# The effect of anisotropy on the formation of heavy quarkonium bound states

Margaret E. Carrington<EMAIL_ADDRESS>Department of Physics, Brandon
University, Brandon, MB R7A 6A9, Canada  Winnipeg Institute for Theoretical
Physics, Winnipeg, MB R3T 2N2, Canada  Gabor Kunstatter
<EMAIL_ADDRESS>Winnipeg Institute for Theoretical Physics, Winnipeg,
MB R3T 2N2, Canada  Department of Physics, University of Winnipeg, Winnipeg,
MB R3M 2E9, Canada  Arghya Mukherjee<EMAIL_ADDRESS>Ramakrishna Mission
Residential College (Autonomous), Narendrapur, Kolkata-700103, India

(May 09, 2024)

###### Abstract

We study the real part of the static potential of a heavy quark-antiquark
system in an anisotropic plasma medium. We use a quasi-particle approach where
the collective dynamics of the plasma constituents is described using hard-
loop perturbation theory. The parton distribution function is characterized by
a set of parameters that can accurately describe the anisotropy of the plasma
produced in a heavy ion collision. We calculate the potential numerically in
strongly anisotropic systems and study the angular dependence of the
distortion of the potential relative to the isotropic one. We obtain an
analytic expression for the real part of the heavy quark potential in the
limit of weak anisotropy using a model that expresses the potential in terms
of effective screening masses that depend on the anisotropy parameters and the
orientation of the quark-antiquark pair. A 1-dimensional potential is
formulated in terms of angle averaged screening masses that incorporate the
anisotropy of the medium into a radial coordinate. We solve the corresponding
Schrödinger equation and show that the magnitude of the binding energy
typically increases with anisotropy. Anisotropy can play an important role,
especially in states with non-zero angular momentum. This means that the
number of bound states that are formed could depend on specific
characteristics of the anisotropy of the plasma. Our study suggests that
plasma anisotropy plays an important role in the dynamics of heavy quarkonium
and motivates further study.

##  I Introduction

The dissociation of heavy quarkonium has been extensively studied as a probe
of the properties of quark-gluon plasma. Due to the large mass of the
constituent quarks, heavy quarkonia can be treated non-relativistically. One
can calculate the heavy quark potential and obtain information about
quarkonium binding energies and decay rates by solving a non-relativistic
Schrödinger equation. In equilibrium plasma there have been many calculations
of the heavy quark potential and the properties of heavy quarkonium (see, for
example, [1, 2, 3, 4]). In recent years progress has been made on the
development of methods to go beyond the equilibrium limit [5, 6, 7, 8, 9, 10].

The heavy-quark potential is a complex function and its real/imaginary parts
give information on the binding energies / decay widths. In this work we
calculate the real part of the potential in a chiral plasma using hard-loop
(HL) resummed perturbation theory with an anisotropic distribution function.
Our distribution function includes a parameter that produces a spheroidal
distortion of the isotropic distribution, as originally introduced in [11]. We
include additional anisotropy parameters that give more general momentum
distributions and provide a more realistic description of a quark-gluon plasma
[12]. We find numerical solutions in the chirally symmetric limit. When the
distribution is spheroidal the potential becomes deeper (more strongly
binding) as the anisotropy increases, and the directional dependence of the
potential shows that the quark-antiquark pair attracts more strongly when they
are aligned in the direction of the anisotropy. When more realistic anisotropy
is introduced this simple picture is destroyed and a much richer structure
develops. We obtain an analytic expression for the potential in the limit of
weak anisotropies. We reformulate the result in a more useful way using a
model that expresses the potential in terms of effective screening masses that
depend on the anisotropy parameters and the orientation of the quark-antiquark
pair [6]. We construct a 1-dimensional potential in terms of an angle averaged
screening mass that incorporates the anisotropy into a radial coordinate [8,
9, 10]. Some physics is necessarily lost when this averaging procedure is
used, but the huge advantage is that the corresponding Schrödinger equation
becomes 1-dimensional and is straightforward to solve numerically. We study
the dependence of the binding energy on the anisotropy parameters and show
that magnitude of the binding energy typically increases with anisotropy. We
show that non-spheroidal anisotropy can play an important role, especially in
states with non-zero angular momentum.

This paper is organized as follows. In section II we describe the analytic
calculation of the potential. Sections II.1-II.3 explain how we calculate the
temporal gluon propagator in an anisotropic system and construct the static
potential from it. In section II.4 we give an analytic result for the
potential in the limit of weak anisotropy. Analytic expressions for the six
dressing functions that are used to calculate the potential are given in
Appendix A. In section II.5 we discuss the ansatz we use to model the
potential in terms of an effective screening mass, and how to reduce it to a
function of one radial variable. We also explain the method we use to solve
the resulting 1-dimensional Schrödinger equation. In section III we present
some numerical results and in section IV we make some concluding remarks.

We use natural units where $\hbar=c=1$. The indices $i,j,k=1,2,3$ and
$\mu,\nu=0,1,2,3$ label, respectively, the Cartesian spatial coordinates and
those of Minkowski space. Our metric is mostly minus
$g_{\mu\nu}=(1,-1,-1,-1)_{\rm diag}$. We use capital letters for four-vectors
so that, for example, $P^{2}=p_{0}^{2}-\vec{p}\cdot\vec{p}=p_{0}^{2}-p^{2}$.

##  II Formalism

At leading order the heavy quark potential is obtained from the Fourier
transform of the temporal component of the retarded gluon effective propagator
(${\cal D}_{\mu\nu}$) in the static limit as

$\displaystyle V(\vec{r})$
$\displaystyle=-g^{2}C_{F}\int\frac{d^{3}p}{(2\pi)^{3}}\,(e^{i\vec{p}\cdot\vec{r}}-1)\mathcal{D}_{00}(p^{0}\rightarrow
0,\vec{p})\,$ (1)

where $g$ is the strong coupling constant and $C_{F}=\frac{4}{3}$ is the
quadratic Casimir of the colour SU(3) group. The inverse of the effective
propagator is related to the gluon polarization tensor ($\Pi_{\mu\nu}$)
through the Dyson-Schwinger equation

$\displaystyle\mathcal{D}_{\mu\nu}^{-1}(p_{0},\vec{p})$ $\displaystyle=$
$\displaystyle(\mathcal{D}_{0})_{\mu\nu}^{-1}(p_{0},\vec{p})+\Pi_{\mu\nu}(p_{0},\vec{p})\,.$
(2)

We work in covariant gauge where
$(\mathcal{D}_{0})_{\mu\nu}^{-1}(p_{0},\vec{p})$ is the inverse of the free
gluon propagator

$\displaystyle\mathcal{D}_{0}^{\mu\nu}(p_{0},\vec{p})$ $\displaystyle=$
$\displaystyle-\frac{g^{\mu\nu}}{P^{2}}+(1-\chi)\frac{P^{\mu}P^{\nu}}{P^{4}}\,$
(3)

and use Landau gauge which corresponds to the choice $\chi=0$. If the medium
is isotropic the temporal component of the effective propagator depends only
on the magnitude of $\vec{p}$ and the corresponding potential is spherically
symmetric. The medium produced in a heavy ion collision is not isotropic, it
is strongly anisotropic. The goal of this work is to study how these
anisotropies affect the heavy quark potential, and the binding energies and
decay rates of quarkonium. The main steps of the work presented in this paper
are:

  1. A

Calculate the gluon polarization tensor in an anisotropic medium in the HL
approximation.

  2. B

Invert the Dyson equation (2) to find the temporal component of the
propagator.

  3. C

Calculate the static potential using equation (1).

  4. D

Represent the anisotropic static potential using an ansatz that models the
effect of the anisotropy through effective screening masses. Further
simplification is obtained by averaging over angles to get an anisotropic
1-dimensional potential.

  5. E

Solve the Schrödinger equation using the 1-dimensional anisotropic potential.

For the first two parts of the calculation we consider a medium that can be
both anisotropic and chirally asymmetric. In the third part we look at plasmas
with zero chemical potential and obtain an analytic result for the potential
in the limit of weak anisotropy. For our calculation to be valid perturbation
theory must work. The non-relativistic approach that we use to study
quarkonium is valid for bound states with large quark masses111We do not
include a non-perturbative string tension term..

###  II.1 The polarization tensor

In this section we discuss the calculation of the polarization tensor in an
anisotropic system using a HL effective theory, following the method of ref.
[12]. We start with a brief discussion of the finite temperature calculation
for a thermal QED plasma. Time-like axial gauge (TAG) is particularly useful
because the gauge condition is imposed in the heat bath rest frame. In TAG
only the spatial components of the propagator are non-zero. Our notation for
the equilibrium fermion distribution function is

$\displaystyle
n_{f}(k)=\frac{1}{e^{\beta(k-\mu)}+1}\text{~{}~{}and~{}~{}}\bar{n}_{f}(k)=\frac{1}{e^{\beta(k+\mu)}+1}\,$
(4)

and the 1-loop photon polarization tensor in the HTL approximation is

$\displaystyle\Pi^{ij}(p_{0},\vec{p})=\Pi^{ij}_{{\rm
even}}(p_{0},\vec{p})+\Pi^{ij}_{{\rm odd}}(p_{0},\vec{p})$ (5)
$\displaystyle\Pi^{ij}_{{\rm
even}}(p_{0},\vec{p})=2g^{2}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{n(k)+\bar{n}(k)}{k}\left(\delta^{ij}+\frac{v^{i}p^{j}+p^{i}v^{j}}{P\cdot
V+i\epsilon}-\frac{P^{2}v^{i}v^{j}}{(P\cdot V+i\epsilon)^{2}}\right)$ (6)
$\displaystyle\Pi^{ij}_{{\rm
odd}}(p_{0},\vec{p})=ig^{2}P^{2}\epsilon^{ijm}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{n(k)-\bar{n}(k)}{k^{2}}\frac{(p_{0}v^{m}-p^{m})}{(P\cdot
V+i\epsilon)^{2}}\,.$ (7)

We use $P\cdot V=p_{0}-\vec{p}\cdot\vec{v}$ and
$\vec{k}/\sqrt{k^{2}+m^{2}}\approx\hat{k}\equiv\vec{v}$ since massless
fermions are consistent with the HTL (and HL) approximation. In a chirally
asymmetric system there are two different chemical potentials, for right and
left handed fermions.

To include momentum anisotropy we modify the distribution function in (6, 7)
using

$\displaystyle n(k)\to n(\vec{k})=C_{\xi}\,n\big{(}kH_{\xi}(\vec{v})\big{)}$
(8)

and similarly for $\bar{n}(k)$. The subscript $\xi$ indicates dependence on a
set of anisotropy parameters that can be used to construct a distribution that
is deformed relative to the isotropic one. The factor $C_{\xi}$ is a
normalization and can be defined in different ways depending on the
calculation being done, as explained at the end of this section. We can
construct a completely general expression for the function $H_{\xi}(\vec{v})$
as a sum of terms that are products of anisotropy parameters and dot products
of the vector $\vec{v}$ with two perpendicular unit vectors. For these unit
vectors we use $\hat{n}_{3}$ along the beam axis and $\hat{n}_{1}$ gives the
direction of transverse anisotropy. We restrict to functions that satisfy the
condition $H_{\xi}(\vec{v})=H_{\xi}(-\vec{v})$ and use an expression of the
form222Note that $\xi_{0}$ in this paper is defined equal to $\xi_{0}-1$ in
[12].

$\displaystyle H^{2}_{\xi}(\vec{v})$ $\displaystyle=$
$\displaystyle(1+\xi_{0})+\xi_{2}(\vec{n}_{1}\cdot\vec{v})^{2}+\xi_{9}(\vec{n}_{3}\cdot\vec{v})^{2}+\xi_{6}(\vec{n}_{1}\cdot\vec{v})(\vec{n}_{3}\cdot\vec{v})$
$\displaystyle+\xi_{4}(\vec{n}_{1}\cdot\vec{v})^{4}+\xi_{8}(\vec{n}_{1}\cdot\vec{v})^{3}(\vec{n}_{3}\cdot\vec{v})+\xi_{11}(\vec{n}_{1}\cdot\vec{v})^{2}(\vec{n}_{3}\cdot\vec{v})^{2}+\xi_{13}(\vec{n}_{1}\cdot\vec{v})(\vec{n}_{3}\cdot\vec{v})^{3}+\xi_{14}(\vec{n}_{3}\cdot\vec{v})^{4}\,.$

The values of the anisotropy parameters $\xi_{i}$ must be chosen so that
$H^{2}_{\xi}(\vec{v})$ is positive for all orientations of the vector
$\vec{v}$, which is equivalent to the requirement that $H_{\xi}(\vec{v})$, and
therefore the argument of the distribution function, is real and positive. For
a given choice of the anisotropy parameters, the isotropic distribution is
expanded in the direction of $\vec{v}$ if $H_{\xi}(\vec{v})<1$, and contracted
if $H_{\xi}(\vec{v})>1$.

