# Integrable scattering theory with higher derivative Hamiltonians

Andreas Fring and Bethan Turner
Department of Mathematics, City, University of London, Northampton Square,
London EC1V 0HB, UK
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract:

We discuss how a standard scattering theory a of multi-particle theory
generalises to systems based on Hamiltonians that involve higher-order
derivatives in their quantum mechanical formulation. As concrete examples, we
consider Hamiltonian systems built from higher-order charges of Calogero and
Calogero-Moser systems. Exploiting the integrability of these systems, we
compute the classical phase shifts and briefly comment on the quantum versions
of these types of theories.

††conference: Integrable scattering theory with higher derivative Hamiltonians

##  1 Introduction

The vast majority of Hamiltonians describing physical systems contain kinetic
terms that are quadratic in momenta in their classical versions, and have
second-order derivatives in their quantum mechanical and quantum field
theoretical formulations. In general, higher order derivative descriptions are
discarded as classical version contain singularities that can be reached in
finite time and quantum versions of these type of theories inevitable
introduce so-called ghosts states that possess negative norms, that lead to a
collapse and/or a violation of unitarity [1]. Despite these severe
deficiencies higher derivative quantum field theoretical models are known to
possess the very appealing feature of being renormalizable and are therefore
frequently considered as promising candidates for theories of everything (TOE)
that include gravity besides all the other known fundamental forces. In that
context the theories in the conventional (3+1)-dimensional space need to be
embedded into a higher dimensional space, which in turn calls for higher
derivative terms. Over the years higher order derivative theories have also
been considered sporadically in other areas. They have been proposed as a
resolution of the cosmological singularity problem [2] with some of their
black holes solutions constructed [3]. In some quantum field theories their
BRST symmetries have been identified [4, 5], they were explored in a massless
particle description of bosons and fermions [6] and also some supersymmetric
versions have been investigated [7]. Classical and quantum stability
properties of higher derivative dynamics were investigated in [8, 9, 10, 11].
As discussed in [12, 13, 14, 15], the undesired features of singularities and
ghost states may be dealt with in individual cases with appropriate techniques
when they are of benign type, i.e. when in their classical versions the
unavoidable singularities can not be reached in finite time.

While there are proposals to deal with some of the conceptual issues in
particular cases, the selection of appropriate models has been rather ad hoc,
with the Pais-Uhlenbeck oscillator [16] being the most popular toy model. It
remains unclear which type or class of higher derivative theories might be
most appropriate and which models contain benign and which contain malevolent
ghost states. Recently Smilga [13] suggested that Hamiltonians build from
higher charges of integrable systems could be a suitable class of benign
systems. Indeed, starting at first with classical theories that have kinetic
terms with higher order momenta build from charges of affine Toda lattice
theories the present authors [17] identified under which circumstance those
theories converge or diverge in phase space.

Motivated by the reasons mentioned above, and the success obtained by taking
integrable systems as suitable candidates for higher derivative theories we
continue here the investigation of these models. In particular, we address the
new question of how one might consistently formulate a scattering theory. We
will answer the question in the context of integrable systems, taking
Calogero-Moser-Sutherland (CMS)-models [18, 19, 20, 21, 22] as suitable
candidates to study.

Thus, our general starting point is a system in the form of standard
Hamiltonians of CMS type

$H=\frac{1}{2}\sum_{i=1}^{\ell}p_{i}^{2}+\sum_{\alpha\in\Delta_{\mathbf{g}}}c_{\alpha}V\left(\alpha\cdot
q\right).$ (1)

The sum in the potential extends over all roots $\alpha$ in the root space
$\Delta_{\mathbf{g}}\in\mathbb{R}^{\ell}$ associated to the Lie algebra
$\mathbf{g}$, $c_{\alpha}$ are real coupling constants,
$p=(p_{1},\ldots,p_{\ell})$ and $q=(q_{1},\ldots,q_{\ell})$ are the $\ell$
momenta and coordinates of the system, respectively. Requiring integrability,
the potential function $V(x)$ may be any function that factorises and
satisfies the following functional relation [23]

$V(x)=-f(x)f(-x),\qquad\text{with}\quad
f(x+y)=\frac{f(x)f^{\prime}(y)-f^{\prime}(x)f(y)}{V(x)-V(y)}.$ (2)

Solutions to these equations are $f(x)=1/x$, $f(x)=1/\sin(x)$,
$f(x)=1/\sinh(x)$ and $f(x)=1/sn(x)$, giving rise to the Calogero-Moser-
Sutherland potentials [23] $V(x)=1/x^{2}$, $V(x)=1/\sin^{2}(x)$,
$V(x)=1/\sinh^{2}(x)$, $V(x)=1/sn^{2}(x)$, respectively. Here we will focus
our attention on the theories that allow for scattering states, i.e.
$V(x)=1/x^{2}$ and $V(x)=1/\sinh^{2}(x)$.

Our manuscript is organised as follows: In section 2 we make some general
comments on how higher order charges can be constructed from Lax pairs
corresponding to the higher order flows in the CMS models. In section 3, we
formulate the scattering theory for higher charge Hamiltonian theories in the
context of integrable systems. In particular, we derive explicit formulas for
the two particle phase shifts. In section 4 and 5 we will discuss in detail
the higher order scattering theories of Calogero and Calogero-Moser systems
associated to the $A_{2}$ and $A_{6}$ Lie algebras, respectively. In section 6
we comment on the quantum mechanical scattering theory. Our conclusions are
stated in section 7.

##  2 Conserved charges in $A_{n}$-Calogero-Moser-Sutherland models

The cornerstone of our construction are the higher charges of integrable
multi-particle systems. A standard way to construct them is in the form of
their Lax pair representation, which implies their classical Liouville
integrability. For CMS-models different versions of Lax pairs can be found in
the literature for specific algebras, concrete choices of the representations
of the roots and functions $f(x)$, but also in more generic terms see e.g.
[24].

We start with the $A_{2}$-CMS theory with a representation independent
formulation for generic functions $f(x)$ subject to the equations (2). A
possible Lax pair is given as

$\displaystyle L$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}p_{1}&i\sqrt{g}f\left(\alpha_{1}\cdot
q\right)&i\sqrt{g}f\left(\alpha_{3}\cdot q\right)\\\
-i\sqrt{g}f\left(\alpha_{1}\cdot
q\right)&p_{2}&i\sqrt{g}f\left(\alpha_{2}\cdot q\right)\\\
-i\sqrt{g}f\left(\alpha_{3}\cdot q\right)&-i\sqrt{g}f\left(\alpha_{2}\cdot
q\right)&p_{3}\\\ \end{array}\right),$ (6) $\displaystyle M$ $\displaystyle=$
$\displaystyle i\sqrt{g}\left(\begin{array}[]{ccc}\frac{f\left(\alpha_{3}\cdot
q\right)f^{\prime}\left(\alpha_{2}\cdot q\right)+f\left(\alpha_{2}\cdot
q\right)f^{\prime}\left(\alpha_{3}\cdot q\right)}{f\left(\alpha_{1}\cdot
q\right)}&f^{\prime}\left(\alpha_{1}\cdot
q\right)&f^{\prime}\left(\alpha_{3}\cdot q\right)\\\
f^{\prime}\left(\alpha_{1}\cdot q\right)&0&f^{\prime}\left(\alpha_{2}\cdot
q\right)\\\ f^{\prime}\left(\alpha_{3}\cdot
q\right)&f^{\prime}\left(\alpha_{2}\cdot q\right)&\frac{f\left(\alpha_{3}\cdot
q\right)f^{\prime}\left(\alpha_{1}\cdot q\right)+f\left(\alpha_{1}\cdot
q\right)f^{\prime}\left(\alpha_{3}\cdot q\right)}{f\left(\alpha_{2}\cdot
q\right)}\\\ \end{array}\right),\,\,\quad$ (10)

so that the Lax equation $\dot{L}=[L,M]$, [25], becomes equivalent to the
equations of motion resulting from the Hamilton $H$ in equation (1)

$\dot{x}_{i}=p_{i},\qquad\dot{p}_{i}=-2g\sum_{j=1}^{3}\alpha_{j}^{i}f\left(\alpha_{j}\cdot
q\right)f^{\prime}\left(\alpha_{j}\cdot q\right),\quad i=1,2,3.$ (11)

The conserved charges are then calculated in the standard fashion from the
trace of products of the $L$-operator

$\displaystyle Q_{1}$ $\displaystyle=$
$\displaystyle\text{tr}(L)=p_{1}+p_{2}+p_{3},$ (12) $\displaystyle Q_{2}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\text{tr}(L^{2})=\frac{1}{2}\left(p_{1}^{2}+p_{2}^{2}+p_{3}^{2}\right)+g\left[f^{2}\left(\alpha_{1}\cdot
q\right)+f^{2}\left(\alpha_{2}\cdot q\right)+f^{2}\left(\alpha_{3}\cdot
q\right)\right],$ (13) $\displaystyle Q_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\text{tr}(L^{3})=\frac{1}{3}\left(p_{1}^{3}+p_{2}^{3}+p_{3}^{3}\right)+g\left[p_{12}f^{2}\left(\alpha_{1}\cdot
q\right)+p_{23}f^{2}\left(\alpha_{2}\cdot
q\right)+p_{13}f^{2}\left(\alpha_{3}\cdot q\right)\right]\\!,\quad\,\,\,$ (14)
$\displaystyle Q_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\text{tr}(L^{4})=\frac{Q_{1}^{4}}{24}-\frac{Q_{1}^{2}Q_{2}}{2}+Q_{1}Q_{3}+\frac{Q_{2}^{2}}{2},$
(15) $\displaystyle Q_{5}$ $\displaystyle=$
$\displaystyle\frac{1}{5}\text{tr}(L^{5})=\frac{Q_{1}^{5}}{80}-\frac{Q_{1}^{3}Q_{2}}{12}-\frac{Q_{1}Q_{2}^{2}}{4}+\frac{Q_{1}Q_{4}}{2}+Q_{2}Q_{3}.$
(16)

We used the abbreviation $p_{ij}:=p_{i}+p_{j}$. As indicated for the first
examples, the charges $Q_{i}$ with $i>3$ can be build from combinations of the
first three independent ones. Identifying $Q_{2}$ with the Hamiltonian in the
standard way, the $Q_{2}$-flow generated by Hamilton’s equations
$\dot{q}_{i}=\partial Q_{2}/\partial p_{i}$, $\dot{p}_{i}=-\partial
Q_{2}/\partial q_{i}$ yields the same equations of motion as the Lax pair,
i.e. (11). The mutual Poisson brackets of the charges vanish

$\left\\{Q_{i},Q_{j}\right\\}:=\sum_{k=1}^{3}\frac{\partial Q_{i}}{\partial
q_{k}}\frac{\partial Q_{j}}{\partial p_{k}}-\frac{\partial Q_{i}}{\partial
p_{k}}\frac{\partial Q_{j}}{\partial q_{k}}=0,\qquad\text{for}\,i,j=1,2,3,$
(17)

i.e. they are in involution. Furthermore, we find formally the same Poisson
bracket relations between the centre-of-mass coordinate
$\chi=q_{1}+q_{2}+q_{3}$ and the conserved charges as in the case of the
affine Toda lattice theories [17], that is

