Open Superstring First Mass Level Effective Lagrangian:
Massive Spin-2 in an Electromagnetic Background

Karim Benakli1,a<EMAIL_ADDRESS>Cassiano A. Daniel2,b
<EMAIL_ADDRESS>and Wenqi Ke1,c<EMAIL_ADDRESS>
_1 Sorbonne Université, CNRS,
Laboratoire de Physique Théorique et Hautes Energies, LPTHE, F-75005 Paris,
France. _

_2 ICTP South American Institute for Fundamental Research
Instituto de Física Teórica, Universidade Estadual Paulista
Rua Dr. Bento Teobaldo Ferraz 271, 01140-070, São Paulo - SP, Brasil. _

###### Abstract

Minimal coupling leads to problems such as loss of causality if one wants to
describe charged particles of spin greater than one propagating in a constant
electromagnetic background. Regge trajectories in string theory contain such
states, so their study may allow us to investigate possible avenues to remedy
the pathologies. We present here two explicit forms, related by field
redefinitions, of the Lagrangian describing the bosonic states in the first
massive level of open superstrings in four dimensions. The first one reduces,
when the electromagnetic field is set to zero, to the Fierz-Pauli Lagrangian
for the spin-2 mode. The second one is a more compact form which simplifies
the derivation of a Fierz-Pauli system of equations of motion and constraints.

##  1 Introduction

There are several motivations to study the propagation of spin-2 states in an
electromagnetic field [1, 2]. This includes the possibility to (roughly)
approximate hadronic resonances at sufficiently low energies, but also as a
problem of mathematical interest. It is remarkable that the content of the
first massive level of bosonic open strings can be reduced to a spin-2 state.
The other string oscillation modes give rise to Stückelberg fields that can be
gauged away. Therefore, the study of the propagation of the first massive
level open bosonic string states in an electromagnetic background allowed
Argyres and Nappi to write the first, and only available, consistent causal
Lagrangian for a charged spin-2 [3]. Although this Lagrangian is causal only
in $D=26$ dimensions, the corresponding form of the equations of motion and
the constraints give an a priori consistent Fierz-Pauli system in arbitrary
dimension, as pointed out by [4, 5]. The analysis of the equations of motion
derived from the Virasoro algebra of open bosonic strings allows to generalize
this system for any integer spin state bigger than one [4]. String field
theory has also been used to study the second massive level of bosonic open
strings by [6] and leads to the action of a charged spin-3 state coupled to
states with lower spins.

In this work, we are interested instead in the massive states of the four-
dimensional superstring. One question we address here is: what is the action
describing the massive bosonic modes of the first excited level of the theory?
In the absence of electromagnetic background, we know that the theory is
described by a Fierz-Pauli action for the spin-2 state, and similar free
actions for the other lower spin fields. But because of the presence of many
states, the theory is more complicated than that of the bosonic string.
Indeed, a superspace action has been derived in [7], and equations of motion
have been obtained in the Lorenz gauge, generalizing the early work [8, 9]. In
particular, it shows couplings induced by the electromagnetic background
between the fields of different spins present at this mass level. One thus
expects from [7], a complicated Lagrangian for physical states, and it has not
yet been explicitly written. How can the action of the bosonic modes be
written as a deformation trivially reducing to Fierz-Pauli in the absence of
an electromagnetic background? In fact, the Lagrangian is not unique as
different choices of physical fields can be made. We will present two
versions. The first one allows, by setting the electromagnetic field to zero,
to directly find the Fierz-Pauli Lagrangian for the massive spin-2, as well as
the free Lagrangian for other fields. The second one is a more compact form
which allows to find the same spin-2 equation of motion and constraints.

Another question we address concerns the equations of motion and constraints.
In [7], these equations couple fields of different spins. Is it possible to
decouple these fields and obtain the equations corresponding to each field
separately? We will see that this is possible at the cost of a complicated
redefinition of the field describing the spin-2. In this work, we will not
decouple the different fields at the Lagrangian level. In the superspace
action with which we start, the lower spin fields play also a role in imposing
constraints that allow to keep the right number of degrees of freedom in the
presence of the electromagnetic background.

Many of the derivation steps of the Lagrangian, the equations of motion and
constraints are long and tedious. They are not included in order to simplify
the presentation and allow the reader to access the most important results.
These calculations will be presented in detail elsewhere [10].

##  2 A deformation of the Fierz-Pauli Lagrangian

An open string carries two charges $q_{0}$ and $q_{\pi}$ located at its ends.
For the low energy effective field theory, only the total charge
$Q=q_{0}+q_{\pi}$ appears. We consider a charged string propagating in a
constant electromagnetic background $F_{mn}$. The corresponding covariant
derivative reads

$\displaystyle D_{n}=(\partial-iQF\cdot X)_{n}$ (2.1)

where $X$ is a space-time coordinate. The extended nature of strings causes
the electromagnetic field to enter all our formulas through the matrix [11]

$\displaystyle\epsilon=\frac{\Lambda^{2}}{\pi}\left[\operatorname{arctanh}\left(\frac{\pi
q_{0}F}{\Lambda^{2}}\right)+\operatorname{arctanh}\left(\frac{\pi
q_{\pi}F}{\Lambda^{2}}\right)\right]$ (2.2)

where $\Lambda$ stands for the fundamental string scale and the covariant
derivative (2.1) is traded for

$\displaystyle\mathfrak{D}_{m}=-\mathrm{i}\,\mathfrak{M}_{mn}\,D^{n},\qquad\left[\mathfrak{D}_{m},\mathfrak{D}_{n}\right]=\mathrm{i}\epsilon_{mn}$
(2.3)

with $\mathfrak{M}$ a matrix that satisfies:

$\displaystyle\mathfrak{M}\cdot\mathfrak{M}^{T}=\frac{\epsilon}{QF}$ (2.4)

It is important to note that in all the manipulations we perform here, the
only property of $\epsilon$ that is used is that it is an anti-symmetric
tensor. The $F_{mn}$-dependence of $\epsilon$ is never explicitly invoked.
Therefore, one could in particular take $\epsilon_{mn}\rightarrow QF_{mn}$ and
$\mathfrak{D}_{m}\rightarrow D_{m}$. This means forgetting about the string
origin of our Lagrangian and the resulting equations, and take the point-like
particle limit.

At the first mass level of the superstring, the bosonic sector has 12 degrees
of freedom (d.o.f.) of mass $M$, among which a spin-2 field described by the
rank-2 tensor $h_{mn}$. The symmetric, traceless and vanishing divergence
conditions leave on-shell 5 degrees of freedom in $h_{mn}$. There is also a
massive vector field $\mathcal{C}_{m}$ counting 3 d.o.f.. The 4 remaining
d.o.f. consist in four scalar fields $\mathcal{M}_{1},\mathcal{N}_{1}$, $A$
and $B$. We will write the Lagrangian as a sum of two pieces
$\mathcal{L}=\mathcal{L}_{1}+\mathcal{L}_{2}$.

