# Splitting of differential Quaternion algebras

Parul Gupta<EMAIL_ADDRESS>,  Yashpreet Kaur<EMAIL_ADDRESS>and  Anupam Singh<EMAIL_ADDRESS>IISER Pune, Dr. Homi Bhabha Road,
Pashan, Pune 411 008, India

###### Abstract.

We study differential splitting fields of quaternion algebras with
derivations. A quaternion algebra over a field $k$ is always split by a
quadratic extension of $k$. However, a differential quaternion algebra need
not be split over any algebraic extension of $k$. We use solutions of certain
Riccati equations to provide bounds on the transcendence degree of splitting
fields of a differential quaternion algebra.

###### Key words and phrases:

Derivations, Differential quaternion algebras, Differential splitting fields,
Rational function fields

###### 2010 Mathematics Subject Classification:

12H05, 16H05, 16W25

The second named author is supported by NBHM (DAE, Govt. of India):
0204/16(33)/2020/R$\&$D-II/26. The third named author is funded by an NBHM
research project grant and partly by SERB CRG/2019/000271 for this research.

##  1\. Introduction

In this paper, we study differential splitting fields of quaternion algebras
with derivations. Quaternion algebras are 4-dimensional central simple
algebras. Derivations on central simple algebras are studied by Hochschild
[4], Hoechsmann[5], Juan and Magid [6], Kulshrestha and Srinivasan [8], Gupta,
Kaur and Singh [3].

A notion of splitting fields for a differential central simple algebra was
introduced in [6]. Authors also proved the existence of a finitely generated
differential field extensions that split the given differential algebra. It is
clear from [6, Proposition 2] that every $m^{2}$-dimensional differential
central simple algebra is split by a differential field extension of
transcendence degree at most $m^{2}$. In particular, every differential
quaternion algebra is split by a differential field extension of transcendence
degree at most $4$ (also see [8, Theorem 4.1]). This raises a question:

######  Question 1.1.

What is the optimal bound on transcendence degree of a splitting field?

We answer the above question for quaternion algebras over a field $k$ with
$\mathrm{char}(k)\neq 2$. In Proposition 4.1, we provide a construction of
differential splitting fields of a differential quaternion algebra using
differential fields containing solutions of a certain Riccati equation. We
then use this proposition to show the transcendence degree of such a splitting
field is at most $3$ (see Theorem 4.3). Moreover, for $0\leq r\leq 3$, we give
examples of differential quaternion algebras whose splitting fields have
transcendence degree at least $r$ over $k$ (see Example 5.2 and Theorem 6.5).
Thus, the bound $3$ is optimal. In Section 6, we consider a particular class
of differential quaternion algebras that are split over a transcendence degree
at most one extension of $k$.

Let $k$ be a field containing a primitive $m$th root of unity, say $\omega$.
An $m^{2}$-dimensional central simple algebra over $k$ is a symbol algebra if
it is generated by two elements $u,v$ such that $u^{m},v^{m}\in{k}^{\times}$
and $vu=\omega uv$. For generators $u,v$ of a symbol algebra $A$, there is a
derivation on $A$ that restricts to a given derivation on $k$ such that
$d(k(u))\subseteq k(u)$ and $d(k(v))\subseteq k(v)$; such a derivation is
named as standard derivation. It was shown in [3, Theorem 5.4] that a
$m^{2}$-dimensional symbol algebra with standard derivation is split by a
finite field extension of degree at most $m^{2}$ if $m$ is odd and $2m^{2}$ if
$m$ is even. It is natural to ask the following questions.

######  Question 1.2.

  1. (i)

Can one classify all differential symbol algebras over a differential field
that are split by finite field extensions?

  2. (ii)

Can one classify all standard derivations on a given symbol algebra?

If $\mathrm{char}(k)\neq 2$ then any quaternion algebra is a symbol algebra.
Thus, in particular, a quaternion algebra with standard derivation is split by
a finite field extension of degree at most $8$ (also see [8, Theorem 4.1]). We
show that its converse is not true (see Example 7.1) In the case of quaternion
algebras, we show that the above questions are inter-related. In Theorem 7.3,
we prove that a differential quaternion algebra is split by a finite
differential field extension if and only if the derivation on the differential
split algebra is standard. We hope to extend these results to an arbitrary
symbol algebra in future.

### Notation

Throughout this paper we fix a field $k$ with $\mathrm{char}(k)\neq 2$, an
algebraic closure $\overline{k}$ of $k$. Let ${k}^{\times}$ denote the
multiplicative group of $k$. For $\alpha,\beta\in{k}^{\times}$, we fix the
quaternion algebra $Q=(\alpha,\beta)_{k}$ with generators $u,v$ such that
$u^{2}=\alpha,v^{2}=\beta$ and $vu=-uv$.

##  2\. Differential splitting of central simple algebras

A ring (resp. algebra or field) $R$ together with an additive map $d\colon
R\rightarrow R$ satisfying $d(xy)=xd(y)+d(x)y$ for all $x,y\in R$, is called
differential ring (resp. algebra or field) and is denoted by $(R,d)$. The map
$d$ on $R$ is called a derivation on $R$. The set $C_{(R,d)}=\\{x\in R\mid
d(x)=0\\}$ is called the set of constants of $(R,d)$.

For any element $\vartheta\in R$, there is an inner derivation
$\partial_{\vartheta}$ on $R$ given by
$\partial_{\vartheta}(x)=x\vartheta-\vartheta x$ for $x\in R$. Note that
$C_{(R,\partial_{\vartheta})}$ is equal to the centralizer of $\vartheta$ in
$R$.

A central simple algebra over $k$ is a finite dimensional $k$-algebra with
center $k$ and no non-trivial two-sided ideals. A central simple $k$-algebra
$A$ splits over a field extension $L/k$ if $A\otimes_{k}L\simeq M_{m}(L)$ as
$L$-algebras and $L$ is called a splitting field of $A$.

Let $(k,\,^{\prime})$ be a differential field. For a field extension $L/k$,
there is a derivation on $L$ that restricts to ′ on $k$ (see [9, Section 3]).
Moreover, when $L/k$ is a finite field extension, there is a unique extension
of ′ on $L$. For convenience, we denote the derivations on field extensions by
′ whenever the derivation is clear from the context.

A central simple $k$-algebra $A$ together with a derivation $d$ that restricts
to ′ on k is called differential central simple algebra over $k$, and is
denoted by $(A,d)$. The set of all derivations on $A$ that are extensions of ′
is denoted by $Der(A/(k,\,^{\prime}))$.

We say that $(A_{1},d_{1})$ and $(A_{2},d_{2})$ are isomorphic as differential
$k$-algebras if there exists a $k$-algebra isomorphism
$\phi:(A_{1},d_{1})\rightarrow(A_{2},d_{2})$ such that $\phi\circ
d_{1}=d_{2}\circ\phi$; and $\phi$ is called differential isomorphism.

The derivation ′ on the matrix algebra $M_{m}(k)$ defined by
$(a_{ij})^{\prime}=(a^{\prime}_{ij})$ is called the coordinate-wise derivation
on $M_{m}(k)$. For the matrix algebra $M_{m}(k)$, we have
$Der(M_{m}(k)/(k,\,^{\prime}))=\\{d_{P}=\,^{\prime}+\partial_{P}\mid\ P\in
M_{m}(k)\\}$ (see [1, Theorem 2]).

In [6], the authors introduced the notion of differential splitting fields. A
differential central simple algebra $(A,d)$ over $(k,\,^{\prime})$ splits over
a differential field $(L,\,^{\prime})\supseteq(k,\,^{\prime})$ if
$(A\otimes_{k}L,d^{\ast})\simeq(M_{m}(L),\,^{\prime})$ as differential
$L$-algebras, where $d^{\ast}:=d\otimes\,^{\prime}=d\otimes
id_{L}+id_{A}\otimes\,^{\prime}$. In [6], the authors proved the existence of
differential splitting fields using the theory of Picard-Vessiot extensions.
The following characterization of isomorphic differential matrix algebras was
crucial in constructing differential splitting fields.

######  Proposition 2.1.

[6, Proposition 2] Let $P\in M_{m}(k)\setminus\\{0\\}$. Then
$(M_{m}(k),d_{P})$ and $(M_{m}(k),\,^{\prime})$ are isomorphic as differential
$k$-algebras if and only if there exists $F\in GL_{m}(k)$ such that
$F^{\prime}=PF$.

For a quaternion algebra $Q$ over $k$, a derivation $d\in
Der(Q/(k,\,^{\prime}))$ is called a standard derivation if there exists
$u,v\in Q^{0}$ such that $u^{2},v^{2}\in{k}^{\times},\ vu=-uv,\ d(u)\in k(u)$
and $d(v)\in k(v)$, and we denote this derivation by $d_{(u,v)}$. In [8,
Proposition 3.2], it was shown that for every derivation $d\in
Der(Q/(k,\,^{\prime}))$, there exists a unique $\vartheta\in Q^{0}$ such that
$d=d_{(u,v)}+\partial_{\vartheta}$. Furthermore, this characterization was
used to construct splitting fields of quaternion algebras with derivations. We
recall results related to differential splitting of quaternion algebras.

######  Theorem 2.2.

Let $(k,\,^{\prime})$ be a differential field with $\mathrm{char}(k)=0$. Let
$Q=(\alpha,\beta)_{k}$ be a quaternion algebra.

  1. (1)

[8, Theorem 4.1] The differential algebra $(Q,d)$ is split by a finitely
generated differential field extension of $(k,\,^{\prime})$. The algebra
$(Q,d_{(u,v)})$ is split by a finite extension of $(k,\,^{\prime})$.

  2. (2)

[8, Theorem 4.5] If $\,{}^{\prime}$ is the zero derivation and $Q$ is a
division algebra with non-zero derivation $d$, then $(Q,d)$ is split by a
field extension of transcendence degree $1$ and is not split by any algebraic
extension of $k$.

We are now interested in constructing differential splitting fields for an
arbitrary differential quaternion algebra $(Q,d)$.

##  3\. Solutions of Riccati equations

In the process of finding the splitting fields of a differential quaternion
algebra, we come across a certain types of first order differential equations
with variable coefficients. In this section, we will note some essential
results concerning homogeneous linear differential equations and Riccati
equations.

A first order differential equation over $k$ of the form

$X^{\prime}=\alpha_{0}+\alpha_{1}X+\alpha_{2}X^{2},$

where $\alpha_{0},\alpha_{2}\neq 0$, is called Riccati equation. If
$\alpha_{0}=\alpha_{2}=0$ then equation becomes a homogenous linear
differential equation. In general, solutions of a Riccati equation are not
known, but there are methods to solve particular cases. We will use Riccati
equations whose solutions are known to construct examples of differential
quaternion algebras that split over fields with fixed transcendence degrees.
For this purpose, in the following results, we relate solutions of Riccati
equations to that of a certain linear differential equation.

######  Proposition 3.1.

Let $a\in k$ and let $\theta\in\overline{k}$ be a solution of $Y^{\prime}=aY$
such that $n:=[k[\theta]:k]$ is minimal. Then $\theta^{n}\in k$.

