#  $\mathbb{Z}/r\mathbb{Z}$-equivariant covers of $\mathbb{P}^{1}$ with moving
ramification

Carl Lian  Institut für Mathematik, Humboldt-Universität zu Berlin, 12489
Berlin, Germany<EMAIL_ADDRESS>and  Riccardo Moschetti  Dipartimento
di Matematica “F. Casorati”, Università degli Studi di Pavia, Via Ferrata 5,
27100 Pavia, Italy<EMAIL_ADDRESS>
###### Abstract.

Let $X\to\mathbb{P}^{1}$ be a general cyclic cover. We give a simple formula
for the number of equivariant meromorphic functions on $X$ subject to
ramification conditions at variable points. This generalizes and gives a new
proof of a recent result of the second author and Pirola on hyperelliptic odd
covers.

##  1\. Introduction

Let $\mathcal{H}$ be a moduli space of branched covers of $\mathbb{P}^{1}$,
and let $\overline{\mathcal{H}}$ be its compactification by Harris-Mumford
admissible covers, see [HM82]. Let
$\phi:\overline{\mathcal{H}}\to\overline{\mathcal{M}}_{g,n}$ be the map
remembering the source of a branched cover, possibly with additional marked
points (e.g. ramification points).

Pushing forward the fundamental class by $\phi$ yields a cycle class on
$\overline{\mathcal{M}}_{g,n}$, which, by a theorem of Faber-Pandharipande,
lies in the tautological ring, see [FP05]. While an algorithm is given to
compute this class, it is in general quite impractical to implement, and few
explicit formulas are known.

In the case where $\dim(\mathcal{H})=\dim(\mathcal{M}_{g,n})$, so that $\phi$
is generically finite, computing this cycle class amounts to computing the
degree of $\phi$, for which formulas were given (assuming the ramification
points lie in distinct fibers) in [L19] using limit linear series and Schubert
calculus techniques. By definition, these degrees count maps
$h:X\to\mathbb{P}^{1}$ out of a fixed curve with ramification conditions
imposed at possibly moving (variable) points.

A variant of this problem has been studied in [MP20], in which $X$ is assumed
to be hyperelliptic and $h$ is required to be equivariant for the
hyperelliptic action. More precisely, it is proven that:

######  Theorem 1.1.

[MP20] Let $X$ be a general hyperelliptic curve of genus $g$ with Weierstraß
points $P_{1},\ldots,P_{2g+2}$. Fix integers
$n_{1},\ldots,n_{2g+2}\in\mathbb{Z}_{\geq 0}$ such that
$\sum_{i=1}^{2g+2}n_{i}=g-1$. Consider all covers $h:X\to\mathbb{P}^{1}$ of
degree $4g$ such that:

  * •

$h$ is equivariant for the hyperelliptic involution on $X$ and the involution
$z\mapsto-z$ on $\mathbb{P}^{1}$,

  * •

$h^{-1}(\infty)=\sum_{i=1}^{2g+2}(2n_{i}+1)P_{i}$,

  * •

$h$ is unramified over $0\in\mathbb{P}^{1}$, and

  * •

all remaining ramification points of $h$ are triple points.

Then, for any choice of $n_{i}$, the number of such covers (called
_hyperelliptic odd covers_) is $2^{2g}$.

Here, we have ramification conditions imposed at the fixed points $P_{i}$, in
addition to moving (unspecified) triple ramification points away from
$0,\infty$. The enumerative count may also be expressed as the degree of the
morphism
$\phi_{\text{HOC}}:\mathcal{H}^{\text{HOC}}_{g,n_{i}}\to\mathcal{H}_{g}$,
where $\mathcal{H}^{\text{HOC}}_{g,n_{i}}$ is the moduli space of
hyperelliptic odd covers for the given choice of integers $n_{i}$,
$\mathcal{H}_{g}$ is the stack of hyperelliptic curves, and
$\phi_{\text{HOC}}$ remembers the source curve. The proof expresses such
covers as solutions to a certain differential equation and studies these
solutions in terms of periods on $X$.

In this work, we employ a more straightforward technique to give a natural
generalization of this problem to arbitrary cyclic covers. Fix a general
cyclic cover $\sigma:X\to\mathbb{P}^{1}$ of degree $r$ with marked
ramification points $P_{i,j}\in X$; the $\mathbb{Z}/r\mathbb{Z}$-orbits of the
marked points are the sets $\\{P_{i,j}\\}$ where $i$ is fixed and $j$ varies.
We will denote by $\mathcal{H}_{g,\mathbb{Z}/r\mathbb{Z},\xi}$ the moduli
space parametrizing such $\sigma$, where $\xi$ is a collection of elements in
$\mathbb{Z}/r\mathbb{Z}$ describing the monodromy over the branch points of
$\sigma$, see §2.1.

We enumerate $\mathbb{Z}/r\mathbb{Z}$-equivariant maps $h:X\to\mathbb{P}^{1}$,
where $\mathbb{Z}/r\mathbb{Z}$ acts on the target by multiplication of a
coordinate $z$ by $r$-th roots of unity. We require $h$ to have two types of
ramification points:

  * •

ramification at $P_{i,j}$ over $0,\infty\in\mathbb{P}^{1}$, with common
ramification indices (for fixed $i$) given by integers $|a_{i}|>0$; we take
$a_{i}>0$ whenever $h(P_{i,j})=0$ and $a_{i}<0$ otherwise.

  * •

ramification away from $0,\infty\in\mathbb{P}^{1}$ at unspecified (and
unordered) points of $X$ occurring in $k$ orbits of size $r$, with
ramification indices given by the multiset $B=\\{b_{1},\ldots,b_{k}\\}$.

We will denote by $\mathcal{H}_{g,\mathbb{Z}/r\mathbb{Z},\xi,B}$ the moduli
space parametrizing such $h$, see §2.2.

Counting $\mathbb{Z}/r\mathbb{Z}$-equivariant covers amounts to computing the
degrees of the forgetful maps
$\phi_{B}:\mathcal{H}_{g,\mathbb{Z}/r\mathbb{Z},\xi,B}\to\mathcal{H}_{g,\mathbb{Z}/r\mathbb{Z},\xi}$.
Under appropriate conditions detailed in §2, these maps are generically quasi-
finite and the spaces are non-empty. Our main theorem is as follows (see also
Theorem 2.3):

######  Theorem 1.2.

Under appropriate numerical constraints ensuring that
$\mathcal{H}_{g,\mathbb{Z}/r\mathbb{Z},\xi,B}$ and
$\mathcal{H}_{g,\mathbb{Z}/r\mathbb{Z},\xi}$ are non-empty of the same
dimension (see (1), (2), (3), (4), (5), and Corollary 2.2), the number of
$\mathbb{Z}/r\mathbb{Z}$-equivariant covers $h:X\to\mathbb{P}^{1}$ of degree
$d$ with ramification indices $|a_{i}|$ over $0,\infty$ is:

$\binom{(t_{0}+t_{\infty})/r}{t_{0}/r}\cdot\prod_{j=1}^{k}(b_{j}-1)\cdot\frac{k!}{\prod_{\ell\in
L^{\prime}}c^{\prime}_{\ell}!}$

where:

  * •

$t_{0},t_{\infty}$ are the numbers of points in $h^{-1}(0),h^{-1}(\infty)$,
respectively, that are not among the marked ramification points $P_{i,j}$ of
$\sigma$, and

  * •

$L^{\prime}$ is the set of distinct ramification indices $b_{j}$ occurring
away from $0,\infty\in\mathbb{P}^{1}$, and for $\ell\in L^{\prime}$,
$c^{\prime}_{\ell}$ is the number of orbits of ramification points with
ramification index $\ell$.111We will see later that it is somewhat more
natural to consider the set $L$ of distinct integers among $b_{j}-1$.

Among the required numerical constraints is that the (signed) ramification
indices $a_{i}$ of $h$ above $0,\infty$ satisfy a certain congruence condition
in terms of the monodromy of $\sigma$, see Corollary 2.2. As long as this
condition is satisfied, the final formula does not depend on the $a_{i}$; this
is also the case in Theorem 1.1.

Theorem 1.1 is the special case of Theorem 1.2 where $r=2$, $t_{0}=4g$,
$t_{\infty}=0$, $k=2g$, and the $b_{i}$ are all equal to 3. In particular,
this covers also the case of elliptic curves with an odd spin structure in
degree four, which has been addressed in [FMNP20] and [L19].

For the proof of Theorem 1.2, the key observation is that the equivariant
covers in question may be enumerated after passing to the scheme-theoretic
quotients by the $\mathbb{Z}/r\mathbb{Z}$-actions on the source and target, so
we reduce to a question of counting maps
$h^{\prime}:\mathbb{P}^{1}\to\mathbb{P}^{1}$ with certain ramification
profiles. We address this question in §3: the moving ramification conditions
are imposed by way of Segre and Veronese embeddings, and we conclude our main
result by analyzing the intersection that arises.

Lastly, we remark that we expect the method to work for equivariant covers for
more general finite subgroups of $\text{PGL}_{2}(\mathbb{C})$, though we have
not pursued this here.

###  1.1. Acknowledgments

C.L. was supported by an NSF Postdoctoral Fellowship, grant DMS-2001976. R.M.
was supported by MIUR: Dipartimenti di Eccellenza Program (2018-2022)-Dept. of
Math. Univ. of Pavia and by PRIN Project _Moduli spaces and Lie theory_
(2017). We thank Gavril Farkas, Pietro Pirola and Johannes Schmitt for helpful
comments and correspondence.

##  2\. Setup

In this section, we introduce the various moduli spaces and the relations
between them. We work over $\mathbb{C}$.

###  2.1. Cyclic covers

Let $X$ be a smooth, projective, and connected curve of genus $g$, and let
$\sigma:X\to\mathbb{P}^{1}$ be a cyclic cover of degree $r$, ramified over
$Q_{1},\ldots,Q_{m}\in\mathbb{P}^{1}$. Let
$\xi=(\xi_{1},\ldots,\xi_{m})\in(\mathbb{Z}/r\mathbb{Z})^{m}$ be the tuple of
monodromy elements over the $Q_{i}$, and let
$\mathcal{H}_{g,\mathbb{Z}/r\mathbb{Z},\xi}$ be the moduli space of such
covers, which is non-empty and smooth of dimension $m-3$ as long as
$\sum_{i}\xi_{i}=0$ in $\mathbb{Z}/r\mathbb{Z}$ and
$\gcd(\\{\xi_{1},\ldots,\xi_{m},r\\})=1$, see [SvZ18, Theorem 3.7 and
Proposition 3.20]. We will assume throughout that $m\geq 3$, but the results
still hold in the case $m=2$, with minor modifications. Taking
$\xi_{i}\in\\{0,1,2,\ldots,r-1\\}$, we may express $X$ birationally as the
curve given by the equation
$y^{r}=(x-\lambda_{1})^{\xi_{1}}\cdots(x-\lambda_{m})^{\xi_{m}}$, and $\sigma$
as the projection to the $x$-coordinate.

