# The origin of the H$\alpha$ line profiles in simulated disc galaxies

Timmy Ejdetjärn1, Oscar Agertz2, Göran Östlin1, Martin P. Rey3, Florent
Renaud4,5
1Oskar Klein Centre, Department of Astronomy, Stockholm University, 106 91
Stockholm, Sweden
2Lund Observatory, Division of Astrophysics, Department of Physics, Lund
University, Box 43, SE-221 00 Lund, Sweden
3Sub-department of Astrophysics, University of Oxford, DWB, Keble Road,
Oxford, OX1 3RH, UK
4Observatoire Astronomique de Strasbourg, Université de Strasbourg, CNRS UMR
7550, F-67000 Strasbourg, France
5University of Strasbourg Institute for Advanced Study, 5 allée du Général
Rouvillois, F-67083 Strasbourg, France E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Observations of ionised H$\alpha$ gas in disc galaxies with high star
formation rates have ubiquitous and significant line broadening with widths
$\sigma_{\rm H\alpha}\gtrsim 50-100\,{\rm km\,s^{-1}}$. To understand whether
this broadening reflects gas turbulence within the interstellar medium (ISM)
of galactic discs, or arises from off-the-plane emission in mass-loaded
galactic winds, we perform radiation hydrodynamic (RHD) simulations of
isolated Milky Way-mass disc galaxies in a gas-poor (low-redshift) and gas
rich (high-redshift) condition and create mock H$\alpha$ emission line
profiles. We find that the vast majority of the ${\rm H\alpha}$ emission is
confined within the ISM, with extraplanar gas contributing mainly to the
extended profile wings. This substantiates the ${\rm H\alpha}$ emission line
as a tracer of mid-plane disc dynamics. We investigate the relative
contribution of diffuse and dense ${\rm H\alpha}$ emitting gas, corresponding
to DIG ($\rho\lesssim 0.1\,{\rm cm^{-3}}$, $T\sim 8\,000$ K) and HII regions
($\rho\gtrsim 10\,{\rm cm^{-3}}$, $T\sim 10\,000$ K), respectively, and find
that DIG contributes $\lesssim 10\,\%$ of the total LHα. However, the DIG can
reach upwards of $\sigma_{\rm H\alpha}\sim 60-80\,{\rm km\,s^{-1}}$ while the
HII regions are much less turbulent $\sigma_{\rm H\alpha}\sim 10-40\,{\rm
km\,s^{-1}}$. This implies that the $\sigma_{\rm H\alpha}$ observed using the
full ${\rm H\alpha}$ emission line is dependent on the relative ${\rm
H\alpha}$ contribution from DIG/HII regions and a larger $f_{\rm DIG}$ would
shift $\sigma_{\rm H\alpha}$ to higher values. Finally, we show that
$\sigma_{\rm H\alpha}$ evolves, in both the DIG and HII regions, with the
galaxy gas fraction. Our high-redshift equivalent galaxy is roughly twice as
turbulent, except for in the DIG which has a more shallow evolution.

###### keywords:

galaxies: disc – galaxies: star formation – ISM: kinematics and dynamics –
ISM: evolution – turbulence – methods: numerical

††pubyear: 2024††pagerange: The origin of the H$\alpha$ line profiles in
simulated disc galaxies–A

##  1 Introduction

The gas inside galactic discs, the interstellar medium (ISM), consist of
various gas phases spanning several order of magnitude of densities and
temperatures, from dense molecular clouds, where stars are formed, to warm
ionised gas, which makes up a majority of the total gas volume but only a
fraction of the mass (McKee & Ostriker, 1977). In particular, the ionised gas
phase is prominent in the disc, outflows and extraplanar gas (Mac Low &
Klessen, 2004; Haffner et al., 2009), and has several formation channels, e.g.
ionising radiation from massive stars, supernovae shock winds, strong stellar
winds, black hole feedback (Mac Low & Klessen, 2004; Haffner et al., 2009;
Heckman & Best, 2014; Somerville & Davé, 2015). The nature and formation of
these gas phases is not fully understood and is relevant for the reprocessing
of gas between phases, but also for observations which use emission lines that
trace ionised gas in order to interpret the dynamics and physical state of
galaxies.

A prominent difference between gas phases is their kinematics. Gas turbulence,
often quantified as the velocity dispersion $\sigma_{\rm g}$, is a fundamental
property of galactic discs, with supersonic motions driving crucial physical
processes such as star formation (Larson, 1981; Mac Low & Klessen, 2004;
Renaud et al., 2012; Federrath, 2018), metal mixing (Yang & Krumholz, 2012;
Armillotta et al., 2018), and density distributions (McKee & Ostriker, 2007).
The ionised gas in local galaxies has been observed to be more turbulent,
$\sigma_{\rm g}$ $\sim 15-30$${\,\rm{km\,s^{-1}}}$, (Epinat et al., 2010;
Moiseev et al., 2015; Green et al., 2014; Varidel et al., 2016; Law et al.,
2022) than both neutral (Tamburro et al., 2009; Wisnioski et al., 2011) and
molecular gas (Nguyen-Luong et al., 2016; Levy et al., 2018), $\sigma_{\rm g}$
$\sim 10$${\,\rm{km\,s^{-1}}}$. Importantly, the thermal energy difference is
not enough to explain the offset in turbulence between these gas phases. The
driver of this supersonic, multi-phase turbulence is not fully understood (see
the reviews by Elmegreen & Scalo, 2004; Mac Low & Klessen, 2004; Glazebrook,
2013), but it is believed that a combination of stellar feedback and
gravitational instability could inject a majority of the turbulent energy
(e.g. Dib et al., 2006; Agertz et al., 2009; Faucher-Giguère et al., 2011;
Hayward & Hopkins, 2017; Krumholz et al., 2018; Orr et al., 2020; Ejdetjärn et
al., 2022; Forbes et al., 2023).

The problem is further compounded, with velocity dispersions being reported
much higher at higher redshift ($z\sim 2$), where line widths on the order of
$\sigma_{\rm g}$ $\sim 30-100$${\,\rm{km\,s^{-1}}}$are observed (Law et al.,
2009; Cresci et al., 2009; Genzel et al., 2011; Alcorn et al., 2018; Übler et
al., 2019; Girard et al., 2021; Lelli et al., 2023). However, high redshift
observations suffer from poor spatial resolutions, which increases the risk of
beam smearing and, in combination with uncertain galaxy inclination
corrections, can raise the velocity dispersion by a factor of 3-5 (see e.g. Di
Teodoro & Fraternali, 2015; Rizzo et al., 2022; Ejdetjärn et al., 2022;
Kohandel et al., 2023). Furthermore, galaxies undergoing interactions or
mergers, e.g. local (Ultra) Luminous Infrared Galaxies (Armus et al., 1987;
Sanders et al., 1988; Carpineti et al., 2015), have been observed to feature
very high levels of turbulence, with broad emission lines ($\sigma_{\rm
g}\gtrsim 100\,{\rm km\,s^{-1}}$) even in the cold molecular gas phase
(Veilleux et al., 2009; Renaud et al., 2014; Scoville et al., 2015, 2017).
Similarly, local low-mass starburst galaxies, which are dispersion-dominated
and likely interacting, have been found with $\sigma_{\rm g}\sim
30-100$${\,\rm{km\,s^{-1}}}$(e.g. Östlin et al., 2001; Green et al., 2014;
Herenz et al., 2016; Bik et al., 2022). Disentangling the effects between
interactions and secular effects on $\sigma_{\rm g}$ is a difficult task,
especially for high-redshift observations with low spatial and spectral
resolution.

One of the most common emission lines used to infer properties of warm ionised
gas in both local and high-redshift galaxies is the H$\alpha$ line, due to its
strength, ubiquity at all redshifts, and straight-forward dust correction
through the Balmer decrement. The largest H$\alpha$ flux within galaxies comes
from HII regions, which are the warm $T\sim 10^{4}$ K, dense $n_{\rm e}\sim
10-10^{3}\,{\rm cm^{-3}}$ gas regions with recent star formation, and are
almost fully ionised due to the ionising flux from massive OB stars (e.g.
Osterbrock & Ferland, 2006). But several studies have found that diffuse
ionised gas (DIG; also referred to as the warm ionised medium, WIM), with
densities $n_{\rm e}\lesssim 0.08\,{\rm cm^{-3}}$ and temperatures $T\sim
8000$ K (see Haffner et al., 2009, for a review), may be the dominant
contributor ($f_{\rm DIG}\gtrsim 50\,\%$) towards to total H$\alpha$
luminosity in local galaxies (e.g. Ferguson et al., 1996; Hoopes et al., 1996;
Oey et al., 2007; Kreckel et al., 2016; Belfiore et al., 2022). However, this
conclusion is debated, given the large spread of $f_{\rm DIG}$ with values as
low as $f_{\rm DIG}\sim 10\%$ in some galaxies (e.g. Oey et al., 2007; Blanc
et al., 2009; Della Bruna et al., 2022; Micheva et al., 2022). This adds more
complexity when trying to discern the underlying origin of the H$\alpha$ line.

Furthermore, the DIG and HII regions inhabit different environments and are
ionised by different mechanisms. A large fraction of the ionisation for the
DIG is believed to come from leaking HII regions (Haffner et al., 2009;
Weilbacher et al., 2018) but many unrelated mechanisms likely contribute, e.g.
supernovae shocks (Collins & Rand, 2001), cosmic rays (Vandenbroucke et al.,
2018), and dissipation of turbulence (Minter & Spangler, 1997; Binette et al.,
2009). In the scenario that the mid-planar DIG originates from ionising
photons leaking from HII regions, it is possible these phases would share
kinematics. However, observations of H$\alpha$ in low-redshift galaxies
indicate that HII regions have $\sigma_{\rm H\alpha}\sim
10-20$${\,\rm{km\,s^{-1}}}$while DIG reaches upwards of $\sigma_{\rm
H\alpha}\sim 50$${\,\rm{km\,s^{-1}}}$ (Della Bruna et al., 2020; den Brok et
al., 2020; Law et al., 2022). The distinct kinematic separation between these
phases may lead to a spread in the observed $\sigma_{\rm H\alpha}$ in
galaxies, as the H$\alpha$ profile might be dominated either by HII regions or
DIG. Disentangling the impact of these environments requires either highly
resolved analysis of the gas properties or detailed profile-fitting of the
different components that make up the H$\alpha$ line.

Galactic outflows are a natural part of disc galaxies and are multi-phase in
nature, with two key drivers suggested in literature: stellar feedback
(primarily supernova) and radio jets from active galactic nuclei (see Heckman
& Best, 2014; Somerville & Davé, 2015, for a recent review). The majority of
extraplanar ionised gas is believed to come from extended DIG (eDIG) a few kpc
from the disc (e.g. Rand, 1996; Rossa et al., 2004; Haffner et al., 2009; Ho
et al., 2016). This gas exists either as a semi-static corona (perpetually
heated by ionising photons from massive stars in the disc or runaway stars) or
as part of ongoing galactic outflows (Zurita et al., 2002; Haffner et al.,
2009; Barnes et al., 2015; Dirks et al., 2023). This suggests that the DIG, at
least partially, inhabits an environment untouched by HII regions and, thus,
contributes the majority of extraplanar H$\alpha$ emission.