###### Proof.

Consider the polynomial ring $k[X]$ with derivation $D$ such that $D(X)=aX$.
Let $p(X)=X^{n}+a_{n-1}X^{n-1}+\dots+a_{1}X+a_{0}$ be the minimal polynomial
of $\theta$ over $k$. Consider the differential homomorphism $\phi:k[X]\to
k[\theta]$ defined by $\phi(X)=\theta$. Then the kernel of $\phi$ is the prime
ideal generated by $p(X)$. Since $D(p)(\theta)=0$, $p$ divides $D(p)$ and
whereby $D(p)=nap$. On comparing the coefficients, we obtain
$D(a_{i})=(n-i)aa_{i}$ for $0\leq i\leq n-1$. This implies $\sqrt[n-i]{a_{i}}$
is a solution of $Y^{\prime}=aY$. Since $n$ is the minimal degree for such a
solution, we obtain $a_{i}=0$ for $1\leq i\leq n-1$. Therefore,
$p(X)=X^{n}+a_{0}$. ∎

######  Lemma 3.2.

Let $k(t)$ be the rational function field with derivation
$t^{\prime}=\alpha_{0}+\alpha_{1}t+\alpha_{2}t^{2}$ where
$\alpha_{0},\alpha_{1},\alpha_{2}\in k$ and $\alpha_{2}\neq 0$. Let
$n\in\mathbb{N}$ and $f\in k(t)$ be a solution of the differential equation
$Y^{\prime}=n\left(\frac{\alpha_{1}}{2}+\alpha_{2}t\right)Y$. Then each zero
and pole of $f$ is a solution of
$X^{\prime}=\alpha_{0}+\alpha_{1}X+\alpha_{2}X^{2}$.

###### Proof.

Let $p=\frac{\alpha_{1}}{2}+\alpha_{2}t$. Since $p\notin k$, the solution
$f\notin k$. Let $\gamma_{1},\ldots,\gamma_{r}\in\overline{k}$, $\eta\in k$
and $m_{1},\ldots,m_{r}\in\mathbb{Z}$ be such that
$f=\eta\prod_{i=1}^{r}(t-\gamma_{i})^{m_{i}}$. Then

$f^{\prime}=\eta\prod_{i=1}^{r}(t-\gamma_{i})^{m_{i}-1}\cdot\left(\sum_{i=1}^{r}\Big{(}(t^{\prime}-\gamma_{i}^{\prime})\prod_{j\neq
i}(t-\gamma_{j})\Big{)}\right)+\frac{\eta^{\prime}}{\eta}f.$

Since $f^{\prime}=npf$, for $1\leq i\leq r$, we have $(t-\gamma_{i})$ divides
$(t^{\prime}-\gamma_{i}^{\prime})=\alpha_{0}+\alpha_{1}t+\alpha_{2}t^{2}-\gamma_{i}^{\prime}$,
which implies that each $\gamma_{i}$ is a solution of
$X^{\prime}=\alpha_{0}+\alpha_{1}X+\alpha_{2}X^{2}$. ∎

######  Lemma 3.3.

Assume that the non-linear differential equation

(3.1) $X^{\prime}=\alpha_{0}+\alpha_{1}X+\alpha_{2}X^{2}$

over $k$ has no algebraic solution and $\lambda_{1},\lambda_{2}$ be its
distinct transcendental solutions. Then the differential equation
$Y^{\prime}=\left(\frac{\alpha_{1}}{2}+\alpha_{2}\lambda_{1}\right)Y$ has no
algebraic solution over $k(\lambda_{1},\lambda_{2})$.

###### Proof.

Set $L=k(\lambda_{1},\lambda_{2})$ and
$p=\frac{\alpha_{1}}{2}+\alpha_{2}\lambda_{1}$. Suppose that $Y^{\prime}=pY$
has a solution in $\overline{L}$.

First assume that $\lambda_{2}\in\overline{k(\lambda_{1})}$. Then
$\overline{L}=\overline{k(\lambda_{1})}$. Let $\theta\in\overline{L}$ be a
solution of $Y^{\prime}=pY$ such that
$n=[k(\lambda_{1},\theta):k(\lambda_{1})]$ is minimal. By Proposition 3.1, it
follows that $\theta^{n}=f\in k(\lambda_{1})$. Then $f^{\prime}=npf$. Since
Equation 3.1 has no solution in $\overline{k}$, by Lemma 3.2 $f\in k$. This
implies $p\in k$, which is a contradiction.

Now assume that $\lambda_{2}\notin\overline{k(\lambda_{1})}$. Set
$K=k(\lambda_{2})$. Note that $\lambda_{2}$ is the only solution of Equation
3.1 in $\overline{K}$. Let $\theta\in\overline{L}$ be a solution of
$Y^{\prime}=pY$ such that $n=[K(\lambda_{1},\theta):K(\lambda_{1})]$ is
minimal. Then, by Proposition 3.1, $\theta^{n}=f\in K(\lambda_{1})$ and
$f^{\prime}=npf$. By Lemma 3.2, we have that
$f=\eta(\lambda_{1}-\lambda_{2})^{m}$ for some $m\in\mathbb{Z}$ and $\eta\in
K$.

Substituting $f=\eta(\lambda_{1}-\lambda_{2})^{m}$ in $f^{\prime}=npf$, we
further get

$\frac{\eta^{\prime}}{n\eta}+\frac{m}{n}\big{(}\alpha_{1}+\alpha_{2}(\lambda_{1}-\lambda_{2})\big{)}=\frac{\alpha_{1}}{2}+\alpha_{2}\lambda_{1}\,.$

On comparing the coefficients, we obtain $m=n$ and
$(\eta^{-1})^{\prime}=n\left(\frac{\alpha_{1}}{2}+\alpha_{2}\lambda_{2}\right)\eta^{-1}$.
Since Equation 3.1 has no solution in $\overline{k}$, using Lemma 3.2, we
arrive at a contradiction as above . ∎

##  4\. Splitting of differential quaternion algebras

Let $Q=(\alpha,\beta)_{k}$ be a quaternion algebra with generators $u,v$. For
$\xi\in\overline{k}$ such that $\xi^{2}=\alpha$, the homomorphism
$\mathbf{\Phi_{Q,\xi}}:Q\otimes_{k}k(\xi)\rightarrow M_{2}(k(\xi))$ determined
by

$u\otimes 1\mapsto A=\left(\begin{matrix}\xi&\\\ &-\xi\\\
\end{matrix}\right);\,\,\,\,v\otimes 1\mapsto B=\left(\begin{matrix}&\beta\\\
1&\\\ \end{matrix}\right),$

and $\lambda\otimes r\mapsto\lambda rI_{2}$ where $\lambda\in k$ and $r\in
k(\xi)$ is an isomorphism of $k(\xi)$-algebras. Let
$d=d_{(u,v)}+\partial_{\vartheta}$, where $\vartheta=a_{1}u+a_{2}v+a_{3}uv\in
Q^{0}$, be a derivation in $Der(Q/(k,\,^{\prime}))$. The isomorphism
$\Phi_{Q,\xi}$ induces a differential isomorphism
$(Q\otimes_{k}k(\xi),d^{\ast})\rightarrow(M_{2}(k(\xi)),d_{P})$ where

(4.1)
$P=\left(\begin{matrix}a_{1}\xi+\frac{\beta^{\prime}}{4\beta}&(a_{2}+a_{3}\xi)\beta\\\
a_{2}-a_{3}\xi&-a_{1}\xi-\frac{\beta^{\prime}}{4\beta}\\\
\end{matrix}\right),$

see [8, Theorem 4.1]. By Proposition 2.1, the differential algebra
$(M_{2}(k(\xi)),d_{P})$ is split over $(L,\,^{\prime})$ if and only if there
exists $F\in GL_{2}(L)$ such that $F^{\prime}=PF$. Write $F=(f_{ij})$ with
$f_{ij}\in L$. By comparing the matrix entries in the equation $F^{\prime}=PF$
we obtain

(4.2) $\displaystyle f_{11}^{\prime}$
$\displaystyle=\left(a_{1}\xi+\frac{\beta^{\prime}}{4\beta}\right)f_{11}+(a_{2}+a_{3}\xi)\beta
f_{21},$ (4.3) $\displaystyle f_{21}^{\prime}$
$\displaystyle=(a_{2}-a_{3}\xi)f_{11}-\left(a_{1}\xi+\frac{\beta^{\prime}}{4\beta}\right)f_{21},$
(4.4) $\displaystyle f_{12}^{\prime}$
$\displaystyle=\left(a_{1}\xi+\frac{\beta^{\prime}}{4\beta}\right)f_{12}+(a_{2}+a_{3}\xi)\beta
f_{22},$ (4.5) $\displaystyle f_{22}^{\prime}$
$\displaystyle=(a_{2}-a_{3}\xi)f_{12}-\left(a_{1}\xi+\frac{\beta^{\prime}}{4\beta}\right)f_{22}.$

Taking $L=k(\xi)(f_{11},f_{12},f_{21},f_{22})$, it is clear that $(Q,d)$ is
always split by a differential field extension of transcendence degree at most
$4$.

Here, we will show that a differential quaternion algebra is split by a
differential field extension of transcendence degree at most $3$.

######  Proposition 4.1.

Let $\xi\in\overline{k}$ be such that $\xi^{2}=\alpha$. Let $(L,\,^{\prime})$
be a differential field extension of $(k(\xi),\,^{\prime})$.

  1. $(a)$

If $(Q,d)$ is split by $(L,\,^{\prime})$, then the differential equation

(4.6)
$X^{\prime}=(a_{2}+a_{3}\xi)\beta+2\left(a_{1}\xi+\frac{\beta^{\prime}}{4\beta}\right)X-(a_{2}-a_{3}\xi)X^{2}$

has at least two distinct solutions in $L$.

  2. $(b)$

If Equation 4.6 has at least two distinct solutions (say
$\lambda_{1},\lambda_{2}$) in $L$, then $(Q,d)$ is split by a simple extension
$(L(\mu),\,^{\prime})$ where $\mu$ is a solution of the differential equation

(4.7)
$\frac{Y^{\prime}}{Y}=(a_{2}-a_{3}\xi)\lambda_{1}-\left(a_{1}\xi+\frac{\beta^{\prime}}{4\beta}\right).$

###### Proof.

  1. $(a)$

Assume that $(Q,d)$ is split by $(L,\,^{\prime})$. Then the algebras
$(Q\otimes k(\xi),d^{\ast})\simeq(M_{2}(k(\xi)),d_{P}))$ are also split by
$(L,\,^{\prime})$. From Proposition 2.1, there exists $F=(f_{ij})\in
GL_{2}(L)$ such that $F^{\prime}=PF$. First suppose that $a_{2}=a_{3}=0$. From
Equations (4.2) to (4.5), we get that $f_{11},f_{12},f_{21}^{-1}$ and
$f_{22}^{-1}$ are solutions of the Equation 4.6,
$X^{\prime}=\left(a_{1}\xi+\frac{\beta^{\prime}}{4\beta}\right)X$.

Now suppose $a_{2}\neq 0$ or $a_{3}\neq 0$. Since $F\in GL_{2}(L)$, from
Equations (4.2) to (4.5), we conclude that $f_{ij}\neq 0$ for all $i,j$.
Considering the equations
$\frac{(\ref{f11})}{f_{11}}-\frac{(\ref{f21})}{f_{21}}$ and
$\frac{(\ref{f12})}{f_{12}}-\frac{(\ref{f22})}{f_{22}}$, we obtain that
$\frac{f_{11}}{f_{21}}$ and $\frac{f_{12}}{f_{22}}$ are solutions of Equation
4.6. Since $F\in GL_{2}(L)$, we have that
$\frac{f_{11}}{f_{21}}\neq\frac{f_{12}}{f_{22}}$.

  2. $(b)$

Set

$F=\left(\begin{matrix}\lambda_{1}\mu&\frac{\lambda_{2}}{\mu(\lambda_{1}-\lambda_{2})}\\\
\mu&\frac{1}{\mu(\lambda_{1}-\lambda_{2})}\\\ \end{matrix}\right).$

Note that $\det(F)=1$ and $F^{\prime}=PF$. Thus, by Proposition 2.1
$(M_{2}(L(\mu)),d_{P})\simeq(M_{2}(L(\mu)),\,^{\prime})$. Hence, $(Q,d)$
splits over $(L(\mu),\,^{\prime})$.

∎

Using this proposition, we can easily construct an example where differential
splitting field of a quaternion algebra is of transcendence degree at most
$1$.