The polarization tensor has in general 9 independent components. We introduce
a complete tensor basis of nine projection operators and decompose the
polarization tensor in terms of nine scalar functions. From the 3-vectors
$\vec{p}$, $\hat{n}_{1}$ and $\hat{n}_{3}$ we construct three ortho-normal
vectors $(\hat{p},~{}n_{f},~{}m_{F})$:

$\displaystyle\hat{p}=\frac{\vec{p}}{p}$ $\displaystyle
n_{f}=\frac{\tilde{n}_{f}}{\sqrt{\tilde{n}_{f}\cdot\tilde{n}_{f}}}\text{~{}~{}with~{}~{}}\tilde{n}_{f}=n_{3}-(n_{3}\cdot\hat{p})\hat{p}\,~{}~{}$
$\displaystyle
m_{F}=\frac{\tilde{m}_{F}}{\sqrt{\tilde{m}_{F}\cdot\tilde{m}_{F}}}\text{~{}~{}with~{}~{}}\tilde{m}_{F}=\tilde{m}_{f}-(n_{f}\cdot\tilde{m}_{f})\,n_{f}\text{~{}~{}and~{}~{}}\tilde{m}_{f}=n_{1}-(n_{1}\cdot\hat{p})\hat{p}\,.$
(10)

The projection operators are defined as

$\displaystyle
P_{1}^{ij}=m_{F}^{i}m_{F}^{j}\,,~{}~{}P_{2}^{ij}=\hat{p}^{i}\hat{p}^{j}\,,~{}~{}P_{3}^{ij}=n_{f}^{i}n_{f}^{j}$
$\displaystyle
P_{4}^{ij}=\hat{p}^{i}n_{f}^{j}+n_{f}^{i}\hat{p}^{j}\,,~{}~{}P_{5}^{ij}=\hat{p}^{i}m_{F}^{j}+m_{F}^{i}\hat{p}^{j}\,,~{}~{}P_{6}^{ij}=n_{f}^{i}m_{F}^{j}+m_{F}^{i}n_{f}^{j}$
$\displaystyle
P_{7}^{ij}=n_{f}^{i}\hat{p}^{j}-\hat{p}^{i}n_{f}^{j}\,,~{}~{}P_{8}^{ij}=n_{f}^{i}m_{F}^{j}-m_{F}^{i}n_{f}^{j}\,,~{}~{}P_{9}^{ij}=\hat{p}^{i}m_{F}^{j}-m_{F}^{i}\hat{p}^{j}\,$
(11)

and the polarization tensor is decomposed as

$\displaystyle\Pi^{ij}=\sum_{i=1}^{6}\pi_{i}P_{i}^{ij}+\sum_{i=7}^{9}\pi_{i}P_{i}^{ij}$
(12)

where we have omitted the functional arguments to shorten the notation. The
last three projection operators are anti-symmetric in their indices, and the
corresponding dressing functions can only be non-zero if the chiral chemical
potential $\mu_{5}\equiv(\mu_{R}-\mu_{L})/2$ is non-zero.

When $\xi_{i}=0$ we have $H^{2}_{\xi}(\vec{v})=1$ and the distribution is
isotropic and thermal. At zero chemical potential there are only two different
non-zero components in equation (12) which are the familiar HTL transverse and
longitudinal functions

$\displaystyle\pi_{1}=\pi_{3}=\Pi_{T}(p_{0},\vec{p})=m_{D}^{2}\frac{p_{0}^{2}}{2p^{2}}\left(1-\frac{P^{2}}{2p_{0}p}\ln\left(\frac{p_{0}+p+i\epsilon}{p_{0}-p+i\epsilon}\right)\right)$
$\displaystyle\pi_{2}=\Pi_{L}(p_{0},\vec{p})=-m_{D}^{2}\frac{p_{0}^{2}}{p^{2}}\left(1-\frac{p_{0}}{2p}\ln\left(\frac{p_{0}+p+i\epsilon}{p_{0}-p+i\epsilon}\right)\right)$
(13)

where the Debye mass is

$\displaystyle
m_{D}^{2}=2g^{2}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{1}{k}\big{[}4n_{f}(k)\big{]}=\frac{g^{2}T^{2}}{3}\,.$
(14)

The corresponding expression in QCD is obtained by replacing $4n_{f}(k)$ with
$2(N_{f}n_{f}(k)+N_{c}n_{b}(k))$ where $n_{b}(k)=1/(e^{k\beta}-1)$, which
gives $[m_{D}^{2}]_{\rm qcd}=g^{2}T^{2}(N_{c}+N_{f}/2)/3$.

Now we discuss the normalization factor $C_{\xi}$ in equation (8). We use a
normalization that leaves the Debye mass invariant under a change of the
anisotropy parameters. We define a parameter that corresponds to the Debye
mass (14) in an anisotropic system

$\displaystyle[m_{D}^{2}]_{\xi}\equiv 8g^{2}\frac{C_{\xi}}{(2\pi)^{3}}\int
d\Omega\int dkk^{2}\frac{1}{k}n_{f}(kH)\,$ (15)

and $C_{\xi}$ is determined by requiring $[m_{D}^{2}]_{\xi}\equiv m_{D}^{2}$.
We make the change of variable $\tilde{k}=kH$ and find the condition that
determines $C_{\xi}$

$\displaystyle 1=C_{\xi}\int\frac{d\Omega}{4\pi}\frac{1}{H^{2}}\,.$ (16)

The motivation for this choice of normalization is as follows. To calculate
the binding energies of the quarkonium system we want to set the threshold
(the value of the potential as $r$ approaches infinity) to zero. The
normalization of the distribution should be chosen so that this threshold
value is independent of the set of anisotropy parameters that is used. This
ensures that when we compare the binding energies produced by different
distributions we are seeing the effect of the distortion of the distribution
that is produced by anisotropy, and not just an overall shift of the
potential. In sec. II.4 we work in the limit of weak anisotropy and verify
analytically that the value of the potential in the limit $r\to\infty$ is
independent of the choice of the anisotropy parameters. We have also
considered several sets of anisotropy parameters with large values and
verified numerically that the threshold of the potential does not change.

To calculate the real part of the static potential we only need the leading
order contributions to the dressing functions in the limit $p_{0}\to 0$. We
will calculate the potential for the special case of a chirally symmetric
plasma where only the first six dressing functions are non-zero. From the HL
integrals (6) one can show that the real parts of these dressing functions are
even in ${p}_{0}$ and the imaginary parts are odd. The leading order terms for
each dressing function have the form

$\displaystyle\pi_{1}$ $\displaystyle=m_{D}^{2}\bar{\pi}_{1R}^{(0)}+\cdots\,,$
$\displaystyle\pi_{2}$
$\displaystyle=m_{D}^{2}\bar{\pi}_{2R}^{(2)}\bar{p}_{0}^{2}+\cdots\,,$
$\displaystyle\pi_{3}$ $\displaystyle=m_{D}^{2}\bar{\pi}_{3R}^{(0)}+\cdots\,,$
$\displaystyle\pi_{4}$
$\displaystyle=im_{D}^{2}\bar{\pi}_{4I}^{(1)}\bar{p}_{0}+\cdots\,,$
$\displaystyle\pi_{5}$
$\displaystyle=im_{D}^{2}\bar{\pi}_{5I}^{(1)}\bar{p}_{0}+\cdots\,,$
$\displaystyle\pi_{6}$ $\displaystyle=m_{D}^{2}\bar{\pi}_{6R}^{(0)}+\cdots\,$
(17)

where $\bar{p}_{0}=p_{0}/m_{D}$. In the HL effective theory the masses of the
plasma partons are neglected and the coefficients on the right side of (17)
depend only on the direction of the vector $\vec{p}$. They must be calculated
numerically except in special cases. To obtain analytic results we work in the
weak anisotropy limit by expanding in the parameters $\xi_{i}$. The resulting
expressions for the dressing functions are given in Appendix A.

###  II.2 Inversion of the propagator

To calculate the static potential we need to invert the covariant inverse
propagator (2) and extract the temporal component. To do this we use a
covariant basis which can be constructed following the method of ref. [13]. We
start with the four 4-vectors $P^{\mu}$, $b^{\mu}=(1,0,0,0)$,
$c^{\mu}=(0,\hat{n}_{3})$ and $a^{\mu}=(0,\hat{n}_{1})$ and construct four
perpendicular 4-vectors as follows. First we define $\tilde{b}^{\mu}$ which is
perpendicular to $P^{\mu}$:

$\displaystyle V^{\mu\nu}=g^{\mu\nu}-\frac{P^{\mu}P^{\nu}}{P^{2}}$
$\displaystyle\tilde{b}^{\mu}=V^{\mu\nu}b_{\nu}=b^{\mu}-\frac{b\cdot
P}{P^{2}}P^{\mu}\,.$ (18)

Then we construct $\tilde{c}$ perpendicular to both $\tilde{b}$ and $P^{\mu}$

$\displaystyle B^{\mu\nu}=\tilde{b}^{\mu}\tilde{b}^{\nu}$ $\displaystyle
V_{b}^{\mu\nu}=V^{\mu\nu}-\frac{B^{\mu\nu}}{\tilde{b}^{2}}\,,$
$\displaystyle\tilde{c}^{\mu}=V_{b}^{\mu\nu}c_{\nu}=c^{\mu}-\frac{c\cdot
P}{P^{2}}P^{\mu}-\frac{c\cdot\tilde{b}}{\tilde{b}^{2}}\tilde{b}^{\mu}\,.$ (19)

Finally we define $\tilde{a}$

$\displaystyle C^{\mu\nu}=\tilde{c}^{\mu}\tilde{c}^{\nu}$ $\displaystyle
V_{bc}^{\mu\nu}=V_{b}^{\mu\nu}-\frac{C^{\mu\nu}}{\tilde{c}^{2}}=V^{\mu\nu}-\frac{B^{\mu\nu}}{\tilde{b}^{2}}-\frac{C^{\mu\nu}}{\tilde{c}^{2}}\,$
$\displaystyle\tilde{a}^{\mu}=V_{bc}^{\mu\nu}a_{\nu}=a^{\mu}-\frac{a\cdot
P}{P^{2}}P^{\mu}-\frac{a\cdot\tilde{b}}{\tilde{b}^{2}}\tilde{b}^{\mu}-\frac{a\cdot\tilde{c}}{\tilde{c}^{2}}\tilde{c}^{\mu}\,.$
(20)

We use the following nine projection operators

$\displaystyle P_{1}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\tilde{a}^{\mu}\tilde{a}^{\nu}\,$ $\displaystyle P_{2}^{\mu\nu}$
$\displaystyle=$ $\displaystyle\tilde{b}^{\mu}\tilde{b}^{\nu}\,$
$\displaystyle P_{3}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\tilde{c}^{\mu}\tilde{c}^{\nu}\,$ $\displaystyle P_{4}^{\mu\nu}$
$\displaystyle=$
$\displaystyle\tilde{b}^{\mu}\tilde{c}^{\nu}+\tilde{b}^{\nu}\tilde{c}^{\mu}\,$
$\displaystyle P_{5}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\tilde{b}^{\mu}\tilde{a}^{\nu}+\tilde{b}^{\nu}\tilde{a}^{\mu}\,$
$\displaystyle P_{6}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\tilde{c}^{\mu}\tilde{a}^{\nu}+\tilde{c}^{\nu}\tilde{a}^{\mu}\,$
$\displaystyle P_{7}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\tilde{b}^{\mu}\tilde{c}^{\nu}-\tilde{b}^{\nu}\tilde{c}^{\mu}\,$
$\displaystyle P_{8}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\tilde{b}^{\mu}\tilde{a}^{\nu}-\tilde{b}^{\nu}\tilde{a}^{\mu}\,$
$\displaystyle P_{9}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\tilde{c}^{\mu}\tilde{a}^{\nu}-\tilde{c}^{\nu}\tilde{a}^{\mu}\,$
(21)

and decompose the polarization tensor

$\displaystyle\Pi^{\mu\nu}=\sum_{i=1}^{9}\Pi_{i}P_{i}^{\mu\nu}\,$ (22)

where we have supressed the functional arguments. Now we write the covariant
gauge propagator in the form

