lambda\lambda^{\prime\prime}}(p,p^{\prime\prime},\gamma)}{d_{\lambda^{\prime\prime}\lambda}^{s}(\gamma)}\sum_{\lambda^{*}=-s}^{+s}e^{i\lambda^{*}(\phi-\phi^{\prime\prime})}d_{\lambda^{*}\lambda}^{s}(x)\
d_{\lambda^{*}\lambda^{\prime\prime}}^{s}(x^{\prime\prime}),$ (18)

where $\gamma=xx^{\prime\prime}+\sqrt{1-x^{2}}\sqrt{1-x^{\prime\prime
2}}\cos{(\phi-\phi^{\prime\prime})}$. By these considerations, Eqs. (12) and
(13) can be written as

$\displaystyle\frac{4mV^{\pi
0t}_{nr,00}(p,p^{\prime},x)}{\omega(p)+\omega(p^{\prime})}$ $\displaystyle=$
$\displaystyle{\cal V}^{\pi
0t}_{r,00}(p,p^{\prime},x)+\frac{1}{\omega(p)+\omega(p^{\prime})}\frac{1}{4}\int_{0}^{\infty}dp^{\prime\prime}p^{\prime\prime
2}\int_{-1}^{+1}dx^{\prime\prime}\ {\cal V}^{{\pi
0t},0}_{r,00}(p,p^{\prime\prime},x,x^{\prime\prime})\ {\cal V}^{\pi
0t}_{r,00}(p^{\prime\prime},p^{\prime},x^{\prime\prime}),\quad\quad$ (19)
$\displaystyle\frac{4mV^{\pi
1t}_{nr,\lambda\lambda^{\prime}}(p,p^{\prime},x)}{\omega(p)+\omega(p^{\prime})}$
$\displaystyle=$ $\displaystyle{\cal V}^{\pi
1t}_{r,\lambda\lambda^{\prime}}(p,p^{\prime},x)$ (21) $\displaystyle+$
$\displaystyle\frac{1}{\omega(p)+\omega(p^{\prime})}\int_{0}^{\infty}dp^{\prime\prime}p^{\prime\prime
2}\int_{-1}^{+1}dx^{\prime\prime}$ (22) $\displaystyle\times$
$\displaystyle\biggl{\\{}\frac{1}{4}{\cal V}^{{\pi
1t,\lambda^{\prime}}}_{r,\lambda 0}(p,p^{\prime\prime},x,x^{\prime\prime})\
{\cal V}^{\pi
1t}_{r,0\lambda^{\prime}}(p^{\prime\prime},p^{\prime},x^{\prime\prime})+\frac{1}{2}{\cal
V}^{{\pi 1t,\lambda^{\prime}}}_{r,\lambda
1}(p,p^{\prime\prime},x,x^{\prime\prime})\ {\cal V}^{\pi
1t}_{r,1\lambda^{\prime}}(p^{\prime\prime},p^{\prime},x^{\prime\prime})\biggr{\\}},$
(23)

where

$\displaystyle{\cal V}^{{\pi
st,\lambda^{\prime}}}_{r,\lambda\lambda^{\prime\prime}}(p,p^{\prime\prime},x,x^{\prime\prime})$
$\displaystyle=$
$\displaystyle\sum_{\lambda^{*}=-s}^{+s}d_{\lambda^{*}\lambda}^{s}(x)\
d_{\lambda^{*}\lambda^{\prime\prime}}^{s}(x^{\prime\prime})\int_{0}^{2\pi}d\phi^{\prime\prime}\
e^{i(\lambda^{*}-\lambda^{\prime})(\phi-\phi^{\prime\prime})}\frac{{\cal
V}^{\pi
st}_{r,\lambda\lambda^{\prime\prime}}(p,p^{\prime\prime},\gamma)}{d_{\lambda^{\prime\prime}\lambda}^{s}(\gamma)}.$
(24)

The $\phi^{\prime\prime}$ integration in the interval $[0,2\pi]$ can be
simplified to $[0,\pi/2]$ interval by