Numerical simulations that capture the multi-phase nature of the ISM confirm
that the warm, ionised gas is more turbulent than the colder, neutral or
molecular gas phases (Orr et al., 2020; Ejdetjärn et al., 2022; Rathjen et
al., 2023; Kohandel et al., 2023). To accurately capture the ionised gas
structure in galaxies, in particular the dense HII regions, very high spatial
resolution with an explicit treatment of radiative transfer effects is
required (see e.g. Deng et al., 2023). Recent radiation hydrodynamic (RHD)
simulations of disc galaxies environments (entire discs or ISM patches) have
investigated the H$\alpha$ emission (e.g. Katz et al., 2019; Tacchella et al.,
2022; Kohandel et al., 2023), with a few RHD simulations investigating the
nature of DIG and/or HII regions (Vandenbroucke & Wood, 2019; Tacchella et
al., 2022). Notably, Tacchella et al. (2022) simulated an isolated Milky Way-
like galaxy and performed mock observations of the H$\alpha$ emission with
post-processing ray tracing. They decomposed the contribution of DIG and HII
regions to the H$\alpha$ luminosity, finding an agreement with observations
$f_{\rm DIG}\sim 50\,\%$. However, no RHD simulation has yet to investigate
the kinematic offset between the two gas phases, as has been observed with the
H$\alpha$ line.

In this paper, we study how H$\alpha$ emission from DIG and HII regions
changes as we go from Milky Way-like galaxies, to more gas-rich, rapidly star
forming galaxies typical of high redshift ($z\sim 1-3$) conditions. Using
hydrodynamical simulations of entire disc galaxies, with an explicit treatment
of multi-frequency radiative transfer, we study to what degree
extraplanar/outflowing gas, as opposed to the ISM, contributes to H$\alpha$
line profiles in such galaxies.

This paper is organised as follows. In Section 2, we outline the setup of our
models and the calculation of relevant parameters, such as the velocity
dispersion $\sigma_{\rm g}$. In Section 3, we visualise the simulations and
create mock H$\alpha$ line profiles to highlight strong H$\alpha$ emitting
regions within the disc and outflows. Furthermore, we show how the turbulence
of the H$\alpha$ line changes between environments. In Section 4, we quantify
contribution of outflows to the H$\alpha$ mock line profiles and resulting
turbulence. Finally, in Section 5 we summarise our conclusions.

##  2 Numerical method

The simulations presented in this work were made with the radiation
hydrodynamics (RHD) code RAMSES-RT (Rosdahl et al., 2013; Rosdahl & Teyssier,
2015), which is a module of the $N$-body and hydrodynamical Adaptive Mesh
Refinement (AMR) code RAMSES (Teyssier, 2002). The code uses an HLLC Riemann
solver (Toro et al., 1994) to solve the Euler fluid equations for the
dynamical interaction of gas with dark matter and stellar populations, using a
second-order Godunov scheme, assuming an ideal mono-atomic gas with adiabatic
index $\gamma=5/3$. Gas interacts through gravity, hydrodynamics, radiation
transfer (RT), and non-equilibrium cooling/heating. Dark matter and stars are
represented as collisionless particles, their dynamics solved with a particle-
mesh solver and cloud-in-cell interpolation (Guillet & Teyssier, 2011).

In the coming sections, we outline the most relevant of the physics routines
within our simulations, but refer to Agertz et al. (2013) for a more detailed
description of the feedback routines, and Rosdahl et al. (2013, see also ) for
the RT routines (see also Agertz et al., 2020, for details on how RT is
applied in our simulations).

Table 1: Properties of the simulated galaxies used in this work. These values are calculated 200 Myr after the last refinement step of each galaxy (see Section 2.3). The H$\alpha$ properties are calculated as described in Section 2.4. Name of run |  $f_{\rm g}$ (%) | SFR (${\rm M_{\odot}\,yr^{-1}}$) | L${}_{\rm H\alpha}\ [10^{41}$ erg s-1 cm-3] |  $\sigma_{\rm H\alpha}$ [km s-1] |  $\sigma_{\rm H\alpha,\,DIG}$ [km s-1] |  $\sigma_{\rm H\alpha,\,HII\,regions}$ [km s-1]
---|---|---|---|---|---|---
fg10 | 9 | 2 | $4.5\pm 1$ | $15\pm 2$ | $43\pm 4$ | $10\pm 1$
fg30 | 33 | 39 | $45\pm 5$ | $35\pm 5$ | $54\pm 5$ | $29\pm 3$

###  2.1 Star formation and feedback physics

In our simulations, star formation events are treated as stochastic events
with star particles of masses $10^{3}\,{\rm M_{\odot}}$. Each particle
represents a single-age stellar population with a Chabrier (2003) initial mass
function. The number of star particles formed are sampled from a discrete
Poisson distribution. Star formation follows a Schmidt law (Schmidt, 1959;
Kennicutt, 1998) of the form

$\displaystyle\dot{\rho}_{*}=\epsilon_{\rm ff}\frac{\rho_{\rm g}}{t_{\rm
ff}},{\rm\ for\ }\rho_{\rm g}>\rho_{\rm*}.$ (1)

The gas density threshold for star formation is chosen to be
$\rho_{\rm*}=100\,m{\rm{}_{H}\,cm^{-3}}$ (above which most of the gas is
molecular, see Gnedin et al., 2009, and references therein). The free-fall
time is $t_{\rm ff}=\sqrt{3\pi/32G\rho}$ and the star formation efficiency per
free-fall time is adopted as $\epsilon_{\rm ff}=10\,\%$. This choice of
$\epsilon_{\rm ff}$ has been shown to reproduce the observed low efficiency
$\epsilon_{\rm ff}\sim 1\,\%$ inside giant molecular clouds (Krumholz & Tan,
2007) and averaged over kpc scales (Bigiel et al., 2008), when paired with a
realistic stellar feedback treatment (Agertz & Kravtsov, 2016; Grisdale et
al., 2018, 2019).

We track the injection of momentum, energy, and metals of supernovae Type Ia
and Type II, and stellar winds through subgrid recipes (see Agertz et al.,
2013; Agertz & Kravtsov, 2015), including the technique of Kim & Ostriker
(2015) for capturing the terminal momentum injected by individual supernova
explosions (for implementation details, see Agertz et al., 2021). The
radiation feedback is self-consistently treated by the RT extension, see
Section 2.2. Metal abundances are advected as scalars with the gas, for which
we track iron and oxygen separately. These metallicities are combined,
assuming a relative solar abundance (Asplund et al., 2009), into a total
metallicity, which is used for cooling, heating, and RT.

###  2.2 Radiative transfer

Stellar radiation is handled with moment-based radiative transfer using the M1
closure approach for the Eddington tensor (Levermore, 1984). This allows for
RT to be described as a set of conservation laws and radiation can then be
diffused via the AMR grid. Intercell flux is calculated with the Global Lax-
Friedrichs function (see Rosdahl et al., 2013). In order to avoid long
computational times due to light propagation, we adopt a reduced speed of
light approximation, $\bar{c}=0.01\,c$ (Rosdahl et al., 2013). The spectral
energy distributions for stellar particles are taken from Bruzual & Charlot
(2003), assuming single stellar population. Radiation is emitted in photon
groups with a set range of frequencies, chosen to represent crucial physical
processes. We adopt six photon groups that represent the infrared, optical,
ionising radiation of HI, HeI, HeII, and photo-disassociation of molecular gas
(see Agertz et al., 2020, for details).

The non-equilibrium chemistry and radiative cooling is solved for five ion
species, neutral & ionised hydrogen and helium, and we track their fractions
($x_{\rm H\small I}$, $x_{\rm H\small II}$, $x_{\rm He\small I}$, $x_{\rm
He\small II}$, $x_{\rm He\small III}$) along with the photon fluxes in every
cell. For gas $T>10^{4}$ K, metal cooling uses tabulated cooling rates from
CLOUDY (Ferland et al., 1998) and $T\leq 10^{4}$ K gas use the fine structure
cooling rates from Rosen & Bregman (1995). Molecular hydrogen H2 is modelled
following Nickerson et al. (2019) and accounts for formation, advection,
destruction, and cooling. Self-shielding of the gas against ionising radiation
is treated self-consistently by the RT module.

###  2.3 Simulation suite

The simulation data used in our analysis are isolated disc galaxies
constructed to be Milky Way-like galaxies, in terms of their mass and size, at
different redshifts. By changing the gas fraction between simulations, each
galaxy represents a Milky Way-like galaxy at a different stage of its
evolution. The initial conditions used here are the same as in Ejdetjärn et
al. (2022, hereafter Paper I) and are based on the isolated discs part of the
AGORA project (Kim et al., 2014, 2016). Briefly, the dark matter halo follows
a NFW-profile (Navarro et al., 1996) with virial mass $M_{200}=1.1\cdot
10^{12}\,{\rm M_{\odot}}$ within $R_{200}=205$ kpc, and a dark matter
concentration of $c=10$. The baryonic disc has a mass of $M_{\rm d}=4.5\cdot
10^{10}\,{\rm M_{\odot}}$ and is represented with an exponential profile with
scale length $r_{\rm d}=3.4$ kpc and scale height $h=0.1\,r_{\rm d}$. The
stellar bulge has a Hernquist-profile (Hernquist, 1990) with scale radius
$r_{\rm b}=0.1\,r_{\rm d}$. The dark matter halo and stellar disc are
represented by $10^{6}$ particles and the bulge by $10^{5}$ particles. The
galaxy is surrounded by a circumgalactic medium with density
$\rho=10^{-5}\,{\rm m_{H}\,cm^{-3}}$ and with an ultraviolet background
following Faucher-Giguère et al. (2009)

We model the galaxy at different redshifts by changing the gas fraction
$f_{\rm g}=M_{\rm g}/(M_{\rm g}+M_{*})$ of the galaxy, keeping the total mass
the same. The fg30 galaxy corresponds to the Milky Way at $z\sim 1.5$ (based
on SFR matching for Milky Way progenitor galaxies van Dokkum et al., 2013),
with initial gas fraction $f_{\rm g}=50\%$. The fg10 galaxy is the Milky Way
at $z\sim 0$ and started with $f_{\rm g}=10\%$. The galaxies were initialised
at coarser spatial resolutions for 100-200 Myr, to allow the gas and particles
to reach an equilibrium, and subsequently refined to a maximum spatial
resolution $\Delta x=6$ pc. We analyse the galaxies 150-200 Myr after the last
refinement step. The time between simulation outputs is 5 Myr and we analyse
10 outputs from each gas fraction. Thus, the calculated properties are time-
averaged over 50 Myr unless otherwise stated.

Figure 1: Colour diagrams of gas density-temperature inside the simulated
galaxies. Each pixel represents all of the cells with those particular
densities and temperatures. Left plots show the H$\alpha$ emissivity weighted-
mean of each cell while the right plots are coloured after the volume of cells
in each pixel. Figure 2: Projection maps of the gas and stellar properties
within the simulated galaxies. Each map is projected both face- and edge-on.