$\displaystyle{\cal
D}^{\mu\nu}(p_{0},\vec{p})=-\frac{g^{\mu\nu}}{P^{2}}+(1-\chi)\frac{P^{\mu}P^{\nu}}{P^{4}}-\sum{\cal
C}_{i}(p_{0},\vec{p})P_{i}^{\mu\nu}$ (23)

where the ${\cal C}_{i}$ are a set of scalar coefficients that can be
expressed in terms of the components of the self-energy using equations (2, 3,
22, 23) by solving ${\cal D}^{-1}_{\mu\lambda}{\cal
D}^{\lambda\nu}=g_{\mu}^{~{}\nu}$. We also need to rewrite the components of
the polarization tensor in the 4-dimensional covariant basis, denoted
$\Pi_{i}$, in terms of the components in the time-like-axial gauge, denoted
$\pi_{i}$, which have been calculated in the HL approximation in sec. II.1 and
Appendix A. Comparing equations (12, 22) it is straightforward to show

$\displaystyle\Pi_{1}$
$\displaystyle=-\frac{p_{\perp}^{2}}{p_{y}^{2}}\pi_{1}\,,\,\Pi_{2}=-\frac{P^{4}}{p_{0}^{2}p^{2}}\pi_{2}\,,\,\Pi_{3}=-\frac{p^{2}}{p_{\perp}^{2}}\pi_{3}\,,\,\Pi_{4}=\frac{P^{2}}{p_{0}\sqrt{p_{\perp}^{2}}}\pi_{4}\,,\,\Pi_{5}=\frac{P^{2}\sqrt{p_{\perp}^{2}}}{p_{0}p\sqrt{p_{y}^{2}}}\pi_{5}\,,\,$
$\displaystyle\Pi_{6}$
$\displaystyle=-\frac{p}{\sqrt{p_{y}^{2}}}\pi_{6}\,,\,\Pi_{7}=-\frac{P^{2}}{p_{0}\sqrt{p_{\perp}^{2}}}i\pi_{7}\,,\,\Pi_{8}=\frac{P^{2}\sqrt{p_{\perp}^{2}}}{p_{0}p\sqrt{p_{y}^{2}}}i\pi_{9}\,,\,\Pi_{9}=-\frac{p}{\sqrt{p_{y}^{2}}}i\pi_{8}$
(24)

where we have defined $p_{\perp}^{2}=p_{x}^{2}+p_{y}^{2}$. The final
expression for the temporal component of the propagator is

$\displaystyle\mathcal{D}^{00}(p_{0},\vec{p})$ $\displaystyle=$
$\displaystyle-\Big{[}(P^{2}-\pi_{3})(P^{2}-\pi_{1})-(\pi^{2}_{6}+\pi^{2}_{8})\Big{]}/\Big{[}(P^{2}-\pi_{1})(P^{2}-\frac{P^{2}}{p_{0}^{2}}\pi_{2})(P^{2}-\pi_{3})$
(25)
$\displaystyle~{}~{}~{}-\frac{P^{2}}{p_{0}^{2}}(\pi_{4}^{2}+\pi_{7}^{2})(P^{2}-\pi_{1})-(\pi_{6}^{2}+\pi_{8}^{2})(P^{2}-\frac{P^{2}}{p_{0}^{2}}\pi_{2})-\frac{P^{2}}{p_{0}^{2}}(\pi_{5}^{2}+\pi_{9}^{2})(P^{2}-\pi_{3})$
$\displaystyle~{}~{}~{}-2\frac{P^{2}}{p_{0}^{2}}(\pi_{4}\pi_{5}\pi_{6}-\pi_{6}\pi_{7}\pi_{9}+\pi_{5}\pi_{7}\pi_{8}+\pi_{4}\pi_{8}\pi_{9})\Big{]}\,.$

Next we discuss the static limit of the temporal propagator in equation (25).
For a chirally symmetric plasma $\pi_{7}=\pi_{8}=\pi_{9}=0$ and equations (17,
25) give an expression for the static temporal propagator. We use the notation
$\bar{p}=p/m_{D}$ and $\hat{p}=\vec{p}/p$ and define
$\mathcal{\bar{D}}_{00}(\bar{p},\hat{p})=m_{D}^{2}\lim_{p_{0}\to
0}\mathcal{{D}}_{00}(p_{0},\vec{p})$. We will also use the notation
$\bar{\pi}_{i}=\pi_{i}/m_{D}^{2}$ for each component of the polarization
tensor. The temporal component of the propagator in the static limit is

$\displaystyle\mathcal{\bar{D}}_{00}(\bar{p},\hat{p})$
$\displaystyle=\frac{(\bar{p}^{2}+\bar{\pi}_{1R}^{(0)})(\bar{p}^{2}+\bar{\pi}_{3R}^{(0)})-(\bar{\pi}_{6R}^{(0)})^{2}}{a(\bar{p}^{2})^{3}+b(\bar{p}^{2})^{2}+c\bar{p}^{2}+d}$
(26)

where

$\displaystyle a$ $\displaystyle=1\,,$ $\displaystyle b$
$\displaystyle=\bar{\pi}_{1R}^{(0)}-\bar{\pi}_{2R}^{(2)}+\bar{\pi}_{3R}^{(0)}\,,$
$\displaystyle c$
$\displaystyle=-\bar{\pi}_{1R}^{(0)}\bar{\pi}_{2R}^{(2)}-\bar{\pi}_{2R}^{(2)}\bar{\pi}_{3R}^{(0)}+\bar{\pi}_{3R}^{(0)}\bar{\pi}_{1R}^{(0)}-(\bar{\pi}_{4I}^{(1)})^{2}-(\bar{\pi}_{5I}^{(1)})^{2}-(\bar{\pi}_{6R}^{(0)})^{2}\,,$
$\displaystyle d$
$\displaystyle=-\bar{\pi}_{1R}^{(0)}\bar{\pi}_{2R}^{(2)}\bar{\pi}_{3R}^{(0)}+2\bar{\pi}_{4I}^{(1)}\bar{\pi}_{5I}^{(1)}\bar{\pi}_{6R}^{(0)}-\bar{\pi}_{1R}^{(0)}(\bar{\pi}_{4I}^{(1)})^{2}-\bar{\pi}_{3R}^{(0)}(\bar{\pi}_{5I}^{(1)})^{2}+\bar{\pi}_{2R}^{(2)}(\bar{\pi}_{6R}^{(0)})^{2}\,.$
(27)

In a spheroidally symmetric system, which can be described with one anisotropy
parameter, $\pi_{i}=0$ for $i\geq 5$ and the static temporal propagator takes
the simple form

$\displaystyle\mathcal{\bar{D}}^{\rm
spheroidal}_{00}(\bar{p},\hat{p})=\frac{\big{(}\bar{p}^{2}+\bar{\pi}^{(0)}_{3R}\big{)}}{\big{(}\bar{p}^{2}-\bar{\pi}^{(2)}_{2R}\big{)}\big{(}\bar{p}^{2}+\bar{\pi}^{(0)}_{3R}\big{)}+\big{(}\bar{\pi}^{(1)}_{4I}\big{)}^{2}}\,.$
(28)

In the isotropic limit $\bar{\pi}^{(1)}_{4I}=0$ and $\bar{\pi}_{2R}^{(2)}=-1$
so we recover the Debye screening form

$\displaystyle\mathcal{\bar{D}}^{\rm
iso}_{00}(\bar{p})=\frac{1}{\bar{p}^{2}+1}\,.$ (29)

The denominator of (26) is a cubic equation in $\bar{p}^{2}$ and therefore has
three roots. As explained under equation (17), the coefficients of the
polarization tensor depend only on the direction of the momentum vector
$\vec{p}$ and therefore we denote the roots of the denominator
$\bar{m}_{i}^{2}(\hat{p})$ with $i\in(1,3)$. One can write the static
propagator in terms of these roots as

$\displaystyle\mathcal{\bar{D}}_{00}(\bar{p},\hat{p})=1+\sum_{i=1}^{3}\frac{{\cal
A}_{i}(\hat{p})}{\bar{p}^{2}+\bar{m}_{i}^{2}(\hat{p})}$ (30)

where

$\displaystyle{\cal
A}_{1}(\hat{p})=\frac{\bar{m}_{1}^{2}}{(\bar{m}_{1}^{2}-\bar{m}_{2}^{2})(\bar{m}_{1}^{2}-\bar{m}_{3}^{2})}\left[\bar{m}_{1}^{2}\big{(}\bar{\pi}_{1R}^{(0)}+\bar{\pi}_{3R}^{(0)}\big{)}-\bar{m}_{1}^{4}+\big{(}\bar{\pi}_{6R}^{(0)}\big{)}^{2}-\bar{\pi}_{1R}^{(0)}\bar{\pi}_{3R}^{(0)}\right]$
(31)

and ${\cal A}_{2}(\hat{p})$ and ${\cal A}_{3}(\hat{p})$ are obtained from
${\cal A}_{1}(\hat{p})$ by cyclically rotating
$(\bar{m}_{1},\bar{m}_{2},\bar{m}_{3})$. The roots
$(\bar{m}_{1},\bar{m}_{2},\bar{m}_{3})$ are dimensionless mass-like parameters
that depend only on $\hat{p}$ but they have a complicated general form and can
be complex.

###  II.3 The static potential

The static potential is obtained from the temporal component of the gluon
propagator using equation (1). We use

$\displaystyle\hat{p}=(\sin\theta_{p}\cos\phi_{p},\sin\theta_{p}\sin\phi_{p},\cos\theta_{p})\,$
$\displaystyle\vec{r}=r(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)\,$
(32)

and we define $\alpha=(g^{2}C_{F})/(4\pi)$ and $x_{p}=\cos\theta_{p}$ and use
the dimensionless radial variable $\bar{r}=rm_{D}$. Equation (1) then takes
the form

$\displaystyle V(\bar{r},\theta,\phi)$ $\displaystyle=$
$\displaystyle-\frac{4\pi\alpha
m_{D}}{(2\pi)^{3}}\int_{-1}^{1}dx_{p}\,\int_{0}^{\infty}d\bar{p}\,\bar{p}^{2}\int_{0}^{2\pi}\,d\phi_{p}\,(e^{i\,\vec{p}\cdot\vec{r}}-1)\mathcal{\bar{D}}_{00}(\bar{p},x_{p},\phi_{p})\,.$
(33)

In the isotropic limit the propagator (29) gives the Debye screened Yukawa
potential

$\displaystyle V_{\rm iso}(\bar{r})=-\alpha
m_{D}\left(1+\frac{e^{-\bar{r}}}{\bar{r}}\right)\,.$ (34)

In the limit $\bar{r}\to 0$ the isotropic potential reduces to the Coulomb
potential which verifies that medium effects disappear at very small
distances. In the asymptotic limit $\lim_{\bar{r}\to\infty}V_{\rm
iso}(\bar{r})=-\alpha m_{D}$.

The potential in an anisotropic medium can be obtained from equations (26, 27,
33). For arbitrary sets of anisotropy parameters the dressing functions must
be calculated numerically and therefore we are also only able to find the
potential numerically. In section III.1 we show some results for the potential
in strongly anisotropic systems.

###  II.4 Weak anisotropy

It is useful to study the limit of weak anisotropy where we have analytic
results for the dressing functions (see Appendix A). It will be convenient to
work in the coordinate system defined by the vectors

$\displaystyle\hat{r}$ $\displaystyle=(0,0,1)\,,$ $\displaystyle\hat{n}_{1}$
$\displaystyle=(\sin\phi,\cos\theta\cos\phi,\sin\theta\cos\phi)\,,$
$\displaystyle\hat{n}_{3}$ $\displaystyle=(0,-\sin\theta,\cos\theta)\,.$ (35)

In the weakly anisotropic limit the static propagator has the form

$\displaystyle\mathcal{\bar{D}}_{00}(\bar{p},x,\phi)$
$\displaystyle=\mathcal{\bar{D}}_{00}^{(1)}(\bar{p},x,\phi)-\frac{1}{\left(\bar{p}^{2}+1\right)^{2}}(C_{\xi}-1)\,$
(36)

where the factor $C_{\xi}$ is defined in equation (16). To linear order in the
anisotropy parameters the normalization $C_{\xi}$ is

$\displaystyle C_{\xi}=1+\sum_{i}\xi_{i}g_{i}\,$ (37)

with

$\displaystyle(g_{0},g_{2},g_{4},g_{9},g_{11},g_{14})=\left(1,\frac{1}{3},\frac{1}{5},\frac{1}{3},\frac{1}{15},\frac{1}{5}\right)\,.$
(38)