$\mathcal{L}_{1}$ contains the decoupled complex scalars
$\mathcal{M}_{1},\mathcal{N}_{1}$ and the massive vector $\mathcal{C}_{m}$:

$\displaystyle\mathcal{L}_{1}=$
$\displaystyle\bar{\mathcal{C}}^{m}\left(\mathfrak{D}^{2}-M^{2}\right)\mathcal{C}_{m}+\mathfrak{D}^{m}\bar{\mathcal{C}}_{m}\mathfrak{D}^{n}{\mathcal{C}}_{n}+2\mathrm{i}\epsilon_{mn}\bar{\mathcal{C}}^{m}\mathcal{C}^{n}$
(2.5)
$\displaystyle+{\mathcal{\bar{M}}}_{1}\left(\mathfrak{D}^{2}-M^{2}\right)\mathcal{M}_{1}+\bar{\mathcal{N}}_{1}\left(\mathfrak{D}^{2}-M^{2}\right)\mathcal{N}_{1}$

From this Lagrangian, we get the equations for the scalar fields:

$\displaystyle\left(\mathfrak{D}^{2}-M^{2}\right)\mathcal{M}_{1}=0,\quad\left(\mathfrak{D}^{2}-M^{2}\right)\mathcal{N}_{1}=0$
(2.6)

and for the massive vector boson:

$\displaystyle\left(\mathfrak{D}^{2}-M^{2}\right)\mathcal{C}_{m}-\mathfrak{D}_{m}\mathfrak{D}_{n}\mathcal{C}^{n}+2\mathrm{i}\epsilon_{mn}\mathcal{C}^{n}=0,\quad\mathfrak{D}^{m}\mathcal{C}_{m}=0.$
(2.7)

On the other hand, the spin-2 state ${h}_{mn}$ appears coupled to the complex
scalars $A$ and $B$ in the second part of the Lagrangian. We define the re-
scaled field:

$\displaystyle\mathcal{H}_{mn}\equiv\left(\eta_{mk}-\mathrm{i}\frac{2}{M^{2}}\epsilon_{mk}\right)h^{k}{}_{n}$
(2.8)

and denote $\left(\epsilon\epsilon\right)\equiv\epsilon^{mn}\epsilon_{mn}$,
$\left(\epsilon\tilde{\epsilon}\right)\equiv\epsilon^{mn}\tilde{\epsilon}_{mn}$,
with the dual field strength
$\tilde{\epsilon}_{mn}\equiv\varepsilon_{mnkl}\epsilon^{kl}/2$. The Lagrangian
of the remaining physical states $\\{h_{mn},A,B\\}$ reads:

$\displaystyle\mathcal{L}_{2}=$
$\displaystyle\left[\bar{A}\mathfrak{D}_{m}-\frac{\mathrm{i}}{M^{2}}\tilde{\epsilon}_{mb}\bar{B}\mathfrak{D}^{b}+\frac{1}{2M^{4}}\left(\epsilon\tilde{\epsilon}\right)\bar{B}\mathfrak{D}_{m}-\frac{1}{2}\varepsilon_{mabc}\mathcal{\bar{H}}^{bc}\mathfrak{D}^{a}-\frac{\mathrm{i}}{M^{2}}\tilde{\epsilon}_{ma}\mathcal{\bar{H}}^{ba}\mathfrak{D}_{b}+\frac{\mathrm{i}}{M^{2}}\tilde{\epsilon}_{mb}\bar{\mathcal{H}}\mathfrak{D}^{b}\right]$
(2.9)
$\displaystyle\quad\times\left(\eta^{mn}-\frac{\mathrm{i}}{M^{2}}\epsilon^{mn}-\frac{2}{M^{4}}\tilde{\epsilon}^{mk}\tilde{\epsilon}_{k}{}^{n}\right)^{-1}$
$\displaystyle\times\left[\mathfrak{D}_{n}A+\frac{\mathrm{i}}{M^{2}}\tilde{\epsilon}_{nl}\mathfrak{D}^{l}B+\frac{1}{2M^{4}}\left(\epsilon\tilde{\epsilon}\right)\mathfrak{D}_{n}B-\frac{1}{2}\varepsilon_{nlpq}\mathfrak{D}^{l}\mathcal{H}^{pq}+\frac{\mathrm{i}}{M^{2}}\tilde{\epsilon}_{nl}\mathfrak{D}_{p}\mathcal{H}^{pl}-\frac{\mathrm{i}}{M^{2}}\tilde{\epsilon}_{nl}\mathfrak{D}^{l}\mathcal{H}\right]$
$\displaystyle-M^{2}\bar{A}A+\bar{B}\left(\mathfrak{D}^{2}-M^{2}\right)B-\frac{2}{M^{4}}\epsilon_{mn}\epsilon^{mk}\bar{B}\mathfrak{D}^{n}\mathfrak{D}_{k}B+\frac{1}{2}\bar{\mathcal{H}}_{(mn)}\mathfrak{D}^{2}h^{mn}+\frac{1}{2}\mathfrak{D}^{n}\bar{\mathcal{H}}_{mn}\mathfrak{D}_{k}h^{mk}$
$\displaystyle-M^{2}\bar{\mathcal{H}}^{(mn)}\mathcal{H}_{(mn)}+\frac{M^{2}}{2}\bar{\mathcal{H}}^{(mn)}{h}_{mn}-\frac{1}{2}\bar{\mathcal{H}}\left(\mathfrak{D}^{2}-M^{2}\right)h+\frac{1}{2}\mathfrak{D}^{n}\bar{\mathcal{H}}_{nm}\mathfrak{D}_{k}\mathcal{H}^{km}$
$\displaystyle+\frac{1}{2}\left(\bar{\mathcal{H}}^{mn}\mathfrak{D}_{m}\mathfrak{D}_{n}h+\text{h.c.}\right)+\frac{1}{2M^{2}}\left[2\mathrm{i}\left(\mathfrak{D}^{n}\bar{\mathcal{H}}_{nm}\epsilon^{mk}\mathfrak{D}_{k}B\right)-\left(\epsilon\epsilon\right)\bar{\mathcal{H}}B+\text{h.c.}\right]$
$\displaystyle+\frac{M^{2}}{2}\left(\bar{\mathcal{H}}^{[mn]}+\frac{1}{M^{2}}\mathrm{i}\epsilon^{mn}\bar{B}\right)\left(\mathcal{H}_{[mn]}-\frac{1}{M^{2}}\mathrm{i}\epsilon_{mn}B\right)$

Despite the complicated couplings in (2.9), the different fields can be
decoupled in the equations of motion. Compact expressions of the latter are
obtained after some algebraic manipulations, which do not involve redefining
the fields, of the Euler-Lagrange equations. From (2.9), we can thus derive
the following equations for the spin-0 fields:

$\displaystyle\left(\mathfrak{D}^{2}-M^{2}\right)A=0,\quad\left(\mathfrak{D}^{2}-M^{2}\right)B=0,$
(2.10)

and for massive spin-2 field:

$\left(\mathfrak{D}^{2}-M^{2}\right){h}_{mn}+2\mathrm{i}\left(\epsilon_{m}{}^{k}{h}_{nk}+\epsilon_{n}{}^{k}{h}_{mk}\right)=0,\qquad
h=0$ (2.11)

Unlike the equations of motion, the divergence constraint of the symmetric
rank-2 tensor $h_{mn}$ is coupled to the scalar fields. The divergence
constraint reads:

$\displaystyle\left[M^{4}-\frac{1}{2}\left(\epsilon\epsilon\right)\right]\mathfrak{D}^{n}h_{mn}-{\mathrm{i}}M^{2}\epsilon_{mk}\mathfrak{D}_{n}h^{nk}-{\mathrm{i}}M^{2}\epsilon^{nk}\mathfrak{D}_{n}h_{mk}+{2}\epsilon_{mk}\epsilon_{nl}\mathfrak{D}^{l}h^{nk}-\epsilon_{mk}\epsilon^{kn}\mathfrak{D}^{l}h_{nl}$
(2.12)
$\displaystyle\quad-\epsilon^{kn}\epsilon_{kl}\mathfrak{D}^{l}h_{mn}+\epsilon^{kn}\epsilon_{k}{}^{l}\mathfrak{D}_{m}h_{nl}+\frac{1}{2}\left(\epsilon\epsilon\right)\mathfrak{D}_{m}B+{\mathrm{i}}M^{2}\epsilon_{mn}\mathfrak{D}^{n}B+{2}\epsilon_{mk}\epsilon^{kn}\mathfrak{D}_{n}B-\mathrm{i}M^{2}\tilde{\epsilon}_{mn}\mathfrak{D}^{n}A$
$\displaystyle=0$

which for $\epsilon=0$ becomes $\partial^{n}h_{mn}=0$. Eq. (2.12) includes
many $\epsilon$-dependent terms and couplings to the scalars, however, there
is an on-shell redefinition of the spin-2:

$\displaystyle\mathfrak{h}_{mn}\equiv$
$\displaystyle\frac{2}{3}h_{mn}-\frac{\mathrm{i}}{M^{2}}\epsilon_{m}{}^{k}h_{kn}-\frac{1}{3M^{2}}\mathfrak{D}_{m}\mathfrak{D}_{k}h^{k}{}_{n}$
(2.13)
$\displaystyle+\frac{\mathrm{i}}{M^{4}}\left(\epsilon_{mk}\mathfrak{D}^{l}\mathfrak{D}^{k}h_{nl}-\epsilon_{mk}\mathfrak{D}_{l}\mathfrak{D}_{n}h^{kl}-\frac{1}{2}\eta_{mn}\epsilon^{kl}\mathfrak{D}^{p}\mathfrak{D}_{l}h_{kp}\right)$
$\displaystyle+\frac{1}{2M^{4}-4\epsilon\epsilon}\left[\frac{4}{3M^{2}}\left(\epsilon\epsilon\right)\mathfrak{D}_{m}\mathfrak{D}_{n}B+\left(1-\frac{4}{M^{4}}\epsilon\epsilon\right)\epsilon_{m}{}^{k}\epsilon_{kn}B+{2\mathrm{i}}\epsilon_{mk}\mathfrak{D}^{k}\mathfrak{D}_{n}B\right.$
$\displaystyle\quad\qquad\qquad\qquad\left.-\left(\frac{1}{3}+\frac{1}{M^{4}}\epsilon\epsilon\right)\left(\epsilon\epsilon\right)\eta_{mn}B\right]$
$\displaystyle+\frac{1}{2M^{4}+4\epsilon\epsilon}\left[-{2\mathrm{i}}\tilde{\epsilon}_{mk}\mathfrak{D}^{k}\mathfrak{D}_{n}A+\frac{5}{4}\left(\epsilon\tilde{\epsilon}\right)\eta_{mn}A-\frac{2}{M^{2}}\left(\epsilon\tilde{\epsilon}\right)\mathfrak{D}_{m}\mathfrak{D}_{n}A+\frac{8}{M^{2}}\tilde{\epsilon}_{mk}\epsilon_{ln}\mathfrak{D}^{k}\mathfrak{D}^{l}A\right]$
$\displaystyle+\left(m\leftrightarrow n\right)$

which decouples the divergence constraint to give the deformed Fierz-Pauli
system:

$\displaystyle\left(\mathfrak{D}^{2}-M^{2}\right)\mathfrak{h}_{mn}+2\mathrm{i}\left(\epsilon_{m}{}^{k}\mathfrak{h}_{nk}+\epsilon_{n}{}^{k}\mathfrak{h}_{mk}\right)=0$
(2.14) $\displaystyle\mathfrak{D}^{n}\mathfrak{h}_{mn}=0,\quad\mathfrak{h}=0$

Therefore, in $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ one starts with 18
complex degrees of freedom. Taking into account the constraints on the spin-2
and spin-1 fields, this leaves us with 12 of them on shell. Introducing the
redefinition (2.13), all equations of motion and constraints are decoupled.
When $\epsilon=0$, i.e. without background, $\mathcal{L}_{2}$ is decoupled,
and we recover two scalars $A$, $B$ as well as a spin-2 $h_{mn}$ described by
the usual Fierz-Pauli Lagrangian.

##  3 A compact Lagrangian

With $\mathcal{L}_{1}$ unchanged, we present here an equivalent, but more
compact version of $\mathcal{L}_{2}$, which allows a simpler derivation of the
equations of motion and constraints. The following Lagrangian involves two
vectors $\\{a_{m},c_{m}\\}$ instead of the scalars $\\{A,B\\}$. There is
indeed a duality relation between the two sets of fields [12], because only
the longitudinal components of the vectors are physical degrees of freedom,
the other components being projected out by the constraints. The Lagrangian
reads:

$\displaystyle\mathcal{L}_{2}=$
$\displaystyle\bar{a}^{m}\left(M^{2}\eta_{mn}-\mathrm{i}\epsilon_{mn}\right)a^{n}+\mathfrak{D}^{m}\bar{a}_{m}\mathfrak{D}^{n}a_{n}-M^{2}\bar{c}^{m}c_{m}-\frac{2}{5}\mathfrak{D}^{m}\bar{c}_{m}\mathfrak{D}^{n}c_{n}$
(3.1)
$\displaystyle+\frac{1}{\sqrt{2}}\left[M\bar{c}^{m}\left(-\frac{2}{5}\mathfrak{D}_{m}\mathcal{H}+\mathfrak{D}^{n}\mathcal{H}_{nm}\right)+\bar{\tilde{F}}^{mn}(a)\left(F_{mn}(c)-\frac{M}{\sqrt{2}}\mathcal{H}_{[mn]}\right)+\text{h.c.}\right]$
$\displaystyle+\frac{1}{2}\bar{\mathcal{H}}_{mn}\mathfrak{D}^{2}h^{mn}+\frac{1}{2}\mathfrak{D}^{n}\bar{\mathcal{H}}_{mn}\mathfrak{D}_{k}h^{mk}-\frac{M^{2}}{2}\bar{\mathcal{H}}^{(mn)}\mathcal{H}_{(mn)}+\frac{M^{2}}{20}\bar{\mathcal{H}}\mathcal{H}+\mathrm{i}\epsilon^{nk}\bar{\mathcal{H}}_{mn}h_{k}{}^{m}$

with the (dual) field strengths given by

$F_{mn}(a)\equiv\mathfrak{D}_{m}a_{n}-\mathfrak{D}_{n}a_{m},\quad\tilde{{F}}_{mn}(a)\equiv\frac{1}{2}\varepsilon_{mnpq}F^{pq}(a)$
(3.2)

and similarly for $F_{mn}(c),\tilde{F}_{mn}(c)$.

We will derive here the equations of motion and the constraints on the fields.
To analyze the Lagrangian, we first decompose $h_{mn}$ into its trace $h$ and
a traceless part $v_{mn}$: $h_{mn}=\frac{1}{4}\eta_{mn}h+v_{mn}$. Combining
the equations of motion of $h$ and $v_{mn}$, we obtain:

$5\mathfrak{D}^{2}h=3M^{2}h-8\sqrt{2}M\mathfrak{D}^{m}c_{m}$ (3.3)

The Euler-Lagrange equations of $a_{m}$, $c_{m}$ read

$\displaystyle
M^{2}a_{m}-\mathrm{i}\epsilon_{mn}a^{n}-\mathfrak{D}_{m}\mathfrak{D}_{n}a^{n}+\sqrt{2}\mathrm{i}\tilde{\epsilon}_{mn}c^{n}-\frac{\mathrm{i}}{M}\epsilon^{nk}\varepsilon_{mkpq}\mathfrak{D}^{q}h_{n}{}^{p}=0$
(3.4)
$\displaystyle-M^{2}c_{m}+\frac{2}{5}\mathfrak{D}_{m}\mathfrak{D}_{n}c^{n}+\sqrt{2}\mathrm{i}\tilde{\epsilon}_{mn}a^{n}-\frac{\sqrt{2}}{5}M\mathfrak{D}_{m}h+\frac{1}{\sqrt{2}M}\left(M^{2}\eta^{nk}-2\mathrm{i}\epsilon^{nk}\right)\mathfrak{D}_{n}h_{mk}=0$

It is useful to compute their divergences

$\displaystyle 0=$ $\displaystyle
M^{2}\mathfrak{D}^{m}a_{m}-\mathrm{i}\epsilon_{mn}\mathfrak{D}^{m}a^{n}-\mathfrak{D}^{2}\mathfrak{D}_{m}a^{m}+\sqrt{2}\mathrm{i}\tilde{\epsilon}_{mn}\mathfrak{D}^{m}c^{n}-\frac{1}{4M}\left(\epsilon\tilde{\epsilon}\right)h$
(3.5) $\displaystyle 0=$
$\displaystyle-M^{2}\mathfrak{D}^{m}c_{m}+\frac{2}{5}\mathfrak{D}^{2}\mathfrak{D}_{m}c^{m}+\sqrt{2}\mathrm{i}\tilde{\epsilon}_{mn}\mathfrak{D}^{m}a^{n}-\frac{\sqrt{2}}{5}M\mathfrak{D}^{2}h$
$\displaystyle+\frac{1}{\sqrt{2}M}\left(M^{2}\eta^{nk}-2\mathrm{i}\epsilon^{nk}\right)\mathfrak{D}^{m}\mathfrak{D}_{n}h_{mk}$