$\left\\{\chi,Q_{n+1}\right\\}=nQ_{n},\qquad n=1,2,\ldots.$ (18)

Interpreting now instead of $Q_{2}$ the charge $Q_{3}$ as the Hamiltonian,
i.e. considering the $Q_{3}$-flow rather than the $Q_{2}$-flow, Hamilton’s
equations of motion $\dot{q}_{i}=\partial Q_{3}/\partial p_{i}$,
$\dot{p}_{i}=-\partial Q_{3}/\partial q_{i}$ become

$\displaystyle\dot{q}_{i}$ $\displaystyle=$ $\displaystyle
p_{i}^{2}+g\left[f^{2}\left(\alpha_{i-1}\cdot
q\right)+f^{2}\left(\alpha_{i}\cdot q\right)\right],\qquad i=1,2,3,\,\mod 3$
(19) $\displaystyle\dot{p}_{i}$ $\displaystyle=$
$\displaystyle-2g\sum_{j=1}^{3}\alpha_{j}^{i}\left(p_{j}+p_{j+1}\right)f\left(\alpha_{j}\cdot
q\right)f^{\prime}\left(\alpha_{j}\cdot q\right).$ (20)

Also these equation of motion may be associated to a Lax pair. We present this
here for our two main theories of interest. As we do not expect to find new
charges for this system we keep the $L$-operator as in (6), and solely modify
the M-operator to

$M=i\sqrt{g}\left(\begin{array}[]{ccc}m_{1}&f^{\prime}\left(\alpha_{1}\cdot
q\right)(p_{1}+p_{2})-Z_{3}&f^{\prime}\left(\alpha_{3}\cdot
q\right)(p_{1}+p_{3})+Z_{2}\\\ f^{\prime}\left(\alpha_{1}\cdot
q\right)(p_{1}+p_{2})+Z_{3}&m_{2}&f^{\prime}\left(\alpha_{2}\cdot
q\right)(p_{2}+p_{3})-Z_{1}\\\ f^{\prime}\left(\alpha_{3}\cdot
q\right)(p_{1}+p_{3})-Z_{2}&f^{\prime}\left(\alpha_{2}\cdot
q\right)(p_{2}+p_{3})+Z_{1}&m_{3}\\\ \end{array}\right),$ (21)

where for the Calogero model when $f(x)=1/x$ we have

$m_{i}=-i\sqrt{g}\\!\sum_{\begin{subarray}{c}j=1,2,3\\\ j\neq
i\end{subarray}}\\!\\!\\!h(q_{i}-q_{j})(p_{i}+p_{j}),\quad\text{and}\quad
Z_{i}=Z=gf(q_{1}-q_{2})f(q_{1}-q_{3})f(q_{2}-q_{3}),$ (22)

and for the Calogero-Moser model when $f(x)=1/\sinh(x)$ we need

$\displaystyle m_{1}$ $\displaystyle=$ $\displaystyle
m_{2}+i\sqrt{g}\left\\{\frac{h^{2}(q_{1}-q_{3})}{f^{2}(q_{1}-q_{3})}p_{1}-\frac{h^{2}(q_{2}-q_{3})}{f^{2}(q_{2}-q_{3})}p_{2}+\left[f^{2}(q_{1}-q_{3})-f^{2}(q_{2}-q_{3})\right]p_{3}\right\\},\qquad$
(23) $\displaystyle m_{3}$ $\displaystyle=$ $\displaystyle
m_{2}+i\sqrt{g}\left\\{\frac{h^{2}(q_{1}-q_{3})}{f^{2}(q_{1}-q_{3})}p_{3}-\frac{h^{2}(q_{1}-q_{2})}{f^{2}(q_{1}-q_{2})}p_{2}+\left[f^{2}(q_{1}-q_{2})-f^{2}(q_{1}-q_{3})\right]p_{1}\right\\},$
(24) $\displaystyle Z_{i}$ $\displaystyle=$ $\displaystyle
Z\prod_{\begin{subarray}{c}j=1,2,3\\\ j\neq
i\end{subarray}}\cosh(q_{i}-q_{j}).$ (25)

Using the different $M$-operators in the Lax pair is then equivalent to the
equations of motion or the $Q_{2}$-flow (11) and the $Q_{3}$-flow (19), (20),
respectively.

The generalization of the Lax pairs to the $A_{n}$-higher-charge theories is
straightforward. We will not report this here, but appeal below to the fact
that this is possible.

##  3 Integrable scattering in higher charge Hamiltonian theories

When the function $f(x)$ is chosen in such a way that we obtain a scattering
theory, that is for the Calogero and Calogero-Moser system, we will encounter
classical phase shifts in the scattering process by comparing the asymptotes
of the in-state with those of the out-state. In general, the classical phase
shifts $\delta_{i}$, are defined as the asymptotic difference in the
coordinates when comparing the free particle motion with an interacting
particle of the same momentum. For integrable systems there exists a second
possibility leading to the same overall shift by adding the consecutive two-
particle phase shifts with the appropriate signs.

Next, we discuss how the standard scattering theory generalises to a
scattering theory based on higher charge Hamiltonians. The potentials we
consider here are impenetrable and repulsive, preventing any particle from
catching up with another during the scattering process. Consequently, the
spatial ordering of the particles on the line remains unchanged at all moments
in time. In a $H=Q_{m}$-theories for $m>2$ we no longer have pure potentials,
that is terms depending only on the coordinates. Instead, the potentials are
multiplied by momenta, which are, however, finite. Thus, it remains still
impossible to cross the singularities in the potentials to re-arrange for a
new ordering. This means the spacial ordering remains fixed for all of these
theories too. For definiteness, we select her for this argument the ordering

$q_{i}(t)>q_{i+1}(t),\qquad\forall t,\,i=1,2,\ldots,n+1,$ (26)

with $n$ being the rank of the $A_{n}$ Lie algebra in the concrete example to
be considered below.

Next, we consider the behaviour of the momenta in the asymptotic in and out-
states, $p_{i}^{-}$ and $p_{i}^{+}$, respectively. We exploit the fact that
the theories are classically integrable and posses a Lax pair formulation. As
we have seen above for higher charge Hamiltonian systems this can also be
achieved by keeping the $L$-operator unchanged and modifying the $M$-operator
appropriately. We recall now the well-known fact that the eigenvalues of the
$L$-operator are preserved, due to the form of the solution to the Lax
equation $L(t)=\exp(-Mt)L(0)\exp(Mt)$. In our concrete case we have
$\lim_{t\rightarrow\pm\infty}L_{ij}=p_{i}^{\pm}\delta_{ij}$ for all
$H=Q_{m}$-theories. Thus the two sets of asymptotic momenta
$\\{p_{1}^{-},p_{2}^{-},\ldots,p_{n+1}^{-}\\}$ and
$\\{p_{1}^{+},p_{2}^{+},\ldots,p_{n+1}^{+}\\}$, which are the eigenvalues of
$L$, must be identical. The only possible change can be a re-ordering when
associating them to the coordinates, which as we argued remain unchanged
throughout the entire scattering process. We stress, that since this reasoning
does not depend on the form of the $M$-operator, but solely on the fact that
the equations of motion can be written equivalently as a Lax pair, it also
holds when $H=Q_{m}$ with $m>2$. The spacial ordering is governed by the
velocities with $\dot{q}_{i}^{-}<\dot{q}_{i+1}^{-}$ in the in-state and
$\dot{q}_{i}^{+}>\dot{q}_{i+1}^{+}$ in the out-state. Given our choice (26),
any other ordering would not be asymptotic. Given the general form of the
charges $Q_{m}$, the asymptotic velocities behave as
$\lim_{t\rightarrow\pm\infty}\dot{q}_{i}\sim(p_{i}^{\pm})^{m-1}$. Thus, for a
$H=Q_{m}$-theory we have the orderings

$\left(p_{i}^{-}\right)^{m}<\left(p_{i+1}^{-}\right)^{m},\qquad\text{and}\qquad\left(p_{i}^{+}\right)^{m}>\left(p_{i+1}^{+}\right)^{m},\qquad\,i=1,2,\ldots,n+1.$
(27)

Evidently this distinguishes a standard Hamiltonian theory, $H=Q_{2}$, from
the higher charge Hamiltonian theories, where in the former of the momenta
govern the ordering, whereas in the latter we have to take higher powers into
account.

Next we discuss the classical phase shifts for the two-particle scattering
between particle $i$ and $j$. For the standard Hamiltonian theory they have
been computed for the $A_{n}$-Calogero and the Calogero-Moser models, see [26,
27]. Translating the results into our notations and conventions they read

$\Delta_{ij}^{C}=0,\qquad\text{and}\qquad\Delta_{ij}^{CM}=\frac{1}{2}\ln\left[1+\frac{4g}{(p_{i}^{-}-p_{j}^{-})^{2}}\right],\qquad\,i,j=1,2,\ldots,n+1,$
(28)

where $p_{i}^{-},p_{j}^{-}$ are the asymptotic momenta of the respective
particles mentioned above for $t\rightarrow-\infty$. We will now show that one
obtains the same two-particle phase shifts when taking higher charges as
Hamiltonians, but the overall shift differs in a multi-particle scattering
event. In general, for a $H=Q_{m}$-theory with repulsive potentials the
overall phase shift of particle $i$ is

$\delta_{i}^{m}=\sum_{j}\text{sign}\left[(p_{i}^{-})^{m-1}-(p_{j}^{-})^{m-1}\right]\Delta_{ij},$
(29)

where the sum over $j$ extends over all particles $i$ scatters with and
$\text{sign}(x)$ denotes the standard signum function, i.e. $\text{sign}(x)=1$
for $x>0$ and $\text{sign}(x)=-1$ for $x<0$. We may think of
$\text{sign}\left[(p_{i}^{-})^{m-1}-(p_{j}^{-})^{m-1}\right]$ as
$\text{sign}\left(q_{i}-q_{j}\right)$ so that this formula would be universal
with no explicit reference made to the charge that corresponds to the
Hamiltonian. However, the ordering in the coordinates for an $H=Q_{m}$-theory
is governed by the asymptotic velocities as
$\lim_{t\rightarrow\pm\infty}\dot{q}_{i}\sim(p_{i}^{\pm})^{m-1}$ rather than
just by the momenta.