###  2.4 Calculating H$\alpha$ kinematic properties

In this section we detail the calculations of various properties for the gas
traced by the H$\alpha$ emission line. We calculate the H$\alpha$ signal
directly from the gas in each cell by applying analytical equations of the
recombination and collisional rate of the H$\alpha$ emissivity $\epsilon_{\rm
H\alpha}$ in each simulation cell. The details of how these equations are
derived and calculated is described in Ejdetjärn et al. (2022, see their
Appendix C)111We note that by using the energy of the H$\alpha$ transition in
their Eq. C2, it is indirectly and incorrectly assumed that the majority of HI
is constantly in the excited 2s state, rather than emitting back to the ground
state. This can be easily corrected for by switching $hv$ in this equation
with $13.6\,{\rm eV}(1^{2}-\ell_{\rm max}^{-2})=12.1\,{\rm eV}$. As
collisional excitation contributes around 10-20% to the total H$\alpha$ signal
($\leq 10\%$ in the case of Tacchella et al., 2022), this does not affect the
results in Paper I.. This approach does not account for absorption and
scattering of the H$\alpha$ line on dust. Thus, the H$\alpha$ emission we
analyse here is the inherent (or, equivalently, the optically thin/absorption-
corrected) emission from the gas.

However, even if we assume our approach is comparable to observations of dust-
corrected H$\alpha$ emission, the exact effects of dust-scattering on the
H$\alpha$ emission line is complex. In the RHD simulation by Tacchella et al.
(2022), the authors performed post-processing of the H$\alpha$ emission and
showed that scattering allows 20-40% of their H$\alpha$ emission to escape
dust-obscured regions, which increases the H$\alpha$ contribution from dusty
HII regions and the relative H$\alpha$ fraction between DIG and HII regions.
Furthermore, they show that scattering boosts the H$\alpha$ signal in
extraplanar DIG gas. We discuss the potential impact of not including
H$\alpha$ dust scattering on our results in Section 4.1 and 4.2.

In order to adequately reproduce the H$\alpha$ signal, the spatial resolution
has to be high enough to capture the smallest HII regions. Deng et al. (2023)
showed how unresolved HII regions in RHD simulations warm up the surrounding
gas and lead to an unrealistic mix of cold and warm gas, resulting in a
partially ionised and warm gas ($\sim$8 000 K) which is then overproducing
H$\alpha$-bright gas, causing an over-representation of dense HII regions to
the total H$\alpha$ signal. In Appendix A, we carry out a post-process
analysis of how well resolved, in terms of the Strömgren radius (${\rm
R_{SG}}$), photoionised gas is, and investigate the impact of resolution
requirements in HII regions. We find that unresolved, hence unrealistic HII
regions can be filtered away by requiring ${\rm R_{SG}}$ to be resolved by at
least 10 cells (in line with the criterion identified by Deng et al., 2023),
which we only enforce in cells with recent star formation, to avoid filtering
less H$\alpha$ bright, non-HII regions. Cells which fit this criterion have
their $\epsilon_{\rm H\alpha}$ reduced to the mean value of resolved HII
regions in our simulations, see Appendix A for details.

After filtering the unresolved HII regions, we achieve a good match for the
global H$\alpha$ luminosity ($L_{\rm H\alpha}$, see Table 1) compared to what
is predicted from their SFR; by adopting the $L_{\rm H\alpha}$-SFR relation

$\displaystyle L_{\rm H\alpha}\,[{\rm erg\,s^{-1}}]=1.26\cdot
10^{41}\,\times\,{\rm SFR\,[M_{\odot}\,yr^{-1}]}$ (2)

from Kennicutt (1998). The predicted values for fg10 and fg30 are $2.5\cdot
10^{41}\ {\rm erg\,s^{-1}cm^{-3}}$ and $5\cdot 10^{42}\ {\rm
erg\,s^{-1}cm^{-3}}$, respectively.

In order to illustrate how our H$\alpha$ emissivity calculations perform, and
to highlight the gas environment in which the H$\alpha$ signal is the
strongest, we show in Figure 1 temperature-density diagrams for each
simulation, using one output 200 Myr after the last refinement (see Section
2.4). The left plots are coloured by the H$\alpha$ emissivity and the right
plots show the total volume. The left plot can be thought of as a
visualisation of the weighting mask that is applied to the data when we
calculate H$\alpha$ variables. The strongest H$\alpha$ signal coincides with
the temperature- and density-range expected inside HII regions ($T\sim
10\,000$ K, $10{\rm\,m_{H}\,cm^{-3}}\leq\rho\leq 1\,000)\,{\rm
m_{H}\,cm^{-3}}$). The diffuse gas ($T\sim 8\,000$ K, $\rho\lesssim
0.1\,{\rm\,m_{H}\,cm^{-3}}$) has notably smaller $\epsilon_{\rm H\alpha}$ but
fill a much larger volume and so it still has a significant contribution to
the total LHα.

The H$\alpha$ properties were calculated by dividing the galaxy into a square
grid with 1 x 1 kpc2 patches (see e.g. Figure 4). The gas velocity dispersion
in H$\alpha$, $\sigma_{\rm H\alpha}$, was calculated as the H$\alpha$
weighted-mean of all simulation cells within each grid patch. We computed mock
line profiles of the H$\alpha$ signal within these grids by binning the gas
velocity, from $-200$${\,\rm{km\,s^{-1}}}$to $+200$${\,\rm{km\,s^{-1}}}$, and
summing up the total H$\alpha$ luminosity in each velocity bin.

The variables presented in Table 1 are time-averaged values for the entire
galaxy. We calculated $\sigma_{\rm H\alpha}$ by first calculating $\sigma_{\rm
H\alpha}$ within each grid patch and then computing an H$\alpha$ weighted-mean
value of these values, as is common to do in observations when the galaxy is
resolved. The grid size adopted is 1 kpc, unless otherwise stated, and we
briefly explore the impact of grid size on $\sigma_{\rm H\alpha}$ in Section
4.1.

Figure 3: Projection maps, edge- and face-on, of the H$\alpha$ emissivity for
the fg30 simulation 200 Myr after the lat refinement step (see Section 2.4).
The blue lines in each grid patch is the mock line profile, with velocity
range $-200$${\,\rm{km\,s^{-1}}}$ to $+200$${\,\rm{km\,s^{-1}}}$, of the
H$\alpha$ emission viewed along the $z$-direction. The line centre (0 km s-1)
is annotated with a weak white line in each grid. The edge-on projection shows
clear blue/red-shifted profiles above the gas disc, which is indicative of
accelerating outflows.

##  3 Results

In Figure 2 we show the gas density, gas temperature, and stellar mass density
maps for our simulations. A significant difference between the gas fractions
is their gas and stellar structure. The lower gas fraction has its cold, dense
gas (and therefore also star formation) confined within spiral arms while the
higher gas fraction shows more clumpy gas and stellar structure, as found in
other theoretical work on gas rich galaxies (e.g. Agertz et al., 2009;
Bournaud et al., 2009; Renaud et al., 2021) and observed in high redshift disc
galaxies (e.g. Elmegreen et al., 2007; Genzel et al., 2011; Zanella et al.,
2019). The maps also reveal that the higher gas fraction galaxy has a more
prominent vertical structure, with a high covering fraction of warm gas
($10^{4}~{}{\rm K}<T<10^{5}~{}{\rm K}$) several kiloparsecs above the disc
plane, and significant amounts of outflows due to stellar feedback processes
(see e.g. Veilleux et al., 2005, for a review). In the next section, we look
at the H$\alpha$ bright gas and show, again, how hot, ionised, gas is ejected
from the disc. In Section 4.2, we investigate in more detail the gas vertical
disc structure and outflows.

Figure 4: The H$\alpha$ surface density maps, viewed from three angles, of the
fg10 (left) and fg30 (right) simulations with mock H$\alpha$ emission line
profiles plotted as blue lines on top of them. The line profiles in all
projection maps are along the vertical, $z$-direction, integrated along line-
of-sight of the grid patches. The total H$\alpha$ profile of the simulations
is seen in the lower right (black line), which we also decompose into several
components: extra-planar gas, DIG, and HII regions. The dashed lines are
profiles normalised to the same max value as the total line. See Section 2.4
for how these mock profiles are calculated. Figure 5: Contours of the relation
between the H$\alpha$ velocity dispersion and the H$\alpha$ luminosity. The
values were calculated within 1 kpc grids, as described in Section 2.4, and
the contours show 90 % and 68 % of the data. The crosses are the LHα-weighted
mean. The gas is divided into diffuse ($\rho_{\rm g}\leq 0.1\,{\rm cm^{-3}}$)
and dense gas ($\rho_{\rm g}\geq 10\,{\rm cm^{-3}}$), which creates two
distinct region in this parameter space; highlighted with a dashed line. This
indicates a clear separation in H$\alpha$ luminosity and kinematics between
(diffuse) DIG and (dense) HII regions. The histograms show the distribution of
the diffuse and dense gas, with line-histograms dividing it into the two
simulations.

###  3.1 The H$\alpha$ signal within the galaxy

In this section, we present mock H$\alpha$ line profiles (see Section 2.4 for
details) for different spatial locations, inside and outside of the disc, to
better understand the origins of profile shapes. In Figure 3 we show the fg30
simulation at 250 Myr projected face- and edge-on ($z$ and $x$ projection)
with a 2 x 2 kpc2 grid of line-profiles normalised to the same maximum line
peak. Both projections show the line profiles in the vertical $z$ direction
for all gas within that grid. In the face-on projection (left), there is no
clear spatial correlation with the line-width of the profile and the H$\alpha$
bright regions. This is likely due to the grid being too coarse and the
projection plots presented next will show a higher spatial resolution. The
edge-on projection (right), shows that extraplanar line profiles are much
broader than those inside the disc. Furthermore, extraplanar line profiles
become increasingly blue- and red-shifted with increasing distance away from
the disc, which is due to the accelerating nature of the outflows. This
highlights that even if the outflows are not contributing significantly to the
total LHα (as we argue below), they shape the wings of the observable profile.

The non-normalised line profiles inside of the fg10 and fg30 galaxies can be
seen in Figure 4. In each figure, the galaxy is projected along three
different axes, but the line profiles shown for the edge-on projections are
all along the $z$-axis, i.e. as if observed face-on. This method allows us to
quantify the contribution to the face-on H$\alpha$ line profile from gas at
different heights and environments within the galaxy. Each figure also contain
the total H$\alpha$ line profile of the galaxy as a thick black line, as well
as divided into specific components: the extraplanar gas ($|z|>0.5$ kpc), HII
regions, and the DIG. The dashed lines are the profiles of the component
corresponding to that colour, but normalised to the peak value of the total
H$\alpha$ profile. The face-on projections show that regions with the highest
levels of $\Sigma_{\rm H\alpha}$ coincide with strong emission lines, as
expected. The side-projections reveal that the total line profile is
completely dominated by emission from warm gas residing in the midplane of the
disc, even for the gas rich galaxy which features vigorous outflows. We
further quantify the H$\alpha$ emission and line-width variation with height
in Section 4.2 and discuss the strength of the outflows in our simulations.

The line profile decomposition of each galaxy further highlights that gas
outside of the disc $|z|>0.5$ kpc provide an insignificant portion of the
total H$\alpha$ line profile shape. However, it does have a broader
distribution, as can be seen from the normalised (dashed) line, and thus has
notably higher $\sigma_{\rm H\alpha}$. Furthermore, in the high gas fraction
galaxy, the extraplanar H$\alpha$ profile is stronger, which is expected from
a thicker gas disc or more massive gas outflows; although, it is also more
narrow due to the disc thickness. We quantify this turbulence increase at
higher vertical distance in Section 4.2. Additionally, as the extraplanar gas
only contributes a small fraction to the total H$\alpha$ profile, a
significant amount of the H$\alpha$ bright, high-velocity gas resides within
the disc. This could be explained by the scenario in which gas gets
accelerated to high velocities within the disc (e.g. due to supernovae), but
has yet to be ejected.