$\displaystyle\int_{0}^{2\pi}d\phi^{\prime\prime}\
e^{im(\phi-\phi^{\prime\prime})}{\cal
F}\bigl{(}\cos(\phi-\phi^{\prime\prime})\bigr{)}$ $\displaystyle=$
$\displaystyle\int_{0}^{2\pi}d\phi^{\prime\prime}\
e^{im\phi^{\prime\prime}}{\cal F}\bigl{(}\cos\phi^{\prime\prime}\bigr{)}$ (25)
$\displaystyle=$ $\displaystyle 2\int_{0}^{\pi/2}d\phi^{\prime\prime}\cos
m\phi^{\prime\prime}\biggl{[}{\cal
F}\bigl{(}\cos\phi^{\prime\prime}\bigr{)}+(-)^{m}{\cal
F}\bigl{(}-\cos\phi^{\prime\prime}\bigr{)}\biggr{]}.$ (26)

##  3 Calculation of Relativistic $NN$ Interactions

To calculate the matrix elements of relativistic potentials for different
spin-isospin $(s,t)$ channels, we solve the integral equations (19) and (21)
using an iterative method proposed by Kamada and Glöckle [9]. The iteration
starts with ${\cal V}^{(0){\pi
st}}_{r,\lambda\lambda^{\prime}}(p,p^{\prime},x)=\dfrac{4mV^{\pi
st}_{nr,\lambda\lambda^{\prime}}(p,p^{\prime},x)}{\omega(p)+\omega(p^{\prime})}$
and stops when the maximal difference between the matrix elements of the
relativistic potential ${\cal V}^{{\pi
st}}_{r,\lambda\lambda^{\prime}}(p,p^{\prime},x)$ obtained from two successive
iterations drops below $10^{-6}$ MeV fm3. After each iteration, to obtain the
matrix elements ${\cal V}^{{\pi
st,\lambda^{\prime}}}_{r,\lambda\lambda^{\prime\prime}}(p,p^{\prime\prime},x,x^{\prime\prime})$
that appears in the kernel of Eqs. (19) and (21), we need to perform the
azimuthal angle integration of Eq. (24). To speed up the convergence of the
iteration in solving Eqs. (19) and (21), in some $(s,t)$ channels even to be
able to reach the convergence, we use the weighted average of relativistic
potential obtained from two successive iterations as

$\displaystyle{\cal V}_{r}^{(n)}\longrightarrow\frac{\alpha{\cal
V}_{r}^{(n)}+\beta{\cal V}_{r}^{(n-1)}}{\alpha+\beta};\ n=1,2,...$ (27)

The input to our calculations is a 3D form of CD-Bonn potential in momentum
helicity representation obtained from the summation of partial wave matrix
elements of the potential up to total angular momentum $j_{max}=20$. For the
discretization of the continuous momentum and angle variables, we use the
Gauss-Legendre quadrature. A combination of hyperbolic and linear mapping with
120 mesh points is used for the momentum variables, and for azimuthal and
polar angle variables, a linear mapping with 40 mesh points is used. In our
calculations, we use the nucleon mass
$m=\frac{2m_{p}m_{n}}{m_{p}+m_{n}}=938.91852$ MeV and the conversion factor
$\hslash c=197.327053\ \text{MeV fm}=1$, where proton and neutron masses are
$m_{p}=938.27231$ MeV and $m_{n}=939.56563$ MeV [12]. The number of iterations
needed to reach convergence in the matrix elements of relativistic potential
calculations in different spin-isospin channels, for different values of the
weight averaging parameters $\alpha$ and $\beta$, is shown in Table 1. Our
numerical analysis indicates that for $(s=0,\ t=0)$ channel, the convergence
can be reached only for $\alpha=\beta=1$, whereas for $(s=0,\ t=1)$ and
$(s=1,\ t=0)$ the fastest convergence can be reached by $\alpha=2,\ \beta=1$.
For $(s=1,\ t=1)$ the fastest convergence can be reached by $\alpha=3,\
\beta=1$. It should be mentioned that Kamada and Glöckle have used
$\alpha=\beta=1$ in their calculations to obtain relativistic potentials from
AV18, CD-Bonn, and Nijm I, II potentials [9].

