text:
\section{Introduction}

Let $R$ be a reduced root system in a
 finite-dimensional Euclidean space
 $(\frak a, \langle\,.\,,\,.\,\rangle)$ with  finite Coxeter group $W.$ Fix a  multiplicity function, i.e., a
  $W$-invariant function $k:R \to [0,\infty[$ and denote by $\{T_\xi(k), \xi\in \frak a\}$ the associated commuting family of
 rational Dunkl operators as introduced in \cite{D1}, see also \cite{D2, DX}.
Then there is a
 unique isomorphism on the vector space $\mathbb C[\frak a]$
  of polynomial functions on $\frak a$ which preserves the degree of homogeneity and satisfies
$$ V_k(1)=1, \quad T_\xi(k) V_k = V_k \partial_\xi \quad \text{ for all }
  \xi \in \frak a.$$
 By \cite{R1}, $V_k$ is  positive on $\mathbb C[\frak a],$ i.e. for $p\in\mathbb C[\frak a]$   with
 $p\geq 0$ we have $V_kp\geq 0$ on $\frak a$. This is equivalent to the fact that
 for each $x\in \frak a$ there is a unique compactly supported probability measure
$\mu_{x}^k\in M^1(\frak a)$ such that the Dunkl kernel $E_k$ associated with $R$ and $k$ has the positive integral representation
\begin{equation}\label{Dunkl_kernel_int} E_k(x,z) = \int_{\frak a} e^{\langle \xi,z\rangle } d\mu_x^k(\xi), \quad \forall \,
x \in \frak a, z\in \frak a_{\mathbb C}.
\end{equation}
The support of $\mu_x^k$ is contained in the convex hull  of
the $W$-orbit of $x$.
Eq.~\eqref{Dunkl_kernel_int} generalizes the Harish-Chandra integral representation of spherical functions  on symmetric spaces of Euclidean type (Ch.IV of \cite{Hel}), as
 for
certain  multiplicities $k$, the Bessel functions
\begin{equation}\label{Def-bessel-function}J_k(x,z) := \frac{1}{|W|}\sum_{w\in W} E_k(wx,z), \quad z\in \frak a_{\mathbb C},
\end{equation}
coincide with the spherical functions of a Cartan motion group, where $R$ and $k$ depend on the root space data of the
underlying symmetric space, see \cite{O, dJ2} for details. The Bessel function $J_k$ is $W$-invariant in both arguments, and in the
 geometric cases, the integral formula for $J_k$ obtained from
\eqref{Dunkl_kernel_int} by taking $W$-means is just the Harish-Chandra formula \cite[Prop. IV.4.8]{Hel}.

\medskip

We now consider two multiplicities $k, k^\prime$ on $R$
with  $k^\prime \geq k\geq 0$, i.e.,
$k^\prime(\alpha) \geq k(\alpha)\geq 0$ for all $\alpha \in R$.
In \cite{RV3} we studied the operator
$$ V_{k^\prime\!,k} := V_{k^\prime} \circ V_k^{-1} $$
which
 intertwines the
Dunkl operators  with multiplicities $k$ and $k^\prime$:
$$ T_\xi(k^\prime) V_{k^\prime\!, k} = V_{k^\prime\!, k} T_\xi(k) \quad (\xi \in \frak a).$$
It had been conjectured until recently that
 $V_{k^\prime\!, k}$ is also positive. This is equivalent
 to the statement that for each $x\in \frak a$, there is
a unique compactly supported probability measure $\mu_x^{k^\prime\!,k}\in M^1(\frak a)$ such that the Sonine formula
\begin{equation}\label{Sonine_E}  E_{k^\prime}(x,z) = \int_{\frak a} E_k(\xi, z)\, d\mu_x^{k^\prime\!,k}(\xi)
\quad \text{for all } z\in \frak a_{\mathbb C}
\end{equation}
holds.
Notice  that $\eqref{Sonine_E}$ yields a corresponding formula for the Bessel function:
\begin{equation}\label{Sonine_B}  J_{k^\prime}(x,z) = \int_{\frak a} J_k(\xi,z) \,d\widetilde \mu_x^{k^\prime\!,k}(\xi)
\quad (z\in \frak a_{\mathbb C})
 \end{equation}
with some $W$-invariant probability measure $\widetilde \mu_x^{k^\prime\!,k}.$ Denote by $\widetilde V_{k^\prime\!, k}$ the restriction of $V_{k^\prime\!, k}$ to $W$-invariant functions. As in \cite{RV3}, it is easy to see that
the existence of a positive Sonine formula \eqref{Sonine_B} for the Bessel functions is equivalent to the positivity of $\widetilde V_{k^\prime\!, k}$.
\medskip

In the rank-one case with
$R= \{\pm 1\}\subset \mathbb R$, we have
$\, J_k(x,y) = j_{k-1/2}(ixy) \,$
with the one-dimensional Bessel function
\begin{equation}\label{B_eindim} j_\alpha(z)  = \, _0F_1(\alpha+1;-z^2/4) \quad\quad
(\alpha\in\mathbb C\setminus\{-1,-2,\ldots\}).\end{equation}
Here \eqref{Sonine_B} is just the well-known classical
Sonine formula  (see e.g.~\cite{A}):
\begin{equation}\label{intrep-1-dim}
j_{\alpha+\beta}(z)= 2\frac{\Gamma(\alpha+\beta+1)}{\Gamma(\alpha+1)\Gamma(\beta)}
\int_{0}^1j_{\alpha}(zx)  x^{2\alpha+1}(1-x^2)^{\beta-1} dx
\end{equation}
for  $\alpha\in]-1,\infty[$ and $\beta \in ]0,\infty[$.
It was proven in \cite{X} that the operator $V_{k^\prime\!, k}$ with $k^\prime > k \geq 0$  is also  positive in the rank-one setting.

On the other hand, for the root systems
$$ B_n = \{ \pm e_i, \, \pm e_i \pm e_j, \, 1\leq i < j \leq n\} \subset \mathbb R^n$$
with $n\ge2$ and certain parameters $k^\prime \geq k \geq 0, $
\eqref{Sonine_E} and  \eqref{Sonine_B} were disproved
in \cite{RV3}. For these root systems we write the  multiplicities as $k=(k_1,k_2)$ with $k_1, k_2$ the values of $k$ on
 the roots $\,\pm e_i$ and $\,\pm e_i\pm e_j,$
respectively. The following result is shown in Section 3 of \cite{RV3} for the Bessel functions $J_{k}^B$ of type $B_n$.

\begin{theorem}\label{main_Bessel} Let $k=(k_1, k_2)$ with  $k_1 \geq 0, \> k_2 >0$, and consider $k^\prime = (k_1+h,k_2)$ with
  $h>0$. Assume that there exists
 a probability measure $m\in M_b(\mathbb R^n)$ such that the restricted Sonine formula
\begin{equation}\label{Sonine_1} J_{k^\prime}^B({\bf 1},iy) = \int_{\mathbb R^n}  J_{k}^B(\xi,iy)\> dm(\xi)
\quad\text{for all }y \in \mathbb R^n
\end{equation}
holds, where ${\bf 1} = (1, \ldots, 1) \in \mathbb R^n$. Then
$h$ is contained in the set
$$ \Sigma(k_2):= \,]k_2(n-1), \infty [ \, \cup \, \bigl(\{0, k_2, \ldots, k_2(n-1)\} -\mathbb Z_+\bigr).$$
\end{theorem}

Therefore, positivity of $\widetilde V_{k^\prime\!, k}$ or $V_{k^\prime\!, k}$ requires that  $h\in\Sigma(k_2)$.  Theorem \ref{main_Bessel}
 is related  to a classical result of Gindikin \cite{G} about  the Wallach set which parametrizes those Riesz distributions on a Euclidean Jordan algebra which are actually positive measures,
 see \cite{FK}.
 Related  results for Riesz and Beta
 distributions  on symmetric cones and in the Dunkl setting are given in \cite{RV2, R3}.

On the other hand, there are some positive Sonine formulas for $J_{k^\prime}^B$ in terms of $J_k^B$  with $k^\prime=(k_1+h,k_2)$ if $k_1 \geq 0$ and $h$ is large enough; these will be treated in Section \ref{large}.
 Namely, if $k_2\in\{1/2,1,2\}$ and $h>k_2(n-1)$, then  a Sonine formula
 follows from the fact that in this case
 the Bessel functions $J_{k}^B$ are closely related to Bessel functions
on matrix cones over $\mathbb R, \mathbb C, \mathbb H$
where  positive Sonine formulas are available by \cite{H, RV2}. Moreover, a restricted explicit Sonine formula of the form \eqref{Sonine_1}
was obtained in \cite{RV3} for arbitrary $k_1\geq 0, k_2 >0$  and $h>k_2(n-1)$ as a consequence of Kadell's \cite{Ka} generalization of the Selberg integral.

The main results of this paper are contained in Sections \ref{discrete} and \ref{infty}. In particular, in Theorem \ref{existence-pos-int-rep-lag} we prove that
 the Bessel function $J_{k}^B$ satisfies a Sonine formula for all   $k_1, k_2>0$ and $\,h\in k_2\cdot\mathbb Z_+$, which includes all parameters $h$ in the discrete part of the generalized Wallach set
 $$\{0,k_2,\ldots, k_2(n-1)\}\, \cup \, [k_2(n-1), \infty[\,.$$
 The proof of this result is based on a formula of Baker and Forrester \cite{BF1} on positive connection coefficients between multivariate Laguerre polynomial systems
and an approximation of Bessel functions by Laguerre polynomials.  Finally in Section \ref{infty}, we consider the limit $h\to\infty$
 under which the Bessel functions of type B tend to those of type A. This leads to a positive integral representation of Bessel functions of type $A$ in terms of such of type $B$.

\section{Basic facts on  Bessel functions}

We start with some background on  rational Dunkl theory  from  \cite{D1, D2, DJO, R1}.
Let  $R$ be  a reduced root system in a finite-dimensional Euclidean space $(\frak a, \langle\,.\,,\,.\,\rangle)$  and
$W$
the associated finite Coxeter group.
The Dunkl operators
 associated with $R$ and multiplicity  $k$ are
 $$ T_\xi(k) = \partial_\xi  + \frac{1}{2}\sum_{\alpha\in R}
 k(\alpha) \langle \alpha, \xi\rangle \frac{1}{\langle \alpha,\,.\,\rangle}(1-\sigma_\alpha) , \quad \xi \in \frak a
 $$
 where the action of $W$ on functions $f: \frak a \to \mathbb C$  is given by $w.f(x) = f(w^{-1}x).$
 It was shown in \cite{D1} that the  $T_\xi(k), \, \xi \in \frak a$,   commute.
 Multiplicities $k\ge0$ are regular, i.e.,
the joint kernel of the $T_\xi(k)$, considered as linear operators on
 polynomials, consists of the constants only.  This is equivalent to the existence of a necessarily unique
 intertwining operator
 $V_k$ as described in the introduction; see \cite{DJO}.
 Moreover, for each  $y \in \frak a_\mathbb C,$ there is a unique solution $f = E_k(\,.\,,y)$ of the
joint eigenvalue problem
$$ T_\xi(k)f = \langle \xi,y\rangle f\quad \forall \, \xi \in \frak a, \,\, f(0)=1.$$
The function $E_k$ is called the Dunkl kernel. The mapping $(x,y)\mapsto E_k(x,y)$ is
 analytic on $ \frak a_\mathbb C\times \frak a_\mathbb C$ with $ E_k(x,y) = E_k(y,x)$, $ E_k(x,0)=1$,  and
 $$   E_k(\lambda x,y) = E_k(x, \lambda y), \quad E_k(wx, wy) = E_k(x,y)\quad (\lambda \in \mathbb C, w \in W).$$

In this paper, we mainly consider the Dunkl kernel $E_k^B$ and the Bessel function $J_k^B$ associated with the root system
$B_n$ on  $\mathbb R^n$ with its usual inner product, c.f. \eqref{Def-bessel-function}.
The associated reflection group is the hyperoctahedral group $W(B_n)= S_n \ltimes \mathbb Z_2^n, $ and
the multiplicity has the form $k=(k_1, k_2)$ as explained in the introduction. In Section \ref{infty}, we shall also consider the Bessel function $J_k^A$ associated with the root system $$A_{n-1} = \{\pm(e_i-e_j), 1 \leq i < j \leq n\}\subset \mathbb R^n.$$ Here the multiplicity is given by a single parameter $k\in [0, \infty[.$
The Bessel functions $J_k^B$ and $J_k^A$ have simple expressions in terms of multivariate hypergeometric functions in the sense of \cite{K, M}. To recall these we need some more notation.
Let
$$\Lambda_n^+ =\{ \lambda\in \mathbb Z_+^n: \lambda_1 \geq \cdots \geq \lambda_n\}$$
be the set of partitions of length at most $n$. We denote
by $C_\lambda^\alpha, \, \lambda \in \Lambda_n^+$  the Jack polynomials of index $\alpha>0$ in $n$ variables (see \cite{Sta}),
normalized such that
\begin{equation}\label{normal}
(z_1 + \cdots + z_n)^m = \sum_{|\lambda|=m} C_\lambda^\alpha(z)\quad (m \in \mathbb Z_+).
\end{equation}