Furthermore, the HII profile follows the total H$\alpha$ profile very closely,
meaning that HII regions constitutes the majority of H$\alpha$ emission. Thus,
the DIG profile, in both galaxies, is rather weak and contribute only roughly
$f_{\rm DIG}\leq 10\%$ of the total LHα. We discuss this result in comparison
with other observational and numerical work in Section 4.1. However, from the
normalised DIG line it is clear that the H$\alpha$ emission line from the DIG
is much broader. In the coming section, we separate the gas into DIG and HII
regions and quantify the H$\alpha$ line profiles in terms of their LHα and
velocity dispersion $\sigma_{\rm H\alpha}$, as a proxy for line-width.

###  3.2 The turbulent nature of H$\alpha$ emitting gas

Observations of the ionised gas with the H$\alpha$ line have shown that disc
galaxies are highly turbulent systems (see discussion and references in
Section 1). However, it is not clear how gas with different physical states
and local environments contribute to the observed broad line profiles. We
investigate the multi-phase nature of the H$\alpha$ line by dividing the gas
into dense ($\rho_{\rm g}\geq 10\,{\rm cm^{-3}}$) and diffuse gas ($\rho_{\rm
g}<0.1\,{\rm cm^{-3}}$), to represent HII regions and the DIG, respectively.
In Figure 5, we plot $\sigma_{\rm H\alpha}$ as a function of LHα in each grid
patch of our two simulations. The plot is divided into two distinct regions
based on their density, which is highlighted with a dashed line. Additionally,
the figure shows histograms of the LHα and $\sigma_{\rm H\alpha}$ distribution
for the diffuse and dense gas. The diffuse gas is able to reach velocity
dispersions as high as $\sigma_{\rm H\alpha}$ $\lesssim
80$${\,\rm{km\,s^{-1}}}$within individual patches and has a mean value of
$\sigma_{\rm H\alpha}$ $\sim 50$${\,\rm{km\,s^{-1}}}$. The denser, more
luminous, gas is less turbulent with $\sigma_{\rm H\alpha}\lesssim
40$${\,\rm{km\,s^{-1}}}$and an average value of $\sigma_{\rm H\alpha}\sim
29$${\,\rm{km\,s^{-1}}}$. There is also an evolution with increased gas
fraction, with the gas rich galaxy ($f_{\rm g}=33\%$) being more luminous and
turbulent than the gas poor case ($f_{\rm g}=9\%$) in _both_ the diffuse and
dense ionised gas.

As this figure illustrates, the H$\alpha$ emission originates from, at least,
two different gas phases with distinct LHα and $\sigma_{\rm H\alpha}$.
Separating the gas into these phases is important for interpreting the
H$\alpha$ emission line in observational studies. In Section 4.1, we go into
more detail about how $\sigma_{\rm H\alpha}$, for the diffuse and dense
ionised gas, varies in different density-temperature environments.

##  4 Discussion

In this Section, we quantify the amount of outflows, using the mass loading
factor, and explore the effect of outflows on the total H$\alpha$ line
profile. We investigate in more detail the variation in $\sigma_{\rm H\alpha}$
in regions with diverse temperatures and densities.

Figure 6: Temperature-density phase diagrams coloured by the H$\alpha$
velocity dispersion in our simulations. The kinematics (but not the
temperature or density) were calculated by separating the H$\alpha$ gas into
diffuse gas (DIG) and dense gas (HII regions). The temperature $\langle
T\rangle_{\rm H\alpha}$ and density $\langle\rho\rangle_{\rm H\alpha}$
presented in this figure are $\epsilon_{\rm H\alpha}$-weighted averages of all
the simulation cells within each grid.

###  4.1 The phase structure of turbulent H$\alpha$ emitting gas

The contribution of H$\alpha$ emission from different gas phases can be
understood from the left plots of Figure 1, which indicate that the majority
of the H$\alpha$ emission comes from a temperature range around $T\sim
10^{3}-10^{5}$ K, and a wide range of gas densities $\rho\sim
10^{-2}-10^{3}\,{\rm cm^{-3}}$. In Section 3.2 we divided the data into
diffuse and dense gas to represent the DIG and HII regions, respectively, and
highlighted how the dense ionised gas is less turbulent than the diffuse gas.
While the dense HII regions are overall more luminous they also suspend a
small volume of the ISM and might thus not represent the overall ionised gas
kinematics. Here, we explore how $\sigma_{\rm H\alpha}$varies across these
wide temperature and density ranges.

In order to investigate the physical state and turbulence levels of H$\alpha$
emitting gas, we show the gas velocity dispersions, binned in temperature-
density phase space, in Figure 6. The temperatures and densities presented in
this figure is the $\epsilon_{\rm H\alpha}$-weighted averaged of all the
simulations cells in each grid patch. Notably, this means that the (average)
gas density is shifted to higher densities compared to the phase diagram of
individual simulation cells in Figure 1. We calculate $\sigma_{\rm H\alpha}$
for the diffuse and dense gas separately, but in the same environment; i.e.
the temperatures and densities represent mean values of _all_ the gas, but
$\sigma_{\rm H\alpha}$ is calculated only for the diffuse or dense gas
($\rho\leq 0.1\,{\rm cm^{-3}}$ and $\rho\geq 10\,{\rm cm^{-3}}$,
respectively).

The dense gas exhibits turbulence levels of $\sigma_{\rm H\alpha}\sim
10-20$${\,\rm{km\,s^{-1}}}$, except for in very high density patches in the
high gas fraction galaxy, in which it can reach up to $\sigma_{\rm
H\alpha}\lesssim 40$${\,\rm{km\,s^{-1}}}$. These high-density patches likely
contain H$\alpha$ luminous HII regions, which boost $\sigma_{\rm H\alpha}$,
and are confined in small regions. On the other hand, the diffuse gas is
ubiquitous throughout (and outside) the disc and is highly turbulent even in
the dense environments, i.e. around the HII regions. The diffuse gas in the
high gas fraction galaxy has $\sigma_{\rm H\alpha}\sim
50-60$${\,\rm{km\,s^{-1}}}$in most environments and, in the more gas rich
galaxy, can reach $\sigma_{\rm H\alpha}\lesssim 80$${\,\rm{km\,s^{-1}}}$in
patches with hotter, and more tenuous, gas. Furthermore, the DIG has been
observed to be warmer ($T\sim 8\,000$ K) than HII regions ($T\sim 10\,000$ K),
which can be seen in this figure. Notably, $\sigma_{\rm H\alpha}$ is slightly
larger at the DIG temperatures and lower densities. This highlights the
distinct kinematic offset between these two gas phases.

The spatial proximity between the DIG and HII region gas phases, as is seen
from observations (e.g. Zurita et al., 2002), is beyond the investigations of
this project, but we do note that the strongest H$\alpha$ signals from the
diffuse gas are close to the star forming regions. Additionally, the DIG
turbulence near the densities of star forming regions ($\rho\sim
100-1000{\rm\,cm^{-3}}$) seem to show slightly higher $\sigma_{\rm H\alpha}$
than the DIG mean (see Figure 6). Furthermore, we check the effects of varying
the patch size from which we calculate each parameter. Notably, when using
larger patch sizes $\sigma_{\rm H\alpha}$ becomes more skewed towards the
lower value associated with HII regions as these dense regions dominate the
H$\alpha$ luminosity. If the contribution of LHα from DIG regions was higher
in our galaxies, $\sigma_{\rm H\alpha}$ might instead increase at lower
spatial resolutions.

In Figure 5, there is a clear correlation between LHα and $\sigma_{\rm
H\alpha}$. These are the most luminous HII regions, which reach upwards of
$\sigma_{\rm H\alpha}$ $\approx 40$${\,\rm{km\,s^{-1}}}$, and are regions of
recent massive star formation. This connection between SFR and $\sigma_{\rm
g}$ is something we explored in detail in Paper I. The driver of global
turbulence within galactic disc is still under discussion (see e.g. Krumholz
et al., 2018; Ejdetjärn et al., 2022; Ginzburg et al., 2022, and references
therein), but it is clear from this figure that the turbulence traced with
H$\alpha$ can be strongly tied to these dense HII regions. This implies
observations of disc galaxies similar in mass and age to those we present
here, with $\sigma_{\rm H\alpha}$ $\gtrsim 40$${\,\rm{km\,s^{-1}}}$require a
greater LHα contribution from the DIG. Indeed, a similar cut-off has been
found for $\sigma_{\rm H\alpha}$ in DIG and HII regions in large-scale surveys
(Law et al., 2022). As HII regions are shrouded within molecular clouds, these
dense gas phases likely feature similar gas kinematics. Indeed, the offset in
$\sigma_{\rm H\alpha}$ between DIG and HII regions in our simulations is
reminiscent of the offsets in turbulence levels reported by Girard et al.
(2021) for dense molecular gas (as traced by CO) and H$\alpha$ traced ionised
gas. We leave the investigation of the relative levels of turbulence between
ionised and molecular gas phases to future work (Ejdetjärn et al., in prep).

As discussed in Section 2.4, our approach to calculate H$\alpha$ emission
assumes a perfect dust-correction. Accounting for dust absorption in HII
regions in our simulations would reduce the amount of H$\alpha$ emission from
HII regions and favour the contribution from the DIG, resulting in a boost in
the relative DIG/HII region contribution. This, in turn, would shift total
$\sigma_{\rm H\alpha}$ towards higher values (i.e. closer to $\sigma_{\rm
H\alpha}$ of the DIG). Tacchella et al. (2022) find a 3-5 times higher DIG
contribution $f_{\rm DIG}$ than our work, which could possibly be explained by
their post-process method of propagating the H$\alpha$ emission through the
ISM, which takes into account absorption and scattering of the line. As post-
processing would require notably more analysis, we leave dust scattering for
future work.

Figure 7: The H$\alpha$ luminosity and velocity dispersion as a function of
the height from the disc. The values are time-averaged from several simulation
outputs with a 1$\sigma$ errorbar. A sensitivity limit, based on the
capabilities of MUSE/JWST within the local/high-redshift universe (comparable
to the low/high gas fraction galaxy), is plotted as a dashed/dotted grey line
(see the details of how these are determined in Section 4.2).

###  4.2 Is the H$\alpha$ signal coming from outflows or the disc?

In Section 3.1 we showed that outflows only have a minor contribution to the
total H$\alpha$ line profile. In Figure 7 we quantify this by calculating the
average LHα surface density ($\Sigma_{\rm H\alpha}$) and $\sigma_{\rm
H\alpha}$ at different heights. There is a clear, steady decline in the
H$\alpha$ brightness for both simulation, which is steeper for the lower gas
fraction disc. On the other hand, there is a steady increase in $\sigma_{\rm
H\alpha}$ as one moves further from the disc, which can be attributed to fast-
speed outflows from the disc with a broad velocity distribution. In
particular, fg10 has a sharp increase in $\sigma_{\rm H\alpha}$ just above the
disc $\sim$1 kpc, which is due to the disc being thin so outflows dominate the
H$\alpha$ gas close to the disc (compared to fg30 with a more extended gas
disc height). Due to the low average H$\alpha$ surface brightness above the
disc, the H$\alpha$ gas in fg10 would not be observable beyond 1.5-2 kpc if
observed with MUSE (assuming an integration time of a couple of hours and
galaxy-wide homogeneous slabs with 1 kpc height). This applies to a object in
the local universe, where a gas fraction of $\sim 10\%$ is appropriate.

