# G-Families of Singquandles

Mohd Ibrahim Sheikh  Department of Mathematics, Graduate School of Natural
Sciences Pusan National University, Busan 46241, Republic of Korea
<EMAIL_ADDRESS>,  Mohamed Elhamdadi  University of South Florida,
Tampa, Florida, USA<EMAIL_ADDRESS>and  DANISH ALI  Department of
Mathematics, Dalian University of Technology, China
<EMAIL_ADDRESS>
# A G-Family of Singquandles and Invariants of Dichromatic Singular links

Mohd Ibrahim Sheikh  Department of Mathematics, Graduate School of Natural
Sciences Pusan National University, Busan 46241, Republic of Korea
<EMAIL_ADDRESS>,  Mohamed Elhamdadi  University of South Florida,
Tampa, Florida, USA<EMAIL_ADDRESS>and  DANISH ALI  Department of
Mathematics, Dalian University of Technology, China
<EMAIL_ADDRESS>
###### Abstract.

We introduce and investigate dichromatic singular links. We also construct
$\mathnormal{G}$-Family of singquandles and use them to define counting
invariants for unoriented dichromatic singular links. We provide some examples
to show that these invariants distinguish some dichromatic singular links.

###### Contents

  1. 1 Introduction
  2. 2 Singular links, Singquandles and Dichromatic Links
  3. 3 Dichromatic Singular Links
  4. 4 $\mathnormal{G}$-Family of Singquandles (Disingquandles)
  5. 5 Computable Invariants for Unoriented Dichromatic Singular Links

Mathematics Subject Classifications (2020): 57M25, 57M27.
Key words and Phrases: Knot; Link; Singular knot; Singular link; Dichromatic
link; Dichromatic singular link; Quandle; Singquandle; Disingquandle;
Disingquandle counting invariant.

##  1\. Introduction

A knot is a simple closed curve in three dimensional space $\mathbb{R}^{3}$
and a disjoint union of two or more knots forms a link with two or more
components [EN]. Knots and links are categorised in many ways. One way is to
use the crossing type as a tool to define a knot or link type. Classical,
virtual and singular knots and links serve as examples as they are all
recognised by the type of crossing they contain. The other way to define link
types is by labelling the components of a classical link. Dichromatic links
are defined by using this technique as their components are either labelled by
$``1"$ or $``2"$ [HK, HP, K, 1, 2]. A singular link is a link with at least
one singular crossing. In this paper we use such labelling technique for
singular links and define a new type of links which we call dichromatic
singular links.

A quandle is an algebraic structure satisfying some axioms that result from
the Reidemeister moves for oriented classical knots and links. If furthermore
all right multiplications by fixed elements of the quandle are involutions
then such structures are called involutory quandles or Kei’s They are used to
investigate unoriented knots and links. Quandles were independently introduced
by Joyce and Matveev [J, M]. Since then they have been used to construct
invariants of knots and links [CCE1, CCEH1, NOS]. Quandles have been also used
to define new algebraic systems by taking a family of quandles at a time. Such
systems are called $\mathnormal{G}$-Family of quandles and this notion was
introduced in $2013$ by Ishii, Iwakiri, Jang and Oshiro [IMJO]. A
$\mathnormal{G}$-Family of quandles were used to define invariants for
handlebody-knots. Also in [LS] Lee and Sheikh used $\mathbb{Z}_{2}$-Family of
quandles to construct algebraic invariants for oriented dichromatic links.

In this paper, we introduce the notions of $\mathnormal{G}$-Family of
singquandles and dichromatic singular links. A dichromatic singular link is an
n component singular link with each of its component labelled as $``1"$ or
$``2"$. A singquandle is an algebraic system whose axioms are motivated by
Reidemeister moves of unoriented singular knots. By taking a family of such
algebaraic systems (Singquandles), we define a new algebraic system which we
call $\mathnormal{G}$-Family of singquandles or disingquandle. The axioms of
the latter are motivated by generalized Reidemeister moves of unoriented
dichromatic singular links. We discuss various examples and some properties of
$\mathnormal{G}$-Family of singquandles, and also show that a
$\mathnormal{G}$-Family of singquandles $\mathnormal{X}$ enables us to
distinguish unoriented dichromatic singular links by computing their sets of
all $\mathnormal{X}$-colorings and proving that these sets are different when
their arcs are colored by the elements of $\mathnormal{X}$.

This paper is organized as follows. Section 2 reviews some preliminaries about
singular links, singquandles as well as dichromatic links and their
generalized Reidemeister moves. In Section 3 we introduce the notion of
dichromatic singular links with some typical examples of unoriented
dichromatic singular links. Section 4 introduces the notion of
$\mathnormal{G}$-Family of singquandles (disingquandles) with some typical
examples of $\mathnormal{G}$-Family of singquandles. Section 5 discusses how
$\mathnormal{G}$-Family of singquandles is related to unoriented dichromatic
singular links and develop computable invariants for unoriented dichromatic
singular links. We discuss some examples which show how the invariants
distinguish unoriented dichromatic singular links, and especially how they
detect the change of component labelings.

##  2\. Singular links, Singquandles and Dichromatic Links

In this section we review some preliminaries about singular links,
singquandles and dichromatic links. Most of the terminologies of this section
can be found in [LS, HN, CEHN]. We begin with the definition of a singular
link.

######  Definition 2.1.

A singular link in $\mathbb{S}^{3}$ is the image of a smooth immersion of $n$
circles in $\mathbb{S}^{3}$ that has finitely many double points, called
singular points.

A singular link in $\mathbb{R}^{3}$ is represented by a singular link diagram
in the plane $\mathbb{R}^{2}$, which is a classical link diagram with one or
more singularities. A singularity is a rigid vertex where a link is glued to
itself. Figure 1 gives two examples of singular links.

Figure 1. Singular Links

Two singular links $\mathnormal{L_{1}}$ and $\mathnormal{L_{2}}$ are isotopy
equivalent if one can be obtained from the other by a finite sequence
generalized Reidemeister moves for singular links as shown in the following
figure. Let $\mathnormal{D_{1}}$ and $\mathnormal{D_{2}}$ be two singular link
diagrams in $\mathbb{R}^{2}$ representing $\mathnormal{L_{1}}$ and
$\mathnormal{L_{2}}$, respectively. Then $\mathnormal{L_{1}}$ and
$\mathnormal{L_{2}}$ are equivalent if and only if $\mathnormal{D_{1}}$ and
$\mathnormal{D_{2}}$ can be transformed into each other by a finite sequence
of classical and singular Reidemeister moves shown in Figure 2.

Figure 2. Classical and Singular Reidemeister Moves

######  Definition 2.2.

[CEHN] Let $(\mathnormal{X},*)$ be an involutive quandle. Let $\mathbf{R_{1}}$
and $\mathbf{R_{2}}$ be two maps from $\mathnormal{X}\times\mathnormal{X}$ to
$\mathnormal{X}$. The quadruple
$(\mathnormal{X},*,\mathbf{R_{1}},\mathbf{R_{2}})$ is called a singquandle if
the following axioms are satisfied

$x=\mathbf{R_{1}}(y,\mathbf{R_{2}}(x,y))=\mathbf{R_{2}}(\mathbf{R_{2}}(x,y),\mathbf{R_{1}}(x,y)),$
$y=\mathbf{R_{2}}(\mathbf{R_{2}}(x,y),x)=\mathbf{R_{1}}(\mathbf{R_{2}}(x,y),\mathbf{R_{1}}(x,y)),$
$\mathbf{R}(x,y)=(\mathbf{R_{1}}(y,\mathbf{R_{2}}(x,y)),\mathbf{R_{2}}(\mathbf{R_{2}}(x,y),x)),$
$(y*z)*\mathbf{R_{2}}(x,z)=(y*x)*\mathbf{R_{1}}(x,z),$
$\mathbf{R_{1}}(x,y)=\mathbf{R_{2}}(y*x,x),$
$\mathbf{R_{2}}(x,y)=\mathbf{R_{1}}(y*x,x)*\mathbf{R_{2}}(y*x,x),$
$\mathbf{R_{1}}(x*y,z)*y=\mathbf{R_{1}}(x,z*y),$
$\mathbf{R_{2}}(x*y,z)=\mathbf{R_{2}}(x,z*y)*y.$

We remind the reader that the singquandle axioms come from the generalized
Reidemeister moves for unoriented singular knots. Singquandles were introduced
as a ramification of quandles with the purpose of studying singular links, see
for example [CCE1, NOS, CEHN].

The following are few typical examples of singquandles.

  * •

For an involutive quandle $(X,*)$ with $x*y=2y-x$ and $X=\mathbb{Z}_{n}$, the
quadruple $(\mathnormal{X},*,\mathbf{R_{1}},\mathbf{R_{2}})$ forms a
singquandle if and only if the following conditions are satisfied:

    1. (1)

$\mathbf{R_{2}}(x,y)=\mathbf{R_{1}}(x,y)+y-x$,

    2. (2)

$\mathbf{R_{1}}(x,y)=\mathbf{R_{1}}(2x-y,x)+y-x$,

    3. (3)

$\mathbf{R_{1}}(x,2y-z)=2y-\mathbf{R_{1}}(2y-x,z)$,

    4. (4)

$\mathbf{R_{2}}(2y-x,z)=2y-\mathbf{R_{2}}(x,2x-z)$.

  * •

For an involutive quandle $(X,*)$ where $X$ is a group $G$ and $x*y=yx^{-1}y$,
the quadruple $(\mathnormal{X},*,\mathbf{R_{1}},\mathbf{R_{2}})$ forms a
singquandle if and only if the following conditions are satisfied:

    1. (1)

$\mathbf{R_{2}}(x,z)z^{-1}yz^{-1}\mathbf{R_{2}}(x,z)=\mathbf{R_{1}}(x,z)x^{-1}yx^{-1}\mathbf{R_{1}}(x,z)$,

    2. (2)

$\mathbf{R_{1}}(x,y)=\mathbf{R_{2}}(xy^{-1}x,x)$,

    3. (3)

$\mathbf{R_{2}}(x,y)=\mathbf{R_{2}}(xy^{-1}x,x)[\mathbf{R_{1}}(xy^{-1}x,x)]^{-1}\mathbf{R_{2}}(xy^{-1}x,x)$,

    4. (4)

$y[\mathbf{R_{1}}(yx^{-1}y,z)]^{-1}y=\mathbf{R_{1}}(x,yz^{-1}y)$,

    5. (5)

$\mathbf{R_{1}}(yx^{-1}y,z)=y[\mathbf{R_{2}}(x,yz^{-1}y)]^{-1}y$.