Next, we consider the Euler-Lagrange equation of $\mathcal{H}_{mn}$:

$\displaystyle\mathcal{R}_{mn}\equiv$
$\displaystyle-\frac{1}{M}\varepsilon_{mnkl}\mathfrak{D}^{k}a^{l}+\frac{2\sqrt{2}}{5M}\eta_{mn}\mathfrak{D}^{k}c_{k}-\frac{\sqrt{2}}{M}\mathfrak{D}_{m}c_{n}+\frac{1}{10}\eta_{mn}h+\frac{1}{M^{2}}\mathfrak{D}^{2}h_{mn}$
(3.6)
$\displaystyle-\frac{1}{M^{2}}\mathfrak{D}_{n}\mathfrak{D}^{k}h_{mk}-h_{mn}-\frac{3}{M^{2}}\mathrm{i}\epsilon_{kn}h^{k}{}_{m}+\frac{1}{M^{2}}\mathrm{i}\epsilon_{m}{}^{k}h_{kn}$
$\displaystyle=0$

whose trace gives

$\displaystyle\mathcal{R}^{m}{}_{m}=\frac{3\sqrt{2}}{5M}\mathfrak{D}^{m}c_{m}-\frac{3}{5}h+\frac{1}{M^{2}}\mathfrak{D}^{2}h-\frac{1}{M^{2}}\mathfrak{D}^{m}\mathfrak{D}^{n}h_{mn}=0$
(3.7)

and using the relation (3.3), it implies

$\mathfrak{D}^{m}c_{m}+\frac{1}{\sqrt{2}M}\mathfrak{D}^{m}\mathfrak{D}^{n}h_{mn}=0$
(3.8)

Having (3.3), (3.5) and (3.8) at hand, the trace constraint of $h_{mn}$ can be
now inferred from the sum
$\left(\mathrm{i}\epsilon^{mn}\mathcal{R}_{mn}+\mathfrak{D}^{m}\mathfrak{D}^{n}\mathcal{R}_{mn}\right)$:

$Mh+4\sqrt{2}\mathfrak{D}^{m}c_{m}=0$ (3.9)

The next step involves calculating the divergence constraint of $h_{mn}$. To
this end, let us contract $\mathcal{R}_{mn}$ with $\mathfrak{D}^{n}$, and
simplify the result using the equation of motion of $c_{m}$:

$\displaystyle 0=\mathfrak{D}^{n}\mathcal{R}_{mn}=$
$\displaystyle-\frac{M}{\sqrt{2}}c_{m}-\frac{2\sqrt{2}}{5M}\mathfrak{D}_{m}\mathfrak{D}_{n}c^{n}+\mathrm{i}\frac{\sqrt{2}}{M}\epsilon_{mn}c^{n}$
(3.10)
$\displaystyle-\frac{1}{10}\mathfrak{D}_{m}h-\frac{1}{2}\mathfrak{D}^{n}h_{mn}+\frac{1}{M^{2}}\mathrm{i}\epsilon_{mk}\mathfrak{D}_{n}h^{kn}$

Moreover, thanks to the equation (3.9), the term
$\mathfrak{D}_{m}\mathfrak{D}_{n}c^{n}$ above can be replaced by
$-\frac{M}{4\sqrt{2}}\mathfrak{D}_{m}h$, which results in

$\displaystyle\mathfrak{D}^{n}\mathcal{R}_{mn}=$
$\displaystyle-\frac{M}{\sqrt{2}}\left(\eta_{mn}-\frac{2\mathrm{i}}{M^{2}}\epsilon_{mn}\right)c^{n}-\frac{1}{2}\left(\eta_{mk}-\frac{2\mathrm{i}}{M^{2}}\epsilon_{mk}\right)\mathfrak{D}_{n}h^{kn}$
(3.11) $\displaystyle=0$

Applying the inverse matrix
$\left(\eta_{mn}-2\mathrm{i}\epsilon_{mn}/M^{2}\right)^{-1}$, this equation
implies the divergence constraint

$\displaystyle c_{m}+\frac{1}{\sqrt{2}M}\mathfrak{D}^{n}h_{mn}=0$ (3.12)

Coming back to the original equation of motion $\mathcal{R}_{mn}$, its
symmetric and antisymmetric parts can now be rewritten as, respectively:

$\displaystyle 2M^{2}\mathcal{R}_{(mn)}=$
$\displaystyle\left(\mathfrak{D}^{2}-M^{2}\right)h_{mn}-2\mathrm{i}\left(\epsilon_{km}h^{k}{}_{n}+\epsilon_{kn}h^{k}{}_{m}\right)=0$
(3.13) $\displaystyle 2M^{2}\mathcal{R}_{[mn]}=$
$\displaystyle-{2}{M}\varepsilon_{mnkl}\mathfrak{D}^{k}a^{l}-{2\sqrt{2}}{M}\left(\mathfrak{D}_{m}c_{n}-\mathfrak{D}_{n}c_{m}\right)+{2\mathrm{i}}\left(\epsilon_{km}h^{k}{}_{n}-\epsilon_{kn}h^{k}{}_{m}\right)=0$

The first line in (3.13) is precisely the equation of motion of the Fierz-
Pauli system and it is the four-dimensional version of the same form obtained
in [3] for $D=26$ dimensions. The second line establishes a duality relation
between the field strength of $a_{m}$, and the sum of the field strength of
$c_{m}$ and
$\mathcal{H}_{[mn]}=\frac{\mathrm{i}}{M^{2}}\left(\epsilon_{km}h^{k}{}_{n}-\epsilon_{kn}h^{k}{}_{m}\right)$.

To summarize, for the symmetric tensor $h_{mn}$, we have derived the following
equations of motion and constraints

$\displaystyle\left(\mathfrak{D}^{2}-M^{2}\right)h_{mn}-2\mathrm{i}\left(\epsilon_{km}h^{k}{}_{n}+\epsilon_{kn}h^{k}{}_{m}\right)=0$
(3.14) $\displaystyle\mathfrak{D}^{n}h_{mn}+\sqrt{2}Mc_{m}=0,\quad
Mh=-4\sqrt{2}\mathfrak{D}^{m}c_{m}$

whereas the equations of motion of the vectors $\\{a_{m},c_{m}\\}$, given by
(3.4), are still coupled. By considering the independent constraints obtained
so far, the d.o.f. counting will reveal 12 complex d.o.f.’s on shell. To make
this manifest, we start by decoupling the equations of motion of
$\\{a_{m},c_{m}\\}$. With the on-shell redefinitions

$\displaystyle a_{m}^{\prime}\equiv
a_{m}-\frac{\mathrm{i}}{M^{2}}\epsilon_{mn}a^{n}-\frac{\mathrm{i}}{M^{3}}\tilde{\epsilon}^{nk}\mathfrak{D}_{k}h_{mn}+\frac{\mathrm{i}}{M^{3}}\tilde{\epsilon}_{mn}\mathfrak{D}^{n}h+\frac{2\sqrt{2}}{M^{2}}\mathrm{i}\tilde{\epsilon}_{mn}c^{n}$
(3.15) $\displaystyle c_{m}^{\prime}\equiv
c_{m}-\frac{\sqrt{2}\mathrm{i}}{2M^{2}}\tilde{\epsilon}_{mn}a^{n}+\frac{\mathrm{i}}{\sqrt{2}M^{3}}\epsilon^{nk}\mathfrak{D}_{n}h_{mk}$

their equations of motion become

$\mathfrak{D}_{m}\mathfrak{D}_{n}a^{\prime
n}=M^{2}a^{\prime}_{m},\quad\mathfrak{D}_{m}\mathfrak{D}_{n}c^{\prime
n}=M^{2}c^{\prime}_{m}$ (3.16)

indicating that $a_{m}^{\prime}$, $c_{m}^{\prime}$ are the gradient of a
scalar and as such have only one (longitudinal) d.o.f. on shell. More
precisely, they are dual to the massive scalars $\\{A,B\\}$ in the previous
section, in the same manner as described in [12].