However, the extraplanar H$\alpha$ luminosity is dominated by few bright
patches (from bright filaments), and such bright individual 1 kpc2 patches
would be detectable to larger heights when present. The higher gas fraction
galaxy is much brighter but is assumed to be further away, as it is modelled
to represent the same type of galaxy at a redshift of $z\sim 1-2$. Using JWST
NIRSPEC/IFU for $\sim$10 hours would allow the high-redshift galaxy to be
observable in slabs to heights 2-3 kpc, but again the brightest individual
kpc2 sized cells would be detectable to larger heights. Hence, the predictions
of our models are in principle testable with deep VLT or JWST observations.

The mass loading factor $\eta=\dot{M}/{\rm SFR}$, which quantifies the amount
of gas mass flowing out from the galaxy relative to its star formation rate,
is presented in Figure 8 as a function of height from the disc. Our high gas
fraction simulation has $\eta\sim 0.5-1$ at a height of a few kpc.
Observations of local galaxies with similar stellar masses, $M_{*}\sim
10^{10}$, are in agreement with ours but there is a large spread $\eta\sim
0.01-10$ (Chisholm et al., 2017, 2018; Schroetter et al., 2019). On the other
hand, our low gas fraction galaxy, equivalent to a local disc, has $\eta\sim
0.1-0.2$ and is thus on the lower limit. Importantly, our galaxies have
outflows that are within the expectations from observations and still the
amount of H$\alpha$ emission in the extraplanar gas is low. This confirms that
our simulations have realistic outflows and that the weak extraplanar
H$\alpha$ emission is indeed a property of these types of disc galaxies.

We note that our simulations are not cosmological, meaning that any gas above
the disc originated either from disc outflows or from the circumgalactic
medium that was initiated with the galaxy. It is possible that inflows or
minor mergers could contribute to strip more gas from the disc to these
heights, to increase the disc thickness, or allow for more shocked, warm, gas.
More massive or gas-rich disc galaxies with more bursty SFRs could possibly
have more volatile outflows, resulting in higher $\eta$ and H$\alpha$ above
the disc. This could then strengthen the H$\alpha$ signal at these heights. As
we are not using post-processing to trace emission line propagation, the
H$\alpha$ line is unaffected by dust absorption and scattering. Extraplanar
H$\alpha$ emission from dust scattering has been found to contribute $\sim
5-20\%$ (Wood & Reynolds, 1999; Barnes et al., 2015) of the total luminosity.
This was also numerically shown by (Tacchella et al., 2022), who found that
dust scattering contributed 10-30% of their H$\alpha$ emission above the disc.
Furthermore, our models do not include magnetic fields or black hole feedback
which could help drive outflows and increase the H$\alpha$ emission and
turbulence outside of the disc (and in the DIG).

Figure 8: The mass loading factor as a function of the height from the disc.
The values are averaged from several simulation outputs with a 1$\sigma$
errorbar.

##  5 Conclusions

The H$\alpha$ emission line is a versatile observational tool for deriving
various properties in both low- and high-redshift galaxies. However, H$\alpha$
is emitted by both dense and diffuse gas, which contribute differently to the
H$\alpha$ emission line profile. Furthermore, H$\alpha$ bright diffuse ionised
gas is ubiquitous inside and around the galactic disc. Untangling the various
contributions to the H$\alpha$ line is important for interpreting the
underlying, complex physical processes.

In this paper, we present radiation hydrodynamic simulations of isolated disc
galaxies in order to evaluate the strength and kinematics of H$\alpha$
emitting gas in outflows, diffuse disc gas region, and dense gas regions. We
model Milky Way-like galaxies, in terms of mass and size, with different gas
fractions ($f_{\rm g}=9\%$ and 33%) as a proxy of the galaxy at different
epochs of its evolution. We carry out mock observation of the H$\alpha$
emission line, which, after careful numerical treatment of unresolved HII
regions (Appendix A), match empirical LHα-SFR relations (Kennicutt, 1998).

The key findings in this work are summarised below:

  1. 1.

We highlight a distinct difference in the H$\alpha$ luminosity and kinematic
signature between the diffuse and dense gas emitting strongly in H$\alpha$, as
is also found between observations of DIG and dense HII region gas (see e.g.
Law et al., 2022, and references therein). In the gas rich galaxy ($f_{\rm
g}=33\%$), the dense gas reaches at most $\sigma_{\rm H\alpha}$ $\lesssim
40$${\,\rm{km\,s^{-1}}}$, with a mean value of 29${\,\rm{km\,s^{-1}}}$, while
the diffuse gas reaches $\sigma_{\rm H\alpha}\lesssim 80\,{\rm km\,s^{-1}}$,
with a mean value of 54${\,\rm{km\,s^{-1}}}$.

  2. 2.

The H$\alpha$ emission line is broader in the younger, i.e. higher gas
fraction, galaxy. The low gas fraction galaxy has an H$\alpha$ emission line
(ionised gas) roughly half as broad (turbulent) when looking at the total
H$\alpha$ emission. This is also true for H$\alpha$ emission from dense, HII
regions. However, the H$\alpha$ emission in the DIG has a more shallow
evolution.

  3. 3.

The contribution of the diffuse gas to LHα is approximated to be $\lesssim
10\,\%$ for H$\alpha$ emitting gas below $\rho_{\rm H}\lesssim 0.1\,{\rm
cm^{-3}}$. This matches with the lower-limits found by observations of the DIG
(see the discussion in Section 1). However, several studies suggests the DIG
can constitute the majority of LHαand numerical work with similar setup to
ours, by Tacchella et al. (2022), agrees with this high DIG fraction. The
major difference with our setup is that we do not perform post-processing to
trace the H$\alpha$ emission line. Our study encourages further work into
post-processing of H$\alpha$ emission line propagation in RHD simulations, to
capture the impact of dust scattering and absorption on the observed H$\alpha$
line profile.

  4. 4.

Gas outside of the galactic disc contribute very little to the total H$\alpha$
signal, despite that the mass outflow in our high gas-fraction galaxy match
with observed values of galaxies with similar stellar mass. Thus, the overall
galaxy disc dynamics could be well-constrained with observations of H$\alpha$,
taking into consideration any kinematic offset between the DIG and HII regions
(see points above).

  5. 5.

The $\sigma_{\rm H\alpha}$ of gas that exists several kpc above the disc is
much more turbulent compared to the mid-plane, increasing by a factor of 2-4
depending on the simulation. However, the luminosity at these heights
corresponds to only a few percent of the total LHα, which has an H$\alpha$
surface density below the sensitivity limit of MUSE in local galaxies and JWST
at redshift $\sim 1.5$ (see Section 4.2). This suggests that the H$\alpha$
emission can be readily observed a few kpc above the disc and the predictions
of our simulations on the H$\alpha$ vertical structure can be tested.

## Acknowledgements

The computations and data storage were enabled by resources provided by LUNARC
- The Centre for Scientific and Technical Computing at Lund University. OA
acknowledges support from the Knut and Alice Wallenberg Foundation, and from
the Swedish Research Council (grant 2019-04659). MPR is supported by the
Beecroft Fellowship funded by Adrian Beecroft. FR acknowledges support
provided by the University of Strasbourg Institute for Advanced Study (USIAS),
within the French national programme Investment for the Future (Excellence
Initiative) IdEx-Unistra.

Simulation outputs were analysed using tools from the YT project (Turk et al.,
2011), numpy (Harris et al., 2020), and MATPLOTLIB for PYTHON (Hunter, 2007).

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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##  Appendix A The impact of unresolved HII regions on H$\alpha$ parameters

Deng et al. (2023) showed that to properly resolve the HII regions in
numerical simulations, the Strömgren radius needs to be resolved by 10-100
simulation cells. Not resolving the HII regions causes these cells to be
heated up by the ionising radiation from the bright stars within, which
results in an unrealistic gas mix of partially-ionised, warm ($T\sim 8\,000$
K), dense ($\rho\geq 100\,{\rm cm^{-3}}$) gas. These artificial conditions are
prime for H$\alpha$ emission and can contaminate analysis of H$\alpha$
quantities. Here we evaluate the impact of unresolved HII regions on the
H$\alpha$ emission and kinematics computed in this paper.

The size of an idealised HII region can approximated by the Strömgren radius.
The equation describes the size of a spherical ionised bubble formed as one or
more very young and luminous stars ionise the surrounding gas, and can be
written as

$\displaystyle R_{\rm SG}=\bigg{(}{\frac{3Q_{0}}{4\pi n_{\rm H}^{2}\alpha_{\rm
B}}}\bigg{)}^{1/3}.$ (3)

Here, $Q_{0}$ is the rate of ionising photons emitted by the stars, $n_{\rm
H}$ is the hydrogen density, and $\alpha_{\rm B}$ is the effective hydrogen
recombination rate, which can be described as a function of gas temperature

$\displaystyle\alpha_{\rm B}(T)=2.753\cdot
10^{-14}\,\bigg{(}\frac{315614}{T}\bigg{)}^{1.5}\Bigg{(}1+\bigg{(}\frac{315614}{T}\bigg{)}^{0.407}\Bigg{)}^{-2.42}.$
(4)

This expression is a fit made by Hui & Gnedin (1997), using observational data
from Ferland et al. (1992).

We implement a criteria for the number of resolved Strömgren radii ($R_{\rm
SG}/\Delta x$) in our simulations by reducing the H$\alpha$ emissivity to the
mean value of resolved HII regions in our galaxies ($\epsilon_{\rm
H\alpha}=2\cdot 10^{-20}{\rm erg\,s^{-1}\,cm^{-3}}$ for fg30 and
$\epsilon_{\rm H\alpha}=7\cdot 10^{-21}{\rm erg\,s^{-1}\,cm^{-3}}$ for fg10)
for the regions unresolved by a certain amount of cells. Figure 9 shows LHα
and $\sigma_{\rm H\alpha}$ as a function of how many cells $R_{\rm SG}$ is
resolved. As can be seen, there is a strong decrease in the H$\alpha$
luminosity at 1-10 cells resolved, as most of the highly luminous, and
unresolved, regions are filtered away. When $R_{\rm SG}/\Delta x\approx 10$
unresolved cells are filtered, the decrease in luminosity is much more
shallow, indicating that the most problematic cells have been corrected. The
expected luminosity from the LHα-SFR relation (Kennicutt, 1998) for these SFRs
is around $2.5\cdot 10^{41}{\rm erg\,s^{-1}\,cm^{-3}}$ and $5\cdot 10^{42}{\rm
erg\,s^{-1}\,cm^{-3}}$ for the low- and high-gas fraction simulations,
respectively. All of this indicates that by filtering cells that do not
resolve the Strömgren sphere with at least 10 cells, we correct for most of
the unresolved HII regions and achieve a much better match with LHα predicted.

Furthermore, the lower plot in this figure suggests that the H$\alpha$
velocity dispersion for the entire galaxy remains largely unaffected by this
filtering process. However, this $\sigma_{\rm H\alpha}$ is integrated over the
entire galaxy (comparable to observations of unresolved galaxies), rather than
computing $\sigma_{\rm H\alpha}$ in each grid patch and then calculating a
mean value from that, which is the common approach for resolved galaxy
observations. Therefore, it is not directly comparable to $\sigma_{\rm
H\alpha}$ we compute in the main body of the text, but, as discussed in
Section 4.1, $\sigma_{\rm H\alpha}$ remains largely unaffected by altering the
grid patch size.