For each $i$, let $f_{i}=\gcd(\xi_{i},r)$, and denote by $P_{i,j}\in X$, for
$j=1,2,\ldots,f_{i}$, the pre-images of $Q_{i}$ under $\sigma$. Let
$e_{i}=\frac{r}{f_{i}}$ be the common ramification index of the $P_{i,j}$. By
Riemann-Hurwitz, we have

(1) $2g-2=\sum_{i=1}^{m}f_{i}(e_{i}-1)-2r=r(m-2)-\sum_{i=1}^{m}f_{i}.$

We remark that the hyperelliptic locus $\mathcal{H}_{g}\subset\mathcal{M}_{g}$
is the image of the map
$\mathcal{H}_{g,\mathbb{Z}/2\mathbb{Z},(1^{2g+2})}\to\mathcal{M}_{g}$
remembering the source of the cover, see [SvZ18, §4.4].

###  2.2. $\mathbb{Z}/r\mathbb{Z}$-equivariant covers

Let $g:\mathbb{P}^{1}\to\mathbb{P}^{1}$ be the degree $r$ map totally ramified
over $0,\infty$. Consider degree $d$ covers $h:X\to\mathbb{P}^{1}$ equivariant
under the $\mathbb{Z}/r\mathbb{Z}$-actions induced by $\sigma$ and $g$, so
that we have a commutative diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{h}$$\textstyle{\mathbb{P}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{\prime}}$$\textstyle{\mathbb{P}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{\mathbb{P}^{1}}$

where $h^{\prime}:\mathbb{P}^{1}\to\mathbb{P}^{1}$ also has degree $d$, and
$\sigma:X\to\mathbb{P}^{1}$ is a cyclic cover of degree $r$ as defined in the
previous section.

For each $i$, the ramification points $P_{i,j}$ of $\sigma$ must all map to
either 0 or $\infty$ under $h$; let $S_{0}$ be the set of $i$ such that
$h(P_{i,j})=0$ and similarly define $S_{\infty}$. For $i\in S_{0}$, let
$a_{i}>0$ be the common ramification index of the $P_{i,j}$ over 0, and for
$i\in S_{\infty}$, let $-a_{i}>0$ be the ramification index of $P_{i,j}$ over
$\infty$.

Suppose in addition that there are $t_{0}$ points not among the $P_{i,j}$
mapping to $0$ under $h$ and similarly define $t_{\infty}$. We assume that all
such points are unramified. The sets of pre-images of $0,\infty$ outside of
the $P_{i,j}$ are disjoint unions of free $\mathbb{Z}/r\mathbb{Z}$-orbits, so
in particular $r|t_{0},t_{\infty}$. Note that we must have

(2) $d=t_{0}+\sum_{i\in S_{0}}a_{i}f_{i}=t_{\infty}+\sum_{i\in
S_{\infty}}(-a_{i}f_{i}).$

Suppose now that $h$ has $K=kr$ additional points of ramification, which
decompose into $k$ orbits of size $r$, and that no two such points appear in
the same fiber of $h$. Let $B=\\{b_{1},\ldots,b_{k}\\}$ denote the (unordered)
multiset of ramification indices (greater than 1) that appear, and let
$c_{\ell}$ be the number of the $b_{1},\ldots,b_{k}$ equal to $\ell+1$. Let
$L$ be the set of $\ell$ for which $c_{\ell}\neq 0$.

Let $\mathcal{H}_{g,\mathbb{Z}/r\mathbb{Z},\xi,B}$ be the space of
$\mathbb{Z}/r\mathbb{Z}$-equivariant covers as described above; we suppress
the data of the $a_{i}$ and the values of $t_{0},t_{\infty}$ for notational
convenience. We have a quasi-finite and étale map
$\delta_{B}:\mathcal{H}_{g,\mathbb{Z}/r\mathbb{Z},\xi,B}\to\mathcal{P}_{k,r}$,
where $\mathcal{P}_{k,r}$ is the smooth variety parametrizing unordered
collections of $k$ distinct $\mathbb{Z}/r\mathbb{Z}$ orbits on
$\mathbb{P}^{1}$, up to the $\mathbb{C}^{*}$ action preserving $0,\infty$. In
particular, $\mathcal{H}_{g,\mathbb{Z}/r\mathbb{Z},\xi,B}$ is smooth of
dimension $k-1$. We are interested in computing degrees of the forgetful maps
$\phi_{B}:\mathcal{H}_{g,\mathbb{Z}/r\mathbb{Z},\xi,B}\to\mathcal{H}_{g,\mathbb{Z}/r\mathbb{Z},\xi}$;
in order for $\phi_{B}$ to expect a non-zero answer, we require

(3) $k=m-2.$

Set $t=t_{0}+t_{\infty}$. By Riemann-Hurwitz, we have

(4) $2g+2d-2=\sum_{i=1}^{m}(|a_{i}|-1)f_{i}+\sum_{j=1}^{k}r(b_{j}-1).$

Substituting $2d=t+\sum_{i=1}^{m}|a_{i}|f_{i}$ and
$2g-2=r(m-2)-\sum_{i=1}^{m}f_{i}$, and $k=m-2$, we find that

(5) $t=r\sum_{j=1}^{k}(b_{j}-2).$

###  2.3. Congruence condition on $a_{i}$

In this section, we show that in order for equivariant covers $h$ to exist,
the vanishing orders $a_{i}$ need to satisfy a certain congruence condition.
We give two perspectives, one algebraic and one topological.

Consider the map $h^{\prime}$. The fiber over 0 consists of the points $Q_{i}$
previously defined for $i\in S_{0}$, with ramification index $a_{i}$, and
$t_{0}/r$ points of ramification index $r$ each. Similarly, the fiber over
$\infty$ consists of $Q_{i}$ for $i\in S_{\infty}$ with ramification index
$-a_{i}$ and $t_{\infty}/r$ points of ramification index $r$ each. Moreover,
it is ramified at $k$ additional (unlabelled) points with ramification index
$b_{j}$.

We may therefore represent the map $h^{\prime}$ by the meromorphic function

$h^{\prime}(x)=\prod_{i=1}^{m}(x-\lambda_{i})^{a_{i}}\cdot\left(\frac{P_{0}(x)}{P_{\infty}(x)}\right)^{r}$

where $P_{0},P_{\infty}$ are non-zero polynomials of degree
$t_{0}/r,t_{\infty}/r$, respectively. We also allow one of $P_{0},P_{\infty}$
to have strictly smaller degree, which occurs when $h^{\prime}$ has a pole or
zero at $x=\infty$.

######  Lemma 2.1.

$h^{\prime}$ lifts to a $\mathbb{Z}/r\mathbb{Z}$-equivariant cover
$h:X\to\mathbb{P}^{1}$ if and only if $a_{i}\equiv\xi_{i}\pmod{r}$ for all
$i$. If this is the case, the lift is unique, up to composition with the
$\mathbb{Z}/r\mathbb{Z}$-action on the target.

###### Proof.

In order to lift such an $h^{\prime}$, the meromorphic function $h$ must send
a general point $(x,y)$ on $X$ to an $r$-th root of $h^{\prime}(x)$, that is,
such an $r$-th root must exist in the function field
$\mathbb{C}(X)=\mathbb{C}(x)[\sqrt[r]{\prod_{i=1}^{m}(x-\lambda_{i})^{\xi_{i}}}]$
of $X$. The field extension $\mathbb{C}(X)/\mathbb{C}(x)$ is Galois with
Galois group $G\cong\mathbb{Z}/r\mathbb{Z}$, corresponding to the cyclic cover
$\sigma$. Write $f=(x-\lambda_{i})^{\xi_{i}}$ and let $\tau\in G$ be a
generator such that

$\tau(\sqrt[r]{f})=\omega\cdot\sqrt[r]{f},$

for some primitive root of unity $\omega\in\mathbb{C}$, so that

$\tau((\sqrt[r]{f})^{a})=\omega^{a}\cdot(\sqrt[r]{f})^{a},$

for $a=0,1,\ldots,r-1$.

Suppose now that $h^{\prime}(x)$ has an $r$-th root $\alpha\in\mathbb{C}(X)$,
so that $\alpha^{r}\in\mathbb{C}(x)$. Then,
$\tau(\alpha)^{r}=\tau(\alpha^{r})\in\mathbb{C}(x)$, so
$\tau(\alpha)^{r}=\alpha^{r}$. Thus, $\tau(\alpha)=\omega\cdot\alpha$ for some
$r$-th root of unity $\omega$. This immediately implies that $\alpha$ is of
the form $f_{0}\cdot(\sqrt[r]{f})^{a}$ for some $f_{0}\in\mathbb{C}(x)$, so
$h^{\prime}=f_{0}^{r}\cdot f^{a}$. In particular, modulo $r$, the $m$-tuple
$(a_{i})$ of exponents appearing in $h^{\prime}$ on the factors
$(x-\lambda_{i})$ must be a multiple of the $m$-tuple $(\xi_{i})$ of exponents
appearing in $f$.

