# Effect of surface H2 on molecular hydrogen formation on interstellar grains

Gang Zhao 1, Qiang Chang1, Xia Zhang2, Donghui Quan3, Yong Zhang 4, Xiao-Hu
Li2
1School of Physics and Optoeletronic Engineering, Shandong University of
Technology, Zibo, 255000, China
2Xinjiang Astronomical Observatory, Chinese Academy of Sciences, 150 Science
1-Street, Urumqi 830011, China
3Department of Chemistry, Eastern Kentucky University, Richmond, KY, USA
4School of Physics & Astronomy, Sun Yat-Sen University, Zhuhai 519082, China
<EMAIL_ADDRESS>
###### Abstract

We investigate how the existence of hydrogen molecules on grain surfaces may
affect H2 formation efficiency in diffuse and translucent clouds. Hydrogen
molecules are able to reduce the desorption energy of H atoms on grain
surfaces in models. The detailed microscopic Monte Carlo method is used to
perform model simulations. We found that the impact of the existence of H2 on
H2 formation efficiency strongly depends on the diffusion barriers of H2 on
grain surfaces. Diffuse cloud models that do not consider surface H2 predict
that H atom recombination efficiency is above 0.5 over a grain temperature (T)
range 10 K and 14 K. The adopted H2 diffusion barriers in diffuse cloud models
that consider surface H2 are 80$\%$ H2 desorption energies so that H2 can be
trapped in stronger binding sites. Depending on model parameters, these
diffuse cloud models predict that the recombination efficiency is between
nearly 0 and 0.5 at 10 K $\leq$ T $\leq$ 14 K. Translucent cloud model results
show that H2 formation efficiency is not affected by the existence of surface
H2 if the adopted average H2 diffusion barrier on grain surfaces is low (194
K) so that H2 can diffuse rapidly on grain surfaces. However, the
recombination efficiency can drop to below 0.002 at T $\geq$ 10 K if higher
average H2 diffusion barrier is used (255 K) in translucent cloud models.

###### keywords:

astrochemistry – molecular processes – ISM: molecules

##  1 Introduction

Molecular hydrogen is the most abundant molecules in astronomical sources.
Moreover, gas-phase H2 is crucial for the subsequent formation of the
molecular ion H${}^{+}_{3}$, which triggers the rich gas-phase interstellar
chemistry (Williams, 1998; Herbst, 2001). Therefore, the H2 formation
mechanism has been an interesting and important research topic since its
discovery. Although almost all interstellar molecules were believed to be
formed in the gas phase in astrochemical models, H2 formation in the gas phase
has been found to be inefficient while interstellar dust grains can play an
important role in increasing the efficiency with which H atoms convert to H2
molecules (Gould & Salpeter, 1963).

However, it is not straightforward to explain H2 formation in astronomical
sources even when the catalytic roles of dust grains are introduced into
models. Interstellar species are believed to be formed on cold grain surfaces
via the so called Langmuir-Hinshelwood mechanism (Watson & Salpeter, 1972;
Pickles & Williams, 1977; Hasegawa et al., 1992). To form H2, H atoms accrete
on dust grains and then bind weakly with surfaces, which is known as
physisorption. They can overcome the diffusion barrier and move on the grain
surfaces via quantum tunneling or thermal hopping. However, laboratory and
theorectical studies showed that quantum tunneling does not contribute much to
the mobility of H atoms on silicate, carbonaceous or solid amorphous water
(ASW) surfaces (Pirronello et al., 1997, 1999; Katz et al., 1999; Nyman,
2021). If H atoms encounter other H atoms, then H2 molecules are formed. But H
atoms can also desorb and leave grain surfaces. A hydrogen atom must reside on
a grain long enough to find a partner H atom to form H2. As the dust
temperature increases, the H atom desorption and diffusion rates also
increase. So the temperature of grain surfaces must be sufficiently low so
that a H atom can encounter another one before it desorbs. On the other hand,
the temperature of grain surfaces must be high enough so that H atoms can
diffuse on the grain surface. The parameter that measures how strongly species
are to bound to grain surfaces is called desorption energy. It was found that
if we assume a single H desorption energy in models, the dust temperature
range over which efficient H2 formation occurs is narrow (6-10 K for olivine
grains) (Katz et al., 1999). Moreover, it was found that the highest dust
temperature at which H2 can be formed efficiently is less than 17 K (Katz et
al., 1999). However, the grain surface temperature in the unshielded diffuse
clouds, where hydrogen molecules are believed to be efficiently formed, is
around 20 K (Li & Draine, 2001).

In order to make the dust temperature range over which efficient H2 formation
occurs wider, H2 formation models adopted at least two types of sites, the
stronger and weaker binding sites, on grain surfaces (Hollenbach & Salpeter,
1971; Chang et al., 2005; Cuppen & Herbst, 2005; Iqbal et al., 2012). In these
models, the stronger binding sites are able to hold H atoms on grain surfaces
long enough so that other H atoms can encounter them to form H2 before they
desorb even when the dust temperatures are high. On the other hand, H atoms in
the weaker binding sites can diffuse to encounter other H atoms even when the
dust temperatures are low. These models are able to significantly enlarge the
dust temperature range over which efficient H2 formation occurs. The upper
limit of dust temperatures at which the efficient conversion from H to H2
occurs depend on the strongest binding sites on grain surfaces. As the
desorption energy of the strongest binding sites increases, this upper limit
also increases. The recent review by Wakelam et al. (2017a) further summarizes
H2 formation models, experiments and observations.

Because H2 molecules cannot react with H atoms on grain surfaces, the impact
of the existence of surface H2 was not considered in earlier H2 formation
models (Hollenbach & Salpeter, 1971; Biham et al., 2001; Chang et al., 2005;
Cuppen & Herbst, 2005; Iqbal et al., 2012). However, studies using
astrochemical models for the cold ($\sim 10$ K) and high density (H nucleus
densities $>10^{12}$ cm-3) sources showed that in order for the modeling
results to be physical, the H2 desorption energies in binding sites occupied
by H2 must be much lower than that on water ice (Garrod & Pauly, 2011;
Hincelin et al., 2015). If a chemical model adopts a single H2 desorption
energy and the H2 desorption energy on water ice is used in the model, then
gas-phase hydrogen molecules would be heavily depleted onto dust grains in the
chemical model (Hincelin et al., 2015), which has not been observed. To solve
this problem, Hincelin et al. (2015) suggested the encounter desorption (ED)
mechanism in their models. The ED mechanism assumes that when a hydrogen
molecule hops into a site occupied by another H2 molecule, the desorption
energy for the hopping H2 molecule becomes the desorption energy of H2 on H2
substrate, which is much lower than that on water ice surfaces. So, surface H2
can desorb much more efficiently even around 10 K if the ED mechanism is
included in models. As a result, models predict that the abundances of surface
H2 in cold and high density sources are only no more than a few monolayers
(Hincelin et al., 2015). In a recent study, Chang et al. (2021) extended the
H2 ED to the H and H2 ED, which assumes that when H atoms encounter H2 on
grain surface, their desorption energy becomes the H atom desorption energy on
H2 substrate, which is much lower than that on water ice surfaces. So, the H
and H2 ED mechanism enhances the desorption rate of surface H, thus, it
becomes more difficult to hydrogenate CO on grain surfaces. So, the production
of a few surface species such as methanol was strongly reduced if the model
adopted the H and H2 ED mechanism.

The desorption rate of H atoms should also increase if we consider H and H2 ED
mechanism in H2 formation models because the desorption energy of H on H2
substrate is much lower than that on bare dust grain surfaces. On the other
hand, the existence of H2 on grain surfaces may affect H2 formation in another
way. The efficient H2 formation at higher temperatures relies on the existence
of stronger binding sites on grain surfaces. If we consider physisorption
only, the stronger binding sites are those where surface species can see more
horizontal neighbors, which are the upward step edges on rough surfaces
(Cuppen & Herbst, 2005). Therefore, a stronger H binding site must also be a
stronger H2 binding site if we consider physisorption only. After a hydrogen
molecule occupies a stronger binding site, this site becomes a weaker binding
site for two reasons. First, for the H atom in this site, the vertical
neighbor below it becomes H2. Secondly, the H atom may not see horizontal
neighbors, thus loses lateral bonds with it neighbors because a H2 molecule
fills the vacancy. We refer to Cuppen & Herbst (2005) for details of how
horizontal and vertical neighbors affect the desorption energies of surface
species.

The impact of the existence of surface deuterium molecules on D2 formation has
been studied in laboratory (Gavilan et al., 2012). It was found that D2
molecules formed more efficiently on silicate surfaces covered by D2 than on
bare silicate surfaces. Gavilan et al. (2012) argued that the reason was that
D atoms diffuse more quickly on surfaces covered by D2. However, the flux of D
atoms in their experiments was more than seven orders of magnitude larger than
the H atom flux in diffuse clouds (Biham et al., 2001). As the H atom
accretion flux significantly decreases, the H atom population on grain
surfaces also drops a lot. Because H2 formation requires at least two H atoms
on grain surfaces, it is not clear if the existence of surface H2 can still
help to increase H2 formation efficiency on grain surfaces.

In this paper, for the first time, we investigate H2 formation by the
recombination of physisorbed H atoms on interstellar dust grains using models
that consider the existence of H2 on grain surfaces. Although our major
purpose is to study H2 formation in diffuse clouds, we also investigate H2
formation in translucent clouds, where typically $25\%$ H nucleus are in the
form of H atoms (Zuo et al., 2018). These H atoms will also be converted to H2
as translucent clouds gradually become denser and form dense dark clouds (Zuo
et al., 2018).

The organization of the paper are the follows. Our models are explained in
Section 2 while the numerical method is introduced in Section 3. Our results
are shown in Section 4. Finally, Section 5 is the section for the discussions
and conclusions.