######  Definition 2.3.

For a positive integer $n\geq 1$. A dichromatic link is a smooth imbedding of
$n$ circles in $\mathbb{R}^{3}$ such that each component is labeled as $``1"$
or $``2"$.

In $\mathbb{R}^{2}$ every dichromatic link is represented by a dichromatic
link diagram which is a classical link diagram with each component labelled
either $``1"$ or $``2"$. For example, see Figure 3.

Figure 3. Dichromatic Links

Two dichromatic links $\mathnormal{L_{1}}$ and $\mathnormal{L_{2}}$ are
isotopy equivalent if one can be obtained from the other by a finite sequence
of generalized Reidemeister moves for the dichromatic links as shown in the
figure 4. Let $\mathnormal{D_{1}}$ and $\mathnormal{D_{2}}$ be two dichromatic
link diagrams in $\mathbb{R}^{2}$ representing $\mathnormal{L_{1}}$ and
$\mathnormal{L_{2}}$, respectively. Then $\mathnormal{L_{1}}$ and
$\mathnormal{L_{2}}$ are equivalent if and only $\mathnormal{D_{1}}$ and
$\mathnormal{D_{2}}$ can be transformed into each other by a finite sequence
of generalized Reidemeister moves shown in the following Figure 4.

Figure 4. Generalized Reidemeister Moves for Dichromatic Links

##  3\. Dichromatic Singular Links

This section is devoted to dichromatic singular links which is a
generalization of singular links. To generate a dichromatic singular link we
label a singular link’s components with $``1"$ or $``2"$. Thus We have the
following definition.

######  Definition 3.1.

A singular link $\mathnormal{L}$ in $\mathbb{R}^{3}$ whose each component is
colored (labelled) by either $``1"$ or $``2"$ is called a dichromatic singular
link.

A dichromatic singular link $\mathnormal{L}$ in $\mathbb{R}^{3}$ is
represented by a dichromatic singular link diagram $\mathnormal{D}$ in
$\mathbb{R}^{2}$ in which each component is labelled $``1"$ or $``2"$. Figure
5 shows two examples of unoriented dichromatic singular link diagrams.

Figure 5. Dichromatic Singular Links

Two dichromatic singular links $\mathnormal{L_{1}}$ and $\mathnormal{L_{2}}$
in $\mathbb{R}^{3}$ are ambient isotopic if there exists a self homeomorphism
$h:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ that takes one link to the other
and preserves the singularities as well as the labels $``1",``2"$ such that
$h(\mathnormal{L_{1}})=\mathnormal{L_{2}}$. Thus two singular dichromatic
links $\mathnormal{L_{1}}$ and $\mathnormal{L_{2}}$ are equivalent if one can
be obtained from the other by a finite sequence of generalized dichromatic
singular Reidemeister moves preserving the label of each component as shown in
the Figure 6. Let $\mathnormal{D_{1}}$ and $\mathnormal{D_{2}}$ be two
dichromatic singular link diagrams in $\mathbb{R}^{2}$ representing
$\mathnormal{L_{1}}$ and $\mathnormal{L_{2}}$, respectively. Then
$\mathnormal{L_{1}}$ and $\mathnormal{L_{2}}$ are equivalent if and only if
$\mathnormal{D_{1}}$ and $\mathnormal{D_{2}}$ can be transformed into each
other by a finite sequence of generalized dichromatic singular Reidemeister
moves shown in the following Figure 6 where ${i,j,k}\in\\{1,2\\}$.

Figure 6. Regular Dichromatic Reidemeister Moves $RI,RII$ and $RIII$ on the
top and Dichromatic Singular Reidemeister Moves $RIVa,RIVb$ and $RV$ in the
middle and on the bottom.

A dichromatic singular link with $n$ components is called as an $n$-component
dichromatic singular link. Thus an $n$-component dichromatic singular link in
$\mathbb{R}^{3}$ can be defined as $L=K_{1}\cup\cdots\cup K_{n}$. Taking
$n=2$, we obtain $2$-component dichromatic singular links. Some $2$-component
unoriented dichromatic singular link diagrams (see p 814 of [Oyamaguchi]) are
shown in Figure 12.

######  Proposition 3.2.

Let $\mathnormal{L_{1}}$ and $\mathnormal{L_{2}}$ be two unoriented
dichromatic singular links in $\mathbb{R}^{3}$ and let $\mathnormal{D_{1}}$
and $\mathnormal{D_{2}}$ be two unoriented dichromatic singular link diagrams
in $\mathbb{R}^{2}$ representing $\mathnormal{L_{1}}$ and
$\mathnormal{L_{2}}$, respectively. Then $\mathnormal{L_{1}}$ and
$\mathnormal{L_{2}}$ are equivalent if and only if $\mathnormal{D_{1}}$ and
$\mathnormal{D_{2}}$ are transformed into each other by a finite sequence of
generalized Reidemeister moves for unoriented dichromatic singular links which
preserve the singularities and the label of each component as shown in the
Fig. 6 where ${i,j,k}\in\\{1,2\\}$ and ambient isotopies of $\mathbb{R}^{2}$.

##  4\. $\mathnormal{G}$-Family of Singquandles (Disingquandles)

Before introducing the notion of $\mathnormal{G}$-Family of Singquandles, we
first recall the definition of $G$-family of quandles from [IMJO].

######  Definition 4.1.

Given a group $G$ and a set $X$, a $G$-family of quandles, denoted by $(G,X)$,
is a choice of quandle operation $*^{g}$ on the set $X$ for each element $g\in
G$ such that the following axioms are satisfied

  1. (1)

For all $g\in G$ and for all $x\in X$, $x*^{g}x=x$,

  2. (2)

For all $g,h\in G$ and for all $x,y\in X$, $(x*^{g}y)*^{h}y=x*^{gh}y$,

  3. (3)

For all $x,y\in X$, $x*^{e}x=x$, where $e$ is the identity element of $G$,

  4. (4)

For all $x,y,z\in X$, $(x*^{g}y)*^{h}z=(x*^{h}z)*^{h^{-1}gh}(y*^{h}z)$

The following are two examples of $G$-families of quandles.

  * •

For any group $G$ and any set $X$, defining $x*^{g}y=x$ for all $x,y\in X$ and
all $g\in G$. This gives a $G$-family of quandles called the _trivial_
$G$-family of quandles.

  * •

Let $(X,*)$ be a quandle of _cyclic type_ [Tamaru] with cardinality $n$. Let
$R_{x}$ denotes the right multiplication by $x$, thus by definition
${R_{x}}^{(n-1)}$ is the identity map. Then define $x*^{i}y={R_{y}}^{i}(x)$
then it is shown in Proposition 2.3 of [IMJO] that $(\mathbb{Z},X)$ is a
$\mathbb{Z}$-family of quandles and also $\mathbb{Z}_{(n-1)}$-family of
quandles.

A $G$-family of quandles $(G,X)$ induces a quandle operation on the set
$G\times X$ by

$(g,x)*(h,y)=(h^{-1}gx,x*^{h}y).$

The notion of $G$-family of quandles was introduced by Ishii, Iwakiri, Jang
and Oshiro in 2013 in [IMJO] in order to produce invariants of handlebody
knots. They defined coloring invariants and cocycle invariants of handlebody
knots. They used these invariants to detect chirality of some handlebody
knots. Later in $2015$, Ishii independently studied the notion of $G$-family
of quandles in connection with the multiple conjugation quandle and showed
that the later one can be obtained from the first one. In $2017$ and $2018$
Ishii, Nelson and Ishii, Iwakiri, Kamada, Kim, Matsuzaki, Oshiro respectively,
used this work and introduced the notions of partially multiplicative
biquandles and multiple conjugation biquandle. In $2021$ Lee and Sheikh
jointly used $G$-family of quandles to construct algebraic invariants for
oriented dichromatic links [LS]. We introduce the following definition.

######  Definition 4.2.

Let $X$ be a set equipped with two binary operations $*_{1}$ and $*_{2}$ such
that both $(X,*_{1}),(X,*_{2})$ are involutive quandles. Let
$\mathbf{R_{1}},\mathbf{R_{2}}$ be two maps from
$\mathnormal{X}\times\mathnormal{X}$ to $\mathnormal{X}$ such that the
quadruples $(\mathnormal{X},*_{1},\mathbf{R_{1}},\mathbf{R_{2}})$ and
$(\mathnormal{X},*_{2},\mathbf{R_{1}},\mathbf{R_{2}})$ are singquandles. Then
the quintuple $(\mathnormal{X},*_{1},*_{2},\mathbf{R_{1}},\mathbf{R_{2}})$ is
called a disingquandle or $\mathbb{Z}_{2}$-family of singquandles if the
following axioms are satisfied

$(x*_{1}y)*_{2}z=(x*_{2}z)*_{1}(y*_{2}z),$
$(x*_{2}y)*_{1}z=(x*_{1}z)*_{2}(y*_{1}z),$
$(y*_{1}z)*_{2}\mathbf{R_{2}}(x,z)=(y*_{2}x)*_{1}\mathbf{R_{1}}(x,z),$
$(y*_{2}z)*_{1}\mathbf{R_{2}}(x,z)=(y*_{1}x)*_{2}\mathbf{R_{1}}(x,z),$
$\mathbf{R_{2}}(x,y)=\mathbf{R_{1}}(y*_{1}x,x)*_{2}\mathbf{R_{2}}(y*_{1}x,x),$
$\mathbf{R_{2}}(x,y)=\mathbf{R_{1}}(y*_{2}x,x)*_{1}\mathbf{R_{2}}(y*_{2}x,x),$

The above axioms of a disingquandle come from the generalized dichromatic
singular Reidemeister moves shown in Figure 6 when we take the coloring rule
shown in Figure 7 under consideration.

Figure 7. Coloring by a disingquandle

The following lemma is motivated by the above construction.

######  Lemma 4.3.

The set of colorings of a dichromatic singular link by a disingquandle does
not change by the dichromatic singular Reidemeister moves shown in Figure 6.

###### Proof.

As in the case of classical and singular knot theories, there is one to one
correspondence between colorings before and after each of the dichromatic
singular Reidemeister moves. The invariance follows directly from the
equations 4.2, 4.2, 4.2, 4.2, 4.2 and 4.2 given in Definition 4.2. ∎

######  Example 4.4.

Let $(\mathnormal{X},*_{1},\mathbf{R_{1}},\mathbf{R_{2}})$ and
$(\mathnormal{X},*_{2},\mathbf{R_{1}},\mathbf{R_{2}})$ be two singquandles
such that such that $x\;*_{1}\;y=x=x\;*_{2}\;y$ and
$\mathbf{R_{1}}(x,y)=\mathbf{R_{2}}(x,y)$, then
$(\mathnormal{X},*_{1},*_{2},\mathbf{R_{1}},\mathbf{R_{2}})$ forms a
disingquandle.