######  Example 4.2.

For $a\in k$, let $d=d_{(u,v)}+\partial_{av}$ be a derivation on the
quaternion algebra $Q=(\alpha,\beta)_{k}$. Then the Equation 4.6 becomes
$X^{\prime}=a(\beta-X^{2})+\frac{\beta^{\prime}}{2\beta}X$. Let
$\eta\in\overline{k}$ such that $\eta^{2}=\beta$, then $\eta,-\eta$ are two
distinct solutions of
$X^{\prime}=a(\beta-X^{2})+\frac{\beta^{\prime}}{2\beta}X$. From Proposition
4.1(b), $(Q,d)$ is split by $k(\eta,\mu)$ where
$\mu^{\prime}=(a\eta-\frac{\beta^{\prime}}{4\beta})\mu$. Note that the
transcendence degree of $k(\eta,\mu)/k$ is at most $1$.

Since the generators $u,v$ and $uv$ are symmetric in the sense that
$d_{(u,v)}=d_{(v,u)}=d_{(u,uv)}$, we have that $(Q,d)$, where
$d=d_{(u,v)}+\partial_{w}$, is split by a differential field of transcendence
degree at most $1$ if $w$ lies in $k(u)$, $k(v)$ or $k(uv)$. This generalizes
[8, Example 4.2].

Now we prove the main theorem of this section.

######  Theorem 4.3.

$(Q,d)$ is split by a differential field extension of $(k,\,^{\prime})$ of
transcendence degree at most $3$.

###### Proof.

It follows from Proposition 4.1(b) that $(Q,d)$ is split by the differential
field extension $L:=k(\xi,\lambda_{1},\lambda_{2},\mu)$ where
$\lambda_{1},\lambda_{2}$ are solutions of Equation 4.6 and $\mu$ is a
solution of Equation 4.7. Clearly, $\mathrm{tr.deg}_{k}(L)\leq 3$. ∎

##  5\. Examples of splitting fields with transcendence degree $2$ and $3$

In this section, we construct examples of differential splitting fields with
transcendence degree at least $2$ and $3$. We first observe the following
result, which is crucial in the construction of such splitting fields.

######  Proposition 5.1.

The differential quaternion algebra $(Q,d)$ is split by a field extension of
transcendence degree at least $2$ if the non-linear differential equation
(4.6)

$X^{\prime}=(a_{2}+a_{3}\xi)\beta+2\left(a_{1}\xi+\frac{\beta^{\prime}}{4\beta}\right)X-(a_{2}-a_{3}\xi)X^{2}$

has no algebraic solutions over $k$. Moreover, if any two solutions of
Equation 4.6 are algebraically independent over $k$, then $(Q,d)$ is split by
a field extension of transcendence degree at least $3$.

###### Proof.

Let $(L,\,^{\prime})$ be a differential field extension of
$(k(\xi),\,^{\prime})$ containing two distinct roots of Equation 4.6. Let
$\lambda\in L$ be a solution of Equation 4.6. By Proposition 4.1(b), $(Q,d)$
is split by an extension $L(\mu)$, where
$\mu^{\prime}=(a_{2}-a_{3}\xi)\lambda-\left(a_{1}\xi+\frac{\beta^{\prime}}{4\beta}\right)\mu$.
Since Equation 4.6 has no algebraic solution over $k$, by Lemma 3.3, it
follows that $\mathrm{tr.deg}_{k}{L(\mu)}=\mathrm{tr.deg}_{k}{L}+1$. ∎

######  Example 5.2.

Let $(k,\,^{\prime})$ be the differential field where $k=\mathbb{Q}(t)$ and
$t^{\prime}=1$. Let $Q=(1,t)_{k}$, $u,v\in Q^{0}$ such that $u^{2}=1$,
$v^{2}=t$ and $vu=-uv$.

(i) Consider the derivation $d=d_{(u,v)}+\partial_{\vartheta}$ on $Q$ with
$\vartheta=\frac{1}{4t}(-u-2v+2uv)$. Then the Equation 4.6 becomes
$X^{\prime}=\frac{1}{t}X^{2}$. The general solution of this differential
equation is given by $\frac{-1}{\log(t)+c}$, where $c$ is an arbitrary
constant. This implies, $X^{\prime}=\frac{1}{t}X^{2}$ has no solution in
$\overline{k}$ and infinitely many solutions in $k(\log(t))$. Since the
transcendence degree of $k(\log(t))$ over $k$ is $1$, from Proposition 5.1 we
have the transcendence degree of splitting fields of $(Q,d)$ is at least $2$.

(ii) Consider the derivation $d=d_{(u,v)}+\partial_{\vartheta}$ on $Q$ with
$\vartheta=-\frac{1}{4t}u-v$ Then the Equation 4.6 becomes
$X^{\prime}=X^{2}-t$, which is a Riccati differential equation that is not
solvable by any algebraic function over $k$ (see [7, Example p.42-43]).

Let $K=k(\lambda)$ be a rational function field where
$\lambda^{\prime}=\lambda^{2}-t$ and $\lambda+\frac{1}{\theta}$ be another
solution of $X^{\prime}=X^{2}-t$ in $\overline{K}$, i.e
$\theta\in\overline{K}$. Then it is easy to observe that
$\theta^{\prime}=-2\lambda\theta-1$. Let
$p(X)=X^{m}+a_{m-1}X^{m-1}+\dots+a_{1}X+a_{0}$ be the minimal polynomial of
$\theta$ over $K$. Consider a differential homomorphism $\phi:K[X]\to
K[\theta]$ defined by $\phi(X)=\theta$. Then the kernel of $\phi$ is the prime
ideal generated by $p(X)$. Since $p^{\prime}(\theta)=0$, $p$ divides
$p^{\prime}$ and whereby $p(X)^{\prime}=-2m\lambda p(X)$. On comparing the
coefficients of $X^{m-1}$, we obtain $a_{m-1}$ is a solution of the
differential equation

(5.1) $Y^{\prime}=-2\lambda Y+c$

over $K$, when $c=m$. Let $m\neq 0$ then $a_{m-1}\neq 0$. Write
$a_{m-1}=\frac{f}{g}$ where $f,g$ are coprime polynomials in $k[\lambda]$ and
$g$ is monic. Then $f^{\prime}g-fg^{\prime}=-2\lambda fg+mg^{2}$ implies $g$
divides $g^{\prime}$.

Let $\gamma_{1},\ldots,\gamma_{r}\in\overline{k}$ and
$m_{1},\ldots,m_{r}\in\mathbb{Z}$ be such that
$g=\prod_{i=1}^{r}(\lambda-\gamma_{i})^{m_{i}}$. Then

$g^{\prime}=\prod_{i=1}^{r}(\lambda-\gamma_{i})^{m_{i}-1}\cdot\left(\sum_{i=1}^{r}\Big{(}(\lambda^{\prime}-\gamma_{i}^{\prime})\prod_{j\neq
i}(\lambda-\gamma_{j})\Big{)}\right).$

Since $g$ divides $g^{\prime}$, for $1\leq i\leq r$, we have
$(\lambda-\gamma_{i})$ divides
$(\lambda^{\prime}-\gamma_{i}^{\prime})=\lambda^{2}-t-\gamma_{i}^{\prime}$,
which implies that each $\gamma_{i}$ is a solution of $X^{\prime}=X^{2}-t$.
Therefore, each $m_{i}=0$ and $g=1$.

Write $f=\sum_{i=0}^{s}f_{i}\lambda^{i}$, where $f_{i}\in k$, $f_{s}\neq 0$.
Then $f^{\prime}=-2\lambda f+c$ implies $s=-2$, which is a contradiction.
Therefore, $m=0$ and $\theta\in K$ is a solution of Equation 5.1 for $c=-1$,
which is again not possible. Thus, any field extension of $k$ containing two
distinct solutions of $X^{\prime}=X^{2}-t$ must be of transcendence degree at
least $2$. By Proposition 5.1, splitting fields of $((1,t)_{k},d)$ have
transcendence degree at least $3$ over $(k,\,^{\prime})$.

##  6\. Splitting fields with transcendence degree $1$

In Example 4.2, we have seen that for derivations of the form
$d=d_{(u,v)}+\partial_{au}$ where $a\in k$, the transcendence degree of
splitting field is at most one. We will further analyze these derivations to
have the transcendence degree equal to $1$.

######  Proposition 6.1.

Let $(Q,d)$ be a differential quaternion algebra where
$d=d_{(u,v)}+\partial_{au}$ for $a\in k$. Then $(Q,d)$ is split by a finite
extension of $(k,\,^{\prime})$, if and only if

$a\in\left\\{\frac{\theta^{\prime}}{n\xi\theta}\mid\theta\in
k(\xi),n\in\mathbb{N}\right\\}.$

###### Proof.

By Proposition 4.1, $(Q,d)$ is split by a finite extension $(L,\,^{\prime})$
of $(k,\,^{\prime})$ if and only if $L$ contains a solution of the equation

(6.1) $X^{\prime}=\left(a\xi+\frac{\beta^{\prime}}{4\beta}\right)X,$

that is, if and only if $L(\sqrt[4]{\beta})$ contains a solution of the
equation $X^{\prime}=a\xi X$. Let $\lambda\in\overline{k(\xi)}$ be a solution
of Equation 6.1 and $\theta=N_{k(\xi)(\lambda)/k(\xi)}(\lambda)$. For
$n=[k(\xi)(\lambda):k(\xi)]$, we have $\theta^{\prime}=na\xi\theta$. ∎

######  Lemma 6.2.

For $\theta\in k(\xi)$, $\frac{\theta^{\prime}}{\theta}\in\xi k$ if and only
if $N_{k(\xi)/k}(\theta)\in C_{k}$.

###### Proof.

Let $\overline{\theta}$ be the conjugate of $\theta$. Note that
$\overline{\theta}^{\prime}=\overline{\theta^{\prime}}$. Then
$\frac{\theta^{\prime}}{\theta}\in\xi k$ if and only if
$\frac{\theta^{\prime}}{\theta}=-\overline{\frac{\theta^{\prime}}{\theta}}$,
that is,
$(N_{k(\xi)/k}(\theta))^{\prime}=\theta^{\prime}\cdot\overline{\theta}+\theta\cdot(\overline{\theta})^{\prime}=0$.
∎

######  Remark 6.3.

To describe the set of all $a\in k$ such that $(Q,d)$ is split by a finite
extension of $(k,\,^{\prime})$ we need to find the set $S=\\{\theta\in
k(\xi)\setminus k\mid N_{k(\xi)/k}(\theta)\in C_{k}\\}$. Consider the non-
empty subset $\tilde{S}=\\{\theta\in k(\xi)\setminus k\mid
N_{k(\xi)/k}(\theta)\in{C}^{\times 2}_{k}\\}$ of $S$. By Hilbert’s Theorem 90
[2, Example 2.3.4], we have

$\tilde{S}=C_{k}\cdot\left\\{\frac{\gamma}{\overline{\gamma}}\mid\gamma\in{k(\xi)}^{\times}\right\\}.$

######  Example 6.4.

Let $(k,\,^{\prime})$ be the differential field where $k=\mathbb{Q}(t)$ and
$t^{\prime}=1$. Let $\alpha\in\mathbb{Q}[t]$ be an odd degree polynomial. Then
$S=\tilde{S}$ as in Remark 6.3. For the derivation $d=d_{(u,v)}+\partial_{au}$
where $a\in k$, $(Q,d)$ is split by a finite extension of $(k,\,^{\prime})$ if
and only if

$a\in\left\\{\frac{\theta^{\prime}}{n\xi\theta}\mid\theta\in\tilde{S},n\in\mathbb{N}\right\\}.$

######  Theorem 6.5.

Let $(k,\,^{\prime})$ be the differential field where $k=\mathbb{Q}(t)$ and
$t^{\prime}=1$. Let $\alpha\in\mathbb{Q}[t]$ be an odd degree polynomial.
Consider the quaternion algebra $Q=(\alpha,\beta)$ with derivation
$d=d_{(u,v)}+\partial_{au}$ where $a=\frac{f}{g}\in\mathbb{Q}(t)$. Then
$(Q,d)$ is not split by any algebraic extension of $k$, if

  1. $(a)$

$deg(g)-deg(f)<\frac{deg(\alpha)+3}{2}$.