Table 1: The number of iterations $N_{iter}$ needed to reach convergence in Eqs. (19) and (21) for the calculation of relativistic potential, obtained from CD-Bonn potential, as a function of the weight averaging parameters $\alpha$ and $\beta$, which are defined in Eq. (27). $\alpha$ |  | $\beta$ |  | $N_{iter}$
---|---|---|---|---
|  |  |  | $(s=0,\ t=0)$ |  | $(s=0,\ t=1)$ |  | $(s=1,\ t=0)$ |  | $(s=1,\ t=1)$
1 |  | 0 |  | - |  | - |  | - |  | 27
1 |  | 1 |  | 76 |  | 18 |  | 21 |  | 19
2 |  | 1 |  | - |  | 13 |  | 21 |  | 12
3 |  | 1 |  | - |  | 19 |  | 34 |  | 10
4 |  | 1 |  | - |  | 25 |  | 50 |  | 12
5 |  | 1 |  | - |  | 30 |  | 69 |  | 14

In Figs. 1-4, the matrix elements of relativistic potential obtained from CD-
Bonn potential in different spin-isospin channels are compared with
corresponding nonrelativistic potentials. The differences between
nonrelativistic and relativistic potentials are also shown. As we can see,
while the nonrelativistic and relativistic potentials show similar structures,
the difference between them is significant and is in the same order of
magnitude as the potentials.

Figure 1: The matrix elements of the relativistic $NN$ potential (left panel),
the nonrelativistic $NN$ potential (middle panel), and their differences
(right panel), calculated for CD-Bonn potential in $(s=0,\ t=0)$ channel, as a
function of $2N$ relative momenta $p=p^{\prime}$ and the cosine of the angle
between them $x$. Figure 2: The same as Fig. 1, but for $(s=0,\ t=1)$ channel.
Figure 3: The same as Fig. 1, but for $(s=1,\ t=0)$ channel. Figure 4: The
same as Fig. 1, but for $(s=1,\ t=1)$ channel.

##  4 Numerical tests for the relativistic $NN$ potentials

In this section, we present two numerical tests for NN bound and scattering
states, which show the validity of our formalism and the accuracy of
calculated relativistic potentials in the 3D scheme.

###  4.1 Deuteron binding energy and wave function

To test the accuracy of the calculated relativistic $NN$ potential in $(s=1,\
t=0)$ channel, we calculate the deuteron binding energy and wave function for
both nonrelativistic CD-Bonn and the obtained relativistic potentials. The
relativistic form of the homogenous LS equation for describing deuteron
binding energy $E_{d}=m_{d}-2m$ and wave function $\psi_{M_{d}}(p)$ is given
by the following coupled integral equations for wave function components
$\psi_{0}$ and $\psi_{1}$

$\psi_{M_{d}}(p)=\frac{2\pi}{m_{d}-\omega(p)}\int_{0}^{\infty}dp^{\prime}\
p^{\prime 2}\left\\{\frac{1}{2}{\cal
V}_{r,M_{d}1}^{+110}(p,p^{\prime})\psi_{1}(p^{\prime})+\frac{1}{4}{\cal
V}_{r,M_{d}0}^{+110}(p,p^{\prime})\psi_{0}(p^{\prime})\right\\},$ (28)

where

${\cal V}_{r,M_{d}\Lambda^{\prime}}^{+110}(p,p^{\prime})=\int_{-1}^{1}dx\
{\cal V}_{r,M_{d}\Lambda^{\prime}}^{+110}(p,p^{\prime},x)\
d^{1}_{M_{d}\Lambda^{\prime}}(x).$ (29)

In the nonrelativistic form, the free propagator is replaced by
$(E_{d}-\frac{p^{2}}{m})^{-1}$ and the relativistic $NN$ potential is replaced
by the nonrelativistic potential
$V_{nr,M_{d}\Lambda^{\prime}}^{+110}(p,p^{\prime})$ [17]. In Table 2, we
present our numerical results for deuteron binding energies obtained from both
CD-Bonn and relativistic potentials. The relative percentage difference of
$0.06\ \%$ indicates an excellent agreement between relativistic and
nonrelativistic deuteron binding energies. In Fig. 5, we show the deuteron
wave function components calculated for both relativistic and nonrelativistic
CD-Bonn potentials. As we can see, the constructed relativistic potential
reproduces the deuteron binding energy and wave function obtained by CD-Bonn
potential with high accuracy.