Following \cite{K}, we define for $\mu \in \mathbb C$ with
$\text{Re}\, \mu > \frac{1}{\alpha}(n-1)$ and arguments $z,w \in \mathbb C^n$ the hypergeometric function

\begin{equation}\label{def-hypergeometric-function}
    _0F_1^\alpha(\mu; z,w):= \sum_{\lambda\in \Lambda_n^+} \frac{1}{[\mu]_\lambda^\alpha\, |\lambda|!} \cdot \frac{C_\lambda^\alpha(z)
C_\lambda^\alpha(w)}{C_\lambda^\alpha({\bf 1})}, \quad {\bf 1} = (1, \ldots, 1) \in \mathbb R^n,\end{equation}
with
 the generalized Pochhammer symbol
\begin{equation}\label{general_Pochhammer}
 [\mu]_\lambda^\alpha = \prod_{j=1}^n \bigl(\mu - \frac{1}{\alpha} (j-1)\bigr)_{\lambda_j}.
\end{equation}
If  $n=1$, then the $C_\lambda^\alpha$ are independent of $\alpha$ and given by  $C_\lambda^\alpha(z) = z^\lambda, \, \lambda\in \mathbb Z_+$, and
$$ _0F_1^\alpha(\mu;-\frac{z^2}{4},1) = j_{\mu-1}(z).$$
Let $k=(k_1,k_2)$ on $B_n$ with $k_1\geq 0$ and $k_2>0$. Then according to \cite[Prop. 4.5]{R2} (c.f. also \cite[Sect. 6]{BF1}),
\begin{equation}\label{ident_Bessel_B} J_k^B(z,w) = \,_0F_1^\alpha\Bigl(\mu; \frac{z^2}{2}, \frac{w^2}{2}\Bigr)
	\end{equation}
with $\, \alpha = \frac{1}{k_2}\, $ and $\, \displaystyle  \mu=\mu(k)= k_1+k_2(n-1)+ \frac{1}{2}.$

Moreover,  for $n \geq 2$ and $k\in \,]0,\infty[,$ the Bessel function of type $A_{n-1}$ has the following hypergeometric expansion (\cite[Sect. 3]{BF2}):
 \begin{equation}\label{Bessel_A} J_k^A(z,w) = \,_0F_0^\alpha(z,w) = \sum_{\lambda \in \Lambda_n^+} \frac{1}{|\lambda|!}\cdot
\frac{C_\lambda^\alpha(z) C_\lambda^\alpha(w)}{C_\lambda^\alpha({\bf
    1})}
\quad \text{with }\, \alpha =1/k\,. \end{equation}

\section{Explicit Sonine formulas for large parameters}\label{large}

We first recapitulate a restricted Sonine formula of the form \eqref{Sonine_1} for  $J_k^B$ from \cite{RV3}.
 Let $k = (k_1, k_2)$ with $\,k_1 \geq 0$ and $k_2>0.$
For a real parameter  $h> k_2(n-1)$ consider the probability density
$$ f_{k,h}(x) := c_{k,h}^{-1}\cdot \prod_{i=1}^n (x_i^2)^{k_1} (1-x_i^2)^{h-k_2(n-1)-1}
\prod_{i<j} |x_i^2-x_j^2|^{2k_2}$$
on $[0,1]^n$ with the normalization constant (a Selberg-type integral)
$$ c_{k,h} = \int_{[0,1]^n} \prod_{i=1}^n (x_i^2)^{k_1} (1-x_i^2)^{h-k_2(n-1)-1}
\prod_{i<j} |x_i^2-x_j^2|^{2k_2}dx.$$
Then according to Eq. (3.4) in  \cite{RV3},
\begin{equation}\label{density} J_{(k_1+h, k_2)}^B({\bf 1},z) = \int_{[0,1]^n} J_k^B(x,z) f_{k,h}(x)dx\quad (z \in \mathbb C^n). \end{equation}
The proof of this formula  is based on \eqref{ident_Bessel_B} and Kadell's \cite{Ka} generalization of the Selberg integral, which implies a restricted Sonine formula for $\,_0F_1^\alpha.$
 The restricted Sonine formula
 \eqref{density} can be extended to a complete  Sonine formula, i.e. a Sonine formula for Bessel functions with arbitrary second argument, in the cases $k_2=1/2,1,2\,$ via
 Bessel functions on matrix cones.
 To explain this, we recapitulate some notations from \cite{FK, R2, RV2}.
For the (skew) fields $\mathbb F:= \mathbb R, \mathbb C, \mathbb H$
 with real dimension $d=1,2,4$  consider the vector spaces $H_n:=H_n(\mathbb F)$ of all Hermitian
$n\times n$ matrices over  $\mathbb F$ as well as the
cones
$\Pi_n:=\Pi_n(\mathbb F)\subset H_n$
 of all positive semidefinite matrices. Let $\Omega_n$ be the interior of $\Pi_n$ consisting of all strictly positive
definite matrices.
The Bessel functions associated with the symmetric cone $\Omega_n$ (in the sense of \cite{FK}) are defined in terms of the spherical polynomials
\[ \Phi_\lambda (a) = \int_{U_n} \Delta_\lambda(uau^{-1})du \quad (\lambda \in \Lambda_n^+, \, \, a\in H_n).\]
Here
$du$ is the normalized Haar measure of the compact group $\,U_n=U_n(\mathbb F)$ and $\Delta_\lambda$ denotes the power function
\[ \Delta_\lambda(a) := \Delta_1(a)^{\lambda_1-\lambda_2}
\Delta_2(a)^{\lambda_2-\lambda_3} \cdot\ldots\cdot \Delta_q(a)^{\lambda_q}\]
 with the principal minors $\Delta_i(a)$.
We renormalize the spherical polynomials according to
$\, Z_\lambda := c_\lambda \Phi_\lambda$
with  $c_\lambda >0$
such that
$$(tr \,a)^k \,=\, \sum_{|\lambda|=k} Z_\lambda(a)
\quad\quad\text{for}\quad
 k\in \mathbb N_0\,;$$
c.f. Section XI.5.~of \cite {FK}.
The $Z_\lambda$
 depend only on the eigenvalues of their argument and are given in terms of Jack polynomials: for $a \in H_n$ with eigenvalues $x = (x_1, \ldots, x_n) \in \mathbb R^n$,
$$Z_\lambda(a) =
C_\lambda^\alpha(x) \quad \text{with } \alpha = \frac{2}{d}\,,$$
see  \cite {FK, M2}. Let
$$ \mu_0:= \frac{d}{2}(n-1).$$
Then for $\mu > \mu_0$, the Bessel function $J_\mu$ associated with $\Omega_n$
is defined according to  \cite{FK} as the $\, _0F_1$-hypergeometric series \begin{equation}\label{power-j}
 J_\mu(a) =
 \sum_{\lambda\in \Lambda_n^+} \frac{(-1)^{|\lambda|}}{[\mu]_\lambda^{2/d}\,|\lambda|!}\,
Z_\lambda(a) \quad (a\in H_n).
\end{equation}
Notice that here
$[\mu]_\lambda^{2/d} \not=
0$ for
all  $\lambda$. Comparing formulas \eqref{power-j} and \eqref{ident_Bessel_B}, one obtains
 for $a\in H_n$ with eigenvalues $x \in \mathbb R^n$ and $\mu\geq \mu_0 + \frac{1}{2}$ the identity
 \begin{equation}\label{D_identity} J_\mu(a^2) = J_k^B(2ix,{\bf 1}) \quad \text{ with } \,k=k(\mu,d) =
\bigl(\mu - \mu_0 - \frac{1}{2}, \frac{d}{2}\bigr); \end{equation}
see \cite[Cor. 4.4]{R2}. Recall that $J_k^B$ is invariant under $W(B_n)$ in both variables. We denote the  associated standard Weyl chamber
by
 $$ C_n^B:= \{ x \in \mathbb R^n: x_1 \geq \ldots \geq x_n\geq 0\}.$$
Now consider $r,s\in \Pi_n$ with ordered eigenvalues $x=(x_1,\ldots, x_n), \, y = (y_1, \ldots, y_n) \in C_n^B$. Then
 by Corollary 4.6. of \cite{R2},
\begin{equation}\label{D_int} J_k^B(2ix,y)=\int_{U_n} J_\mu(r us^2u^{-1}r)\, du \quad \text{ with } k=k(\mu,d).\end{equation}
For indices  $\mu, \nu > \mu_0$
we introduce the Beta-Riesz probability measures
\begin{equation}\label{def-beta-dist}
d\beta_{\mu,\nu}(r):= \frac{1}{ B_\Omega(\mu,\nu)}  \Delta(r)^{\mu-1-\mu_0}\Delta(I-r)^{\nu-1-\mu_0}
dr\Bigl|_{\Omega_n^I} \end{equation}
on the relatively compact set $\Omega_n^I:=\{r\in \Omega_n: \> I_n-r\in  \Omega_n\},$ where
 $$ B_\Omega(\mu,\nu):=\int_{\Omega_n^I} \Delta(r)^{\mu-1-\mu_0}\Delta(I-r)^{\nu-1-\mu_0}
dr. $$
 We recall the following Sonine formula from
 \cite[Theorem 1]{RV2}; see also (2.6') of \cite{H} for $\mathbb F=\mathbb R$.

\begin{proposition}\label{int-rep-classical}
For all $\mu,\nu >\mu_0$  and $r\in\Pi_n$,
$$J_{\mu+\nu}(r)=
\int_{ \Omega_n^I } J_\mu(r s) d\beta_{\mu,\nu}(s).$$
\end{proposition}

Here we adopt a common notation in the literature and consider the Bessel function $J_\mu$ as a function of the eigenvalues of the (not necessarily Hermitian) matrix $rs$, which
coincide with those of the Hermitian matrices $\sqrt s \,r\sqrt s$ and
$\sqrt r \,s\sqrt r.$

From Proposition \ref{int-rep-classical} and formula \eqref{D_int} we now derive a Sonine representation for $J_k^B$.
For this, consider
the map $\,\sigma:\Pi_n\to C_n^B$ where $\sigma(r)$ is the ordered spectrum of $r\in \Pi_n$. We shall identify $x\in \mathbb R^n$ with the diagonal matrix $\text{diag}(x_1, \ldots, x_n)\in H_n$.
 Theorem VI.2.3 of \cite{FK} states that
\begin{equation}\label{polar} \int_{\Pi_n} f(r) \,dr\,=\,c_0 \int_{U_n}\int_{ C_n^B} f(u x u^{-1})\cdot
\prod_{i<j} (x_i-x_j)^d \,dx\, du \end{equation}
for integrable functions $f:\Pi_n\to\mathbb C, $ with some constant $c_0>0$. Let
$$C_{n,1}^B:=\{x\in C_n^B:\> x_1\le1\} = \{ x \in \mathbb R^n: 1 \geq x_1 \geq \cdots \geq x_n \geq 0\}.$$
Then by \eqref{polar}, the image measure of $\beta_{\mu,\nu}$ under $\sigma$
is  the probability measure
\begin{equation}
d\rho_{k_1,k_2,h}(x)= \frac{1}{B_{k_1,k_2,h}}  \prod_{i=1}^n x_i^{k_1-1/2}(1-x_i)^{h-1-\mu_0} \cdot \prod_{i<j}(x_i-x_j)^{2k_2}
dt\Bigl|_{C_{n,1}^B} \end{equation}
on $C_{n,1}^B$
with the
normalization constant $B_{k_1,k_2,h}=B_\Omega(\mu,\nu)/c_0$ where $(k_1,k_2)=k(\mu,d)$ as in \eqref{D_identity} and $h = \nu$. We obtain

\begin{theorem}\label{int-rep-classical-chamber} Let $\,\mathbb F \in \{\mathbb R, \mathbb C, \mathbb H\}$ and $k_2 = d/2$ with $d= \text{dim}_{\mathbb R} \mathbb F.$
Let further $k_1\ge 0$ and $h> \mu_0=\frac{d}{2}(n-1)$.
Then for all $z\in \mathbb C^n$ and $x\in C_n^B$,
$$ J_{(k_1+h,k_2)}^B(x,z) = \int_{C_{n,1}^B} \int_{U_n(\mathbb F)} J_{(k_1,k_2)}^B
\bigl(\sqrt{\sigma(x u\xi u^{-1}x)}\,,z\bigr)\,du\,
   d\rho_{k_1,k_2,h}(\xi).$$
\end{theorem}

\begin{proof}  Let $x,y\in  C_n^B$ and put $\mu=k_1+\mu_0+\frac{1}{2}.$ Formulas \eqref{D_int}, \eqref{polar}  and Proposition \ref{int-rep-classical} imply that
\begin{align*}
J_{(k_1+h,k_2)}^B(x,2iy) \,&
=\,  \int_{U_n}\int_{\Omega_n^I} J_{\mu}\bigl(
x u y^2 u^{-1}xs\bigr)\, d\beta_{\mu,h}(s)\,du \\ &= \,
\int_{U_n}\int_{\Omega_n^I} J_{\mu}\bigl(
 u y^2 u^{-1}x sx \bigr)\, d\beta_{\mu,h}(s) \,du\\
 &= \,
\int_{U_n}\int_{U_n}\int_{C_{n,1}^B} J_\mu\bigl(uy^2u^{-1}\cdot  x v\xi v^{-1}x\bigr) \,d\rho_{k_1,k_2, h}(\xi)\,dv\, du \\
&=\,\int_{C_{n,1}^B} \int_{U_n} J_{(k_1,k_2)}^B
\bigl(\sqrt{\sigma(x v  \xi v^{-1}x)}\,,2iy\bigr)\,dv\,
   d\rho_{k_1,k_2,h}(\xi).
   \end{align*}
 Analytic extension in the first argument now yields the assertion.
\end{proof}

Note that for  $k_2 =1/2,1,2$ and $\eta={\bf 1}$, the integral representation in Theorem \ref{int-rep-classical-chamber}
coincides with formula \eqref{density}.