##  2 Models

Based on the work by Biham et al. (2001), we assume in our models that the H
nucleus density of diffuse clouds is 10 cm-3 while the gas-phase temperature
is 100 K. Although both olivine and carbon grains exist in diffuse clouds, we
only study the H2 formation efficiency on silicate-composed olivine grains in
diffuse clouds. The effect of H2 existence on molecular hydrogen formation on
carbon grains will be briefly discussed in Section 5. Assuming the gas-phase
species is composed of H atoms only, the H atom accretion flux is
$f1=1.8\times 10^{-9}$ ML s-1 in diffuse cloud models (Biham et al., 2001).
Diffuse clouds are those in which more than $10\%$ H nuclei have been
converted to H2 (Wakelam et al., 2017a), so, we assume $75\%$ or $50\%$ H
nuclei exist in the form of H atoms in diffuse cloud models. The purpose of
this work is to study how the existence of surface H2 may affect only the
physisorbed H atom recombination efficiency. Therefore, H2 formation scenarios
involving chemisorption are ignored in models.

In translucent clouds, the gas temperature is 15-40 K, the H nucleus density
is between 500 and 5000 cm-3 while dust grains are believed to be covered by
water ice (Wakelam et al., 2017a). In translucent cloud models, we therefore
assume that the H nucleus density is 1000 cm-3 and the gas temperature is 30
K. In a typical translucent cloud, $75\%$ of the H nuclei have been converted
to H2 (Zuo et al., 2018). The H atom accretion flux in these translucent
clouds is found to be $f2=1.3\times 10^{-8}$ ML s-1.

Diffusion barrier and desorption energy are the key parameters that determine
the rates of surface species diffusion and desorption respectively. The
desorption energy of H2 on amorphous silicate surfaces, $E_{D1_{H_{2}}}$, was
found to follow a continuous distribution based on experimental analysis (He
et al., 2011). On the other hand, He et al. (2011) found the value of
$E_{D1_{H_{2}}}$ can be represented by three discrete values, 35, 53 and 75
mev. The lowest value, 35 mev is about $40\%$ lower than the intermediate
value, 53 mev while the highest value, 75 mev is about $40\%$ higher than the
intermediate value. Hama et al. (2012) experimentally studied the diffusion
barrier of H atom, $E_{b2_{H}}$ on ASW surfaces and found that $E_{b2_{H}}$
should be represented by at least three discrete values, $<18$ mev, 22 mev and
$>30$ mev. The lowest value, $<18$ mev is more than $18\%$ lower than the
intermediate value, 22 mev while the the highest value, $>30$ mev, is about
$40\%$ higher than the intermediate value.

Since the aforementioned experimental findings suggest that three categories
of binding sites can represent all the various binding sites on surfaces, we
assume the desorption energies of a surface species on dust grain without H2
existence always take three discrete values in models. The lowest value is
$40\%$ lower than the intermediate one while the highest value is $40\%$
higher than the intermediate one. Hereafter, following nomenclature by Hama et
al. (2012), the binding sites with the lowest, intermediate and highest
desorption energy are called the shallow, middle and deep potential sites. For
simplicity, the number of shallow, middle and deep potential sites are assumed
to be equal on grain surfaces in our models. So, the desorption energy of the
middle potential sites is the average desorption energy of all binding sites
on grain surfaces. The three discrete desorption energies are taken from
literature if they are available. For instance, we adopt the three discrete H2
desorption energies on silicate surfaces, which were suggested by He et al.
(2011) in our models. On the other hand, if only a single desorption energy
for a surface species is available in literature, we assume that single value
to be the desorption energy in the middle potential sites.

In a recent review (Wakelam et al., 2017a), the suggested desorption energies
of H atoms on silicate and ASW surfaces are 44 and 50 mev respectively. These
two values are adopted for the H atom desorption energies in the middle
potential sites for the silicate and water ice surfaces respectively. The
desorption energy of H2 on water ice surfaces varies much in literature. We
adopt the commonly used value of $E_{D2_{H_{2}}}=440$ K (Cuppen & Herbst,
2007). Moreover, in order to study the effect of $E_{D2_{H_{2}}}$ on
recombination, we also adopt a much larger value,
$E_{D2_{H_{2}}}^{{}^{\prime}}=800$ K (Wakelam et al., 2017b). These two values
are assigned to the desorption energy of H2 in the middle potential sites on
water ice surfaces in translucent cloud models.

The suggested diffusion barrier of H atoms on the silicate surfaces is 35 mev
(406 K) (Wakelam et al., 2017a). So, the ratio of the H atom diffusion barrier
to desorption energy is $R=35/44\sim 0.8$. We assume R is fixed to be 0.8 for
the shallow, middle and deep potential sites on silicate surfaces. Moreover,
because the diffusion barrier of H2, $E_{b1_{H_{2}}}$, on the silicate
surfaces is not available in literature, we assume
$E_{b1_{H_{2}}}=0.8E_{D1_{H_{2}}}$. Therefore, R is 0.8 for all species on
silicate surfaces in our models.

Hama et al. (2012) found that diffusion barrier of H atoms in the middle
potential sites on ASW surfaces is 22 mev (255 K). So, on the water ice middle
potential sites, R is $22/50\sim 0.44$. We assume R=0.44 for both H and H2 on
all water ice binding sites in our models. The H atom diffusion barrier on the
shallow water ice sites is $40\%$ lower than that in the middle potential
sites. The adopted energy value is 13.2 mev (153 K), which is lower than its
upper bound suggested by Hama et al. (2012). The adopted H atom diffusion
barrier on the deep potential sites is 30.8 mev (357 K), which is larger than
its lower bound suggested by Hama et al. (2012). The diffusion barrier of H2
on water ice middle potential sites is, $E_{b2_{H_{2}}}=0.44E_{D2_{H_{2}}}$ or
$E_{b2_{H_{2}}}^{{}^{\prime}}=0.44E_{D2_{H_{2}}}^{{}^{\prime}}$.

Table LABEL:table1 summarizes the desorption energies and diffusion barriers
on the shallow, middle and deep potential sites, which are not occupied by H2,
in our models.

Table 1: Surface species desorption energies and diffusion barriers on
silicate and water ice surfaces

| H2 on silicates | H2 on water ice1 | H2 on water ice1 | H on silicates | H on water ice
---|---|---|---|---|---
desorption energy (K) | (406, 615, 870) | (264, 440, 616) | (480, 800, 1120) | (306, 510, 715) | (348, 580, 812)
diffusion barrier (K) | (325, 492, 696) | (116, 194, 271) | (211, 352, 493) | (244, 406, 568) | (153, 255, 357)

Notes.
All energy values are in K. 1 mev = 11.6 K. (a, b, c) are desorption energies
or desorption barriers for the shallow, middle and deep potential sites
respectively. 1 These two sets of energy values are used in different models,
which will be explained in detail later in this section.

The desorption energies of H2 and H on H2 substrate are much lower than these
on silicate or ASW surfaces. At 10 K, the desorption energy of H on H2
substrate was calculated to be $E_{D_{HH_{2}}}=45$ K (Pierre et al., 1985;
Vidali, 1991; Cuppen & Herbst, 2007). Cuppen & Herbst (2007) estimated that
the desorption energy of H2 on H2 substrate is, $E_{D_{H_{2}H_{2}}}=23$ K.
Recently, by performing quantum chemical calculations, Das et al. (2021) found
that $E_{D_{H_{2}H_{2}}}$ is between 67 K and 79 K while $E_{D_{HH_{2}}}$ is
between 23 K and 25 K. Moreover, if a site has been occupied by H2, a H atom
in this site may not be able to see horizontal neighbors, thus cannot form
lateral bonds with its horizontal neighbors. So, the desorption energy of the
surface species is further reduced when the existence of H2 is considered.
Fig. 1 illustrates lateral bonds formed between H atoms and their horizontal
neighbors in different binding sites.

Figure 1: Lateral bonds formed by H atoms and their neighbors in different
sites. In sites other than A and B, H atoms can form lateral bonds with their
horizontal neighbors. Site A is occupied by H2, so the H atom in this site
cannot form lateral bonds with its horizontal neighbors. Site B is a binding
site in which a H atom cannot see horizontal neighbors, thus, no lateral bonds
can be formed between the H atom and its horizontal neighbors.

Not only H2 molecules but also H atoms can enhance the desorption rates of H2.
Cuppen & Herbst (2007) estimated that the desorption energy of H2 on H
substrate is, $E_{D_{H_{2}H}}=30$ K, which is also significantly lower than
that on silicate surfaces. In all our models, if H2 molecules resides in sites
occupied by H atoms, the desorption energy of H2 on these sites become 30 K.

Models M01-M06 are used to study H2 formation on the olivine grains in diffuse
clouds. Model M01 is a reference model that is similar to earlier models
(Chang et al., 2005; Cuppen & Herbst, 2005). Gas-phase H2 are allowed to
accrete on grain surfaces in models M02-M06, but not in model M01. When a
exothermic chemical reaction occurs on grain surfaces, products may desorb
immediately, which is known as chemical desorption (Garrod et al., 2007). The
chemical desorption coefficient, $\mu$, measures the percentage of products
that desorb immediately after a reaction fires. In the reference model M01,
$\mu$ is set to be 1. In model M02, the desorption energies of H2 and H on H2
substrate are 23 K and 45 K respectively. Moreover we assume a surface species
loses all lateral bonds with its horizontal neighbors if the surface species
is in a site already occupied by H2. The chemical desorption coefficient is
set to be 1 in model M02. To study the effect of $\mu$ value on H2 formation
efficiency, we simulate model M03, which is similar to model M02, but $\mu$ is
set to be 0.