Now, suppose that $a_{i}\equiv a\xi_{i}\pmod{r}$ for all $i$. Then, up to
multiplication by an $r$-th root of unity (corresponding to the cyclic action
on $\mathbb{P}^{1}$), we need
$h(x,y)=y^{a}\cdot\prod_{i=1}^{m}(x-\lambda_{i})^{(a_{i}-a\xi_{i})/r}\cdot\left(\frac{P_{0}(x)}{P_{\infty}(x)}\right)$.
However, in order for such an $h$ to be $\mathbb{Z}/r\mathbb{Z}$-equivariant,
we need $a\equiv 1\pmod{r}$. ∎

We may also see the necessary condition $a_{i}\equiv\xi_{i}\pmod{r}$ from the
point of view of monodromy as follows. Suppose first that $a_{i}>0$, and let
$\gamma_{z}$ be a small loop with basepoint $z_{0}\in\mathbb{P}^{1}$ winding
$a_{i}$ times counterclockwise around 0; the orientation of $\mathbb{P}^{1}$
is given by the complex structure. Then, given a basepoint
$y_{0}\in\mathbb{P}^{1}$ such that $h^{\prime}(y_{0})=z_{0}$, $\gamma_{z}$
lifts under $h^{\prime}$ uniquely to a loop $\gamma_{y}$ winding once around
$\lambda_{i}$. Next, given a basepoint $x_{0}\in X$ such that
$\sigma(x_{0})=y_{0}$, $\gamma_{y}$ lifts uniquely to a path in $X$ travelling
from $x_{0}$ to $\tau^{\xi_{i}}\cdot x_{0}$, where
$\tau\in\mathbb{Z}/r\mathbb{Z}$ is the generator acting on $T_{x_{0}}X$ by
$e^{2\pi i/r}$.

On the other hand, we may also lift $\gamma_{z}$ to $X$ via the maps $g$ and
$h$: letting $y^{\prime}_{0}=h(x_{0})$, we find that $\gamma_{z}$ lifts under
$g$ uniquely to a path $\gamma^{\prime}_{y}$ in $\mathbb{P}^{1}$ travelling
from $y^{\prime}_{0}$ to $\tau^{a_{i}}\cdot y^{\prime}_{0}$. Now, we must have
$h(\tau^{\xi_{i}}\cdot x_{0})=\tau^{a_{i}}\cdot y^{\prime}_{0}$, from which we
conclude $a_{i}\equiv\xi_{i}\pmod{r}$ from the equivariance of $h$.

If $a_{i}<0$ instead, we may follow the same argument, except now $\gamma_{z}$
will be a path with winding number $-a_{i}$ around $\infty$, and the monodromy
of $g$ around $\infty$ is the opposite of that around 0.

We have therefore shown that:

######  Corollary 2.2.

The space of equivariant covers $\mathcal{H}_{g,\mathbb{Z}/r\mathbb{Z},\xi,B}$
is empty unless $a_{i}\equiv\xi_{i}\pmod{r}$ for all $i$.

Henceforth we assume this congruence condition on the $a_{i}$.

###  2.4. Reduction to $h^{\prime}$

The main idea which allows us to count the number of
$\mathbb{Z}/r\mathbb{Z}$-equivariant covers $h:X\to\mathbb{P}^{1}$, is that we
can count maps $h^{\prime}:\mathbb{P}^{1}\to\mathbb{P}^{1}$ such that
$h^{\prime}\circ\sigma=g\circ h$ instead. In this section we formulate this
statement in terms of moduli spaces.

Let $\mathcal{H}_{\operatorname{rat}}$ be the moduli space of covers
$h^{\prime}:\mathbb{P}^{1}\to\mathbb{P}^{1}$ with ramification profiles
dictated by $h$ upon passing to the scheme-theoretic
$\mathbb{Z}/r\mathbb{Z}$-quotients. That is, the zeroes and poles of
$h^{\prime}$ are given by the $\lambda_{i}$ and the roots of
$P_{0},P_{\infty}$ and have orders exactly $|a_{i}|,r$, respectively, and
$h^{\prime}$ is branched over $k$ additional distinct points with orders
governed by $B$.

The source $\mathbb{P}^{1}$ has a fixed choice of coordinate $x$, whereas the
target $\mathbb{P}^{1}$ has fixed points $0,\infty$, but the functions
$h^{\prime}$ are taken only up to a constant factor. That is, two covers are
considered isomorphic if they agree up to scaling on the target.

Now, we have a quasi-finite and étale morphism
$\delta_{\operatorname{rat}}:\mathcal{H}_{\operatorname{rat}}\to\mathcal{P}_{k}$,
where $\mathcal{P}_{k}$ is the space of collections of $k$ unordered branch
points on the target, up to scaling. Thus, $\mathcal{H}_{\operatorname{rat}}$
is smooth of dimension $k-1=m-3$.

On the other hand, we have a morphism
$\phi_{\operatorname{rat}}:\mathcal{H}_{\operatorname{rat}}\to(\mathbb{P}^{1})^{m-3}$
remembering the $\lambda_{i}$ for $i\geq 4$. The source and target have the
same dimension, and by generic smoothness, the general fiber of
$\phi_{\operatorname{rat}}$ consists of a finite number of reduced points
equal to the degree of $\phi_{\operatorname{rat}}$. The above discussion shows
that $\deg(\phi_{\operatorname{rat}})=\deg(\phi_{B})$.

###  2.5. Main result

We now restate the main theorem.

######  Theorem 2.3.

Assume the numerical conditions (1), (2), (3), (4), (5), as well as the
congruence condition given in Corollary 2.2, so that the map
$\phi_{B}:\mathcal{H}_{g,\mathbb{Z}/r\mathbb{Z},\xi,B}\to\mathcal{H}_{g,\mathbb{Z}/r\mathbb{Z},\xi}$
has non-empty source and target of the same dimension. Then, the degree of
$\phi_{B}$ is

$\binom{t/r}{t_{0}/r}\cdot\prod_{j=1}^{k}(b_{j}-1)\cdot\frac{k!}{\prod_{\ell\in
L}c_{\ell}!}.$

In the case of hyperelliptic odd covers for which all Weierstraß points live
over $\infty$ (with fixed ramification indices $a_{i}$), we have $r=2$,
$m=2g+2$, $\xi_{i}\equiv a_{i}\equiv 1\pmod{2}$ for all $i$, $d=t=t_{0}=4g$,
$k=2g$, and $b_{j}=3$ for all $j$. Thus, $L=\\{2\\}$ and $c_{2}=2g$, so we
find that $\deg(\phi)=2^{2g}$, recovering the main result of [MP20].

Suppose instead that $r$ is arbitrary and $B=\\{2,\ldots,2\\}$, that is, the
moving ramification conditions are generic (and thus vacuous), we must have
$t=0$ by (5). Then, in the notation of §2.3, we must have simply
$h^{\prime}(x)=\prod_{i=1}^{m}(x-\lambda_{i})^{a_{i}}$, and indeed Theorem 2.3
gives $\deg(\phi_{B})=1$.

###  2.6. Theta characteristics

We comment here on one additional aspect of the case $r=2$ which plays an
important role in the computation of [MP20], namely that of the theta
characteristic associated with a hyperelliptic odd cover, and how the
situation changes in our more general setting.

Recall that if $h:X\to\mathbb{P}^{1}$ is any cover for which all ramification
indices are odd, one may naturally associate to $h$ a theta characteristic on
$X$, that is, a square root of the canonical bundle (see [ACGH85, Appendix B]
and [Mum71] for the basic theory of theta characteristics). Namely, we have

$\omega_{X}\cong
h^{*}\mathcal{O}_{\mathbb{P}^{1}}(-2)\otimes\mathcal{O}_{X}(2R).$

where $2R$ denotes the ramification divisor of $h$, which by assumption is a
square. Thus, the divisor

$\Theta_{h}:=h^{*}\mathcal{O}_{\mathbb{P}^{1}}(-1)\otimes\mathcal{O}_{X}(R)$

is a theta characteristic on $X$.

Now, suppose $X$ is hyperelliptic and $h$ is equivariant for the hyperelliptic
action, that is, $h$ is a point of
$\mathcal{H}_{g,\mathbb{Z}/2\mathbb{Z},(1^{2g+2}),B}$. Assume further that the
$|a_{i}|$ are all odd and that all $b_{j}$ are equal to 3. This situation
generalizes that of [MP20] in that Weierstraß points are allowed over 0.

In this case, we can write

$\displaystyle h^{-1}(0)$ $\displaystyle=\sum_{i\in
S_{0}}(2n_{i}+1)P_{i}+\frac{t_{0}}{2}H,$ $\displaystyle h^{-1}(\infty)$
$\displaystyle=\sum_{i\in S_{\infty}}(2n_{i}+1)P_{i}+\frac{t_{\infty}}{2}H$

where $H$ is the hyperelliptic divisor on $X$.

If we denote by $R^{\prime}$ the ramification divisor outside $h^{-1}(0)$ and
$h^{-1}(\infty)$, we have $\deg(R^{\prime})=4k$, where as before, $k$ is the
number of $\mathbb{Z}/r\mathbb{Z}$-orbits of branch points of outside $0$ and
$\infty$. In particular, $R^{\prime}=2kH$. Thus, we find, up to linear
equivalence,

$\displaystyle\Theta_{h}$ $\displaystyle=kH+\sum_{i\in
S_{0},S_{\infty}}n_{i}P_{i}-\sum_{i\in S_{0}}(2n_{i}+1)P_{i}-\frac{t_{0}}{2}H$
$\displaystyle=\left(k-\frac{t_{0}}{2}\right)H+\sum_{i\in
S_{\infty}}n_{i}P_{i}-\sum_{i\in S_{0}}(n_{i}+1)P_{i}.$

In the covers of [MP20], we have $k=2g$, $t_{0}=4g$, $t_{\infty}=0$, and
$S_{0}=\emptyset$, so in fact $\Theta_{h}$ is effective. In our more general
situation in which $S_{0}$ may be non-empty, note that even if we retain the
conditions $(t_{0},t_{\infty})=(4g,0)$ and $k=2g$, any (possibly non-
effective) theta characteristic of $X$ arises as above. Indeed, if

$\Theta=\sum_{i=1}^{2g+2}c_{i}P_{i},$

then we may take $S_{\infty}$ to be the set of $i$ such that $c_{i}\geq 0$,
and $n_{i}=c_{i}$ for $i\in S_{\infty}$ and $n_{i}=-c_{i}-1$ for $i\in S_{0}$.

##  3\. Counting meromorphic functions on $\mathbb{P}^{1}$

In this section, we carry out the proof of Theorem 2.3. We have reduced to
counting the possible meromorphic functions

$h^{\prime}(x)=\prod_{i=1}^{m}(x-\lambda_{i})^{a_{i}}\cdot\left(\frac{P_{0}(x)}{P_{\infty}(x)}\right)^{r},$

where we impose the condition that, away from the $\lambda_{i}$ and the roots
of $P_{0},P_{\infty}$, the cover $h^{\prime}$ has ramification indices (at
distinct branch points) given by the multi-set $B$. We parametrize choices of
$P_{0},P_{\infty}$ by a product of projective spaces, and impose the
ramification conditions from $B$ in terms of a combination of Segre and
Veronese embeddings.