\section{Positive Sonine formulas for discrete parameters}\label{discrete}

Besides the  Sonine formulas in Theorem \ref{int-rep-classical-chamber} and Eq. \eqref{density},
Theorem \ref{main_Bessel} admits the following  partial converse statement:

\begin{theorem}\label{existence-pos-int-rep-lag} Let $k=(k_1,k_2)$ with $k_1\geq 0, k_2 >0$ and $\,h\in k_2\cdot\mathbb Z_+ =\{0, k_2, 2k_2,\ldots\}$.
 Then
  for each $x\in C_n^B$ there exists a unique probability measure $m_x\in M^1(C_n^B)$ such that
\begin{equation}\label{Sonine_1-allg} J_{(k_1+h,k_2)}^B(x,z) =
 \int_{C_n^B}  J_{k}^B(\xi,z) \,dm_x(\xi)\end{equation}
for all $z\in\mathbb C^n.$ The support of $m_x$ is contained in
$[0,x]:= [0,x_1]\times\ldots\times [0,x_n].$
\end{theorem}

The uniqueness statement in  Theorem \ref{existence-pos-int-rep-lag} follows from the injectivity of the Dunkl transform of bounded measures, see \cite[Theorem 2.6]{RV0}.
For the proof of the existence part, it suffices to consider $ z= iy$ with $y \in \mathbb R^n$ and prove the case $h=k_2$.
In this case, the theorem can be derived from an explicit formula of Baker and Forrester \cite[formula (4.23)]{BF1}
on the connection coefficients between multivariate Laguerre polynomial systems. To start with, recall that according to \cite[Proposition 4.3]{BF1},
the Laguerre polynomials in $n$ variables associated with parameters $\alpha >0$ and $a > -1$ are given by
\begin{equation}\label{def-Laguerre}
L_\kappa^{a}(x; \alpha)=\frac{[a+q]_\kappa^{\alpha}}{|\kappa|!}
\sum_{\lambda\subseteq \kappa} \binom{\kappa}{\lambda}
\frac{(-1)^{|\lambda|}}{[a+q]_\lambda^{\alpha}}\,\frac{C_\lambda^\alpha(x)}{ C_\lambda^\alpha({\bf 1}) }\quad (\kappa \in \Lambda^+_n\,, \, \, x\in \mathbb R^n)
\end{equation}
where $ q = 1 + (n-1)/\alpha,\,$ the notion $\lambda\subseteq \kappa$ means that the diagram of $\lambda$ is contained in that of $\kappa$, i.e.
$\lambda_i \leq \kappa_i$ for all $i=1, \ldots, n,$ and the generalized binomial coefficients $\,\binom{\kappa}{\lambda}=\binom{\kappa}{\lambda}_\alpha $ are defined by
the binomial formula for the Jack polynomials,
$$ \frac{C_\kappa^\alpha (x +{\bf 1})}{C_\kappa^\alpha({\bf 1})} = \sum_{\lambda \subseteq \kappa}\binom{\kappa}{\lambda}\,
\frac{C_\lambda^\alpha(x)}{C_\lambda^\alpha({\bf 1})}.$$
We consider the renormalized Laguerre polynomials
$$ \widetilde L_\kappa^a(x;\alpha) := \frac{L_\kappa^a(x;\alpha)}{L_\kappa^a(0;\alpha)} = \sum_{\lambda\subseteq \kappa}
\binom{\kappa}{\lambda}\,
\frac{(-1)^{|\lambda|}}{[a+q]_\lambda^{\alpha}}\,\frac{C_\lambda^\alpha(x)}{ C_\lambda^\alpha({\bf 1}) }$$
and choose the parameters according to those in
\eqref{ident_Bessel_B}, namely $\,\alpha:= \frac{1}{k_2},$ $\, a := k_1 -\frac{1}{2}.$
Then $\, a+q = k_1 + k_2(n-1) + \frac{1}{2} = \mu(k) =: \mu.$

\smallskip

The first step towards the proof of Theorem \ref{existence-pos-int-rep-lag}
is the
 following limit result for  Laguerre polynomials which generalizes  Proposition 3.3 of \cite{Far}
for $\alpha = 1.$ In the one-dimensional case $n=1$, this limit transition is well-known.

\begin{lemma}\label{laguerre-to-bessel}
 Let $k= (k_1,k_2)$ with $k_1\geq 0, k_2 >0, $ put $ \, \alpha:= \frac{1}{k_2}\,, \,a:= k_1-\frac{1}{2}$ and fix $\,y\in \mathbb R^n$. For $x \in C_n^B\,$
 consider the sequence of partitions $(\lambda_j(x))_{j\in \mathbb N} \subseteq \Lambda^+_n$ with
$\lambda_j(x):= \lfloor j\cdot x\rfloor$ where Gaussian brackets are taken componentwise.
 Then
 $\lim_{j\to\infty} \lambda_j(x)/j=x$, and
$$\lim_{j\to\infty}\widetilde L_{\lambda_j(x)}^{a}(y^2/j;\alpha) =\,J_k^B(iy,2\sqrt{x}\,)$$
locally  uniformly in $x$ (here the square root is also taken componentwise).
\end{lemma}

\begin{proof} We adopt the method from \cite{Far}. First, recall the monic renormalization of the Jack polynomials
$C_\lambda^\alpha$, which is  determined by
$$ P_\lambda^\alpha = m_\lambda + \sum_{\mu < \lambda} d_{\lambda\mu}(\alpha) m_\mu $$
where the $m_\lambda$ are the monomial symmetric functions $\, m_\lambda(z) = \sum_{w\in S_n} z^{w.\lambda} $ and $\mu < \nu$ means that
$\mu$ is strictly lower than $\nu$ in the usual dominance order on $\Lambda_n^+$.
According to \cite[formula (16)]{K} and the identities on p. 15 of \cite{KS},
 $$ P_\lambda^\alpha = \frac{c_\lambda^\prime}{\alpha^{|\lambda|}|\lambda|!}\, C_\lambda^\alpha \quad\text{with} \quad
  c_\lambda^\prime = \prod_{s\in \lambda} (\alpha(a_\lambda(s) +1) + l_\lambda(s)),$$
 where $a_\lambda(s)$ and $l_\lambda(s)$ denote the arm-length and leg-length of $s\in \lambda$, c.f. \cite{KS, Sah} for the definitions.
 According to \cite{OO}, the  generalized binomial coefficients $ \binom{\kappa}{\lambda}$  can be written in terms of the so-called shifted
 Jack polynomials
 $P_\lambda^*(x,1/\alpha)$ as
 $$ \binom{\kappa}{\lambda} = \frac{P_\lambda^*(\kappa;1/\alpha)}{P_\lambda^*(\lambda;1/\alpha)}.$$
 Moreover,
 $$ P_\lambda^*(\lambda; 1/\alpha) = \frac{1}{\alpha^{|\lambda|}} c_\lambda^\prime.$$
The shifted Jack polynomials are of the form (c.f. \cite{OO})
$$ P_\lambda^*(z;1/\alpha) = P_\lambda^\alpha (z) + \text{terms of lower degree in dominance order.} $$
Therefore
\begin{equation}\label{limitbinom} \lim_{j\to\infty} \frac{1}{j^{|\lambda|}}\binom{\lambda_j(x)}{\lambda}  = \,\frac{\alpha^{|\lambda|}}{c_\lambda^\prime} \cdot
\lim_{j\to \infty} P_\lambda^\alpha\bigl(\frac{\lambda_j(x)}{j}\bigr) = \frac{\alpha^{|\lambda|}}{c_\lambda^\prime} \cdot
P_\lambda^\alpha(x) = \frac{C_\lambda^\alpha(x)}{|\lambda|!},
 \end{equation}
and the convergence is locally uniform in $x$.
With $\alpha = \frac{1}{k_2}, \,  a= k_1-\frac{1}{2}$ and $\mu=\mu(k)= k_1+k_2(n-1) + \frac{1}{2}\,$ we thus obtain
\begin{align} \label{limit_laguerre}\lim_{j\to \infty} \widetilde L^a_{\lambda_j(x)} \bigl(y^2/j;\alpha\bigr) &=
\lim_{j\to\infty} \sum_{\lambda\subseteq \lambda_j(x)}  \frac{1}{j^{|\lambda|}}\binom{\lambda_j(x)}{\lambda}
\frac{(-1)^{|\lambda|}}{[\mu]_\lambda^\alpha} \frac{C_\lambda^\alpha(y^2)}{C_\lambda^\alpha({\bf 1})} \notag\\
& = \sum_{\lambda\in \Lambda_n^+}
\frac{(-1)^{|\lambda|}}{[\mu]_\lambda^\alpha} \cdot \frac{C_\lambda^\alpha(x)\,C_\lambda^\alpha(y^2)}{|\lambda|!\,
C_\lambda^\alpha({\bf 1})}\, = \, J_k^B(iy, 2\sqrt{x}).
\end{align}
For the uniformity statement, we proceed as in \cite{V}. We use that the binomial coefficients satisfy
$ \binom{\kappa}{\lambda} \geq 0 $ for $\lambda \subseteq \kappa$ (Theorem 5 of \cite{S2}) as well as
$$ \sum_{|\lambda|=m} \binom{\kappa}{\lambda} = \binom{|\kappa|}{m}, \quad m \in \mathbb Z_+ $$
from \cite{L1}. Note also that $\, [\mu]_\lambda^\alpha \geq \, (\frac{1}{2})^{|\lambda|}.$ As the coefficients of the $C_\lambda^\alpha$ in their monomial expansion are nonnegative, we may estimate
$$ \Big\vert\frac{C_\lambda^\alpha(y^2)}{C_\lambda^\alpha({\bf 1})}\Big\vert \leq \|y\|_\infty^{\,2|\lambda|}\,.$$
We therefore obtain
\begin{align*} \sum_{\lambda\subseteq \lambda_j(x)}  \frac{1}{j^{|\lambda|}}\binom{\lambda_j(x)}{\lambda} &
\frac{1}{[\mu]_\lambda^\alpha}\, \Big\vert\frac{C_\lambda^\alpha(y^2)}{C_\lambda^\alpha({\bf 1})}\Big\vert \, \leq \,
 \sum_{m=0}^\infty \,\frac{2^m}{j^m} \,\|y\|_\infty^{2m}\cdot \sum_{|\lambda|=m} \binom{\lambda_j(x)}{\lambda}\, \notag\\
&=\,\sum_{m=0}^\infty \Bigl(\frac{2\|y\|_\infty^2}{j}\Bigr)^{\!m} \cdot \!\binom{|\lambda_j(x)|}{m}\, \leq \,\sum_{m=0}^\infty
\frac{(2n\|y\|_\infty^2\|x\|_\infty)^m}{m!}\,.
\end{align*}
This estimate shows that the convergence in \eqref{limit_laguerre} is locally uniform in $x\in C_n^B.$
\end{proof}

The following observation is the central ingredient for the proof of
Theorem \ref{existence-pos-int-rep-lag}. It follows immediately from   identity (4.23) of \cite{BF1}.

\begin{lemma}\label{pos-connection-Lagu}
For each partition $\kappa\in \Lambda^+_n$ there exist connection coefficients $c_{\kappa,\lambda}= c_{\kappa,\lambda}(a,\alpha)\ge0$ with
$\,\sum_{\lambda\subseteq\kappa}c_{\kappa,\lambda}=1\,$
such that
$$\widetilde L_{\kappa}^{a+ 1/\alpha}(x; \alpha) \,= \,
\sum_{\lambda\subseteq\kappa}c_{\kappa,\lambda}\,\widetilde L_{\lambda}^{a}(x;\alpha)  \quad\quad(x\in \mathbb R^n) .$$
\end{lemma}

 \begin{proof}[Proof of Theorem \ref{existence-pos-int-rep-lag}]
Let $k=(k_1,k_2), \,a:= k_1-\frac{1}{2}$ and  $\alpha = \frac{1}{k_2}.$  Fix $y\in \mathbb R^n.$ For
$x\in C_n^B\,$  consider the sequence of partitions
$(\lambda_j= \lambda_j(x))_{j\in \mathbb N}$
   as described in Lemma \ref{laguerre-to-bessel} with
 $\lim_{j\to\infty} \lambda_j/j=x$.
 We
 proceed  as in \cite{RV3} and introduce the discrete probability measures
 $$  \mu_j:= \mu_j(x):=\sum_{\lambda \subseteq \lambda_j}c_{\lambda_{j},\lambda}\,\delta_{\lambda/j}
 \in M^1(\mathbb R^n), \quad j \in \mathbb N;$$
 here $\delta_\xi$ denotes the point measure in $\xi\in \mathbb R^n.$
The supports of the $\mu_j$  are contained in the compact set $\,[0, x] \cap C_n^B\,.$
In terms of these measures,   Lemma \ref{pos-connection-Lagu} implies that
  \[
\widetilde L_{\lambda_{j}}^{a+k_2}(\xi;\alpha)\, = \, \int_{[0,\eta]}
\widetilde L_{jw}^{a}(\xi;\alpha) d\mu_j(w), \quad \xi \in \mathbb R^n.
 \]
By Prohorov's theorem (see e.g.~\cite{Kal}),
  the set $\{\mu_j:\> j\in  \mathbb N\}$ is relatively sequentially compact.
  After passing to a subsequence if necessary, we obtain that the $\mu_j$ tend weakly to a probability measure
  $m_x\in M^1([0,x])$  as $j\to \infty.$
Using Lemma \ref{laguerre-to-bessel}, we conclude that
 \begin{align*}
 J_{(k_1+k_2, k_2)}^B\bigl(iy, 2\sqrt{x}\,\bigr) &=\lim_{j\to\infty}\widetilde L_{\lambda_{j}}^{a+k_2}(y^2/j;\alpha)\,  =\,
  \lim_{j\to\infty}
 \int_{[0,x]} \widetilde L_{jw}^a(y^2/j;\alpha)\, d\mu_j(w)
 \\
& =\int_{[0,x]}  J_{k}^B\bigl(iy,2\sqrt{w}\,\bigr)\,dm_x(w).
\notag
\end{align*}
This readily implies the assertion.
\end{proof}

\begin{remark} Our results on complete Sonine formulas for $J_k^B$ in Theorems \ref{int-rep-classical-chamber} and \ref{existence-pos-int-rep-lag} do not cover the case $k=(k_1, k_2)$ with  arbitrary $k_1 \geq 0, k_2 >0$ and $h >k_2(n-1).$ We conjecture that in this case, a positive Sonine formula exists as well.