Because both $E_{D_{H_{2}H_{2}}}$ and $E_{D_{HH_{2}}}$ are not well known yet,
we simulate models M04 and M05 to study how the value of $E_{D_{H_{2}H_{2}}}$
and $E_{D_{HH_{2}}}$ may affect the H2 formation efficiency. These two models
are similar to model M02. The difference between models M02 and M04 is that
the desorption energies of H and H2 on H2 substrate are set to be 24 K and 73
K respectively in model M04 based on the energy values suggested by Das et al.
(2021). Since the residence time of H atoms on dust grains increases with
$E_{D_{HH_{2}}}$, H2 formation efficiency at higher temperatures should
increase as $E_{D_{HH_{2}}}$ becomes larger. We would like to set the value of
$E_{D_{HH_{2}}}$ to be as high as possible in model M05. However, to the best
of our knowledge, the upper bound of $E_{D_{HH_{2}}}$ is not available in
literature. We argue that the desorption energy of H atoms on the silicate
shallow potential sites may be a upper bound for $E_{D_{HH_{2}}}$ for the
following reasons. Hydrogen atoms in sites occupied by H2 can hardly form
lateral bonds with their horizontal neighbors in sites occupied by H2 as shown
in Fig. 1. So, even if the vertical bond formed between H atoms and H2
substrate is as strong as that between H atoms and silicate surfaces,
$E_{D_{HH_{2}}}$ cannot be higher than the desorption energy of H atoms on the
silicate shallow potential sites. Similarly, the desorption energy of H2 on
the silicate shallow potential sites may be a upper bound for
$E_{D_{H_{2}H_{2}}}$. In model M05, both $E_{D_{H_{2}H_{2}}}$ and
$E_{D_{HH_{2}}}$ are set to be their upper bounds.

In models M02-M05, deep H atom potential sites must also be deep H2 potential
sites. This assumption is based on our current understanding about how
stronger binding sites are formed (Cuppen & Herbst, 2005), which might be
limited. Therefore, we also simulated model M06, in which deep potential sites
for H atoms cannot be deep potential sites for H2 molecules. The deep H atom
potential site can be a middle or shallow potential site for H2 molecules with
equal probability. Similarly, it is equally likely that a deep H2 molecule
potential sites is a middle or shallow potential site for H atoms. In models
M02-M06, the ratio of a surface species diffusion barrier to its desorption
energy on sites occupied by H2, R${}_{H_{2}}$ is fixed to be 0.8. Moreover,
the ratio of H2 diffusion barrier to its desorption energy on sites occupied
by H atoms, RH is also 0.8.

Models M11-M15 are used to study H2 formation on dust grains in translucent
clouds. In all these models, $\mu$ is set to be 1. The model M11 is a
reference model in which H2 cannot accrete while H2 are allowed to accrete in
models M12-M15. The desorption energies of H and H2 on sites occupied by H2
are 45 K and 23 K respectively in models M12 and M13. In model M12,
R${}_{H_{2}}$ and RH are fixed to be 0.44, which is the same as the value of R
on water ice binding sites without H2. These two ratios are 0.8 in model M13
to study how R${}_{H_{2}}$ and RH may affect the recombination efficiency. The
H2 desorption energy on water ice is 440 K in models M11-M13. To study how the
H2 desorption energy on water ice may affect the recombination efficiency on
grain surfaces, we simulated models M14 and M15. The H2 desorption energy is
800 K in these two models. In model M14, R${}_{H_{2}}$ and RH are 0.44 while
in model M15, these two ratios are 0.8.

Our purpose is to make the reference models to be as close to the earlier
models (Chang et al., 2005; Cuppen & Herbst, 2007) as possible for comparison,
so, some of the assumptions in the earlier models are kept in our models. The
H atom quantum tunneling effect was not considered in the earlier models
because H atoms diffuse mainly by thermal hopping on silicate or ASW surfaces,
so this effect is also ignored in our models. The rate of hopping is
$\nu\exp(-E_{b}/T)$, where $E_{b}$ is the diffusion barrier, T is the dust
grain temperature, the pre-exponential factor $\nu$ is $10^{12}$ s-1. The rate
of desorption is $\nu\exp(-E_{D}/T)$ where $E_{D}$ is the desorption energy.
When H atoms land on empty sites or sites occupied by H2, the sticking
coefficient is 1. However, H atoms are not allowed to accrete on sites already
occupied by H atoms. On the other hand, we assume the H2 sticking coefficient
is always 1 regardless of the landing sites.

Table LABEL:table2 summarizes models in this work.

Table 2: Summary of models

| astronomical | H2 | R | R${}_{H_{2}}$=RH | H2 desorption | H atom desorption | H2 desorption | $\mu$ | same?1
---|---|---|---|---|---|---|---|---|---
| sources | accretion? |  |  | energy on H2 | energy on H2 | energy on middle potential |  |
|  |  |  |  | substrates (K) | substrates (K) | water ice sites (K) |  |
M01 | diffuse | no | 0.8 | N.A. | N.A. | N.A. | N.A. | 1 | yes
| clouds |  |  |  |  |  |  |  |
M02 | diffuse | yes | 0.8 | 0.8 | 23 | 45 | N.A. | 1 | yes
| clouds |  |  |  |  |  |  |  |
M03 | diffuse | yes | 0.8 | 0.8 | 23 | 45 | N.A. | 0 | yes
| clouds |  |  |  |  |  |  |  |
M04 | diffuse | yes | 0.8 | 0.8 | 73 | 24 | N.A. | 1 | yes
| clouds |  |  |  |  |  |  |  |
M05 | diffuse | yes | 0.8 | 0.8 | 406 | 306 | N.A. | 1 | yes
| clouds |  |  |  |  |  |  |  |
M06 | diffuse | yes | 0.8 | 0.8 | 23 | 45 | N.A. | 1 | no
| clouds |  |  |  |  |  |  |  |
M11 | translucent | no | 0.44 | N.A. | N.A. | N.A. | N.A. | 1 | yes
| clouds |  |  |  |  |  |  |  |
M12 | translucent | yes | 0.44 | 0.44 | 23 | 45 | 440 | 1 | yes
| clouds |  |  |  |  |  |  |  |
M13 | translucent | yes | 0.44 | 0.8 | 23 | 45 | 440 | 1 | yes
| clouds |  |  |  |  |  |  |  |
M14 | translucent | yes | 0.44 | 0.44 | 23 | 45 | 800 | 1 | yes
| clouds |  |  |  |  |  |  |  |
M15 | translucent | yes | 0.44 | 0.8 | 23 | 45 | 800 | 1 | yes
| clouds |  |  |  |  |  |  |  |

Notes.
1 same means deep H atom potential sites are also deep H2 potential sites.
N.A. means the parameter is not applicable to the model.

##  3 Numerical Method

In this work, we use the microscopic Monte Carlo (MMC) method to perform model
simulations. The MMC method is a rigorous approach that can treat the finite
size effect. In this section, we only briefly introduce this method and refer
to Chang et al. (2005) and Chang & Herbst (2012) for details.

To perform MMC simulations, binding sites on grains are put on a $L\times L$
square lattice, where L is the square root of the total number of sites on the
grain. Previous studies show that L=100 is already large enough to eliminate
finite size effect (Chang et al., 2005), so in this work, L is fixed to be
100. Each site has four nearest neighbor sites on the square lattice. Gas-
phase species can accrete and become surface species on the lattice. We keep
track of the position of each surface species, which hops from one site to one
of its nearest neighbor. A chemical reaction occurs when two reactive species
are in the same site on the lattice. Other than hopping, surface species can
also desorb. To decide whether a surface species hops or desorbs, we generate
a random number, X, that is uniformly distributed between 0 and 1. Assuming
the hopping and desorption rates of the surface species are $b_{1}$ and
$b_{2}$ respectively, the species hops if $X<\frac{b_{1}}{b_{1}+b_{2}}$;
otherwise, it desorbs. There are three types of events on the lattice,
accretion, hopping and desorption. We keep track of the absolute time when
each event occurs. The smallest absolute time is sorted out and the event that
occurs at this time is executed.

The absolute time when each event happens are calculated as the follows. In
MMC simulations, after an event occurs, the same event will occur again after
a “waiting time”, $\tau=-ln(Y)/k$, where Y is another random number uniformly
distributed within (0, 1) while k is the rate of the event. So, assuming a
surface species hops from one site to another at time t, the absolute time it
hops again or desorbs is calculated as,
$t^{{}^{\prime}}=t-ln(Y)/(b_{1}+b_{2})$. Similarly, the absolute time when an
accretion event of a gas-phase species occurs is,
$t_{acc}=t_{acc}^{{}^{\prime}}-ln(X^{{}^{\prime}})/k_{acc}$, where
$X^{{}^{\prime}}$ is a random number uniformly distributed within (0, 1),
$t_{acc}^{{}^{\prime}}$ is the absolute time when last accretion event occurs
(it is 0 in the beginning of simulations) while $k_{acc}$ is the accretion
rate of the species.

In the beginning of model simulations, the population of surface H and H2
fluctuates but also increases. After some time, the steady state condition is
reached when both H and H2 population fluctuates around the converged value.
After steady states have been reached, we choose a time period $\tau^{,}$ to
calculate the recombination efficiency, $\eta$, which measures how efficiently
H2 are formed on grain surfaces. We count the number of H2 formed, $N_{H_{2}}$
and the number of H atom accretion events, $N_{H}$ during $\tau^{,}$. The
recombination efficiency is calculated as, $\eta=\frac{2N_{H_{2}}}{N_{H}}$. We
set $\tau^{,}$ large enough so that the value of $\eta$ converges.

##  4 Results

###  4.1 Diffuse clouds

The H2 molecules were observed to be formed with rate coefficient 3-4$\times
10^{-17}$ cm3 s-1 in the diffuse interstellar medium (Gry et al., 2002;
Wakelam et al., 2017a). The H2 formation rate coefficient 3$\times 10^{-17}$
cm3 s-1 corresponds to a recombination efficiency of 0.5 on dust grains (Gry
et al., 2002; Hollenbach et al., 1971). So, a recombination efficiency of 0.5
is the minimum to explain observations toward diffuse clouds. Hereafter,
efficient H2 formation means $\eta\geq 0.5$ in this work.