###  3.1. The map $\psi$

Let $\mathbb{P}^{t_{0}/r},\mathbb{P}^{t_{\infty}/r}$ be the projective spaces
parametrizing non-zero polynomials $P_{0},P_{\infty}$ of degrees (at most)
$t_{0}/r,t_{\infty}/r$, up to scaling, and let
$U\subset\mathbb{P}^{t_{0}/r}\times\mathbb{P}^{t_{\infty}/r}$ be the open
locus consisting of polynomials $P_{0},P_{\infty}$ satisfying the following
properties:

  * •

$P_{0},P_{\infty}$ have no roots in common,

  * •

$P_{0},P_{\infty}$ do not have roots at any of the $\lambda_{i}$,

  * •

$P_{0},P_{\infty}$ have no double roots.

Here, we regard $x=\infty$ as a possible root of $P_{0},P_{\infty}$. Thus, the
statement that $P_{0},P_{\infty}$ have no roots in common includes the
statement that at most one of them has zero leading coefficient, and the
statement that they have no double roots includes the statement that the first
two coefficients of either polynomial do not both vanish.

If $(P_{0},P_{\infty})\in U$, the ramification divisor for $h^{\prime}$ away
from the $\lambda_{i}$ and the roots of $P_{0},P_{\infty}$ is given by the
roots of the polynomial

$\displaystyle\psi_{h^{\prime}}(x):$
$\displaystyle=\frac{1}{\prod_{i=1}^{m}(x-\lambda_{i})^{a_{i}-1}}\cdot\frac{P_{\infty}^{r+1}}{P_{0}^{r-1}}\cdot\frac{d}{dx}h^{\prime}(x)$
$\displaystyle=\sum_{i=1}^{m}a_{i}\cdot\frac{\prod_{i=1}^{m}(x-\lambda_{i})}{x-\lambda_{i}}\cdot
P_{0}\cdot
P_{\infty}+r\prod_{i=1}^{m}(x-\lambda_{i})\cdot(P^{\prime}_{0}P_{\infty}-P_{0}P^{\prime}_{\infty})$

The degree of $\psi_{h^{\prime}}(x)$ is $b=\sum_{j}(b_{j}-1)$ (note that the
leading terms of the two summands cancel), unless $\infty$ is among the
additional ramification points of $h^{\prime}$, in which the ramification
index at $\infty$ is $b-\deg(\psi_{h^{\prime}})+1$.

Note that the definition of $\psi_{h^{\prime}}(x)$ makes sense even for
$(P_{0},P_{\infty})\notin U$, and that the roots of $\psi_{h^{\prime}}(x)$
away from $\lambda_{i}$ and the roots of $P_{0},P_{\infty}$, taken with
multiplicity, still control the ramification of $h^{\prime}$. In this case,
$\psi_{h^{\prime}}(x)$ will in general have extraneous roots among
$\lambda_{i}$ and the roots of $P_{0},P_{\infty}$.

We have always that $\psi_{h^{\prime}}(x)$ is not identically zero, or else
$h^{\prime}$ is constant, which is impossible even for $(P_{0},P_{\infty})\in
U$ because the $a_{i}$ are not divisible by $r$. Therefore, we may make the
following definition.

######  Definition 3.1.

We define
$\psi:\mathbb{P}^{t_{0}/r}\times\mathbb{P}^{t_{\infty}/r}\to\mathbb{P}^{b}$ by
$\psi((P_{0},P_{\infty}))=\psi_{h^{\prime}}(x)$.

The map $\psi$ is the composition of a Segre embedding followed by a linear
projection whose base locus does not meet the image of
$\mathbb{P}^{t_{0}/r}\times\mathbb{P}^{t_{\infty}/r}$, so its degree is equal
to that of the Segre embedding, which is $\binom{t/r}{t_{0}/r}$.

###  3.2. The map $\rho$

We want to intersect $\psi$ with the locus of polynomials that arise as a
product of the form $\prod_{\ell\in L}R_{\ell}(x)^{\ell}$, where $R_{\ell}$
has degree $c_{\ell}$. This is the image of the map defined below:

######  Definition 3.2.

We define the map

$\rho:\prod_{\ell\in L}\mathbb{P}^{c_{\ell}}\to\mathbb{P}^{b}$

by sending $(R_{\ell})_{\ell\in L}\mapsto\prod_{\ell\in
L}R_{\ell}^{c_{\ell}}$.

The map $\rho$ is the composition of the $\ell$-Veronese embedding on the
factor corresponding to $\ell\in L$, followed by a Segre embedding, followed
by a linear projection. Because $\rho$ has no base locus, its degree is the
degree of the composition of the first two maps, which is

$\prod_{\ell\in L}\ell^{c_{\ell}}\cdot\frac{(\sum_{\ell\in
L}c_{\ell})!}{\prod_{\ell\in
L}c_{\ell}!}=\prod_{j=1}^{k}(b_{j}-1)\cdot\frac{k!}{\prod_{\ell\in
L}c_{\ell}!}.$

###  3.3. Intersection of $\psi$ and $\rho$

Theorem 2.3 now follows from Bézout’s Theorem, assuming that, for
$\lambda_{i}$ general, the intersections of $\psi$ and $\rho$ correspond
exactly to points in the corresponding fiber of $\mathcal{H}_{rat}$, and that
this intersection is transverse. These statements are verified below,
completing the proof of the main result.

######  Proposition 3.3.

Suppose the $\lambda_{i}\in\mathbb{C}$ are general and consider the
intersection of $\psi,\rho$ in $\mathbb{P}^{b}$. Then, any cover
$h^{\prime}:\mathbb{P}^{1}\to\mathbb{P}^{1}$ defined by a point of
intersection between $\psi,\rho$ is a point of
$\mathcal{H}_{\operatorname{rat}}$.

###### Proof.

Suppose we have
$h^{\prime}(x)=\prod_{i=1}^{m}(x-\lambda_{i})^{a_{i}}\cdot\left(\frac{P_{0}(x)}{P_{\infty}(x)}\right)^{r}$
defining a point of intersection of $\psi,\rho$ outside of
$\mathcal{H}_{\operatorname{rat}}$. We claim that the corresponding map
$h^{\prime}:\mathbb{P}^{1}\to\mathbb{P}^{1}$, possibly of degree less than
$d$, has strictly fewer than $k$ branch points away from $0,\infty$, and thus
varies in a family of dimension strictly less than that of
$\mathcal{H}_{\operatorname{rat}}$, so is avoided by a general choice of
$\lambda_{i}$. (If $k=1$, then we find that $h^{\prime}$ may only have two
branch points, and hence two ramification points, but we see that this is
impossible when $m\geq 3$, as $r\neq a_{i}$.)

To see this, note first that if $h^{\prime}$ has $k$ distinct ramification
points away from $0,\infty$, then the ramification indices must be given
exactly by the elements of $B$. Now, we claim that any choices of
$P_{0},P_{\infty}$ that define a cover $h^{\prime}$ outside of
$\mathcal{H}_{\operatorname{rat}}$ increases the total amount of ramification
of $h^{\prime}$ over 0 and $\infty$ compared to the change in degree, which
cannot happen if the ramification away from $0,\infty$ is unchanged.

On one hand, suppose that the degree of $h^{\prime}$ is $d$, so that
$P_{0},P_{\infty}$ have no roots in common and no roots at the $\lambda_{i}$.
Then, the only way $h^{\prime}$ can fail to define a point of
$\mathcal{H}_{\operatorname{rat}}$ is if some of the ramification imposed by
the $\lambda_{i}$ and $P_{0},P_{\infty}$ coalesces, either by allowing
$P_{0},P_{\infty}$ to have multiple roots or roots at $\lambda_{i}$. However,
this increases the total amount of ramification of $h^{\prime}$.

On the other hand, if $P_{0},P_{\infty}$ are chosen to decrease the degree of
$h^{\prime}$, any such choice will decrease the degree of $h^{\prime}$
strictly more than the amount of ramification lost. For example, if
$P_{0},P_{\infty}$ have a linear factor in common, the degree of $h^{\prime}$
drops by $r$, which must decrease the total ramification by $2r$, but the
amount of ramification lost over $0,\infty$ is only $2r-2$. Similarly, if
$P_{0}$ or $P_{\infty}$ has a root at $\lambda$ canceling a factor of
$(x-\lambda)$, we get a similar contradiction.

Therefore, if the $\lambda_{i}$ are chosen generally, the ramification for all
covers in the intersection of $\psi,\rho$ is as expected. ∎

######  Proposition 3.4.

Suppose the $\lambda_{i}\in\mathbb{C}$ are general. Then, the intersection of
$\psi,\rho$ is transverse.

###### Proof.

Suppose $\psi,\rho$ have a tangent vector. We then obtain a 1-parameter
deformation

$\widetilde{h^{\prime}}=\prod_{i=1}^{m}(x-\lambda_{i})^{a_{i}}\cdot\left(\frac{P_{0}+\epsilon
Q_{0}}{P_{\infty}+\epsilon Q_{\infty}}\right)^{r}$

of the meromorphic function $h^{\prime}$, which yields a 1-parameter
deformation as a point of $\mathcal{H}_{\operatorname{rat}}$. By construction,
$\widetilde{h^{\prime}}$ keeps the $\lambda_{i}$ constant on the target.
Because the $\lambda_{i}$ are general and $\phi_{\operatorname{rat}}$ is
generically unramified, $\widetilde{h^{\prime}}$ must is in fact be the
trivial deformation on $\mathcal{H}_{\operatorname{rat}}$.