\end{remark}

We conclude this section with an immediate consequence of
Theorem \ref{main_Bessel}, Lemma \ref{laguerre-to-bessel}, and the proof of Theorem \ref{existence-pos-int-rep-lag}:

\begin{corollary}\label{main_Laguerre} Let $a\geq -1/2,\, \, \alpha >0$  and $h>0.$  Assume that
for each partition $\kappa\in \Lambda_n^+$ there exist nonnegative connection coefficients $c_{\kappa,\lambda}\ge0$ such that
$$L_{\kappa}^{a+h}(x;\alpha)=
\sum_{\lambda\subseteq\kappa}c_{\kappa,\lambda} L_{\lambda}^{a}(x;\alpha) \quad\quad  (x\in \mathbb R^n).$$
Then  $h$ is contained in the set
$ \,\Sigma(\alpha^{-1})$ of Theorem \ref{main_Bessel}.
\end{corollary}

\section{The limit $h\to\infty$}\label{infty}

We finally turn to some application of Theorem \ref{existence-pos-int-rep-lag} for $h\to\infty$ which is based on the fact that
the Bessel function $ J_{(k_1,k_2)}^B$ of type $B_n$ tends to a Bessel function of type $A_{n-1}$ as $k_1\to \infty.$

Indeed, the following limit relation follows easily from a comparison of the coefficients in \eqref{Bessel_A} and
 \eqref{def-hypergeometric-function} and is well-known;
 see e.g.~\cite{RV1} where also estimations for the rate of convergence are given.

\begin{lemma}\label{b-to-a}
Let $n\geq 2$ and $k_2> 0$. Then, locally uniformly in $x,y\in\mathbb R^n$,
$$\lim_{k_1\to+ \infty} J_{(k_1,k_2)}^B(2\sqrt{k_1}\,x, iy)= J_{k_2}^A(x^2,-y^2).$$
 \end{lemma}

Using Prohorov's theorem
as in the proof of Theorem  \ref{existence-pos-int-rep-lag},
 we obtain the following  integral representation from Lemma \ref{b-to-a} and  Theorem  \ref{existence-pos-int-rep-lag}:

\begin{theorem} \label{intrep-a-to-b}
Let $k=(k_1,k_2)$ with $k_1,k_2>0$. Then
  for each $x\in C_n^B$ there exists a unique probability measure $\mu_x = \mu_x(k) \in M^1(  C_n^B)$ such that
\begin{equation}\label{Sonine_limit-allg}
 J_{k_2}^A(x^2,-y^2) =
 \int_{ C_n^B}  J_{(k_1,k_2)}^B(\xi,iy)\, d\mu_x(\xi) \quad\text{for all } \, y\in\mathbb R^n .\end{equation}
\end{theorem}

\begin{proof} The uniqueness again follows from the injectivity of the Dunkl transform of measures. For the existence, fix $k_1,k_2$ as well as $x\in C_n^B$. Theorem \ref{existence-pos-int-rep-lag}
shows that for each $j\in\mathbb N$ there is a probability measure $\mu_j\in M^1(C_n^B)$ such that
\begin{equation}\label{int-rep-j}
 J_{(k_1+jk_2,k_2)}^B\bigl(2\sqrt{k_1+jk_2}\cdot x\,, iy\bigr)\,=
\int_{ C_n^B}  J_{(k_1,k_2)}^B(\xi,iy) \,d\mu_j(\xi) \quad \forall \,y\in\mathbb R^n,
\end{equation}
where the support of $\mu_j$ is contained in  $\,\big[0,\,2\sqrt{k_1+jk_2}\cdot x\big]$. We prove that the sequence $(\mu_j)$
is tight, which implies that it has a subsequence which converges weakly to some probability measure $\mu_x\in M^1(C_n^B)$;
for tightness and the existence of convergent subsequences we refer to Section 4  of \cite{Kal}. The arguments at the end of the proof of
 Theorem  \ref{existence-pos-int-rep-lag} in combination with Lemma \ref{b-to-a} will then lead to
$$  J_{k_2}^A(x^2, -y^2) =
 \int_{ C_n^B}  J_{(k_1,k_2)}^B(\xi,iy)\, d\mu_x(\xi)$$
 as claimed.
In order to check the tightness of $(\mu_j)$, we recapitulate from formulas \eqref{def-hypergeometric-function} and  \eqref{ident_Bessel_B}
that for $k=(k_1,k_2),$
\begin{equation}
 J_{k}^B(z,w) = \,\sum_{\lambda\in \Lambda_n^+} \frac{1}{[\mu(k)]_\lambda^\alpha \,|\lambda|!\,4^{|\lambda|}} \cdot \frac{C_\lambda^\alpha(z^2)
C_\lambda^\alpha(w^2)}{C_\lambda^\alpha({\bf 1})} \quad (z,w\in \mathbb C^n)
\end{equation}
with $\mu(k)=k_1+k_2(n-1)+\frac{1}{2}$. Hence, by (\ref{int-rep-j}),
\begin{align}
\sum_{\lambda\in \Lambda_n^+} &\frac{(k_1+jk_2)^{|\lambda|}}{[\mu(k_1+jk_2,k_2)]_\lambda^\alpha\, |\lambda|!}\cdot
 \frac{C_\lambda^\alpha(x^2)\,
C_\lambda^\alpha(-y^2)}{C_\lambda^\alpha({\bf 1})}\notag\\
=& \sum_{\lambda\in \Lambda_n^+} \frac{1}{[\mu(k)]_\lambda^\alpha \,|\lambda|!\,4^{|\lambda|}} \cdot
\frac{C_\lambda^\alpha(-y^2)}{C_\lambda^\alpha({\bf 1})} \cdot \int_{C_n^B} C_\lambda^\alpha(\xi^2) \> d\mu_j(\xi)\notag
\end{align}
for all $y\in [0,\infty[^n$. If we compare the coefficients of this power series in $y$ for $\lambda=(1,0,\ldots,0)$
and use that $C_{(1,0,\ldots,0)}^\alpha(z)=z_1+\ldots+z_n\,$ (c.f. the normalization \eqref{normal}),
we obtain from (\ref{general_Pochhammer}) and a straightforward computation that
\begin{equation}\label{second-moment}
\int_{C_n^B}(\xi_1^2+\ldots+\xi_n^2)\,d\mu_j(\xi)\,= \,\frac{4(k_1+jk_2)(k_1+k_2(n-1)+1/2)}{k_1+k_2(n+j-1)+1/2}
\end{equation}
which remains bounded as $j\to\infty$. By Exercise 4 in Section 4 of \cite{Kal}  this implies the tightness of $(\mu_j)$ and thus the claim.
\end{proof}

We  have no general explicit formula for the measures $\mu_\eta$ in Theorem \ref{intrep-a-to-b}.
 However, in certain cases explicit formulas are known.
 For example, Lemma \ref{b-to-a} and Theorem \ref{int-rep-classical-chamber} lead for $k_2=d/2\in \{1/2,1,2\}$ to the following

\begin{corollary}
Let $\,\mathbb F\in \{\mathbb R, \mathbb C, \mathbb H\}$ and $d = \text{dim}_{\mathbb R}\mathbb F$.
Then for $k_1\ge0$, $x\in C_n^B$ and all $z\in \mathbb C^n$,
\begin{equation}\label{Sonine_limit-2}
 J_{d/2}^A(x^2, -z^2) = \int_{ C_n^B} \int_{U_n(\mathbb F)} J_{(k_1,d/2)}^B
\bigl(\sqrt{\sigma(x u\xi u^{-1}x)}\,,z\bigr)\,du\,
   d\mu_{k_1,d}(\xi) \end{equation}
with the probability measure
$$ d\mu_{k_1,d}(\xi)=c_{k_1,d}\prod_{i=1}^n \xi_i^{k_1-1/2}  \prod_{i<j} (\xi_i-\xi_j)^{d}\cdot
e^{-(\xi_1+\ldots+\xi_n)/2} \> d\xi$$
with suitable normalizing constant $c_{k_1,d}>0$.
\end{corollary}

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}

Jet quenching, as anticipated by Bjorken 25 years ago \cite{bj}, is a prominent
signature of the intriguing state of matter created at the Relativistic Heavy
Ion Collider. The suppression of {\it light} hadron spectra at large transverse momentum $p_{\perp}$ \cite{phenix,star} can be explained -- at least qualitatively -- by attributing the hadron attenuation to the radiative energy loss of the parent parton (light quark or gluon) induced by its rescatterings in the hot or dense medium.
On the other hand, recent experimental data on heavy flavour quenching
\cite{Adler:2005xv,Bielcik:2005wu}, measured indirectly via the $p_{\perp}$-spectra of electrons from $D$ and $B$ meson decays, suggest that the radiative energy loss of {\it heavy} quarks might be insufficient to explain the observed attenuation\footnote{It is stressed in Ref.~\cite{Armesto:2005mz} that this statement might be somewhat premature, since the theoretical calculation of heavy quark production suffers from large uncertainties already in proton-proton collisions, and also because the contributions to the electron spectra from charm and beauty
are not separated experimentally.}.
This renewed the interest in the {\it collisional} part $-\Delta E_{coll}$
of the parton energy loss \cite{Wicks:2005gt}, which in the case of a heavy quark might not be negligible -- contrary to what has often been assumed.
A basic quantity required to estimate collisional quenching is the rate of energy loss per unit distance, $-dE/dx$, of a parton produced in the remote past and travelling in a large size medium, as studied in Refs.~\cite{bj,TG,BTqcd}.
For heavy ion collisions, where a parton initially produced in a hard subprocess crosses a medium of finite size $L$, we expect deviations from the linear law $-\Delta E_{coll}(L) = (-dE/dx) \cdot L$ \cite{PGG,Djordjevic:2006tw,Wang:2006qr,Gossiaux:2006yr,Gossiaux:2007gd}. However, the knowledge of $-dE/dx$ is a prerequisite before attempting any evaluation of $-\Delta E_{coll}$.

So far, the most detailed calculation of $-dE/dx$ for a heavy quark in the quark gluon plasma is done by Braaten and Thoma \cite{BTqcd}, and is based on their previous evaluation of muon collisional energy loss in QED \cite{BT}.
Here we will reconsider the latter calculation, which appears to suffer from an incorrect assumption on the magnitude of the momentum exchange in elastic scattering.
As in Ref.~\cite{BT} (referred to as BT in the following) we study the propagation of a muon of mass $M$ and momentum $P = (E, \bm p)$ in an $e^\pm\gamma$ plasma at a temperature $T \ll M$, but large enough to neglect the electron mass.
The muon can be considered as a test particle, losing (or gaining) energy due to elastic ($2\rightarrow2$) scattering off thermal particles of momentum $K$. The latter can be electrons and positrons ($t$-channel scattering, see Fig.~\ref{fig:SE}a), or thermal photons (Compton scattering, see Fig.~\ref{fig:compton}).
The Mandelstam invariants of the elastic processes are defined as
\begin{equation}
  \label{mandelstam}
  s = (P+K)^2 \, , \ \ t = (P-P')^2 \, , \ \ u =(P-K')^2 \, \ .
\eeq
In the high energy limit\footnote{When $E$ becomes large, $s = M^2 +2 PK \simeq 2 PK \sim \morder{ET}$, since $K \lesssim} \newcommand{\mathrel{\rlap{\lower0.2em\hbox{$\sim$}}\raise0.2em\hbox{$>$}}}{\gtrsim \morder{T}$ is constrained by a thermal distribution. The high energy limit $s \gg M^2$ is thus equivalent to $E \gg M^2/T$. For convenience it will often be referred to as the $v \equiv p/E \to 1$ limit.}, the BT calculation {\it incorrectly} assumes that the (spacelike) momentum exchange $P-P' \equiv Q = (\omega,\bm q)$ is always small compared to the incoming muon energy, namely $|\omega| \leq q \equiv |\bm q| \ll E$.
We mention that this problem has also been noted recently in the QCD context of Refs.~\cite{Zakharov,Qin}\footnote{Those
references indeed mention the need for a careful treatment of the kinematics, including the region of large
transfers $q \sim \morder{E}$. However, those studies seem to focus on $t$-channel scattering. In the present paper
we also stress (in the case of QED), that including Compton scattering is required for consistency.}.

As a first consequence of the BT assumption $q \ll E$, the Compton contribution $-dE_{\gamma}/dx$ was neglected in Ref.~\cite{BT}.
Second, in the $v \to 1$ limit the result for the energy loss $-dE_e/dx$ due to scattering off electrons (and positrons) found in \cite{BT},
\begin{equation}
\left. -\frac{dE_e}{dx} \right|_{{\rm BT}}^{v\to 1} = \frac{e^4 T^2}{48 \pi}
\left[ \ln{\frac{2 E}{e^2 T}}  + 2.031  \right] ,
\label{BTfull}
\eeq
has an incorrect constant next to the leading logarithm.
For a heuristic argument showing the importance of large momentum exchange $q$, we note that the logarithmic energy dependence in \eq{BTfull} arises from an integral $\int_T^{q_{\rm max}} dq /q$, with $q_{\rm max} \simeq E$ at high energy.
While the region $q \sim q_{\rm max}$ does not contribute to the leading logarithm, it {\it does} contribute to the constant next to it (\eg, the interval $[q_{\rm max}/2, q_{\rm max}]$ yields $\ln{2}$).
From this simple observation we infer that the approximation $q \ll E$ used in the BT calculation is legitimate only at logarithmic accuracy, but not to calculate the constant term in \eq{BTfull}. The evaluation of this constant requires an accurate treatment of the very hard region $q \sim q_{\rm max}$. For the $t$-channel contribution we will derive the following analytic result correcting \eq{BTfull},
\beqa
\left. -\frac{dE_e}{dx} \right|^{v\to 1} &=& \frac{e^4 T^2}{48 \pi}
\left[ \ln{\frac{2 E}{e^2T}}  + \underbrace{\ln{24}  - \gamma + \frac{\zeta'(2)}{\zeta(2)}} - \frac{3}{4} \right]
\simeq \frac{e^4 T^2}{48 \pi} \left[ \ln{\frac{2 E}{e^2T}}  +  1.281 \right]  \label{PPfull} \, , \nn \\
&& \hskip 4cm 2.031
\end{eqnarray}
where $\gamma \simeq 0.577$ is Euler's constant.