  2. $(b)$

for a factor $h$ of $\alpha$, $h^{2}$ divides $g$.

###### Proof.

Suppose $(Q,d)$ is split by an algebraic extension of $k$. Then, by Example
6.4, there exists $\theta\in\tilde{S}$ such that
$a=\frac{\theta^{\prime}}{n\xi\theta}$. It follows from Remark 6.3 that
$\theta=c\frac{\gamma_{0}+\xi\gamma_{1}}{\gamma_{0}-\xi\gamma_{1}}$ where
$c\in\mathbb{Q}$ and $\gamma_{0},\gamma_{1}\in\mathbb{Q}[t]$ are coprime
polynomials. Observe that

$\frac{\theta^{\prime}}{n\xi\theta}=\frac{2\alpha(\gamma_{0}\gamma_{1}^{\prime}-\gamma_{0}^{\prime}\gamma_{1})+\alpha^{\prime}\gamma_{0}\gamma_{1}}{n\alpha(\gamma_{0}^{2}-\alpha\gamma_{1}^{2})}\in\mathbb{Q}(t).$

Set
$p=2\alpha(\gamma_{0}\gamma_{1}^{\prime}-\gamma_{0}^{\prime}\gamma_{1})+\alpha^{\prime}\gamma_{0}\gamma_{1}$
and $q=n\alpha(\gamma_{0}^{2}-\alpha\gamma_{1}^{2})$.

  1. $(a)$

Let $m=deg(\alpha),m_{0}=deg(\gamma_{0})$ and $m_{1}=deg(\gamma_{1})$. We have
$deg(p)\leq m+m_{0}+m_{1}-1$ and $deg(q)=m+\max(2m_{0},2m_{1}+m)$ and
calculating further, we get that $deg(q)-deg(p)\geq\frac{m+3}{2}$.

  2. $(b)$

Let $h\in\mathbb{Q}[t]$ be an irreducible factor of $\alpha$. Write
$\alpha=h\tilde{\alpha}$ with $\tilde{\alpha}\in\mathbb{Q}[t]$. If
$h\nmid\gamma_{0}$ then $h^{2}\nmid q$, whereby $h^{2}\nmid g$. If
$h\mid\gamma_{0}$ then $h^{2}|q$, $h^{3}\nmid q$ and $h|p$, whereby
$h^{2}\nmid g$.

∎

##  7\. Standard derivations

Let $(k,\,^{\prime})$ be a differential field and let $(Q,d)$ be a
differential quaternion algebra over $(k,\,^{\prime})$. Recall that $d$ is a
standard derivation on $Q$ if there exist $u,v\in Q^{0}$ such that
$u^{2},v^{2}\in{k}^{\times},uv=-vu,d(u)\in k(u)$ and $d(v)\in k(v)$. For a
differential field extension $(L,\,^{\prime})$ over $(k,\,^{\prime})$, we say
that $d$ becomes standard over $L$ if $d^{\ast}=d\otimes\,^{\prime}$ is
standard derivation on $Q\otimes_{k}L$. A quaternion algebra with standard
derivation is split by a finite field extension of degree at most $8$ (see [8,
Theorem 4.1]), but the following example shows that the converse need not be
true.

######  Example 7.1.

Let $(k,\,^{\prime})$ be the differential field where $k=\mathbb{Q}(t)$ and
$t^{\prime}=1$. Let $Q=(1,t)_{k}$ with $u,v\in Q$ such that $u^{2}=1$,
$v^{2}=t$ and $vu=-uv$. Consider the derivation
$d=d_{(u,v)}+\partial_{-\frac{1}{8t}u}$ on $Q$. Then Equations (4.6) and (4.7)
will become $X^{\prime}=\frac{1}{4t}X$ and $Y^{\prime}=-\frac{1}{8t}Y$,
respectively, and it follows from Proposition 4.1 that a field extension
$(L,\,^{\prime})$ splits $(Q,d)$ if and only if $L$ contains solutions of
these equations, that is, if and only if $L$ contains $k(\sqrt[8]{t})$. Thus,
$[L:k]\geq 8$.

We now show that $d$ is not a standard derivation on $Q$. Let
$\tilde{u}=a_{1}u+a_{2}v+a_{3}uv$ such that ${\tilde{u}}^{2}\in{k}^{\times}$
and $d(\tilde{u})\in k\tilde{u}$. Then $d(\tilde{u})\in k\tilde{u}$ implies
that $d(\tilde{u})=\frac{N(\tilde{u})^{\prime}}{2N(\tilde{u})}\tilde{u}$ and
we get the following equations:

(7.1) $\displaystyle\frac{N(\tilde{u})^{\prime}}{2N(\tilde{u})}a_{1}$
$\displaystyle=a_{1}^{\prime}$ (7.2)
$\displaystyle\frac{N(\tilde{u})^{\prime}}{2N(\tilde{u})}a_{2}$
$\displaystyle=a_{2}^{\prime}+\frac{1}{2t}a_{2}+\frac{a_{3}}{4t}$ (7.3)
$\displaystyle\frac{N(\tilde{u})^{\prime}}{2N(\tilde{u})}a_{3}$
$\displaystyle=a_{3}^{\prime}+\frac{1}{2t}a_{3}+\frac{a_{2}}{4t}$

Note that $a_{2}=0$ if and only if $a_{3}=0$. Suppose that
$a_{2},a_{3}\in{k}^{\times}$. From Equations (7.2) and (7.3), we obtain that
$\frac{a_{2}}{a_{3}}$ is a solution of the Riccati Equation
$X^{\prime}=\frac{1}{4t}(X^{2}-1)$. Note that $\pm 1$ are solutions of this
equation and using separation of variables method, we observe that the general
solution other than $\pm 1$ is $X=\frac{1+c\sqrt{t}}{1-c\sqrt{t}}$ with
$c^{\prime}=0$. Since $a_{2},a_{3}\in k$, we get that $\frac{a_{2}}{a_{3}}=\pm
1$. Thus $N(\tilde{u})=a_{1}^{2}\neq 0$, whereby $a_{1}\in{k}^{\times}$. If
$a_{2}=a_{3}$, it follows from Equations (7.1) and (7.2) that
$\frac{a_{1}}{a_{2}}$ is a solution of the linear equation
$Y^{\prime}=\frac{3}{4t}Y$ and hence $\frac{a_{1}}{a_{2}}$ is a constant
multiple of $t^{\frac{3}{4}}$, which is a contradiction. Similarly, if
$a_{2}=-a_{3}$, we get that $\frac{a_{1}}{a_{2}}$ is a solution of
$Y^{\prime}=\frac{1}{4t}Y$, whereby $\frac{a_{1}}{a_{2}}$ is a constant
multiple of $t^{\frac{1}{4}}$, which is again a contradiction. Thus
$a_{2}=a_{3}=0$, and $\tilde{u}\in ku$.

Thus any $\tilde{u}\in Q^{0}$ such that $d(\tilde{u})\in k\tilde{u}$ is an
element of $ku$. Since any pair of such elements cannot generate $Q$ over $k$,
$d$ is not a standard derivation on $Q$.

Now, we show that a derivation on a quaternion algebra becomes standard over
its differential splitting fields.

######  Proposition 7.2.

Let $(Q,d)$ be a differential quaternion algebra over $(k,\,^{\prime})$ and
let $(L,\,^{\prime})$ be a differential splitting field. Then $d$ becomes
standard over $L$.

###### Proof.

Since $(Q\otimes_{k}L,d^{*})\simeq(M_{2}(L),d_{P})$ for a trace zero matrix
$P$, we have $(M_{2}(L),d_{P})\simeq(M_{2}(L),\,^{\prime})$. By Proposition
2.1, there exists $F\in GL_{2}(L)$ such that $F^{\prime}=PF$. By Proposition
4.1, $L$ contains a solution of Equation 4.7, say $\theta$. Set

$U=F\left(\begin{matrix}\theta&0\\\
0&-\theta\end{matrix}\right)F^{-1}\mbox{and
}V=F\left(\begin{matrix}0&\theta\\\ \theta&0\end{matrix}\right)F^{-1}.$

Then $U^{2}=V^{2}=\theta^{2}$, $VU=-UV$,
$d_{P}(U)=\frac{\theta^{\prime}}{\theta}U$ and
$d_{P}(V)=\frac{\theta^{\prime}}{\theta}V$. Let $\tilde{u}$ and $\tilde{v}$ be
pre-images of $U$ and $V$ respectively, in $Q\otimes_{k}L$. Then $\tilde{u}$
and $\tilde{v}$ are generators of $Q\otimes_{k}L$ with
$d^{*}(\tilde{u})=\frac{\theta^{\prime}}{\theta}\tilde{u}$ and
$d^{*}(\tilde{v})=\frac{\theta^{\prime}}{\theta}\tilde{v}$. Thus,
$(Q\otimes_{k}L,d^{*})=(Q\otimes_{k}L,d_{(\tilde{u},\tilde{v})})$. ∎

######  Theorem 7.3.

Let $(Q,d)$ be a differential quaternion algebra over $(k,\,^{\prime})$. Then
$d$ becomes standard over a finite extension of $k$ if and only if $(Q,d)$ is
split by a finite extension of $k$.

###### Proof.

Suppose $d$ becomes standard over a finite extension of $k$, then using
Theorem 2.2(1) we get that $(Q,d)$ is split by a finite extension of $k$.

Conversely, let $(Q,d)$ be split by a finite extension $(L,\,^{\prime})$ of
$(k,\,^{\prime})$. Then by Proposition 7.2, $d$ becomes standard over $L$. ∎

######  Remark 7.4.

The class of differential quaternion algebras over $\mathbb{Q}(t)$ defined in
Theorem 6.5 provides examples of derivations that do not become standard over
any finite extension of $\mathbb{Q}(t)$.

### Acknowledgement

The authors wish to thank Amit Kulshrestha and Varadharaj R. Srinivasan for
interesting discussions on this topic and raising 1.2.

## References

  * [1] S. A. Amitsur, _Extensions of derivations to central simple algebras_ , Commun. Algebra, vol. 10 issue 8 (1982), 797–803.
  * [2] P. Gille and T. Szamuely, _Central simple algebras and Galois cohomology_ , Cambridge Studies in Advanced Mathematics, vol. 101, Cambridge University Press, 2006.
  * [3] P. Gupta, Y. Kaur, A. Singh, _Splitting fields of differential symbol algebras_ , J. Pure Appl. Algebra vol. 227 Issue: 5 (2023), 1-16.
  * [4] G. Hochschild, _Restricted Lie algebras and simple associative algebras of characteristic $p$_ Trans. Am. Math. Soc. 80 (1955), 135–147.
  * [5] K. Hoechsmann, _Simple algebras and derivations_ , Trans. Amer. Math. Soc. 108 no. 1 (1963), 1–12.
  * [6] L. Juan and A. R. Magid, _Differential central simple algebras and Picard-Vessiot representations_ , Proc. Amer. Math. Soc. vol. 136, no. 6 (2008), 1911–1918.
  * [7] I. Kaplansky, An introduction to differential algebra, Actualités Sci. Ind., 1251, Publ. Inst. Math. Univ. Nancago, vol. 5, Hermann, Paris, 1957.
  * [8] A. Kulshrestha and V. R. Srinivasan, _Quaternion algebras with derivations_ , J. Pure Appl. Algebra vol. 226 issue 2 (2022), 1–14.
  * [9] M. Rosenlicht, _Integration in finite terms_ , Am. Math. Mon. vol. 79 issue 9 (1972), 963–972.