This means that, as meromorphic functions on
$\mathbb{P}^{1}\times\operatorname{Spec}\mathbb{C}[\epsilon]/\epsilon^{2}$, we
have

$\prod_{i=1}^{m}(x-\lambda_{i})^{a_{i}}\cdot\left(\frac{P_{0}+\epsilon
Q_{0}}{P_{\infty}+\epsilon
Q_{\infty}}\right)^{r}=\mu\prod_{i=1}^{m}(x-\lambda_{i})^{a_{i}}\cdot\left(\frac{P_{0}}{P_{\infty}}\right)^{r}$

for a scalar $\mu\in\mathbb{C}[\epsilon]/\epsilon^{2}$. We find that
$Q_{0},Q_{\infty}$ must be scalar multiples of $P_{0},P_{\infty}$,
corresponding to the trivial tangent vector on $\psi$, and the ramification
remains unchanged to first order, corresponding to the trivial tangent vector
on $\rho$. This completes the proof. ∎

## References

  * [ACGH85] Enrico Arbarello, Maurizio Cornalba, Phillip A. Griffiths and Joseph D. Harris, _Geometry of algebraic curves. Vol. I._ , Springer-Verlag, New York (1985).
  * [FP05] Carel Faber and Rahul Pandharipande, _Relative maps and tautological classes_ , J. Eur. Math. Soc. 7 (2005), 13-49.
  * [FMNP20] Gavril Farkas, Riccardo Moschetti, Carlos Naranjo, Gian Pietro Pirola, _Alternating Catalan numbers and curves with triple ramification_ , Ann. Sc. Norm. Super. Pisa, to appear.
  * [HM82] Joe Harris, David Mumford, _On the Kodaira dimension of the moduli space of curves_. Invent. Math. 67 (1982), 23-86.
  * [L19] Carl Lian, _Enumerating pencils with moving ramification on curves_ , J. Algebraic Geom., to appear.
  * [MP20] Riccardo Moschetti and Gian Pietro Pirola, _Hyperelliptic odd coverings_ , Israel J. Math., to appear.
  * [Mum71] David Mumford, _Theta characteristics of an algebraic curve_ , Ann. Sci. Éc. Norm. Supér. 4 (1971), 181-192.
  * [SvZ18] Johannes Schmitt and Jason van Zelm, _Intersection of loci of admissible covers with tautological classes_ , Selecta Math. (N.S.) 26, Article Nr. 79 (2020).


# Quantum and thermal pressures from light scalar fields

Hauke Fischer<EMAIL_ADDRESS>Atominstitut, Technische Universität
Wien, Stadionallee 2, 1020 Vienna, Austria  Christian Käding
<EMAIL_ADDRESS>Atominstitut, Technische Universität Wien,
Stadionallee 2, 1020 Vienna, Austria  Mario Pitschmann
<EMAIL_ADDRESS>Atominstitut, Technische Universität Wien,
Stadionallee 2, 1020 Vienna, Austria

###### Abstract

Light scalar fields play a variety of roles in modern physics, especially in
cosmology and modified theories of gravity. For this reason, there is a zoo of
experiments actively trying to find evidence for many scalar field models that
have been proposed in theoretical considerations. Among those are setups in
which the pressures expected to be induced by light scalar fields between two
parallel plates are studied, for example, Casimir force experiments. While it
is known that classical and quantum pressures caused by light scalar fields
could have significant impacts on such experiments, in this article, we show
that this can also be the case for thermal pressure. More specifically, we
derive expressions for the quantum and thermal pressures induced by exchanges
of light scalar field fluctuations between two thin parallel plates. As
particular examples, we then look at screened scalar fields. For chameleon,
symmetron and environment-dependent dilaton models, we find large regions in
their parameter spaces that allow for thermal pressures to equal or exceed the
quantum pressures. By comparing with earlier constraints from quantum pressure
calculations, we conclude that thermal pressures induced by chameleons are
actually of experimental significance.

dark energy; screened scalar fields; quantum pressure; thermal pressure

##  I Introduction

Light scalar fields appear in many contexts in modern cosmology or
modifications of general relativity, for example, in scalar-tensor theories of
gravity [1]. They often serve as the basis for proposed solutions of some of
modern physics’ greatest open problems, including dark energy (DE) and dark
matter (DM) [2, 3]. Many of such theories give rise to a universal coupling
between the scalar degree of freedom and Standard Model matter, leading to an
additional force of Nature. Such a fifth force, however, is tightly
constrained within our Solar System [4, 5, 6].

Screening mechanisms were devised to provide a way of circumventing these
constraints while also offering a rich phenomenology. They strongly suppress
(screen) scalar fifth forces in environments of sufficiently high mass
densities, e.g., our Solar System, but enable them to develop their full
strengths in regions of lower densities. There are a variety of different
light scalar field models with screening mechanisms, so-called screened scalar
fields, some of which are even being discussed as presenting an alternative to
particle DM through their fifth forces [7, 8, 9, 10]. Some of the most popular
screened scalar field models are chameleons [11, 12], symmetrons [13, 14, 15,
16, 17, 18, 19, 20], and environment-dependent dilatons [15, 21, 22, 23, 24,
25, 26]. They are the examples used in the present article, and have already
been tested and discussed in multiple experimental setups; see, e.g., Refs.
[27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45,
46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62]. For an
overview of current constraints on the parameter spaces of these models, see
Refs. [63, 64]. While screened scalar fields are usually treated as classical
fields, there have already been first attempts to describe them in quantum
field theoretical frameworks [65, 66, 67, 68] and from the perspective of
particle physics [69]. The present article will add to this.

Casimir force experiments like the upcoming Casimir And Non Newtonian force
EXperiment (Cannex) [70, 59] are expected to further constrain the parameter
spaces of light scalar field models [71, 72, 73, 74, 55]. This makes this type
of experiments an interesting subject for theoretical analyses and motivates
the investigation in the present article. Ref. [65] initiated the discussion
of quantum pressure induced between two parallel plates, as can be found in
Cannex, by interactions with quantum fluctuations of screened scalar fields.
For this, the plates were approximated by having infinite widths and infinite
thicknesses. Using this approximation, Ref. [65] showed that the quantum
pressures of chameleons and symmetrons can actually be quite significant and
lead to tighter constraints on these models’ parameter spaces. In the present
article, we consider two equal, infinitely wide, but very thin plates and also
study the quantum pressures induced by light scalar fields, while considering
chameleons, symmetrons, and environment-dependent dilatons as representative
examples. In order to obtain analytically tractable results, we restrict our
analysis to parameter regimes in which these screened scalar fields have
Compton wavelengths much larger than the thickness of one plate. This means
that throughout our investigation, we can safely assume the scalar fields to
have constant masses defined by the density of the vacuum surrounding the
plates. Consequently, the general parts of our results are applicable to any
type of scalar field model that exhibits a constant mass in the considered
situation. Motivated by the results of Refs. [66, 67, 68], which indicated
that finite-temperature effects of light scalar fields could be relevant for
experiments, we also compute expressions for the pressures induced by thermal
scalar fluctuations. Comparing the results for quantum and thermal pressures,
we find that both can be of comparable magnitudes in some situations. Due to
the proven significance of quantum pressures according to Ref. [65], we
consequently conclude that thermal pressures are likely also relevant for
screened scalar field models in some experimental setups and must therefore be
taken into account.

This article is structured as follows: At first, in Sec. II, we shortly
introduce the three considered screened scalar field models and discuss their
couplings to fermions. Next, in Sec. III, we compute the quantum and thermal
potentials, and derive and discuss the resulting pressures between two
parallel plates. Finally, we draw our conclusions in Sec. IV.

##  II Scalar field models

While it should be noted that some of the main results in this article are
applicable to more general types of light scalar fields, we will use three
popular types of screened scalar fields models as examples for our discussion.
In this way, we are able to make comparisons with the results of Ref. [65].
Therefore, in this section, we first give a short overview of the considered
screened scalar field models and provide all necessary formulas. Afterwards,
we discuss the couplings of the scalar fields to fermions since they are
required for the computations done in this article.

###  II.1 Models

As examples, we consider three screened scalar field models: chameleons,
symmetrons, and environment-dependent dilatons. Each of them can be generally
described by the Einstein frame action

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{4}x\,\sqrt{-g}\left(-\frac{m_{\text{Pl}}^{2}}{2}\,R+\frac{1}{2}(\partial\varphi)^{2}-V_{X}(\varphi)\right)+\int
d^{4}x\,\sqrt{-\tilde{g}^{X}}\,\mathcal{L}_{\text{SM}}(\tilde{g}_{\mu\nu}^{X},\psi_{i})~{},$
(1)

in which $m_{\text{Pl}}$ is the reduced Planck mass; $\varphi$ represents the
scalar field; $X\in\\{C,S,D\\}$ labels the scalar field model, i.e., chameleon
($C$), symmetron ($S$), or dilaton ($D$); $V_{X}(\varphi)$ is the scalar’s
potential; and $\mathcal{L}_{\text{SM}}$ denotes a Lagrangian describing the
Standard Model fields $\psi_{i}$. For notational convenience, in Eq. (1), we
kept the Jordan frame metric defined in terms of the conformal factor
$A_{X}(\varphi)$ as $\tilde{g}^{X}_{\mu\nu}=A_{X}^{2}(\varphi)g_{\mu\nu}$. Due
to the conformal factor, the scalar $\varphi$ couples to the field $\psi_{i}$.
In turn, this leads to an effective potential for the scalar:

$\displaystyle V_{X;\text{eff}}(\varphi)$ $\displaystyle:=$ $\displaystyle
V_{X}(\varphi)+A_{X}(\varphi)T^{\mu}_{\phantom{\mu}\mu}$ (2)

with $T^{\mu}_{\phantom{\mu}\mu}$ as the trace of the energy momentum tensor
of $\psi_{i}$.

For chameleons, we have

$\displaystyle V_{C}$ $\displaystyle=$
$\displaystyle\frac{\Lambda^{n+4}}{\varphi^{n}}~{},~{}A_{C}(\varphi)\,=\,\exp\left(\frac{\varphi}{M_{C}}\right)\,\approx\,1+\frac{\varphi}{M_{C}}+\frac{\varphi^{2}}{2M_{C}^{2}}~{},$
(3)

where $n\in\mathbb{Z}^{+}\cup 2\mathbb{Z}^{-}\setminus\\{-2\\}$ defines the
exact chameleon model; $\Lambda$ is a mass scale determining the chameleon’s
self-interaction, except for the case $n=-4$, in which case it must actually
be a dimensionless constant; and $M_{C}$ is another mass scale, which controls
the chameleon coupling to matter. Here, we have assumed $\varphi\ll M_{C}$ and
kept the second order term in order to allow for comparisons with symmetron
and dilaton models. If we only consider the conformal factor up to the first
order in $\varphi/M_{C}$, the effective potential defined by Eqs. (2) and (3)
gives rise to an effective chameleon mass

$\displaystyle m_{C}^{2}$ $\displaystyle=$
$\displaystyle\frac{n(n+1)\Lambda^{n+4}}{\varphi_{C}^{n+2}}$ (4)

with the chameleon vacuum expectation value (VEV) given by

$\displaystyle\varphi_{C}$ $\displaystyle=$
$\displaystyle\left(n\Lambda^{n+4}\frac{M_{C}}{T^{\mu}_{\phantom{\mu}\mu}}\right)^{1/(n+1)}~{}.$
(5)

While we have kept the second order term of the conformal factor in Eq. (3) in
order to enable us to discuss the exchange of two chameleon field
fluctuations, the second order corrections to Eqs. (4) and (5) are not
relevant in the following discussions. Clearly, for any permitted value of
$n$, a larger $T^{\mu}_{\phantom{\mu}\mu}$ implies a larger $m_{C}$ in Eq.
(4). Considering the example of non-relativistic matter, this means that a
quantum of a chameleon field is the heavier the denser its surrounding
environment it couples to. A heavier chameleon induces a shorter-ranged fifth
force. This is the essence of the chameleon screening mechanism [11, 12].