Equations (26, 27, 63, 65) give

$\displaystyle\mathcal{\bar{D}}_{00}^{(1)}(\bar{p},x,\phi)$
$\displaystyle=\frac{1}{\bar{p}^{2}+1}+\frac{\xi_{0}}{\left(\bar{p}^{2}+1\right)^{2}}+\frac{\xi_{9}\left(2-3\left(\hat{p}\cdot\hat{n}_{3}\right){}^{2}\right)}{3\left(\bar{p}^{2}+1\right)^{2}}+\frac{\xi_{2}\left(2-3\left(\hat{p}\cdot\hat{n}_{1}\right){}^{2}\right)}{3\left(\bar{p}^{2}+1\right)^{2}}$
$\displaystyle+\frac{\xi_{4}\left(20\left(\hat{p}\cdot\hat{n}_{1}\right){}^{4}-30\left(\hat{p}\cdot\hat{n}_{1}\right){}^{2}+9\right)}{15\left(\bar{p}^{2}+1\right)^{2}}+\frac{\xi_{8}\left(\hat{p}\cdot\hat{n}_{1}\right)\left(4\left(\hat{p}\cdot\hat{n}_{1}\right){}^{2}-3\right)\left(\hat{p}\cdot\hat{n}_{3}\right)}{3\left(\bar{p}^{2}+1\right)^{2}}$
$\displaystyle+\frac{\xi_{11}\left(5\left(4\left(\hat{p}\cdot\hat{n}_{3}\right){}^{2}-1\right)\left(\hat{p}\cdot\hat{n}_{1}\right){}^{2}-5\left(\hat{p}\cdot\hat{n}_{3}\right){}^{2}+3\right)}{15\left(\bar{p}^{2}+1\right)^{2}}-\frac{\xi_{6}\left(\hat{p}\cdot\hat{n}_{1}\right)\left(\hat{p}\cdot\hat{n}_{3}\right)}{\left(\bar{p}^{2}+1\right)^{2}}$
$\displaystyle+\frac{\xi_{13}\left(\hat{p}\cdot\hat{n}_{1}\right)\left(\hat{p}\cdot\hat{n}_{3}\right)\left(4\left(\hat{p}\cdot\hat{n}_{3}\right){}^{2}-3\right)}{3\left(\bar{p}^{2}+1\right)^{2}}+\frac{\xi_{14}\left(20\left(\hat{p}\cdot\hat{n}_{3}\right){}^{4}-30\left(\hat{p}\cdot\hat{n}_{3}\right){}^{2}+9\right)}{15\left(\bar{p}^{2}+1\right)^{2}}\,.$
(39)

Equations (36, 39) are valid only in the limit of weak anisotropy, to linear
order in the anisotropy parameters, but to simplify the notation we do not
introduce an additional subscript to indicate this.

To find the potential in the limit of weak aniostropy we substitute (36, 39)
into (33) and perform the angular integrals. The result for the potential has
the form

$\displaystyle V_{\rm aniso}(\bar{r},\theta,\phi)=V_{\rm
aniso}^{(1)}(\bar{r},\theta,\phi)-\frac{\alpha
m_{D}}{2}(1-e^{-\bar{r}})(C_{\xi}-1)\,$ (40)

where

$\displaystyle V_{\rm aniso}^{(1)}(\bar{r},\theta,\phi)=-\frac{\alpha
m_{D}}{\bar{r}}e^{-\bar{r}}(1-\sum_{i}\xi_{i}f_{i}(\bar{r},\theta,\phi))-\alpha\bar{m}_{D}$
(41)

with

$\displaystyle\bar{m}_{D}=m_{D}(1-\sum_{i}\xi_{i}h_{i})\,.$ (42)

The non-zero $h_{i}$ coefficients are

$\displaystyle(h_{0},h_{2},h_{4},h_{9},h_{11},h_{14})=\left(\frac{1}{2},\frac{1}{6},\frac{1}{10},\frac{1}{6},\frac{1}{30},\frac{1}{10}\right)\,.$
(43)

The $f_{i}$ coefficients depend on the orientation of the position vector
($\vec{r}$) but can be written in a frame independent way. The coefficients
$f_{0}$, $f_{9}$, $f_{2}$, $f_{6}$ have the fairly simple form

$\displaystyle f_{0}=-\frac{\bar{r}}{2}$ $\displaystyle
f_{9}=\frac{1}{6\bar{r}^{2}}\Big{[}3\left(\bar{r}^{3}+3\bar{r}^{2}+6\bar{r}-6e^{\bar{r}}+6\right)\left(\hat{r}\cdot\hat{n}_{3}\right){}^{2}-2\bar{r}^{3}-3\bar{r}^{2}-6\bar{r}+6\left(e^{\bar{r}}-1\right)\Big{]}~{},$
$\displaystyle
f_{2}=\frac{1}{6\bar{r}^{2}}\Big{[}3\left(\bar{r}^{3}+3\bar{r}^{2}+6\bar{r}-6e^{\bar{r}}+6\right)\left(\hat{r}\cdot\hat{n}_{1}\right){}^{2}-2\bar{r}^{3}-3\bar{r}^{2}-6\bar{r}+6\left(e^{\bar{r}}-1\right)\Big{]}~{},$
$\displaystyle
f_{6}=\frac{1}{2\bar{r}^{2}}\left(\bar{r}^{3}+3\bar{r}^{2}+6\bar{r}-6e^{\bar{r}}+6\right)\left(\hat{r}\cdot\hat{n}_{1}\right)\left(\hat{r}\cdot\hat{n}_{3}\right)~{}.$
(44)

As expected from the structure of equation (II.1), the coefficient $f_{2}$ can
be obtained from $f_{9}$ by replacing $\hat{r}\cdot\hat{n}_{3}$ with
$\hat{r}\cdot\hat{n}_{1}$. The frame independent expressions for the
coefficients $f_{4}$, $f_{8}$ and $f_{11}$ are

$\displaystyle f_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{\bar{r}^{4}}\Big{[}-\frac{2}{3}\left(\hat{r}\cdot\hat{n}_{1}\right){}^{4}f_{4}^{(4,0)}+\left(\hat{r}\cdot\hat{n}_{1}\right){}^{2}f_{4}^{(2,0)}+\frac{1}{10}f_{4}^{(0,0)}\Big{]}$
$\displaystyle f_{8}$ $\displaystyle=$
$\displaystyle\frac{1}{\bar{r}^{4}}\Big{[}\frac{1}{2}\left(\hat{r}\cdot\hat{n}_{1}\right)\left(\hat{r}\cdot\hat{n}_{3}\right)f_{8}^{(1,1)}-\frac{2}{3}\left(\hat{r}\cdot\hat{n}_{1}\right)^{3}\left(\hat{r}\cdot\hat{n}_{3}\right)f_{8}^{(3,1)}\Big{]}$
$\displaystyle f_{11}$ $\displaystyle=$
$\displaystyle\frac{1}{\bar{r}^{4}}\Big{[}\frac{1}{6}\left(\hat{r}\cdot\hat{n}_{1}\right){}^{2}f_{11}^{(2,0)}-\frac{2}{3}\left(\hat{r}\cdot\hat{n}_{1}\right){}^{2}\left(\hat{r}\cdot\hat{n}_{3}\right){}^{2}f_{11}^{(2,2)}+\frac{1}{6}\left(\hat{r}\cdot\hat{n}_{3}\right){}^{2}f_{11}^{(0,2)}+\frac{1}{30}f_{11}^{(0,0)}\Big{]}$

where

$\displaystyle f_{4}^{(4,0)}$ $\displaystyle=$ $\displaystyle
f_{8}^{(3,1)}=f_{11}^{(2,2)}=\bar{r}^{5}+10\bar{r}^{4}+55\bar{r}^{3}+15\left(e^{\bar{r}}+13\right)\bar{r}^{2}+420\bar{r}-420\left(e^{\bar{r}}-1\right)$
$\displaystyle f_{4}^{(2,0)}$ $\displaystyle=$ $\displaystyle
f_{8}^{(1,1)}=f_{11}^{(2,0)}=f_{11}^{(0,2)}=\bar{r}^{5}+7\bar{r}^{4}+34\bar{r}^{3}+6\left(e^{\bar{r}}+19\right)\bar{r}^{2}+240\bar{r}-240\left(e^{\bar{r}}-1\right)$
$\displaystyle f_{4}^{(0,0)}$ $\displaystyle=$
$\displaystyle=f_{11}^{(0,0)}=-3\bar{r}^{5}-10\bar{r}^{4}-40\bar{r}^{3}-120\bar{r}^{2}-240\bar{r}+240\left(e^{\bar{r}}-1\right)\,.$
(46)

The coefficients $f_{14}$ and $f_{13}$ can be obtained from $f_{4}$ and
$f_{8}$ respectively by interchanging $\hat{r}\cdot\hat{n}_{3}$ and
$\hat{r}\cdot\hat{n}_{1}$.

We consider several limits of the anisotropic potential in equation (40). It
is straightforward to see that in the limit $\xi_{i}\to 0$ the anisotropic
potential reduces to the isotropic result in equation (34). From equations
(44, LABEL:hard, 46) one can show that the factor $e^{-\bar{r}}f_{i}/\bar{r}$
goes to zero in the limit $\bar{r}\to\infty$ for all values of $i$. The
threshold of the potential (40) is therefore
$-\alpha[\bar{m}_{D}+m_{D}(C_{\xi}-1)/2]$ and from (38, 42, 43) it is easy to
verify this is the isotropic threshold value $-\alpha m_{D}$. One can also
show that the anisotropic potential (40) reduces to the Coulomb potential in
the limit $\bar{r}\to 0$. To see this we use equations (43, 44, LABEL:hard,
46) to show that for $i\in\\{0,2,4,9,11,14\\}$ the functions
$f_{i}(\bar{r},\theta,\phi)$ and constants $h_{i}$ satisfy the relation

$\displaystyle\lim_{\bar{r}\to 0}f_{i}(\bar{r},\theta,\phi)=-\bar{r}h_{i}\,$
(47)

and for $i\in\\{6,8,13\\}$

$\displaystyle\lim_{\bar{r}\to 0}f_{i}(\bar{r},\theta,\phi)={\cal
O}(\bar{r}^{2})\,.$ (48)

###  II.5 Ansatz for the anisotropic potential

We would like to solve the 3-dimensional Schrödinger equation constructed from
the anisotropic potential in equation (40). Although this calculation is
straightforward to formulate numerically it is difficult to carry out in
practice. We therefore concentrate on a simpler approach [6, 8, 9, 10] which
will let us assess the importance of the anisotropy parameters we have
introduced. The basic idea is to introduce an ansatz for the anisotropic
potential that allows us to model the effects of the anisotropy in terms of an
effective screening mass that is position dependent, and then work with an
angle averaged screening mass that depends on only the radial coordinate.

####  II.5.1 Screening masses

To begin we consider the simple ansatz

$\displaystyle\tilde{V}_{\rm
ansatz}(\bar{r},\theta,\phi)=-\alpha\bar{m}_{D}-\frac{\alpha}{r}e^{-r\tilde{m}_{D}(\bar{r},\theta,\phi)}\,$
(49)

where $\bar{m}_{D}$ is the constant mass scale given in equations (42, 44,
LABEL:hard) and $\tilde{m}_{D}$ is an effective screening mass that depends on
the anisotropy parameters and position. The physical motivation for the ansatz
in equation (49) is easy to understand. The effective screening mass should be
the inverse of the length scale where screening effects become important in an
anisotropic medium. It should therefore be the inverse of a screening length
scale, denoted $r_{s}$, defined through an equation of the form

$\displaystyle\frac{|r\tilde{V}_{\rm ansatz}(r)|_{r\to 0}}{|r\tilde{V}_{\rm
ansatz}(r)|_{r\to r_{s}}}=\frac{1}{e}\,.$ (50)

The numerator in equation (50) is $-\alpha$. If we drop the constant in (49),
which is equivalent to setting the threshold to zero, the denominator is
$-\alpha e^{-\tilde{m}r_{s}}$ and eq. (50) gives $r_{s}\sim 1/\tilde{m}_{D}$.

At very small distances we should find that medium effects disappear and we
recover the Coulomb limit. This physical property of the static potential is
not satisfied by the ansatz (49) which gives

$\displaystyle\lim_{r\to 0}\tilde{V}_{\rm
ansatz}(\vec{r})=-\frac{\alpha}{r}-\alpha(\bar{m}_{D}-\tilde{m}_{D})\,.$ (51)

We use a modified ansatz that preserves the Coulomb limit [8]

$\displaystyle V_{\rm
ansatz}(\vec{r})=-\alpha\bar{m}_{D}-\frac{\bar{m}_{D}}{\tilde{m}_{D}}\frac{\alpha}{r}e^{-r\tilde{m}_{D}}+\frac{\alpha}{r}\left(\frac{\bar{m}_{D}}{\tilde{m}_{D}}-1\right)\,$
(52)

where

$\displaystyle\tilde{m}_{D}\equiv
m_{D}\big{(}1+\sum_{i}\xi_{i}c_{i}(\bar{r},\theta,\phi)\big{)}\,.$ (53)

The potential (52) reduces to the isotropic form when the anisotropy
parameters are set to zero and gives the Coulomb potential in the limit that
$r$ goes to zero. The coefficients $c_{i}$ are determined by matching (52) to
our result (40) with the threshold values subtracted. This gives

$\displaystyle
c_{i}(\bar{r},\theta,\phi)=\frac{2f_{i}(\bar{r},\theta,\phi)+2h_{i}(e^{\bar{r}}-1)+g_{i}{\bar{r}}}{2(1-e^{\bar{r}}+\bar{r})}\,$
(54)

where the coefficients $f_{i}(\bar{r},\theta,\phi)$ and $h_{i}$ and $g_{i}$
are given in equations (38, 43, 44, LABEL:hard, 46).