# NN scattering with $N\Delta$ coupling: Dibaryon resonances without
”dibaryons”?

J.A. Niskanen  Helsinki Institute of Physics, PO Box 64, FIN-00014 University
of Helsinki, Finland<EMAIL_ADDRESS>
###### Abstract

It is shown by coupled-channel $NN$ scattering calculations that intermediate
two-baryon $N\Delta$ states produce resonance-like structures in some isospin
one states, often interpreted as more exotic manifest six-quark states.
Optimal conditions for such appearances involve a decrease of the orbital
angular momentum in the transition $NN\rightarrow N\Delta$. Detailed complex
phase shift results are presented for low isospin one partial waves up to
$J=4$, where $N\Delta$ excitation is still found comparable to or even larger
than one pion exchange. Actually, its importance persists at least to $J=6$ or
further.

##  I Introduction

Shortly after the appearance of the quark model in 1964 gellmann  it was
suggested dyson ; jaffe  the possible existence also of states with six quarks
(somewhat like six-quark bags). Although this idea concerned only colour
singlet involving all three quark flavors (known at the time), as a follow-up
it was further suggested that this kind of objects would play an important
role as resonant $s$-channel intermediate states in the interaction dynamics
of two nucleons (for a balanced review see e.g. Ref. dibaryn ). Searches have
since continued for indications in experimental data (for an early review in
the $NN$ sector see e.g. Ref. yokosawa ). An interest in such activity has
been sustained over decades and got new wind from observations at the
WASA@COSY detector of Forschungszentrum Jülich in two-pion production
reactions, where a resonant structure with $I(J^{P})=0(3^{+})$, called
$d^{\prime}(2380)$, was seen at 2380 MeV with the width of 70 MeV adlar106 .

However, there is another possibility for strong resonance-like energy
dependencies, namely two-pion exchange with the excitation of an intermediate
state with a $\Delta(1232)$ resonance. It was realized early in pion physics
that pion-nucleon interaction, both elastic scattering and reactions, in the
region of a few hundred MeV is dominated by this spin-$\frac{3}{2}$ and
isospin-$\frac{3}{2}$ particle watson . Pion exchange can generate such an
excitation, transforming one of the nucleons or both into a $\Delta$, and its
consequent decay may end up in pion production reaction or (if reabsorbed by
the other nucleon) yield strong attractive two-pion exchange, which is not
reducible to two-nucleon intermediate states appearing in iteration of a
normal $NN$ potential green ; arenhovel . This mechanism acts in the same
energy region as several of the conjectured dibaryons and may compete with
their effects or be interpreted as those. In the 1970’s and 1980’s the
attractive effect was included for example in the so called Bonn boson
exchange potentials machleidt ; elster , well fitted to phase shift analyses
in the elastic energy region. The aim of the present paper is to elaborate the
coupled channels more in the inelastic region.

The next section sets out the basic formalism for generating the $N\Delta$
configurations in the isovector $NN$ scattering and discusses their main
properties. The numerical results for $NN$ phase shifts are presented and
discussed then in Sec. III.

##  II $N\Delta$ states

In this work (as in its predecessors since Ref. ppdpi ) the $N\Delta$
component is generated from the incident $NN$ wave by a coupled system of
Schrödinger equations involving as the new element the transition potential
$NN\leftrightarrow\Delta N$ due to pion exchange

$V_{\rm
tr}=\frac{\mu}{3}\frac{ff^{\ast}}{4\pi}{\bm{T}}_{1}\cdot{\bm{\tau}}_{2}\,[S_{12}\,V_{\rm
T}(r)+{\bm{S}}_{1}\cdot{\bm{\sigma}}_{2}\,V_{\rm SS}(r)]+(1\leftrightarrow
2)\;.$ (1)

Here $f^{2}/4\pi=0.076$ and $f^{\ast 2}/4\pi=0.35$ are the pion (pseudovector)
coupling constants to the nucleon and $N\leftrightarrow\Delta$ vertices, and
$\mu$ is the pion mass. $\bm{S}$ and ${\bm{S}}^{\dagger}$ are the transition
spin operators brown  (analogously $\bf T$ and ${\bm{T}}^{\dagger}$ for the
isospin) defined e.g. by their reduced matrix elements
$<\Delta||{\bm{S}}||N>=2$ in the convention of Refs. edmonds ; deshalit ;
messiah . The standard tensor operator

$S_{12}=3{\bm{\sigma}}_{1}\cdot\hat{\bm{r}}{\bm{\sigma}}_{2}\cdot\hat{\bm{r}}-{\bm{\sigma}}_{1}\cdot{\bm{\sigma}}_{2}$
(2)

can be generalized to

$S_{12}=3{\bm{S}}_{1}\cdot\hat{\bm{r}}{\bm{\sigma}}_{2}\cdot\hat{\bm{r}}-{\bm{S}}_{1}\cdot{\bm{\sigma}}_{2}+(1\leftrightarrow
2)$ (3)

in the case of $NN\rightarrow\Delta N$ transition. The radial functions are

$V_{\rm SS}(r)=\exp(-\mu r)/(\mu r)\,,\qquad V_{\rm T}(r)=[1+3/(\mu r)+3/(\mu
r)^{2}]\,V_{\rm SS}(r)$ (4)

familiar from pion exchange between two nucleons. The $NN$ equation is then
coupled by $V_{\rm tr}$ to the $N\Delta$ equations including (in addition to
their possible interactions not discussed here) the $\Delta-N$ mass difference
$\Delta M$ and (above the inelastic threshold) the effect of the width
$-i\Gamma/2$. This procedure yields an admixture of $N\Delta$ configurations
in principle on the same footing as the $NN$ (apart from the mass difference)
into the wave function, which must necessarily be there due to the nucleon and
and pion coupling to the $\Delta$. The imaginary term from the width gives
naturally also inelasticity above the pion threshold. As a sideline it is
important to note that the width, extending to infinity, causes the wave
function to shrink making the $N\Delta$ channel asymptotically closed, not
unlike a bound state. The method was applied relatively successfully to
calculate both total and differential cross sections and analyzing powers
ppdpi  and other spin observables spinobs ; deutpol  for the reaction
$pp\rightarrow d\pi^{+}$. It may be useful to have in mind that the background
of this paper lies in this reaction, and some argumentation to follow largely
arises and and is adopted from that work and its later developments.

At this stage it is also worth commenting that, following the suggestion by
Durso et al. durso , originally two time orderings taking into account the
different $\Delta$ mass was used for the range in Ref. ppdpi . However,
inclusion of all time orderings gives back the “normal” static OPE range
$1/\mu$ in Eq. (4) used later since Ref. improv . The transition potential
presently also includes $\rho$ exchange and dipole form factors with cut-off
masses $\Lambda_{\pi}=1000$ MeV and $\Lambda_{\rho}=1050$ MeV. Interfering
destructively with pion exchange, the former acts in the dominant tensor term
similarly to a long range cut-off. For practical purposes, it is enough to get
the strength of the tensor part from the top of the peak of the $pp\rightarrow
d\pi^{+}$ total cross section hoftiezer  as the only adjusted parameter
relevant for $N\Delta$ and arranged with the above form factors, well in
accord with the nucleon size and mesonic corrections to the free $\Delta$
width pindcpl . The spin-spin part of the transition potential is a much
weaker effect.

Heavy mesons like the $\rho$ may be frowned in more recent chiral effective
field theories (EFT), composed as systematic pion exchange perturbation series
including $\pi N$ and $\pi\pi$ contact terms with few low-energy parameters.
(See e.g. eft  for recent progress and extensive bibliography.) However, so
far this approach has been limited to lower $NN$ momenta below inelasticity.
Therefore, a more phenomenological approach consisting of fewer plausible
iterative diagrams, but extending higher in energy, may have value and
interest. Concerning the role of $\rho$ exchange, it might be of some interest
in future work to compare spin observables of $pp\rightarrow d\pi^{+}$
including and omitting $\rho$ exchange to test its justification, since,
contrary to the tensor, its spin-spin term adds constructively with pion
exchange, which should give differing spin dependence.

Apart from the explicit generation and use of the $N\Delta$ wave function, as
iterated this exchange produces a strong attraction at least below the formal
$N\Delta$ threshold, which must be accounted for in the overall effective two-
nucleon force. Namely, starting with phenomenological data-fitted $NN$
potentials, such as e.g. the Reid potential reid , this additional effect must
be subtracted to avoid doubly counting its influence. In Ref. ppdpi  this was
done by assuming a closure approximation for the iterated transition and using
additive repulsion

$V_{\rm clos}(r)=[V_{\rm tr}(r)]^{2}/\Delta E$ (5)

with the energy denominator $\Delta E$ suitably adjusted at each energy. It
should be stressed that at that stage the purpose was not to calculate or
publish $NN$ scattering and its phases but only to generate dependable wave
functions for computing the pion production matrix elements necessary in the
reaction $pp\rightarrow d\pi^{+}$. It is still of interest to note that to
produce phase equivalent interactions with and without the $N\Delta$ admixture
by this procedure, $\Delta E$ predicted a rotational series dibarmass  $\Delta
E\approx{\rm const}+40L_{N\Delta}(L_{N\Delta}+1)$ MeV for the effective
intermediate state (isospin one “dibaryon”) masses, well in accord with the
contemporary experimental situation yokosawa . This meant that the $N\Delta$
state wave function acted as if it were approximately concentrated at the
distance of about 1 fm.

Later the double-counting modification has been performed by inclusion of
energy-independent short and intermediate range Yukawa terms csb ; width  into
the Reid potential reid  and its extension to higher partial waves with $J=3$
day . This will be the topic of this paper: to get from a seemingly energy-
independent potential strong energy dependencies, often attributed to and
parametrized and fitted with explicitly energy-dependent “dibaryons”.

Also it is worth realizing that, as different as the above described methods
to deal with double counting may appear, their principal application to the
reaction $pp\rightarrow d\pi^{+}$ is relatively independent of these details.
In particular, its total cross section is practically independent of the $NN$
phase shifts. However, dibaryon fits have often only considered this
observable. For example Kamo and Watari kamo  fitted the total cross section
of $pp\rightarrow d\pi^{+}$ with three dibaryons (i.e. six parameters),
assuming a constant background from the $\Delta$. However, the $N\Delta$
threshold effect does not behave like a constant background and, in fact, the
coupled $N\Delta$ model reproduced both the total cross section and also
differential and spin observables quite well, even better than Ref. kamo ,
particularly after some improvements in the treatment of pion $s$-wave
rescattering and the $N\Delta$ width improv ; width . (To my knowledge
dibaryon fits have not often been applied to differential observables in pion
production, though Ref. kamo  does give predictions or fits with also these.)

It might appear that, in the second order iteration of pion exchange in $NN$
scattering and first order in pion production, the effect of the intermediate
$\Delta$ could be obtained simply by inserting into the perturbation theory
energy denominator the $\Delta\\!-\\!N$ mass difference $\Delta M$ and width
giving $E-\Delta M+i\Gamma/2$. Although this looks like a trivial way to get a
resonant structure, however, it is open to some questions. It was pointed out
above that already the extremely phenomenological numerical calculation
dibarmass  indicates some more subtle structure in the energy denominator,
e.g. from the centrifugal effects discussed in more detail in Ref. centrif .
This is due to the fact that in asymptotically closed $N\Delta$ channels the
expectation value of the centrifugal potential is well defined and quantized
as argued above. Furthermore, the orbital angular momentum of the excitation
can be $L_{NN}\pm 2$ in addition to $L_{NN}$, obviously favoring $L_{NN}-2$
and strongly suppressing $L_{NN}+2$. Even the definition of the width in the
two-baryon system needs special attention. Quite obviously it cannot be the
free width improv ; width , but consideration of the different parts of the
$N\Delta$ kinetic energy expectation values should be addressed apart from the
internal $\Delta$ particle excitation.