We now show how (29) is obtained in general, by first deriving the two-
particle phase shift (28) from the two-particle theory for higher charge
Hamiltonians along the lines of [26, 27]. The conserved charges in involution
are in this case

$\displaystyle Q_{1}$ $\displaystyle=$ $\displaystyle p_{1}+p_{2},$ (30)
$\displaystyle Q_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(p_{1}^{2}+p_{2}^{2}\right)+gV(q),$ (31)
$\displaystyle Q_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\left(p_{1}^{3}+p_{2}^{3}\right)+g\left(p_{1}+p_{2}\right)V(q)=Q_{1}Q_{2}-\frac{1}{6}Q_{1}^{3},$
$\displaystyle\vdots$ $\displaystyle Q_{n}$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{N}c_{l}Q_{1}^{a_{l}}Q_{2}^{\frac{1}{2}(n-a_{l})},\quad\left\\{\begin{array}[]{
l c l l }N&=&\frac{1}{2}(n-1),a_{l}=2l+1&\quad\text{for}\,n\,\text{odd}\\\
N&=&\frac{1}{2}n,a_{l}=2l&\quad\text{for}\,n\,\text{even}\end{array}\right.,$
(35)

where for definiteness we chose $q=q_{2}-q_{1}>0$ and $c_{l}\in\mathbb{R}$ are
arbitrary constants at this point. Since these charges are conserved they
imply the equalities

$\displaystyle p_{1}+p_{2}$ $\displaystyle=$ $\displaystyle
p_{1}^{-}+p_{2}^{-},$ (36)
$\displaystyle\frac{1}{2}\left(p_{1}^{2}+p_{2}^{2}\right)+gV(q)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left[(p_{1}^{-})^{2}+(p_{2}^{-})^{2}\right],$
$\displaystyle\vdots$ $\displaystyle Q_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{n}\left[(p_{1}^{-})^{n}+(p_{2}^{-})^{n}\right],$ (38)

where on the right hand side of (3), (38) we have taken into account that
$\lim_{t\rightarrow-\infty}V(q)=0$, i.e. all particles move infinitely far
away from each other in the in and out-states and fixed the constants $c_{l}$
to obtain the right hand side of (38). Solving (36), (3) or in general (36),
(38) we obtain in each case

$\left.\begin{tabular}[]{ l l }$p_{1}(t)$
\\!\\!\\!&$\\!\\!\\!\\!\\!=\frac{1}{2}\left[p_{1}^{-}+p_{2}^{-}\pm\sqrt{k^{2}-4gV(q)}\right]$\\\
$p_{2}(t)$
\\!\\!\\!&$\\!\\!\\!\\!\\!=\frac{1}{2}\left[p_{1}^{-}+p_{2}^{-}\mp\sqrt{k^{2}-4gV(q)}\right]$\end{tabular}\right\\}\quad\text{for
}\,t\,\begin{subarray}{c}<\\\ >\end{subarray}\,t_{0}.$ (39)

We abbreviated here $k:=p_{1}^{-}-p_{2}^{-}>0$ with the positivity being
implied by our ordering in the coordinates. The time at the turning point when
$k^{2}=4gV(q_{0})$ is denoted as $t_{0}$, so that the upper sign refers to the
time before the scattering process and the lower sign to the situation
afterwards.

Up to this point, there are no discernible differences among the various
scenarios when interpreting different charges as Hamiltonians. However, now we
will use the general equation of motions $\dot{q}_{i}=\partial Q_{m}/\partial
p_{i}$ rather than (11) for the evolution of the coordinate difference

$\dot{q}=\dot{q}_{2}-\dot{q}_{1}=\mp\frac{1}{2}\sum^{N}_{l=0}c_{l}(n-a_{l})(p_{2}-p_{1})Q_{1}^{a_{l}}Q_{2}^{\frac{1}{2}(n-(a_{l}+1))},\quad\text{for
}\,t\,\begin{subarray}{c}<\\\ >\end{subarray}\,t_{0}.$ (40)

Notice that the explicit $p$ and $q$ is reduced to the factor $(p_{2}-p_{1})$.
Within the charges this dependence can be replaced by their asymptotic values.
Replacing now the momenta using (39) we obtain

$\dot{q}=\mp K\sqrt{1-\frac{4gV(q)}{k^{2}}},\quad\text{for
}\,t\,\begin{subarray}{c}<\\\ >\end{subarray}\,t_{0},$ (41)

with constant
$K:=\frac{1}{2}\sum^{N}_{l=0}c_{l}(n-a_{l})Q_{1}^{a_{l}}Q_{2}^{\frac{1}{2}(n-(a_{l}+1))}$.
The equations are easily integrated out by separation of variables. For the
Calogero model and Calogero-Moser model we obtain

$\displaystyle t$ $\displaystyle=$
$\displaystyle\mp\frac{1}{K}\sqrt{q^{2}-\frac{4g}{k^{2}}}+a_{C}^{\mp},\qquad\qquad\qquad\qquad\qquad\,\,\,\,\text{for
}V(q)=\frac{1}{q^{2}},$ (42) $\displaystyle t$ $\displaystyle=$
$\displaystyle\mp\frac{1}{K}\ln\left[\cosh
q+\frac{1}{k}\sqrt{k^{2}\sinh^{2}q-4g}\right]+a_{CM}^{\mp},\quad\text{for
}V(q)=\frac{1}{\sinh^{2}q},\qquad$ (43)

respectively. Where $a_{C}^{\mp}$ and $a_{CM}^{\mp}$ are integration
constants. A relation between the constants for $t<t_{0}$ and $t>t_{0}$ is
obtained by demanding that both solutions coincide at the turning point
($t_{0}$,$q_{0}$). This requirement yields

$a_{C}^{-}=a_{C}^{+},\qquad\text{and}\qquad a_{CM}^{-}=\frac{2\ln\left(\cosh
q_{0}\right)}{K}+a_{CM}^{+}=\frac{1}{K}\ln\left(1+\frac{4g}{k^{2}}\right)+a_{CM}^{+}.$
(44)

Further constraints on the constants arise from matching with the asymptotic
behaviour arising directly from the equations of motion

$q(t)=Kt+q_{2}^{\mp}-q_{1}^{\mp},\quad\text{for }\,t\,\begin{subarray}{c}<\\\
>\end{subarray}\,t_{0}.$ (45)

This leads to the identifications

$a_{C}^{-}=\frac{q_{2}^{-}-q_{1}^{-}}{K}=a_{C}^{+}=\frac{q_{1}^{+}-q_{2}^{+}}{K},\qquad
a_{CM}^{-}=\frac{q_{2}^{-}-q_{1}^{-}}{K}=a_{CM}^{+}=\frac{q_{1}^{+}-q_{2}^{+}}{K},$
(46)

which when combined with (44) and the fact that $\delta_{1}+\delta_{2}=0$
leads to

$\displaystyle\delta_{1}^{C}$ $\displaystyle=$ $\displaystyle
q_{1}^{+}-q_{2}^{-}=q_{2}^{+}-q_{1}^{-}=\delta_{2}^{C},\qquad\Rightarrow\qquad\delta_{1}^{C}=\delta_{2}^{C}=0,$
(47) $\displaystyle\delta_{2}^{CM}$ $\displaystyle=$
$\displaystyle\ln\left(1+\frac{4g}{k^{2}}\right)+\delta_{1}^{CM}\qquad\qquad\,\,\Rightarrow\qquad\delta_{2}^{CM}=-\delta_{1}^{CM}=\frac{1}{2}\ln\left(1+\frac{4g}{k^{2}}\right).\,\,\,\,\,\,$
(48)

Notice that the sum in (29) only includes particle $2$. Next, we illustrate
the above general statements with concrete examples.

##  4 A higher charge $A_{2}$-Calogero model

For the $A_{2}$-Calogero model with $f(x)=1/x$, i.e. $V(x)=1/x^{2}$, the
coupling constants set equal $c_{\alpha}=g$ and the simple roots taken in the
standard three dimensional representation $\alpha_{1}=(1,-1,0)$,
$\alpha_{2}=(0,1,-1)$, $\alpha_{3}=\alpha_{1}+\alpha_{2}=(1,0,-1)$, see e.g.
[28], an explicit analytical solution to the equations of motion (11) with
$Q_{2}$ taken as Hamiltonian

$\dot{q}_{i}=p_{i},\qquad\dot{p}_{i}=2g\sum_{\begin{subarray}{c}k=1,2,3\\\
i\neq k\end{subarray}}\frac{1}{(q_{i}-q_{k})^{3}},$ (49)

was already obtained in [29] by separating variables

$\displaystyle q_{1}(t)$ $\displaystyle=$ $\displaystyle
R_{0}+R_{1}t+\frac{1}{\sqrt{6}}r(t)\cos[\phi(t)]+\frac{1}{\sqrt{2}}r(t)\sin[\phi(t)],$
(50) $\displaystyle q_{2}(t)$ $\displaystyle=$ $\displaystyle
R_{0}+R_{1}t+\frac{1}{\sqrt{6}}r(t)\cos[\phi(t)]-\frac{1}{\sqrt{2}}r(t)\sin[\phi(t)],$
(51) $\displaystyle q_{3}(t)$ $\displaystyle=$ $\displaystyle
R_{0}+R_{1}t-\frac{2}{\sqrt{6}}r(t)\cos[\phi(t)].$ (52)

The functions and the constant

$\displaystyle\phi(t)$ $\displaystyle:=$
$\displaystyle\frac{1}{3}\arccos\left\\{\kappa\sin\left[\arcsin\left(\frac{\cos\left(3\phi_{0}\right)}{\kappa}\right)-3\arctan\left(\frac{\sqrt{2}\text{E}\left(t-t_{0}\right)}{B}\right)\right]\right\\},$
(53) $\displaystyle r(t)$ $\displaystyle:=$
$\displaystyle\sqrt{\frac{B^{2}}{\text{E}}+2\text{E}\left(t-t_{0}\right)^{2}},\qquad\kappa:=\sqrt{1-\frac{9g}{2B^{2}}},$
(54)

depend on the parameters $E,B,R_{0},R_{1},g,\phi_{0},t_{0}\in\mathbb{R}$, with
$E$ being the energy and $B$ an angular constant of motion. The dependent
parameters may be expressed entirely in terms of the initial conditions
$q_{i}(t_{0})=:q_{i}^{0}$ as

$\phi_{0}=\arccos\left(\frac{q_{1}^{0}+q_{2}^{0}-2q_{3}^{0}}{2\lambda}\right),\qquad\text{and}\qquad
B=\lambda\sqrt{\frac{2}{3}E},$ (55)

with

$\lambda:=\sqrt{\left(q_{1}^{0}\right)^{2}+\left(q_{2}^{0}\right)^{2}+\left(q_{3}^{0}\right)^{2}-q_{1}^{0}q_{2}^{0}-q_{1}^{0}q_{3}^{0}-q_{2}^{0}q_{3}^{0}},$
(56)

which is useful for the numerical considerations. The asymptotic behaviour for
these solutions is found to be

$\displaystyle\lim_{t\rightarrow+\infty}q_{1}(t)\sim
q_{1}^{a}+p_{1}^{a}t,\quad\lim_{t\rightarrow+\infty}q_{2}(t)\sim
q_{2}^{a}+p_{2}^{a}t,\quad\lim_{t\rightarrow+\infty}q_{3}(t)\sim
q_{3}^{a}+p_{3}^{a}t,$ (57)
$\displaystyle\lim_{t\rightarrow-\infty}q_{1}(t)\sim
q_{3}^{a}+p_{3}^{a}t,\quad\lim_{t\rightarrow-\infty}q_{2}(t)\sim
q_{2}^{a}+p_{2}^{a}t,\quad\lim_{t\rightarrow-\infty}q_{3}(t)\sim
q_{1}^{a}+p_{1}^{a}t.$ (58)

We compute the values for the asymptotic momenta $p_{i}^{a}$ and the crossing
values of the asymptotes with the $y$-axis $q_{i}^{a}$ from