Fig. 2 shows the calculated recombination efficiency, $\eta$ as a function of
grain surface temperature, T in the reference model M01. Similar to previous
models (Chang et al., 2005; Cuppen & Herbst, 2007), H atoms can be trapped in
the stronger binding sites, so H2 can be efficiently formed on grains over a
temperature range between 10 K and 14 K in this model. At temperatures between
10 and 13 K, $\eta$ increases with T because surface H atom population
decreases at higher temperatures, so more H atoms are able to accrete on grain
surface instead of being rejected to the gas-phase. This phenomenon was also
reported by previous studies (Chang et al., 2005). At temperatures above 13 K,
$\eta$ decreases as T increases because of the enhanced desorption rate of H
atoms. In model M01, we assume three different values of
$\theta=n_{H}/(2n_{H_{2}}+n_{H})$, where $n_{H_{2}}$ and $n_{H}$ are the
number densities of gas-phase H and H2 respectively. We can see that the value
of $\theta$ does not affect $\eta$ much in the model M01.

The existence of H2 on grain surface in model M02 can significantly reduce the
recombination efficiency as show in Fig. 3. The decrease is especially severe
at around 12 K, where the recombination efficiency drops to nearly 0.
Efficient H2 formation is not possible for all T $\geq 10$ K in model M02. In
this model, the desorption rate of H atoms increases due to surface H2
although H atoms cannot react with H2, so $\eta$ in model M02 is more than one
order of magnitude lower than that in model M01. On the other hand, we can see
that $\eta$ as a function of T in model M02 is more complex than in model M01.
The reason is the following. Higher T can increase H2 desorption rates, so the
population of H2 decreases as the T increases. So, H atoms are less likely to
encounter H2 and then desorb at higher T. Moreover, there are more stronger
sites available on grain surfaces for H atoms to occupy and then be trapped as
the H2 population decreases. Therefore, $\eta$ should increase if we only
consider the decrease of surface H2 population at higher T. On the other hand,
higher T can also increase the thermal desorption rate of H atoms, which
decreases $\eta$. Fig. 3 shows that the net effect of increasing T on $\eta$
is dependent on T. At grain temperatures lower than 12 K, $\eta$ decreases as
T increases. However, at grain temperatures between 12 K and 15 K, $\eta$
increases with T. As T increases beyond 15 K, $\eta$ decreases as T increases.
Finally, we assumed two values of $\theta$, 0.75 and 0.5 in the model M02.
Fig. 2 shows that the recombination efficiency becomes larger as $\theta$
becomes larger in the model M01. The reason is that the accretion rate of H2
is smaller when $\theta$ is larger, so the population of surface H2 decreases,
which helps H atoms to be trapped on grain surfaces.

Fig. 4 shows the recombination efficiency predicted by model M03 as a function
of T. By comparing models M02 and M03 results, we see that the chemical
desorption coefficient, $\mu$ can hardly affect $\eta$ for both $\theta=0.5$
and 0.75. The reason is that the major source of surface H2 is the accretion
of H2 from gas phase. Therefore, we assume $\mu$ to be 1 for all models other
than M03.

Fig. 5 shows the recombination efficiency predicted by model M04 as a function
of grain temperature. The model M04 adopts the more up-to-date desorption
energies of H atoms and H2 molecules on H2 substrates. Recombination
efficiency in model M04 is typically lower than model M02 for both
$\theta=0.75$ and 0.5. For instance, at 15 K, $\eta$ in model M04 is about
half of that in model M02. The reason is that the desorption energy of H atoms
on H2, $E_{D_{HH_{2}}}$ in the model M04 is lower than that in model M02, so H
atoms desorb more quickly when they encounter H2. Moreover, because of the
higher desorption energy of H2 molecules on H2 in model M04, the population of
H2 in the model M04 is larger than the model M02, so H atoms are more likely
to encounter H2 and then desorb in the model M04.

Fig. 6 shows the recombination efficiency predicted by model M05 as a function
of T. Compared with models M02-M04 results, the recombination efficiency
predicted by model M05 increased significantly because of the much larger H
atom desorption energy on sites occupied by H2. However, efficient H2
formation is still not possible on dust grains if T is above 10 K for both
$\theta=0.5$ and 0.75.

Fig. 7 shows the recombination efficiency predicted by model M06 as a function
of T. Hydrogen atom can still efficiently desorb when they encounter H2 on
grain surfaces, although there are more deep potential sites available for
them to occupy in this model. Therefore, over all, the recombination
efficiency predicted by this model is lower than model M01, but larger than by
models M03-M05. Moreover, at T above 12 K, $\eta$ predicted by model M06
decreases more quickly than by model M02. So, in model M06, efficient H2
formation occurs only around T=11 K for both $\theta=0.5$ and 0.75.

Figure 2: The calculated recombination efficiency, $\eta$ as a function of
grain temperature, T using model M01. Circles and squares represent
$\theta=0.75$ and 0.5 respectively.  Figure 3: The calculated recombination
efficiency, $\eta$ as a function of grain temperature, T using model M02.
Circles and squares represent $\theta=0.75$ and 0.5 respectively.  Figure 4:
The calculated recombination efficiency, $\eta$ as a function of grain
temperature, T using model M03. Circles and squares represent $\theta=0.75$
and 0.5 respectively.  Figure 5: The calculated recombination efficiency,
$\eta$ as a function of grain temperature, T using model M04. Circles and
squares represent $\theta=0.75$ and 0.5 respectively.  Figure 6: The
calculated recombination efficiency, $\eta$ as a function of grain
temperature, T using model M05. Circles and squares represent $\theta=0.75$
and 0.5 respectively.  Figure 7: The calculated recombination efficiency,
$\eta$ as a function of grain temperature, T using model M06. Circles and
squares represent $\theta=0.75$ and 0.5 respectively.

###  4.2 Translucent clouds

Fig. 8 shows the recombination efficiency predicted by models M11-M13 as a
function of grain temperature. In the reference model M11, H2 can be
efficiently synthesized until T is close to 15 K. The recombination efficiency
predicted by model M12 is almost the same as that by the reference model M11.
The reason is the following. First, the diffusion barriers of H2 on water ice
sites are low enough so that H2 cannot be trapped by the deep potential sites
while H atoms can be trapped in the deep potential sites in model M12.
Secondly, when H2 molecules diffuse and encounter H atoms, their desorption
energies significantly drops, so H2 can desorb quickly in this model. So,
there are only a few H2 molecules on the lattice. Therefore, H atoms can
hardly desorb by encountering H2 on the lattice, thus, $\eta$ predicted by
models M11 and M12 is almost identical. On the other hand, we can see that the
recombination efficiency can hardly be affected by the variation of
R${}_{H_{2}}$ and RH values in models M13. The small H2 population on the
lattice makes it rare for H atoms to have opportunity to encounter H2
molecules. So, the value of R${}_{H_{2}}$ or RH is not important in model M14.

Fig. 9 shows the recombination efficiency predicted by models M14-M15 as a
function of T. The diffusion barriers on water ice sites adopted by these
models are almost a factor of two larger than those in models M11-M13. The
enhanced diffusion barriers enable H2 to be trapped by the deep potential
sites, which significantly increases the H and H2 interaction on grain
surfaces. The recombination efficiency predicted by both models is less than
0.002 and is much lower than that by models M11-M13. Molecular hydrogen cannot
be efficiently synthesized at temperatures above 10 K in both models.

The recombination efficiency predicted by model M15 is lower than that by
model M14. The diffusion barrier of H atoms on H2 in model M14 is lower than
that in model M15. In the MMC simulations as introduced in Section 3, if the
desorption energy of a species is constant, the probability that a surface
species desorb decreases as the diffusion barrier of that species decreases.
Therefore, H atoms in sites occupied by H2 are more likely to diffuse out of
these sites instead of desorb in model M14 than in model M15. So model M14
predicts larger recombination efficiency.

Figure 8: The calculated recombination efficiency as a function of grain
temperature using models M11-M13. 1-$\theta=$0.75 in all models.  Figure 9:
The calculated recombination efficiency as a function of grain temperature
using models M14 and M15. 1-$\theta=$0.75 in all models

##  5 Summary and conclusions

Although H2 molecules may account for a large fraction of gas phase species in
the diffuse and translucent clouds, their existence on grain surfaces has been
ignored by almost all previous H2 formation models. To the best of our
knowledge, this work is the first time that the detailed MMC method has been
used to investigate how the existence of H2 may affect the efficiency of
molecular hydrogen formation on cold interstellar dust grains in diffuse and
translucent clouds. We found that the impact of the existence of H2 on
recombination efficiency mainly depends on the diffusion barriers of H2 on
grain surfaces.

In the diffuse cloud models M02-M06, because of the high diffusion barriers on
olivine grains, H2 molecules can be trapped by the deep potential sites. Site
occupied by H2 become much weaker binding sites for H in these models. So, in
models M02-M05, it becomes more difficult for H atoms to become trapped in
these sites already occupied by H2. Hydrogen atoms can also efficiently desorb
when they encounter H2 molecules in models M02-M05, leading to a lower
predicted recombination efficiency in models M02-M05 than in the reference
model M01, which does not allow H2 to exist on grain surfaces. Moreover,
models M02-M05 all fail to predict the minimum recombination efficiency (0.5)
to explain observations.

The recombination efficiency increases with the value of $E_{D_{HH_{2}}}$,
which is, however, not well known yet. The adopted value of $E_{D_{HH_{2}}}$
in models M02-M04 is much lower than the desorption energy of H atoms on
silicate surfaces while model M05 adopts a upper bound of $E_{D_{HH_{2}}}$.
Future more rigorous study might suggest higher values of $E_{D_{HH_{2}}}$ and
$E_{D_{H_{2}H_{2}}}$ than these used in models M02-M04. However, because even
model M05 predicts significant reduction of the recombination efficiency, we
argue that even if these higher values of $E_{D_{HH_{2}}}$ and
$E_{D_{H_{2}H_{2}}}$ are adopted by models M02-M04, these models may still
predict lower recombination efficiency than the reference model M01 does.