To decouple the trace constraint of $h_{mn}$, a natural guess would be:

$\displaystyle\mathcal{H}_{mn}^{\prime}$
$\displaystyle\equiv\mathcal{H}_{(mn)}+\frac{\sqrt{2}}{M}\eta_{mn}\mathfrak{D}^{k}c_{k}$
(3.17)

where $\mathcal{H}_{(mn)}$ is the symmetric part of the rescaled $h_{mn}$,
defined in (2.8). This new spin-2 satisfies

$\displaystyle\left(\mathfrak{D}^{2}-M^{2}\right)\mathcal{H}^{\prime}_{mn}+2\mathrm{i}\left[\left(\epsilon\cdot\mathcal{H}^{\prime}\right)_{mn}-\left(\mathcal{H}^{\prime}\cdot\epsilon\right)_{mn}\right]=0,$
(3.18)
$\displaystyle\mathfrak{D}^{n}\mathcal{H}^{\prime}_{mn}=-\frac{\mathrm{i}}{M}\tilde{\epsilon}_{mn}a^{n}+\mathrm{i}\frac{\sqrt{2}}{M}\epsilon_{mn}c^{n},\quad\mathcal{H}^{\prime}=0$

In the free case, $\epsilon=0$, the above equations reduce to the Fierz-Pauli
system. The divergence constraint of $\mathcal{H}^{\prime}_{mn}$ is coupled to
$\\{a_{m},c_{m}\\}$ with the field strength $\epsilon_{mn}$, which resembles
the result in [7], Eq (6.72), but $a_{m}$ in this reference is a massive
vector boson having 3 d.o.f.’s.

We may also attempt to use the redefinition in [3], where the only physical
field is the charged spin-2. It was convenient to introduce the double
rescaling:

$\mathcal{H}^{\prime\prime}_{mn}=\left(\eta_{mk}-\mathrm{i}\frac{2}{M^{2}}\epsilon_{mk}\right)\left(\eta_{nl}-\mathrm{i}\frac{2}{M^{2}}\epsilon_{nl}\right)h^{kl}$
(3.19)

This ansatz can be modified to have a vanishing trace in our case:

$\displaystyle\mathcal{H}^{\prime\prime}_{mn}=$
$\displaystyle\left(\eta_{mk}-\mathrm{i}\frac{2}{M^{2}}\epsilon_{mk}\right)\left(\eta_{nl}-\mathrm{i}\frac{2}{M^{2}}\epsilon_{nl}\right)h^{kl}$
(3.20)
$\displaystyle+\frac{\sqrt{2}}{2M}\eta_{mn}\left(2\mathfrak{D}^{k}c_{k}+\mathrm{i}\frac{\sqrt{2}}{M^{2}}\tilde{\epsilon}^{kl}\mathfrak{D}_{k}a_{l}+\frac{2}{M^{2}}\mathrm{i}\epsilon^{kl}\mathfrak{D}_{k}c_{l}\right),$
$\displaystyle\mathcal{H}^{\prime\prime}=$ $\displaystyle 0$

But again, the divergence constraint is non-vanishing. And to absorb the
latter, other terms of order $\mathcal{O}(\epsilon^{2})$ and higher
derivatives shall be included in the new spin-2 definition. Beginning with a
general ansatz that has such higher order terms, and imposing the divergence
as well as the trace constraint, we find a redefinition that satisfies these
requirements:

$\displaystyle\mathfrak{h}_{mn}\equiv$
$\displaystyle\frac{2}{3}h_{mn}-\frac{1}{6}\eta_{mn}h-\frac{\mathrm{i}}{M^{2}}\epsilon_{m}{}^{k}h_{kn}+\frac{\sqrt{2}}{3M}\mathfrak{D}_{m}c_{n}-\frac{1}{M^{4}}\left(\epsilon_{mk}\epsilon^{lk}h_{nl}+\epsilon_{mk}\epsilon_{nl}h^{kl}-\frac{1}{2}\eta_{mn}\epsilon^{kl}\epsilon^{p}{}_{l}h_{kp}\right)$
(3.21)
$\displaystyle-\frac{\mathrm{i}\sqrt{2}}{M^{3}}\left(\epsilon_{mk}\mathfrak{D}^{k}c_{n}-\epsilon_{mk}\mathfrak{D}_{n}c^{k}+\frac{1}{2}\eta_{mn}\epsilon^{kl}\mathfrak{D}_{k}c_{l}\right)$
$\displaystyle-\frac{1}{2M^{4}+4\epsilon\epsilon}\left[-\frac{2\mathrm{i}}{M}\tilde{\epsilon}_{mk}\mathfrak{D}^{k}\mathfrak{D}_{n}\mathfrak{D}_{l}a^{l}+\frac{5}{4M}\left(\epsilon\tilde{\epsilon}\right)\eta_{mn}\mathfrak{D}^{k}a_{k}-\frac{2}{M^{3}}\left(\epsilon\tilde{\epsilon}\right)\mathfrak{D}_{m}\mathfrak{D}_{n}\mathfrak{D}_{k}a^{k}\right.$
$\displaystyle\qquad\qquad\qquad\left.+\frac{8}{M^{3}}\tilde{\epsilon}_{mk}\epsilon_{ln}\mathfrak{D}^{k}\mathfrak{D}^{l}\mathfrak{D}^{p}a_{p}\right]$
$\displaystyle+\frac{1}{2M^{4}-4\epsilon\epsilon}\left[\frac{M^{2}}{6}\mathfrak{D}_{m}\mathfrak{D}_{n}h-\frac{1}{4}\epsilon_{mk}\epsilon^{k}{}_{n}h+\frac{\mathrm{i}}{2}\epsilon_{mk}\mathfrak{D}^{k}\mathfrak{D}_{n}h-\frac{5\epsilon\epsilon}{24}\eta_{mn}h\right]+\left(m\leftrightarrow
n\right)$

One can check that the above definition yields a decoupled Fierz-Pauli system.
To recapitulate, with the redefined fields (3.15) and (3.21), the Lagrangian
(3.1) gives on-shell the following equations of motion and constraints:

$\displaystyle\left(\mathfrak{D}^{2}-M^{2}\right)\mathfrak{h}_{mn}=2\mathrm{i}\left(\epsilon_{km}\mathfrak{h}^{k}{}_{n}+\epsilon_{kn}\mathfrak{h}^{k}{}_{m}\right)$
(3.22) $\displaystyle\mathfrak{D}^{n}\mathfrak{h}_{mn}=0,\quad\mathfrak{h}=0$
$\displaystyle\mathfrak{D}_{m}\mathfrak{D}_{n}a^{\prime
n}=M^{2}a^{\prime}_{m},\quad\mathfrak{D}_{m}\mathfrak{D}_{n}c^{\prime
n}=M^{2}c^{\prime}_{m}$

Finally, we can proceed to an immediate count of the degrees of freedom. We
find the presence of 12 on-shell bosonic degrees of freedom described by
$\mathcal{L}_{1}+\mathcal{L}_{2}$. Although, contrary to (2.9),
$\mathcal{L}_{2}$ given by (3.1) involves extra non-propagating fields, the
transverse components of $\\{a_{m}^{\prime},c_{m}^{\prime}\\}$, it has the
advantage of a more compact form and allows a simpler derivation of equations
of motion and constraints.

##  4 Conclusions

We have presented here for the first time the explicit expression of the two-
derivative Lagrangian describing the states of the first excited level of the
open superstring in the presence of a constant electromagnetic background. In
fact, we have given two different forms, each with its own advantages.