We stressed that working {\it beyond} logarithmic accuracy requires
considering the kinematic region of very hard transfers $q \sim E$. Since $\bm k' = \bm k + \bm q$ and $k \lesssim} \newcommand{\mathrel{\rlap{\lower0.2em\hbox{$\sim$}}\raise0.2em\hbox{$>$}}}{\gtrsim \morder{T}$, this corresponds to $k' \sim E$. Using
\begin{equation}
s = (P'+K')^2 = 2 P'\cdot K' + M^2 \mathop{\simeq}_{v\to 1} 2E' k' (1 -\cos{\theta_{p'k'}}) \, ,
\label{sestimate}
\eeq
we infer from $s \sim ET$ and $k' \sim E$ that the angle $\theta_{p'k'}$ between
$\bm p'$ and $\bm k'$ must be small, and that the constant next to the leading logarithm (partly) arises from the angular region
\begin{equation}
\theta_{p'k'} \sim \sqrt{T/E} \ll 1 \; .
\label{angles}
\eeq
Here is an essential point. When $E \to \infty$, the constant is sensitive to scatterings where $\bm p'$ and $\bm k'$ are collinear. But collinear outgoing particles should be associated within the same `jet' of particles, and such collinear
configurations should thus be removed from the definition of {\it observable} collisional energy loss.
Hence, we conclude that {\it when $E \to \infty$, the constant next to the leading logarithm in $-dE/dx$ is not an observable}, only the leading logarithmic term of \eq{PPfull} is meaningful. Strictly speaking, in the asymptotic limit $E \to \infty$ we can only state
\begin{equation}
 -\left. \frac{dE_e}{dx} \right|^{E\to \infty} \rightarrow \frac{e^4 T^2}{48 \pi}\,  \left[ \ln{\frac{E}{e^2 T}}  + \morder{1} \right] ,
\label{interpretation1}
\eeq
where the constant $\sim \morder{1}$ depends on the details of the jet definition.

For finite $E$, however, $\theta_{p'k'} \sim \sqrt{T/E}$ is a non-zero angle\footnote{For instance, for $E = 10 \,{\rm GeV}$
and $T= 500 \,{\rm MeV}$ we have $\sqrt{T/E} \simeq 0.22\,{\rm rad} \simeq 13^\circ$.}. Depending on the experimental angular resolution, processes transferring a large fraction of the incoming energy $E$ to the particle $K'$ (forming with $P'$ the angle $\theta_{p'k'} \neq 0$) might be counted as observable energy loss. In the present study we assume the angular resolution to be much better than $\sqrt{T/E}$.
Then it is meaningful to include the (correct) constant next to the leading logarithm, as done in \eq{PPfull}.

Comparing now \eq{BTfull} and \eq{PPfull}, it might seem that the difference in the constant term is only of minor importance.
However, the incorrect approximation of Ref.~\cite{BT} has also been used to calculate the collisional energy loss of a heavy quark in QCD \cite{BTqcd}, and the latter must thus also be corrected.
Moreover, the relevance of the region $q \sim E$ in the $t$-channel contribution suggests that $s$ and $u$-channels (Compton scattering) are important, contrary to what is assumed in \cite{BT}.
Indeed, we find in the present case of QED (see section 3) that the Compton scattering contribution
precisely arises from the domain \eq{angles}\footnote{The presence of $s$ and $u$-channel contributions arising from the region $q \sim E$ will also affect the QCD results obtained in \cite{BTqcd}.}.
Since quasi-collinear configurations \eq{angles} already contribute to the $t$-channel contribution \eq{PPfull},
Compton scattering cannot be dropped by invoking some collinearity argument.
In fact, our calculation reveals that Compton scattering not only contributes to a new constant, but more crucially to
an additional logarithmic term,
\begin{equation}
\left. -\frac{dE_{\gamma}}{dx} \right|^{v\to 1} = \frac{e^4 T^2}{96 \pi}
\left[ \ln{\frac{4TE}{M^2}}  - \frac{5}{6} - \gamma + \frac{\zeta'(2)}{\zeta(2)} \right]  .
\label{loss-us}
\eeq
This logarithm is of collinear origin\footnote{Such
potentially large logarithms were mentioned in Ref.~\cite{alam} in the case of the collisional energy loss of light partons.}, as is
obvious from its divergence in the formal $M \to 0$ limit.

In summary, if we aim to control the constant next to the leading logarithm in the $t$-channel contribution \eq{PPfull}, $s$ and $u$-channels must be included {\it for consistency}. This, in turn, brings an additional potentially large logarithm.
For definiteness we state our complete result for the muon energy loss in the
$v\to 1$ limit, obtained by adding \eq{loss-us} to \eq{PPfull},
\beqa
\left. -\frac{dE}{dx} \right|^{v\to 1} &=& \frac{e^4 T^2}{48 \pi}
\left[ \ln{\frac{2 E}{e^2T}}  + \frac{1}{2} \ln{\frac{TE}{M^2}} +\ln{48} + \frac{3}{2} \left( \frac{\zeta'(2)}{\zeta(2)} -\gamma \right) - \frac{7}{6} \right] \nn \\
&\simeq& \frac{e^4 T^2}{48 \pi} \left[ \ln{\frac{2 E}{e^2T}}  + \frac{1}{2} \ln{\frac{TE}{M^2}} + 0.984 \right]
\label{PPstu} \, .
\end{eqnarray}
This corrects the BT result \eq{BTfull}.

Our paper is organized as follows. In section 2 we focus on the contribution $-dE_e/dx$ from $t$-channel exchange to the muon energy loss. In section 2.1 we present an exact relation between $-dE_e/dx$
and the muon self-energy. In section 2.2 we argue that the phase space can be conveniently decomposed
in terms of the Lorentz invariant momentum exchange $t$, into the regions $|t| < |t^\star|$ and
$|t| > |t^\star|$, where the cut-off $t^\star$ satisfies $e^2T^2 \ll |t^\star| \ll T^2$, but otherwise is arbitrary. The contributions from the two regions are evaluated in sections 2.3 and 2.4 and summed in section
2.5, where we derive the result \eq{PPfull} and discuss the incorrect assumption made in the BT calculation \cite{BT}. (For completeness, we repeat the calculation of \eq{PPfull} in Appendices B and C by following the procedure used in Ref.~\cite{BT}, {\it i.e.}}  \newcommand{\eg}{{\it e.g.}\ by decomposing the phase space into a soft and a hard domain with respect to $q^\star = |t^\star|^{1/2}$. We can thus precisely see where the approximation $q \ll E$ used in \cite{BT} fails when evaluating the hard contribution from $q > q^\star$.)
Section 3 is devoted to the contribution from Compton scattering $-dE_\gamma/dx$ quoted in \eq{loss-us}, and our results are summarized in section 4.

\section{Scattering off electrons}
\label{sec2}

\subsection{An exact relation: energy loss from self-energy}

Among the two processes contributing to the muon energy loss, namely scattering off thermal photons and scattering off thermal electrons (or positrons), we focus here on the latter mechanism, even though the idea of calculating the energy loss of a test particle from its self-energy is more general.
Our discussion below follows the BT calculation \cite{BT} of the soft contribution to the energy loss,
which we generalize appropriately.

Let us start by recalling that the collisional energy loss of a test particle is closely related to its interaction
rate $\Gamma$, the latter being obtained from the imaginary part of the particle's self-energy $\Sigma$ evaluated at the energy
$p_0 = E+i\epsilon$ \cite{Weldon:1983jn}. In the case of a muon,
\begin{equation}
	\Gamma(E)
	=
	-\frac1{2E}\, \left( 1-n_F(E) \right)
	{\rm tr}\left[ (P\hskip-0.7em/ \, + M)\, {\rm Im}\,\Sigma(P)\right] \, .
\label{eq:Weldons formula}
\end{equation}
For the $t$-channel interaction with electrons and positrons (involving a single photon exchange), the corresponding self-energy is the 1-loop graph depicted in Fig.~\ref{fig:SE}b, with a resummed photon propagator for reasons to be explained shortly.
\begin{figure}[b]
\bigskip
\centerline{\raisebox{-2mm}{\includegraphics[width=2.5cm]{scattering_mu_e}}
			\quad \quad
			\raisebox{-2mm}{\includegraphics[width=3.5cm]{selfenergy_muon}}}
\caption[*]{(a) $t$-channel scattering amplitude off electrons contributing to the muon interaction rate.
(b) The dressed muon self-energy.}
\label{fig:SE}
\end{figure}
Since the on-shell self-energy is gauge invariant, one may choose a convenient gauge in order to evaluate \eq{eq:Weldons formula}.
In Coulomb gauge, and without further approximation, the trace in \eq{eq:Weldons formula} reads \cite{BT}
\begin{eqnarray}
\label{eq:tr}
    && {\rm tr}\left[ (P\hskip-0.7em/ \, + M)\, {\rm Im}\Sigma(P)\right]
	\;=\;
	-4\pi e^2 \left( 1+e^{-E/T} \right)
	\int_q \int_{-\infty}^\infty d\omega \left( 1+n_B(\omega) \right)
	\frac{{\cal A\, B}}{2E'} \, ,
	\\  && \quad
	{\cal A}
	\;=\;
	\rho_L(\omega,q)\left( 2E^2-E\omega-\bm{pq} \right)
	+2\rho_T(\omega,q)\left( p^2-E\omega+\bm{pq}-(\bm{pq}/q)^2 \right),
	\nonumber \\[1mm]  && \quad
	{\cal B}
	\;=\;
	(1-n_F(E'))\, \delta(E-E'-\omega) - n_F(E')\, \delta(E+E'-\omega) \, ,
	\nonumber
\end{eqnarray}
where $E' = \sqrt{(\bm p - \bm q)^2+M^2}$, $Q = (\omega, \bm q)$ is the photon momentum, and
$n_{B,F}$ denote the Bose-Einstein and Fermi-Dirac thermal distributions. We also use the shorthand notation
\begin{equation}
\int_q  \equiv \int \frac{d^3 \bm q}{(2\pi)^3}
\eeq
and the spectral functions of the longitudinal and transverse photons \cite{BT}
\begin{equation}
\rho_{L,T}(\omega,q) \equiv
-\frac1{\pi}\, {\rm Im}\left[ \Delta_{L,T}(\omega+i\epsilon,q) \right] \, ,
\label{eq:rho}
\end{equation}
where $\Delta_{L,T}$ are the longitudinal and transverse photon propagators.

According to the assumption $M \gg T$, we have $n_F(E') \ll 1$ and the factor ${\cal B}$ in (\ref{eq:tr}) reduces to
$\delta(E-E'-\omega)$.
Performing the angular integral in (\ref{eq:tr}) yields
\begin{eqnarray}
	\Gamma_e(E)
	&=&
	\frac{e^2}{2\pi v} \int_0^\infty dq\,q
	\int_{\omega_-}^{\omega_+} d\omega \left( 1+n_B(\omega) \right)
	\left\{ \rho_L(\omega,q) \left[ 1 - \left( \frac\omega{E} - \frac{t}{4E^2} \right) \right] \right.
	\nonumber \\
	&& \hskip 2cm
	+ \left.
	 \rho_T(\omega,q) \left[ v^2 - \frac{\omega^2}{q^2}
			+\left( \frac{t\,\omega}{E q^2} - \frac{t}{2E^2} - \frac{t^2}{(2Eq)^2} \right)
 	\right] \right\} .
\label{eq:Gamma}
\end{eqnarray}
Comparing to the analogous BT result, we observe that relaxing their assumption
$|\omega|, q \lesssim} \newcommand{\mathrel{\rlap{\lower0.2em\hbox{$\sim$}}\raise0.2em\hbox{$>$}}}{\gtrsim T$ yields the additional terms put in between parentheses, and requires using the exact expression $\omega_\pm(q) = E - \sqrt{(p \mp q)^2+M^2}$ for the bounds instead of the approximation $\pm vq$.

The energy loss per unit length is then obtained by weighting the differential interaction rate by $\omega/v$,
\begin{eqnarray}
	-\frac{dE_e}{dx}
	&=&
	\frac{e^2}{2\pi v^2} \int_0^\infty dq\,q
	\int_{\omega_-}^{\omega_+} d\omega\, \omega \left( 1+n_B(\omega) \right)
	\left\{ \rho_L(\omega,q) \left[ 1 - \left( \frac\omega{E} - \frac{t}{4E^2} \right) \right] \right.
	\nonumber \\
	&& \hskip 2cm
	+ \left.
	 \rho_T(\omega,q) \left[ v^2 - \frac{\omega^2}{q^2}
			+\left( \frac{t\,\omega}{E q^2} - \frac{t}{2E^2} - \frac{t^2}{(2Eq)^2} \right)
 	\right] \right\} .
\label{eq:dEdx_full}
\end{eqnarray}
As discussed in Ref.~\cite{BT}, the additional factor of $\omega$ is improving the infrared behavior of the integral (\ref{eq:dEdx_full}) compared to (\ref{eq:Gamma}). In fact, evaluating $\Gamma_e$ from Eq.~(\ref{eq:Gamma}) with spectral functions obtained as the discontinuity of the 1-loop resummed propagators $\Delta^{\rm 1-loop} = (\Delta_0^{-1} - \Pi^{\rm 1-loop})^{-1}$, yields an infrared divergence due to soft transverse exchanges.
In contrast, weighting the integrand of $\Gamma_e$ by $\omega$ leads to a well-defined energy loss \eq{eq:dEdx_full}.