######  Example 4.5.

Let $(\mathnormal{X},*_{1},\mathbf{R_{1}},\mathbf{R_{2}})$ and
$(\mathnormal{X},*_{2},\mathbf{R_{1}},\mathbf{R_{2}})$ be two singquandles. If
for all $x,y\in X$ we have $x*_{1}y=x*_{2}y$ then
$(\mathnormal{X},*_{1},*_{2},\mathbf{R_{1}},\mathbf{R_{2}})$ forms a
disingquandle.

Now Example 4.5 combined with Proposition 3.6 in [CEHN] gives the following
example.

######  Example 4.6.

Let $\Lambda=\mathbb{Z}[t,B]/(t^{2}-1,B(1+t),t-(1-B)^{2})$ and $X$ be an
$\Lambda$-module. Define $x*_{1}y=tx+(1-t)y,R_{1}(x,y)=(1-t-b)x+(t+b)y$ and
$R_{2}(x,y)=(1-B)x+By$, then by setting $*_{2}=*_{1}$, then one obtains that
$(\mathnormal{X},*_{1},*_{2},\mathbf{R_{1}},\mathbf{R_{2}})$ forms a
disingquandle.

######  Example 4.7.

Let $X$ be a module over $\Lambda=\mathbb{Z}[t]$. Define
$x*_{1}y=x*_{2}y=tx+(1-t)y,R_{1}(x,y)=(1-t-B)x+(t+B)y$ and
$R_{2}(x,y)=(1-B)x+By$. Setting $t=-1$ and $X=\mathbb{Z}_{7}$, then the
quintuple $(\mathnormal{X},*_{1},*_{2},\mathbf{R_{1}},\mathbf{R_{2}})$ forms a
disingquandle if $B=4$ or if $B=5$.

This example can be generalized to $\mathbb{Z}_{p}$, where $p$ is a prime as
follows.

######  Example 4.8.

Let $p$ be an odd prime and let $B\in\mathbb{Z}_{p}$. Consider
$\mathbb{Z}_{p}$ with $x*_{1}y=x*_{2}y=-x+2y,R_{1}(x,y)=(2-B)x+(-1+B)y$ and
$R_{2}(x,y)=(1-B)x+By$. Let $\zeta$ be a primitive root of unity in
$\mathbb{Z}_{p}$ so that $\zeta^{\frac{p-1}{2}}=-1$. By choosing
$1-B=\zeta^{\frac{p-1}{2}}$ we obtain that
$(\mathbb{Z}_{p},*_{1},*_{2},\mathbf{R_{1}},\mathbf{R_{2}})$ forms a
disingquandle.

######  Example 4.9.

Let $X=G$ be a multiplicative group with the involutive quandle operations
$x*_{1}y=x*_{2}y=yx^{-1}y$ (core quandle on $G$), then a direct computation
gives the fact that the quintuple
$(\mathnormal{X},*_{1},*_{2},\mathbf{R_{1}},\mathbf{R_{2}})$ forms a
disingquandle if and only if $\mathbf{R_{1}}$ and $\mathbf{R_{2}}$ satisfies
the following equations:

$\mathbf{R_{2}}(x,z)z^{-1}yz^{-1}\mathbf{R_{2}}(x,z)=\mathbf{R_{1}}(x,z)x^{-1}yx^{-1}\mathbf{R_{1}}(x,z),$
$\mathbf{R_{2}}(x,y)=\mathbf{R_{2}}(xy^{-1}x,x)[\mathbf{R_{1}}(xy^{-1}x,x)]^{-1}\mathbf{R_{2}}(xy^{-1}x,x),$

A straightforward computation gives the following solution

$\mathbf{R_{1}}(x,y)=x\;\textit{and}\;\mathbf{R_{2}}(x,y)=y,\;\textit{for
all}\;x,y,z\in G.$

Now assume that $G$ is an abelian group without $2$-torsion, so that
$x*y=-x+2y$, then $(\mathnormal{X},*_{1},*_{2},\mathbf{R_{1}},\mathbf{R_{2}})$
forms a disingquandle if and only if $R_{2}(x,y)=R_{1}(x,y)+y-x,$ where
$R_{1}$ satisfies the identity $R_{1}(x,y)=R_{1}(-x+2y,x)+y-x$. For example
for any integer $m$, the map $R_{1}(x,y)=mx+(2m+1)y$ give a solution. Thus we
have a family of solutions parametrized by the integer $m$:

$R_{1}(x,y)=mx+(2m+1)y,\;R_{2}(x,y)=(m-1)x+2(m+1)y.$

######  Definition 4.10.

A map $f:X\rightarrow Y$ is called a homomorphism of disingquandle
$(\mathnormal{X},*_{1},*_{2},\mathbf{R_{1}},\mathbf{R_{2}})$ and
$(\mathnormal{Y},*^{\prime}_{1},*^{\prime}_{2},\mathbf{R^{\prime}_{1}},\mathbf{R^{\prime}_{2}})$
if the following conditions are satisfied for all $x,y,z\in X$

  1. (i)

$f(x*_{1}y)=f(x)*^{\prime}_{1}f(y)$,

  2. (ii)

$f(x*_{2}y)=f(x)*^{\prime}_{2}f(y)$,

  3. (iii)

$f(\mathbf{R_{1}}(x,y))=\mathbf{R^{\prime}_{1}}(f(x),f(y))$,

  4. (iv)

$f(\mathbf{R_{2}}(x,y))=\mathbf{R^{\prime}_{1}}(f(x),f(y))$.

If a homomorphism of disingquandle is bijective, then it is called an
isomorphism of disingquandle. We say that two $\mathbb{Z}_{2}$-families of
singquandles are isomorphic if there exists an ismorphism of disingquandle
between them.

######  Definition 4.11.

Let $(\mathnormal{X},*_{1},*_{2},\mathbf{R_{1}},\mathbf{R_{2}})$ be a
disingquandle. A subset $Y\subset X$ is called a sub-disingquandle if
$(\mathnormal{Y},*_{1},*_{2},\mathbf{R_{1}},\mathbf{R_{2}})$ is itself a
disingquandle.

######  Example 4.12.

We use Example 4.9 to get the following 2 examples:

  * •

Let $X=\mathbb{Z}_{9}$ be the dihedral quandle with $x*y=-x+2y$,
$R_{1}(x,y)=mx+(2m+1)y,\;R_{2}(x,y)=(m-1)x+2(m+1)y.$. Then
$(\mathnormal{Y},*_{1},*_{2},\mathbf{R_{1}},\mathbf{R_{2}})$ is itself a
disingquandle with $Y=\mathbb{Z}_{3}$.

  * •

Let $X=\mathbb{Z}_{25}$ be the dihedral quandle with $x*y=-x+2y$,
$R_{1}(x,y)=mx+(2m+1)y,\;R_{2}(x,y)=(m-1)x+2(m+1)y.$. Then
$(\mathnormal{Y},*_{1},*_{2},\mathbf{R_{1}},\mathbf{R_{2}})$ is itself a
disingquandle with $Y=\mathbb{Z}_{5}$.

Given a homomorphism of disingquandles, we obtain the following lemma.

######  Lemma 4.13.

The image Im(f) of any homomorphism of disingquandle $f$ defined from
$(\mathnormal{X},*_{1},*_{2},\mathbf{R_{1}},\mathbf{R_{2}})$ to
$(\mathnormal{Y},*^{\prime}_{1},*^{\prime}_{2},\mathbf{R^{\prime}_{1}},\mathbf{R^{\prime}_{2}})$
is always a sub-disingquandle.

###### Proof.

Given that
$f:(\mathnormal{X},*_{1},*_{2},\mathbf{R_{1}},\mathbf{R_{2}})\rightarrow(\mathnormal{Y},*^{\prime}_{1},*^{\prime}_{2},\mathbf{R^{\prime}_{1}},\mathbf{R^{\prime}_{2}})$
is a homomorphism. Then the equations $(i),(ii),(iii)$ and $(iv)$ of
Definition 4.10 imply that Im(f) is closed under $*_{1},*_{2},\mathbf{R_{1}}$
and $\mathbf{R_{2}}$. Then the axioms of disingquandle are satisfied in $Y$.
Hence they are automatically satisfied in Im(f). This ends the proof of the
lemma. ∎

Now we introduce the notion of fundamental disingquandle of an unoriented
dichromatic singular link and provide an illustrative example. Let $D$ be a
diagram of an unoriented dichromatic singular link $L$ in $\mathbb{R}^{2}$. We
define the fundamental disingquandle of $D$, denoted by $\mathcal{DSQ}(D)$, as
the set of equivalence classes of disingquandle words $W$-$\mathcal{DSQ}(D)$
under the equivalence relation generated by the axioms of disingquandle and
the crossing relations shown in Figure 7, where $W$-$\mathcal{DSQ}(D)$ are
defined by taking a set of generators $X=\\{x_{1},x_{2},x_{3},.....,x_{n}\\}$
which corresponds bijectively with the semi arcs in $D$, recursively by the
following two rules:

  1. (1)

$X\subset$ $W$-$\mathcal{DSQ}(D)$,

  2. (2)

If $x,y\in$ $W$-$\mathcal{DSQ}(D)$, then
$x*_{1}y,x*_{2}y,\mathbf{R_{1}}(x,y),\mathbf{R_{2}}(x,y)\in$
$W$-$\mathcal{DSQ}(D)$.

######  Example 4.14.

Consider the following unoriented dichromatic singular link $L$.

Figure 8. Fundamental Disingquandle of Unoriented Dichromatic Singular Links

The fundamental disingquandle of $L$ is given by

$\mathcal{DSQ}(L)=\langle
x,y,z,u,v|\;z=x*_{2}y;u=y*_{1}z;v=z*_{2}u;x=R_{1}(u,v);y=R_{2}(u,v)\rangle.$

This presentation can be simplified to the following presentation of
$\mathcal{DSQ}(L)$

$\langle x,y|\;x=R_{1}(y*_{1}(x*_{2}y),(x*_{2}y)*_{2}(y*_{1}(x*_{2}y)));\\\
y=R_{2}(y*_{1}(x*_{2}y),(x*_{2}y)*_{2}(y*_{1}(x*_{2}y)))\rangle.$

##  5\. Computable Invariants for Unoriented Dichromatic Singular Links

Let $D$ be an unoriented dichromatic singular link diagram and let
$\mathcal{A}(D)$ denote the set of all arcs of $D$. Let
$(X,*_{1},*_{2},\mathbf{R_{1}},\mathbf{R_{2}})$ be a disingquandle. A
disingquandle coloring of $D$ by $X$, or simply disingquandle $X$-coloring of
$D$, is a map $\mathcal{C}:\mathcal{A}(D)\rightarrow X$ such that at every
classical and singular crossing, the relations depicted in Figure 7 hold. The
disingquandle element $\mathcal{C}(s)$ is called a color of the arc $s$ and
the pair $(D,\mathcal{C})$ is called the $X$-colored unoriented dichromatic
singular link diagram by $\mathcal{C}$. The set of all disingquandle
$X$-colorings of $D$ is denoted by ${\rm Col}^{dsq}_{X}(D)$. Then we have the
following:

######  Lemma 5.1.

Let $D$ and $D^{\prime}$ be two unoriented dichromatic singular link diagramss
in $\mathbb{R}^{2}$ that can be transformed into each other by unoriented
generalized dichromatic singular Reidemeister moves as shown in the Figure 6.
Then for any finite disingquandle $X$, there is a one-to-one correspondence
between ${\rm Col}^{dsq}_{X}(D)$ and ${\rm Col}^{dsq}_{X}(D^{\prime})$.