Moreover, there is a perhaps deeper problem of the applicability of
perturbation theory in this context: Perturbation theory needs the matrix
element of the perturbation between the initial and intermediate (or final)
state. But what is the “unperturbed” wave function of the $N\Delta$, which has
yet to be computed to get this matrix element? This is a problem in old
perturbation calculations such as Ref. brack , which can reproduce the total
cross section of $pp\rightarrow d\pi^{+}$ well by straightforward substitution
of the above energy denominator but not the differential one or spin
observables chai ; maxwell1 ; maxwell2 .

The question of the applicability of perturbation theory may not be as
relevant in $NN$ scattering for the fits with hypothesized dibaryons. The
resonant structure as substituted by hand is then trivial and automatic. The
positions and widths are free to choose in this procedure. With the $N\Delta$
coupled channels there is no such luxury for these. The position and magnitude
come directly from the coupled Schrödinger equation as a threshold cusp (with
the $\Delta M$ and centrifugal forces as necessary and unavoidable parts of
the equation), and the width is calculated self-consistently for each channel
along the lines of Ref. width . Although in the coordinate space equations the
resonant structure is not manifest, in the equivalent momentum (or energy)
representation, of course, it must be and, after calculation, becomes visible
for observables also in the coordinate representation. So, from this on we’ll
stick to this representation, which is also the most transparent framework to
incorporate the necessary centrifugal barriers.

##  III Results

###  III.1 ${}^{1}D_{2}$ wave

Figure 1: The ${}^{1}D_{2}$ phase shift. The solid line is the calculation
with ${}^{5}S_{2}(N\Delta)$, ${}^{5}D_{2}(N\Delta)$ and ${}^{5}G_{2}(N\Delta)$
admixtures and the filled circles and triangles are the experimental real and
imaginary component of the phase shift extracted from the energy-dependent fit
to $pp$ data arndt , respectively.

First Fig. 1 shows the behavior of the ${}^{1}D_{2}$ phase shift $\Re\delta$
together with its inelasticity $\Im\delta$ including the possible
${}^{5}S_{2}(N\Delta)$, ${}^{5}D_{2}(N\Delta)$ and ${}^{5}G_{2}(N\Delta)$
components in the calculation. Here the data have been fitted below
inelasticity (below 300 MeV), first starting with the phenomenological Reid
potential reid  and then making the modification csb  to avoid the double
counting of the $N\Delta$ as explained in the previous section.111One can
follow these steps for the ${}^{3}F_{3}$ wave in detail from Fig. 6 of Ref.
width . Then the resonance-like energy dependence will arise above 400 MeV due
to the coupling to the isobar excitations, most pronouncedly to
${}^{5}S_{2}(N\Delta)$, not by any further fitting. It can be seen that,
compared with the energy-dependent analysis of $pp$ scattering arndt , above
300 MeV the results even slightly exaggerate resonant behavior still after the
smoothing effect of the width inclusion. The structure follows clearly
perturbative argumentation: attraction below the threshold at about 600 MeV
and then a change of the sign into repulsion.

In general, the $S$-wave $N\Delta$ component is the most important in
reactions in this energy regime and it is favored also by the decrease of the
angular momentum dibarmass  in the $D$\- to $S$-wave transition as noted in
Sec. II. Incidentally, at distances most relevant for pion exchange this
decrease is associated with an effective decrease in the centrifugal potential
comparable to the threshold $\Delta M$, making the $N\Delta$ excitation
effectively rather degenerate with the initial $NN$ state, and therefore
strongly boosting this transition.

The $p$-wave pions associated with the initial ${}^{1}D_{2}(NN)$ state account
for about 80% of the $pp\rightarrow d\pi^{+}$ total cross section in the peak
region around 580 MeV (laboratory energy) hoftiezer , and this affluence
arises mainly from the ${}^{5}S_{2}(N\Delta)$ admixture. The missing
attraction at the highest energies might possibly be achieved by inclusion of
some higher nucleon resonance ($N(1520)$ or $N(1535)$) or double $\Delta$
excitation.

###  III.2 ${}^{3}F_{3}$ wave

Figure 2: As Fig. 1 for the ${}^{3}F_{3}$ phase shift with
${}^{5}P_{3}(N\Delta)$, ${}^{5}F_{3}(N\Delta)$ and ${}^{3}F_{3}(N\Delta)$
admixtures. An earlier calculation with only the ${}^{5}P_{3}(N\Delta)$
component was presented in width  emphasizing its role as a candidate
dibaryon.

Fig. 2 shows the second important dibaryon candidate ${}^{3}F_{3}$ starting
from the $NN$ potential extension day  of the Reid potential. Here the
included $N\Delta$ configurations are ${}^{5}P_{3}(N\Delta)$,
${}^{5}F_{3}(N\Delta)$ and ${}^{3}F_{3}(N\Delta)$ with the first one being
dominant due to the decrease of the orbital angular momentum and consequently
the smaller centrifugal potential in the $N\Delta$ system favoring it as
argued already in Ref. dibarmass . This plot is very similar to Fig. 6 of Ref.
width  where, however, only the “dibaryonic” state ${}^{5}P_{3}(N\Delta)$ was
included to emphasize its special influence. The higher angular momentum
states give minor attraction of 0.5–1.0 degrees above 500 MeV and, in
comparison with Ref. width , perhaps one might also argue a very slight
broadening upwards in energy due to the stronger centrifugal effect in them.

This wave was found absolutely essential in successful production of
differential observables in $pp\rightarrow d\pi^{+}$ ppdpi ; improv  and it
also accounts for 10-20% of its total cross section in the peak region.
Probably the two dibaryonic states discussed so far dominate also its isospin-
cousin component in $pp\rightarrow np\pi^{+}$ at intermediate and high
energies. Furthermore, ${}^{3}F_{3}$ appears to give the largest individual
contribution at least to the total cross section of the reaction
$pp\rightarrow pp\pi^{0}$, with the final isotriplet $NN$ state, although
overall this is significantly smaller than the isosinglet.

###  III.3 ${}^{1}G_{4}$ wave

The next potential $N\Delta$ dibaryon-like state, as argued in Refs. dibarmass
; centrif  and also experimentally suggested in Yokosawa’s review yokosawa ,
${}^{1}G_{4}(NN)$ is discussed in Fig. 3 with the ${}^{5}D_{4}(N\Delta)$,
${}^{5}G_{4}(N\Delta)$ and ${}^{5}I_{4}(N\Delta)$ admixtures. Again,
computationally a clear broad resonance-like maximum is seen in the phase
shift at about 800 MeV laboratory energy, while in the data the peak is
missing. Also inelasticity is overestimated in the calculated
scattering.222For small inelasticity in the presentation of this work as
$\exp(-2\,\Im\delta)$ in the $S$-matrix the imaginary part of the phase
$\Im\delta\propto\rho^{2}$ can be more sensitive to variations than the $\rho$
parameter of Ref. arndt .

Figure 3: Solid curves and data as in Fig. 1 for the ${}^{1}G_{4}$ phase shift
including ${}^{5}D_{4}(N\Delta)$, ${}^{5}G_{4}(N\Delta)$ and
${}^{5}I_{4}(N\Delta)$ admixtures. Dashed: OPE used for the diagonal $NN$
potential but with $N\Delta$ admixture. Dashdot: only elastic OPE used
overall.

Except for the ${}^{1}D_{2}$, in two-nucleon scattering the Argand diagrams of
these states do not present any clear resonance behavior from phase shifts.
Only the ${}^{1}D_{2}(NN)$ amplitude crosses the imaginary axis slightly above
700 MeV (lab energy) but with rather strong inelasticity width . On the other
hand, the ${}^{3}F_{3}$ amplitude remains on the left-hand side of the
imaginary axis at all energies width . The ${}^{1}G_{4}$ state has a
calculated bump of $\Re\delta$ not seen in the data. However, although the
latter $NN$ amplitudes may not show particular drastic features in actual pure
phase shift Argand diagrams, interestingly the $N\Delta$ wave functions do
suggest resonant structures, both in magnitude and phase displayed for example
in two Argand-like diagrams designed for the ${}^{5}D_{4}(N\Delta)$ component
in Ref. centrif , which may show up in reactions sensitive to $N\Delta$ and
involving overlaps with $N\Delta$ configurations. Apparently the maximum in
the phase shift here is due to constructive interference of the $N\Delta$
effect with the attractive background from the $NN$ diagonal potential (a
modified ${}^{1}D_{2}$ Reid potential very similar to that used for Fig. 1 was
employed).

To study a little bit further this interplay in the ${}^{1}G_{4}$ wave, the
dashed curves utilize instead of the modified Reid potential as the diagonal
$NN$ interaction the one pion exchange potential (OPE), expected to dominate
peripheral waves. The overall phase $\Re\delta$ for this is slightly more
attractive, increasingly so for higher energies, and the maximum moves to
higher energy. This probably follows from the lack of the short range
repulsion present in the phenomenological potential. But more importantly, it
is very interesting and intriguing that the $N\Delta$ excitation is still very
active at such a high value of $L$ (dashed curves vs. dash-dot). This is
probably due to the long (OPE) range of the transition potential itself and
the favourable decrease of the centrifugal repulsion in the change
${}^{1}G_{4}(NN)\rightarrow\ ^{5}D_{4}(N\Delta)$. It might be argued that in
the asymptotically free $NN$ wave that repulsion vanishes at large distances.
However, to have any strong interaction effect at all, the nucleons must be
close enough to feel the action of OPE, in the region where the centrifugal
$NN$ repulsion of the $G$-wave is in the GeV magnitude. That is also
essentially the range of the $N\Delta$ wave function centrif . Again, due to
the difference in the centrifugal potentials the $N\Delta$ admixture may not
be practically an excitation at all. The coupling effect looks even more
drastic when comparing with the pure OPE in the $NN$ wave without the
$\Delta$-resonance generation (dash-dot curve). Clearly, the $G$-wave is not
peripheral enough for the pure OPE to dominate alone at least above 400 MeV.
Below the inelastic threshold 300 MeV, the result here (dashed vs. dash-dot)
is numerically in perfect agreement with the EFT calculation eft  in next-to-
next-to-leading order (NNLO) vs. leading order (LO).

The dipole form factor with the range parameter $\Lambda_{\pi}=1000$ MeV has
been included in the OPE’s all the time.

###  III.4 $J=0$ waves

Figure 4: $J=0$ phase shifts: solid curves ${}^{1}S_{0}$ and dash-dot
${}^{3}P_{0}$. The circles show the analysis of Arndt et al. arndt  for
${}^{1}S_{0}$ and the squares for ${}^{3}P_{0}$. The filled symbols present
the real parts and the hollow ones the imaginary parts. For clarity, all
imaginary parts $\Im\delta$ have been multiplied by 10. The dotted curve shows
the ${}^{1}S_{0}$ phase calculated with the double-counting correction but
without the $N\Delta$ coupling.

After these somewhat peripheral waves it may be in order to study similarly
the opposite situation with the $J=0$ states, where the transition into
dibaryonic states with decreasing $L$ is not possible. Quite contrary, the
${}^{1}S_{0}$ state is coupled only to the $D$-wave ${}^{5}D_{0}(N\Delta)$,
which should involve some 300 MeV extra centrifugal energy (plus some radial
addition, too centrif ), bringing its effective excitation energy very high.