In model M06, because H atoms and H2 molecules do not share common stronger
binding sites, the stronger H atom binding sites can still trap H atoms when
H2 molecules are trapped on grain surfaces. However, H atoms can still desorb
efficiently when they encounter H2. Model M06 results show that as long as H2
are trapped by the strong binding sites and the desorption energy of H atoms
on H2 substrate is much lower than that on silicate grain surfaces, efficient
H2 formation ($\eta\geq 0.5$) occurs only when the dust surface temperature is
around 11 K.

We used olivine grains as an example in the diffuse cloud models. The
diffusion barrier of H2 on carbon grain surfaces is even larger that on
silicate surfaces. So, H2 molecules could be more tightly trapped on grain
surfaces. The tightly trapped H2 molecules should also make H2 formation on
carbon grains less efficient.

In the translucent clouds models M12 and M13, the diffusion barriers of H2 on
grains covered by water ice are low enough so that H2 molecules diffuse even
faster than H atoms and cannot be trapped by the deep potential sites.
Moreover, H2 molecules can efficiently desorb when they encounter H atoms on
grain surfaces. So H2 population in these two models are very low so that the
enhanced H atom desorption by encountering H2 molecules is minimal. Therefore,
the existence of H2 on grain surfaces can hardly affect the recombination
efficiency. On the other hand, in models M14 and M15, the diffusion barriers
of H2 are almost a factor of two larger than these in models M12 and M13, so
H2 can be trapped by the deep potential sites. In models M14 and M15,
efficient H2 formation is not possible when the temperatures are above 10 K.

Based on the translucent cloud model results, we can conclude that lower H2
diffusion barriers on bare dust grains can enhance the recombination
efficiency. This conclusion suggests that $\eta$ may also increase if lower H2
diffusion barriers is used in diffuse models. In order to find out how lower
H2 diffusion barriers may affect diffuse cloud model results, we simulate a
test diffuse cloud model M07 using reduced H2 diffusion barriers, which are
$50\%$ of the H2 desorption energies on silicate surfaces. The H2 diffusion
barriers in the shallow, middle and deep silicate potential sites are 203 K,
308 K and 435 K respectively in model M07. Other than the H2 diffusion
barriers in silicate sites, parameters for model M07 are the same as these for
model M02. Fig. 10 shows the recombination efficiency as a function of T
predicted by the test model M07 and the reference model M01. We can see that
H2 can be efficiently synthesized ($\eta\geq 0.5$) when the grain temperatures
are between 10 K and 14 K in model M07, which agrees well with model M01
predictions. So, lower H2 diffusion barriers can also enhance recombination
efficiency in diffuse clouds. However, to the best our knowledge, the H2
diffusion barriers on silicate surfaces are poorly known. More work should be
done to gauge this energy value in order to better understand H2 formation in
diffuse clouds.

Figure 10: The calculated recombination efficiency as a function of grain
temperature using models M01 and M07. $\theta=0.5$ in both models.

Since the classical work by Hollenbach & Salpeter (1971), models that adopted
grain surfaces whose binding sites have various desorption energies and
diffusion barriers have been successfully used to enhance the H atom
recombination efficiency at higher temperatures. The key to the success at
higher temperatures is the existence of stronger H atom binding sites in these
models. Ironically, these stronger sites may become much weaker H atom binding
sites after H2 molecules occupy them. Therefore, the success may not be as
plausible as it seemed to be if we consider surface H2 in models. Our model
results show that H2 molecules must diffuse fast enough on grain surfaces so
that they are not trapped on grain surfaces in order that H2 can be formed
efficiently.

We consider physisorption only in our models while H atoms can also be
chemisorbed on silicate or carbon surfaces (Cazaux & Tielens, 2004). It is not
clear whether models that involve chemisorption can explain H2 formation in
diffuse clouds or not if the existence of H2 on grain surface is considered.
More study should be done in order to unveil the mystery of H2 formation in
diffuse clouds.

Hydrogen atoms can only be physisorbed on water ice surfaces (Wakelam et al.,
2017a). Our translucent cloud model results further show that the conversion
from physisorbed H atoms to H2 molecules can be very slow on grain surfaces if
the diffusion barrier of H2 on water ice surface is large enough. Therefore,
this diffusion barrier could be important for the molecular evolution of
translucent clouds. However, because surface chemical reactions that involve
H2 typically have a large reaction barrier, this diffusion barrier is not well
studied so far. More rigorous research should be done to gauge this energy
value.

## Acknowledgments

This work was funded by the National Natural Science Foundation of China
(grant Nos. 11973099, 11973075 and 12173023). We thank our referee for
constructive comments to improve the quality of the manuscript. We would like
to thank Professor Laura Rowe for her help with proofreading and scientific
English writing. We thank Di Li for helpful discussions.

## Data availability

No new data were generated or analysed in support of this research.

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11institutetext: Daniel Grumiller 22institutetext: Institute for Theoretical
Physics, TU Wien, Wiedner Hauptstrasse 8-10, A-1040 Vienna, Austria, 22email:
<EMAIL_ADDRESS>33institutetext: Martin Laihartinger
44institutetext: Institute for Theoretical Physics, TU Wien, Wiedner
Hauptstrasse 8-10, A-1040 Vienna, Austria, 44email:
<EMAIL_ADDRESS>55institutetext: Romain Ruzziconi
66institutetext: Institute for Theoretical Physics, TU Wien, Wiedner
Hauptstrasse 8-10, A-1040 Vienna, Austria, 66email:
<EMAIL_ADDRESS>
# Minkowski and (A)dS ground states in general 2d dilaton gravity

Daniel Grumiller  Martin Laihartinger and Romain Ruzziconi

###### Abstract

We reorganize the derivative expansion of general (power-counting non-
renormalizable) 2d dilaton gravity such that the mass function is integrable.
As an example, we consider a three-parameter family of models and provide
conditions on the parameters such that the ground state is either Minkowski,
Rindler, or (A)dS.

##  1 Introduction to general 2d dilaton gravity

Two is the lowest spacetime dimension with enough room for space and time.
Consequently, it is the lowest dimension that allows for lightcones, horizons,
and black holes. Moreover, two is the lowest dimension permitting non-trivial
intrinsic curvature and topology. For all these reasons, two-dimensional (2d)
theories of gravity serve as minimal toy models to elucidate conceptual
aspects of classical and quantum gravity, including black hole evaporation and
holographic correspondences.

Since Einstein gravity does not exist in 2d, one has to resort to some
alternative gravity theory. All known approaches111 This includes intrinsic
models Jackiw:1985je ; Teitelboim:1983ux ; Callan:1992rs , higher
curvature/torsion models Katanaev:1986wk , gravity as linear Isler:1989hq ;
Chamseddine:1989yz ; Verlinde:1991rf  or non-linear gauge theory Ikeda:1993fh
; Schaller:1994es , the target-space action of non-critical strings in 2d
Mandal:1991tz ; Elitzur:1991cb ; Witten:1991yr ; Dijkgraaf:1992ba , a scaling
limit of gravity in $2+\epsilon$ dimensions Mann:1992ar ; Lemos:1993py ;
Grumiller:2007wb  and various dimensional reductions to 2d, see
Grumiller:2002nm  and additional refs. therein. lead to the same class of
theories (possibly after some field redefinitions and/or integrating in/out
auxiliary fields), namely 2d dilaton gravity, a scalar-tensor theory. In its
most general version, the bulk action

$I[g_{\mu\nu},\,X]=-\frac{\kappa}{4\pi}\,\int\operatorname{d}\\!^{2}x\sqrt{-g}\,\Big{(}XR-2{\cal
V}\big{(}X,\,-(\partial X)^{2}\big{)}\Big{)}$ (1)

contains an arbitrary function $\cal V$ of two variables, the dilaton $X$ and
its kinetic term $(\partial X)^{2}$. We adapted signs consistently with
Grumiller:2021cwg . Special cases often considered in the literature are the
restriction to power-counting renormalizable models Odintsov:1991qu ;
Russo:1992yg  or to potentials that are independent from the kinetic term
Louis-Martinez:1993bge ; Witten:2020ert . In our work, we do not impose such
restrictions.

While the action (1) has appeared, e.g., in Strobl:1999wv ; Grumiller:2002nm ;
Grumiller:2002md ; Kunstatter:2015vxa ; Takahashi:2018yzc  (also known as
“kinetic gravity braiding” Deffayet:2010qz  or “Horndeski theory”
Horndeski:1974wa ), besides the class of dilaton scale invariant models
studied in detail in Grumiller:2021cwg  no charting of the full model space
was attempted yet.

In this proceedings contribution, we make the first steps in this direction.
In section 2 we review how to obtain all classical solutions, focussing on the
linear dilaton sector, up to the integration of a conservation equation. In
section 3 we reorganize the way that models are formulated by demanding the
conservation equation to be integrable and providing some function therein as
input, from which the potential $\cal V$ is then deduced. In section 4 we
focus on a three-parameter family as an example and discuss under which
conditions the models exhibit ground states that are maximally symmetric
(Minkowski, Rindler, or (A)dS). In section 5 we conclude.