###### Proof.

It suffices to prove the assertion for the case that $D^{\prime}$ is obtained
from $D$ by a single an unoriented generalized dichromatic singular
Reidemeister move. Let $E$ be an open disk in $\mathbb{R}^{2}$ where the
unoriented generalized dichromatic singular Reidemeister move under
consideration is applied. Then
$D\cap(\mathbb{R}^{2}-E)=D^{\prime}\cap(\mathbb{R}^{2}-E)$. Now let
$\mathcal{C}$ be a disingquandle $X$-coloring of $D$. Since
$(X,*_{1},\mathbf{R_{1}},\mathbf{R_{2}})$ and
$(X,*_{2},\mathbf{R_{1}},\mathbf{R_{2}})$ are both singquandles by the
disingquandle definition 4.2, it is obviously seen from the Figure 6 that the
restriction of $\mathcal{C}$ to
$D\cap(\mathbb{R}^{2}-E)(=D^{\prime}\cap(\mathbb{R}^{2}-E))$ can be extended
to a unique disingquandle $X$-coloring of $D^{\prime}$ for unoriented
generalized dichromatic singular Reidemeister moves $RI,RII$ and $RIII$. Also,
using the disingquandle axioms 4.2 to 4.2, it is easily seen from the Figure 6
that the restriction of $\mathcal{C}$ to
$D\cap(\mathbb{R}^{2}-E)(=D^{\prime}\cap(\mathbb{R}^{2}-E))$ can be extended
to a unique disingquandle $X$-coloring of $D^{\prime}$ for an unoriented
generalized dichromatic Reidemeister moves $RIVa,RIVb$ and $RV$. This
completes the proof. ∎

In an $X$-colored unoriented dichromatic singular link diagram
$(D,\mathcal{C})$, we think of elements of a disingquandle $X$ as labels for
the arcs in $D$ with different operations at crossings as shown in Figure 7.
Then it is seen from Lemma 5.1 that the disingquandle axioms of Definition 4.2
are transcriptions of a generating set of unoriented generalized Reidemeister
moves for unoriented dichromatic singular links which are sufficient to
generate any other unoriented generalized dichromatic singular Reidemeister
moves. That is, the axioms 4.2 and 4.2 come from the unoriented generalized
dichromatic singular Reidemeister move $RIVa$, the axioms 2.2 and 2.2 come
from the unoriented generalized dichromatic singular Reidemeister move $RIVb$
and the axioms 2.2 and 2.2 come from the unoriented generalized dichromatic
singular Reidemeister move $RV$ as seen in Figure 6.

######  Theorem 5.2.

Let $L$ be an unoriented dichromatic singular link in $\mathbb{R}^{3}$ and let
$D$ be a diagram of $L$. Then for any finite disingquandle $X$, the
cardinality $\sharp{\rm Col}^{dsq}_{X}(L)$ is an invariant of $L$.

###### Proof.

Let $D^{\prime}$ be any other unoriented dichromatic singular link diagram of
$L$ obtained from $D$ by applying a finite number of unoriented generalized
dichromatic singular Reidemeister moves. Then it is direct from Lemma 5.1 that
$\sharp{\rm Col}^{dsq}_{X}(D^{\prime})=\sharp{\rm Col}^{dsq}_{X}(D)$. This
completes the proof. ∎

If $X$ is a finite disingquandle, we call the cardinality $\sharp{\rm
Col}^{dsq}_{X}(D)$ the disingquandle $X$-coloring number or the disingquandle
counting invariant of $L$, and denote it by $\mathbb{Z}^{dsq}_{X}(L)$, i.e.,
$\mathbb{Z}^{dsq}_{X}(L)=\sharp{\rm Col}^{dsq}_{X}(D).$

######  Theorem 5.3.

Let $L$ be an unoriented dichromatic singular link and let $X$ be a
disingquandle. Then there is a one-to-one correspondence between ${\rm
Col}^{dsq}_{X}(L)$ and ${\rm Hom}(\mathcal{DSQ}(L),X)$. Consequently,
$\mathbb{Z}^{dsq}_{X}(L)=\sharp{\rm Hom}(\mathcal{DSQ}(L),X).$

###### Proof.

Since the disingquandle $X$-colorings of $L$ generate the fundamental
disingquandle $\mathcal{DSQ}(L)$ of a link $L$ which is generated by its arc
labels. Also each arc of $L$ is assigned an element of $X$, for a
disingquandle $X$-coloring of $L$, so we can associate each coloring a map
$f:\mathcal{DSQ}(L)\rightarrow X$ where if an arc is labelled $a$ in the
fundamental disingquandle and is assigned the color $x\in X$, then $f(a)=x$.
This completes the proof. ∎

Now we give an example.

######  Example 5.4.

Now, we give an explicit example of _three unoriented dichromatic singular
links_ $L_{1}$, $L_{2}$ and $L_{3}$ and show that the coloring invariant
distinguishes them from each other. Consider the singquandle
$(\mathnormal{X},*,\mathbf{R_{1}},\mathbf{R_{2}})$, where $X=\mathbb{Z}_{6}$,
$x*_{1}y=x*_{2}y=-x+2y=x*y$, $R_{1}(x,y)=x+3,$ and $R_{2}(x,y)=3x^{2}+3x+y+3$
(see page 9 of [CCEH]). By checking directly that the equations of Definition
4.2 hold we obtain that the quintuple
$(\mathnormal{X},*_{1},*_{2},\mathbf{R_{1}},\mathbf{R_{2}})$ form a
disingquandle. Now coloring the two top arcs of link $L_{1}$ by $x$ and $y$ as
in the figure 9 below gives that the coloring equations are:

$x=R_{1}(R_{1}(x,y),R_{2}(x,y))\quad\mbox{and}\quad
y=R_{2}(R_{1}(x,y),R_{2}(x,y)).$

Figure 9. Unoriented Dichromatic Singular Link($L_{1}$)

One then gets the system, $\begin{cases}x=3+3+x,\\\
y=3+3(3+x)+3(3+x)^{2}+(3+3x+3x^{2}).\end{cases}$
Any pair $(x,y)$ gives a solution to this system over $\mathbb{Z}_{6}$ and
thus the set ${\rm Col}^{dsq}_{X}(L_{1})$ is equal to $\mathbb{Z}_{6}^{2}$.
Now coloring the link $L_{2}$ as in the figure 10 below gives that the
coloring equations are:

$R_{1}(R_{1}(x,y),x*R_{1}(x,y))=R_{2}(x,y)*y\quad\mbox{and}\quad
R_{2}(R_{1}(x,y),x*R_{1}(x,y))=y.$

Figure 10. Unoriented Dichromatic Singular Link($L_{2}$)

One then obtain that the solution is given by $y=3x^{2}+4x+3$, thus the ${\rm
Col}^{dsq}_{X}(L_{2})$ is

$\\{(0,3),(1,4),(2,5),(3,0),(4,1),(5,2)\\}.$

Now we consider the link $L_{3}$ (dichromatic singular Whitehead) as in the
following figure 11.

Figure 11. Unoriented Dichromatic Singular Link($L_{3}$)

The coloring equations are:

$R_{2}(y*R_{1}(x,y),x*(y*R_{1}(x,y)))=R_{1}(x,y),$

and

$R_{1}(y*R_{1}(x,y),x*(y*R_{1}(x,y)))=R_{2}(x,y)*y.$

The system of these two equations reduces to $\begin{cases}0=3y^{2}+y+2x,\\\
0=3x^{2}+x+2y,\end{cases}$
and thus we obtain that $2(y-x)=0$ giving $x=y$ or $y=x+3$.

Then ${\rm Col}^{dsq}_{X}(L_{3})=\\{(x,x),x\in X\\}\;\cup\;\\{(x,x+3),x\in
X\\}$.

Thus the _three_ links $L_{1}$, $L_{2}$ and $L_{3}$ are _pairwise_ distinct.

######  Example 5.5.

Let
$1^{2}_{1},3^{2}_{1},4^{2}_{1},5^{2}_{1},5^{2}_{2},5^{2}_{3},6^{2}_{1},6^{2}_{2},6^{2}_{3},6^{2}_{4},6^{2}_{5},6^{2}_{6},6^{2}_{7},6^{2}_{8},6^{2}_{9},6^{2}_{10},6^{2}_{11},$
and $6^{2}_{12}$ be the eighteen unoriented dichromatic singular links in
Figure 12 and let $X$ be the disingquandle in Example 5.4. By similar
calculations as in the example, we obtain the following table:

${\begin{array}[]{|c|c|}\hline\cr L&\\#{\rm Col}^{dsq}_{X}(L)\\\ \hline\cr
6^{2}_{2}&0\\\ \hline\cr 6^{2}_{6}&2\\\ \hline\cr 4^{2}_{1},6^{2}_{12}&18\\\
\hline\cr
1^{2}_{1},3^{2}_{1},5^{2}_{1},5^{2}_{2},5^{2}_{3},6^{2}_{1},6^{2}_{3},6^{2}_{4},6^{2}_{5},6^{2}_{7},6^{2}_{8},6^{2}_{9},6^{2}_{10},6^{2}_{11}&6\\\
\hline\cr\end{array}}$

This table shows that the disingquandle counting invariant
$\mathbb{Z}^{dsq}_{X}(L)$ distinguishes some of these eighteen unoriented
dichromatic singular links.

Figure 12. Table of Unoriented Dichromatic Singular Links

## Acknowledgement

Mohamed Elhamdadi was partially supported by Simons Foundation collaboration
grant 712462.