Fig. 4 presents the results for this wave by circles and solid curves and for
${}^{3}P_{0}$ by squares and dash-dot curves. A good agreement can be found
for the real part in both waves up to 800 MeV (well, up to 500-600 MeV for the
${}^{3}P_{0}$ wave). In the $S$-wave the $N\Delta$ coupling is complemented
with the repulsive double counting correction $770\exp(-4\mu r)/(\mu r)$ MeV
to the Reid potential reid  \- the phase has been fitted between 50 and 300
MeV (the solid curve vs. filled circles). However, consistently with the
arguments given earlier, the isobar excitation produces very little
inelasticity - hardly distinguishable from the energy axis even after
multiplying by ten. This is due to the minuteness of the
${}^{5}D_{0}(N\Delta)$ width calculated with the prescription of Refs. improv
; width . Also the ${}^{1}S_{0}$ contribution to $pp\rightarrow d\pi^{+}$
cross section is very small, in part due to destructive interference between
the $NN$ and $N\Delta$ components. The same seems to be true for
$pp\rightarrow pp\pi^{0}$ $s$-wave production (with ${}^{3}P_{0}$ final state
nucleons). To make the imaginary parts of the $NN$ phase shifts comparable in
the scale, all of them have been multiplied by 10.

As a sideline, it is worth noting that, as a check, it was verified that in
the low-energy limit the phase turns down at about 1 MeV laboratory energy
yielding for the singlet scattering length a negative value $a_{\rm
S}\approx-38$ fm. (Of course, in this context only the sign matters.)
Therefore, the possible presence of a bound state and the ensuing extra node
in the $NN$ wave function should not be the source for the smallness of the
width. As another sideline one might also note that, in spite of the
${}^{5}D_{0}(N\Delta)$ state being unfavored in reactions such as
$pp\rightarrow d\pi^{+}$ (due to destructive $NN$ and $N\Delta$ interference),
its attractive effect may, however, be surprisingly strong. A quick
calculation indicated that, with the direct $NN$ interaction totally
discarded, the iteration of the $N\Delta$ transition gives a positive
scattering length $a_{\rm S}\approx 2.3$ fm suggesting that the $N\Delta$
component alone could support a bound state with a binding energy of $E_{\rm
B}\approx 8$ MeV. To further illustrate the strength and importance of the
$N\Delta$ configurations even in elastic scattering, the dotted curve still
shows the ${}^{1}S_{0}$ phase shift using the above double-counting repulsion
correction but without including the ${}^{5}D_{0}(N\Delta)$ coupled channel.
The gap between the two results (solid vs. dotted) is massive and is one way
of viewing the significance of the coupling effect.

For the ${}^{3}P_{0}(NN)$ wave (coupled to ${}^{3}P_{0}(N\Delta)$), also with
an attempted fit up to 300 MeV, $\Re\delta$ agrees well with data up to about
600 MeV. Its inelasticity agrees roughly with data in magnitude but would peak
at higher energy. As noted before, this paper includes the inelasticity as the
imaginary part $\Im\delta$ of the phase shift, which causes a different energy
dependence from the $\rho$ parameter of arndt  for small values. It may be of
some interest to note that, especially for the ${}^{3}P_{0}$ state, also the
large real part of the phase shift can have a strong effect in the
inelasticity (with the notation of Ref. arndt )

$1-\eta^{2}=\frac{4\tan^{2}\rho}{(1+\tan^{2}\rho)^{2}+\tan^{2}\delta}\;,$ (6)

bringing it down in spite of the seemingly very large values of $\rho$.

The $J=0$ state pion reactions are somewhat superficially related by their
possibility to $s$-wave pion production by transitions
${}^{1}S_{0}\rightarrow\\!^{3}P_{0}s$ and
${}^{3}P_{0}\rightarrow\\!^{1}S_{0}s$. However, an explicit calculation
suggests these reaction channels to be significantly smaller than production
in other $NN$ states discussed here. Further, in their comparison, the former
is nearly an order of magnitude smaller than the latter.

###  III.5 ${}^{3}P_{1}$ wave

Figure 5: ${}^{3}P_{1}$ phase shift: solid curves as described in Ref. width
and earlier figures. The circles and triangles show the analysis of Arndt et
al. arndt . The dot-dash curve shows the modification described in the text,
and the dotted one is 100 times the sole contribution from $pp\rightarrow
d\pi^{+}$ to inelasticity, ($s+d$)-wave pions.

Another $NN$ wave without a possibility of the favorable decrease of $L$ in
the $N\Delta$ transition is ${}^{3}P_{1}$, which has particular interest
because of its significance in pion production. Especially, due to $s$-wave
pion rescattering in $s$-wave pion production, it dominates $pp\rightarrow
d\pi^{+}$ and $pp\rightarrow np\pi^{+}$ at threshold. Fig. 5 shows the results
for this wave (solid line) with a global fit to the phases of Ref. arndt  deep
into pion production region. Generally there is a fair agreement of
$\Re\delta$ with the smooth data behavior, quite enough for the originally
intended calculation of pion production reactions and isospin mixings.
However, one may envision a tendency to slight similarity with the
${}^{3}F_{3}$ wave (Fig. 2). To emphasize this tendency one might try to
improve the already good fit by mimicking the curvature in the elastic region
below 300 MeV (dash-dot line). Rather clearly now the latter curve has as well
a wide shoulder in the energy region 600–800 MeV, similar to the case of
${}^{3}F_{3}$ also with $P$-wave $N\Delta$’s, but not really visible in the
data for this wave.

The data indicate small and remarkably constant inelasticity $\Im\delta\approx
4^{\circ}-5^{\circ}$ above 600 MeV, whereas the calculated results increase
rather linearly with energy exaggerating it above 600 MeV. It is probably
useful to remember the behavior of $s$-wave pion production in $pp\rightarrow
d\pi^{+}$. At threshold this $NN$ wave dominates, the cross section being
linear in the pion c.m. momentum $q_{\pi}$, that is square root of the excess
energy as is usual for the two-body final state phase space. However, this
two-body reaction dominates the inelasticity of ${}^{3}P_{1}(NN)$ only up to
420 MeV, but above this energy the scattering inelasticity calculated from the
absorptive width quickly takes over so that already by 500 MeV two-body pion
production $pp\rightarrow d\pi^{+}$ is less than 10% of the total. Moreover,
due to destructive interference between direct production and pion $s$-wave
rescattering the cross section from this amplitude virtually vanishes around
600 MeV ppdpi . Above that energy direct production from the axial charge part
of the pion-baryon vertex wins and the amplitude changes sign, but the
$s$-wave pion cross section remains still very small as seen already in Ref.
ppdpi . This wave is illustrated by the dotted curve in Fig. 5 (multiplied by
the factor of 100, including also $d$-wave pions). In the width prescription
of Ref. width  the phase space integral over intermediate $N\Delta$ momenta is
augmented by the $pp\rightarrow d\pi^{+}$ cross section.

To check the validity of this procedure and to avoid problems with possible
double counting in explicit consideration of this reaction the dash-dot curve
now includes only the phase-space integrated width in the calculation of the
scattering inelasticity (and the elastic part of the phase shift). The
$pp\rightarrow d\pi^{+}$ cross section is then added explicitly by hand to
yield the total absorption cross section and finally to calculate the
inelastic phase shift for the dash-dot curve from

$1-\eta^{2}=\frac{k^{2}}{\pi(2J+1)}\;\sigma_{\rm abs}$ (7)

with $k$ the nucleon c.m. momentum. In spite of the dominance of the
$d\pi^{+}$ final state at threshold (due to different phase space), its
overall effect is very little for the present purpose, since actually also the
two-body reaction cross section is very small there. Above 600 MeV the
inelasticity from this pion production is less than 1% of the scattering
inelasticity so that its effect is very small whether it is included in the
width (imaginary potential) or explicitly added afterwards by Eq. (7) into the
total absorption cross section to calculate $\Im\delta$. Namely the dotted
curve shows 100 times its contribution. It might still be noted that except
for the minimum around 600 MeV this ${}^{3}P_{1}$-wave contribution to
$pp\rightarrow d\pi^{+}$is still definitively dominated by $s$-wave pions over
the also possible $d$-wave.

Is there a way to obtain the observed constancy of the inelasticity showing in
the data? The structure of the ${}^{3}P_{1}$ contribution to $pp\rightarrow
d\pi^{+}$ is intriguing as a total change from monotonously growing cross
section at threshold. In spite of this, the view of the $pp\rightarrow
d\pi^{+}$ possibly changing the trend of $\Im\delta$ towards a more constant
value seems hopeless due to the smallness of its ${}^{3}P_{1}$-wave
contribution. However, quite probably the importance of $s$-wave pion
rescattering at threshold also holds in the isospin related three-particle
final state in $pp\rightarrow np\pi^{+}$. Furthermore, also the destructive
competition between the axial charge contributions could remain true producing
a similar strong pion minimum for wave functions at any given final $NN$
energy. This might reduce the smeared final state phase space integral of the
cross section to resemble better the constant inelastic data. True enough, in
the three-body $pp\rightarrow pp\pi^{0}$ reaction there is a strong minimum in
$s$-wave production around 550 MeV (however originating from the initial
${}^{3}P_{0}(NN)$ state), giving considerable constancy of the cross section
up to near 600 MeV pppi0 ; meyer ; impact  (though the cross section in this
final state isospin arrangement is much smaller). The present phase
calculation, taking the width from integration over intermediate $N\Delta$
momenta (and using as input for this the free $\Delta$ width) to simulate the
three-body reaction width , does not have these effects in. It does not
require much imagination to consider how much the general outlook and the
average of the dotted curve (Fig. 5) would differ, if the minimum caused by
the destructive interference would be smeared over in the phase space
integral. However, it would require a totally independent and different
calculation of the three-body reaction. Some hints in this direction may be
found in Ref. budzanowski .

###  III.6 ${}^{3}P_{2}-\\!^{3}F_{2}$ waves

Figure 6: ${}^{3}P_{2}-\\!^{3}F_{2}$ phase shifts: solid curves the real and
imaginary parts of the $P$-wave shift, the dashed curve for the $F$ wave.
Respectively, the circles and squares show the analysis of Arndt et al. arndt
transformed to the Stapp parametrization (8) (filled symbols for $\Re\delta$,
hollow symbols for $\Im\delta$). The $F$-wave experimental inelasticity is
multiplied by ten.

The tensor coupled case ${}^{3}P_{2}-{{}^{3}F_{2}}$ is significantly more
involved arndt . In the elastic region it is standard to express the $2\times
2$ $S$-matrix in the Stapp form stapp

$S=\begin{pmatrix}\cos(2\epsilon)\exp(2i\delta_{-})&i\sin(2\epsilon)\exp[i(\delta_{-}+\delta_{+})]\\\
i\sin(2\epsilon)\exp[i(\delta_{-}+\delta_{+})]&\cos(2\epsilon)\exp(2i\delta_{+})\end{pmatrix}\;,$
(8)

where, in addition to the phase shifts $\delta_{\pm}$ in the respective
$\pm=J\pm 1$ $NN$ channels treatable as before in this paper, the mixing
parameter $\epsilon$ opens the possibility of their mixing. In this paper
inelasticity has been implemented by complex phase shifts, which automatically
arise for complex potentials. To avoid difficulties in analyses Ref. arndt
accommodates inelasticity instead by adding a symmetric imaginary $2\times 2$
component into the otherwise real elastic $K$-matrix

$K=i(1-S)(1+S)^{-1}.$ (9)

Thus, keeping this as the real part of the $K$-matrix, it is generalized to
the complex form

$K=\Re
K+i\,\begin{pmatrix}\tan^{2}\rho_{-}&\tan\rho_{-}\tan\rho_{+}\cos\phi\\\
\tan\rho_{-}\tan\rho_{+}\cos\phi&\tan^{2}\rho_{+}\end{pmatrix}\;.$ (10)

However, I proceed in the present work by forcing inelasticity into the Stapp
form. Then, even the mixing parameter may become complex, in contrast to the
choice of arndt .