$\displaystyle p_{i}^{a}$ $\displaystyle=$ $\displaystyle
p_{i}^{+}=\lim_{t\rightarrow+\infty}\dot{q}_{i}(t),\qquad\quad\text{and}\qquad
q_{i}^{a}=\lim_{t\rightarrow+\infty}\left[q_{i}(t)-p_{i}^{a}t\right],$ (59)
$\displaystyle p_{i}^{a}$ $\displaystyle=$ $\displaystyle
p_{4-i}^{-}=\lim_{t\rightarrow-\infty}\dot{q}_{4-i}(t),\qquad\text{and}\qquad
q_{i}^{a}=\lim_{t\rightarrow-\infty}\left[q_{4-i}(t)-p_{4-i}^{a}t\right].$
(60)

Setting $R_{0}=R_{1}=0$ in the explicit solutions, we obtain

$\displaystyle p_{1/2}^{a}$ $\displaystyle=$
$\displaystyle\sqrt{E}\left[\frac{1}{\sqrt{3}}\cos(\varphi)\pm\sin(\varphi)\right],\quad
p_{3}^{a}=-\sqrt{E}\frac{2}{\sqrt{3}}\cos(\varphi),$ (61) $\displaystyle
q_{1/2}^{a}$ $\displaystyle=$
$\displaystyle\frac{B\cos\left(3\phi_{0}\right)\left(3\cos(\varphi)-\sqrt{3}\sin(\varphi)\right)}{3\sqrt{E}\sqrt{1+\frac{9g}{B^{2}}+\cos\left(6\phi_{0}\right)}},\quad
q_{3}^{a}=\frac{2B\sin(\varphi)\cos\left(3\phi_{0}\right)}{3\sqrt{E}\sqrt{\frac{9g}{B^{2}}+\cos\left(6\phi_{0}\right)+1}},\qquad$
(62) $\displaystyle\varphi$ $\displaystyle:=$
$\displaystyle\frac{1}{3}\arccos\left(\sqrt{\kappa-\cos^{2}\left(3\phi_{0}\right)}\right).$
(63)

In figure 1 panel (a) we depict these solutions for the ordering chosen in
(26), i.e. $q_{1}>q_{2}>q_{3}$, together with their asymptotes.

Figure 1: Coordinates as functions of time for the $A_{2}$-Calogero model with
$Q_{2}=H$ (panel a) and $Q_{3}=H$ (panel b) (solid lines) together with their
asymptotes (dashed lines). The initial conditions respect $Q_{1}=\chi=0$ and
are taken to $q_{1}(0)=2.63998$, $q_{2}(0)=-0.994087$, $q_{3}(0)=-1.64589$,
$p_{1}(0)=0.152777$, $p_{2}(0)=-1.00456$, $p_{3}(0)=0.851788$ corresponding to
$E=1.5$, $B=4$, $g=0.25$, $\phi_{0}=-3$, $t_{0}=0$, $R_{0}=0$, $R_{1}=0$. For
these choices we have $q_{1}^{a}=-1.50968$, $q_{2}^{a}=2.5458$ and
$q_{3}^{a}=-1.03612$ for the values of the asymptotes at $t=0$ . The coupling
constant is taken to $g=1/4$.

Next we compare these results with the numerical solutions for the system
(19), (20) when $Q_{3}$ is taken to be the Hamiltonian with the same function
$f(x)$ and the same representation for the simple roots. In this case the
equations of motion are

$\dot{q}_{i}=p_{i}^{2}+g\sum_{\begin{subarray}{c}k=1,2,3\\\ i\neq
k\end{subarray}}\frac{1}{(q_{i}-q_{k})^{2}},\quad\dot{p}_{i}=2g\sum_{\begin{subarray}{c}k=1,2,3\\\
i\neq k\end{subarray}}\frac{p_{i}+p_{k}}{(q_{i}-q_{k})^{3}}.$ (64)

From figures 1 and 2 panel (a) we observe that while the asymptotic behaviour
for the coordinates is different in both cases, the asymptotic values for the
momenta are identical up to permutations of the particle type. The
inequalities for the ordering of the powers of the momenta in (26) and (27)
are confirmed.

However, as we will see next while the scattering is governed by the ordering
of the momenta when $H=Q_{2}$, it is governed by the ordering of momenta
squared when $H=Q_{3}$.

Figure 2: Momenta as functions of time for the $A_{2}$-Calogero model for
$Q_{2}=H$ (solid lines) and $Q_{3}=H$ (dashed lines) in panel (a) and squared
momenta as functions of time in panel (b). The initial conditions and coupling
constant are the same as in figure 1. When $H=Q_{2}$ the values for the
asymptotic momenta are $p_{1}^{-}=p_{3}^{+}=-1.24871$,
$p_{2}^{-}=p_{2}^{+}=0.15298$, $p_{3}^{-}=p_{1}^{+}=1.09573$, which when
$H=Q_{3}$ are re-ordered to $p_{3}^{-}=p_{1}^{+}=-1.24871$,
$p_{1}^{-}=p_{3}^{+}=0.15298$, $p_{2}^{-}=p_{2}^{+}=1.09573$.

Indeed, when $H=Q_{2}$ we observe that the ordering of the momenta in the in-
state $p_{3}^{-}>p_{2}^{-}>0>p_{1}^{-}$ at $t\rightarrow-\infty$ changes for
$t\rightarrow\infty$ into $p_{1}^{+}<0<p_{2}^{+}<p_{3}^{+}$ in the out-state,
i.e. particle 1 undergoes a head on collision with particle 2 and 3, whereas
particle 3 scatters with particle 1 by overtaking it due to its larger
momentum in the same direction. After the scattering event all particles move
away from each other.

In contrast, when $H=Q_{3}$ this interpretation does not hold. As we observe
in figure 2 panel (a), in this case we have the ordering
$p_{3}^{-}<0<p_{1}^{-}<p_{2}^{-}$ in the in-state, which if the scattering
were governed by the momenta would suggest for the given spacial ordering that
particle 3 would simple move away to the left without scattering with any
other particles and the only scattering process would be between particle 1
and 2 with the latter overtaking the former. Clearly this is not what we
observe in figure 2 panel (a). Instead, we see from the $H=Q_{3}$-equations of
motion (64) that $\lim_{t\rightarrow\pm\infty}\dot{q}_{i}(t)\sim p_{i}^{2}$,
since $\lim_{t\rightarrow\pm\infty}|q_{i}(t)-q_{j}(t)|\sim\infty$, so that the
squared momenta govern the asymptotic behaviour. This feature is precisely
confirmed by figure 2 panel (b), where we observe that the ordering
$(p_{3}^{-})^{2}>(p_{2}^{-})^{2}>0>(p_{1}^{-})^{2}$ changes into
$(p_{1}^{+})^{2}>(p_{2}^{+})^{2}>0>(p_{3}^{+})^{2}$, i.e. particle 3 is
overtaking particle 1 and 2 whereas particle 2 is overtaking particle 1.

Next we compute the classical phase shifts $\delta_{i}$, defined as the
asymptotic shift in the coordinates when comparing the free particle with a
Calogero particle of the same momentum for the two different Hamiltonians.

When $H=Q_{2}$ they are computed from the analytical expressions derived above
to

$\displaystyle\delta_{1}$ $\displaystyle=$
$\displaystyle\lim_{t\rightarrow\infty}\left[q_{1}(t)-q_{3}(-t)-2p_{1}^{a}t\right]=q_{1}^{a}-q_{1}^{a}=0,$
(65) $\displaystyle\delta_{2}$ $\displaystyle=$
$\displaystyle\lim_{t\rightarrow\infty}\left[q_{2}(t)-q_{2}(-t)-2p_{2}^{a}t\right]=q_{2}^{a}-q_{2}^{a}=0,$
(66) $\displaystyle\delta_{3}$ $\displaystyle=$
$\displaystyle\lim_{t\rightarrow\infty}\left[q_{3}(t)-q_{1}(-t)-2p_{3}^{a}t\right]=q_{3}^{a}-q_{3}^{a}=0.$
(67)

This zero phase shift is indeed what we expect, as the classical phase shift
in any two-particle scattering process in the Calogero model is zero, as we
have seen.

When $H=Q_{3}$ we compute the asymptotic behaviour numerically, obtaining

$\displaystyle\lim_{t\rightarrow+\infty}q_{1}(t)\sim
q_{3}^{a}+\left(p_{3}^{a}\right)^{2}t,\quad\lim_{t\rightarrow+\infty}q_{2}(t)\sim
q_{1}^{a}+\left(p_{1}^{a}\right)^{2}t,\quad\lim_{t\rightarrow+\infty}q_{3}(t)\sim
q_{2}^{a}+\left(p_{2}^{a}\right)^{2}t,\qquad\quad$ (68)
$\displaystyle\lim_{t\rightarrow-\infty}q_{1}(t)\sim
q_{2}^{a}+\left(p_{2}^{a}\right)^{2}t,\quad\lim_{t\rightarrow-\infty}q_{2}(t)\sim
q_{1}^{a}+\left(p_{1}^{a}\right)^{2}t,\quad\lim_{t\rightarrow-\infty}q_{3}(t)\sim
q_{3}^{a}+\left(p_{3}^{a}\right)^{2}t.$ (69)

The numerical values for the asymptotics expressions of $q_{i}$ and $p_{i}$
are reported in the captions of figures 1 and 2 and coincide precisely with
those obtained from the analytical expressions (61) and (62) for the given
initial conditions.

The classical phase shifts are then

$\displaystyle\delta_{1}$ $\displaystyle=$
$\displaystyle\lim_{t\rightarrow\infty}\left[q_{1}(t)-q_{3}(-t)-2\left(p_{2}^{a}\right)^{2}t\right]=q_{3}^{a}-q_{3}^{a}=0,$
(70) $\displaystyle\delta_{2}$ $\displaystyle=$
$\displaystyle\lim_{t\rightarrow\infty}\left[q_{2}(t)-q_{2}(-t)-2\left(p_{1}^{a}\right)^{2}t\right]=q_{1}^{a}-q_{1}^{a}=0,$
(71) $\displaystyle\delta_{3}$ $\displaystyle=$
$\displaystyle\lim_{t\rightarrow\infty}\left[q_{3}(t)-q_{1}(-t)-2\left(p_{2}^{a}\right)^{2}t\right]=q_{2}^{a}-q_{2}^{a}=0.$
(72)

Thus, while the asymptotic behaviour for the coordinates is different when
taking either $Q_{2}$ or $Q_{3}$ as the Hamiltonian, the overall classical
phase shifts are identical to zero in both cases.

We conclude this section with a brief remark on the sensitivity with regard to
the chosen initial conditions. For the coordinates the choice was made to
implement the centre-of-mass condition. For the momenta we took the $p_{i}(0)$
such that $Q_{1}=0$. From the Poisson bracket relations (18) follows that
different choices with $Q_{1}\neq 0$ will simply add this value to all momenta
for the standard $H=Q_{2}$-theory and for a general $H=Q_{m}$-theory it will
add to each velocity an overall constant $mQ_{m}$, as this is non-vanishing.
One might eliminate this shift by different choices of the initial momenta.
However, for the sake of comparison, we opted to use the same initial
conditions in all cases. Thus, unlike as in bounded motions [17], where one
finds a strong sensitivity in regard to the initial conditions with non-
vanishing right hand sides in (18) leading to divergencies, for the scattering
theory the overall shifts do not produce any qualitative change in behaviour.
We will use these type of initial conditions throughout the manuscript.