The symmetron model is defined by

$\displaystyle V_{S}$ $\displaystyle=$
$\displaystyle-\frac{\mu^{2}}{2}\,\varphi^{2}+\frac{\lambda_{S}}{4}\,\varphi^{4}~{},~{}A_{S}(\varphi)\,=\,1+\frac{\varphi^{2}}{2M_{S}^{2}}$
(6)

with the tachyonic mass $\mu$, the dimensionless self-coupling parameter
$\lambda_{S}$, and the mass scale $M_{S}$ defining the symmetron’s coupling to
matter. It is screened by the Damour-Polyakov mechanism [15], which means that
the symmetron’s effective coupling to matter varies with the density of the
environment. In the case $T^{\mu}_{\phantom{\mu}\mu}\geq\mu^{2}M_{S}^{2}$, the
symmetron VEV must vanish, which results in a complete decoupling from matter
if quantum or thermal fluctuations of the symmetron are small enough to be
ignored. However, if $T^{\mu}_{\phantom{\mu}\mu}<\mu^{2}M_{S}^{2}$, the
symmetron has a non-vanishing VEV

$\displaystyle\varphi_{S}$ $\displaystyle=$
$\displaystyle\pm\sqrt{\frac{1}{\lambda_{S}}\left(\mu^{2}-\frac{T^{\mu}_{\phantom{\mu}\mu}}{M_{S}^{2}}\right)}~{},$
(7)

which implies that the symmetron fifth force is unscreened. The symmetron mass
is given by

$\displaystyle m_{S}^{2}$ $\displaystyle=$
$\displaystyle\begin{cases}\frac{T^{\mu}_{\phantom{\mu}\mu}}{\mathcal{M}^{2}}-\mu^{2}&,~{}T^{\mu}_{\phantom{\mu}\mu}\geq\mu^{2}M_{S}^{2}\\\
2\left(\mu^{2}-\frac{T^{\mu}_{\phantom{\mu}\mu}}{\mathcal{M}^{2}}\right)&,~{}T^{\mu}_{\phantom{\mu}\mu}<\mu^{2}M_{S}^{2}\end{cases}~{}.$
(8)

Note that the symmetron also has an environment-dependent mass and is
therefore, technically, a chameleon as well. However, the variation of the
effective mass usually has only little influence on the fifth force screening
of this model.

Finally, the environment-dependent dilaton has

$\displaystyle V_{D}$ $\displaystyle=$ $\displaystyle
V_{0}\,{e}^{-\lambda\varphi/m_{\text{Pl}}}~{},~{}A_{D}(\varphi)\,=\,1+A_{2}\,\frac{\varphi^{2}}{2m_{\text{Pl}}^{2}}~{},$
(9)

where $V_{0}$ is a constant energy density, $\lambda$ denotes the dilaton’s
dimensionless self-coupling constant, and $A_{2}$ is a dimensionless constant
determining the coupling to matter. Resulting from this, the dilaton has a VEV
[26]

$\displaystyle\varphi_{D}$ $\displaystyle=$
$\displaystyle\frac{m_{\text{Pl}}}{\lambda}\,W\bigg{(}\frac{\lambda^{2}V_{0}}{A_{2}T^{\mu}_{\phantom{\mu}\mu}}\bigg{)}$
(10)

with the Lambert $W$-function

$\displaystyle W(x)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\frac{(-n)^{n-1}}{n!}\,x^{n}~{}.$ (11)

Consequently, its mass is given by

$\displaystyle m_{D}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{m_{\text{Pl}}^{2}}\left(\lambda^{2}V_{0}\,e^{-\lambda\varphi_{D}/m_{\text{Pl}}}+A_{2}T^{\mu}_{\phantom{\mu}\mu}\right)~{}.$
(12)

As can be inferred from Eq. (10), the dilaton VEV is the smaller the larger
$T^{\mu}_{\phantom{\mu}\mu}$. This means that, as long as any dilaton
fluctuations are negligible, the coupling between dilaton and matter is small
in dense environments. In turn, this leads to the dilaton fifth being rendered
weak in such environments. Initially, it was thought that the dilaton was only
screened by this realisation of the Damour-Polyakov mechanism. However, Ref.
[55] showed that this is only partially true. In fact, there are only certain
parts of the dilaton’s parameter space in which the Damour-Polyakov mechanism
is the dominant reason for fifth force screening. In other parameter regions,
the dilaton fith force is mainly screened by the chameleon mechanism.

###  II.2 Couplings to fermions

For the computations in this article, we will consider that the screened
scalar fields couple to the nucleons in the two parallel plates. Consequently,
we must describe the interactions between screened scalar fields and fermions.

From the effective actions of the considered models, we can conclude for the
following interaction Lagrangians:

$\displaystyle\mathcal{L}_{C;\text{int}}$ $\displaystyle=$
$\displaystyle-\left(\frac{\varphi}{M_{C}}+\frac{\varphi^{2}}{2M_{C}^{2}}\right)T^{\mu}_{\phantom{\mu}\mu}~{},~{}\mathcal{L}_{S;\text{int}}\,=\,-\frac{\varphi^{2}}{2M_{S}^{2}}T^{\mu}_{\phantom{\mu}\mu}~{},~{}\mathcal{L}_{D;\text{int}}\,=\,-A_{2}\frac{\varphi^{2}}{2m^{2}_{\text{Pl}}}T^{\mu}_{\phantom{\mu}\mu}~{}.$
(13)

We consider an interaction with a Dirac particle of mass $m$ with stress-
energy tensor

$\displaystyle T^{\mu}_{\phantom{\mu}\nu}$ $\displaystyle=$
$\displaystyle\bar{\psi}\mathrm{i}\gamma^{\mu}\partial_{\nu}\psi-\bar{\psi}\mathrm{i}\not{\partial}\psi\delta^{\mu}_{\phantom{\mu}\nu}+m\bar{\psi}\psi\delta^{\mu}_{\phantom{\mu}\nu}~{}.$
(14)

From this follows the trace

$\displaystyle T^{\mu}_{\phantom{\mu}\mu}$ $\displaystyle=$
$\displaystyle-3\bar{\psi}\mathrm{i}\not{\partial}\psi+4m\bar{\psi}\psi$ (15)
$\displaystyle=$ $\displaystyle m\bar{\psi}\psi~{},$

where we have used the Dirac equation in the last step. After expanding the
fields in terms of their VEVs and fluctuations $\phi$, such that
$\varphi=\varphi_{X}+\phi$, we find that the Lagrangians in Eq. (13) give rise
to interactions of fermions with one or two scalar fluctuations. This can be
summarised in terms of two general interaction Lagrangians:

$\displaystyle\mathcal{L}_{X;\text{int}}^{(1)}$ $\displaystyle=$
$\displaystyle-
g_{X}^{(1)}\phi\bar{\psi}\psi~{},~{}\mathcal{L}_{X;\text{int}}^{(2)}\,=\,-g_{X}^{(2)}\phi^{2}\bar{\psi}\psi~{}.$
(16)

The dimensionless $g_{X}^{(1)}$ are given by

$\displaystyle g_{C}^{(1)}$ $\displaystyle\approx$
$\displaystyle\frac{m}{M_{C}}~{},~{}g_{S}^{(1)}\,=\,\frac{\varphi_{S}m}{M_{S}^{2}}~{},~{}g_{D}^{(1)}\,=\,\frac{A_{2}\varphi_{D}m}{m_{\text{Pl}}^{2}}~{},$
(17)

while the $g_{X}^{(2)}$ have dimensions of an inverse mass and are

$\displaystyle g_{C}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{m}{2M_{C}^{2}}~{},~{}g_{S}^{(2)}\,=\,\frac{m}{2M_{S}^{2}}~{},~{}g_{D}^{(2)}\,=\,\frac{A_{2}m}{2m_{\text{Pl}}^{2}}~{}.$
(18)

In Eq. (17), we have dropped the second order term of $g_{C}^{(1)}$ since it
will always be much smaller than the first order term. However, in order to
also allow for a two-scalar exchange, we have kept the second order term that
defines $g_{C}^{(2)}$ in Eq. (18).

##  III Quantum and thermal pressures

In this section, we compute the quantum and thermal pressures induced by light
scalar fields between two infinitely wide, but very thin plates. For this, we
first derive the quantum and thermal potentials from the interaction
Lagrangians in Eq. (16). From those we then obtain the corresponding
pressures. Finally, we compare quantum and thermal pressures in order to
identify conditions for which both are of comparable magnitude. The obtained
results are generally applicable to any scalar field model that has, in the
considered experimental situation, a constant mass and couples to fermions as
in Eq. (16).

Note that, for our examples of screened scalar fields, we assume a constant
mass $m_{X}$ throughout this discussion. This mass is defined only by the
constant density of the vacuum or gas surrounding the two parallel plates.
Such an assumption is well-justified as long as the Compton wavelength of a
screened scalar field within one of the plates is much larger than the plate’s
thickness. In this case, the scalar field will not be able to minimize its
potential within the plate. Consequently, the density of the plate will only
lead to small perturbations of the scalar’s VEV and mass between the plates,
which we can safely ignore. Certainly, this restricts the validity of the
following discussion only to regions of the model parameter spaces that can
comply with the requirements of our assumption.