Table 2: Calculated deuteron binding energy for relativistic and nonrelativistic CD-Bonn potentials and their relative percentage difference. $E_{d}^{r}$ (MeV) |  |  $E_{d}^{nr}$ (MeV) |  | $\left|\dfrac{(E_{d}^{r}-E_{d}^{nr})}{E_{d}^{nr}}\right|\times 100\%$
---|---|---|---|---
$-2.22463$ |  | $-2.22325$ |  | $0.06207$
Figure 5: The absolute value of deuteron wave function components $\psi_{0}$
and $\psi_{1}$, calculated for relativistic (r) and nonrelativistic (nr) CD-
Bonn potentials, as a function of the relative momentum $p$.

###  4.2 $np$ elastic scattering

For the second numerical test, we calculate the differential and total cross-
section of $np$ elastic scattering for the relativistic potential constructed
from CD-Bonn potential. To describe the relativistic $np$ elastic scattering
in momentum helicity space, the relativistic form of inhomogeneous LS
equations for $2N$ $t-$matrices in singlet and triplet spin states can be
obtained as

$\displaystyle T^{\pi 0t}_{00}(p,p^{\prime},x)$ $\displaystyle=$
$\displaystyle{\cal V}^{\pi 0t}_{r,00}(p,p^{\prime},x)$ (30) $\displaystyle+$
$\displaystyle\frac{1}{4}\int_{0}^{\infty}dp^{\prime\prime}p^{\prime\prime
2}\int_{-1}^{+1}dx^{\prime\prime}\ \frac{{\cal V}^{\pi
0t,0}_{r,00}(p,p^{\prime\prime},x,x^{\prime\prime})\ T^{\pi
0t}_{00}(p^{\prime\prime},p^{\prime},x^{\prime\prime})}{\omega(p_{0})-\omega(p^{\prime\prime})+i\epsilon},$
(31) $\displaystyle T^{\pi 1t}_{\lambda\lambda^{\prime}}(p,p^{\prime},x)$
$\displaystyle=$ $\displaystyle{\cal V}^{\pi
1t}_{r,\lambda\lambda^{\prime}}(p,p^{\prime},x)$ (33) $\displaystyle+$
$\displaystyle\frac{1}{2}\int_{0}^{\infty}dp^{\prime\prime}p^{\prime\prime
2}\int_{-1}^{+1}dx^{\prime\prime}\ \frac{{\cal V}^{\pi
1t,\lambda^{\prime}}_{r,\lambda 1}(p,p^{\prime\prime},x,x^{\prime\prime})\
T^{\pi
1t}_{1\lambda^{\prime}}(p^{\prime\prime},p^{\prime},x^{\prime\prime})}{\omega(p_{0})-\omega(p^{\prime\prime})+i\epsilon}$
(34) $\displaystyle+$
$\displaystyle\frac{1}{4}\int_{0}^{\infty}dp^{\prime\prime}p^{\prime\prime
2}\int_{-1}^{+1}dx^{\prime\prime}\ \frac{{\cal V}^{\pi
1t,\lambda^{\prime}}_{r,\lambda 0}(p,p^{\prime\prime},x,x^{\prime\prime})\
T^{\pi
1t}_{0\lambda^{\prime}}(p^{\prime\prime},p^{\prime},x^{\prime\prime})}{\omega(p_{0})-\omega(p^{\prime\prime})+i\epsilon}.$
(35)