One result of our work that we want to comment on is that the equations of
motion and constraints obtained are those expected. In particular, for the
massive spin-2, we find at the end a Fierz-Pauli system of the form described
by Argyres-Nappi, which is what would be expected if it were a dimensional
reduction. We want to emphasize, however, that the intermediate calculations
are very different. First of all, at the computational level, let us note that
unlike the bosonic case, for superstrings the mass does not depend on the
electromagnetic background. Then, more importantly, the first mass level
contains other physical fields than the massive spin-2 and the equations of
motion directly derived from the Lagrangian couple the different fields. But
we have explicitly shown how we can decouple them at the level of these
equations, which is not trivial. This required a redefinition of the fields
which can be implemented at the level of the Lagrangian to obtain directly the
decoupled equations. We perform this procedure in another work devoted to
fermionic partners, but here the expressions become long and complicated,
which is not desirable. Let us finally note that for all Lagrangians,
presented here and in the previous literature, the Euler-Lagrange equations do
not give directly the equations in their final compact forms but algebraic
manipulations are necessary to reduce them to these forms. This also
highlights the difficulty of guessing a four-dimensional Lagrangian for the
charged massive spin-2 from the equations of motion.

## Acknowledgements

We are grateful to Nathan Berkovits for useful discussions. CAD acknowledges
FAPESP grant numbers 2020/10183-9 and 2022/14599-0 for financial support. The
work of WK is supported by the Contrat Doctoral Spécifique Normalien (CDSN) of
Ecole Normale Supérieure – PSL.

## References

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# Dilepton polarization as a signature of plasma anisotropy

Maurice Coquet<EMAIL_ADDRESS>Michael Winn
<EMAIL_ADDRESS>Université Paris-Saclay, Centre d’Etudes de Saclay (CEA),
IRFU
Département de Physique Nucléaire (DPhN), Saclay, France  Xiaojian Du
<EMAIL_ADDRESS>Instituto Galego de Fisica de Altas Enerxias (IGFAE),
Universidade de Santiago de Compostela, E-15782 Galicia, Spain  Jean-Yves
Ollitrault<EMAIL_ADDRESS>Université Paris-Saclay, CNRS, CEA,
Institut de physique théorique, 91191 Gif-sur-Yvette, France  Sören
Schlichting<EMAIL_ADDRESS>Fakultät für Physik,
Universität Bielefeld, D-33615 Bielefeld, Germany

###### Abstract

We propose the angular distribution of lepton pairs produced in
ultrarelativistic heavy-ion collisions as a probe of thermalization of the
quark-gluon plasma. We focus on dileptons with invariant masses large enough
that they are produced through quark-antiquark annihilation in the early
stages of the collision. The angular distribution of the lepton in the rest
frame of the pair then reflects the angular distribution of quark momenta. At
early times, the transverse pressure of the quark-gluon plasma is larger than
its longitudinal pressure as a result of the fast longitudinal expansion,
which results in an oblate lepton distribution. By constrast, direct (Drell-
Yan) production by quarks and antiquarks from incoming nuclei, whose momenta
are essentially longitudinal, results in a prolate distribution. As the
invariant mass increases, Drell-Yan gradually becomes the dominant source of
dilepton production, and the lepton distribution evolves from oblate to
prolate. The invariant mass at which the transition occurs is highly sensitive
to the equilibration time of the quark-gluon plasma or, equivalently, the
shear viscosity over entropy ratio $\eta/s$ in the early stages of the
collision.

Introduction. — Ultrarelativistic heavy-ion collisions produce a quark-gluon
plasma which gradually thermalizes as it expands into the vacuum Busza:2018rrf
. There is wide consensus on the theory side Schlichting:2019abc ;
Berges:2020fwq  that before it approaches local thermal equilibrium, there is
a pre-equilibrium phase during which the longitudinal pressure is smaller than
the transverse pressure, as a consequence of the fast longitudinal expansion
Bjorken:1982qr . The pressure anisotropy is the standard quantity which
characterizes the deviation from local equilibrium Baier:2000sb ;
Epelbaum:2013ekf ; Berges:2013fga ; Kurkela:2015qoa ; Kurkela:2018vqr , but
its experimental signatures have remained elusive so far Alqahtani:2017mhy .
We propose to detect it by making use of the kinematic properties of electron-
positron or muon-antimuon pairs from the collision, known as dileptons.

Dileptons are produced throughout the collision, and the vast majority reach
the detectors without undergoing further interactions, so that they are golden
signatures of the early history, unlike hadrons. The invariant mass $M$ of a
dilepton gives a handle on its production time. On average, the larger $M$,
the earlier it is produced. The chronology is as follows. First comes
production through the annihilation of a quark and antiquark belonging to
incoming nuclei, known as the Drell-Yan process Drell:1970wh , which is the
dominant dilepton source for the largest values of $M$, apart from narrow
resonances (e.g., bottomonium decays). Then follows quark-antiquark
annihilation in the quark-gluon plasma, which itself starts with pre-
equilibrium emission McLerran:1984ay . This source, which is the focus of our
study, is expected to exceed Drell-Yan production for masses $M\lesssim$3-4
GeV Coquet:2021lca , depending on how fast the quark-gluon plasma thermalizes.
Later emission in the hadronic phase is expected to be significant only for
much lower masses, $M\lesssim 1.2$ GeV Rapp:2013nxa ; Song:2018xca .
Therefore, our main focus is the intermediate mass region $1.2<M<5$ GeV,
excluding the narrow resonances from charmonium decays Andronic:2015wma .

This source of dilepton production has not yet been isolated in ongoing
experiments, because of the large background created by weak semileptonic
decays of hadrons containing a charm quark, which dominates at LHC energies
ALICE:2018ael . With the future detector projects LHCb Upgrade 2 LHCb:2018roe
and ALICE 3 Adamova:2019vkf , rejection of this background should be within
reach, by measuring the slight displacement, by a fraction of a millimeter or
in the millimeter range for large Lorentz boosts at forward rapidity, of the
vertex of the weak decay relative to the primary vertex of the collision.

Figure 1: Illustration of the typical kinematic configuration of leading-order
dilepton production from the Drell-Yan process (left) and from the quark-gluon
plasma in the pre-equilibrium stage (right). The dotted arrow represents the
longitudinal $z$-axis, parallel to the beam line. The polar angle in the
Collins-Soper frame is labeled $\theta$.

The kinematics of lepton pair production can be used to probe the anisotropy
of the quark momentum distribution, which itself reflects the pressure
anisotropy in the quark-gluon plasma Baym:2017qxy . The leading-order process
is $q\bar{q}\to l^{+}l^{-}$, where $l^{-}$ and $l^{+}$ denote the lepton and
its antiparticle. In the center-of-mass frame, if one neglects the lepton mass
(which is an excellent approximation for both muons and electrons in the
considered invariant mass range), the distribution of leptons per solid angle
is proportional to $1+\cos^{2}\theta$, where $\theta$ denotes the angle
between quark and lepton momenta.111This is due to the fact that the
interaction is mediated by a spin-1 particle, hence the term of “polarization”
traditionally used to characterize this effect Speranza:2018osi , even though
it refers to the unpolarized cross section. Therefore, emission parallel to
the quark is more probable by a factor two than perpendicular to the quark.
The Drell-Yan process, where quark momenta are mostly longitudinal, will
therefore result in preferential emission of longitudinal leptons, while pre-
equilibrium emission in the quark-gluon plasma, where quark momenta are mostly
transverse in the rest frame of the fluid, will result in preferential
emission of transverse leptons. This is illustrated schematically in Fig. 1.