In the phases of Fig. 6 the elastic region fit of $\Re\delta$ continues as
essentially perfect somewhat above 400 MeV, but above 500 MeV deviates
strongly especially for the ${}^{3}P_{2}$-wave being far too attractive (solid
curve vs. full circles). However, inelasticity in this incident wave is
excellent (the imaginary part of the phase, solid curve vs. hollow circles).
The experimental phase parameters are extracted from the data of Ref. arndt
using their prescription for transforming the $K$-matrix to the $S$-matrix
elements. In contrast, for the incident ${}^{3}F_{2}$-wave the elastic phase
remains reasonable, but the imaginary part is tenfold as compared with the
data. However, one should note the earlier warning about the different
sensitivity of the two representations for small parameters ($\Im\delta$ vs.
$\rho$) in this case.

Figure 7: The mixing angle $\epsilon$ as defined by Stapp et al. stapp
extended to inelastic region. The circles ($\Re\epsilon$) and squares
($\Im\epsilon$) show the analysis of Arndt et al. arndt  forced into the Stapp
form. The solid curves show the results originating from the incident $P$-wave
and dashed curves from the $F$-wave.

Fig. 7 shows the $J=2$ mixing parameter $\epsilon$. In form $\Re\epsilon$ is
very reasonable in comparison with data (the lower curves vs. circles).
However, the imaginary part (squares) may be surprising, not present in the
original definition (10) of Ref. arndt . Nevertheless, from the inverse of Eq.
9

$S=(1+iK)(1-iK)^{-1}$ (11)

one can get the off-diagonal elements related by (Eq. (7) of Ref. arndt )

$S_{0}=2iK_{0}/D_{K}\;,$ (12)

where $S_{0}$ (resp. $K_{0}$) is the off-diagonal element of the $S$\- and
$K$-matrix and $D_{K}$ is the determinant of $(1-iK)$. By definition (5) of
Ref. arndt  above inelastic threshold $K$ acquires both real and imaginary
parts, and it is perfectly plausible that the mixing parameter $\epsilon$ is
complex. It is just a consequence of the different definitions, and
$\Im\epsilon$ may be rather directly (and simply) linked to $\rho$ at least
for small mixing angles. In comparisons between theory and experiment, the
difference between the upper curves and the squares is about the most relevant
at this point in consideration of how inelasticity is dynamically embedded in
theory and shows up in $\Im\epsilon$.

There is another perhaps strange point of interest and suspect. Numerically it
was found that the exact values of $\epsilon$ depend on which of the two waves
is the incident state (see solid vs. dashed curves in Fig. 7). This means that
the $S$-matrix is not perfectly symmetric as normally expected.333The
$K$-matrix of Ref. arndt  defined in Eq. (10) is symmetric by construction.
Apparently this feature has to do with the numerical finding reported earlier
in Ref. width  that, in addition to the rather natural dependence on the
relative orbital angular momentum between the $\Delta$ and the nucleon
themselves in the intermediate state, also the external incident $NN$ angular
momentum matters in the calculation and the value of the effective $N\Delta$
width. As noted already in Ref. width , the treatment of inelasticity did not
start from explicitly Hermitian first principles, but used existing
inelasticity (width) of the $\pi N$ resonance, the $\Delta$, as the starting
assumption. However, this may be a practicable procedure for implementing
intuitively apparently physical effects in the system. Thus there would be
somewhat different absorption for the two different initial waves. If this
difference was taken off, the $S$-matrix came out as symmetric (as in the
formulation of Ref. arndt ). Also, for the applicability of the theory it may
be happy that this effect is actually small as compared even with the (already
rather small) total values of $\epsilon$.

Figure 8: The inelasticity parameter $\rho_{\pm}$ as defined by Arndt et al.
arndt . The $P$-wave solid vs. circles, the $F$-wave dashed vs. squares. This
may be compared with the $\Im\delta$’s of Fig. 6 in the Stapp representation.

Finally, just for completeness and easier comparisons Fig. 8 presents the
inelasticity parameter $\rho$ as defined by Arndt et al. arndt  easily
obtainable by its definition (9) from the calculated $S$-matrix as

$\tan^{2}\rho_{-}=\Re(1+S_{+}-S_{-}-S_{-}S_{+}+{S_{0}^{-}}^{2})/D_{S}\;.$ (13)

Here, in an awkward notation $S_{0}^{-}$ is the off-diagonal $S$-matrix
element originating from the initial $-$ channel (respectively
$+\leftrightarrow-$) and $D_{S}$ is the determinant of $(1+S)$. The real phase
parameters $\Re\delta_{\pm}$ are practically the same in both representations,
Stapp or Arndt.

###  III.7 More peripheral waves

Figure 9: Solid curves and circles show the ${}^{1}I_{6}$ partial wave phase
from the coupled channels calculation and from arndt  (only $\Re\delta$ given
for data). The dashed curves and triangles present the ${}^{5}H_{5}$ wave, and
the dotted curves display the pure OPE result for both.

In Subsec. III.3 the persistent prominence of the attractive $N\Delta$
contribution even in the high ${}^{1}G_{4}$ partial wave is very suggestive
that this behavior may extend even higher. Fig. 9 presents results for two
more peripheral waves, making possible a comparison of ${}^{3}H_{5}$ and
${}^{1}I_{6}$ waves including OPE together with $N\Delta$ or OPE alone. The
dashed curves can be contrasted with the data of Arndt et al. arndt  in the
case of ${}^{3}H_{5}$. The model gives quite similar behavior as seen for
${}^{3}F_{3}$, only at about 300 MeV higher energy. The trend may also be seen
in the data (negative triangles), though the rise of the data at high energy
is more pronounced and pointing towards even higher energies. The predicted
inelasticity is also tolerable. The solid curve vs. circles, displaying the
real part of the phase shift for the ${}^{1}I_{6}$ wave, is perfectly
satisfactory with coupled channels. Also the minor imaginary part $\Im\delta$
is acceptable, though the data are negligibly small and not shown.

One can make a very interesting comparison with the pure OPE calculation
(dotted curves). Clearly at and above the nominal $\Delta$ threshold isobar
excitation is still well comparable with OPE in such high angular momentum
states. The effect is especially outstanding and manifest in the destructive
interference with the repulsive triplet OPE both in ${}^{3}F_{3}$ and
${}^{3}H_{5}$ (also with a hint in ${}^{3}P_{1}$), challenging the old wisdom
of the OPE dominance at high $L$.

Similarly to the ${}^{1}G_{4}$ wave, below 300 MeV the deviation from OPE to
full coupled channels (solid vs. dense dots for ${}^{1}I_{6}$ and dashed vs.
sparse dots for ${}^{3}H_{5}$) is similar to the EFT results of Ref. eft  from
LO to NNLO.

##  IV Conclusion

The aim of the present paper has been to set up a framework to produce
resonant structures in $NN$ scattering without explicit input of energy
dependencies, for reference and benchmarks to confront and compare with
explicit energy dependent fits, such as those with hypothesized dibaryons.
Much of the necessary physics can be incorporated by a formulation of coupled
channels in terms of excitation of intermediate $N\Delta$ components. As
already seen at several points, this is not a new idea sugawara  and, in
principle, it has strong foundations in pion exchange: the pion coupling to
nucleons and $\Delta$’s is well established and should be reconciled parallel
with $NN$ scattering dynamics. Only after considering this is it meaningful to
involve other dibaryons.

The model shortly outlined in Sec. II is substantiated by references to older
work, in particular on pion production, which is a natural association and a
relative to the present work. The relation to “dibaryon” fits is first to
obtain as good a fit to the data as possible below inelasticity and then leave
the rest to depend on $N\Delta$ dynamics above the pion threshold with no
supplemental degrees of freedom. The desired features in the $\Delta$
excitation energy region are successfully predicted for the $NN$ waves which
have also the most important resonant amplitudes in pion production in
$pp\rightarrow d\pi^{+}$ (${}^{1}D_{2}$ and ${}^{3}F_{2}$, Figs. 1 and 2).

The more peripheral wave ${}^{1}G_{4}$ is seen to obtain still a major
contribution from the $N\Delta$ coupling, even overpowering OPE, normally
expected to overwhelmingly dominate (Fig. 3). This may be understood by the
interplay with centrifugal effects: effectively the intermediate
${}^{5}D_{4}(N\Delta)$ state is not energetically higher than the $NN$
centrifugal barrier within the OPE range and thus not particularly an
excitation. The calculated phase shift $\Re\delta_{4}$ displays rather a clear
and wide maximum not seen in the data. The inelasticity $\Im\delta_{4}$ is
somewhat overestimated in comparison with Ref. arndt , but this may be
exaggerated for small phases by the different parametrization.

In the $J=0$ states the phases come out rather good and also $\Im\delta$ for
the ${}^{3}P_{0}$ is reasonable though not perfect (Fig. 4). The ${}^{1}S_{0}$
inelasticity does not agree with data, being much smaller, essentially
negligible. Calculationally the reason is the very small calculated $N\Delta$
width and the large centrifugal barrier in the intermediate
${}^{5}D_{0}(N\Delta)$ state (both are interrelated with each other). Also the
phase shift analysis result for $\Im\delta$ is exceedingly small for such a
low partial wave, and one might again remind on the sensitivity of the
different parametrizations for small phases.

In the ${}^{3}P_{1}$ incident wave a slight shoulder is predicted in
$\Re\delta_{1}$, not seen in the smooth data (Fig. 5). Also the present
calculation overestimates the small and constant $\Im\delta_{1}$. However,
possible improvements are suggested in the text from explicit experience in
pion production (e.g. destructively interfering pion $s$-wave rescattering
missing in the present calculation). After all, the inelasticities are
predominantly pion production.

In the tensor-coupled system ${}^{3}P_{2}-\\!^{3}F_{2}$ the real phases are
reasonable, except for the $P$-wave $\Re\delta$ above 600 MeV, which is much
too attractive (Fig. 6). The real part of the mixing angle $\Re\epsilon_{2}$
compares reasonably with the data (which only give the real part) (Fig. 7).
However, the dynamic coupled channels model calculation also predicts an
imaginary part for this. An interesting peculiarity (due to the different
definitions) is getting an imaginary part for the Stapp mixing angle from the
real Arndt parameter (only about half of it being of dynamical origin, the
other half coming from the transformation between the two representations).
Another interesting, perhaps troublesome, result was that, parallel to earlier
findings improv ; width , also the angular momentum of the incident external
$NN$ state influences the widths calculated in the $N\Delta$ intermediate
states and the $S$-matrix becomes slightly asymmetric.

Furthermore, even in higher peripheral partial waves ${}^{3}H_{5}$ and
${}^{1}I_{6}$ the $N\Delta$ was seen to contribute to the elastic parts of the
phase shifts comparably with OPE and to inelasticity perhaps alone by
construction. One might raise the question whether this behavior continues
without end, thus challenging the idea of neglecting anything else but OPE in
peripheral waves. The results for these peripheral waves agree numerically
with those of Ref. eft  using EFT.

Obviously much of resonances or resonance-like structures can be obtained by
coupled channels involving $N\Delta$ excitations, even slight indications
arose in waves where they are not seen in data. The method appears to be a
very reasonable, even successful starting point for producing realistic wave
functions to use in further calculations. The main shortcoming may be that
some additional interactions in strongly interacting medium are not
automatically included which affect inelasticity, such as e.g. $s$-wave
rescattering of pions in the ${}^{3}P_{1}$ wave. But this is just what the
“further calculations” mean, with some references pointed out in the text.
However, most of the $p$-wave pion effects come already in with the dominant
$\Delta$. In spite of the aforementioned shortcomings, apart possibly from
excessive attraction in the ${}^{3}P_{2}$ wave above 500 MeV, remaining
distractions from data do not show particularly systematic and pressing need
for extraneous dibaryons.

###### Acknowledgements.

I thank M. E. Sainio for useful comments.

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