\subsection{Phase space decomposition}
\label{sec:PS decomposition}

The necessity of expressing the energy loss in terms of a {\it dressed} exchanged photon propagator (obtained by resumming the 1-loop photon self-energy) arises from the long-range nature of the Coulomb interaction.
However, this infrared dynamics becomes unimportant for short-range interactions.
This motivates, on physical grounds, a decomposition of the phase space in Eq.~(\ref{eq:dEdx_full}).

In the BT calculation of the energy loss, the matching procedure developed by Braaten and Yuan
\cite{Braaten:1991dd} is used. A momentum scale $q^\star$, chosen as $e T \ll q^\star \ll T$ but otherwise arbitrary, is introduced to separate soft interactions, with $q < q^\star$, from hard ones  with $q > q^\star$, see Fig.~\ref{fig:decompose phase space}a.
\begin{figure}[ht]
\centerline{\includegraphics[width=5cm]{HTL_BraatenYuan_prescription.eps}\quad\quad
			\includegraphics[width=5cm]{HTL_t_prescription.eps}}
\caption[*]{Two ways of decomposing the exchanged photon phase space. (a) Braaten-Yuan prescription,
by introducing a cut-off $q^\star$ separating a soft region (shaded) from a hard one, as applied in \cite{BT}.
(b) Our approach, using a cut-off $t^\star \equiv -(q^\star)^2$ in the invariant momentum transfer.}
\label{fig:decompose phase space}
\end{figure}
The soft contribution to $dE/dx$ is evaluated using the `hard thermal loop' (HTL) approximation \cite{pisarski,BI} for the dressed photon propagator.
On the other hand, the hard $q > q^\star$ contribution in (\ref{eq:dEdx_full}) is obtained by keeping only the leading term in the expansion $\rho \propto {\rm Im}\left[ (\Delta_0^{-1} - \Pi)^{-1} \right] = \Delta_0^2\, {\rm Im}\left[ \Pi \right] + {\cal O}(\Pi^2)$. This approximation corresponds to evaluating the elastic scattering amplitude in Fig.~\ref{fig:SE}a with the tree-level photon propagator
(which would yield an infrared divergent result for the energy loss in the absence of the cut-off $q^\star$).
The sum of the soft and hard contributions should, of course, be independent of the arbitrary scale $q^\star$, as was verified in \cite{BT}.
However, this consistency check could not reveal that BT's result for the hard contribution is incomplete beyond logarithmic accuracy, as we will show in the following.

For this purpose it will be convenient to decompose the phase space with respect to a cut-off $t^\star \equiv -(q^\star)^2$ in the invariant momentum transfer, as illustrated in Fig.~\ref{fig:decompose phase space}b.
This choice is motivated by two facts.
First, in the region where $|t| < |t^\star|$, the HTL approximation for the photon propagator is known to be valid \cite{Peshier:1998dy} although $\omega$ and $q$ can be individually large. In fact, while the HTL approximation is usually derived under the assumption $\omega,q \ll T$ (implying $\omega^2+q^2 \ll T^2$), it actually holds if the Minkowski `norm'
$|\omega^2-q^2|$ is small compared to $T^2$ \cite{Peshier:1998dy}.
Secondly, the calculation of the contribution where $|t| > |t^\star|$ is more transparent (see section \ref{sec:tchannelhard}),
since in this region the squared scattering amplitude is a function of the Mandelstam invariants only.
We stress that when $|t| > |t^\star|$
we can indeed neglect the `medium modifications' to the matrix elements since the region $|t| > |t^\star|$ is
contained in the hard $q > q^\star$ Braaten-Yuan region.

\subsection{Contribution from $|t| < |t^\star|$}
\label{sec:tchannelsoft}

We calculate this contribution to $dE_e/dx$ from \eq{eq:dEdx_full} by changing variables to $t = \omega^2-q^2$ and $x = \omega/q$. Since $|t| < |t^\star|$, we can omit the terms of order ${\cal O}(|t|^{1/2}/E)$ and approximate the bounds on $x$ by $\pm v$. The term $\sim \omega/E$ in the integrand of \eq{eq:dEdx_full} is easily checked to be exponentially suppressed. Using the HTL approximation for the spectral function we obtain
\begin{equation}
-\left. \frac{dE_e}{dx} \right|_{|t| < |t^\star|}
	=
	\frac{e^2}{4\pi v^2}
	\int_{t^\star}^0 dt\, (-t)
	\int_{-v}^{v} dx\, \frac{x}{(1-x^2)^2}
	\left( 1+n_B(\omega) \right)
	\left[ \rho_L + (v^2 - x^2) \rho_T \right] ,
\eeq
where $\omega = x \sqrt{-t/(1-x^2)}$. From a simple parity argument, we can replace the factor $1+n_B(\omega)$ by its
even part, {\it i.e.}}  \newcommand{\eg}{{\it e.g.}\ $1+n_B(\omega) \rightarrow \frac12$, and we find
\begin{equation}
-\left. \frac{dE_e}{dx} \right|_{|t| < |t^\star|}
	=
	\frac{e^2}{8\pi v^2}
	\int_{-v}^{v} dx\, \frac{x}{(1-x^2)^2}
	\int_{t^\star}^0 dt\, (-t)
	\left[ \rho_L + (v^2 - x^2) \rho_T \right] .
\label{tsoft2}
\eeq
Using the HTL longitudinal and transverse photon propagators
\begin{equation}
\Delta_L(\omega, q) = \frac{1}{q^2+\Pi_L(x)}  \, , \quad
\Delta_T(\omega, q) = \frac{1}{\omega^2-q^2-\Pi_T(x)}
\label{propagators}
\eeq
with the self-energies \cite{BI}
\beqa
\Pi_L(x) &=& m_D^2 \left[ 1-Q(x) \right] \, , \nn \\
\Pi_T(x) &=& \frac{m_D^2}{2} \left[ x^2+(1-x^2)Q(x) \right]
		  = \frac{m_D^2}{2}\, x (1-x^2) Q'(x) \, , \label{self} \\
Q(x) &\equiv& \frac{x}{2}\ln\frac{x+1}{x-1} \, ,\nn
\end{eqnarray}
where $m_D = eT/\sqrt{3}$ is the Debye mass in the QED plasma, the expression \eq{tsoft2} becomes
\beqa
-\left. \frac{dE_e}{dx} \right|_{|t| < |t^\star|} &=& \frac{e^2}{8\pi^2 v^2} \int_{-v}^{v} dx\, \frac{x}{1-x^2}
\int^{t^\star}_0 dt\, {\rm Im} \left[ \frac{t}{t-{\tilde \Pi}_L(x)} -\frac{v^2 - x^2}{1-x^2} \frac{t}{t-\Pi_T(x)}\right]  \nn \\
&=& \frac{e^2}{8\pi^2 v^2} \int_{-v}^{v} dx\, \frac{x}{1-x^2} {\rm Im}
\left[ {\tilde \Pi}_L \ln{\frac{|t^\star|+{\tilde \Pi}_L}{{\tilde \Pi}_L}}
-\frac{v^2 - x^2}{1-x^2} \Pi_T \ln{\frac{|t^\star|+\Pi_T}{\Pi_T}}  \right] \, . \nn \\
\label{tsoft3}
\end{eqnarray}
We used \eq{eq:rho} and introduced ${\tilde \Pi}_L(x) \equiv (1-x^2)\Pi_L(x)$.
We now take advantage of $|t^\star| \gg |{\tilde \Pi}_L(x)|, |\Pi_T(x)|$, and extract the $|t^\star|$-dependence, writing for instance
\begin{equation}
\ln{\frac{|t^\star|}{\Pi_T}} = \ln{\frac{2|t^\star|}{m_D^2}} + \ln{\frac{m_D^2}{2 \Pi_T}} \, .
\eeq
We arrive at
\beqa
-\left. \frac{dE_e}{dx} \right|_{|t| < |t^\star|} &=& \frac{e^4T^2}{48\pi v} \left[ 1 - \frac{1-v^2}{2v}\,\ln\frac{1+v}{1-v} \right]
\ln\left( \frac{2 |t^\star|}{m_D^2} \right) \nn \\
&-& \frac{e^4T^2}{48\pi^2 v^2} \int_{-v}^{v} dx\, \frac{x}{1-x^2} {\rm Im}
\left[ f_L \ln f_L - \frac{v^2-x^2}{1-x^2} f_T \ln f_T \right] \, ,
\label{tsoft4}
\end{eqnarray}
where
\beqa
f_L(x) &=& 2 \, {\tilde \Pi}_L(x) /m_D^2 = 2 (1-x^2)\left( 1-Q(x) \right) , \nn \\
f_T(x) &=& 2 \, \Pi_T(x) /m_D^2 = x (1-x^2) Q'(x) \, .
\end{eqnarray}
The integral in \eq{tsoft4} seems difficult to evaluate analytically for arbitrary $v$. However, in the limit $v \to 1$ we are interested in, the result is surprisingly simple: the integral vanishes, as can be checked numerically. Hence,
\begin{equation}
-\left. \frac{dE_e}{dx} \right|_{|t| < |t^\star|}^{v \to 1} = \frac{e^4T^2}{48\pi}\, \ln{\frac{6 |t^\star|}{e^2 T^2}} \, .
\label{softtstar}
\end{equation}
We end this section by presenting an alternative and fully analytical way to obtain the result \eq{softtstar}
-- which proves indirectly that the integral in \eq{tsoft4} indeed vanishes for $v = 1$.
Referring to Fig.~\ref{fig:decompose phase space}, the kinematic region $|t| < |t^\star|$ is obviously given by the reunion of the soft region
$q^2 < (q^\star)^2 = |t^\star|$ and the region where $q^2 > |t^\star|$ and $|t| < |t^\star|$, or
equivalently  $(1-x^2)|t^\star| < |t| < |t^\star|$, since $|t| = (1-x^2)q^2$. The contribution
from the latter region is easy to calculate along the lines which led to \eq{tsoft3}. When $v=1$ we get
\beqa
-\left. \frac{dE_e}{dx} \right|_{|t| < |t^\star|,\, q > q^\star} &=& \frac{e^2}{8\pi^2} \int_{-1}^{1} dx\, \frac{x}{1-x^2}
\int^{t^\star}_{(1-x^2)t^\star} dt\, {\rm Im} \left[ \frac{{\tilde \Pi}_L}{t - {\tilde \Pi}_L}- \frac{\Pi_T}{t - \Pi_T}  \right]  \nn \\
&=& \frac{e^4 T^2}{48\pi} \int_{-1}^{1} dx\, \frac{3x^2}{2} \ln{\left( \frac{1}{1-x^2} \right)}
= \frac{e^4 T^2}{48\pi}  \left[ \frac{8}{3} - \ln 4 \right] \, ,
\label{interregion}
\end{eqnarray}
where we again used $|t^\star| \gg |{\tilde \Pi}_L(x)|, |\Pi_T(x)|$. The contribution from
$q^2 < |t^\star|$ is precisely the BT `soft' contribution, which reads for $v=1$ \cite{BT}
\begin{equation}
-\left. \frac{dE_e}{dx} \right|_{q < q^\star}= \frac{e^4 T^2}{24 \pi}\left[ \ln{\frac{q^\star}{e T}} + 0.256 \right]  \, .
\label{BTsofta}
\eeq
In Appendix B we rederive this result and determine the constant analytically,
\begin{equation}
-\left. \frac{dE_e}{dx} \right|_{q < q^\star}=  \frac{e^4 T^2}{24 \pi}\left[ \ln{\frac{q^\star}{e T}} +\frac{\ln 24}{2} -\frac{4}{3} \right] \, .
\label{PPsoftb}
\eeq
By adding \eq{interregion} and \eq{PPsoftb} we confirm the result \eq{softtstar} for the contribution
from the domain $|t| < |t^\star|$.

\subsection{Contribution from $|t| > |t^\star|$}
\label{sec:tchannelhard}

As already argued in section \ref{sec:PS decomposition}, when $|t| > |t^\star|$ thermal corrections to the exchanged photon propagator can be ignored. In this kinematic domain the energy loss can thus be obtained from the general relation \cite{BT}
\beqa
  - \frac{dE_i}{dx}
  &=&
  \frac{1}{2Ev}
	\int_k \frac{n_i(k)}{2k}
	\int_{k'} \frac{\bar n_i(k')}{2k'}\,
	\int_{p'} \frac1{2E'}\,	(2\pi)^4\delta^{(4)}(P+K-P'-K')\,
	\frac{1}{d}\sum_{\rm spins}\left| {\cal M}_i\right|^2\, \omega  \nn \\
\label{eq:dEdx0}
\end{eqnarray}
by inserting a factor $\Theta(|t| - |t^\star|)$ in the integrand. The tree-level matrix element ${\cal M}_i$ describes the scattering off a target particle of type $i$.
Summing $|{\cal M}_i|^2$ over initial and final spin states and dividing by the degeneracy factor $d$ of the incoming test particle gives its energy loss $- dE_i/dx$ averaged over spin states.
Furthermore, $n_i(k) = (\exp(k/T) \pm 1)^{-1}$ is the thermal distribution of the target particles, and $\bar n_i = 1 \pm n_i$ accounts for the Bose enhancement or Pauli blocking for the scattered state.
In line with our previous considerations, the target particles are assumed to be massless.

The tree-level matrix elements squared depend only on the Mandelstam in\-va\-riants $s$ and $t$.
Then, as derived in Appendix A, the phase space integral in \eq{eq:dEdx0} can be reduced to
\begin{equation}
 - \frac{dE_i}{dx}
  = \frac{1}{v}\, d_i
	\int_k \frac{n_i(k)}{2k}
	\left(	1 - \frac{s+M^2}{s-M^2}\frac{k}E \right)
	\int_{t_{\rm min}}^0 dt\, (-t)\, \frac{d\sigma_i}{dt} \, ,
\label{eq:dEdx}
\eeq
where only the approximation $\bar n_i \rightarrow 1$ has been made, which is also justified in Appendix A.
Apart from this simplification, the expression \eq{eq:dEdx} is exact, in particular with respect to the kinematics of the scattering process.
The bound in the $t$-integral is
\begin{equation}
	t_{\rm min} = -\frac{(s-M^2)^2}s \, ,
\eeq
and we introduced the differential cross section
\begin{equation}
	\frac{d\sigma_i}{dt}
	=
	\frac1{16\pi (s-M^2)^2}\,
	\frac1{d\, d_i} \sum_{\rm spins}\left| {\cal M}_i \right|^2  \ \ ,
\label{crossdef}
\eeq
where $d_i$ is the spin degeneracy of the target particles.