The natural observable to quantify this effect is the distribution of the
angle between the positive lepton and the $z$-axis in the rest frame of the
dilepton. This axis corresponds to the beam direction222In general, the two
colliding beams are not exactly collinear, and the ”beam direction” is defined
as the bisector between the two beams.. The defined angle is referred to as
the polar angle of the Collins-Soper reference frame Collins:1977iv , which we
denote by $\theta$. Neglecting the lepton masses, the cosine of this angle is
computed from the momenta ${\bf p}_{3}$ and ${\bf p}_{4}$ and energies
$E_{3}=p_{3}$ and $E_{4}=p_{4}$ of $l^{+}$ and $l^{-}$ in the laboratory frame
using the standard formula:

$\cos\theta=\left.\frac{p_{4}^{z}}{|{\bf p}_{4}|}\right|_{\text{CS
frame}}=\frac{2\left(E_{3}p_{4}^{z}-E_{4}p_{3}^{z}\right)}{M\sqrt{M^{2}+p_{T}^{2}}},$
(1)

where $p_{T}$ in the denominator denotes the transverse momentum of the
dilepton, $p_{T}\equiv\left|{\bf p}_{T3}+{\bf p}_{T4}\right|$.

We first evaluate the distribution of $\cos\theta$ for dileptons emitted by
the quark-gluon plasma, including the pre-equilibrium stage. This calculation
is done along the lines of our previous works Coquet:2021gms ; Coquet:2021lca
. We then carry out a similar calculation for the Drell-Yan process. We
finally present our results for the sum of these two contributions, and show
how it evolves as a function of the invariant mass $M$.

Dilepton emission by the quark-gluon plasma. — We calculate the production
rate of dileptons to leading order in perturbation theory, that is, the rate
of quark-anti-quark annihilation, without any additional gluon in the initial
or final stage. We denote by $f_{q}(\mathbf{p}_{1},x)$ and
$f_{\bar{q}}(\mathbf{p}_{2},x)$ the phase-space distributions of quarks and
anti-quarks, where ${\bf p}_{1}$ and ${\bf p}_{2}$ are their momenta, and $x$
denotes the space-time coordinate where annihilation occurs. The production
rate of a lepton pair with momenta ${\bf p}_{3}$ and ${\bf p}_{4}$ is obtained
by summing the cross section over ${\bf p}_{1}$ and ${\bf p}_{2}$ :

$\displaystyle\frac{dN}{d^{4}xd^{3}{\bf p}_{3}d^{3}{\bf p}_{4}}$
$\displaystyle=\frac{e^{4}}{M^{4}}\frac{\sum
q_{f}^{2}}{(2\pi)^{3}(2p_{3})(2\pi)^{3}(2p_{4})}$
$\displaystyle\times\int\frac{d^{3}{\bf
p}_{1}}{(2\pi)^{3}(2p_{1})}\frac{d^{3}{\bf
p}_{2}}{(2\pi)^{3}(2p_{2})}f_{q}({\bf p}_{1})f_{\bar{q}}({\bf p}_{2})$
$\displaystyle\times
l_{\mu\nu}\Pi^{\mu\nu}(2\pi)^{4}\delta^{(4)}\left(p_{1}+p_{2}-p_{3}-p_{4}\right),$
(2)

where the squared matrix element $l_{\mu\nu}\Pi^{\mu\nu}$ is given by
Schwartz:2014sze :

$\displaystyle l_{\mu\nu}\Pi^{\mu\nu}$ $\displaystyle=$ $\displaystyle
32N_{c}\left[(p_{1}.p_{3})(p_{2}.p_{4})+(p_{1}.p_{4})(p_{2}.p_{3})\right].$
(3)

In the rest frame of the dilepton, where ${\bf p}_{2}=-{\bf p}_{1}$ and ${\bf
p}_{4}=-{\bf p}_{3}$, Eq. (3) yields the usual $1+\cos^{2}\theta$
distribution, where $\theta$ is the angle between ${\bf p}_{4}$ and ${\bf
p}_{2}$.

The next step is to model the phase-space distributions $f_{q,\bar{q}}({\bf
p},x)$ of quarks and antiquarks. The calculation closely follows the steps of
our previous works Coquet:2021gms ; Coquet:2021lca , and we only recall the
main points, referring to these earlier works for technical details.

We first discuss the momentum dependence. In thermal equilibrium, the phase-
space distribution would be a Fermi-Dirac distribution at temperature $T(x)$
in the local rest frame. In order to describe the pre-equilibrium stage, two
modifications must be brought to this equilibrium distribution.

The first modification is the momentum anisotropy. Due to the rapid
longitudinal expansion of the medium at early times, longitudinal momenta are
strongly redshifted in the rest frame of a fluid cell. This implies that the
quark distributions favor large transverse momenta over longitudinal momenta.
We encapsulate this property by making the following replacement in the
expression for the quark/anti-quark distributions:
$|\mathbf{p}|\rightarrow\sqrt{p_{t}^{2}+\xi^{2}p_{z}^{2}}$, where $\xi>1$ is
an anisotropy parameter Martinez:2007pjh . Our main goal in this paper is to
see this momentum anisotropy through the produced leptons.

The second modification that one must include is the underpopulation of quarks
with respect to the equilibrium value. Indeed, the initial state of heavy-ion
collisions is mainly comprised of gluons, which dominate the phase space
Lappi:2006fp . The production of quarks and the approach towards chemical
equilibrium only develops incrementally through partonic interactions
Kurkela:2018oqw ; Kurkela:2018xxd . This effect is taken into account by
multiplying the momentum distributions by a global “quark suppression” factor
$q_{s}$, which is smaller than unity.

We now explain how the space-time dependence is modeled. We assume for
simplicity that the quark-gluon plasma undergoes uniform longitudinal
expansion Bjorken:1982qr . We neglect transverse flow, which develops linearly
as a function of time Vredevoogd:2008id , and is therefore negligible at the
early times which we consider. This is likely to be a poor approximation for
the lowest invariant masses considered in this paper, but the effect of
transverse flow on dilepton polarization for these low masses has already been
studied elsewhere Speranza:2018osi . In addition, we assume that the
transverse density is uniform. We have shown previously that taking into
account the inhomogeneity of the density profile amounts to a modest overall
rescaling Coquet:2021gms . Thanks to these simplifications, the integral of
the production rate over space-time coordinates reduces to an integral over
time Coquet:2021gms .

Both the anisotropy parameter $\xi$ and the quark suppression factor $q_{s}$
are computed as functions of time using QCD kinetic theory Giacalone:2019ldn ;
Du:2020dvp ; Du:2020zqg . This model provides the evolution of pressure
anisotropy and of the fraction of energy density carried by quarks as a
function of time. The control parameter which determines how fast the pressure
isotropizes is the ratio of the shear viscosity over entropy of the quark-
gluon plasma, $\eta/s$ Kovtun:2004de ; Romatschke:2007mq ; Heinz:2013th ,
which is assumed to be constant in our calculation, of which it is the only
free parameter. Dimensional analysis shows that the pressure anisotropy $\xi$
and the quark suppression factor $q_{s}$ solely depends on the dimensionless
variable $\tilde{w}=\tau T/(4\pi\eta/s)$ Heller:2016rtz , where $\tau$ is the
time in the rest frame of the fluid, and $T$ the temperature at time $\tau$,
defined from the energy density. This variable $\tilde{w}$ physically
represents the ratio between the time $\tau$ and the equilibration time, which
itself depends on $\tau$ through the temperature $T$. The pressure becomes
isotropic in the limit $\tilde{w}\gg 1$, and is strongly anisotropic if
$\tilde{w}\ll 1$. Thus, the larger $\eta/s$, the slower the quark distribution
approaches isotropy and chemical equilibrium. We match the anisotropy
parameter $\xi$ and the quark suppression factor $q_{s}$ to the results from
QCD kinetic theory, and fix the late-time entropy density using the
multiplicity measured by the ALICE collaboration in Pb–Pb collisions at
$\sqrt{s_{\mathrm{NN}}}=5.02~{}\mathrm{TeV}$ ALICE:2016fbt .

Figure 2: Distribution of $\cos\theta$, defined by Eq. (1), for quark-gluon
plasma production with two values of the shear viscosity over entropy ratio
$\eta/s$, and Drell-Yan production calculated at NLO, (left) in the
$2.5<M<3~{}\mathrm{GeV}/c^{2}$ invariant mass bin and (right) in the
$4.5<M<5~{}\mathrm{GeV}/c^{2}$.