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# Probing the Pomeron spin structure with Coulomb-nuclear interference

B. Z. Kopeliovich1 M. Krelina2,3 I. K. Potashnikova1 1Departamento de Física,
Universidad Técnica Federico Santa María
Avenida España 1680, Valparaíso, Chile  2FNSPE, Czech Technical University in
Prague
Brehova 7, 11519 Prague, Czech Republic  3Physikalisches Institut, University
of Heidelberg
Heidelberg 69120, Germany

###### Abstract

Polarized $pp$ elastic scattering at small angles in the Coulomb-nuclear
interference (CNI) region offers a unique opportunity to study the spin
structure of the Pomeron. Electromagnetic effects in elastic amplitude can be
equivalently treated either as Coulomb corrections to the hadronic amplitude
(Coulomb phase), or as absorption corrections to the Coulomb scattering
amplitude. We perform the first calculation of the Coulomb phase for the spin-
flip amplitude and found it significantly exceeding the widely used non-flip
Coulomb phase. The alternative description in terms of absorption corrections,
though equivalent, turned out to be a more adequate approach for the Coulomb
corrected spin-flip amplitude. Inspired by the recent high statistics
measurements of single-spin asymmetry with the HJET polarimeter at the BNL, we
also performed a Regge analysis of data, aiming at disentangling the Pomeron
contribution. However, in spite of an exceptional accuracy of the data, they
do not allow to single out the Pomeron term, which strongly correlates with
the major sub-leading Reggeons. A stable solution can be accessed only by
making additional ad hoc assumptions, e.g. assuming the Pomeron to be a simple
Regge pole, or fixing some unknown parameters. Otherwise, in addition to the
STAR data at $\sqrt{s}=200\,\mbox{GeV}$ new measurements, say at
$100\,\mbox{GeV}$ or $500\,\mbox{GeV}$, could become decisive.

###### keywords:

single spin asymmetry, Coulomb nuclear interference, Pomeron spin structure

###### PACS:

24.70.+s, 25.40.Cm, 11.55.Jy, 34.80.Nz

††journal: Physics Letters B

##  1 Introduction

The Pomeron has been introduced in the Regge theory as a rightmost singularity
in the complex angular momentum plane, having vacuum quantum numbers and
dominating elastic scattering amplitude at high energies. Originally, having
no dynamical input, for the sake of simplicity, it has been assumed to be a
Regge pole with the intercept $\alpha_{\mathbb{P}}(0)=1$, however later, the
observed rise of the total cross sections with energy led to a higher value of
the intercept $\alpha_{\mathbb{P}}(0)>1$ [1]. Besides, the absorptive
corrections, generating Regge cuts, make the structure of the singularity more
complicated.

With the advent of QCD, it was realized that the Pomeron corresponds to
gluonic exchanges in the $t$-channel, what naturally explains why the cross
section is nearly constant, or slowly rising with energy, and why the elastic
amplitude is predominantly imaginary. The spin structure of the Pomeron
exchange amplitude is related to the helicity conserving quark-gluon vertex,
this is why it has been widely believed that the Pomeron has no spin-flip
component.

Experimental measurement of the hadronic spin-flip amplitude is a challenge.
Indeed, the single-spin asymmetry is proportional to $\sin(\Delta\phi)$, where
$\Delta\phi$ is the relative phase between spin-flip and non-flip amplitudes.
If the Pomeron were a Regge pole, this phase shift would be exact zero.
Otherwise, it is expected to be small, suppressing spin effects in elastic
$pp$ scattering.

A unique opportunity to get a sizable single-spin asymmetry $A_{N}$ is to
arrange Coulomb-nuclear interference (CNI) between nearly real Coulomb and
almost imaginary Pomeron elastic amplitudes. In this case, the relative phase
is optimal for single spin asymmetry. Even if the Pomeron is spin-less, the
Coulomb amplitude does have a known spin-flip part, due to the existence of
the anomalous magnetic moment of the proton, generating a considerable spin-
flip amplitude. This was first proposed in [2], and a peculiar $t$-dependence
of the single-spin asymmetry $A_{N}(t)$ was found (see also [3]) with a
maximum of about $4.5\%$ at $t=t_{max}$ with

$t_{max}=-\sqrt{3}\,\frac{8\pi\alpha_{em}}{\sigma_{tot}^{pp}}\approx-0.0025\,\mbox{GeV}^{2},$
(1)

where $t$ and $s$ are the 4-momentum transfer squared and c.m. energy squared,
respectively.

If, however, the Pomeron also has a spin-flip component, the curve $A_{N}(t)$,
shifts up or down, keeping approximately the same shape, depending on the
phase and magnitude of the hadronic spin-flip. This was proposed in [4] as a
way to measure the Pomeron spin-flip.

The present analysis of data on $A_{N}(t)$ covers a wide energy range, where
dominance of the Pomeron term is not guaranteed. This is why we have to rely
on Regge phenomenology including sub-leading Regge terms. The exceptionally
high accuracy of the fix-target data give a chance to determine the spin-flip
part of the Pomeron amplitude.

##  2 Spin structure of hadronic elastic amplitudes

The elastic $pp$ amplitude is fully described by five independent helicity
amplitudes $\phi_{i}(s,t)$ ($i=1,...5$) defined in [5, 3]. The total and
elastic cross sections and single-spin asymmetry $A_{N}(t)$ are expressed via
these amplitudes,

$\displaystyle\sigma^{pp}_{tot}$ $\displaystyle=$ $\displaystyle
4\pi\,\textrm{Im}(\phi_{1}+\phi_{3})|_{t=0}\equiv
8\pi\,\textrm{Im}\phi_{+}(t=0),$ $\displaystyle\frac{d\sigma^{pp}_{el}}{dt}$
$\displaystyle=$ $\displaystyle
2\pi\left\\{|\phi_{1}|^{2}+|\phi_{2}|^{2}+|\phi_{3}|^{2}+|\phi_{4}|^{2}+4|\phi_{5}|^{2}\right\\},$
$\displaystyle A_{N}\frac{d\sigma^{pp}_{el}}{dt}$ $\displaystyle=$
$\displaystyle-4\pi\,\textrm{Im}\left\\{(\phi_{1}+\phi_{2}+\phi_{3}-\phi_{4})\phi_{5}^{*}\right\\}.$
(2)

The spin amplitudes $\phi_{i}$ contain the hadronic and electromagnetic parts,
as well as their interferences. In what follows we study all of them.

To simplify notations we neglect small amplitudes $\phi_{2,4}$ and relate the
amplitudes $\phi_{i}$ to the spin-flip and non-flip elastic amplitudes,

$\displaystyle\phi_{+}(q_{T})$ $\displaystyle=$ $\displaystyle
f_{\mathbb{P}}^{nf}(q_{T})+f_{\mathbb{R}}^{nf}(q_{T});$
$\displaystyle\phi_{5}(q_{T})$ $\displaystyle=$ $\displaystyle
f_{\mathbb{P}}^{sf}(q_{T})+f_{\mathbb{R}}^{sf}(q_{T}).$ (3)

Here we replaced the 4-momentum transfer squared by its transverse component
squared, $t\equiv-q^{2}\approx-q_{T}^{2}$. The longitudinal momentum transfer
in elastic scattering is vanishingly small at high energies.

Relying on Regge phenomenology we single out two terms in the amplitudes (3).
The first one, dominating at high energies, is usually called Pomeron.
Although the related singularity in the complex angular momentum plane is
expected to have a complicated structure [6, 7, 8], within a restricted energy
range it can be treated as an effective Regge pole with an intercept above
one. The intercept and the amplitude phase might be different for the non-flip
and spin-flip components (e.g. see [9]), in contrast to a real Regge pole.

The second term represents the common contribution of the major sub-leading
Reggeons ($f,\ \omega,\ \rho,\ a_{2}$) having highest intercepts
$\alpha_{\mathbb{R}}(0)\approx 0.5$, which we fix at this value. We have no
tools to disentangle different kinds of Reggeons, because include in the
analysis only elastic $pp$ data. As long as we are hunting for the spin-flip
part of the Pomeron amplitude $f_{\mathbb{P}}^{sf}$, which is presumably
small, the Reggeons might be important even at high energies, because have
large spin-flip component, especially iso-vector Reggeons $\rho$ and $a_{2}$.

###  2.1 Non-flip amplitudes

We parametrize the $s$ and small-$q_{T}$ dependences of the Pomeron and
Reggeon amplitudes as,

$\displaystyle f_{\mathbb{P}}^{nf}(q_{T})$ $\displaystyle=$ $\displaystyle
h_{\mathbb{P}}^{nf}\,e^{-{1\over
2}B_{\mathbb{P}}q_{T}^{2}};\left(\frac{s}{s_{0}}\right)^{\alpha_{\mathbb{P}}^{nf}(0)-1};$
(4) $\displaystyle f_{\mathbb{R}}^{nf}(q_{T})$ $\displaystyle=$ $\displaystyle
h_{\mathbb{R}}^{nf}\,e^{-{1\over
2}B_{\mathbb{R}}^{nf}q_{T}^{2}}\left(\frac{s}{s_{0}}\right)^{\alpha_{\mathbb{R}}^{nf}(0)-1}.$
(5)

We fix the values of the intercepts, which are known [10, 11] and are close to
the values,

$\displaystyle\alpha_{\mathbb{P}}^{nf}(0)$ $\displaystyle=$ $\displaystyle
1.1;$ (6) $\displaystyle\alpha_{\mathbb{R}}^{nf}(0)$ $\displaystyle=$
$\displaystyle 0.5.$ (7)

The energy dependence of the $q_{T}$-slopes is assumed to be logarithmic, in
accordance with the standard Regge-pole form,

$\displaystyle B_{\mathbb{P}}^{nf}$ $\displaystyle=$
$\displaystyle\left(B_{\mathbb{P}}^{0}\right)^{nf}+2\left(\alpha^{\prime}_{\mathbb{P}}\right)^{nf}\ln(s/s_{0}),$
(8) $\displaystyle B_{\mathbb{R}}^{nf}$ $\displaystyle=$
$\displaystyle\left(B_{\mathbb{R}}^{0}\right)^{nf}+2\left(\alpha^{\prime}_{\mathbb{R}}\right)^{nf}\ln(s/s_{0}),$
(9)

which is applicable even if the pole is effective, within a restricted energy
interval.

The slope of meson Regge trajectories
$\alpha^{\prime}_{\mathbb{R}}=0.9\,\mbox{GeV}^{-2}$ is universal, because it
is inversely proportional to the color-triplet string tension [12]. The slope
of the Pomeron trajectory is poorly known, since the glue-balls lying on this
trajectory have not been well identified so far, and the trajectory
$\alpha_{\mathbb{P}}(t)$ at negative $t$ is not linear [13]. Besides,
proximity of the unitarity bound in high-energy partial elastic $pp$
amplitude, leads to a partial amplitude rising with energy only at large
impact parameters. This experimental observation [14] once again emphasizes
that the Pomeron is not a pole and the related parameters,
$\alpha_{\mathbb{P}}(0)$, $\alpha^{\prime}_{\mathbb{P}}$, should be treated as
effective values, which can be used only in a restricted energy range.
Therefore, the interaction radius and the effective
$\alpha^{\prime}_{\mathbb{P}}$ significantly increase in comparison with the
bare Pomeron parameters [15], so it should be adjusted to data, as well as the
constant $B_{\mathbb{P,R}}^{0}$.