Our expression \eq{eq:dEdx} generalizes a formula used by Bjorken \cite{bj} for a massless test particle of infinite energy, to the massive and finite-energy case. It will allow us to calculate in a rather compact way the contribution from $|t| > |t^\star|$ to the collisional energy loss.
Focusing on the large $E$ limit and inserting $\Theta(|t| - |t^\star|)$
in the integrand of \eq{eq:dEdx} we obtain
\begin{equation}
   \left. -\frac{dE_i}{dx}\right|_{|t| > |t^\star|}^{v \to 1}  =  d_i
	\int_k \frac{n_i(k)}{2k}
	\int_{t_{\rm min}}^{t^\star} dt\, (-t)\, \frac{d\sigma_i}{dt} \, .
\label{eq:dEdx_asy}
\eeq
The energy loss $-dE_e/dx$ arising from $|t| > |t^\star|$ of a muon scattering off electrons and positrons\footnote{The contribution of positrons
is identical to that of electrons for an $e^\pm \gamma$ plasma with vanishing chemical potential, and is accounted for by a factor $2$ in Eq.~\eq{tampsquared}.} is obtained from \eq{eq:dEdx_asy} with \eq{crossdef} and
\begin{equation}
2 \sum_{\rm spins} \left| {\cal M}_{e^-} \right|^2
= 32\,e^4 \left[ \frac{(s-M^2)^2}{t^2} + \frac{s}{t} + \frac{1}{2} \right]  .
\label{tampsquared}
\eeq
We mention that the two last terms of \eq{tampsquared} contribute to the $t$-integral in \eq{eq:dEdx_asy} as a constant,
\begin{equation}
\frac{1}{(s-M^2)^2} \int_{t_{\rm min}}^{t^\star} dt\, (-t) \, \left[ \frac{s}{t} + \frac{1}{2} \right] \simeq - \frac{3}{4} \ \ ,
\eeq
where we used $s \gg M^2 \gg |t^\star|$. This results in a non-zero contribution to $-dE_e/dx$,
\begin{equation}
\frac{e^4 T^2}{48 \pi} \cdot \left(-\frac{3}{4} \right) \ \ ,
\label{lastterms}
\eeq
which arises from the very hard region $-t \sim -t_{\rm min} \sim s$, {\it i.e.}}  \newcommand{\eg}{{\it e.g.}\ $q \sim \morder{E}$.
This proves that the assumption $q \ll E$ used in \cite{BT} is inappropriate when calculating $-dE/dx$ beyond logarithmic accuracy.

Only the first term of \eq{tampsquared} is sensitive to the cut-off $t^{\star}$, since for $t^{\star} \to 0$ it yields a logarithmic divergence in the infrared,
\begin{equation}
\frac{1}{(s-M^2)^2} \int_{t_{\rm min}}^{t^\star} dt\, (-t) \, \left[ \frac{(s-M^2)^2}{t^2} \right] = \ln \frac{|t_{\rm min}|}{|t^\star|}
\simeq \ln \frac{s}{|t^\star|} \ \ .
\label{reg}
\eeq
With the help of the definite integrals
\beqa
\int_0^\infty dx \, n_F(x) \, x  &=& \frac{\pi^2}{12} \, , \\
\int_0^\infty dx \, n_F(x) \, x \ln{x} &=& \frac{\pi^2}{12} \left[ 1- \gamma + \ln{2} + \frac{\zeta'(2)}{\zeta(2)}\right] \, ,
\end{eqnarray}
the contribution of the first term of \eq{tampsquared} to \eq{eq:dEdx_asy} is found to be
\begin{equation}
\frac{e^4 T^2}{48 \pi}
\left[ \ln\frac{8TE}{|t^\star|} - \gamma + \frac{\zeta'(2)}{\zeta(2)} \right] .
\label{firstterm}
\eeq
Adding the contributions \eq{lastterms} and \eq{firstterm} we obtain
\begin{equation}
- \left. \frac{dE_e}{dx} \right|_{|t|>|t^\star|}^{v \to 1} = \frac{e^4T^2}{48\pi} \left[ \ln\frac{8ET}{|t^\star|} - \gamma - \frac{3}{4} +\frac{\zeta'(2)}{\zeta(2)} \right]  .
\label{hardtstar}
\eeq

\subsection{Complete $t$-channel result and discussion}

Combining the contributions from $|t|<|t^\star|$ and $|t|>|t^\star|$ given by \eq{softtstar} and \eq{hardtstar},
we find the muon energy loss from scattering off electrons and positrons as
\begin{equation}
\left. -\frac{dE_e}{dx} \right|^{v\to 1} = \frac{e^4 T^2}{48 \pi}
\left[ \ln{\frac{2 E}{e^2T}}  + \ln{24}  - \gamma + \frac{\zeta'(2)}{\zeta(2)} - \frac{3}{4} \right] ,
\label{PPfull2}
\eeq
as already quoted in \eq{PPfull}.

We stress that the leading logarithmic term in \eq{PPfull2} arises from the
region $m_D^2 \ll -t \ll s \sim ET$. Since $-t = - (K-K')^2 = 2 k k' (1 -\cos{\theta_{kk'}})$ and
$k \sim T$, this implies $k' = |{\bm k} + {\bm q}|\ll E$. The approximation
$q \ll E$ used in \cite{BT} is thus legitimate at logarithmic accuracy. Beyond logarithmic accuracy however, the region
contributing to \eq{PPfull2} extends to $m_D^2 \lesssim} \newcommand{\mathrel{\rlap{\lower0.2em\hbox{$\sim$}}\raise0.2em\hbox{$>$}}}{\gtrsim -t \lesssim} \newcommand{\mathrel{\rlap{\lower0.2em\hbox{$\sim$}}\raise0.2em\hbox{$>$}}}{\gtrsim s \sim ET$. In particular
the region of maximal transfers $-t \sim -t_{\rm min} \sim  s \sim ET$
({\it i.e.}}  \newcommand{\eg}{{\it e.g.}, $k' \sim E$) affects the constant. Thus the latter could
not be correctly determined in \cite{BT}, where the approximation $q \ll E$ was used.
As mentioned in the Introduction (see \eq{sestimate} and \eq{angles}), $k' \sim E$ also implies that the angular domain $\theta_{p'k'} \sim \sqrt{T/E} \ll 1$ contributes to the constant.

The constant in our result \eq{PPfull2} differs from that of the BT result \eq{BTfull}.
In order to confirm our result, we present an alternative calculation in Appendices B and C. There we follow the BT approach by using the familiar Braaten-Yuan decomposition of the phase space into a soft $q<q^\star$ and a hard $q>q^\star$ kinematic domain, see Fig.~\ref{fig:decompose phase space}a.
In Appendix B we confirm (see \eq{soft4}) the BT result \eq{BTsofta} for the soft contribution in the $v\to 1$ limit.
In Appendix C, we repeat the BT calculation of the hard $q > q^\star$ contribution {\it without using the approximation $q \ll E$}. As expected, we find that the constant is sensitive to the hard domain $q \sim E$.
As shown explicitly in Appendix C, the approximation $q \ll E$ would amount to neglect terms both in $\left| {\cal M}_e \right|^2 $ and in the $\delta$-function for energy conservation, which are important to determine the constant. Those corrections lead to the result \eq{totalhard}, instead of \eq{BThard} as found by BT\footnote{It is a coincidence that the last term $\sim - 3/4$
of \eq{totalhard} missed by BT is identical to the contribution \eq{lastterms}. Hence, in effect the BT result is the same as what we would obtain by keeping only the term $\sim (s-M^2)^2/t^2$ in \eq{tampsquared}.}.

Adding the soft \eq{soft4} and hard \eq{totalhard} contributions we recover \eq{PPfull2}, found within our decomposition of phase space using an invariant separation scale $t^\star$.
We view this as a corroborating evidence of the correctness of our results.

\section{Compton scattering}
\label{sec:Compton}

The contribution $-dE_{\gamma}/dx$ to the muon energy loss from Compton scattering (see Fig.~\ref{fig:compton}) is dominated, as we will shortly see, by hard transfers $-t \sim s$. Thus it can be obtained from \eq{eq:dEdx_asy} by setting $t^\star =0$ and by using
\beqa
\sum_{\rm spins}\left| {\cal M}_\gamma \right|^2 &=& 8e^4 \left[ \left( \frac{-\tilde u}{\tilde s}+2 M^2\frac{\tilde s+2M^2}{\tilde s^2} \right)
+\left( \frac{\tilde s}{-\tilde u}+2 M^2\frac{\tilde u+2M^2}{\tilde u^2} \right) \right.
\nn \\ && \hskip 5.8cm
+ \left. 2 M^2\frac{\tilde s+\tilde u+4M^2}{\tilde s \tilde u} \right] \, ,
\label{eq:M2Compton}
\end{eqnarray}
where we define $\tilde u \equiv u-M^2$ and $\tilde s \equiv s-M^2$, which satisfy $\tilde s + \tilde u + t = 0$. The three terms in \eq{eq:M2Compton} correspond to the contributions from the $s$ and $u$-channels, and from the interference term.
\begin{figure}[ht]
\centerline{\raisebox{-5mm}{\includegraphics[width=7cm]{scattering_Compton_su}}}
\caption[*]{Amplitude ${\cal M}_\gamma$ for Compton scattering.}
\label{fig:compton}
\end{figure}
Using \eq{eq:M2Compton} in our formula \eq{eq:dEdx_asy} we can easily show that all terms which explicitly depend on $M^2$ in \eq{eq:M2Compton} yield contributions to $-dE_{\gamma}/dx$ which are suppressed by at least one power of $s \sim E T$ when $E \to \infty$. Thus for our purposes \eq{eq:M2Compton} can be approximated by
\begin{equation}
\sum_{\rm spins}\left| {\cal M}_\gamma \right|^2 \simeq 8e^4 \left[ \frac{-\tilde u}{\tilde s} +\frac{\tilde s}{-\tilde u} \right]  \, .
\label{eq:M2Comptonappr}
\eeq
This yields the integral
\beqa
\int_{t_{\rm min}}^0 dt\, (-t)\, \frac{d\sigma_\gamma}{dt} &=& \frac{e^4}{8\pi}
\int_{\tilde u_{\rm min}}^{\tilde u_{\rm max}} d\tilde u\, \frac{\tilde u + \tilde s}{\tilde s^2}
\left[ \frac{-\tilde u}{\tilde s} +\frac{\tilde s}{-\tilde u} \right] + \morder{1/s}
\label{sigmagamma0} \\
&=& \frac{e^4}{8\pi} \left( \ln\frac{s}{M^2} - \frac56 \right) + \morder{1/s} \, ,
\label{sigmagamma}
\end{eqnarray}
where we changed variables from $t$ to $\tilde u$, with the bounds $\tilde u_{\rm min} = -\tilde s$, $\tilde u_{\rm max} = -M^2\, \tilde s/s$.

The leading logarithm arises from the $u$-channel, more specifically from the kinematic region $\tilde u_{\rm min} \ll \tilde u \ll \tilde u_{\rm max}$, {\it i.e.}}  \newcommand{\eg}{{\it e.g.}\ $M^2 \ll -\tilde u \ll s$ when $s \gg M^2$. Since in this region $-t \simeq \tilde s \simeq s$ (recall that $\tilde s + \tilde u + t = 0$), the physical interpretation of the logarithmic enhancement is the same as for the total cross section $\sigma_\gamma = \int dt\, d\sigma_\gamma/dt$. The latter behaves in the high energy limit as $\alpha^2 s^{-1} \ln(s/M^2)$ \cite{PS}, with the logarithm originating from backward scattering of the photons in the center of momentum frame.
We thus infer that the Compton contribution to the energy loss is dominated, at large energies and to leading logarithmic accuracy, by the same mechanism.

Plugging \eq{sigmagamma} into \eq{eq:dEdx_asy}, and using
\beqa
\int_0^\infty dx \, n_B(x) \, x  &=& \frac{\pi^2}{6} \, , \\
\int_0^\infty dx \, n_B(x) \, x \ln{x} &=& \frac{\pi^2}{6} \left[ 1- \gamma + \frac{\zeta'(2)}{\zeta(2)}\right]
\end{eqnarray}
for the integral over $k$, we obtain
\begin{equation}
\left. -\frac{dE_{\gamma}}{dx} \right|^{v\to 1} = \frac{e^4 T^2}{96 \pi}
\left[ \ln{\frac{4 T E}{M^2}} - \frac{5}{6} - \gamma + \frac{\zeta'(2)}{\zeta(2)} \right] \, ,
\label{usfull}
\eeq
as quoted in \eq{loss-us}.

The logarithm in \eq{usfull} arises from $M^2 \ll -\tilde u \ll s$, while
the regions $-\tilde u  \sim M^2$ and $-\tilde u  \sim s$ only contribute to the constant.
Hence the complete expression \eq{usfull} stems from\footnote{Since $M \gg T$, the contribution from $s$ and $u$-channels arises from exchanges where thermal corrections to the muon propagator are suppressed. The calculation using a bare muon propagator in Fig.~\ref{fig:compton} is thus legitimate.} $M^2 \lesssim} \newcommand{\mathrel{\rlap{\lower0.2em\hbox{$\sim$}}\raise0.2em\hbox{$>$}}}{\gtrsim -\tilde u \lesssim} \newcommand{\mathrel{\rlap{\lower0.2em\hbox{$\sim$}}\raise0.2em\hbox{$>$}}}{\gtrsim s$, including also very hard exchanges $-t \sim s \sim ET \gg M^2$.
>From $-t = - (K-K')^2 = 2 k k' (1 -\cos{\theta_{kk'}}) \sim s \sim ET$, the typical values of $k' = k +\omega$ contributing to \eq{usfull} are $k' \sim k'_{\rm max} \sim E$. From \eq{sestimate} we thus find that $s$ and $u$-channel contributions arise from
the angular domain $\theta_{p'k'} \sim \sqrt{T/E} \ll 1$, as anticipated in the Introduction.