Results for the distribution of $\cos\theta$, defined by Eq. (1), are
displayed in Fig. 2 for two ranges of invariant mass $M$, and two values of
the parameter $\eta/s$, at midrapidity for $0-5\%$ central Pb–Pb collisions at
$\sqrt{s_{\rm NN}}=5.02$ TeV. This distribution would be uniform if the
distribution was isotropic. Instead, it is maximum for leptons emitted
perpendicularly to the collision axis, corresponding to $\cos\theta=0$. This
is the expected consequence of the anisotropy of the quark momentum
distribution, which is itself largest for purely transverse momenta. This
maximum is more pronounced for the larger value of $M$: for $\eta/s=0.16$, the
variation of the distribution as a function of $\cos\theta$ is $\sim 16\%$ for
the lower mass range $2.5<M<3~{}\mathrm{GeV}/c^{2}$ (left), and $\sim 30\%$
for the higher mass range $4.5<M<5~{}\mathrm{GeV}/c^{2}$. This is because
higher masses are on average produced at earlier times, where the momentum
anisotropy is larger. When the viscosity over entropy ratio $\eta/s$
increases, the number of produced dileptons decreases, due to the larger quark
suppression Coquet:2021lca . The anisotropy also becomes more pronounced,
because approach to equilibrium is slower. Simple scaling arguments show that
pre-equilibrium effects scale with $\eta/s$ and $M$ like $(\eta/s)M^{2}$
Coquet:2021gms ; Garcia-Montero:2023lrd . Our calculation confirms this
scaling.

Drell-Yan dileptons. — At large invariant masses, dilepton production in
hadron-hadron and nucleus-nucleus collisions at the LHC is dominated by the
Drell-Yan process. The leading-order production process is the same as for
production in the quark-gluon plasma, namely, quark-antiquark annihilation.
Therefore, the Drell-Yan process is usually considered as an experimentally
irreducible background when studying production by a quark-gluon plasma. We
will show that the angular distribution of leptons offers the possibility to
disentangle the two processes.

Drell-Yan production has been studied for several decades. The theory is much
more advanced than that of production in the quark-gluon plasma. The process
is calculated in perturbative QCD using collinear factorization. The advantage
over quark-gluon plasma production is that next-to-leading (NLO) calculations
are available, which we use here.

In addition to the leading-order quark-anti-quark annihilation
$q\bar{q}\rightarrow V\rightarrow l^{+}l^{-}$, NLO processes include virtual
corrections ($q\bar{q}\rightarrow Vg$), as well as Compton scattering
($qg\rightarrow Vq$). These calculations are performed using the DrellYan
Turbo software Camarda:2019zyx , with nuclear parton distributions (nPDFs)
from EPPS Eskola:2016oht .

For invariant masses $M\leq 10~{}\mathrm{GeV}$, the perturbative calculation
exhibits increasingly large uncertainties. These uncertainties mainly stem
from the variation of the renormalization and factorization scales introduced
in the collinear factorization approach. In this framework, the uncertainties
are small at the $Z$-pole, but grow rapidly going towards the intermediate and
low mass regions, as was shown in our previous publications, where the
renormalization and factorization scales were varied by a factor of two
Coquet:2021lca . For illustration purposes, these variations will not be
considered in the following, and both scales are set equal to the dilepton
mass. This does not alter the qualitative message of the present study.

Results for the $\cos\theta$ distribution of Drell-Yan dileptons are shown in
Fig. 2. As expected, the angular distribution is opposite to that of quark-
gluon plasma production, with a peak at $\cos\theta\ =\pm 1$ instead of
$\cos\theta=0$. Note that the dependence on $\cos\theta$ is weaker than the
simple $1+\cos^{2}\theta$ expected from the leading-order Drell-Yan process.
This is the effect of NLO corrections, and of quark momenta not being purely
longitudinal. In the lower mass interval (left), production in the quark-gluon
plasma dominates over Drell-Yan Coquet:2021lca . In the higher mass interval
(right) it can be of similar magnitude or smaller, depending on the value of
$\eta/s$.

Total dilepton emission. — We finally sum the contributions of quark-gluon
plasma production and Drell-Yan, and study the dependence of the sum on
$\cos\theta$. We choose as a measure of the angular asymmetry the average
value of $(3\cos^{2}\theta-1)/2$, which corresponds to the quadrupole moment,
and vanishes for an isotropic distribution:

$\left\langle\frac{3\cos^{2}\theta-1}{2}\right\rangle\equiv\frac{\int_{-1}^{1}d\cos\theta~{}\frac{1}{2}(3\cos^{2}\theta-1)\frac{dN}{d\cos\theta}}{\int_{-1}^{1}d\cos\theta~{}\frac{dN}{d\cos\theta}}.$
(4)

Positive values of the quadrupole moment correspond to a prolate distribution
and negative values to an oblate distribution. The extreme values are $1$,
corresponding to all leptons emitted along the $z$ axis, and $-\frac{1}{2}$,
corresponding to all leptons emitted in the transverse plane. Note that if
$dN/d\cos\theta\propto 1+\cos^{2}\theta$, which is the maximal angular
dependence that one would expect from quark-antiquark annihilation, the
quadrupole moment defined by Eq. (4) is only $\frac{1}{10}$. If the same
distribution is rotated by an angle $\pi/2$, corresponding to emission by
purely transverse quarks, the quadrupole moment is $-\frac{1}{20}$. One
therefore expects the quadrupole moment to lie between $-0.05$ and $0.1$, that
is, in a range ten times smaller than the mathematically-allowed interval
between $-0.5$ and $1$.

Figure 3: Quadrupole moment of the angular distribution of leptons, defined as
$\frac{1}{2}\left\langle 3\cos^{2}(\theta)-1\right\rangle$, for quark-gluon
plasma production and Drell-Yan, as a function of the invariant mass $M$ of
the dilepton pair.

Fig. 3 displays the variation of the quadrupole moment with the invariant mass
$M$ for both processes individually, as well as for the sum. For quark-gluon
plasma production, the quadrupole moment is negative as expected,
corresponding to an oblate distribution. It goes to zero for small values of
$M$, which are produced at late times when the pressure is
anisotropic.333Note, however, that our calculation, which neglects transverse
flow, is not reliable for small $M$ Speranza:2018osi . For large values of
$M$, it tends asymptotically to the lower bound $-0.05$, as expected. The
variation with $\eta/s$ scales with $(\eta/s)M^{2}$, as expected by
dimensional analysis Coquet:2021gms .

For Drell-Yan production, the quadrupole moment is positive, corresponding to
a prolate distribution, and varies weakly with $M$. Note that it is smaller by
a factor $\sim 2$ than the upper bound $\frac{1}{10}$.

If one sums the contributions of quark-gluon plasma production and Drell-Yan
before evaluating the asymmetry, one finds that the asymmetry is negative for
the lower values of $M$, where Drell-Yan is negligible, and becomes positive
for larger values of $M$, where Drell-Yan dominates. The value of $M$ for
which the transition occurs depends on $\eta/s$. The smaller $\eta/s$, the
more copiously the quark-gluon plasma produces lepton pairs, and negative
asymmetries are observed up to $M\sim 4$ GeV.

In summary, we have introduced a new observable, the quadrupole moment of the
angular distribution of leptons, which is negative for dileptons produced in
the quark-gluon plasma, and positive for Drell-Yan dileptons. Measurement of
this observable in future LHC experiments will provide the first experimental
proof that the pressure tensor of the quark-gluon plasma is oblate at early
times, and provide a first direct experimental constraint on the
isotropization time of the quark-gluon plasma.

###### Acknowledgements.

This work is supported in part by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) through the CRC-TR 211 “Strong-interaction matter
under extreme conditions” project number 315477589 – TRR 211 and in part in
the framework of the GLUODYNAMICS project funded by the “P2IO LabEx
(ANR-10-LABX-0038)” in the framework “Investissements d’Avenir”
(ANR-11-IDEX-0003-01) managed by the Agence Nationale de la Recherche (ANR),
France.

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