##  2 Classical solutions

To discuss the classical solutions of general 2d dilaton gravity (1), it is
convenient to work in the first order formulation of the theory Klosch:1995fi
; Klosch:1995qv , whose action

$I[\omega,\,e_{a},\,X,\,X^{a}]=\frac{\kappa}{2\pi}\int\Big{(}X\,\operatorname{d}\\!\omega+X^{a}\,\big{(}\operatorname{d}\\!e_{a}+\epsilon_{a}{}^{b}\,\omega\wedge
e_{b}\big{)}+\epsilon\,{\cal V}(X,\,X^{c}X_{c})\Big{)}$ (2)

depends on the (dualized) Lorentz connection $\omega$, the zweibein $e_{a}$,
the dilaton $X$, and auxiliary scalar fields $X^{a}$. We introduced the volume
form $\epsilon=\frac{1}{2}\epsilon^{ab}e_{a}\wedge e_{b}$. In light-cone
coordinates $(x^{+},x^{-})$ for the Lorentz indices $a,b$, the Minkowski
metric reads as $\eta_{+-}=1$, $\eta_{\pm\pm}=0$. Our sign convention for the
Levi-Civitá symbol is $\epsilon^{\pm}{}_{\pm}=\pm 1$. The equations of motion

$\displaystyle\operatorname{d}\\!X+X^{-}e^{+}-X^{+}e^{-}=0$ (3a)
$\displaystyle(\operatorname{d}\\!\pm\omega)X^{\pm}\pm e^{\pm}\,\mathcal{V}=0$
(3b)
$\displaystyle\operatorname{d}\\!\omega+\epsilon\,\frac{\partial\mathcal{V}}{\partial
X}=0$ (3c)
$\displaystyle(\operatorname{d}\\!\pm\omega)e^{\pm}+\epsilon\,\frac{\partial\mathcal{V}}{\partial
X^{\mp}}=0$ (3d)

are of first order in derivatives. Solving (3c) and (3a) for $\omega$ and
$X^{a}$ and re-injecting these solutions into the action (2) recovers the
second order action (1) (see e.g. Grumiller:2021cwg  for details).

The constant dilaton sector of the theory is given by the solutions of
(3a)-(3d) satisfying

${\cal V}(X,\,X^{a}X_{a})=0=X^{a}$ (4)

These solutions have constant dilaton, constant curvature, and are locally
maximally symmetric. They are covered extensively in the literature (see e.g.
Hartman:2008dq ), so we do not discuss them in this work.

Instead, we focus on the linear dilaton sector of the theory and summarize the
solution algorithm discussed in Grumiller:2021cwg . Combining (3b) and (3a)
obtains

$\operatorname{d}\\!\,Y-\mathcal{V}(X,\,2Y)\,\operatorname{d}\\!X=0\qquad\qquad
Y:=X^{+}X^{-}\,.$ (5)

As discussed below, this relation implies Casimir conservation
$\operatorname{d}\\!C=0$ and can be integrated to yield the Casimir function
$C(X,\,Y)$. Assuming $X^{+}\neq 0$ (essentially without loss of generality)
the equation (3b) implies

$\omega=-\frac{\operatorname{d}\\!X^{+}}{X^{+}}-Z\,\mathcal{V}$ (6)

where $Z=e^{+}/X^{+}$. Similarly, the equation (3a) yields

$e^{-}=\frac{\operatorname{d}\\!X}{X^{+}}+X^{-}Z$ (7)

and the volume element $\epsilon=-Z\wedge\operatorname{d}\\!X$. Combining the
upper signs (3d) and (3b) gives

$\operatorname{d}\\!Z=\left(Z\wedge\operatorname{d}\\!X\right)\frac{\partial{\cal
V}}{\partial Y}\,.$ (8)

Inserting $Z=\operatorname{d}\\!v\,e^{Q(X)}$ into this equation yields

$\frac{\operatorname{d}\\!Q}{\operatorname{d}\\!X}=-\frac{\partial{\cal
V}}{\partial Y}$ (9)

which can be formally integrated as

$Q(X)=-\int^{X}\frac{\partial\cal V}{\partial Y}$ (10)

by virtue of (5). The line element reads as

$\operatorname{d}\\!s^{2}=2e^{+}e^{-}=2e^{Q}\,\operatorname{d}\\!v\operatorname{d}\\!X+2Ye^{2Q}\,\operatorname{d}\\!v^{2}$
(11)

where $Y$ is constrained through (5) and $Q(X)$ is given in (10). Introducing
the radial coordinate $\operatorname{d}\\!r=e^{Q}\operatorname{d}\\!X$, which
can be integrated to give the dilaton as a function of the radius $X(r)$, the
line element (11) takes Eddington–Finkelstein form,

$\operatorname{d}\\!s^{2}=2e^{2Q}Y\operatorname{d}\\!v^{2}+2\operatorname{d}\\!v\operatorname{d}\\!r\,.$
(12)

The solution (12) is completely determined once the potential $\mathcal{V}$ is
provided, up to the integration constant $C$ in (5) corresponding to the
Casimir of the theory.222The constant coming from the integration of (9) is
trivial since it can be fixed by a choice of units. All solutions exhibit at
least one Killing vector, namely $\partial_{v}$.

We highlight three properties of the Casimir $C$: 1. Physically, it encodes
the mass of the solution. 2. Notably, the function $Q(X)$, in general, depends
on the Casimir, except for power-counting renormalizable models. 3. To obtain
explicit solutions, we still need to integrate (5). How to achieve this is the
subject of the next section.

##  3 Integrable conservation equation

In practice, given a potential $\mathcal{V}$, it might be difficult to
integrate the conservation equation (5) and find an explicit solution for $Y$.
Furthermore, it is not a priori obvious to determine which class of potentials
$\mathcal{V}$ will allow us to find explicit solutions and obtain a complete
classification of the ground states. Instead, we will use another route and
start from a class of exact differential equations for (5). They are of the
form

$p(X,\,Y)~{}\operatorname{d}\\!X+q(X,\,Y)~{}\operatorname{d}\\!Y=0$ (13)

where $p(X,\,Y)$ and $q(X,\,Y)$ are functions satisfying the integrablity
condition

$\frac{\partial p}{\partial Y}=\frac{\partial q}{\partial X}\,.$ (14)

With this condition, the conservation equation (13) can be rewitten in terms
of the Casimir $C(X,\,Y)$ as

$\operatorname{d}\\!C=0\qquad\textrm{with}\qquad p=\frac{\partial C}{\partial
X}\qquad q=\frac{\partial C}{\partial Y}$ (15)

which readily can be integrated. Comparing with (5), we identify

$\mathcal{V}=-\frac{p}{q}\,.$ (16)

Our approach is reminiscent of inverse scattering methods: instead of starting
with some potential $\mathcal{V}$ and determining the mass function $C$, we
postulate some suitable mass function $C$, use (15) to get the functions $p$
and $q$, and obtain the potential $\mathcal{V}$ as output by virtue of
(16).333The analog of the integrability condition (14) would be difficult to
formulate if one started from a generic potential $\mathcal{V}$ instead of
starting from equation (13), which justifies our approach. In terms of (16),
equation (10) leads to

$Q(X)=\int^{X}\frac{\partial}{\partial
Y}\left(\frac{p}{q}\right)\operatorname{d}\\!X^{\prime}=\int^{X}\left(\frac{1}{q}\frac{\partial
p}{\partial Y}-\frac{p}{q}~{}\frac{1}{q}\frac{\partial q}{\partial
Y}\right)\operatorname{d}\\!X^{\prime}~{}.$ (17)

The first term in the integral on the right can be rewritten with the relation
(14).

$Q(X)=\int^{X}\left(\frac{1}{q}\frac{\partial q}{\partial
X^{\prime}}+\mathcal{V}~{}\frac{1}{q}\frac{\partial q}{\partial
Y}\right)\operatorname{d}\\!X^{\prime}$ (18)

Performing the change of integration variable
$\operatorname{d}\\!X=\mathcal{V}^{-1}\operatorname{d}\\!Y$ in the second
term, and using $\operatorname{d}\\!q=\frac{\partial q}{\partial
Y}~{}\operatorname{d}\\!Y+\frac{\partial q}{\partial
X}~{}\operatorname{d}\\!X$, we finally obtain

$Q(X)=\int\frac{1}{q}~{}\operatorname{d}\\!q=\ln\,q(X)$ (19)

where, as explained in the previous section, the integration constant does not
contain physical information and has therefore been set to zero. In general,
$Q(X)$ depends on the Casimir $C$ when $q$ depends on $Y$, since we use the
integrated version of (5) to express $Y$ as function of $X$ and the Casimir
$C$. Power-counting renormalizable models, $\mathcal{V}(X,\,2Y)=V(X)-YU(X)$,
are a notable exception: for these models, the integrating factor $Q(X)$ is
independent from the Casimir $C$.

Up to this stage, given $p$, $q$ in equation (13) satisfying the integrability
condition (14), we have shown how to write explicitly the line element (11)
using (19) and the constraint on $Y$ that can be readily obtained from the
Casimir conservation deduced from (13). Now we intend to determine the ground
states of these families of metrics, which are characterized by solutions with
(i) constant curvature and (ii) vanishing Casimir function. The first
condition is by choice — we are mostly interested in models that have a
maximally symmetric ground state, like flat space, Rindler, or (A)dS2. The
second condition is without loss of generality: we can always shift the whole
spectrum in such a way that the value of the Casimir function for the ground
state vanishes.

The Ricci scalar $R$ is expressed in terms of the Killing norm $K$ as444Zeros
of the Killing norm $K$ in (20) correspond to loci of Killing horizons. They
arise either for $q=0$ or $Y=0$. The case of $q=0$ leads to an extremal
Killing horizon, as $q=0$ is a double zero for the Killing norm. The loci of
non-extremal Killing horizons are those for which $Y=0$.

$R=\partial_{r}^{2}K\qquad\qquad K=2q^{2}Y\,.$ (20)

We distinguish three types of vacua, depending on the values of the curvature
and Killing norm:

  * •

Minkowski ground states are characterized by vanishing Ricci scalar, $R=0$,
and constant Killing norm, $K(r)=\rm const.$

  * •

Rindler ground states are given by vanishing Ricci scalar, $R=0$, and linear
Killing norm, $K(r)\propto r$.