It is instructive to write the integral $\sim \int d\tilde u /\tilde u$ appearing in \eq{sigmagamma0} as an integral over the angle between ${\bm p}$ and ${\bm k}'$ by using
\begin{equation}
\tilde u = (P-K')^2 -M^2 = -2 P K' \mathop{\simeq}_{v\to 1} -2E k' (1 -\cos{\theta_{pk'}}) \, .
\label{uestimate}
\eeq
Hence, in order to satisfy
$-\tilde u \lesssim} \newcommand{\mathrel{\rlap{\lower0.2em\hbox{$\sim$}}\raise0.2em\hbox{$>$}}}{\gtrsim s \sim ET$, we must have $\theta_{pk'} \ll 1$ in \eq{uestimate}, and the domain $M^2 \lesssim} \newcommand{\mathrel{\rlap{\lower0.2em\hbox{$\sim$}}\raise0.2em\hbox{$>$}}}{\gtrsim -\tilde u \lesssim} \newcommand{\mathrel{\rlap{\lower0.2em\hbox{$\sim$}}\raise0.2em\hbox{$>$}}}{\gtrsim s$ leading to the logarithm translates into $M/E \lesssim} \newcommand{\mathrel{\rlap{\lower0.2em\hbox{$\sim$}}\raise0.2em\hbox{$>$}}}{\gtrsim \theta_{pk'} \lesssim} \newcommand{\mathrel{\rlap{\lower0.2em\hbox{$\sim$}}\raise0.2em\hbox{$>$}}}{\gtrsim \sqrt{T/E}$.
Consequently, the result \eq{usfull} stems from the angular regions
\begin{equation}
M/E \lesssim} \newcommand{\mathrel{\rlap{\lower0.2em\hbox{$\sim$}}\raise0.2em\hbox{$>$}}}{\gtrsim \theta_{pk'} \lesssim} \newcommand{\mathrel{\rlap{\lower0.2em\hbox{$\sim$}}\raise0.2em\hbox{$>$}}}{\gtrsim \sqrt{T/E} \, , \quad
 \theta_{p'k'} \sim \sqrt{T/E} \, .
\label{suangular}
\eeq

\section{Summary and outlook}

In this study we have reconsidered the Braaten-Thoma calculation \cite{BT} of the muon collisional energy loss in a hot QED plasma. For the $t$-channel contribution (scattering off electrons and positrons), we have shown that controlling the constant next to the leading logarithm requires, in the limit $E \gg M^2/T$, considering the region where the invariant transfer $-t$ is on the order of its maximal value, $-t \sim -t_{\rm min} \simeq s \gg M^2$.
The BT calculation of the $t$-channel contribution, which was based on the invalid assumption $q \ll E$, evaluated the constant incorrectly.
We obtained the corrected result for this contribution in Eq.~\eq{PPfull}.

We showed that the `constant' is sensitive to the angular domain
$\theta_{p'k'} \sim \sqrt{T/E}$, {\it i.e.}}  \newcommand{\eg}{{\it e.g.}\ to collinear configurations when $E \to \infty$.
Thus, for consistency the contribution from $s$ and $u$-channels (Compton scattering), which arises from similar configurations, {\it must} be included in the energy loss.
As already stressed in the Introduction, this is our main message. Removing Compton scattering from the definition of energy loss implies that we have to give up determining the constant next to the leading logarithm in the $t$-channel contribution. In other words, working beyond logarithmic accuracy is meaningful only with Compton scattering taken into account.
The Compton process yields a potentially large `collinear' logarithm $\propto \ln(ET/M^2)$, see Eq.~\eq{loss-us}.
It arises from hard transfers $-t \simeq s$, and was previously neglected in \cite{BT}.

It will be interesting to study the consequences of our findings for the collisional energy loss of a heavy quark in a hot QCD plasma, and to see how the results of \cite{BTqcd} are modified\footnote{A first step in this direction was done in \cite{Peshier:2006hi}.}.
We also point to the necessity, in phenomenological studies, to take into account the finite experimental angular resolution $\delta$. If the latter is of the order of
$\sqrt{T/E}$ (or larger), the final state configurations with $\theta_{p'k'} < \delta$ should be removed from
the definition of energy loss, modifying our full result \eq{PPstu} by introducing a $\delta$-dependence.
Since (quasi)-collinear configurations $\theta_{p'k'} \sim \sqrt{T/E}$ correspond to hard exchanges
$q \sim E$, the angular resolution $\delta$ will actually translate to an upper cut-off in $q$,
above which the elastic processes under consideration will not contribute to an observable energy loss.


text:
\section{Introduction}
ADASS is the major annual conference for those interested in astronomy
software and systems, from architects and developers to users and
managers.
Its 24 year history indicates its usefulness to individuals and
institutions, as a place to showcase our latest work, to learn new
techniques and discover which have been successfully used in our
specialist field; to discuss emerging technologies with colleagues
with similar interests and generally to work towards continually
improving the overall shape of the field of astronomical software.

Success is no reason for complacency, however. Over the last two
decades we have seen the needs of our community change, and indeed our
demographic has itself changed. On the broader astronomical stage
we're seeing our travel budgets shrinking, and many find themselves in
the uncomfortable position of having to decide whether their single
overseas conference for the year will focus on their science or
on the technology aspects of their work
If it is to continue being a useful and worthwhile conference, ADASS
must adapt, keeping up with the needs of the ADASS community and the
astronomical community as a whole, while keeping in mind our changing
demographics.

To this end the ADASS Program Organising Committee (POC) ran a survey
in early 2014 to ascertain the needs of the community and the
direction and shape which future ADASS meetings should take. 140
people participated (as a comparison, a typical ADASS has around 270
attendees), 84\% of whom had attended 2 or more ADASS conferences,
12\% had attended 1 and 14\% had never been to an ADASS.

\section{Results of the 2014 Survey and the BoF Discussion}
\begin{description}
\item[Identity] We advertise the ADASS conference as {\em``a forum for
{\bf \em scientists}, {\bf \em developers} and {\bf \em programmers} working in areas related to algorithms,
software and systems for the acquisition, reduction, analysis, and dissemination
of astronomical data''}.
The 140 survey participants identified themselves as astronomers
(69\%), software engineers (38\%), programmers (30\%), software
architects (29\%), managers (28\%) and computer scientists (22\%),
with a small number of instrument scientists, and database, archive
and data specialists\footnote{Participants were able to choose
  multiple categories, hence the sum of all the categories is greater
  than 100\%.}.
The fact that 31\% of people did not self-identify as astronomers shows that we
have moved from a community of ``astronomers who code'', to a more
diverse one which includes people whose primary expertise is in
engineering. This reflects the trend we see in our institutes and
telescope facilities, and it is important that the focus and program
of the ADASS conference also evolves over time to keep up with the
changing needs of our community.
The very small numbers of postdocs (6) and students (4) who
participated in the survey is compatible with the small number from
these two groups who attend the conference. This was discussed in the
BoF and it was agreed that mechanisms should be put in place to
encourage more students and early career postdocs to come to ADASS.

\item[Themes] Each year's conference is based around a handful of key
  themes. These are chosen by the POC and are intended to reflect the
  current work, trends and needs of our community, including
  suggestions made by the conference attendees\footnote{A list of the
    themes chosen for the last several conferences can be found at
    \url{http://adass.org/docs/ADASSThemeHistory.pdf}}.
The BoF participants agreed that final responsibility for theme selection
should rest with the POC, and also that the community should be
actively encouraged to nominate key themes. The Appgree mobile
application\footnote{\url{http://www.appgree.com}}, which was tested during
this conference, was put forward as a possible way of facilitating participation
and of canvassing community preferences on the submitted topics.
For ADASS XXV the POC plans to use Appgree to receive theme suggestions
and hopes to see wide participation from the community.
Additionally, 88 people submitted key topic suggestions through the survey, and these will
also inform the selection of key themes for the next several conferences.

\item[Conference Content]
Both the survey and the BoF discussion agreed that there is a need
for greater focus on algorithms, statistics, analysis methods, data
science, etc., in invited and contributed talks, and less emphasis on update reports.
ADASS talks should be more about the algorithms and techniques than about
the projects themselves.
There was considerable support from the BoF session for invited review
talks looking at the state of play in various aspects of software.

There were several practical suggestions made during the BoF which had the support of the
room:
Authors should be able to classify their paper under more than one key
theme when submitting an abstract. This would help the LOC when setting up the conference
sessions and ensure that each talk was presented in the most relevant session
from the author's point of view.
There was also a request to allow authors to supply, when submitting
an abstract, an optional link containing more information about the proposed
paper.

Discussion on student attendance saw strong agreement on ensuring that
the conference fee has a student rate, and allowing authors to
indicate whether they are a student both on registration and abstract
submission.
There was also a general feeling that there should be a way to reserve
contributed talk slots for early career attendees (particularly those
who have not previously presented a talk), but the details of how this might be
achieved were not discussed.

Another subject raised was whether some plenary discussion time could
be scheduled during the oral sessions,
at the end of each major topic, at the end of each
day, or at the end of the conference. There was some support for this and
also some concern, from participants who thought it would cut into the
time available and result in fewer talks.

A few people showed interested in a hack day, and there was no
opposition to the idea. This could be scheduled as a half-day at the
end of the conference, or be run concurrently with the tutorial.

\item[Posters]
ADASS has a strong tradition of lively and effective poster sessions. The
posters are displayed in a prominent position (close to where coffee
is served) and a good amount of time is allocated to allow people to
view the posters and for meaningful discussions to occur.
Both the survey results and the BoF discussion expressed support for
maintaining the poster sessions as they are.

A popular suggestion raised at the BoF was to declare two time slots
(one for even-numbered and another for odd-numbered posters)
when people were expected to be in attendance at their posters, to
facilitate discussion.

\item[Birds of a Feather]
BoF sessions at ADASS are an opportunity for
informal, free-flowing discussions among people with a
common interest.
BoFs are scheduled based on proposals submitted by attendees wishing to
organise and chair a session on a given subject.
An average
of 5 BoFs are held each year although the number proposed and held in
any one year varies widely. Some subjects (e.g.\ FITS and other data
formats) have a BoF most years whilst other topics come up once or
twice when they are of interest to the community and then fade away as
their relevance and urgency decreases.

At the XXIII-rd meeting in 2013 informal ``pop-up'' BoFs
were suggested and tried, where the subjects for discussion were
proposed and scheduled during the conference, in the style of an
``unconference''. This was repeated at this year's meeting,
and although the take-up of pop-up BoFs has been small the opinion at the
BoF was that we should continue to offer them.

A suggestion was made to dedicate a session to people's specific
technical problems - ``I've got a problem, does anyone know the
answer?'', harnessing the concentrated expertise of the conference
attendees. It was agreed that pop-up BoFs may be useful here, or
a hack-day, depending on the nature of the problem.

\item[Governance]
The ADASS Program Organising Committee (POC) consists of around 15
people, these being representatives of the ADASS sponsoring
institutions and members of the community with an interest in
ADASS. Members are nominated to the POC and ratified by the
sitting committee. The 2014 POC
consists of 16 people\footnote{The current POC membership
  can be found at \url{http://adass.org/POC.html}}, and includes an executive committee
comprised of the POC Chair, Vice-chair, Secretary and
Treasurer.

The survey indicated that the majority of the ADASS community are
satisfied with the conference organisation and governance (67\%) with
smaller fractions preferring the POC to be directly elected from and by the
community (18\%), or wanting to change the length of the term of POC membership (17\%).

It was noted at the BoF session that the POC is gender-unbalanced at
present, and has been so for all of ADASS's history. Additionally the
wish was expressed for the composition of the POC to be more geographically diverse.

\item[Costs and Sponsorship]
There was agreement at the BoF that we should seek industry
  support for the conference. This was mainly seen as a way to reduce
  costs, and particularly to make it easier for students to
  attend. The practicalities are open to debate, but nobody
  saw it as bad, as long as our independence is preserved.

Once specific idea which saw considerable support was to secure
  sponsorship from a printer company in the form of a printer loan for
  a few days at the start of the meeting.  The posters could then be
  printed at the conference location, avoiding the too-common event of
  posters getting lost or damaged in transit.

The conference fee was discussed, but there was no obvious consensus
on whether lowering the fees would make a difference to people's
ability to attend the meeting, as this is often a fraction of the
travel and accommodation cost.
The comment was made that it is hard for young people to obtain travel
funding to come to ADASS, although it was agreed that it is important
that they do. Student conference rates help slightly as does the
Financial Aid scheme.

Finally, the demonstration fees were deemed to be too high by some survey
respondents, who argued that demo booths are no longer the technological
burden on the LOC that they once were, and that several of the
institutes who wish to have a booth at ADASS are the same ones
who sponsor the conference.
After discussion the POC agreed to reduce the fee
for demonstrations from research institutions, but
maintain the current fee for commercial bodies.

\end{description}

\section{Where do we go from here?}

The discussion must continue. Please make your ideas and suggestions
known. Speak with a member of the POC, or send your thoughts by email
(\url{poc@adass.org}). Alternatively tweet your suggestions to
(\url{@astroADASS}) or use Appgree to participate in
ADASS polls.

\acknowledgments
On behalf of the ADASS POC the authors would like to thank
all those who took part in the survey, and everyone who
participated in the BoF discussion.

\end{document}