####  II.5.2 Angle averaged effective masses

To avoid the numerical difficulties of solving a 3-dimensional differential
equation we use an angle averaged screening mass $\mathcal{M}_{lm}$ defined as

$\displaystyle\mathcal{M}_{lm}(\xi_{i})$ $\displaystyle=\int d\Omega
Y_{lm}^{*}(\theta,\phi)\tilde{m}_{D}(\xi_{i},\theta,\phi)Y_{lm}(\theta,\phi)$
(55)

which depends only on the anisotropy parameters $\xi_{i}$. These screening
masses are

$\displaystyle\mathcal{M}_{lm}$
$\displaystyle=M_{lm}+m_{D}(C_{\xi}-1)\left(\frac{1}{3}-\frac{1}{\bar{r}}-\frac{\bar{r}}{36}\right)$
$\displaystyle M_{00}$
$\displaystyle=m_{D}\Bigg{[}1-\frac{\xi_{0}}{2}-\frac{\xi_{9}}{6}-\frac{\xi_{2}}{6}-\frac{\xi_{11}}{30}-\frac{\xi_{4}}{10}-\frac{\xi_{14}}{10}\Bigg{]}$
$\displaystyle M_{10}$
$\displaystyle=m_{D}\Bigg{[}1-\frac{\xi_{0}}{2}-\frac{\xi_{9}}{10}-\frac{\xi_{2}}{5}-\frac{\xi_{11}}{35}-\frac{9\xi_{4}}{70}-\frac{3\xi_{14}}{70}+\bar{r}\left(-\frac{\xi_{11}}{1575}-\frac{4\xi_{14}}{525}+\frac{\xi_{2}}{225}+\frac{2\xi_{4}}{525}-\frac{2\xi_{9}}{225}\right)\Bigg{]}$
$\displaystyle M_{11}$
$\displaystyle=m_{D}\Bigg{[}1-\frac{\xi_{0}}{2}-\frac{\xi_{9}}{5}-\frac{3\xi_{2}}{20}-\frac{\xi_{11}}{28}-\frac{3\xi_{4}}{35}-\frac{9\xi_{14}}{70}+\bar{r}\left(-\frac{\xi_{2}}{450}+\frac{\xi_{9}}{225}+\frac{\xi_{11}}{3150}+\frac{2\xi_{14}}{525}-\frac{\xi_{4}}{525}\right)\Bigg{]}$
$\displaystyle M_{1-1}$ $\displaystyle=M_{11}\,.$ (56)

These expressions show that the anisotropy parameters that break the
spheroidal symmetry of the distribution can play an important role, especially
for states with non-zero angular momentum quantum numbers.

####  II.5.3 Schrödinger equation and its solutions

It is reasonably straightforward to solve the Schrödinger equation using the
angle averaged effective mass $\mathcal{M}_{lm}$ in the ansatz (52) to find
the wavefunction and binding energy for the state with quantum numbers
$(l,m)$. The numerical difficulty is vastly reduced because the differential
equation that must be solved is 1-dimensional, and the physical effects of the
anisotropy are packaged into the screening masses. The validity of the
procedure is analysed in [8] and verified in [10].

The Schrödinger equation for a potential that depends on one radial coordinate
can be separated in the usual way by writing
$\psi(\vec{r})=Y_{l0}(\theta,\phi)R_{nl}(r)$ and $u_{nl}(r)=rR_{nl}(r)$ so
that the differential equation to be solved has the form

$\displaystyle-\frac{\hbar^{2}}{2m}\left(\frac{d^{2}u(r)}{dr^{2}}-\frac{l(l+1)u(r)}{r^{2}}\right)+V_{lm}(r)u(r)=Eu(r)\,.$
(57)

We use $2m=M_{Q}$ where $M_{Q}$ is the mass of the heavy quark and scale
variables using the Bohr radius $a=4\pi/(g^{2}C_{F}M_{Q})$ defining
$\tilde{r}=r/a$ and $\tilde{V}=M_{Q}a^{2}V$ and $\tilde{E}=M_{Q}a^{2}E$. In
terms of these variables the Schrödinger equation has the form

$\displaystyle-\frac{d^{2}u(\tilde{r})}{d\tilde{r}^{2}}+\tilde{V}_{\rm
eff}(\tilde{r})u(\tilde{r})=\tilde{E}u(\tilde{r})$ (58)

with

$\displaystyle\tilde{V}_{\rm
eff}(\tilde{r})=\left(\tilde{V}(\tilde{r})+\frac{l(l+1)}{\tilde{r}^{2}}\right)\,.$
(59)

We use the method of [14, 15] to solve equations (58, 59). The calculation
uses the fact that $\tilde{V}(\tilde{r})$ must be less singular than
$-1/\tilde{r}^{2}$ for the energy eigenvalue to be bounded from below, and
that if $\tilde{r}^{2}V(\tilde{r})$ is analytic the solution of the
Schrödinger equation must have the form
$u(\tilde{r})\propto\tilde{r}^{l+1}[1+{\cal O}(\tilde{r})]$ at small
$\tilde{r}$. We can therefore solve the second order differential equation
(58) with the boundary conditions $u(\tilde{r}_{\rm min})=\tilde{r}_{\rm
min}^{l+1}$ and $u^{\prime}(\tilde{r}_{\rm min})=(l+1)\tilde{r}_{\rm min}^{l}$
where $\tilde{r}_{\rm min}$ is a small number that must be greater than zero
for numerical reasons (we use $10^{-3}$). The idea of the method is to find
the true eigenvalue by performing a systematic search for the eigenvalue that
produces a normalizable eigenfunction. The state is identified from the number
of nodes.

##  III Results

We define $\hat{V}=V/(g^{2}C_{F}m_{D})$ so that $\hat{V}_{\rm
iso}=-e^{-\bar{r}}/(4\pi\bar{r})$ and $\hat{V}_{\rm vacuum}=-1/(4\pi\bar{r})$.
In all of the cases we discuss, when the value of an anisotropy parameter is
not given it is set to zero. The angles $\theta$ and $\phi$ are defined in
equation (32) and we use $x=\cos(\theta)=\cos(\hat{n}_{3}\cdot\hat{r})$.

###  III.1 Strong anisotropy

In fig. 1 we show the correction to the isotropic potential divided by the
vacuum potential as a function of $x=\cos(\theta)$ and $\phi$ at $\bar{r}=0.8$
for two different sets of parameters. The left panel shows that when the
distribution function has spheroidal symmetry the potential depends only on
the polar angle, and is deepest when the quark-antiquark pair are aligned with
the direction of the anisotropy. The right panel shows the dependence of the
potential on the azimuthal angle when one additional anisotropy parameter is
introduced.

Figure 1: Contour plots of the correction to the isotropic potential divided
by the vacuum potential at $\bar{r}=0.8$. The left panel shows $\xi_{9}=10$
and the right panel is $\xi_{9}=10$ and $\xi_{2}=8$.

In fig. 2 we show the potential divided by the vacuum potential for three
values of $\theta$ at $\phi=\pi/4$ and three values of $\phi$ at
$\theta=\pi/2$. The anisotropy parameters used are
$(\xi_{0},\xi_{2},\xi_{4},\xi_{6},\xi_{8},\xi_{9},\xi_{11},\xi_{13},\xi_{14})=(5,8,37,-21,21,8,15,-28,40)$.

Figure 2: The potential divided by the vacuum potential for three values of
$\theta$ at $\phi=\pi/4$ and three values of $\phi$ at $\theta=\pi/2$ using
the anisotropy parameters
$(\xi_{0},\xi_{2},\xi_{4},\xi_{6},\xi_{8},\xi_{9},\xi_{11},\xi_{13},\xi_{14})=(5,8,37,-21,21,8,15,-28,40)$.

###  III.2 Weak anisotropy

We consider five different sets of parameters. In the first set we use
$\xi_{9}=0.95$ and all other parameters set to zero. The next four sets have
$\xi_{9}=0.95$, one of the parameters ($\xi_{2}$, $\xi_{4}$, $\xi_{14}$,
$\xi_{11}$) set to $0.8$, and all others set to zero. If fig. 3 we show the
ansatz for the potential for the state $(l,m)=(0,0)$ minus the isotropic
potential for these parameters.

Figure 3: The angle averaged ansatz for the anisotropic potential minus the
isotropic potential using ${\cal M}_{00}$. In all cases $\xi_{9}=0.95$, the
labelled parameter(s) is $0.8$, and all other parameters are set to zero.

We study the dependence of the binding energies on the anisotropy parameters.
We solve the 1-dimensional Schrödinger equation (58, 59) using the ansatz
(52), shifted so the threshold is set to zero, and with the screening mass
$\tilde{m}_{D}$ replaced by the angle averaged masses (56). In our
calculations we use $T=196$ MeV and $M_{Q}=2m_{b}$ with $m_{b}=4.676$ GeV. The
Debye mass is $m_{D}=gT\sqrt{(N_{c}+N_{f}/2)/3}$ and we take $N_{c}=3$ and
$N_{f}=3$.

First we consider the ground state $(n,l,m)=(1,0,0)$. We use $g=1.85$ which is
a typical value from the literature [6, 10]. We note this value is larger than
one, which appears to contradict the assumption that a perturbative approach
is valid. Recall that the coordinate space potential is obtained by Fourier
transforming the propagator in momentum space, and even though the
polarization tensor is calculated perturbatively, the potential is explicitly
non-perturbative in the coupling constant. This can be seen already from the
isotropic calculation where the Fourier transform of the HTL propagator
$-1/(p^{2}+m_{D}^{2})$ gives the isotropic potential $-g^{2}C_{F}/(4\pi
r)e^{-rm_{D}}$ with $m_{D}\sim gT$. If we had expanded the propagator in $g$
we would have obtained the Coulomb potential $-g^{2}C_{F}/(4\pi r)$, which is
just the leading term in the expansion in $r$ of the isotropic potential. The
unexpanded isotropic potential is not a polynomial in $g$ and it correctly
includes the thermal screening. These observations are typically used to
justify the use of larger values of $g$.

Table 1 shows the ground state binding energy minus the binding energy of the
isotropic state for each set of parameters. The magnitude of the binding
energy always increases, which means that anisotropy promotes binding in the
ground state, and is consistent with the fact that the ansatz gives a deeper
potential (see fig. 3).

- |  $\xi_{2}$ |  $\xi_{4}$ |  $\xi_{14}$ |  $\xi_{11}$
---|---|---|---|---
-0.617 | -4.30 | -1.52 | -1.52 | -0.866
Table 1: Ground state binding energy minus the binding energy of the isotropic
state. In all cases $\xi_{9}=0.95$ and $g=1.85$. The parameter indicated in
the top row is $0.8$ and all other parameters are set to zero.

A Yukawa potential of the form $-g^{2}C_{F}e^{-m_{D}r}/(4\pi r)$ does not bind
if the screening is too large. The criticallity condition depends on the value
of the quantum number $l$ and has the form $2am_{D}<n_{l}$ where
$a=4\pi/(C_{F}g^{2}M_{Q})$ is the Bohr radius and $n_{0}\approx 1.2$ and
$n_{1}\approx 0.23$. These values agree with those of ref. [16] with the
appropriate conversions of units and the mass parameter. Using our value of
the temperature and quark mass we have $g_{c}\approx 0.41$ for states with
$l=0$ and $g_{c}\approx 2.1$ for $l=1$. This means that we do not have bound
states with $l=1$ using $g=1.85$. However, we have not included a string
tension term in the ansatz that we use for the potential. This term would
modify the long distance form of the potential so that bound states would form
more easily. The purpose of our calculation is only to study the effect of
anisotropy and we can do that without introducing a constant that must be
fitted using information outside the scope of our calculation by simply
increasing the value of $g$. We have therefore also considered $g=2.85$, which
is above the critical value for $l=1$.