We performed a two-parameter fit to data [10] for $pp$ elastic slope with
parametrizations (8), $s_{0}=1\,\mbox{GeV}^{2}$, and found,

$\displaystyle\left(B_{\mathbb{P}}^{0}\right)^{nf}$ $\displaystyle=$
$\displaystyle 8.67\pm 0.34\,\,\mbox{GeV}^{-2};$
$\displaystyle\left(\alpha^{\prime}_{\mathbb{P}}\right)^{nf}$ $\displaystyle=$
$\displaystyle 0.27\pm 0.02\,\,\mbox{GeV}^{-2}.$ (10)

Notice that these slope values do not affect much our analysis of data at
small $|t|<0.02\,\mbox{GeV}^{2}$.

The other parameters in the non-flip amplitudes Eqs. (4), (5) were also fitted
to data on total and differential elastic $pp$ cross section, and the ratio of
real-to-imaginary parts of the forward elastic amplitude [10]. Nonetheless,
the real part of the Pomeron non-flip amplitude, was fixed by the derivative
relation obtained within the eikonal Regge model in [16], or with the general
dispersion approach in [17]. In the approximation of small
$\alpha^{nf}_{\mathbb{P}}(0)-1$ this relation reads

$\frac{{\rm Re}\,h_{\mathbb{P}}^{nf}(0)}{{\rm
Im}\,h_{\mathbb{P}}^{nf}(0)}=\frac{\pi}{2}\,\frac{\partial\ln[{\rm
Im}\,f_{\mathbb{P}}^{nf}(0)]}{\partial\ln s},$ (11)

where $f_{\mathbb{P}}^{nf}$ is given by (4).

Notice that the simplified model of an effective Pomeron pole Eq. (4), we rely
upon for the non-flip amplitude, fails at much higher energies of the LHC,
where data show the cross section rising much faster, as was predicted in
[15]. However, in the restricted energy range below $\sqrt{s}\leq
200\,\mbox{GeV}$, we are interested in, the model of an effective Pomeron pole
describes data well [11].

The sub-leading Reggeons are known to be subject to exchange degeneracy based
on duality of the $t$\- and $s$-channel descriptions for the amplitude. As a
result, among the leading should Reggeons with intercepts $\alpha(0)\approx
0.5$ the pairs of $f-\omega$ and $\rho-a_{2}$, cancel in the imaginary, but
add up in real parts of the $pp$ (also $K^{+}p$) elastic amplitude. In reality
such a symmetry is broken, and a part of the Reggeon contribution shows up as
falling total $pp$ cross section at medium-high energies. We combine here all
Reggeons in an effective one with intercept fixed at
$\alpha^{nf}_{\mathbb{R}}(0)=0.5$, but unknown magnitude. Moreover, the
residue factor of such an effective Regge pole does not have the usual phase
dictated by the value of $\alpha^{nf}_{\mathbb{R}}(0)$, so we fit $\,{\rm
Im}\,h_{\mathbb{R}}^{nf}$ and $\,{\rm Re}\,h_{\mathbb{R}}^{nf}$ separately.

The fit with the effective Pomeron and Reggeon poles includes 3 parameters,

$\displaystyle\,{\rm Im}\,h_{\mathbb{P}}^{nf}(0)$ $\displaystyle=$
$\displaystyle 1.89\pm 0.002\,\mbox{GeV}^{-2};$ $\displaystyle\,{\rm
Im}\,h_{\mathbb{R}}^{nf}(0)$ $\displaystyle=$ $\displaystyle 10.25\pm
0.053\,\mbox{GeV}^{-2};$ $\displaystyle\,{\rm Re}\,h_{\mathbb{R}}^{nf}(0)$
$\displaystyle=$ $\displaystyle-11.69\pm 0.417\,\mbox{GeV}^{-2},$ (12)

while $\,{\rm Re}\,h_{\mathbb{P}}^{nf}(0)$ is determined by the relation (11).

These results for the non-flip amplitudes will be used in the further fit to
data on single-spin asymmetry.

###  2.2 Spin-flip amplitudes

The spin-flip amplitudes are parametrized in analogy to Eqs. (4), (5)

$\displaystyle f_{\mathbb{P}}^{sf}(q_{T})$ $\displaystyle=$
$\displaystyle\frac{q_{T}}{m_{N}}\,h_{\mathbb{P}}^{sf}\,e^{-{1\over
2}B_{\mathbb{P}}^{sf}q_{T}^{2}}\left(\frac{s}{s_{0}}\right)^{\alpha_{\mathbb{P}}^{sf}(0)-1},$
(13) $\displaystyle f_{\mathbb{R}}^{sf}(q_{T})$ $\displaystyle=$
$\displaystyle\frac{q_{T}}{m_{N}}\,h_{\mathbb{R}}^{sf}e^{-{1\over
2}B_{\mathbb{R}}^{sf}q_{T}^{2}}\left(\frac{s}{s_{0}}\right)^{\alpha_{\mathbb{R}}^{sf}(0)-1},$
(14)

where

$\displaystyle B_{\mathbb{P}}^{sf}$ $\displaystyle=$
$\displaystyle\left(B_{\mathbb{P}}^{0}\right)^{sf}+2\left(\alpha^{\prime}_{\mathbb{P}}\right)^{sf}\ln(s/s_{0});$
(15) $\displaystyle B_{\mathbb{R}}^{sf}$ $\displaystyle=$
$\displaystyle\left(B_{\mathbb{R}}^{0}\right)^{sf}+2\left(\alpha^{\prime}_{\mathbb{R}}\right)^{sf}\ln(s/s_{0}).$
(16)

None of the ingredients in Eqs. (13)-(16) are known, in particular, the
Pomeron spin-flip amplitude, which is the main goal of the present study.

If the Reggeons were true Regge poles, the intercepts
$\left(\alpha_{\mathbb{R}}(0)\right)^{sf}$ and slopes
$\left(\alpha^{\prime}_{\mathbb{R}}\right)^{sf}$ should be the same for the
spin-flip and non-flip amplitudes. In fact, that is quite a good
approximation. The $\rho$-Reggeon trajectory has been well measured at
positive and negative $t$, and is perfectly linear with the universal Regge
slope. This shows smallness of corrections from the Regge-cuts, like
$\rho$-${\mathbb{P}}$, which has nearly the same intercept
$\alpha_{\rho{\mathbb{P}}}(0)=\alpha_{\rho}(0)+\alpha_{\mathbb{P}}(0)-1$.
However the Regge slope is almost twice as small as for the $\rho$-pole.
Smallness of the Regge cut corrections is confirmed by calculation in the
eikonal model. Therefore, we fix
$\left(\alpha^{\prime}_{\mathbb{R}}\right)^{sf}=0.9\,\mbox{GeV}^{-2}$.

The first terms in Eqs. (15),(16) are unknown, but should not be very
different from the non-flip values, because are also controlled by the proton
size. Moreover, the further analysis shows that the results are nearly
independent of the $\left(B_{\mathbb{P,R}}^{0}\right)^{\\!sf}$ values, because
of smallness of $t$ in the analyzed data. The fit results hardly vary even if
these spin-flip slopes are reduced down to zero. So a good approximation is to
fix
$\left(B_{\mathbb{P,R}}^{0}\right)^{\\!sf}=\left(B_{\mathbb{P,R}}^{0}\right)^{\\!nf}$.

##  3 Coulomb amplitudes

Small-angle single spin asymmetry $A_{N}$ is mainly due to interference of
nearly real Coulomb spin-flip and almost imaginary non-flip hadronic
amplitudes. Such a large phase difference is optimal according to Eq. (2) to
maximize $A_{N}$. Multiple electromagnetic interactions affect the phases of
both Coulomb and hadronic amplitudes

While the magnitude of the hadronic spin-flip amplitude is still questionable
[3], the spin structure of the electromagnetic amplitude of $pp$ elastic
scattering is well known. The Coulomb spin amplitudes (C) in impact parameter
space have the eikonal form [18], related to the momentum representation by
Fourier transformation,

$\displaystyle\phi_{+}^{em}(q_{T})$ $\displaystyle=$
$\displaystyle\frac{i}{2\pi}\int
d^{2}b\,e^{i\vec{q}_{T}\cdot\vec{b}}\,\left(1-e^{i\chi_{C}^{nf}(b)}\right),$
(17) $\displaystyle\phi_{5}^{em}(q_{T})$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int
d^{2}b\,e^{i\vec{q}_{T}\cdot\vec{b}}\,\,\chi_{C}^{sf}(b)\,e^{i\chi_{C}^{nf}(b)},$
(18)

with the non-flip and spin-flip eikonal phases,

$\displaystyle\chi_{C}^{nf}(b)$ $\displaystyle=$
$\displaystyle-\frac{\alpha_{em}}{2\pi}\int
d^{2}q_{T}\,\frac{F^{2}_{1}(q_{T}^{2})}{q_{T}^{2}+\lambda^{2}}\,e^{-i\vec{q}_{T}\cdot\vec{b}};$
(19) $\displaystyle\chi_{C}^{sf}(b)$ $\displaystyle=$
$\displaystyle-\frac{\alpha_{em}\kappa_{p}}{4\pi m_{p}}\int
d^{2}q_{T}\,\frac{F_{1}(q_{T}^{2})F_{2}(q_{T}^{2})}{q_{T}^{2}+\lambda^{2}}\,\frac{(\vec{q}_{T}\cdot\vec{b})}{b}\,e^{-i\vec{q}_{T}\cdot\vec{b}},$
(20)

respectively. Here $\kappa_{p}=\mu_{p}-1=1.793$ is the anomalous magnetic
moment of the proton. $F_{1}(q_{T}^{2})$ and $F_{2}(q_{T}^{2})$ are the Dirac
and Pauli electromagnetic formfactors, respectively. They are related to the
electric and magnetic fomfactors $(1+\gamma)F_{1}=G_{E}+\gamma G_{M}$;
$(1+\gamma)\kappa\,F_{2}=G_{M}-G_{E}$, where $\gamma=q_{T}^{2}/4m_{p}^{2}$. At
small $q_{T}^{2}\ll 4m_{p}^{2}$, we are interested in, we rely on the
approximation $F_{2}\approx F_{1}$.