Figure 9: The H$\alpha$ luminosity and velocity dispersion above different
criteria for resolving the Strömgren sphere $R_{\rm SG}$ with cells of size
$\Delta x$. ${\rm L_{H\alpha,\,0}}$ here is the un-filtered luminosity of each
simulations, which is $5.5\cdot 10^{41}\,{\rm erg\,s^{-1}}$ and $8.5\cdot
10^{42}\,{\rm erg\,s^{-1}}$ for fg10 and fg30, respectively.


# Massive to Massless
by
Applying a Nonlocal Field Redefinition

Mojtaba Najafizadeh School of Physics, Institute for Research in Fundamental
Sciences (IPM),
P.O.Box 19395-5531, Tehran, Iran  Department of Physics, Faculty of Science,
Ferdowsi University of Mashhad,
P.O.Box 1436, Mashhad, Iran
<EMAIL_ADDRESS>
###### Abstract

The invariance of physical observables like particle’s mass under a local
field redefinition is a well-known and important property of quantum field
theory. In this paper, on the other hand, we investigate nonlocal field
redefinitions eliminating a physical observable from a theory. Indeed, by
giving explicit examples, we present invertible nonlocal field redefinitions
reducing a massive theory to the massless one, and vice versa. The studied
examples are: classical harmonic oscillator, the scalar field theory, and the
Dirac field theory.

###### keywords:

Nonlocal field redefinition, Harmonic oscillator, Scalar field, Dirac field
Preprint code:IPM/P-2023/07

##  1 Introduction

Some of coupling parameters in any action principle can be removed by a
redefinition of the fields, which are known as the redundant parameters [1].
On the other hand, physical observables like particle masses and scattering
amplitudes are independent of field redefinitions which is a well-known and
important property of quantum field theory. In order to clarify the issue, let
us follow Weinberg’s approach [1] and consider the action of a scalar field
theory in the form

$S=\int
d^{4}x\left[-\tfrac{\,1\,}{2}\,Z\,(\partial^{\mu}\mathrm{\Phi}\,\partial_{\mu}\mathrm{\Phi}+m^{2}\mathrm{\Phi}^{2})-\tfrac{1}{4!}\,g\,Z^{2}\mathrm{\Phi}^{4}\right].$
(1)

The constant $Z$ which is called a field renormalization constant is a
redundant coupling because $\delta S\scalebox{0.9}{$/$}\delta Z$ vanishes when
we use the field equation. In other words, the constant $Z$ can be removed by
redefining the field,
$\mathrm{\Phi}\rightarrow(\scalebox{0.85}{${1}/{\sqrt{Z}}$})\,\mathrm{\Phi}$,
while neither the scalar particle mass $m$ nor the coupling $g$ can be
eliminated by a field redefinition. Indeed, in this example, the field
redefinition is continuous and local, but what happens if one applies a
nonlocal field redefinition?

In this paper, we observe by applying nonlocal field redefinition a physical
parameter of a theory may be redundant as well. The physical parameter we aim
to make redundant is angular frequency of the simple harmonic oscillator, as
well as mass of the scalar field and the Dirac field. As we know, in the last
two examples, massive fields of spin-$0$ and spin-$\tfrac{1}{2}$ have the same
degrees of freedom as their corresponding massless fields, that is, in four
dimensions, 1 for the massive  $/$ massless real scalar field and 8 for the
massive  $/$ massless Dirac field. However, except these two fields, other
arbitrary fields of spin $s\,(\,\geqslant 1)$ have different degrees of
freedom depending on being massive or massless111 See e.g. [2] for degrees of
freedom of a massive  $/$ massless arbitrary spin-$s$ field in any spacetime
dimension.. Let us refer to these cases in the conclusion.

Therefore, the present work is devoted to investigating a set of field
redefinitions subject to nonlocality. The layout of the paper is as follows.
In Sec. 2, we study the classical harmonic oscillator which simplifies the
calculations in the next two sections. In Secs. 3 and 4, we investigate the
nonlocality of field redefinitions for the scalar field theory and the Dirac
field theory respectively. We conclude and present open problems in Sec. 5.

We will work in the“mostly plus” signature for the metric and concentrate on
$d$-dimensional Minkowski spacetime. Natural units, $c=1=\hbar$, and shorthand
notation $\partial_{\mu}\mathrel{\mathop{\mathchar
58\relax}}=\partial\scalebox{0.85}{$/$}\partial x^{\mu}$ (with
$\mu=0,1,\ldots,d-1$) will be used.

##  2 Harmonic Oscillator

Although our concentration may lie on the classical  $/$ quantum field
theories, it is beneficial to first study classical harmonic oscillator
shedding light on our nonlocal field redefinitions discussed in Secs. 3 and 4.
Thus, let us consider a Newtonian massive particle whose position only depends
on time and subject to the force law of the usual Hooke’s law. The equation of
motion reads

$\Big{(}\,\frac{d^{2}}{dt^{2}}\,+\,\omega^{2}\,\Big{)}\,\mathrm{X}(t)=0\,,$
(2)

where $\omega=\scalebox{0.9}{$\sqrt{{k}/{m}}$}\,,\,m\neq 0,$ is angular
frequency of a simple harmonic oscillator system. In comparison to the given
example in Sec. 1, here, the angular frequency is not a redundant parameter as
it is a physical observable and cannot be removed by a so-called local wave
function redefinition. Nevertheless, it may be redundant by performing a
nonlocal wave function redefinition. To see this redundancy, we introduce a
new wave function $x(t)$ through

$\displaystyle\mathrm{X}(t)=\bigg{(}\,\,\underset{n=0}{\overset{\infty}{\textstyle\sum}}\,(\,\omega
t\,)^{2n}\,~{}\frac{\scalebox{0.85}{$(-1)$}^{n}}{4^{n}~{}n!~{}(t\,\tfrac{d}{dt}+\tfrac{1}{2})_{n}}\,\,\bigg{)}\,x(t)\,,$
(3)

where $(\cdots)_{n}$ in the denominator stands for the rising Pochhammer
symbol [3] with the argument $a$ of any $n\in\mathbb{N}$

$(a)_{n}=a(a+1)\cdots(a+n-1)\,.$ (4)

By plugging (3) into the equation of motion (2), and doing some calculations
explained below, we shall arrive at

$\Big{(}\,\frac{d^{2}}{dt^{2}}\,\Big{)}\,x(t)=0\,.$ (5)

Multiplying the latter by $m$, $m\neq 0$, it is nothing but a free Newtonian
massive particle subject to no force. As a consequence, the nonlocal wave
function redefinition (3) could drop the physical observable, that is angular
frequency, from the original equation (2). It is remarkable to mention that
relation (3) can be also expressed in terms of the Bessel function of the
first kind (see e.g. [4]). However, we shall keep the above form dealing with
the Pochhammer symbol properties which makes for easy calculations, as we will
see later.

We have found that the above procedure can be invertible. In fact, this time,
we consider the free Newtonian massive particle equation (5) as our starting
point. After that, we introduce a new wave function $\mathrm{X}(t)$ via

$\displaystyle
x(t)=\bigg{(}\,\,\underset{n=0}{\overset{\infty}{\textstyle\sum}}~{}\frac{\scalebox{0.85}{$1$}}{~{}4^{n}\,n!\,(t\,\tfrac{d}{dt}-n-\tfrac{1}{2})_{n}}~{}(\,\omega
t\,)^{2n}\,\bigg{)}\,\mathrm{X}(t)\,.$ (6)

By plugging the latter into (5) and performing some calculations we can reach
the simple harmonic oscillator equation (2). This means as the nonlocal wave
function redefinition could remove the angular frequency from the system,
invertibly, we could restore it to the system by an inverse operation.
Therefore, the angular frequency as a physical observable could be redundant
upon applying a nonlocal wave function redefinition.

Let us now go into the details of the above calculations. First of all, we
note that the appeared operators in (2)-(6), which are ${d^{2}}/{dt^{2}}$,
$t^{2}$ and $t\,{d}/{dt}$, can be packed into the following generators

$\ell_{0}\mathrel{\mathop{\mathchar
58\relax}}=\tfrac{1}{2}\,(t\,\tfrac{d}{dt}+\tfrac{1}{2})\,,\quad\quad\quad\ell_{1}\mathrel{\mathop{\mathchar
58\relax}}=-\,\tfrac{1}{2}\,\tfrac{d^{2}}{dt^{2}}\,,\quad\quad\quad\ell_{-1}\mathrel{\mathop{\mathchar
58\relax}}=-\,\tfrac{1}{2}\,t^{2}\,.$

These generators satisfy the $sl(2,R)$ algebra

$[\,\ell_{m}\,,\,\ell_{n}\,]=(m-n)\,\ell_{m+n}\,,\qquad m,n=0,\pm 1\,.$ (7)

It is then convenient to find the following commutation relation for any
$n\in\mathbb{N}$

$\Big{[}\,\,\frac{d^{2}}{dt^{2}}\,,\,t^{2n}\,\,\Big{]}=4n\,(a-n+1)\,t^{2n-2}\,,$
(8)

where

$a\mathrel{\mathop{\mathchar 58\relax}}=t\,\frac{d}{dt}\,+\frac{1}{2}\,.$ (9)

At this moment, let us define two nonlocal dimensionless (
$[\,t\,]=\mathrm{T}=[\omega]^{-1}$) operators

$\displaystyle\mathrm{P}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\underset{n=0}{\overset{\infty}{\textstyle\sum}}\,(\,\omega
t\,)^{2n}\,~{}\frac{\scalebox{0.85}{$(-1)$}^{n}}{4^{n}~{}n!~{}(a)_{n}}\,,$
(10) $\displaystyle\mathrm{Q}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\underset{n=0}{\overset{\infty}{\textstyle\sum}}\,(\,\omega
t\,)^{2n}\,~{}\frac{\scalebox{0.85}{$(-1)$}^{n}}{4^{n}~{}n!~{}(a+2)_{n}}\,,$
(11)

where $(a)_{n}$ and $a$ were respectively introduced in (4), (9). We recall
that the former operator (10) appeared in (3). At this stage, using the
commutation relation (8), and the property of the Pochhammer symbol [3]

$(a)_{n}=(a+n-1)\,(a)_{n-1}\,,$ (12)

it is straightforward to examine that two operators, $\mathrm{P}$ and
$\mathrm{Q}$, will satisfy the following identity

$\Big{(}\,\frac{d^{2}}{dt^{2}}\,+\,\omega^{2}\,\Big{)}\,\mathrm{P}=\mathrm{Q}\,\Big{(}\,\frac{d^{2}}{dt^{2}}\,\Big{)}\,.$
(13)

As a result, by applying the nonlocal wave function redefinition (3), or
equivalently $\mathrm{X}=\mathrm{P}\,x$, and then using the identity (13), we
immediately realize that Eq. (2) can be reduced to (5), i.e.