Next we consider a model that contains particles that scatter with non-zero
phase shifts.

##  5 Higher charge $A_{n}$-Calogero-Moser models

We compare the results of the previous subsection now with a similar
calculation carried out for the $A_{2}$-Calogero-Moser model with
$f(x)=1/\sinh(x)$, i.e. $V(x)=1/\sinh^{2}(x)$ and the representation of the
simple roots taken to be the same as in the previous section. In this case the
equations of motion (11) with $Q_{2}$ taken as Hamiltonian become

$\dot{q}_{i}=p_{i},\qquad\dot{p}_{i}=2g\sum_{\begin{subarray}{c}k=1,2,3\\\
i\neq k\end{subarray}}\frac{\coth(q_{i}-q_{k})}{\sinh^{2}(q_{i}-q_{k})}.$ (73)

Taking instead $Q_{3}$ to be the Hamiltonian equations of motion become

$\dot{q}_{i}=p_{i}^{2}+g\sum_{\begin{subarray}{c}k=1,2,3\\\ i\neq
k\end{subarray}}\frac{1}{\sinh^{2}(q_{i}-q_{k})},\quad\dot{p}_{i}=2g\sum_{\begin{subarray}{c}k=1,2,3\\\
i\neq
k\end{subarray}}\frac{\coth(q_{i}-q_{k})}{\sinh^{2}(q_{i}-q_{k})}\left(p_{i}+p_{k}\right).$
(74)

Some numerical solutions for $q_{i}(t)$ are depicted in figure 3. We do not
present the solutions for the momenta as they are very similar to those of the
Calogero model seen in figure 2.

###  5.1 $A_{2}$\- higher charge Calogero-Moser model

For the Calogero-Moser systems the two-particle phase shift $\Delta_{ij}$ and
the overall classical phase shifts $\delta_{i}$ can be computed from (28) and
(29), respectively. Starting with the $H=Q_{2}$-theory, for our ordering
$q_{3}>q_{2}>q_{1}$, implying $p_{3}^{-}>p_{2}^{-}>p_{1}^{-}$, the sign in
overall phase shift for particle $i$ are determined from
$\text{sign}(p_{i}^{-}-p_{j}^{-})$. We confirm the general formulas with the
numerical values obtained from solution depicted in figure 3

$\displaystyle\delta_{1}$ $\displaystyle=$
$\displaystyle-\Delta_{12}-\Delta_{13}=(q_{3}^{a})^{+}-(q_{1}^{a})^{-}=-0.28929,$
(75) $\displaystyle\delta_{2}$ $\displaystyle=$
$\displaystyle\Delta_{21}-\Delta_{23}=(q_{2}^{a})^{+}-(q_{2}^{a})^{-}=-0.17121,$
(76) $\displaystyle\delta_{3}$ $\displaystyle=$
$\displaystyle\Delta_{31}+\Delta_{32}=(q_{1}^{a})^{+}-(q_{3}^{a})^{-}=0.46050.$
(77)

The two-particle phase shifts are computed to $\Delta_{12}=0.20572$,
$\Delta_{13}=0.08358$ and $\Delta_{23}=0.37692$ computed from formula (28)
with the values of the asymptotic moment $p_{i}^{-}$ as stated in figure 2 for
$g=1/4$. The numerical values for the intersection of the asymptotes with the
$y$-axis $(q_{i}^{a})^{\pm}$ are reported in figure 3.

Figure 3: Coordinates as functions of time for the $A_{2}$-Calogero-Moser
model with $Q_{2}=H$ (panel a) and $Q_{3}=H$ (panel b) (solid lines) together
with their in-state (dotted lines) and out-state asymptotes (dashed lines).
The initial conditions are the same as in figure 1. For the asymptotes at
$t=0$ we obtain for $H=Q_{2}$ the values $(q_{1}^{a})^{+}=-1.14586$,
$(q_{2}^{a})^{+}=2.26276$, $(q_{3}^{a})^{+}=-1.11690$,
$(q_{1}^{a})^{-}=-0.82760$, $(q_{2}^{a})^{-}=2.43396$,
$(q_{3}^{a})^{-}=-1.60636$ and for $H=Q_{3}$ we have
$(q_{1}^{a})^{+}=-0.82760$, $(q_{2}^{a})^{+}=-1.22944$,
$(q_{3}^{a})^{+}=2.05704$, $(q_{1}^{a})^{-}=2.63968$,
$(q_{2}^{a})^{-}=-1.52278$, $(q_{3}^{a})^{-}=-1.1169$.

When $H=Q_{3}$ the ordering of the scattering is no longer controlled by the
asymptotic momenta, but instead by the momenta squared. Once more, we confirm
the general formulas with the numerical values obtained from our example
depicted in figure 3

$\displaystyle\delta_{1}$ $\displaystyle=$
$\displaystyle-\Delta_{12}-\Delta_{13}=(q_{3}^{a})^{+}-(q_{1}^{a})^{-}=-0.58264,$
(78) $\displaystyle\delta_{2}$ $\displaystyle=$
$\displaystyle\Delta_{21}-\Delta_{23}=(q_{2}^{a})^{+}-(q_{2}^{a})^{-}=0.28929,$
(79) $\displaystyle\delta_{3}$ $\displaystyle=$
$\displaystyle\Delta_{31}+\Delta_{32}=(q_{1}^{a})^{+}-(q_{3}^{a})^{-}=0.29334,$
(80)

where $\Delta_{12}=0.37692$, $\Delta_{13}=0.20572$ and $\Delta_{23}=0.08358$
computed from formula (28) with the values of the asymptotic moment
$p_{i}^{-}$ as stated in figure 2 for $g=1/4$.

###  5.2 $A_{6}$\- higher charge Calogero-Moser model

Next, we demonstrate that the behaviour observed in the previous section is
similar for higher rank $A_{n}$-Calogero theories, exemplified by the
$A_{6}$-theory and also reveal more features that are not observable in the
$A_{2}$-theory with less particle in play.

Figure 4: Coordinates $q_{i}(t)$ as functions of time $t$ for the
$A_{6}$-Calogero model with $H=Q_{2}$ panel (a), $H=Q_{4}$ panel (b),
$H=Q_{6}$ panel (c), $H=Q_{3}$ panel (d), $H=Q_{5}$ panel (e) and $H=Q_{7}$
panel (f). The initial conditions are taken as $q_{1}(0)=10$, $q_{2}(0)=3$,
$q_{3}(0)=2$, $q_{4}(0)=1$, $q_{5}(0)=-2$, $q_{6}(0)=-4$, $q_{7}(0)=-10$,
$p_{1}(0)=1$, $p_{2}(0)=-0.7$, $p_{3}(0)=0.6$, $p_{4}(0)=-1.6$,
$p_{5}(0)=0.3$, $p_{6}(0)=-0.5$, $p_{7}(0)=0.9$ and the coupling constant as
$g=0.05$.

Figure 5: Momenta $p_{i}(t)$ as functions of time $t$ for the $A_{6}$-Calogero
model with $H=Q_{2}$ panel (a), $H=Q_{4}$ panel (b), $H=Q_{6}$ panel (c),
$H=Q_{3}$ panel (d), $H=Q_{5}$ panel (e) and $H=Q_{7}$ panel (f). The initial
conditions and the coupling constant are the same as in figure 4.

In the $q(t)$-plots in figure 4 the most obvious feature that is confirmed is
the fact that the spacial ordering does not change during the entire
scattering process. Recalling that
$\lim_{t\rightarrow\pm\infty}\dot{q}_{i}\sim(p_{i}^{\pm})^{n-1}$ we also
observe that $H=Q_{m}$ theories with $m$ odd can only have positive asymptotic
gradients, whereas those with $m$ even have negative gradients for
$p_{i}^{\pm}<0$. As we argued in general at the beginning of section 3, the
ordering of the particles is governed by the powers of the momenta as
specified in (27). This is confirmed by our observation in figure 5. We see
that in the $H=Q_{2},Q_{4},Q_{6}$-theories, panels (a)-(c), the asymptotic
sets of momenta re-organise as

$p_{1}^{-}<p_{2}^{-}<p_{3}^{-}<p_{4}^{-}<p_{5}^{-}<p_{6}^{-}<p_{7}^{-}\quad\rightarrow\quad
p_{7}^{+}<p_{6}^{+}<p_{5}^{+}<p_{4}^{+}<p_{3}^{+}<p_{2}^{+}<p_{1}^{+},$ (81)

so that the ordering on the line (26) is enforced by $p_{i}^{-}<p_{i+1}^{-}$.
In contrast, for the $H=Q_{3},Q_{5},Q_{7}$-theories, panels (d)-(f), the
ordering is dictated by $(p_{i}^{-})^{2}<(p_{i+1}^{-})^{2}$ with the momenta
permuted as

$\\!\\!p_{7}^{-}<p_{4}^{-}<p_{2}^{-}<p_{1}^{-}<p_{3}^{-}<p_{5}^{-}<p_{6}^{-}\quad\rightarrow\quad
p_{1}^{+}<p_{4}^{+}<p_{6}^{+}<p_{7}^{+}<p_{5}^{+}<p_{3}^{+}<p_{2}^{+}.$ (82)

Using the six occurring asymptotic values of the momenta $p_{1}=-1.621$,
$p_{2}=-0.722$, $p_{3}=-0.505$, $p_{4}=0.305$, $p_{5}=0.643$, $p_{6}=0.900$,
$p_{7}=1.000$, we calculate the two-particle phase shifts (28) by suitably
permuting them according to (81) and (82). We only report here three digits as
this is sufficient for our arguments even though higher accuracy can of course
be achieved. The values for $\Delta_{ij}$ are conveniently reported in a
table:

i / j | 1 | 2 | 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---|---|---
1 | $\star$ | 0.111 | 0.074 | 0.026 | 0.019 | 0.016 | 0.014
2 | 0.111 | $\star$ | 0.836 | 0.087 | 0.051 | 0.037 | 0.033
3 | 0.074 | 0.836 | $\star$ | 0.133 | 0.070 | 0.048 | 0.042
4 | 0.026 | 0.087 | 0.133 | $\star$ | 0.505 | 0.224 | 0.173
5 | 0.019 | 0.051 | 0.070 | 0.505 | $\star$ | 0.698 | 0.472
6 | 0.016 | 0.037 | 0.048 | 0.224 | 0.698 | $\star$ | 1.522
7 | 0.014 | 0.033 | 0.042 | 0.173 | 0.472 | 1.522 | $\star$

Using the table we compare next the two alternative ways to calculate the
overall shift, i.e. directly from the asymptotic values and alternatively via
(28). The phase shifts for particle $i$ in the $H=Q_{2},Q_{4},Q_{6}$-theories
are all identical resulting to