At small $q_{T}$ the formfactor can be approximated by the Gaussian form,

$F_{1}(q_{T})=e^{-\frac{1}{6}\langle r_{em}^{2}\rangle_{p}\,q_{T}^{2}},$ (21)

where $\langle r_{em}^{2}\rangle_{p}$ is the proton mean charge radius
squared. We fix it at the value $\sqrt{\langle
r_{em}^{2}\rangle_{p}}=0.875\,\mbox{fm}$ [10].

Notice that in the parametrization proposed in [3], and used in all following
data analyses, the slopes of elastic $pp$ scattering and of the
electromagnetic formfactor were taken equal, which is an oversimplification.
One of them, the hadronic slope $B_{pp}(s)$, rises with energy, while another
one, in Eq. (21), is energy independent. We rely on the more realistic
parametrization, explained above.

In order to keep the integrals in Eqs. (19) and (20) finite we supply the
photon with a small mass $\lambda$ which disappears from the final
expressions. Notice that the pure Coulomb amplitude has a nonzero phase coming
from the higher order terms in (17)-(18), e.g. two photon exchange amplitude
is imaginary.

##  4 Coulomb-nuclear interference

The long-range Coulomb forces also affect the strong-interaction amplitude.
This is illustrated in Fig. 1, following the consideration of this problem in
[18].

Figure 1: Three types of interaction: pure electromagnetic (a), pure strong
interaction (b), and combined strong and electromagnetic interactions (c).

These graphs can be grouped and interpreted differently. One way, employed in
[18], is to combine the last two graphs, (N) and (CN), and treat it as a
Coulomb modified strong-interaction amplitude. The modification is
approximated by giving an extra phase factor to the hadronic amplitude. This
factor is called in the literature Coulomb phase [19, 20, 18].

###  4.1 Coulomb phase shift of hadronic amplitudes

This effect has been calculated so far [19, 20, 18] only for non-flip
amplitudes, but applied incorrectly to the spin effects. Here we derive the
Coulomb-modified phases for all spin amplitudes.

Using (19) and (20) we can calculate the phases of non-flip and spin-flip
electromagnetic amplitudes, Eq. (17) and (18) respectively, as,

$\displaystyle\delta_{C}^{nf}(q_{T})$ $\displaystyle=$
$\displaystyle\frac{2\pi\,\phi_{+}^{em}(q_{T})}{\int
d^{2}b\,e^{i\vec{q}_{T}\cdot\vec{b}}\,\chi_{C}^{nf}(b)}-1;$ (22)
$\displaystyle\delta_{C}^{sf}(q_{T})$ $\displaystyle=$
$\displaystyle\frac{2\pi\,\phi_{5}^{em}(q_{T})}{\int
d^{2}b\,e^{i\vec{q}_{T}\cdot\vec{b}}\,\chi_{C}^{sf}(b)}-1.$ (23)

If the multiple Coulomb interactions generated the same phase shift for the
electromagnetic (C) and hadronic (N+CN) amplitudes, there would be no effect
on the spin-asymmetry $A_{N}$ at all. However, the Coulomb induced phase
shifts of the two term (C) and (N+CN) depicted in Fig. 1 are different and the
difference is usually called Coulomb phase.

The non-flip phase Eq. (22) was calculated in [18] analytically and our
numerical calculations confirm that result. The spin-flip phase Eq. (23) is
calculated here for the first time.

The hadronic part of the amplitudes includes the two other terms in Fig. 1
combined together, (N)+(NC), which correspond to the the contribution of
strong interactions Eq. (3), modified by Coulomb corrections. The non-flip
amplitude reads [18],

$\left.\phi_{+}(s,q_{T})\right|_{(N)+(NC)}=\frac{i}{2\pi}\int
d^{2}b\,e^{i\vec{q}_{T}\cdot\vec{b}}\,e^{i\chi_{C}^{nf}(b)}\,\gamma_{N}^{nf}(b),$
(24)

where

$\gamma_{N}^{nf}(b)=\frac{i}{2\pi}\int
d^{2}q_{T}\,e^{-i\vec{q}_{T}\cdot\vec{b}}\,\phi_{+}(q_{T}),$ (25)

and $\phi_{+}(q_{T})$ is given by Eq. (3). The phase of this amplitude is
given by,

$\left.\delta^{nf}(q_{T})\right|_{(N)+(NC)}=\frac{\left.\phi_{+}^{h}(q_{T})\right|_{(N)+(NC)}}{\left.\phi_{+}^{h}(q_{T})\right|_{(N)}}-1.$
(26)

We assume here that the phase is small, $\delta\ll 1$, which is justified by
the higher order ($\alpha_{em}^{2}$) corrections, related to the second and
higher terms in the expansion of the exponential $\exp(i\chi_{C}^{nf})$ in Eq.
(17). The smallness of the Coulomb correction allows to represent it as a
small shift of the phase.

Notice that both terms, $(N)$ and $(NC)$, contributing to (24), are controlled
by short-range strong interactions and have a sizeable magnitude only at small
impact parameters, $b^{2}\lesssim 2B_{pp}$. The Coulomb forces nevertheless,
considerably affect the phase of the combined amplitude. The Coulomb phase for
the non-flip hadronic amplitude is given by the difference between (26) and
(22).

The spin-flip amplitude has a structure, analogous to Eq. (24), but the
hadronic non-flip factor $\gamma_{N}^{nf}(b)$ should be replaced by a spin-
flip amplitude, either hadronic, or Coulomb. So we get,

$\left.\phi_{5}(s,q_{T})\right|_{(N)+(NC)}=\frac{i}{2\pi}\int
d^{2}b\,e^{i\vec{q}_{T}\cdot\vec{b}}\,e^{i\chi_{C}^{nf}(b)}\left[\chi_{C}^{sf}(b)\,\gamma_{N}^{nf}(b)+\gamma_{N}^{sf}(b)\right].$
(27)

The first term here is given by Eqs. (20) and (25). The second term is given
by the Fourier transformed hadronic spin-flip amplitude Eq. (3).

So far, the non-flip Coulomb phase shift Eq. (26) has been used for the
hadronic spin-flip amplitude [3, 21, 22, 23]. The Coulomb corrected hadronic
spin-flip amplitude is given by Eq. (27). However converting it to a Coulomb
phase shift might be problematic. As was mentioned above, the relative value
of the Coulomb corrections in the non-flip amplitude, Eq. (12) is suppressed
by $\alpha_{em}$. However, the relative magnitude of the Coulomb correction in
the spin-flip amplitude, given by the first term in Eq. (27), is much larger,
of the order of $\alpha_{em}h_{\mathbb{P}}^{nf}/h_{\mathbb{P}}^{sf}$.

Another source of enhancement of the Coulomb correction is the less singular
behavior of the spin-flip amplitude, $1/q_{T}$, compared with the quadratic
singularity, $1/q_{T}^{2}$, in the non-flip amplitude. So the spin-flip
Coulomb interaction is less peripheral, and is more affected by the
interference with short-range strong interactions.

Such a large Coulomb correction cannot be represented as a phase shift,
because it also affects the absolute value of the amplitude. Therefore, in the
next section, we re-group the graphs in Fig. 1 in a way that the modification
acquires a meaning of hadronic corrections to the Coulomb spin-flip amplitude.
Such a more accurate calculation of Coulomb-nuclear interference is used for
further analysis of data.

###  4.2 Absorptive corrections

One can group the graphs in Fig. 1 differently, so that the result can be
interpreted as absorption corrections to the Coulomb amplitude. Of course, the
final results must remain unchanged, either for the spin-flip, or non-flip
amplitudes, and numerical comparison confirms that.

In all calculations of the CNI contribution to single-spin asymmetry,
performed so far [3, 21, 22, 23], the Coulomb phase applied to the hadronic
spin-flip amplitude, has been taken from spin non-flip calculations [19, 20,
18]. Such a procedure is unjustified, moreover, is quite incorrect, as was
demonstrated in the previous section.

If one combines the graphs (C) and (CN) depicted in Figs. 1a and 1c
respectively, one gets Coulomb amplitude with absorption corrections related
to possibility of strong inelastic interactions, destroying the rapidity gap.
This is why it is also called the amplitude of survival probability of a large
rapidity gap, associated with elastic Coulomb interaction of hadrons.

Absorption corrections are most effectively calculated in impact parameter
representation. One should Fourier transform the $q_{T}$-dependent
electromagnetic amplitudes to $b$-space, like was done in Eqs. (17), (18).
Then introduce the absorptive factor,

$\phi^{em}(b)\Rightarrow\phi^{em}(b)\times S(b),$ (28)

where

$S(b)=1+2i\gamma^{nf}_{N}(b),$ (29)

and $\gamma^{nf}_{N}(b)$ is defined in (25). To avoid a terminological
confusion, notice that the correction, corresponding to the graph in Fig. 1c,
is not pure absorptive, i.e. imaginary, but $\gamma^{nf}_{N}(b)$ contains a
small real part.

Now we are in a position to calculate the absorption corrected
$q_{T}$-dependent electromagnetic helicity amplitudes by making inverse
Fourier transformation to momentum representation,

$\displaystyle\widetilde{\phi}_{+}^{em}(q_{T})$ $\displaystyle=$
$\displaystyle\frac{i}{2\pi}\int
d^{2}b\,e^{i\vec{q}_{T}\cdot\vec{b}}\,\left(1-e^{i\chi_{C}^{nf}(b)}\right)\,S(b)$
(30) $\displaystyle=$ $\displaystyle\phi_{+}^{em}(q_{T})-\frac{i}{2\pi}\int
d^{2}b\,e^{i\vec{q}_{T}\cdot\vec{b}}\,\left(1-e^{i\chi_{C}^{nf}(b)}\right)\,[1-S(b)],$

where $\chi_{C}^{nf}(b)$ is given by (19). The absorptive correction here is
given by the second term here, which does not contain a long-range divergency,
because the factor $1-S(b)$ vanishes at $b^{2}\gg B_{pp}$.