  * •

(A)dS ground states are characterized by positive (negative) Ricci scalar, so
that the Killing norm is quadratic. We can distinguish between Poincaré (A)dS
with $K(r)\propto(-)r^{2}$, global (A)dS with $K(r)\propto((-)r^{2}+1)$ and
(A)dS Rindler where $K(r)\propto(-)r^{2}+Ar+B$ for some constants $A$, $B$. In
the following, when discussing (A)dS ground states, we will only refer to
Poincaré (A)dS ground states.

##  4 Three-parameter family as example

Power-counting renormalizable models involve potentials and Casimirs that are
at most linear in $Y$. The ground states of these models have been extensively
discussed in the literature (see e.g. Katanaev:1996ni ). Here, we focus on the
simplest general dilaton gravity that is not power-counting renormalizable by
considering a Casimir $C(X,Y)$ quadratic in $Y$,

$C(X,Y)=V_{0}(X)+V_{1}(X)\,Y+V_{2}(X)\,Y^{2}$ (21)

where the potentials $V_{0}$, $V_{1}$, $V_{2}$ are taken to be monomials in
$X$.

$V_{0}(X)=A~{}X^{a}\qquad\qquad V_{1}(X)=B~{}X^{b}\qquad\qquad
V_{2}(X)=D~{}X^{c}$ (22)

The parameters $A$, $B$, $D$ and $a$, $b$, $c$ are constant. The condition
(ii) to obtain ground state solutions requires solving $C(X,Y)=0$ for $Y$.
This leads to two branches of solutions.

$Y^{\pm}=\frac{-V_{1}\pm\sqrt{V_{1}^{2}-4V_{2}V_{0}}}{2V_{2}}$ (23)

Evaluating the function $q(X,Y)$ on these solutions yields

$q(X,Y^{\pm})=V_{1}+2V_{2}Y_{\pm}=\pm\sqrt{V_{1}^{2}-4V_{2}V_{0}}~{}.$ (24)

Plugging (23) and (24) into the second equation of (20) gives the ground state
Killing norm

$K^{\pm}_{0}:=K^{\pm}|_{C=0}=(V_{1}^{2}-4V_{2}V_{0})\frac{-V_{1}\pm\sqrt{V_{1}^{2}-4V_{2}V_{0}}}{V_{2}}~{}.$
(25)

For the three-parameter family (22), this reads as

$K^{\pm}_{0}=\big{(}B^{2}X^{2b}-4AD~{}X^{a+c}\big{)}\Big{(}-\frac{B}{D}X^{b-c}\pm\sqrt{\frac{B^{2}}{D^{2}}X^{2b-2c}-4\frac{A}{D}X^{a-c}}\Big{)}~{}.$
(26)

By expanding $Y^{\pm}$ for small values of $D$, one can see that the branch
$Y^{-}$ is continuously connected to the linear solution, while the branch
$Y^{+}$ is not. This constitutes a criterion to select the branch $Y^{-}$
rather than $Y^{+}$, which is what we are going to do.

We now focus on the condition (i) defining ground states. Following the
classification of vacua provided at the end of the previous section, we treat
each case separately. We determine first the conditions on the model
parameters $a,b,c$ such that the ground state of the corresponding model has
the desired properties, then write down the potential associated with the
model and finally display the metric (12) that emerges as solution of the
model, for any value of the Casimir $C$. The relation between dilaton and
radial coordinate is displayed in the appendix. As an abbreviation we often
use the discriminant $\Delta:=B^{2}-4AD$.

###  4.1 Minkowski ground states

Minkowski ground states are found by requiring the expression of the Killing
norm (26) to be constant. Rewriting (26) as

$K^{-}_{0}=X^{3b-c}\,\big{(}B^{2}-4AD~{}X^{a+c-2b}\big{)}\Big{(}-\frac{B}{D}-\sqrt{\frac{B^{2}}{D^{2}}-4\frac{A}{D}X^{a+c-2b}}\Big{)}=\textrm{const}~{},$
(27)

we see that this condition is satisfied for generic $A$, $B$ and $D$ only if
each term in the product is constant, and therefore $a+c-2b=0$ and $3b-c=0$.
Minkowski ground state models then have to obey the relations

$\boxed{\phantom{\Bigg{(}}\textrm{Minkowski\;ground\;state\;models:}\qquad\qquad
a=-b\qquad\qquad c=3b\phantom{\Bigg{)}}}$ (28)

between the parameters, yielding the potential

$\boxed{\phantom{\Bigg{(}}{\cal V}_{\textrm{\tiny Mink}}\big{(}X,\,-(\partial
X)^{2}\big{)}=a\,X^{2a-1}\,\frac{4A+2BX^{-2a}(\partial
X)^{2}-3DX^{-4a}(\partial X)^{4}}{-4B+4DX^{-2a}(\partial
X)^{2}}\,.\phantom{\Bigg{)}}}$ (29)

The metric (12) for Minkowski ground state models simplifies to
($\Delta=B^{2}-4AD$)

$\operatorname{d}\\!s^{2}=2\operatorname{d}\\!v\operatorname{d}\\!r-I_{0}\,\Big{(}\Delta+4CDX^{-a}\Big{)}\Big{(}B+\sqrt{\Delta+4CDX^{-a}}\Big{)}\,\operatorname{d}\\!v^{2}$
(30)

where we labelled by $I_{0}$ the freedom of fixing the integration constant in
the function $Q(X)$ given in (19). Demanding the terms proportional to the
Casimir $C$ to be subleading for large $X$ establishes the convexity condition
$a>0$. For positive $a$ and a convenient choice of $I_{0}$, the metric (30)
has the asymptotically flat expansion

$\operatorname{d}\\!s^{2}=2\operatorname{d}\\!v\operatorname{d}\\!r-\Big{(}1-MX^{-a}+{\cal
O}\big{(}M^{2}X^{-2a}\big{)}\Big{)}\,\operatorname{d}\\!v^{2}$ (31)

where the mass parameter $M$ is proportional to $C$. Demanding that large
values of the radial coordinate also correspond to large values of the dilaton
requires $a\leq 1$. Thus, standard Minkowski ground state models obey
$a\in(0,1]$ in addition to (28).

Non-generic cases for specific values of the constants $A$, $B$, $D$ exist as
well. If $A=0$ and $B\neq 0$, we get the ground state condition $c=3b$ with no
further restriction. In the case of $B=0$ and $A\neq 0$,
$K^{-}_{0}=\textrm{const}$ leads to $c=-3a$. The case $D=0$ leads to a two-
parameter family of power-counting renormalizable models with the conditions
on the parameter $K_{0}=2q^{2}Y=-2AB~{}X^{a+b}=\textrm{const}$, so that
$a=-b$. Comparing with the well-known two-parameter family
$(a^{\prime}b^{\prime})$ (see e.g. Katanaev:1996ni ) where the potential reads
as
$\mathcal{V}_{(a^{\prime}b^{\prime})}(X,2Y)=\frac{a^{\prime}}{X}Y+X^{a^{\prime}+b^{\prime}}$
and using the dictionary $a^{\prime}=-b$ and $b^{\prime}=a-1$, we recover the
known result for Minkowski ground state $a^{\prime}=1+b^{\prime}$, which
constitutes a consistency check of our derivation.

###  4.2 Rindler ground states

With the radial coordinate defined by
$r=\int^{X}q(X^{\prime},Y)\operatorname{d}\\!X^{\prime}$, the condition
$K^{-}_{0}\propto r$ reads as follows

$\displaystyle K^{-}_{0}$
$\displaystyle=\big{(}B^{2}X^{2b}-4AD~{}X^{a+c}\big{)}\Big{(}-\frac{B}{D}X^{b-c}-\sqrt{\frac{B^{2}}{D^{2}}X^{2b-2c}-4\frac{A}{D}X^{a-c}}\Big{)}$
$\displaystyle\propto
X^{b+1}\,\Big{[}\frac{B}{b+1}+\frac{2D}{c+1}\Big{(}-\frac{B}{D}-\sqrt{\frac{B^{2}}{D^{2}}-4\frac{A}{D}X^{a-2b+c}}\Big{)}\Big{]}~{}.$
(32)

This can be rewritten as

$\left(-\frac{B}{D}-\sqrt{\frac{B^{2}}{D^{2}}-4\frac{A}{D}X^{a-2b+c}}\right)\left(-\frac{2D}{c+1}+X^{2b-c-1}(B^{2}-4AD~{}X^{a-2b+c})\right)\propto\frac{B}{b+1}~{}.$
(33)

For generic $A$, $B$ and $D$ this can be solved for the parameters

$\boxed{\phantom{\Bigg{(}}\textrm{Rindler\;ground\;state\;models:}\qquad\qquad
a=1\qquad\qquad c=2b-1\phantom{\Bigg{)}}}$ (34)

yielding the potential

$\boxed{\phantom{\Bigg{(}}{\cal V}_{\textrm{\tiny R}}\big{(}X,-(\partial
X)^{2}\big{)}=X^{-b}\frac{4A-2bBX^{b-1}(\partial
X)^{2}+(2b-1)DX^{2b-2}(\partial X)^{4}}{-4B+4DX^{b-1}(\partial
X)^{2}}\,.\phantom{\Bigg{)}}}$ (35)

The metric (12) for Rindler ground state models simplifies to
($\Delta=B^{2}-4AD$)

$\operatorname{d}\\!s^{2}=2\operatorname{d}\\!v\operatorname{d}\\!r-I_{0}X^{1+b}\Big{(}\Delta+\frac{4CD}{X}\Big{)}\Big{(}B+\sqrt{\Delta+\frac{4CD}{X}}\Big{)}\,\operatorname{d}\\!v^{2}$
(36)

where $I_{0}$ has the same meaning as before. The terms proprotional to the
Casimir $C$ are always subleading at large $X$. Demanding that large values of
the radial coordinate also correspond to large values of the dilaton requires
$b\geq-1$. For these standard Rindler ground state models, the metric (36) has
the asymptotically Rindler expansion

$\operatorname{d}\\!s^{2}=2\operatorname{d}\\!v\operatorname{d}\\!r-\Big{(}r-Mr^{b/(1+b)}+{\cal
O}\big{(}M^{2}r^{(b-1)/(1+b)}\big{)}\Big{)}\,\operatorname{d}\\!v^{2}$ (37)

where the mass parameter $M$ is proportional to $C$.