$\displaystyle\delta_{1}$ $\displaystyle=$
$\displaystyle-\Delta_{12}-\Delta_{13}-\Delta_{14}-\Delta_{15}-\Delta_{16}-\Delta_{17}=(q^{a}_{7})^{+}-(q^{a}_{1})^{-}=-0.260,$
(83) $\displaystyle\delta_{2}$ $\displaystyle=$
$\displaystyle\Delta_{21}-\Delta_{23}-\Delta_{24}-\Delta_{25}-\Delta_{26}-\Delta_{27}=(q^{a}_{6})^{+}-(q^{a}_{2})^{-}=-0.923,$
(84) $\displaystyle\delta_{3}$ $\displaystyle=$
$\displaystyle\Delta_{31}+\Delta_{32}-\Delta_{34}-\Delta_{35}-\Delta_{36}-\Delta_{37}=(q^{a}_{5})^{+}-(q^{a}_{3})^{-}=0.606,$
(85) $\displaystyle\delta_{4}$ $\displaystyle=$
$\displaystyle\Delta_{41}+\Delta_{42}+\Delta_{43}-\Delta_{45}-\Delta_{46}-\Delta_{47}=(q^{a}_{4})^{+}-(q^{a}_{4})^{-}=-0.656,$
(86) $\displaystyle\delta_{5}$ $\displaystyle=$
$\displaystyle\Delta_{51}+\Delta_{52}+\Delta_{53}+\Delta_{54}-\Delta_{56}-\Delta_{57}=(q^{a}_{3})^{+}-(q^{a}_{5})^{-}=-0.525,$
(87) $\displaystyle\delta_{6}$ $\displaystyle=$
$\displaystyle\Delta_{61}+\Delta_{62}+\Delta_{63}+\Delta_{64}+\Delta_{65}-\Delta_{67}=(q^{a}_{2})^{+}-(q^{a}_{6})^{-}=-0.500,$
(88) $\displaystyle\delta_{7}$ $\displaystyle=$
$\displaystyle\Delta_{71}+\Delta_{72}+\Delta_{73}+\Delta_{74}+\Delta_{75}+\Delta_{76}=(q^{a}_{1})^{+}-(q^{a}_{7})^{-}=2.257.$
(89)

Also the phase shifts for particle $i$ in the $H=Q_{3},Q_{5},Q_{7}$-theories
are identical. Taking the permutation of the asymptotic momenta and the
different orderings into account, we obtain

$\displaystyle\delta_{1}$ $\displaystyle=$
$\displaystyle-\Delta_{43}-\Delta_{45}-\Delta_{42}-\Delta_{46}-\Delta_{47}-\Delta_{41}=(q^{a}_{7})^{+}-(q^{a}_{1})^{-}=-1.148,$
(90) $\displaystyle\delta_{2}$ $\displaystyle=$
$\displaystyle\Delta_{34}-\Delta_{35}-\Delta_{32}-\Delta_{36}-\Delta_{37}-\Delta_{31}=(q^{a}_{6})^{+}-(q^{a}_{2})^{-}=-0.929,$
(91) $\displaystyle\delta_{3}$ $\displaystyle=$
$\displaystyle\Delta_{54}+\Delta_{53}-\Delta_{52}-\Delta_{56}-\Delta_{57}-\Delta_{51}=(q^{a}_{5})^{+}-(q^{a}_{3})^{-}=-0.665,$
(92) $\displaystyle\delta_{4}$ $\displaystyle=$
$\displaystyle\Delta_{24}+\Delta_{23}+\Delta_{25}-\Delta_{26}-\Delta_{27}-\Delta_{21}=(q^{a}_{4})^{+}-(q^{a}_{4})^{-}=0.784,$
(93) $\displaystyle\delta_{5}$ $\displaystyle=$
$\displaystyle\Delta_{64}+\Delta_{63}+\Delta_{65}+\Delta_{62}-\Delta_{67}-\Delta_{61}=(q^{a}_{3})^{+}-(q^{a}_{5})^{-}=-0.531,$
(94) $\displaystyle\delta_{6}$ $\displaystyle=$
$\displaystyle\Delta_{74}+\Delta_{73}+\Delta_{75}+\Delta_{72}+\Delta_{76}-\Delta_{71}=(q^{a}_{2})^{+}-(q^{a}_{6})^{-}=2.228,$
(95) $\displaystyle\delta_{7}$ $\displaystyle=$
$\displaystyle\Delta_{14}+\Delta_{13}+\Delta_{15}+\Delta_{12}+\Delta_{16}+\Delta_{17}=(q^{a}_{1})^{+}-(q^{a}_{7})^{-}=0.260.$
(96)

We also verify that in all cases the total overall shift is zero,
$\sum_{i=1}^{7}\delta_{i}=0$. While the $H=Q_{m}$-theories share a lot of
features for even and odd $m$, there are also a number of differences. We
notice, especially from figure 5, that for large $m$ the actual scattering
event becomes more and more squeezed around $t=0$. The reason for this is that
the singularities in potentials acquire higher order and, as a result, become
more short-ranged. Another noteworthy feature is the change in the ordering in
which the particles scatter with respect to the value of $m$. At present, we
lack an explanation that could predict these different scattering sequences.

##  6 A comment on the quantum mechanical scattering

We finish with a brief comment on the quantum theory. In general, we will have
to rely on numerical solutions for the higher particle theories. However, for
the $A_{2}$-Calogero model we can extend the known analytic solution of the
time-independent Schrödinger equation [19, 29] to the $H=Q_{3}$-theory. For
this purpose we first transform our system to the centre-of-mass coordinates
using Jacobi coordinates111When changing the factor $1/3$ in the definition of
$R$ to $1/\sqrt{3}$ the new coordinates simply result from taking a two
dimensional representation for the $A_{2}$ simple roots, see for instance
[17].

$R=\frac{1}{3}(q_{1}+q_{2}+q_{3}),\quad\zeta=\frac{1}{\sqrt{6}}(q_{1}+q_{2}-2q_{3}),\quad\eta=\frac{1}{\sqrt{2}}(q_{1}-q_{2}).$
(97)

Setting $R$ to zero the remaining two dimensional system is further
transformed to the polar coordinates $r$ and $\phi$ with

$\zeta=r\cos(\phi),\qquad\eta=r\sin(\phi).$ (98)

In these variable the second and third charge, (13) and (14) take on the form

$\displaystyle Q_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(-\partial_{rr}-\frac{1}{r}\partial_{r}-\frac{1}{r^{2}}\partial_{\phi\phi}+\frac{9g}{r^{2}\sin^{2}(3\phi)}\right),$
(99) $\displaystyle Q_{3}$ $\displaystyle=$
$\displaystyle\frac{i\sin(3\phi)}{3\sqrt{3}r^{3}}\left\\{\left(9g\csc^{2}(3\phi)+16\right)\partial_{\phi}-2\partial_{\phi\phi\phi}-12\cot(3\phi)\partial_{\phi\phi}+r\left[6r\partial_{rr\phi}-18\partial_{r\phi}\right.\right.$
$\displaystyle\left.\left.-3\cot(3\phi)\left(9g\csc^{2}(3\phi)+2\right)\partial_{r}+6\cot(3\phi)\partial_{r\phi\phi}-2r\cot(3\phi)\left(r\partial_{rrr}-3\partial_{rr}\right)\right]\right\\}.\qquad\quad$

We convince ourselves that these charges commute, i.e. $[Q_{2},Q_{3}]=0$. The
most general eigenfunction $\psi$ to $Q_{2}\psi=E\psi$ was found in [19] as

$\psi(r,\phi,g,E)=\sum_{\ell=0}^{\infty}\psi_{\ell}(r,\phi,g,E)=\sum_{\ell=0}^{\infty}N_{\ell}J_{3a+3\ell+\frac{3}{2}}\left(\sqrt{E}r\right)C_{\ell}^{a+\frac{1}{2}}\left[\cos(3\phi)\right]\sin^{a+\frac{1}{2}}(3\phi),$
(101)

where $J_{n}(z)$ denotes the Bessel function of the first kind, $C_{n}^{m}$ a
Gegenbauer polynomial, $N_{\ell}$ is a normalisation constant and
$a:=\frac{1}{2}\sqrt{2g+1}$. This is indeed a scattering state for the
continuous eigenvalue $E$. Since $Q_{2}$ and $Q_{3}$ commute also $Q_{3}\psi$
must be an eigenstate of $Q_{2}$. Indeed, we find

$Q_{3}\psi=\frac{iE^{3/2}}{3\sqrt{3}}\left[\sum_{\ell=0}^{\infty}a_{\ell}\psi_{\ell+1}+\sum_{\ell=1}^{\infty}b_{\ell}\psi_{\ell-1}\right]=\frac{iE^{3/2}}{3\sqrt{3}}\left[\sum_{\ell=0}^{\infty}\left(a_{\ell-1}+b_{\ell+1}\right)\psi_{\ell}\right],$
(102)

with

$a_{\ell}=(\ell+1)\frac{2\ell+1-\sqrt{2g+1}}{2\ell(\ell+1)-g},\quad
b_{\ell}=a_{\ell}-2.$ (103)

The expressions can be used to extract quantum scattering matrices from the
asymptotic behaviour. We leave these investigation for a separate analysis
[30].

##  7 Conclusion

We generalised the standard integrable scattering theories based on
conventional Hamiltonians quadratic in the momenta to theories involving
higher powers in momenta and additional non-potential terms. As candidates for
these theories we used higher order charges of integrable systems. We
discussed as particular examples the scattering behaviour of Calogero and
Calogero-Moser systems related to the $A_{2}$ and $A_{6}$ Lie algebras. We
derived the overall and two-particle classical scattering phase shifts. Based
on the fact that all higher charge theories allow for a Lax pair formulation
with an $L$-operator in common with the Hamiltonian system, we argued that
they possess the same set of asymptotic momenta. Moreover, these systems also
possess the trademark property of integrable systems of their overall shift
being the sum of all two particle scattering events. However, due to the
different asymptotic dependencies of the velocities on the momenta, the
summation differs for different $m$ in the $H=Q_{m}$-theories.

Naturally there are a number of interesting open questions left to be answered
in future work. At present we can precisely which two-particle scattering
processes take place in dependence on $m$, but not the precise order in which
they take place. It would be interesting to find out how these results
generalise to Calogero and Calogero-Moser models based on other type of
algebras, representations of the root systems, to integrable systems of
different type and to non-integrable systems in general. More detailed
analysis on the non-integrable systems in form of perturbed integrable systems
would reveal more information about the robustness of the features obtained
here. Of further interest is also a more detailed analysis of the quantum
version of higher derivative theories [30].

Acknowledgments: BT is supported by a City, University of London Research
Fellowship.