The on-shell momentum $p_{0}$ for an incident projectile energy $E_{lab}$ is
defined by
$p_{0}=\sqrt{\frac{m_{p}^{2}E_{lab}(E_{lab}+2m_{n})}{(m_{p}+m_{n})^{2}+2m_{p}E_{lab}}}$
[12]. By replacing the relativistic free propagator
$(\omega(p_{0})-\omega(p^{\prime\prime})+i\epsilon)^{-1}$ with
$(\frac{p_{0}^{2}}{m}-\frac{p^{\prime\prime 2}}{m}+i\epsilon)^{-1}$ and
relativistic potential ${\cal V}_{r}$ with the nonrelativistic potential
$V_{nr}$, Eqs. (31) and (35) yield the nonrelativistic form of LS equations of
Ref. [16]. The relativistic differential cross section of $np$ elastic
scattering as a function of the incident projectile energy is given by

$\displaystyle\frac{d\sigma}{d\Omega}=(2\pi)^{4}\left(\frac{\omega(p_{0})}{4}\right)^{2}\frac{1}{4}\sum_{m^{\prime}_{s_{1}}m^{\prime}_{s_{2}}m_{s_{1}}m_{s_{2}}}\left|T^{phys}_{m^{\prime}_{s_{1}}m^{\prime}_{s_{2}}m_{s_{1}}m_{s_{2}}}(p_{0},p_{0},x^{\prime})\right|^{2},$
(36)

where the matrix elements of the on-shell physical $t-$matrices can be
obtained from the on-shell relativistic $NN$ $t-$matrices of Eqs. (31) and
(35) as

$\displaystyle
T^{phys}_{m^{\prime}_{s_{1}}m^{\prime}_{s_{2}}m_{s_{1}}m_{s_{2}}}(p_{0},p_{0},x)$
$\displaystyle=$ $\displaystyle\frac{1}{4}\sum_{{\pi
st}}(1-\eta_{\pi}(-)^{s+t})\
C^{2}\left(\frac{1}{2}\frac{1}{2}t,m_{t_{1}}m_{t_{2}}\right)$ (37)
$\displaystyle\times$ $\displaystyle\ \
C\left(\frac{1}{2}\frac{1}{2}s,m^{\prime}_{s_{1}}m^{\prime}_{s_{2}}\lambda^{\prime}_{0}\right)\
C\left(\frac{1}{2}\frac{1}{2}s,m_{s_{1}}m_{s_{2}}\lambda_{0}\right)\
\sum_{\lambda^{\prime}}d^{s}_{\lambda^{\prime}_{0}\lambda^{\prime}}(x)\
T^{{\pi st}}_{\lambda^{\prime}\lambda_{0}}(p_{0},p_{0},x),$ (38)

where $m_{s_{i}}$ and $m_{t_{i}}$ are the spin and isospin projection of
single nucleons along the quantization $z-$axis, and the coefficients $C$ are
the Clebsch-Gordan coefficients. In Fig. 6, we show the differential cross
sections of $np$ elastic scattering, calculated for relativistic and
nonrelativistic CD-Bonn potentials, for the projectile energies
$E_{lab}=50,96,143$ and $200$ MeV. Total cross-sections can provide a more
detailed comparison between relativistic and nonrelativistic potentials. In
Table 3, we present our numerical results for total cross-sections of $np$
elastic scattering, obtained by relativistic and nonrelativistic CD-Bonn
potentials, as a function of the incident projectile energy $E_{lab}$. As we
can see, the maximum relative percentage difference is less than $0.01\ \%$,
indicating that the relativistic total cross-sections are in excellent
agreement with the corresponding nonrelativistic cross-sections.