Non-generic cases for specific values of the constants $A$, $B$, $D$ exist as
well. For $A=0$, the condition $K^{-}_{0}\propto r$ is satisfied provided
$c=2b-1$. For $B=0$ and $A\neq 0$ we get the solution $a=1$ satisfying
$K^{-}_{0}\propto r$. For $D=0$, the condition $K_{0}^{-}\propto r$ yields
$a=1$ with $b$ arbitrary. In terms of the $(a^{\prime}b^{\prime})$ family, we
get $b^{\prime}=0$, which confirms the known result Katanaev:1996ni .

###  4.3 (A)dS ground states

The (A)dS condition $K^{-}_{0}\propto r^{2}$ is given by

$\displaystyle K^{-}_{0}$
$\displaystyle=X^{3b-c}\,\big{(}B^{2}-4AD~{}X^{a-2b+c}\big{)}\Big{(}-\frac{B}{D}-\sqrt{\frac{B^{2}}{D^{2}}-4\frac{A}{D}X^{a-2b+c}}\Big{)}$
$\displaystyle\propto
X^{2b+2}\left[\left(\frac{B}{b+1}\right)^{2}+\frac{4BD}{(b+1)(c+1)}\left(-\frac{B}{D}-\sqrt{\frac{B^{2}}{D^{2}}-4\frac{A}{D}X^{a-2b+c}}\right)+\right.$
$\displaystyle\left.+\left(\frac{2D}{c+1}\right)^{2}\left(-\frac{B}{D}-\sqrt{\frac{B^{2}}{D^{2}}-4\frac{A}{D}X^{a-2b+c}}\right)^{2}\right]~{}.$
(38)

For generic $A$, $B$ and $D$ this is solved for the parameters obeying

$\boxed{\phantom{\Bigg{(}}\textrm{(A)dS\;ground\;state\;models:}\qquad\qquad
a=b+2\qquad\qquad c=b-2\,.\phantom{\Bigg{)}}}$ (39)

yielding the potential

$\boxed{{\cal V}_{\textrm{\tiny(A)dS}}\big{(}X,-(\partial
X)^{2}\big{)}=X\frac{4aA-2(a-2)BX^{-2}(\partial X)^{2}+(a-4)DX^{-4}(\partial
X)^{4}}{-4B+4DX^{-2}(\partial X)^{2}}\,.}$ (40)

The metric (12) for (A)dS ground state models simplifies to
($\Delta=B^{2}-4AD$)

$\operatorname{d}\\!s^{2}=2\operatorname{d}\\!v\operatorname{d}\\!r+I_{0}X^{2a-2}\Big{(}\Delta+\frac{4CD}{X^{a}}\Big{)}\Big{(}B+\sqrt{\Delta+\frac{4CD}{X^{a}}}\Big{)}\,\operatorname{d}\\!v^{2}$
(41)

where again $I_{0}$ has the same meaning as before. The terms proprotional to
the Casimir $C$ are subleading at large values of the dilaton if $a>0$. If
$a>1$, then large values of the radial coordinate also correspond to large
values of the dilaton. For these standard (A)dS ground state models, the
metric (41) has the asymptotically (A)dS expansion

$\operatorname{d}\\!s^{2}=2\operatorname{d}\\!v\operatorname{d}\\!r+\Big{(}\Lambda\,r^{2}+M\,r^{(a-2)/(a-1)}{\cal
O}\big{(}M^{2}r^{-2/(a-1)}\big{)}\Big{)}\,\operatorname{d}\\!v^{2}$ (42)

where $\Lambda=\Lambda(A,B,D)$ is positive (negative) for (A)dS, and the mass
parameter $M$ is proportional to the Casimir $C$.

Again, some non-generic cases for specific values of the constants $A$, $B$,
$D$ exist. If $A=0$, $K^{-}\propto r^{2}$ yields $c=b-2$. For $B=0$ we get
$c=a+4b$. For $D=0$ we get the result $a=b+2$ from the condition $K_{0}\propto
r^{2}$. This is consistent with the AdS ground states previously found for the
model $\mathcal{V}_{(a^{\prime})}(X,2Y)=\frac{a^{\prime}}{X}Y+X$
Katanaev:1996ni .

##  5 Generalizations

There are three types of generalizations of the standard Minkowski-, Rindler-,
and (A)dS-ground state models discussed in this work.

  1. 1.

Beyond standard models. One can relax the convexity conditions imposed as
standard. For instance, one could consider Minkowski ground state models (29)
where the parameter $a$ lies outside the interval $(0,1]$. Then either the
weak coupling region, $X\to\infty$, no longer corresponds to large values of
the radial coordinate, $r\to\infty$, or the Killing norm is dominated
asymptotically by the terms containing the Casimir $C$. In the latter case,
generic solutions no longer are asymptotically flat. Similar remarks apply to
Rindler and (A)dS ground state models. In the restricted class of power-
counting renormalizable models, this issue was addressed in Grumiller:2014oha
; Bagchi:2014ava .

  2. 2.

Beyond maximally symmetric ground states. One can give up the requirement of
having a maximally symmetric ground state and study all members of the family
of models defined by (21) with (22). It could be interesting to scan this
model space by imposing as constraint reasonable thermodynamics, along the
lines of section 4.2 in Grumiller:2007ju .

  3. 3.

Beyond quadratic ansatz. One can extend the discussion beyond the 3-parameter
family considered in the present work. In particular, generalizing the ansatz
(21) to $C(X,Y)=\sum_{n=0}^{4}V_{n}(X)Y^{n}$ yields explicit (if cumbersome)
expressions, and all our results generalize straightforwardly to this case.
Generalizations beyond these algebraic examples require new techniques or
special choices of $C(X,Y)$.

Besides these generalizations, there are also some evident potential
applications of the models studied in the present work. We list three examples
for possible follow-up work and expect that some of these points will be
addressed in the near future.

  1. 1.

Thermodynamics and holographic renormalization. Presently, it is unclear which
boundary terms have to be added to the bulk action (1) to render the
variational principle well-defined. For the class of power-counting
renormalizable models, these boundary terms were constructed in
Grumiller:2007ju  (using the Hamilton–Jacobi method, see e.g.
Papadimitriou:2010as ). They play an important role in thermodynamics since in
the Euclidean version of the theory the Helmholtz free energy is determined
from the (holographically renormalized) on-shell action. It could be rewarding
to extend the thermodynamical discussion of Grumiller:2007ju  to generic 2d
dilaton gravity models (1), like in section 5 of Grumiller:2021cwg .

  2. 2.

Asymptotic symmetries and boundary excitations. Models with maximally
symmetric ground states could allow for some standard boundary conditions,
using notions of locally asymptotically flat or locally asymptotically (A)dS
spacetimes, along the lines of e.g. Grumiller:2017qao ; Grumiller:2021cwg .

  3. 3.

SYK-like correspondences. Given the relation between Schwarzian actions and
AdS isometries (see Maldacena:2016hyu  and the reviews Mertens:2018fds ;
Sarosi:2017ykf ; Gu:2019jub ) one might expect SYK-like models Kitaev:15ur ;
Sachdev:1992fk  dual to the AdS ground state models. Similarly, by analogy to
Afshar:2019axx  Rindler ground state models could lead to a twisted warped
boundary action.

###### Acknowledgements.

DG thanks the organizers of the MATRIX Event “2D Supersymmetric Theories and
Related Topics” for their efforts. DG and RR were supported by the Austrian
Science Fund (FWF), projects P 30822, P 32581, P 33789, and Y 1447.

## Appendix — Relation between dilaton and radial coordinate

The relation between dilaton and radial coordinate,

$\operatorname{d}\\!r=e^{Q(X)}\,\operatorname{d}\\!X$ (43)

can be integrated in closed form in terms of elementary functions and the
hypergeometric function ${}_{2}F_{1}$ for all models considered in our work.
Dropping the integration constant in $Q(X)$, we list now these results, using
again the definition $\Delta=B^{2}-AD$ and introducing the definition
$M=-4CD/\Delta$.

For Minkowski ground state models, we obtain

$r=\frac{2X\sqrt{\Delta(1-MX^{-a})}}{(a-2)(3a-2)M^{2}}\,\Big{(}\frac{2a}{a+2}\,X^{a}\,_{2}F_{1}(1,1+\tfrac{1}{a},\tfrac{3}{2}+\tfrac{1}{a};\,X^{a}/M)\\\
+M\big{(}a+(2-a)MX^{-a}\big{)}\Big{)}$ (44)

which for large values of $X$ and $a\in(0,1]$ expands as

$r\propto X^{1-a}+\dots$ (45)

For Rindler ground state models, we obtain

$r=\frac{2}{3}\,\sqrt{\Delta}\,\Big{(}\frac{1}{M}\Big{)}^{1/2-b}\,(X-M)^{3/2}\,_{2}F_{1}(\tfrac{3}{2},\tfrac{1}{2}-b,\tfrac{5}{2};\,1-X/M)$
(46)

which for large values of $X$ and $b>-1$ expands as

$r\propto X^{1+b}+\dots$ (47)

For (A)dS ground state models, we obtain

$r=\frac{a\sqrt{\Delta(1-MX^{-a})}X^{a-1}}{(a-2)(a-1)}\,\Big{(}\,_{2}F_{1}(1,1-\tfrac{1}{a},\tfrac{3}{2}-\tfrac{1}{a};\,X^{a}/M)+1-\tfrac{2}{a}\Big{)}$
(48)

which for large values of $X$ and $a>1$ expands as

$r\propto X^{-1+a}+\dots$ (49)

In all cases, the omitted proportionality constants are independent from $M$
and can be adjusted conveniently by fixing the integration constant in $Q(X)$.

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