$\Big{(}\,\frac{d^{2}}{dt^{2}}\,+\,\omega^{2}\,\Big{)}\mathrm{X}=0~{}~{}~{}~{}\rightarrow~{}~{}~{}~{}\Big{(}\,\frac{d^{2}}{dt^{2}}\,+\,\omega^{2}\,\Big{)}\mathrm{P}\,x=0~{}~{}~{}~{}\rightarrow~{}~{}~{}~{}\mathrm{Q}\,\Big{(}\,\frac{d^{2}}{dt^{2}}\,\Big{)}x=0\,.$

Let us end this section by discovering inverse operators. This enables us to
follow the reverse procedure applied above. One way to find the inverse
operators is as follows. Using the fact that
$\mathrm{P}\mathrm{P}^{-1}=1=\mathrm{Q}^{-1}\mathrm{Q}$, we can multiply the
identity (13) by $\mathrm{Q}^{-1}$ to the left and by $\mathrm{P}^{-1}$ to the
right which results in

$\mathrm{Q}^{-1}\Big{(}\,\frac{d^{2}}{dt^{2}}\,+\,\omega^{2}\,\Big{)}=\Big{(}\,\frac{d^{2}}{dt^{2}}\,\Big{)}\mathrm{P}^{-1}\,.$
(14)

Here we ask ourselves if it is possible to figure out inverse operators,
$\mathrm{P}^{-1}$ and $\mathrm{Q}^{-1}$, such that satisfy (14)? The answer is
positive. In fact, if we introduce

$\displaystyle\mathrm{P}^{-1}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\underset{n=0}{\overset{\infty}{\textstyle\sum}}~{}\frac{\scalebox{0.85}{$1$}}{~{}4^{n}~{}n!~{}(a-n-1)_{n}}~{}(\,\omega
t\,)^{2n}\,,$ (15) $\displaystyle\mathrm{Q}^{-1}$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\underset{n=0}{\overset{\infty}{\textstyle\sum}}~{}\frac{\scalebox{0.85}{$1$}}{~{}4^{n}~{}n!~{}(a-n+1)_{n}}~{}(\,\omega
t\,)^{2n}\,,$ (16)

the so-called “inverse identity” (14) will be then satisfied. To check this,
it is sufficient to use (8) and (12). We note that the inverse operator in
(15) is the one that became visible in (6). Therefore, in a reverse way, the
Eq. (5) could be reduced to (2), upon taking into account the nonlocal wave
function redefinition (6), corresponding to $x=\mathrm{P}^{-1}~{}\mathrm{X}$,
and applying the inverse identity (14). Inspired by the harmonic oscillator,
let us begin studying the classical  $/$ quantum field theory in the next two
sections.

##  3 Scalar field theory

We take into account a real scalar field theory in $d$-dimensional Minkowski
spacetime given by the action

$S=\tfrac{1}{2}\int
d^{d}x\,\,\mathrm{\Phi}\,\big{(}\,\partial^{\mu}\partial_{\mu}-m^{2}\,\big{)}\,\mathrm{\Phi}\,,$
(17)

where $m$ is scalar field’s mass and $\mu=0,1,\ldots,d-1$. We note, to
simplify calculations, an integration by part is used by which the standard
field derivative term,
$-\,\partial^{\mu}\mathrm{\Phi}\,\partial_{\mu}\mathrm{\Phi}$
($\,=\mathrm{\Phi}\,\partial^{\mu}\partial_{\mu}\mathrm{\Phi}\,+\,\hbox{total
derivative}$), has been rewritten as the above form. As already mentioned, the
mass parameter in the action (17) is not redundant and does not eliminate by a
local field redefinition [1]. Nevertheless, we can make it redundant if we
introduce a nonlocal field redefinition, which is

$\displaystyle\mathrm{\Phi}=\bigg{(}\,\underset{n=0}{\overset{\infty}{\textstyle\sum}}\,(\,mx\,)^{2n}\,~{}\frac{\scalebox{0.85}{$(-1)$}^{n}}{4^{n}\,n!~{}\big{(}x^{\mu}\,\partial_{\mu}+\tfrac{d}{2}\big{)}_{n}}\,\bigg{)}\,\varphi\,.$
(18)

Indeed, by plugging the latter into (17), the massive scalar field action (17)
reduces to (details of the calculations are given below)

$\mathcal{S}=\tfrac{1}{2}\int
d^{d}x\,\,\varphi\,\partial^{\mu}\partial_{\mu}\,\varphi\,,$ (19)

which is a massless scalar field action. We notice the action (19) is
conformal invariant and the scaling dimension of $d$-dimensional scalar field
is $\mathrm{\Delta}=(d-2)/2$.

Invertibly, we discover the massless scalar field action (19) can be reduced
to the massive one (17) by applying the following nonlocal field redefinition

$\displaystyle\varphi=\bigg{(}\,\underset{n=0}{\overset{\infty}{\textstyle\sum}}~{}\frac{\scalebox{0.85}{$1$}}{~{}4^{n}\,n!~{}\big{(}x^{\mu}\partial_{\mu}-n+\tfrac{d-2}{2}\big{)}_{n}}~{}(mx)^{2n}\,\bigg{)}\,\mathrm{\Phi}\,.$
(20)

Here, we note that the operator $x^{\mu}\partial_{\mu}$ in the denominator,
denoting by $\mathrm{D}$, is nothing but the dilatation (rigid scaling)
operator. Hence, the argument of the Pochhammer symbol in (20) can be taken
into account as $\mathrm{D}-n+\mathrm{\Delta}$. As we will see later, we
expect a similar structure for the Dirac field as well.

In this place, we give useful relations enabling the reader to simply check
the details of calculations. To this end, we first obtain the following
commutation relation, for any $n\in\mathbb{N}$, in any spacetime dimension $d$

$\left[\,\,\partial^{\mu}\partial_{\mu}\,,\,x^{2n}\,\,\right]=4n\,(h-n+1)\,x^{2n-2}\,,$
(21)

where

$h\mathrel{\mathop{\mathchar
58\relax}}=x^{\mu}\,\partial_{\mu}+\frac{d}{2}\,.$ (22)

After that, we define dimensionless ( $[m]=\mathrm{M}=[x]^{-1}$) operators in
$d$ spacetime dimensions

$\displaystyle\mathbf{P}\mathrel{\mathop{\mathchar
58\relax}}=\underset{n=0}{\overset{\infty}{\textstyle\sum}}\,(\,mx\,)^{2n}\,~{}\frac{\scalebox{0.85}{$(-1)$}^{n}}{4^{n}\,n!~{}(h)_{n}}\,,$
(23) $\displaystyle\mathbf{Q}\mathrel{\mathop{\mathchar
58\relax}}=\underset{n=0}{\overset{\infty}{\textstyle\sum}}\,(\,mx\,)^{2n}\,~{}\frac{\scalebox{0.85}{$(-1)$}^{n}}{4^{n}\,n!~{}(h+2)_{n}}\,,$
(24) $\displaystyle\mathbf{P}^{-1}\mathrel{\mathop{\mathchar
58\relax}}=\underset{n=0}{\overset{\infty}{\textstyle\sum}}\,~{}\frac{\scalebox{0.85}{$1$}}{4^{n}\,n!~{}(h-n-1)_{n}}~{}(\,mx\,)^{2n}\,,$
(25) $\displaystyle\mathbf{Q}^{-1}\mathrel{\mathop{\mathchar
58\relax}}=\underset{n=0}{\overset{\infty}{\textstyle\sum}}\,~{}\frac{\scalebox{0.85}{$1$}}{4^{n}\,n!~{}(h-n+1)_{n}}~{}(\,mx\,)^{2n}\,,$
(26)

where the Pochhammer symbol $(h)_{n}$ and $h$ were introduced in (4), (22). We
note that the operators $\mathbf{P}$ and $\mathbf{P}^{-1}$ are those employed
in (18) ($\mathrm{\Phi}=\mathbf{P}\,\varphi$) and (20)
($\varphi=\mathbf{P}^{-1}\,\mathrm{\Phi}$) respectively. Now, using the
commutation relation (21) and the Pochhammer symbol property (12), we can
conveniently demonstrate that the set of operators (23)-(26) will satisfy the
following identities

$\displaystyle\big{(}\,\partial^{\mu}\partial_{\mu}-m^{2}\,\big{)}\,\mathbf{P}$
$\displaystyle=\mathbf{Q}\,\big{(}\,\partial^{\mu}\partial_{\mu}\,\big{)}\,,$
(27)
$\displaystyle\mathbf{Q}^{-1}\,\big{(}\,\partial^{\mu}\partial_{\mu}-m^{2}\,\big{)}$
$\displaystyle=\big{(}\,\partial^{\mu}\partial_{\mu}\,\big{)}\,\mathbf{P}^{-1}\,.$
(28)

Furthermore, since we argue at the level of the action, Hermitian conjugation
of the above operators is needed to calculate. Therefore, we introduce the
Hermitian conjugation rules

$(x^{\mu})^{\dagger}=x^{\mu}\,,\qquad(\partial_{\mu})^{\dagger}=-\,\partial_{\mu}\,,$
(29)

under which, e.g., $h^{\dagger}=-h$. Using (29), we can perform some
calculations and realize that the Hermitian conjugation of (23) and (24)
results in

$\mathbf{P}^{\dagger}=\mathbf{Q}^{-1}\,,\qquad\mathbf{Q}^{\dagger}=\mathbf{P}^{-1}\,.$
(30)

These results could be also obtained without any explicit calculations from
(27), (28). In fact, by taking the Hermitian conjugation of (27), and its
comparison with (28), the relations in (30) will be acquired directly.

Using the above facilities, one can conveniently reduce the massive action to
the massless one and vice versa. For instance, applying the field redefinition
(18), which is corresponding to
$\mathrm{\Phi}=\mathbf{P}\,\varphi=\varphi\,\mathbf{P}^{\dagger}$, the action
(17) reduces to (19)

$\displaystyle S=\tfrac{1}{2}\int
d^{d}x\,\,\varphi\,\mathbf{P}^{\dagger}\big{(}\,\partial^{\mu}\partial_{\mu}-m^{2}\,\big{)}\,\mathbf{P}\,\varphi=\tfrac{1}{2}\int
d^{d}x\,\,\varphi\\!\underbrace{\,\mathbf{P}^{\dagger}\,\mathbf{Q}\,}_{=\,1}\big{(}\,\partial^{\mu}\partial_{\mu}\,\big{)}\,\varphi=\mathcal{S}\,,$

in which relations (27), (30) have been used.

##  4 Dirac field theory

In $d$-dimensional flat spacetime, the massive Dirac field is given by the
action

$S=-\int
d^{d}x\,\bar{\mathrm{\Psi}}\big{(}\,\gamma\cdot\partial+m\,\big{)}\mathrm{\Psi}$
(31)

where the dot stands for the scalar product, $\gamma^{\mu}$ are
$d$-dimensional Dirac gamma matrices, and
$\bar{\mathrm{\Psi}}\mathrel{\mathop{\mathchar
58\relax}}={\mathrm{\Psi}}^{\dagger}\,i\,\gamma^{0}$ is the Dirac adjoint.
Similar to the scalar field, here, we intend to illustrate that the physical
mass parameter $m$ can be redundant, by applying a nonlocal field
redefinition. For this purpose, we introduce a new field $\psi$ by

$\displaystyle\mathrm{\Psi}=\bigg{(}\,\underset{k=0}{\overset{\infty}{\textstyle\sum}}\,\Big{[}(m\,\gamma\cdot
x)^{2k}-2k\,(m\,\gamma\cdot
x)^{2k-1}\Big{]}\,\frac{\scalebox{0.85}{$1$}}{~{}4^{k}~{}k!~{}\big{(}x^{\mu}\,\partial_{\mu}+\tfrac{d}{2}\big{)}_{k}~{}}\bigg{)}\,\psi\,.$
(32)

Plugging the latter into the massive action (31), and doing some calculations
explained below, the mass parameter will be eliminated so as we shall arrive
at

$\mathcal{S}=-\int d^{d}x~{}\bar{\psi}~{}\gamma\cdot\partial~{}{\psi}\,,$ (33)

that is the Dirac action for the massless field $\psi$.