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# Canonical Ensemble vs. Grand Canonical Ensemble in the Description of
Multicomponent Bosonic Systems

D. Anchishkin  Bogolyubov Institute for Theoretical Physics, 03143 Kyiv,
Ukraine  Taras Shevchenko National University of Kyiv, 03022 Kyiv, Ukraine
Frankfurt Institute for Advanced Studies, Ruth-Moufang-Strasse 1, 60438
Frankfurt am Main, Germany  V. Gnatovskyy  Taras Shevchenko National
University of Kyiv, 03022 Kyiv, Ukraine  D. Zhuravel  Bogolyubov Institute for
Theoretical Physics, 03143 Kyiv, Ukraine  V. Karpenko  Taras Shevchenko
National University of Kyiv, 03022 Kyiv, Ukraine  I. Mishustin  Frankfurt
Institute for Advanced Studies, Ruth-Moufang-Strasse 1, 60438 Frankfurt am
Main, Germany  H. Stoecker  Frankfurt Institute for Advanced Studies, Ruth-
Moufang-Strasse 1, 60438 Frankfurt am Main, Germany  Johann Wolfgang Goethe
University, D-60438 Frankfurt am Main, Germany

(September 3, 2024)

###### Abstract

The thermodynamics of a system of interacting bosonic particles and
antiparticles in the presence of the Bose-Einstein condensate is studied in
the framework of the Skyrme-like mean-field model. It is assumed that the
total charge density (isospin density) is conserved at all temperatures. Two
cases are explicitly considered: zero and nonzero isospin charge of the
system. A comparative analysis is carried out using Canonical Ensemble and
Grand Canonical Ensemble. It is shown that the Grand Canonical Ensemble is not
suitable for describing bosonic systems of particles and antiparticles in the
presence of a condensate.

relativistic bosonic system of particles and antiparticles, Bose-Einstein
condensation

The purpose of this letter is to report on the results of the study of the
thermodynamic properties of a bosonic system, in particular, the nature of
phase transitions during Bose-Einstein condensation with a conserved isospin
(charge) density. As was stated by Kerson Huang, the real conservation law
deals with the conserved quantity that is the number of particles minus the
number of antiparticles huang-1987 . That is why, any study of the Bose-
Einstein condensation in the relativistic Bose gas must take antiparticles
into account. Firstly this was discussed in haber-1981 . Our consideration of
the system of interacting bosonic particles and antiparticles at finite
temperatures is carried out within the framework of the thermodynamic mean-
field model, which was introduced in Anch-1992  and further developed in Anch3
. The self-consistent equations for the total particle density $n$ and the
charge density $n_{I}$ in this model are key elements for determining the
thermodynamic state of the system Anch4 ; Anch5 . When particles
($\pi^{-}$-mesons) and antiparticles ($\pi^{+}$-mesons) are in a thermal
(kinetic) phase, the corresponding expressions can be written as

$n=g\int\frac{d^{3}k}{(2\pi)^{3}}\,\left[f\big{(}E(k,n),\mu_{I}\big{)}\,+\,f\big{(}E(k,n),-\mu_{I}\big{)}\right]\,,$
(1)
$n_{I}=g\int\frac{d^{3}k}{(2\pi)^{3}}\,\left[f\big{(}E(k,n),\mu_{I}\big{)}\,-\,f\big{(}E(k,n),-\mu_{I}\big{)}\right]\,.$
(2)

The Bose-Einstein distribution functions $f\big{(}E,\mu_{I}\big{)}$

$f\big{(}E,\mu_{I}\big{)}\,=\,\left[\exp{\left(\frac{E-\mu_{I}}{T}\right)}-1\right]^{-1}\,,$

contains a term that describes the interaction in the system $U(n)$:
$E(k,n)=\omega_{k}+U(n),\,\omega_{k}=\sqrt{m^{2}+\bf k^{2}}$. It is important
to note that in the case of $\mu_{I}=0$, the number of particles and
antiparticles in the system are equal to each other, and the net charge in the
system (see Eq.(2)) is equal to zero, $n_{I}=0$. This case is discussed in
detail in Anch4 . There the system of particles and antiparticles was
considered using Grand Canonical Ensemble.

Below we consider Bose-Einstein condensation in a two-component system say
$\pi^{-}$ and $\pi^{+}$ mesons with non-zero isospin (charge) density
$n_{I}=n^{(-)}-n^{(+)}$. Then we can classify the system phase structure in
accordance with three basic combinations which determine the states of the
system and necessary conditions for developing the condensate, i.e., the
macroscopic population of the ground state ($\boldsymbol{k}=0$):
(a) both components, i.e., particles and antiparticles, are in the thermal
(kinetic) phase
($m+U(n)-\mu_{I}>0,\ m+U(n)+\mu_{I}>0$).
(b) Particles are in the condensate phase and antiparticles are in the thermal
(kinetic) phase
($m+U(n)-\mu_{I}=0,\,m+U(n)+\mu_{I}>0$).
(c) Both components, particles and antiparticles, are in the condensate phase
at $\boldsymbol{k}=0$
($m+U(n)-\mu_{I}=0,\,m+U(n)+\mu_{I}=0$), which is equivalent to the system of
equations

$\displaystyle\ \mu_{I}\,=\,0\,,$ (3) $\displaystyle\ U(n)\,+\,m\,=\,0\,.$ (4)

It is necessary to note, that expression “particles are in the condensate
phase” is, of course, a conventional one, because in the essence it is a
mixture phase. Indeed, at a fixed thermodynamic state, or at a fixed
temperature, some fraction, for instance of $\pi^{-}$ mesons, is in thermal
state, whereas another part of these mesons is in a condensed state.

We assume that the interaction between particles is described by the Skyrme-
like mean-field which depends only on the total particle-number density $n$

$U(n)\,=\,-\,A\,n\,+\,B\,n^{2}\,,$ (5)

where $A$ and $B$ are the positive model parameters. For given mean field (5)
there are two roots of equation (4)

$n_{1,2}\,=\,\sqrt{\frac{m}{B}}\left(\kappa\mp\sqrt{\kappa^{2}-1}\right)\,,$
(6)

where

$\kappa\,\equiv\,\frac{A}{2\,\sqrt{m\,B}}\,.$ (7)

Then, one can parameterize the attraction coefficient as $A=\kappa A_{\rm c}$
with $A_{\rm c}=2\sqrt{mB}$. The parameter $\kappa$ determines the phase
structure of the system, namely the allowed/forbidden particle density domain.
There are no real roots for the values of the parameter $\kappa<1$, and we
name this as a “weak” attraction. The critical value $A_{\rm c}$ is obtained
when both roots coincide, i.e. $\kappa=\kappa_{\rm c}=1$, then $A=A_{\rm
c}=2\sqrt{mB}$. And the second interval corresponds to $\kappa>1$, where eq.
(4) has two real roots. We associate this interval with a “strong” attractive
interaction. The dependence of the total particle density $n$ on the parameter
$\kappa$ for a system with zero charge $n_{I}=0$ is shown in Fig. 1 (left
panel).

Let us consider several possibilities.
(a) Both particles and antiparticles are present in the thermal phase. The
behavior of the particle-antiparticle bosonic system in thermal phase is
determined by the set of two transcendental equations (1), (2) (set (a)),
which should be solved selfconsistently with respect to $n$ and $\mu_{I}$ for
a given temperature $T$ and $n_{I}$. In the present study we consider bosons
with spin zero, i.e., the degeneracy factor $g=1$ for every boson component.
We would like to point out that in fact we consider the many-particle system
in the Canonical Ensemble, where the independent canonical variables are $T$
and $n_{I}$. In this approach the chemical potential $\mu_{I}$ is a
thermodynamic function, which depends on the canonical variables.

(b) Particles are in the condensate phase and antiparticles are in the thermal
phase. When the particles are in the condensate phase and antiparticles are
still in the thermal phase equations (1), (2) should be generalized to include
condensate component of $\pi^{-}$ mesons, $n^{(-)}_{\rm cond}$. It should be
taken into account that the particles (negatively charged component) can be in
a condensed state under the necessary condition

$m\,+\,U(n)\,-\,\mu_{I}\,=\,0\,.$ (8)

Let us look on the evolution of the particle-antiparticle system during
cooling it from high temperatures, where the two components are both in the
thermal phase. When the temperature decreases from high values, the density of
particles $n^{(-)}(T,n_{I})$, see Fig. 1 right panel, first reaches the
critical curve at temperature $T_{\rm c}^{(-)}$, where condition (8) is
fulfilled. This means that the curve $n_{\rm lim}(T)$, defined as

$n_{\rm
lim}(T)\,=\,\int\frac{d^{3}k}{(2\pi)^{3}}\,f\big{(}\omega_{k},\mu_{I}\big{)}\Big{|}_{\mu_{I}=m}\,,$
(9)

is indeed the critical curve for boson particles or for high-density component
because at the same temperature $T$ the both curves correspond to the states
with the same distribution function $f(T,\mu_{I}=m)$. As one can see, when
$\mu_{I}=m$ the Bose-Einstein distribution function gets the maximal value at
a fixed temperature $T$. This means the maximum density of thermal bosons that
can be reached at a fixed temperature $T$ in an ideal gas and in the system of
interacting bosons under consideration.

So, the crossing point of the curves $n^{(-)}(T,n_{I})$ and $n_{\rm lim}(T)$
determines the critical temperature $T_{\rm c}^{(-)}$ for high-density
component of the system. At $T<T_{\rm c}^{(-)}$ the part of $\pi^{-}$-mesons
in the thermal phase, $n^{(-)}_{\rm th}$, is limited from above by the
critical curve (9) that means $n^{(-)}_{\rm th}=n_{\rm lim}(T)$. Thus, the
density $n^{(-)}$ of $\pi^{-}$-mesons (the high-density component of the
gas)consists of two parts, the density of the condensed $\pi^{-}$ mesons,
$n^{(-)}_{\rm cond}$ and the density of thermal $\pi^{-}$ mesons,
$n^{(-)}_{\rm th}$, or $n^{(-)}=n^{(-)}_{\rm cond}(T)+n_{\rm lim}(T)$.
Therefore, we can write the generalization of the set of eqs. (1), (2) (set
(b)) as

$\displaystyle n$ $\displaystyle=$ $\displaystyle n^{(-)}_{\rm cond}(T)+n_{\rm
lim}(T)\,+\,\int\frac{d^{3}k}{(2\pi)^{3}}\,f\big{(}E(k,n),-\mu_{I}\big{)}\,,$
(10) $\displaystyle n_{I}$ $\displaystyle=$ $\displaystyle n^{(-)}_{\rm
cond}(T)+n_{\rm
lim}(T)-\int\frac{d^{3}k}{(2\pi)^{3}}\,f\big{(}E(k,n),-\mu_{I}\big{)}\,.$ (11)

Solution of set (b) for temperatures $T$ from the interval $T<T_{\rm c}^{(-)}$
provides the function $n^{(+)}(T,n_{I})$. Obviously, only thermal $\pi^{+}$
meson contribute to the density $n^{(+)}$. On the other hand, two fractions of
$\pi^{-}$-mesons contribute to the density $n^{(-)}$ : condensed particles
with the particle-number density $n^{(-)}_{\rm cond}(T)$, and thermal
particles with the density $n_{\rm lim}(T)$.