###  III.1 Potentials

We have to consider the exchanges of one or two scalar fluctuations between
two nucleons, one within one plate and one within the other plate. At first,
we look at the single-particle exchange. In order to compute the quantum
potential induced by light scalar fields from the interaction Lagrangian
$\mathcal{L}_{X}^{(1)}$ in Eq. (16), we follow Ref. [75]. Eq. (16) implies the
scattering amplitude

$\displaystyle\mathrm{i}\mathcal{M}_{X;Q}^{(1)}$ $\displaystyle=$
$\displaystyle\left(\mathrm{i}g_{X}^{(1)}\right)^{2}\,\bar{u}(p_{2}^{\prime},\sigma_{2})u(p_{2},\sigma_{2})\,\frac{\mathrm{i}}{q^{2}-m_{X}^{2}}\,\bar{u}(p_{1}^{\prime},\sigma_{1})u(p_{1},\sigma_{1})~{},$
(19)

where the subscript $Q$ indicates that we refer to the amplitude resulting
from the quantum fluctuations of the scalar fields. In the non-relativistic
limit, the amplitude takes the form

$\displaystyle\mathcal{M}_{X;Q}^{(1)}$ $\displaystyle=$
$\displaystyle-\left(\mathrm{i}g_{X}^{(1)}\right)^{2}\,2m\,\frac{1}{\bm{q}^{2}+m_{X}^{2}}\,2m~{}.$
(20)

A comparison with

$\displaystyle\mathcal{M}_{X;Q}^{(1)}$ $\displaystyle=$
$\displaystyle-(2m)^{2}\,V^{(1)}_{X;Q}(\bm{q})~{},$ (21)

where $V^{(1)}_{X;Q}(\bm{q})$ is the Fourier transformed of the quantum
potential $V^{(1)}_{X;Q}(r)$, yields immediately

$\displaystyle V^{(1)}_{X;Q}(\bm{q})$ $\displaystyle=$
$\displaystyle-\left(g_{X}^{(1)}\right)^{2}\frac{1}{\bm{q}^{2}+m_{X}^{2}}~{}.$
(22)

With the inverse Fourier transform

$\displaystyle V^{(1)}_{X;Q}(r)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}q}{(2\pi)^{3}}\,V^{(1)}_{X;Q}(\bm{q})\,e^{\mathrm{i}\bm{q}\cdot\bm{r}}~{},$
(23)

we obtain the quantum potential for the single-scalar exchange:

$\displaystyle V^{(1)}_{X;Q}(r)$ $\displaystyle=$
$\displaystyle-\left(g_{X}^{(1)}\right)^{2}\frac{1}{4\pi r}\,e^{-m_{X}r}~{}.$
(24)

Next, we say that the thin parallel plates and the nucleons within them have a
constant temperature $T$. Further, we follow Refs. [66, 67, 68] in assuming
that the screened scalar field had sufficient time to thermalize with the
plates, such that we can expect thermal scalar fluctuations associated with
the temperature $T$. In this case, we can add a finite temperature
contribution [76] to the propagator. However, as is explained below, the
thermal contribution at tree level vanishes for kinematical reasons. In fact,
the amplitude for the exchange of a single thermal scalar fluctuation between
two nucleons is given by

$\displaystyle\mathrm{i}\mathcal{M}_{X;T}^{(1)}$ $\displaystyle=$
$\displaystyle\left(\mathrm{i}g_{X}^{(1)}\right)^{2}\,\bar{u}(p_{2}^{\prime},\sigma_{2})u(p_{2},\sigma_{2})\,2\pi\delta(q^{2}-m_{X}^{2})n(T,E_{q})\,\bar{u}(p_{1}^{\prime},\sigma_{1})u(p_{1},\sigma_{1})~{},$
(25)

where we use the Boltzmann distribution

$\displaystyle n(T,E_{q})$ $\displaystyle=$
$\displaystyle\exp\left(-E_{q}/T\right)$ (26)

with energy $E_{q}$. Though, since in the rest frame of a nucleon, i.e.
$p_{1}=(m,\bm{0})^{T}$, we have

$\displaystyle q^{2}-m_{X}^{2}$ $\displaystyle=$
$\displaystyle\left(p_{1}-p_{1}^{\prime}\right)^{2}-m_{X}^{2}$ (27)
$\displaystyle=$ $\displaystyle 2m^{2}-2p_{1}\cdot p_{1}^{\prime}-m_{X}^{2}$
$\displaystyle=$ $\displaystyle
2m\left(m-\sqrt{m^{2}+{\bm{p_{1}}^{\prime}}^{2}}\right)-m_{X}^{2}<0~{},$

relativistic invariance implies $\delta(q^{2}-m_{X}^{2})=0$ as well as
$\mathcal{M}_{X;T}^{(1)}=0$, as expected. This means that there are no thermal
potential and pressure from a single-scalar exchange between two on-shell
fermions.

For the quantum and thermal potentials from the two-scalar exchange, we take
and validate the results from Ref. [76], which itself makes use of Refs. [77,
78, 79]. Therefore, for the quantum contribution, we find

$\displaystyle V^{(2)}_{X;Q}(r)$ $\displaystyle=$
$\displaystyle-\left(g_{X}^{(2)}\right)^{2}\frac{m_{X}}{8\pi^{3}r^{2}}K_{1}(2m_{X}r)~{}.$
(28)

Furthermore, for the thermal contribution, we obtain

$\displaystyle V^{(2)}_{X;T}(r)$ $\displaystyle=$
$\displaystyle-\left(g_{X}^{(2)}\right)^{2}\frac{1}{2\pi^{3}}\frac{1}{r}\frac{Tm_{X}}{\sqrt{1+(2rT)^{2}}}K_{1}\left(\frac{m_{X}}{T}\sqrt{1+(2rT)^{2}}\right)~{}.$
(29)

It should be noted that this usual procedure of obtaining potentials
implicitly assumes the nucleons to be on-shell, as is the case for free
asymptotic scattering states. However, in the next section, we consider the
case in which the nucleons are bound within the macroscopic plates. Hence, for
our discussion, such a procedure of obtaining effective macroscopic potentials
is an approximation.

###  III.2 Pressures

Having derived the quantum and thermal potentials between two nucleons, we can
now use these results to compute the pressures between two parallel plates at
separation $\ell$, with thicknesses $D$ and widths $2L$ in both transversal
dimensions; see Fig. 1.

Figure 1: Schematic depiction of the two parallel plates; the plates are
identical in dimensions and material. We consider thin plates, which means
that $D\ll\ell$. Furthermore, we take the limit $L\to\infty$. The plates are
surrounded by vacuum or some kind of gas, and they themselves consist of a
material with a density that leads to $D\ll 1/m_{X;\mathrm{Plate}}$ for the
considered regions of the screened scalar fields models’ parameter spaces.

The induced pressure between the two plates is computed by

$\displaystyle\mathcal{P}^{(Z)}_{X;Y}(\ell)$ $\displaystyle=$
$\displaystyle-\frac{\partial}{\partial\ell}\frac{\mathfrak{U}^{(Z)}_{X;Y}(\ell)}{S}~{},$
(30)

where $\mathfrak{U}^{(Z)}_{X;Y}(\ell)/S$ is the integrated potential per area
$S=4L^{2}$ of both plates with $Y\in\\{Q,T\\}$ and $Z\in\\{1,2\\}$. We
restrict our discussion to the case $D\ll\ell$ and $L\to\infty$, and note that
the results are only valid for parameters where $D\ll 1/m_{X;\mathrm{Plate}}$
is fulfilled. The last assumption corresponds to a field’s Compton wavelength
within a plate being much larger than the plate’s thickness. In this case, the
field cannot reach its VEV, $\varphi_{X;\mathrm{Plate}}$, within the plate but
instead stays close to its VEV in vacuum111Here, the term vacuum also refers
to any residual gases., $\varphi_{X;\mathrm{Vac}}$. Consequently, we can
safely assume that
$\varphi_{X;\mathrm{Plate}}\approx\varphi_{X;\mathrm{Vac}}=:\varphi_{X}$ and
$m_{X;\mathrm{Plate}}\approx m_{X;\mathrm{Vac}}=:m_{X}$. In turn, the
assumptions we make allow us to approximate the potential per area by

$\displaystyle\frac{\mathfrak{U}^{(Z)}_{X;Y}(\ell)}{S}$ $\displaystyle\approx$
$\displaystyle\lim_{L\to\infty}\frac{D^{2}\rho^{2}}{S}\int_{-L}^{L}dx\int_{-L}^{L}dx^{\prime}\int_{-L}^{L}dy\int_{-L}^{L}dy^{\prime}V^{(Z)}_{X;Y}\left(\sqrt{\ell^{2}+(y^{\prime}-y)^{2}+(x^{\prime}-x)^{2}}\right)~{},$
(31)

where the factor $D^{2}$ stems from our approximation of the two integrals
over the longitudinal direction, $\rho$ is the nucleon number density of each
plate222We note, that the electron contributions can be safely neglected at
our level of precision., and we set
$r=\sqrt{\ell^{2}+(y^{\prime}-y)^{2}+(x^{\prime}-x)^{2}}$. We introduce new
coordinates

$\displaystyle x_{\pm}$ $\displaystyle=$ $\displaystyle x^{\prime}\pm
x~{},\qquad y_{\pm}=y^{\prime}\pm y~{}.$ (32)

Hence, we obtain

$\displaystyle\frac{\mathfrak{U}^{(Z)}_{X;Y}(\ell)}{S}$ $\displaystyle\approx$
$\displaystyle\lim_{L\to\infty}\frac{D^{2}\rho^{2}}{4S}\int_{-2L}^{2L}dx_{-}\int_{|x_{-}|-2L}^{2L-|x_{-}|}dx_{+}\int_{-2L}^{2L}dy_{-}\int_{|y_{-}|-2L}^{2L-|y_{-}|}dy_{+}V^{(Z)}_{X;Y}\left(\sqrt{\ell^{2}+y_{-}^{2}+x_{-}^{2}}\right)$
(33) $\displaystyle\approx$
$\displaystyle\lim_{L\to\infty}\frac{D^{2}\rho^{2}}{S}\int_{-2L}^{2L}dx_{-}\int_{-2L}^{2L}dy_{-}\left(2L-|x_{-}|\right)\left(2L-|y_{-}|\right)V^{(Z)}_{X;Y}\left(\sqrt{\ell^{2}+y_{-}^{2}+x_{-}^{2}}\right)~{}.$