As a matter of fact, we realized that this method can be invertible as well.
Indeed, if we introduce the nonlocal field redefinition

$\displaystyle\psi=\bigg{(}\,\underset{k=0}{\overset{\infty}{\textstyle\sum}}\,\frac{\scalebox{0.85}{$(-1)$}^{k}}{4^{k}\,k!\,\big{(}x^{\mu}\,\partial_{\mu}-k+\tfrac{d}{2}\big{)}_{k}~{}}\Big{[}(m\,\gamma\cdot
x)^{2k}-2k\,(m\,\gamma\cdot x)^{2k-1}\Big{]}\bigg{)}\,\mathrm{\Psi},$ (34)

then the mass parameter can be restored to the theory, such that this time by
plugging (34) into the massless action (33), we can arrive back to the massive
action (31). We note that, similar to the scalar field redefinition (20),
here, the argument of the Pochhammer symbol in (34) can be considered as
$\mathrm{D}-k+{\varDelta}+\tfrac{1}{2}$, where $\varDelta=(d-1)/2$ is the
scaling dimension of $d$-dimensional Dirac spinor field.

The step-by-step details of the calculations are very similar to the case of
the scalar field. However, here, we have the Clifford algebra
$\\{\gamma^{\mu},\gamma^{\nu}\\}=2\eta^{\mu\nu}$ that using it one can simply
find $\\{\gamma\cdot\partial,\gamma\cdot x\\}=2(x\cdot\partial)+d$ and
$[\gamma\cdot\partial,(\gamma\cdot x)^{2}]=2(\gamma\cdot x)$. In the same way,
one can find a generalization of odd and even powers of $\gamma\cdot x$.
Therefore, we acquire commutation ($[a,b]=ab-ba$) and anticommutation
($\\{a,b\\}=ab+ba$) relations, in any spacetime dimension $d$, for any
$k\in\mathbb{N}$

$\displaystyle\left[\,\gamma\cdot\partial~{},~{}(\gamma\cdot
x)^{\,2k}\,\right]$ $\displaystyle=$ $\displaystyle 2k\,(\gamma\cdot
x)^{\,2k-1}\,,$ (35)
$\displaystyle\left\\{\gamma\cdot\partial\,,\,(\gamma\cdot
x)^{\,2k-1}\right\\}$ $\displaystyle=$ $\displaystyle
2\,(h-k+1)~{}(\gamma\cdot x)^{\,2k-2}\,,$ (36)

where $h$ defined in (22). After that, we introduce dimensionless operators in
$d$ spacetime dimensions

$\displaystyle\mathbb{P}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\underset{k=0}{\overset{\infty}{\textstyle\sum}}\,\Big{[}(m\,\gamma\cdot
x)^{2k}-2k\,(m\,\gamma\cdot
x)^{2k-1}\Big{]}\frac{\scalebox{0.85}{$1$}}{~{}4^{k}~{}k!~{}(h)_{k}~{}}\,,$
(37) $\displaystyle\mathbb{Q}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\underset{k=0}{\overset{\infty}{\textstyle\sum}}\,\Big{[}(m\,\gamma\cdot
x)^{2k}+2k\,(m\,\gamma\cdot
x)^{2k-1}\Big{]}\frac{\scalebox{0.85}{$1$}}{~{}4^{k}~{}k!~{}(h+1)_{k}~{}}\,,$
(38) $\displaystyle\mathbb{P}^{-1}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\underset{k=0}{\overset{\infty}{\textstyle\sum}}\,\frac{\scalebox{0.85}{$(-1)$}^{k}}{~{}4^{k}~{}k!~{}(h-k)_{k}~{}}\,\Big{[}(m\,\gamma\cdot
x)^{2k}-2k\,(m\,\gamma\cdot x)^{2k-1}\Big{]}\,,$ (39)
$\displaystyle\mathbb{Q}^{-1}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\underset{k=0}{\overset{\infty}{\textstyle\sum}}\,\frac{\scalebox{0.85}{$(-1)$}^{k}}{~{}4^{k}~{}k!~{}(h-k+1)_{k}~{}}\,\Big{[}(m\,\gamma\cdot
x)^{2k}+2k\,(m\,\gamma\cdot x)^{2k-1}\Big{]}\,,$ (40)

with $h$ introduced in (22). We notice that the operators $\mathbb{P}$ and
$\mathbb{P}^{-1}$ are those emerged in (32) ($\mathrm{\Psi}=\mathbb{P}\,\psi$)
and (34) ($\psi=\mathbb{P}^{-1}\,\mathrm{\Psi}$). Using the commutation (35)
and anti-commutation (36) relations as well as the property of the Pochhammer
symbol (12), it follows that the above operators satisfy the following
identities

$\displaystyle\big{(}\,\gamma\cdot\partial+m\,\big{)}\,\mathbb{P}$
$\displaystyle=\mathbb{Q}\,\big{(}\,\gamma\cdot\partial\,\big{)}\,,$ (41)
$\displaystyle\mathbb{Q}^{-1}\,\big{(}\,\gamma\cdot\partial+m\,\big{)}$
$\displaystyle=\big{(}\,\gamma\cdot\partial\,\big{)}\,\mathbb{P}^{-1}\,.$ (42)

To find the Hermitian conjugate of operators, we use (29) accompanied with

$(\gamma^{\mu})^{\dagger}=\gamma^{0}\gamma^{\mu}\gamma^{0}\,,\qquad(\gamma^{0})^{\dagger}=-\,\gamma^{0}\,,\qquad(\gamma^{0})^{2}=-1\,.$
(43)

As a consequence, the Hermitian conjugates of the operators (37)-(40) read

$\displaystyle\mathbb{P}^{\dagger}$
$\displaystyle=-\,\gamma^{0}\,\mathbb{Q}^{-1}\,\gamma^{0}\,,\qquad\qquad(\,\mathbb{P}^{-1})^{\dagger}=-\,\gamma^{0}\,\mathbb{Q}\,\gamma^{0}\,,$
(44) $\displaystyle\mathbb{Q}^{\dagger}$
$\displaystyle=-\,\gamma^{0}\,\mathbb{P}^{-1}\,\gamma^{0}\,,\qquad\qquad(\mathbb{Q}^{-1})^{\dagger}=-\,\gamma^{0}\,\mathbb{P}\,\gamma^{0}\,.$
(45)

At this stage, we are equipped to reduce each actions, (31) or (33), to each
other by applying an appropriate nonlocal field redefinition. For instance,
let us apply $\psi=\mathbb{P}^{-1}\,\mathrm{\Psi}$ (34) (which, using (44),
its Dirac adjoint becomes $\bar{\psi}=\bar{\mathrm{\Psi}}\,\mathbb{Q}$) into
the massless Dirac field action (33) arriving at (31). It reads

$\displaystyle\mathcal{S}$ $\displaystyle=-\int
d^{d}x~{}\bar{\mathrm{\Psi}}\,\mathbb{Q}~{}(\gamma\cdot\partial)~{}\mathbb{P}^{-1}\,\mathrm{\Psi}=-\int
d^{d}x~{}\bar{\mathrm{\Psi}}\,\underbrace{\mathbb{Q}\,\mathbb{Q}^{-1}}_{=\,1}(\gamma\cdot\partial+m)~{}\mathrm{\Psi}=S\,,$

in which relations (42) and (44) have been utilized.

##  5 Conclusions and Discussion

In the quantum field theory, redundant coupling parameters can be removed by
applying a local field redefinition, while physical observables like particle
masses and scattering amplitudes are independent of field redefinitions [1].
In this paper, we investigated the possibility if a physical mass parameter
can be redundant by applying a nonlocal field redefinition. For this purpose,
we presented three explicit examples equipped with invertible nonlocal wave
$/$ field redefinitions. In the case of the classical harmonic oscillator, the
angular frequency became redundant using (3), while it could be recaptured to
the system by (6). For the scalar field theory, by applying (18), the mass
parameter of the spin-$0$ field was removed from the theory, while it could be
reattached to the theory upon (20). In similar fashion, we introduced (32) and
its inverse (34) for the Dirac field theory, using them, the mass parameter of
the spin-$\tfrac{1}{2}$ spinor field could be eliminated and restored
respectively.

The present work opens several avenues for further research. For instance, as
already mentioned in the introduction, except spin-$0$ and spin-$\tfrac{1}{2}$
fields, other arbitrary fields of spin $s\,(\,\geqslant 1)$ have different
degrees of freedom depending on being massive or massless. For example, in $d$
dimensions, the spin-$1$ field has ($d-1$) massive degrees of freedom, while
it has ($d-2$) massless degrees of freedom. Thus, it is of interest to find
invertible operators (if any) making the mass parameter redundant in the Proca
action in which the degrees of freedom will change by mass redundancy. If such
operators exist, another example for future investigations is studying mass
redundancy in the massive Fierz-Pauli action [5] as well as massive higher
spin field theories222 In this case, if one finds inverse operators, then, the
argument of the Pochhammer symbol may be expected to be of the form
$\mathrm{D}-n+{\mathrm{\Delta}}+s$, where $s$ is spin of the field. (see e.g.
[6, 7]), in which it seems that something like the Stueckelberg mechanism
should play a role333 We thank Sylvester James Gates Jr. for bringing our
attention to this point..

Moreover, in this work, we concentrated on the mass redundancy in a free
theory. Further investigation can eliminate the mass parameter in an
interacting theory, for example in (1), for which the limit $g\rightarrow 0$
should recover our results. An even deeper study would remove the coupling
parameter in an interacting theory. For this case, one may refer to [8, 9]
including nonlocal (and nonlinear) transformations that map an interacting
theory to a free one, and where the fermions are needed to match the Jacobian
of the transformation (see also [10] and references therein for a recent
development).

Furthermore, our method may establish a duality between theories in flat and
curved spacetimes. To be more precise, the scalar field theory in anti-de
Sitter spacetime includes the cosmological constant playing a role like mass.
Thus, making use of our nonlocal transformations, one can simply drop the
cosmological constant from the field equation which this in turn yields the
equation of motion in flat spacetime. Therefore, as an application, a suitable
generalization of our nonlocal transformations may be beneficial to construct
a flat spacetime holographic correspondence [11].

In addition, it was shown invertible transformations of dynamical variables
can change the number of dynamical degrees of freedom [12]. In this respect,
it is attractive to compare the result with the present work and find a
relationship (if any). Besides, in the context of the Lorentz-violating
$b^{\mu}$ model, a nonlocal operator was found such that the redefined field
satisfies an equation of motion with a scaled coefficient $(1+\xi)b^{\mu}$
[13]. It would be nice to investigate whether this result can be achieved by
the approach in this work. Finally, since our nonlocal field redefinitions may
be understood as a kind of higher-derivative field redefinitions, thus, it is
appealing to study our method for theories of higher-derivative terms and also
for the nonlocal field theories (see, e.g., [14, 15] and references therein).

## Acknowledgments

We are grateful to Sylvester James Gates Jr. for useful correspondence and
suggestions during the work. The author acknowledges Abasalt Rostami for
fruitful discussions. We are also thankful to Mohammad Reza Garousi, Hermann
Nicolai, and Shahin Sheikh-Jabbari for reading the manuscript and providing
helpful comments, and to Hamid Reza Afshar for his support and encouragement.
We also thank Joseph Buchbinder for comments. This work is partially supported
by IPM funds.

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