Table 2 gives the binding energy divided by the binding energy of the
isotropic state for each set of anisotropy parameters, for several different
states, with $g=2.85$. The results show that $\xi_{2}$ is particularly
important, especially when $l=1$. In most cases the magnitude of the binding
energy increases but for the state $(2,1,1)$ it can slightly decrease.

state |  - |  $\xi_{2}$ |  $\xi_{4}$ |  $\xi_{14}$ |  $\xi_{11}$
---|---|---|---|---|---
(1,0,0) | 1.0041 | 1.0283 | 1.0100 | 1.0100 | 1.0057
(2,0,0) | 1.0409 | 1.2866 | 1.1004 | 1.1004 | 1.0573
(2,1,0) | 1.2477 | 1.5152 | 1.2844 | 1.4795 | 1.287
(2,1,1) | 0.97462 | 1.4527 | 1.1052 | 1.0011 | 0.99507
Table 2: Binding energy divided by the binding energy of the isotropic state
with $g=2.85$ and $\xi_{9}=0.95$.

##  IV Conclusions

We have calculated the real part of the heavy quark potential using hard-loop
(HL) resummed perturbation theory with an anisotropic distribution function.
This function depends on a set of anisotropy parameters that can be chosen so
that the momentum distribution provides a more realistic description of a
quark-gluon plasma than has been used in previous calculations. We have
derived an analytic expression for the heavy quark potential in a chirally
asymmetric plasma, and found numerical solutions in the chirally symmetric
limit. In the simple case of a spheroidally asymmetric plasma the potential
becomes deeper (more strongly binding) as the anisotropy increases and the
quark-antiquark pair attracts more strongly when they are aligned with the
direction of the anisotropy. More realistic momentum distributions produce a
much richer structure.

We have obtained an analytic expression for the potential in the limit of weak
anisotropies. Using an ansatz and an averaging procedure we have constructed a
potential that depends on one radial coordinate but still incorporates some of
the anisotropy of the original system. We have used this 1-dimensional
potential to study the dependence of the binding energy on the anisotropy
parameters and found that the magnitude of the binding energy typically
increases with anisotropy. This result is interesting because it means that
the number of bound states that are formed at a given temperature will depend
on the anisotropy of the momentum distribution.

The imaginary part of potential can be calculated using the techniques we have
developed in this paper. These results are particularly interesting because
they will allow us to study the effect of anisotropy on the dissociation
temperatures of quarkonium bound states.

##  Appendix A The dressing functions for weak anisotropy to leading order in
$p_{0}\to 0$

In this section we give analytic results for the components of the dressing
functions that we use to calculate the static potential. We consider only a
chirally symmetric system and we work in the limit of weak anisotropy. In the
limit $p_{0}\to 0$ the dressing functions can be expanded as in equation (17).
For $C_{\xi}=1$ the coefficients of these expansions that are needed to
calculate the heavy-quark potential are

$\displaystyle\bar{\pi}_{1R}^{(0)}$
$\displaystyle=-\frac{1}{3}\xi_{6}\sqrt{1-x^{2}}x\cos(\phi)-\frac{1}{15}\xi_{8}\sqrt{1-x^{2}}x\cos(\phi)\left(\left(2x^{2}-5\right)\cos(2\phi)+2\left(x^{2}+2\right)\right)$
$\displaystyle+\frac{1}{15}\xi_{13}\left(4x^{2}-3\right)\sqrt{1-x^{2}}x\cos(\phi)+\frac{1}{6}\xi_{2}\left(\left(x^{2}-2\right)\cos(2\phi)+x^{2}\right)+\frac{2}{15}\xi_{14}\left(2x^{2}-3\right)x^{2}$
$\displaystyle-\frac{\xi_{9}x^{2}}{3}+\frac{1}{30}\xi_{4}\left(\left(4x^{4}-2x^{2}-8\right)\cos(2\phi)+\left(x^{4}-5x^{2}+4\right)\cos(4\phi)+3\left(x^{4}+x^{2}\right)\right)$
$\displaystyle+\frac{1}{30}\xi_{11}\left(-4x^{4}+x^{2}+\left(-4x^{4}+7x^{2}-2\right)\cos(2\phi)\right)\,,$
(60) $\displaystyle\bar{\pi}_{2R}^{(2)}$
$\displaystyle=-1+\xi_{0}+\frac{1}{3}\xi_{8}x\sqrt{1-x^{2}}\left(\cos(3\phi)-4x^{2}\cos^{3}(\phi)\right)+\frac{1}{6}\xi_{2}\left(3\left(x^{2}-1\right)\cos(2\phi)+3x^{2}+1\right)$
$\displaystyle-\xi_{6}x\sqrt{1-x^{2}}\cos(\phi)+\frac{1}{3}\xi_{13}x\left(4x^{2}-3\right)\sqrt{1-x^{2}}\cos(\phi)+\xi_{9}\left(\frac{2}{3}-x^{2}\right)$
$\displaystyle+\frac{1}{30}\xi_{4}\left(15x^{4}+5\left(x^{2}-1\right)^{2}\cos(4\phi)+10\left(2x^{4}-x^{2}-1\right)\cos(2\phi)+3\right)$
$\displaystyle+\frac{1}{30}\xi_{11}\left(-20x^{4}+15x^{2}-5\left(4x^{4}-5x^{2}+1\right)\cos(2\phi)+1\right)$
$\displaystyle+\frac{1}{15}\xi_{14}\left(20x^{4}-30x^{2}+9\right)\,,$ (61)
$\displaystyle\bar{\pi}_{3R}^{(0)}$
$\displaystyle=\frac{1}{3}\xi_{2}\left(2x^{2}-1\right)\cos^{2}(\phi)-\frac{2}{3}\xi_{6}x\sqrt{1-x^{2}}\cos(\phi)+\frac{2}{15}\xi_{14}\left(8x^{4}-12x^{2}+3\right)$
$\displaystyle-\frac{1}{15}\xi_{8}x\sqrt{1-x^{2}}\cos(\phi)\left(\left(8x^{2}-5\right)\cos(2\phi)+8x^{2}+1\right)$
$\displaystyle+\frac{4}{15}\xi_{13}x\left(4x^{2}-3\right)\sqrt{1-x^{2}}\cos(\phi)+\xi_{9}\left(\frac{1}{3}-\frac{2x^{2}}{3}\right)$
$\displaystyle+\frac{2}{15}\xi_{4}\cos^{2}(\phi)\left(4x^{4}+x^{2}+\left(4x^{4}-5x^{2}+1\right)\cos(2\phi)-2\right)$
$\displaystyle+\frac{1}{30}\xi_{11}\left(2\left(1-2x^{2}\right)\sin^{2}(\phi)-4\left(8x^{4}-8x^{2}+1\right)\cos^{2}(\phi)\right)\,,$
(62) $\displaystyle\bar{\pi}_{6R}^{(0)}$
$\displaystyle=\frac{1}{6}\xi_{6}\sqrt{1-x^{2}}\sin(\phi)+\frac{1}{30}\xi_{11}x\left(6x^{2}-5\right)\sin(2\phi)+\frac{1}{10}\xi_{13}\left(1-2x^{2}\right)\sqrt{1-x^{2}}\sin(\phi)$
$\displaystyle-\frac{1}{5}\xi_{4}x\sin(2\phi)\left(\left(x^{2}-1\right)\cos(2\phi)+x^{2}\right)+\frac{1}{10}\xi_{8}\sqrt{1-x^{2}}\sin(\phi)\left(\left(3x^{2}-1\right)\cos(2\phi)+3x^{2}\right)$
$\displaystyle-\frac{1}{3}\xi_{2}x\sin(\phi)\cos(\phi)\,.$ (63)
$\displaystyle\bar{\pi}_{4I}^{(1)}$
$\displaystyle=\frac{1}{4}\pi\xi_{2}x\sqrt{1-x^{2}}\cos^{2}(\phi)+\frac{3}{16}\pi\xi_{4}x\sqrt{1-x^{2}}\cos^{2}(\phi)\left(\left(x^{2}-1\right)\cos(2\phi)+x^{2}+1\right)$
$\displaystyle+\frac{1}{8}\pi\xi_{6}\left(2x^{2}-1\right)\cos(\phi)+\frac{1}{16}\pi\xi_{11}x\sqrt{1-x^{2}}\left(\left(2-3x^{2}\right)\cos(2\phi)-3x^{2}+1\right)$
$\displaystyle+\frac{3}{32}\pi\xi_{8}\left(x^{2}\left(4x^{2}-3\right)\cos^{3}(\phi)+\left(2x^{2}-1\right)\sin^{2}(\phi)\cos(\phi)\right)-\frac{1}{8}3\pi\xi_{14}x\left(1-x^{2}\right)^{3/2}$
$\displaystyle-\frac{1}{4}\pi\xi_{9}x\sqrt{1-x^{2}}-\frac{3}{32}\pi\xi_{13}\left(4x^{4}-5x^{2}+1\right)\cos(\phi)\,,$
(64) $\displaystyle\bar{\pi}_{5I}^{(1)}$
$\displaystyle=\frac{1}{32}\pi\xi_{11}\left(3x^{2}-1\right)\sqrt{1-x^{2}}\sin(2\phi)+\frac{3}{32}\pi\xi_{13}x\left(x^{2}-1\right)\sin(\phi)$
$\displaystyle-\frac{3}{32}\pi\xi_{8}x\sin(\phi)\left(\left(3x^{2}-2\right)\cos^{2}(\phi)+\sin^{2}(\phi)\right)-\frac{1}{4}\pi\xi_{2}\sqrt{1-x^{2}}\sin(\phi)\cos(\phi)$
$\displaystyle-\frac{3}{32}\pi\xi_{4}\sqrt{1-x^{2}}\sin(2\phi)\left(\left(x^{2}-1\right)\cos(2\phi)+x^{2}+1\right)-\frac{1}{8}\pi\xi_{6}x\sin(\phi)\,.$
(65)

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# Extending rational models of communication from beliefs to actions

Theodore R. Sumers1, Robert D. Hawkins2, Mark K. Ho1, Thomas L. Griffiths1,2
1Department of Computer Science, Princeton University
2Department of Psychology, Princeton University
{sumers, rdhawkins, mho<EMAIL_ADDRESS>
###### Abstract

Speakers communicate to influence their partner’s _beliefs_ and shape their
_actions_. Belief- and action-based objectives have been explored
independently in recent computational models, but it has been challenging to
explicitly compare or integrate them. Indeed, we find that they are conflated
in standard referential communication tasks. To distinguish these accounts, we
introduce a new paradigm called signaling bandits, generalizing classic Lewis
signaling games to a multi-armed bandit setting where all targets in the
context have some relative value. We develop three speaker models: a belief-
oriented speaker with a purely informative objective; an action-oriented
speaker with an instrumental objective; and a combined speaker which
integrates the two by inducing listener beliefs that generally lead to
desirable actions. We then present a series of simulations demonstrating that
grounding production choices in future listener actions results in relevance
effects and flexible uses of nonliteral language. More broadly, our findings
suggest that language games based on richer decision problems are a promising
avenue for insight into rational communication.

Keywords: Communication, rational speech acts, multi-armed bandits, language
games

## Introduction

> “Language is used for doing things.” (?, ?, p. 3)

But what things? Broadly, accounts of communicative goals have been formulated
in terms of listener _beliefs_ and _actions_ : “Alan is speaking with the aim
of getting Barbara to understand him and to act on that understanding” (?, ?,
p. 11). But how do these aims relate? Is language primarily a tool for
informing others, shaping their actions, or some combination of the two?

Classical accounts have emphasized beliefs, framing communication as
information transfer between speaker and listener (?, ?). This is reflected in
the recent Rational Speech Acts (RSA) framework, which typically defines a
speaker’s utility in terms of epistemic objectives like informativeness (?, ?,
?, ?). A related perspective from the connectionist literature defines meaning
in terms of effects on the listener’s latent representations (?, ?). Yet
beliefs themselves are an unsatisfying objective, as they are imperceptible
and have no real-world consequences. Grice himself suggested informativeness
should be generalized to “influencing or directing the actions of others” (?,
?, p. 47).

Under an alternative action-oriented view, communication is an extension of an
agent’s basic capability to interface with the world, allowing a speaker to
act indirectly through others. This has been explored in game-theoretic
pragmatics (?, ?, ?, ?) and emergent communication (?, ?). Practical natural
language interfaces, including instruction-following (?, ?) and task-oriented
dialogue systems (?, ?), typically learn direct mappings between commands and
agent actions. While such imperative language is effective at inducing
immediate actions, it cannot offer a full account of communication. Humans
clearly employ more sophisticated strategies to “program” others (?, ?).

To reconcile these perspectives, we propose a unified computational model that
integrates both action-based and belief-based objectives: communication
operates by influencing intermediate beliefs, but its objective is to shape
downstream actions in the world. Under this _combined_ model, speakers reason
about the decision problem a listener faces, how utterances may change their
latent beliefs, and finally how those beliefs give rise to actions. They then
choose utterances to induce belief states that maximize value in expectation
over possible actions. Varying the scope of the decision problem allows
reasoning at different time horizons, from “nudges” (?, ?) that encourage
specific actions (e.g. getting someone to close a window by saying “It’s
chilly in here”) to intervening on norms more broadly (?, ?).