(c) Both particles and antiparticles are present in the condensate phase. In
this case, additionally to condition (8) for $\pi^{-}$ mesons, the argument of
the distribution function for $\pi^{+}$ mesons has to satisfy analogous
condition to guarantee that both components of the gas are in the condensate
at the same temperature $T$ and chemical potential $\mu_{I}$. These two
conditions are ($m+U(n)-\mu_{I}=0$, $m+U(n)+\mu_{I}=0$). With account for
these conditions, eqs. (10), (11) should be modified to include condensate
component $n^{(+)}_{\rm cond}$ of $\pi^{+}$ mesons assuming that the density
of thermal $\pi^{+}$ mesons equals now to $n_{\rm lim}(T)$, i.e. the same way
as the density of thermal $\pi^{-}$ mesons. So, when both components are in
the condensate, set (c) of selfconsistent equations reads

$\displaystyle n$ $\displaystyle=$ $\displaystyle n^{(-)}_{\rm
cond}(T)+n^{(+)}_{\rm cond}(T)+2\,n_{\rm lim}(T)\,,$ (12) $\displaystyle
n_{I}$ $\displaystyle=$ $\displaystyle n^{(-)}_{\rm cond}(T)\,-\,n^{(+)}_{\rm
cond}(T)\,.$ (13)

It turns out that solutions to cases (b) and (c) exist in the same temperature
interval. Indeed, in addition to self-consistent solutions of equations (b),
there are two more ”condensate” branches of solutions associated with the
roots $n_{1}$ and $n_{2}$ of equation $U(n)+m=0$, presented in (6). For
example, the corresponding particle densities of $\pi^{-}$ and $\pi^{+}$
mesons, which are the splitting of the total density $n_{2}$, are equal to
$n^{(-)}_{2}=(n_{2}+n_{I})/2$ and $n^{(+)}_{2}=(n_{2}-n_{I})/2$, respectively.
The same split of the total density $n_{1}$ creates another branch. Both of
these pairs of densities, which are a split of the roots $n_{1}$ and $n_{2}$,
satisfy eqs. (12), (13). Meanwhile, in the competition between these two
“condensate” branches, the second branch, created by $n_{2}$ splitting, is
preferable, since the pressure corresponding to these states is higher. On the
other hand, the competition between solutions (b) and (c) is resolved also in
the standard way according to the Gibbs criterion: the states that correspond
to the larger pressure, are predominant in the thermodynamic realization.
Using this rule we find the temperature $T_{\rm cd}$ from equation
$p_{(b)}(T,n_{I})=p_{(c)}(T,n_{I})$, where the pressure $p_{(b)}(T,n_{I})$
corresponds to solutions of the set of equations (b), and $p_{(c)}(T,n_{I})$
to the set of equations (c). For the temperatures larger than $T_{\rm cd}$ the
pressure (c) dominates, i.e. $p_{(c)}(T,n_{I})>p_{(b)}(T,n_{I})$. Therefore,
according to the Gibbs criterion, the transition to another branch of self-
consistent solutions gives rise to a phase transition of the first order at
the temperature $T_{\rm cd}$.

Results of the numerical solution of the sets of equations (a), (b) and (c) at
$\kappa=1.1$ ($B=10mv^{2}_{0}$, $m=140\,{\rm MeV}$, $v_{0}=0.45\,{\rm
fm^{3}}$) for the particle-number densities are depicted in Fig. 1 in the
right panel. The density $n^{(-)}(T)$ of $\pi^{-}$ mesons depicted as a solid
blue curve that consists of several horizontal segments and one vertical
segment, which represents a first-order phase transition. The density
$n^{(+)}(T)$ of $\pi^{+}$ mesons is depicted as a dashed blue line, which also
consists of several horizontal segments and one vertical segment, which also
represents phase transition of the first order. Metastable and forbidden
states belonging to “thermal” solutions are shown in Fig. 1 in both panels as
dashed and dotted segments that look like extensions of thermal branches. It
is seen from the figure that the system is really described at conserved
isospin (charge) density, i.e., for each temperature point on the graph, the
condition $n^{(-)}(T)-n^{(+)}(T)=0.1$ ${\rm fm^{-3}}$ is fulfilled. The
conservation of charge leads to the splitting of the roots (6) shown in the
left panel. It looks like: $n_{1}\to n^{(-)}_{1},\ n^{(+)}_{1}$ and $n_{2}\to
n^{(-)}_{2},\ n^{(+)}_{2}$, where the difference is constant, for example
$n^{(-)}_{2}-n^{(+)}_{2}=n_{I}$.

Figure 1: Left panel: Boson system at zero charge density $n_{\rm I}=0$ (or
$\mu_{I}=0$). Particle density vs. temperature at supercritical attraction
$\kappa=1.1$ is shown as a solid blue line consisting of several segments. The
vertical segment (solid blue line) indicates a phase transition of the first
order with the creation of the condensate. Dependencies of the particle
density on the temperature at ‘weak” attraction $\kappa\leq 1$ are shown as
solid black lines in the thermal phase. A dashed red line is the critical
curve. Right panel: Boson system at finite charge density $n_{\rm I}=0.1$
fm-3. Temperature dependencies of the density of negative $n^{(-)}$ and
positive $n^{(+)}$ particles are shown as a solid blue line consisting of
several segments and a dashed blue line consisting of several segments. The
vertical segment for both dependencies indicates a phase transition of the
first order with the creation of the condensate. In the condensate phase
$\mu_{I}=0$. A dashed red line is the critical curve $n_{\rm lim}(T)$, see eq.
(9).

In both panels of Fig. 1, in the condensate phase, both systems are
represented by zero chemical potential regardless of whether the particle-
antiparticle system described in the right panel has a finite charge density,
i.e., $n_{I}=0.1$ $\rm fm^{-3}$, while the particle-antiparticle system
described in the left panel is characterized by zero charge density, i.e.,
$n_{I}=0$. So if one intends to study both systems, one system with a finite
charge density and another with zero charge density within the Grand Canonical
Ensemble, then the canonical variables should be $(T,\,\mu_{I}=0)$ when
describing the condensate phase in both systems.

It seems that we come to some contradiction, because in the text books it is
written that the chemical potential should reflect the conservation of charge
or conservation of the particle number. This contradiction is resolved by
realizing that the Grand Canonical Ensemble with the canonical variables
$(T,\,\mu)$ is suited for the description of the thermal phase or for a
description of the particles, which are in the kinetic states, but not in the
condensate states. Indeed, it can be seen in Fig. 1 on the right panel in the
temperature interval that corresponds to the condensate phase, that is,
between points 2 and 3 on the graph, that for each temperature from this
interval the density of thermal $\pi^{-}$ mesons is equal to the density of
thermal $\pi^{+}$ mesons, both equal to $n_{\rm lim}(T)$ defined in (9). In
other words, these two densities of thermal mesons, which are characterized by
$\mu_{I}=0$, are equal to the density on the critical curve $n_{\rm lim}$,
which is depicted as the dashed red line in Fig. 1 (right panel). That is why
the charge density, which is determined only by thermal particles, is zero,
i.e., $n_{I}^{\rm(therm)}=n_{\rm lim}(T)-n_{\rm lim}(T)=0$. Therefore, the
chemical potential, which corresponds to the charge of these thermal
particles, is also zero. We can conclude that the chemical potential $\mu_{I}$
is a useful quantity only for describing thermal or kinetic particles.

When describing the same system at a finite charge density in the field-
theoretic approach mishustin-anchishkin-2019 , we encounter exactly the same
paradox. Indeed, for developing of the condensate by both particles and
antiparticles, two conditions must be met: $M^{2}-\mu_{I}=0$ and
$M^{2}+\mu_{I}=0$, where $M$ is the effective mass of quasiparticles. By
complete analogy with case (c) discussed above, these conditions lead to two
equations: $M^{2}=0$ and $\mu_{I}=0$. So, it turns out that the system with a
finite charge density $n_{I}\neq 0$ is characterized by zero value of the
chemical potential. On the other hand, we see that in the presence of
condensate, the density of thermal particles is the same in the negatively and
positively charged components of the system, i.e., $n^{(-)}_{\rm
th}(T)=n^{(+)}_{\rm th}(T)$. Hence, the problem can be resolved by accepting
that the chemical potential is responsible only for thermal (kinetic)
particles.

A similar situation arises when describing an ideal bosonic gas with a
constant density of particles $n$. In this case, for all temperatures in the
condensate phase, the chemical potential is equal to the mass of the particle
$\mu=m$. But this value of the chemical potential determines the maximum
density of only thermal particles for a given temperature $T$. Therefore, it
is not possible to determine the state of the system by the $(T,\,\mu)$
variables in the presence of condensate, since this pair of variables defines
any value of the total particle density $n$. This is due to the fact that the
chemical potential is not a free variable, its values are fixed by the
condition of condensate formation, i.e., $m-\mu=0$.

On the other hand, describing a relativistic ideal boson gas of particles and
antiparticles with conserved charge $n_{I}\neq 0$, we face one essential
limitation on the allowed states of the system, which is caused by the same
reasons. Indeed, if we assume that particles and antiparticles are
simultaneously in the condensate phase, then two conditions must be satisfied
simultaneously: $m-\mu_{I}=0$ and $m+\mu_{I}=0$, where $\mu_{I}$ is the
isospin chemical potential, which corresponds to $n_{I}$. This leads to two
equations: $m=0$ and $\mu_{I}=0$. As we can see, the first equation is not
possible or unphysical. So, we can formulate the theorem: in a relativistic
bosonic ideal gas of particles and antiparticles with a conserved charge
$n_{I}\neq 0$, only one component of the system can form a condensate phase.
The sign of the charge $n_{I}$ determines the answer, which component of the
system, particles or antiparticles, can be in the condensate. In addition, in
the thermodynamic mean-field model considered above, we have similar
conditions for the formation condensate by both components: $m+U(n)-\mu_{I}=0$
and $m+U(n)+\mu_{I}=0$. When the effective interaction in the system vanishes,
i.e., $U(n)\to 0$, these conditions are the reason for the same statement that
only one component of the bosonic particle-antiparticle system can develop a
condensate. And as we saw above, both components can be in a condensed state
only with the help of the attractive mean field, when its value is equal to
the mass of the particle, i.e. $m=|U(n)|$.

Conclusion.
We have shown how the relativistic interacting system of Bose particles and
antiparticles can be described at zero and finite isospin (charge) density. In
both cases the meson system develops a first order phase transition for
sufficiently strong attractive interactions via forming a Bose condensate with
releasing the latent heat. The model predicts that the condensed phase is
characterized by a constant density of particles.

We have demonstrated that the Grand Canonical Ensemble is not suitable for
describing a multi-component system in the presence of condensate phase. In
particular, it cannot describe the condensate state in the system of particles
and antiparticles. The reason is that the chemical potential is not a free
parameter in the condensate phase, its values are determined by the necessary
condition for condensate formation.

Details of the calculations presented in this letter will be published in
universe-2023 .

## Acknowledgements

D.A. is very grateful to J. Steinheimer and O. Philipsen for useful
discussions and comments. D.A. greatly appreciates the warm hospitality and
support provided by the FIAS administration and scientific community. The work
of D.A. and D.Zh. was supported by the Simons Foundation and by Program ”The
structure and dynamics of statistical and quantum-field systems” of the
Department of Physics and Astronomy of the NAS of Ukraine. I.M. thanks FIAS
for support and hospitality. H.St. thanks for support from the J. M. Eisenberg
Professor Laureatus of the Fachbereich Physik.

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