Figure 6: The $np$ differential cross sections calculated for relativistic (solid magenta lines) and nonrelativistic CD-Bonn (dashed red lines) potentials as a function of scattering angle $\theta_{c.m.}$ in the CM frame, for the incident projectile energies $E_{lab}=50,\ 96,\ 143$, and $200$ MeV. The experimental data are from Refs. [42, 43] for $E_{lab}=50$ MeV (Exp. 1 and Exp. 2, respectively), form Refs. [44, 45, 46] for $E_{lab}=96$ MeV (Exp. 1, Exp. 2 and Exp. 3, respectively), from Ref. [47] for $E_{lab}=142.8$ MeV (Exp. 1) and from Refs. [48, 49, 50] for $E_{lab}=200$ MeV (Exp. 1, Exp. 2 and Exp. 3, respectively). Table 3: The total cross section of $np$ elastic scattering as a function of the incident projectile energy $E_{lab}$, calculated for relativistic ($\sigma_{r}$) and nonrelativistic ($\sigma_{nr}$) CD-Bonn potentials. $E_{lab}$ (MeV) |  $\sigma_{r}$ (mb) |  $\sigma_{nr}$ (mb) | $\left|\frac{\sigma_{r}-\sigma_{nr}}{\sigma_{nr}}\right|\times 100\%$
---|---|---|---
0.001 | 20337.1 | 20336.6 | 0.00246
0.01 | 19222.1 | 19221.7 | 0.00208
0.1 | 12786.7 | 12786.6 | 0.00078
1 | 4262.94 | 4262.95 | 0.00023
3 | 2291.39 | 2291.40 | 0.00044
5 | 1632.54 | 1632.55 | 0.00061
10 | 943.252 | 943.253 | 0.00011
20 | 483.491 | 483.491 | 0.00000
50 | 167.926 | 167.925 | 0.00060
100 | 75.0321 | 75.0359 | 0.00506
250 | 37.6465 | 37.6485 | 0.00531
500 | 28.2841 | 28.2869 | 0.00990
750 | 24.4850 | 24.4872 | 0.00898

Kamada and Glöckle have shown in Ref. [9] that the obtained relativistic
potential from AV18 potential, in a PW decomposition, reproduces the
nonrelativistic phase shifts with five significant figures with projectile
energy in the domain $(1-350)$ MeV. As one can see in Table 3, our
nonrelativistic and relativistic cross-section results are also in perfect
agreement with five significant figures with an incident projectile energy in
the broader domain $(0.001-750)$ MeV. So we are convinced that the 3D
formulation and calculations for relativistic potential provide the same
accuracy as a PW calculation.

Moreover, in a prior study for calculation of relativistic potentials from
spin-independent MT potential [10], which has no spin and isospin complexity
of CD-Bonn, we obtained a relative percentage difference of $0.06\%$ between
nonrelativistic and relativistic deuteron binding energies and a maximum
relative percentage difference of $0.007\%$ in two-body total elastic
scattering cross-sections, which can be compared with $0.06\%$ and $0.01\%$
relative percentage differences obtained in this study for CD-Bonn potential.
This comparison indicates calculating relativistic $NN$ interactions from
realistic interactions in a 3D scheme provides almost the same accuracy level
as a spin-independent calculation.

##  5 Conclusion and outlook

In this paper, the quadratic equation, which connects the relativistic and
nonrelativistic $NN$ potentials, is formulated in momentum helicity space as a
single and four coupled three-dimensional integral equations for $2N$ singlet
and triplet spin states. In our numerical calculations, we implement the CD-
Bonn potential to obtain the matrix elements of the relativistic potential as
a function of the magnitude of $2N$ relative momenta, the angle between them,
and spin and isospin quantum numbers. The quadratic integral equations are
solved using an iterative scheme. Our numerical results indicate that
calculated relativistic $NN$ potential from the CD-Bonn potential reproduces
$2N$ observables for deuteron binding energy and the differential and total
cross sections of $np$ elastic scattering with high accuracy. The
implementation of relativistic $NN$ potentials in the relativistic description
of triton binding energy and wave function is currently underway.

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## Acknowledgements

The work of M. R. Hadizadeh was supported by the National Science Foundation
under grant No. NSF-PHY-2000029 with Central State University.

## Author contributions statement

M.R.H. designed and directed the project; M.R.H. and M.R. developed the
theoretical formalism; F.N. calculated the matrix elements of CD-Bonn
potential in a 3D scheme; M.R.H. and M.R. calculated the matrix elements of
relativistic potentials; all authors discussed the results and contributed to
the final manuscript.

## Competing financial interests

The authors declare no competing interests.

## Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.