We first extend the Rational Speech Acts (RSA) framework (?, ?) to account for
action-based objectives. We then introduce a new task paradigm, signaling
bandits, which combines traditional signaling games (?, ?) with multi-armed
bandits (?, ?). In Simulation 1, we show that standard reference games are a
special case of the signaling bandit framework and conflate speaker
objectives. In Simulation 2, we find that action-based speakers provide
decision-relevant information, but employ hyperbolic and false utterances that
improve decisions at the cost of distorting listener beliefs. Finally,
Simulation 3 shows that when the Combined speaker reasons about a larger set
of possible actions, it becomes more truthful and produces utterances that
improve decision-making over the full distribution of contexts the listener
could face. These results suggest that integrated reasoning over listener
belief states and actions creates a pressure to transmit appropriately
generalizable information.

## Action-Grounded Speaker Models

Suppose a friend is about to step into a busy street. We have a brief moment
to say something– but what should we say? The basic principle of
_informativeness_ suggests that we aim to reduce their uncertainty about the
world as much as possible. Without a notion of _relevance_ (?, ?), however, it
would be equally informative to mention a passing cloud or an incoming car.
Recent accounts have formalized relevance as a _question under discussion_
(QUD) that collapses the utility of the listener’s beliefs to a coarse-grained
state, such as the current state of the road (e.g. ?, ?, ?). But our friend
may benefit from a more outcome-grounded notion of relevance, projecting not
to epistemic states but to the _decision problem_ they face (?, ?):
intuitively, regardless of what they know, we want them to cross safely.

A simple way to ground relevance in the decision problem is to maximize the
likelihood that the listener will take a specific action, effectively using
that action as the QUD. This model, which we call the _action_ -oriented
speaker, would suggest something like “Cross after that car passes!” However,
rather than choosing a premeditated action from our own perspective, we could
instead give the listener information that allows them to act optimally from
_their_ perspective. This combined speaker aims to maximize the full expected
utility of the listener’s actions under the induced beliefs. Such a speaker
might say “Look both ways!” which encourages our friend to check for cars
outside of our own field of view and (incidentally) generalizes across
intersections.

Belief-oriented speaker. We formalize these models by first introducing the
Rational Speech Acts framework (?, ?), which instantiates Gricean maxims (?,
?) as recursive social inference. In this framework, speakers have knowledge
of the world state $w$ and choose between utterances $u$ proportionally to
their utility $U(u,w)$, where $\beta_{S}$ is a soft-max parameter controlling
speaker optimality:

$P_{S}(u\mid w)\propto\text{exp}\\{\beta_{S}\cdot U(u,w)\\}$ (1)

Typically, this utility is defined in terms of the listener’s _beliefs_ ,
their information gain about the state $w$:

$U_{\text{Belief}}(u\mid w)=\log P_{L}(w\mid u)$ (2)

This utility requires the speaker to reason about the listener’s expected
beliefs after hearing the utterance:

$P_{L}(w\mid u)\propto\delta_{\llbracket u\rrbracket(w)}P(w)$ (3)

where $\delta_{\llbracket u\rrbracket(w)}$ represents the meaning of $u$,
evaluating to one when utterance $u$ is true of $w$ and zero otherwise. This
formulation optimizes for accurate listener beliefs, but lacks a notion of
relevance.

Action-oriented speaker. Our second model extends the basic RSA framework to
reason about actions a listener could take in the environment. Rather than
adding an additional objective to the epistemic utility, we ground this
objective in a decision problem. Specifically, we re-formulate the listener as
a reinforcement learning (RL) agent (?, ?) with a set of possible actions that
may be taken, $\cal{A}$. At each point in time, a subset of those actions are
available to the listener, which we call an action context
$A\subseteq\cal{A}$. We assume the listener will choose actions to maximize a
reward function, where the scalar reward value associated with each action is
defined by the world state: $R:\mathcal{A}\times W\rightarrow\mathbb{R}$. We
thus define the listener’s policy $\pi_{L}$ in an action context $A$ to be

$\pi_{L}(a\mid u,A)\propto\exp\\{\beta_{L}\cdot R_{L}(a,u)\\}$ (4)

where $\beta_{L}$ is the softmax optimality and $R_{L}$ is the listener’s
expected reward for action $a\in A$ after hearing utterance $u$. We define
expected reward with respect to their posterior beliefs about the likely state
of the world:

$R_{L}(a,u)=\sum_{w\in W}R(a,w)P_{L}(w\mid u)$ (5)

In this work, we assume speakers are cooperative and have access to a ground-
truth world state $w$. Action-based rational speakers reason about how their
utterances will affect listener actions (Eqs. 3, 4, 5) and communicate to
maximize the listener’s reward. We first describe a “pure” _action_ -oriented
speaker, which chooses the highest-reward action $a^{*}\in A$ and optimizes
the probability of the listener taking that action:

$a^{*}\triangleq\operatorname*{arg\,max}_{a\in A}R(a,w)$ (6)
$U_{\text{Action}}(u\mid A,a^{*})=\log[\pi_{L}(a^{*}\mid u,A)]$ (7)

Critically, rather than aiming for high-performing beliefs in general, this
speaker only considers the listener’s beliefs insofar as they are relevant for
producing the pre-selected action. Intuitively, this can be thought of as
imperative language.

Combined speaker. Our final speaker unifies belief- and action-oriented
objectives by optimizing over both, inducing beliefs which are likely to
maximize rewards _in general_ :

$U_{\text{Combined}}(u\mid A,w)=\sum_{a\in A}\pi_{L}(a\mid u,A)R(a,w)$ (8)

Effectively, this utility shifts the locus of decision-making to the listener:
the combined speaker treats them as an independent agent and optimizes their
beliefs, rather than choosing an action for them.111Note our two action-
oriented speakers can be recovered from a more general utility introducing an
additional soft-max over the reward term $R(a,w)$, yielding the Action utility
with $\beta\rightarrow\infty$ and the Combined utility with $\beta=1$. We now
introduce an experimental setting to explore these speaker models.

## Signaling Bandits

We define a new language game which enables study of the speaker models
described above. In this setting, speakers cannot directly signal a unique
correct action because all actions have some relative value. They must instead
supply partial information to guide decision making. We first review the
structure of Lewis signaling games and note their limitations. We then
describe multi-armed bandits, a setting studied in reinforcement learning.
Finally, we combine the two to produce a new game, signaling bandits.

### Lewis Signaling Games

Lewis signaling games are two-player collaborative settings with a speaker and
a listener (?, ?). Following the notation introduced previously, a signaling
game is defined by a world state $w$, action context available to the
listener, $A$, and utterances available to the speaker, $\cal{U}$. There is
one action $a^{*}\in A$ with a positive reward; other actions have zero
reward. The world state $w$ implies the correct action $a^{*}$. The speaker
knows $w$ but the listener does not. During gameplay, the speaker chooses an
utterance $u\in\cal{U}$ and sends it to the listener. The listener updates
their beliefs, $P_{L}(w\mid u)$, and uses the posterior to choose an action,
$\pi_{L}(a\mid u,A)$.

Signaling games formalize the coordination problem underlying communication
(?, ?, ?). However, the interplay of beliefs, actions, and rewards is highly
constrained. The state of the world $w$ is synonymous with the correct action
$a^{*}$, and players are indifferent over other actions. It is thus impossible
to discriminate the three speaker objectives defined above (Eq. 2, 7,
8).222But see ? (?) for evidence in favor of action-oriented speakers. For a
richer decision-making setting, we turn to multi-armed bandits.

### Multi-Armed Bandits

A multi-armed bandit is a single-player sequential game. In each round, the
player takes an action and receives a scalar reward (?, ?). Players seek to
maximize their rewards, but are initially ignorant of the reward structure.
Over multiple rounds, they must balance exploration (choosing a new action to
learn its reward) with exploitation (choosing the most valuable known action).
Because payoffs are scalar, decisions are more nuanced than Lewis games.

Contextual bandits extend this to study learning via abstract information.
Actions are now characterized by features, and rewards are defined with
respect to these features. Formally, a feature function $\phi$ describes
actions: $\phi:A\rightarrow\mathbb{R}^{K}$. Rewards are then defined as a
function of these features: $R:\phi(a)\rightarrow\mathbb{R}$. Thus, rather
than learn about the reward of a specific action, players can learn about the
reward of a feature which applies to many actions. For example, an animal
might learn that ripe yellow bananas are high-reward, while rotten brown
bananas are low-reward. Associating the payoff with the color (a feature)
rather than the banana (a specific action) allows knowledge to transfer to new
settings (the next banana). Contextual bandits have been studied extensively
in reinforcement learning and to a lesser degree for emergent communication
(?, ?). Yet to our knowledge, they have not been used to study communication
with an existing language. In the next section, we introduce a two-player
version of this game.

### Signaling Bandits

We combine the communication of Lewis games with the reward structure of
contextual bandits to create a new game, signaling bandits. Unlike Lewis
games, speakers no longer communicate concrete information (which action is
correct). Instead, they communicate abstract information (how much features
are worth). We now describe basic gameplay.

Figure 1: Signaling bandits combine Lewis signaling games with multi-armed
bandits. (A) The world $w$ is defined by correspondences between features and
rewards (table margins), which combine additively to create possible actions
$\cal{A}$ (table contents). (B) Two contexts $A$ with example utterances. The
first utterance is true and useful (encouraging the listener to avoid the blue
objects). The second is false (circles are worth 1). It is locally useful
(encouraging the listener to choose the circle) but will lead the listener
astray in other contexts.

As in Lewis games, signaling bandits are two-player games with a speaker and a
listener. Each game is defined by a world state $w$, a set of all possible
actions $\cal{A}$ and a set of speaker utterances $\cal{U}$. In each round,
the listener faces an action context $A\subseteq\cal{A}$. However, unlike
Lewis games, there is no single “correct” action. Instead, as in contextual
bandits, each action has a scalar reward defined by the world state $w$. Here,
we assume features are indicator variables over actions:

$\phi:A\rightarrow\\{0,1\\}^{K}$ (9)

Rewards $R$ are linear over these features, parameterized by $w$:

$R(a,w)=w^{\top}\phi(a).$ (10)

Concretely, this means the world state $w$ is a vector encoding the reward
associated with each feature. A listener with full world knowledge (every
element of $w$) can calculate exact rewards using Eq. 10 and thus select the
optimal action in any context. Fig. 1A depicts this visually: $w$ defines the
value of individual shapes and colors (table margins), which in turn
determines the reward for each possible action in $\cal{A}$ (table contents).
The value of an individual feature (one element of $w$) thus constitutes
partial knowledge about the world.

The speaker helps the listener by providing such partial knowledge. Formally,
${\mathcal{U}}$ is a set of tuples of the form
$\langle\mathds{1}_{K},\mathbb{R}\rangle$ which specify a given feature and
scalar value. As shown in Fig. 1, these are messages like $\langle$Blue,
-2$\rangle$ or $\langle$Circle, 2$\rangle$. Speakers choose utterances and
send them to the listener. The (literal) listener updates their beliefs by
setting the corresponding feature to the transmitted scalar value:

$p_{L}(w_{L}\mid u)=p_{L}(w_{L}\mid u_{\mathds{1}_{K}}=u_{\mathbb{R}})$ (11)

and then chooses an action from the available set $A$ according to their
posterior belief over rewards (Eqs. 4 and 5).

Transmitting partial knowledge and constructing different contexts
$A\subseteq\cal{A}$ introduces several important dynamics. First, it induces
relevance effects, since different knowledge will be useful in different
contexts. For example, in Fig. 1B, $\langle$Green, 2$\rangle$ would improve
decision making in both contexts while $\langle$Blue, -2$\rangle$ would only
be relevant for the top one. Second, it accommodates nonliteral language
naturally, as false beliefs can yield good decisions. In the bottom context of
Fig. 1B, a false message $\langle$Circle, 2$\rangle$ maximizes the probability
of the listener choosing the optimal action. Finally, it allows us to explore
generalization: whether a listener’s beliefs facilitate good decision making
over other action contexts constructed from the same world. A bias towards
communicating generalized information is implicated in cultural learning (?,
?, ?); thus, modeling the dynamics of speaker objectives and resulting
generalization performance is of significant theoretical interest.

Signaling bandits creates clean distinctions between listener beliefs
$P_{L}(w\mid u)$, the optimal action $a^{*}$, and the value of individual
actions, $R(a,w)$. This allows for meaningful differences between speaker
models (Eqs. 2, 7, 8). In the following section, we use simulations to
illustrate this. We return to extensions beyond basic gameplay in the General
Discussion.