The absorption corrected spin-flip electromagnetic amplitude Eq. (18) has
analogous form,

$\displaystyle\widetilde{\phi}_{5}^{em}(q_{T})$ $\displaystyle=$
$\displaystyle\frac{i}{2\pi}\int
d^{2}b\,e^{i\vec{q}_{T}\cdot\vec{b}}\,\,\chi_{C}^{sf}(b)\,e^{i\chi_{C}^{nf}(b)}\,S(b)$
(31) $\displaystyle=$ $\displaystyle\phi_{5}^{em}(q_{T})-\frac{i}{2\pi}\int
d^{2}b\,e^{i\vec{q}_{T}\cdot\vec{b}}\,\,\chi_{C}^{sf}(b)\,e^{i\chi_{C}^{nf}(b)}\,[1-S(b)].$

Within the approach used in this Sect. 4.2 the amplitude
$\tilde{\phi}^{em}_{5}$ (l.h.s. of (31)) is given by the sum of Figs. 1a and
1c terms. Thus it should contain also the Pomeron spin-flip contribution
$\gamma^{sf}_{N}$ (similar to the second term in r.h.s. of (27)) modified by
the Coulomb phase $\exp(i\chi^{nf}_{C}(b))$. However here we consider the
small $q_{T}$ region where the asymmetry caused by the interference of the
Pomeron spin-flip amplitude Fig. 1b with the Coulomb non-flip amplitude
$\phi^{em}_{+}$ is enhanced by the singular $1/t$ factor in Coulomb amplitude,
while the $\gamma^{sf}_{N}$ contribution from $\tilde{\phi}^{em}_{5}$ (31) is
enhanced by the $\ln(1/t)$ only. Therefore, in comparison with a much larger,
$\phi^{em}_{+}\times f^{sf}_{P}$ contribution, we neglect this term in (31)
and in our further analysis.

##  5 Data analysis: spin-flip amplitudes

Now we can calculate the single-spin asymmetry $A_{N}(t)$, Eq. (2), since all
the spin-flip amplitudes (13), (14), (31), and non-flip amplitudes (4),
(5)(30), amplitudes are either known, or parametrized.

At small $t$ the dominant contribution to $A_{N}$ comes from the interference
of hadronic and electromagnetic amplitudes. At high energies the former is
expected to be nearly imaginary, while the latter is almost real. Such a large
phase shift allows to maximize the single-spin asymmetry $A_{N}$[2]. Of
course, at medium-high energies, the sub-leading Reggeons with intercepts
$\alpha_{\mathbb{R}}(0)\approx 0.5$ can supply a considerable real part.
Besides, the iso-vector Reggeons ($\rho,a_{2}$) are predominantly spin-flip,
so contribute to the hadronic spin-flip amplitudes, which can interfere with
electromagnetic non-flip component.

We performed a fit simultaneously to all available data for $A_{N}(q_{T})$,
with 5 unknown parameters, the real and imaginary parts of the spin-flip
Pomeron and Reggeon amplitudes in Eq. (13), (14). The fifth parameter is the
Pomeron spin-flip intercept $\alpha_{\mathbb{P}}^{sf}(0)$, which might be
different from the known non-flip value. If the Pomeron was a Regge pole, the
spin-flip and non-flip intercepts would coincide. However, none of the
contemporary dynamic models for the Pomeron support its Regge pole origin,
some even predict a considerably larger value of $\alpha_{\mathbb{P}}^{sf}(0)$
[9].

Our fit to data [22, 23, 24] revealed a strong correlation between
$\alpha_{\mathbb{P}}^{sf}(0)$ and other parameters. The $\chi^{2}$ profile of
this parameter is plotted in Fig. 2.

Figure 2: (Color online) Upper panel: real and imaginary parts of the factor
$h_{\mathbb{P}}^{sf}$, which defines the magnitude of the Pomeron spin-flip
component defined in (13), (14) vs the effective intercept
$\alpha_{\mathbb{P}}^{sf}(0)$. Bottom panel: $\chi^{2}/ndf$ vs
$\alpha_{\mathbb{P}}^{sf}(0)$.

For each fixed value of $\alpha_{\mathbb{P}}^{sf}(0)$ other four parameters
are fitted and their values strongly correlate with the chosen value of
$\alpha_{\mathbb{P}}^{sf}(0)$. One can see that the value of $\chi^{2}$ is
close to the number of degrees of freedom (ndf) within a wide range of
$\alpha_{\mathbb{P}}^{sf}(0)=0.9-1.3$. This shows that preferable value of
$\alpha_{\mathbb{P}}^{sf}(0)$ cannot be reliably localized because all high
statistics data are available only at medium-high energies, where Reggeon
contribution is large and strongly correlates with the Pomeron. The data at
$\sqrt{s}=200\,\mbox{GeV}$ alone, of course cannot determine
$\alpha_{\mathbb{P}}^{sf}(0)$, only together with lower energy HJET data [23],
which suffer of strong correlations. Apparently new data at collider energies
are required.

The real and imaginary parts of the Pomeron spin-flip are plotted with their
error band in Fig. 2 vs the fixed value of $\alpha_{\mathbb{P}}^{sf}(0)$.
Interestingly, the spin-flip Pomeron amplitude turns out to be predominantly
imaginary, like the non-flip one. However, the real-to-imaginary ratio is much
different from what the differential relation (11) would give, if were naively
applied to the spin-flip amplitude (where it has never been proven). In such a
case it would be negative at $\alpha_{\mathbb{P}}^{sf}(0)<1$

We conclude that unfortunately energy dependence of the spin-flip Pomeron
amplitude cannot be determined from available data because of large
correlations with other Reggeons at low energies.

To show an example of full set of other parameters we choose the value of
$\alpha_{\mathbb{P}}^{sf}(0)=1.1$, the same as for non-flip (4).

$\displaystyle\alpha_{\mathbb{P}}^{sf}(0)$ $\displaystyle=$ $\displaystyle
1.1({\rm fixed})$ $\displaystyle\,{\rm Im}\,h_{\mathbb{P}}^{sf}$
$\displaystyle=$ $\displaystyle 0.177\pm 0.0122\,\mbox{GeV}^{-2}$
$\displaystyle\,{\rm Re}\,h_{\mathbb{P}}^{sf}$ $\displaystyle=$ $\displaystyle
0.048\pm 0.002\,\mbox{GeV}^{-2}$ $\displaystyle\,{\rm
Im}\,h_{\mathbb{R}}^{sf}$ $\displaystyle=$ $\displaystyle-4.352\pm
0.370\,\mbox{GeV}^{-2}$ $\displaystyle\,{\rm Re}\,h_{\mathbb{R}}^{sf}$
$\displaystyle=$ $\displaystyle-2.233\pm 0.064\,\mbox{GeV}^{-2}$ (32)
$\displaystyle\chi^{2}/ndf$ $\displaystyle=$ $\displaystyle 315.9/329$

The choice of $\alpha_{\mathbb{P}}^{sf}(0)=\alpha_{\mathbb{P}}^{nf}(0)$ is
made just for convenience, to make the fractional spin-flip of the Pomeron,
$r_{5}^{\mathbb{P}}$ [3] independent of energy,

$r_{5}^{\mathbb{P}}=\frac{m_{N}\,f_{\mathbb{P}}^{sf}(q_{T})}{q_{T}\,{\rm
Im}\,f_{\mathbb{P}}^{sf}(q_{T})}$ (33)

For the above sample of parameters Eq. (32) $r_{5}$ is energy independent,

$\,{\rm Im}\,r_{5}=0.094\pm 0.006.$ (34)

This ratio also can be treated as the anomalous magnetic moment of the Pomeron
$\mu_{\mathbb{P}}=2r_{5}^{\mathbb{P}}$, introduced in [4].

As was mentioned, Fig. 2 also shows a good description of data with
$\chi^{2}\approx ndf$ for a wide range of effective intercepts. To visualize
the quality of description we plotted $A_{N}(t)$ calculated with the
parameters (32) in comparison with STAR [22] and HJET data in Fig. 3.

Figure 3: (Color online) Collider RHIC [22] and fixed-target [23] data vs
calculations with the parameters (32).

##  6 Summary and discussion

We analyzed data on single-spin asymmetry $A_{N}(t)$ in small-angle $pp$
elastic scattering, where it is presumably dominated by Coulomb-nuclear
interference. The main objective of our analysis was the Pomeron spin
structure and its energy dependence.

Electromagnetic corrections to the hadronic amplitude, widely known as Coulomb
phase, have never been derived for the spin-flip component of the hadronic
amplitude, but the non-flip Coulomb phase has been incorrectly applied to the
spin-flip amplitude. We calculated the spin-flip Coulomb phase and found it
significantly exceeding that for the non-flip amplitude. Moreover, typically
the corrections are so large, that hardly can be treated as a phase shift.

Electromagnetic corrections can be equivalently interpreted either as Coulomb
corrections to the hadronic amplitude, or as hadronic (absorption) corrections
to the Coulomb amplitude. Although the latter interpretation is more
traditional, the former is more adequate for the spin-flip amplitude, and
offers easier evaluation.

The wide energy range of currently available data allows to perform a Regge
analysis of the spin-flip hadronic amplitudes, aiming at disentangling the
Pomeron and Reggeon terms. However, in spite of high statistics of data from
the fixed-target HJET measurements, these two contributions in the spin-flip
amplitude cannot be reliably separated because of strong correlations. Fig. 2
demonstrates that the results strongly depend on the value of the unknown
effective intercept $\alpha_{\mathbb{P}}^{sf}(0)$. The Pomeron is not a Regge
pole, so the effective intercept of its spin-flip and non-flip components
might be quite different (e.g. see [9]).

Unfortunately, data at sufficiently high energies, to neglect the Reggeon
contribution, are available only at one energy $\sqrt{s}=200\,\mbox{GeV}$,
what is insufficient for determination of $\alpha_{\mathbb{P}}^{sf}(0)$. One
can fit data well with about the same quality, fixing
$\alpha_{\mathbb{P}}^{sf}(0)$ at different values as is demonstrated in Fig.
2. At the same time the real and imaginary parts of the Pomeron spin-flip
amplitude vary considerably. Additional measurements at a different collider
energy $\sqrt{s}$, e.g. $100\,\mbox{GeV}$ or $500\,\mbox{GeV}$ could solve the
problem.

The recently published alternative Regge analysis [23] of the same data
deserves commenting, since it arrived at quite different conclusions. The
reason is the additional unjustified assumptions made in the analyses. The
Pomeron was assumed to be a ”simple” Regge pole, contradicting any theoretical
expectation (see e.g. [6, 7, 8]). Fig. 2 demonstrates that just one ad hoc
assumption about the value of $\alpha_{\mathbb{P}}^{sf}(0)$ immediately leads
to certain fit results with small errors.

## Acknowledgements

We are thankful to Andrei Poblaguev and Wlodek Guryn for informative
discussions. The work of B.Z.K. and I.K.P. was supported in part by grants
ANID - Chile FONDECYT 1170319, and by ANID PIA/APOYO AFB180002.
The work of M.K. was supported by the project Centre of Advanced Applied
Sciences CZ.T02.1.01/0.0/0.0/16-019/0000778 and by International Mobility of
Researchers - MSCA IF IV at CTU in Prague
CZ.02.2.69/0.0/0.0/20_079/0017983, Czech Republic. It was also supported at
the initial stage by the CONICYT Postdoctorado N.3180085 (Fondecyt Chile).

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