Taking the limit, we find

$\displaystyle\frac{\mathfrak{U}^{(Z)}_{X;Y}(\ell)}{S}$ $\displaystyle\approx$
$\displaystyle
D^{2}\rho^{2}\int_{-\infty}^{\infty}dx_{-}\int_{-\infty}^{\infty}dy_{-}V^{(Z)}_{X;Y}\left(\sqrt{\ell^{2}+y_{-}^{2}+x_{-}^{2}}\right)$
(34) $\displaystyle\approx$ $\displaystyle 2\pi
D^{2}\rho^{2}\int\limits_{0}^{\infty}d\mathfrak{r}\,\mathfrak{r}\,V^{(Z)}_{X;Y}\\!\left(\sqrt{\ell^{2}+\mathfrak{r}^{2}}\right)~{}.$

Substituting $v=\sqrt{\ell^{2}+\mathfrak{r}^{2}}$, we have

$\displaystyle\frac{\mathfrak{U}^{(Z)}_{X;Y}(\ell)}{S}$ $\displaystyle\approx$
$\displaystyle 2\pi
D^{2}\rho^{2}\int\limits_{\ell}^{\infty}dv\,v\,V^{(Z)}_{X;Y}\\!\left(v\right)~{}.$
(35)

After substituting this into Eq. (30), we obtain

$\displaystyle\mathcal{P}^{(Z)}_{X;Y}(\ell)$ $\displaystyle\approx$
$\displaystyle-2\pi
D^{2}\rho^{2}\partial_{\ell}\int\limits_{\ell}^{\infty}dv\,v\,V^{(Z)}_{X;Y}\\!\left(v\right)$
(36) $\displaystyle\approx$ $\displaystyle 2\pi D^{2}\rho^{2}\ell
V^{(Z)}_{X;Y}\\!\left(\ell\right)~{}.$

This means, from Eqs. (24), (28) and (29) we obtain:

$\displaystyle\mathcal{P}^{(1)}_{X;Q}(\ell)$ $\displaystyle\approx$
$\displaystyle-\left(g_{X}^{(1)}\right)^{2}\,\frac{D^{2}\rho^{2}}{2}\,e^{-m_{X}\ell}~{},$
(37) $\displaystyle\mathcal{P}^{(2)}_{X;Q}(\ell)$ $\displaystyle\approx$
$\displaystyle-\left(g_{X}^{(2)}\right)^{2}\frac{m_{X}D^{2}\rho^{2}}{4\pi^{2}\ell}K_{1}(2m_{X}\ell)~{},$
(38) $\displaystyle\mathcal{P}^{(2)}_{X;T}(\ell)$ $\displaystyle\approx$
$\displaystyle-\left(g_{X}^{(2)}\right)^{2}\frac{D^{2}\rho^{2}}{\pi^{2}}\frac{Tm_{X}}{\sqrt{1+(2\ell
T)^{2}}}K_{1}\left(\frac{m_{X}}{T}\sqrt{1+(2\ell T)^{2}}\right)~{}.$ (39)

Consequently, the total pressure, excluding pressures from classical fifth
forces, induced by a scalar field of model $X$ is given by

$\displaystyle\mathcal{P}_{X}(\ell)$ $\displaystyle=$
$\displaystyle\mathcal{P}^{(1)}_{X;Q}(\ell)+\mathcal{P}^{(2)}_{X;Q}(\ell)+\mathcal{P}^{(2)}_{X;T}(\ell)$
(40) $\displaystyle\approx$
$\displaystyle-D^{2}\rho^{2}\bigg{\\{}\left(g_{X}^{(1)}\right)^{2}\frac{1}{2}\,e^{-m_{X}\ell}+\left(g_{X}^{(2)}\right)^{2}\frac{m_{X}}{\pi^{2}}\bigg{[}\frac{1}{4\ell}K_{1}(2m_{X}\ell)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{T}{\sqrt{1+(2\ell
T)^{2}}}K_{1}\left(\frac{m_{X}}{T}\sqrt{1+(2\ell
T)^{2}}\right)\bigg{]}\bigg{\\}}~{}.~{}~{}~{}~{}~{}~{}~{}$

It should be noted, that each of the three pressure contributions, and as such
the total pressure as well, are negative. This corresponds to an attractive
force between the plates.

###  III.3 Discussion

We now discuss the regions of the considered screened scalar field models’
parameter spaces where the induced thermal pressures are at least as large as
the quantum pressures. This means that we are looking for model parameters
that fulfill

$\displaystyle\frac{\pi^{2}}{2}\left(\frac{g_{X}^{(1)}}{g_{X}^{(2)}}\right)^{2}\,\frac{e^{-m_{X}\ell}}{m_{X}}+\frac{1}{4\ell}K_{1}(2m_{X}\ell)\leq\frac{T}{\sqrt{1+(2\ell
T)^{2}}}K_{1}\left(\frac{m_{X}}{T}\sqrt{1+(2\ell T)^{2}}\right)~{}.$ (41)

In order to obtain quantitative results, we consider a plate separation of
$55\,\mu m$, as was also used in Ref. [65]. In addition, we follow Ref. [68]
and assume the parallel plates to be surrounded by a residual hydrogen gas of
temperature $T=300$ K and with a pressure of $9.6\times 10^{-10}$ mbar that is
in thermal equilibrium with the plates and the light scalar fields. The
resulting hydrogen mass density $\rho_{H_{2}}$ determines the VEVs and masses
of the considered screened scalar field models. Using these exemplary
experimental conditions, the parts of the three models’ parameter spaces that
fulfill Eq. (41) are depicted in Fig. 2.

(a)

(b)

(c)

Figure 2: Areas in the model parameter spaces for which Eq. (41) is valid for
a hydrogen gas with $T=300$ K and a pressure of $9.6\times 10^{-10}$ mbar
surrounding the two parallel plates; note that the plots continue where they
touch one of the axes; (a): chameleon model with $n=1$; (b): symmetron in the
symmetry-broken (unscreened) phase for $\lambda_{S}=0.1$; (c): environment-
dependent dilaton with $V_{0}=1\,\text{MeV}^{4}$; note that the plot barely
changes for larger $V_{0}$.

At first, we look at the $n=1$ chameleon. Taking into account $\varphi_{C}\ll
M_{C}$ and Eq. (41), we find that large parts of this model’s parameter space
allow for a thermal pressure at least as strong as the quantum pressure terms;
see Fig. 2(a). Interestingly, the area depicted in Fig. 2(a) is entirely
contained in the region that Ref. [65] constrains by solely using the
chameleon quantum pressure. This means that the thermal pressure induced by
chameleons is of experimental significance and must be taken into account.

Next, we look at the symmetron model. We only consider symmetrons in the
symmetry-broken phase, in which the fifth force is not screened, and which is
determined by the condition $\mu^{2}>\rho_{H_{2}}/M_{S}^{2}$. Furthermore, we
set $\lambda_{S}=0.1$ as Ref. [65] also did. In Fig. 2(b), we see that also
for large parts of the symmetron parameter space, the thermal pressure can be
at least as strong as the quantum pressure. Ref. [65] has already provided
symmetron constraints based on quantum pressure computations. However,
unfortunately, the corresponding constraint plot in Ref. [65] is clearly not
correct since the designations of Broken phase and Unbroken phase have to be
swapped in order for the labelling of the two axes to be consistent with the
conditions that define each phase.

Finally, we look at environment-dependent dilatons. Again, we find that large
parts of the model parameter space allow for a thermal pressure that is
significant in comparison to the quantum pressures. However, as of yet, nobody
has computed quantum pressures for environment-dependent dilatons in specific
experiments. Therefore, we cannot make a statement about the absolute
magnitude of quantum and thermal pressures induced by this model. Such
computations might be subject of a future work that takes into account a more
realistic setup.

##  IV Conclusions

Light scalar fields appear in many discussions throughout modern physics, and
serve as candidates for dark energy or dark matter. While there are many
experiments searching for evidence of such fields, there are physical aspects
that are still not fully considered in experimental and theoretical analyses.
A few years ago, Ref. [65] showed that the pressure induced by quantum
fluctuations of screened scalar fields between two infinitely wide and
infinitely thick plates can be of great experimental importance. However,
thermal pressures were not taken into account.

In the present article, we took first steps in the discussion of thermal
pressures from light scalar fields. For this, we computed quantum and thermal
potentials from one- and two-scalar exchanges between two nucleons. Next, we
integrated these potentials over two thin but infinitely wide parallel plates,
and subsequently derived expressions for quantum and thermal pressures from
light scalar fields. Interestingly, we found that there is no thermal pressure
from a single-particle exchange.

As explicit examples, we looked at three popular screened scalar field models:
chameleons, symmetrons, and environment-dependent dilatons. Since we derived
our general results for scalar fields with constant masses, we had to restrict
our discussion to model parameters and experimental specifications for which
the screened scalar fields cannot reach the minima of their potentials within
each one of the two plates, such that we were allowed to set the scalar’s VEVs
and masses to the values corresponding to the density of the vacuum or gas the
two plates are embedded in. Using an explicit experimental setup as an
example, for all three considered screened scalar field models, we found large
regions in their parameter spaces for which the thermal pressures are at least
as significant as the quantum pressures. The region for the $n=1$ chameleon
model is fully included in the area constrained in Ref. [65], which prompted
us to conclude that chameleon thermal pressures are actually of experimental
significance.

Our findings imply that re-evaluations of the chameleon and symmetron
constraints presented in Ref. [65], that also take into account thermal
pressures, might be necessary. However, this quite likely also applies to many
other constraints on screened scalar fields, including environment-dependent
dilatons, for which thermal effects were not taken into account. Therefore, we
believe that finite-temperature effects must necessarily be discussed in
future analyses of experiments that aim to constrain screened scalars or light
scalar fields in general. For this, we must go beyond the initial steps taken
here. This means that future analyses should make use of less assumptions than
we did in ours. For example, if looking at screened scalar fields, the changes
of VEVs and masses due to the presence of the two plates must be accurately
described. In addition, the impact of the possible presence of a vacuum
chamber or other experimental equipment should be taken into account and the
assumption $D\ll\ell$ might have to be dropped. However, in this case,
analytical solutions can most likely not be obtained, which necessitates the
use of sophisticated numerical methods.

###### Acknowledgements.

This research was funded in whole or in part by the Austrian Science Fund
(FWF)
$[10.55776/\mathrm{P}34240]$, and $[10.55776/\mathrm{PAT}8564023]$, and is
based upon work from COST Action COSMIC WISPers CA21106, supported by COST
(European Cooperation in Science and Technology).

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