## Simulations

We perform three simulations within the signaling bandits framework. We first
describe the general procedure and metrics used to measure speaker behaviors.
For all simulations, we set
$\beta_{L}=3,\beta_{\text{Belief}}=3,\beta_{\text{Action}}=3$, and
$\beta_{\text{Combined}}=2$. Speaker optimality does not affect the
qualitative results; we analyze optimal speakers ($\beta\rightarrow\infty$) at
the end of this section. We assume listeners have uniform priors over feature
rewards, and allow speakers to send only a single message.

Procedure. For each simulation, we define a world state $w$ and set of
allowable utterances $\cal{U}$. For an action context $A$, we first compute
each speaker’s distribution over utterances $u\in\mathcal{U}$ as defined by
Eqs. 2, 7, and 8. We then compute the listener’s resulting distribution over
actions (Eq. 4).

Evaluation Metrics. We use four metrics to summarize speaker behavior. First,
their probability of choosing a true utterance, P(truthful). Second, the
probability of the listener choosing the optimal action in $A$ (Eq. 4), which
we write as $\pi_{L}(a^{*})$ for brevity. Third, the expected reward on the
action context $A$ (Eq. 8), which we write as $R_{S}(A)$. Finally, we want to
know whether speakers are over-optimizing for a particular context. To
evaluate this, we calculate the expected generalization, which indicates
whether the listener’s resulting belief state yields good performance on other
contexts drawn from $\mathcal{A}$. Formally, we compute the expected reward of
the speaker’s utterance $u$ across all possible contexts:

$R_{S}(u,w,\mathcal{A})=\sum_{A\in[\mathcal{A}]^{3}}R_{S}(u,w,A)P(A)$ (12)

where $P(A)$ is the probability of an action context $A$; here, we assume a
uniform distribution over all contexts of size 3. Again, we shorten this to
$R_{S}(\mathcal{A})$ for clarity. If local performance substantially exceeds
generalization, $R_{S}(A)\gg R_{S}(\mathcal{A})$, we say the speaker
generalizes poorly: it is optimizing local decision-making by providing false
or less broadly useful information.

### Simulation 1: Reproducing reference games

Figure 2: Simulation 1 constructs a traditional Lewis signaling game within the signaling bandits framework. Note that two of the possible utterances are false. Speaker |  $P_{S}$(truthful) | $\pi_{L}(a^{*})$ | $R_{S}(A)$ | $R_{S}(\mathcal{A})$
---|---|---|---|---
Belief | 1.00 | .500 | 100 | -
Action | 1.00 | .500 | 100 | -
Combined | 1.00 | .500 | 100 | -
Table 1: Simulation 1 results. Reference games align all three speakers’
objectives and so cannot disambiguate them. Generalization,
$R_{S}(\mathcal{A})$, does not apply to this setting.

Our first simulation constructs a Lewis signaling game as a special case of
our more general class of signaling bandits (Fig. 2). We show that this case
cannot distinguish between our models, motivating Simulations 2 and 3.

Figure 3: Simulation 2 results. Shading indicates probability of speaker choosing that utterance, averaged over all 84 3-action contexts. X’s indicate true utterances. Left: Belief speakers choose true utterances at random. Center/Right: Action and Combined speakers focus on decision-relevant features and exaggerate to improve listener decisions. Speaker |  $P_{S}$(truthful) | $\pi_{L}(a^{*})$ | $R_{S}(A)$ | $R_{S}(\mathcal{A})$
---|---|---|---|---
Belief | 1.00 | .499 | .539 | .539
Action | .330 | .772 | 1.18 | .486
Combined | .360 | .742 | 1.28 | .522
Table 2: Simulation 2 results. Action and Combined frequently send false
messages. They obtain high performance on the local context, $R_{S}(A)$, but
generalize poorly, $R_{S}(\mathcal{A})$.

Setup. To construct a Lewis game with a single target, we define $w$ and
construct a context $A$ containing one action with arbitrarily high reward
(the target) and two with zero reward (the distractors). We restrict
utterances $\mathcal{U}$ to positive messages (corresponding to possible
referential labels).

Results and Discussion. Results are summarized in Table 1. All three speaker
objectives are aligned, and so we find that they behave identically: they
choose between the two literally true messages $\langle$Green, 100$\rangle$ or
$\langle$Circle, 100$\rangle$. No model has any reason to prefer a false
utterance, or to prefer one true utterance over the other.

### Simulation 2: Divergent speaker behaviors

We next explore how different speaker models may diverge for other tasks in
our signaling bandit paradigm (Fig. 1).

Setup. We consider the world state depicted in Fig. 1. We set $\mathcal{U}$ to
all feature-value tuples and evaluate each speaker’s behavior across all
$\binom{9}{3}=84$ possible contexts of 3 actions.

Results and Discussion. We plot each speaker’s probability over individual
utterances in Fig. 3, and summarize the results in Table 2. All metrics are
averaged across the 84 contexts. First, we observe that Belief speakers choose
a true utterance at random, regardless of the action context. This yields
relatively poor performance locally but perfect generalization
($R_{S}(A)=R_{S}(\mathcal{A})=.539$). In contrast, both Action and Combined
speakers lie frequently ($P_{S}$(truthful)$<.5$). Action speakers tailor their
utterances to induce the single optimal action in each set. As a result, they
exaggerate whichever feature values align with the best action in the
immediate context. This strategy succeeds locally: they induce the most-
optimal action a majority of the time ($\pi_{L}(a^{*})=.772$) and obtain more
than twice the reward obtained by the Belief speaker ($R_{S}(A)=1.18$).
However, the resulting distortion in listener beliefs means they generalize
poorly ($R_{S}(\mathcal{A})=.486$). Finally, Combined speakers achieve a
middle ground. They obtain the best outcome less frequently than Action
speakers ($\pi_{L}(a^{*})=.742$), but higher reward locally ($R_{S}(A)=1.28$)
and generally ($R_{S}(\mathcal{A})=.522$). Sensitivity to the rewards of all
three actions leads them to distort beliefs less than Action speakers. We
visualize the divergence between Action and Combined speakers in Fig. 4 in a
single action context to better understand these differences.

Speaker |  $P_{S}$(truthful) |  $\pi_{L}(a^{*}$) | $R_{S}(A)$ | $R_{S}(\mathcal{A})$
---|---|---|---|---
Belief | 1.00 | .499 | - | .539
Action | .440 | .566 | - | .748
Combined | .534 | .627 | - | .949
Table 3: Simulation 3 results. When communicating about a larger action
context, Action and Combined speakers become more truthful and generalization
improves dramatically.

### Simulation 3: Expanding speaker context

Simulation 2 showed that Action and Combined speakers can be myopic: they
produce messages to induce locally-optimal actions at the cost of
generalization. Simulation 3 explores how this changes when they optimize over
the entire action space $\mathcal{A}$. We find that both Action and Combined
speakers become more truthful and generalize better.

Setup. We use the same world as Simulation 2 (shown in Fig. 1). We first
construct a single “global” action context of all 9 actions: $A=\mathcal{A}$.
We compute each speaker’s distribution over utterances for this 9-action
context, then evaluate generalization over 3-action contexts,
$R_{S}(\mathcal{A})$. Because there is no “local” context, we do not compute
$R_{S}(A)$.

Figure 4: Comparing Action and Combined in one action context (Simulation 2).
Shading indicates difference in utterance probabilities between speakers. The
Action speaker maximizes the listener’s probability of choosing the red
circle: $\pi_{L}(a^{*})=.737,R_{S}(A)=.456$. The Combined speaker achieves
higher reward by avoiding the blue square:
$\pi_{L}(a^{*})=.627,R_{S}(A)=.482$. Neither sends messages about green,
demonstrating relevance effects.

Results and Discussion. Results are summarized in Table 3. The Belief speaker
is unchanged. Both action-oriented speakers are more truthful and generalize
better than Simulation 2. However, the Action speaker fixates on the single
best option (the green circle). It sends false messages which exaggerate its
value, e.g. $\langle$Circle, 2$\rangle$, or discourage alternatives, e.g.
$\langle$Red, -2$\rangle$. As a result, it fares poorly whenever a green
circle is not present. This illustrates the brittleness of optimizing to
obtain a specific action. Because the Combined speaker optimizes in
expectation over all actions, it is more likely to send true messages about
extreme values, e.g. $\langle$Green, 2$\rangle$ or $\langle$Blue, -2$\rangle$.
It obtains both higher rewards $R_{S}(\cal{A})$ and the optimal action
$\pi_{L}(a^{*})$ more frequently.

### Effects of speaker optimality

While we fixed the speaker optimality parameter $\beta_{S}$ throughout our
simulations, it may interact in important ways with our model comparison.
First, our Belief speaker is insensitive to $\beta$: since we assumed that the
listener’s prior over $w$ is uniform, all true utterances are equally
valuable. By contrast, the Action and Combined speakers are sensitive to
$\beta$ in different ways, since they take the soft-max over different
quantities (Action over log-probabilities and Combined over expected
utilities). At any given $\beta$, Combined dominated Action on all metrics, so
we tuned $\beta$ to equate them ($\beta_{\text{Action}}=3$,
$\beta_{\text{Combined}}=2$). At the same time, it is informative to explore
their asymptotic behavior as $\beta\rightarrow\infty$, which is summarized in
Table 4. In Simulation 2, we find that Action and Combined still send false
messages and generalize poorly in the limit. Yet when multiple messages are
equally likely to induce an optimal action, the Combined speaker converges to
preferring truthful ones while the Action speaker is indifferent. This
difference results in dramatically better performance in Simulation 3: the
Combined speaker converges on producing the utterance $\langle$Green,
2$\rangle$, while the optimal Action speaker remains ambivalent between
$\langle$Green, 2$\rangle$ and $\langle$Circle, 2$\rangle$. In sum, the
Combined speaker’s sensitivity to underlying reward structure leads it to
consistently produce more truthful and generalizable utterances across
contexts and in the limit of optimality.

Context | Speaker |  $P_{S}$(truthful) |  $\pi_{L}(a^{*}$) | $R_{S}(A)$ | $R_{S}(\mathcal{A})$
---|---|---|---|---|---
Both | Belief | 1.00 | .499 | .539 | .539
Local | Action | .458 | .942 | 1.63 | .690
Local | Combined | .488 | .942 | 1.67 | .710
Global | Action | .500 | .624 | - | .958
Global | Combined | 1.00 | .731 | - | 1.28
Table 4: Speaker behavior as $\beta\rightarrow\infty$. _Local_ context
corresponds to Simulation 2, and _Global_ context to Simulation 3.

## General Discussion

Humans communicate to influence one anothers’ beliefs _and_ actions. Here, we
explored different ways speakers can reason about these objectives. We
introduced two action-oriented speaker models which optimize for a downstream
_decision problem_ , grounding relevance in the listener’s actual decision
context (?, ?). Critically, we proposed that rational “Combined” speakers
should consider both beliefs and actions: they should communicate to induce
belief states that are likely to produce high-value actions. To distinguish
speaker models, we introduced a new communication game, signaling bandits.
Signaling bandits generalizes Lewis signaling games to multi-armed bandits,
formalizing communication in richer decision settings. Simulations show that
the Combined speaker prefers _generalizable_ information that is likely to
produce high-value actions across a distribution of possible future contexts.
This finding raises intriguing connections to belief-oriented accounts of
generics (e.g.“Birds fly”; ?, ?) as well as biases towards generalizable
examples in non-linguistic pedagogy (?, ?, ?). We are thus optimistic that
such speaker models may provide a bridge from communicative principles to
social learning more broadly.

This work represents a small step towards a deeper exploration of action-
grounded models of rational communication. First, human experiments are needed
to validate our simulations. Second, we considered only literal listeners in
collaborative settings. Pragmatic listeners may reason about a speaker’s
objectives and knowledge, as well as the action context the speaker considered
(?, ?). Finally, we explored only single-round gameplay; iterated games would
allow for richer interactions. Speakers could observe listener actions and
infer their beliefs via inverse reinforcement learning (?, ?). A single
message followed by learner actions in multiple contexts would force speakers
to optimize for a distribution over contexts. This would make generalization
an objective rather than an incidental effect, as in optimal reward design (?,
?). Listeners could learn both socially (via speaker messages) and
individually (via their own actions). We hope we have successfully signaled
the high value of research in this paradigm!

## Acknowledgements

We thank our anonymous reviewers for their thoughtful feedback. This work was
supported by NSF grant #1545126 and John Templeton Foundation grant #61454 to
TLG and NSF grant #1911835 to RDH.

Code for simulations available at: https://github.com/tsumers/signaling-
bandits

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