# NEATH II: N2H+ as a tracer of imminent star formation in quiescent high-
density gas

F. D. Priestley1, P. C. Clark1, S. C. O. Glover2, S. E. Ragan1, O. Fehér1, L.
R. Prole3, R. S. Klessen2,4
1School of Physics and Astronomy, Cardiff University, Queen’s Buildings, The
Parade, Cardiff CF24 3AA, UK
2Universität Heidelberg, Zentrum für Astronomie, Institut für Theoretische
Astrophysik, Albert-Ueberle-Straße 2, D-69120 Heidelberg, Germany
3Centre for Astrophysics and Space Science Maynooth, Department of Theoretical
Physics, Maynooth University, W23 F2H6 Maynooth, Ireland
4Universität Heidelberg, Interdisziplinäres Zentrum für Wissenschaftliches
Rechnen, Im Neuenheimer Feld 205, D-69120 Heidelberg, Germany
Email<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Star formation activity in molecular clouds is often found to be correlated
with the amount of material above a column density threshold of $\sim
10^{22}\,{\rm cm}^{-2}$. Attempts to connect this column density threshold to
a volume density above which star formation can occur are limited by the fact
that the volume density of gas is difficult to reliably measure from
observations. We post-process hydrodynamical simulations of molecular clouds
with a time-dependent chemical network, and investigate the connection between
commonly-observed molecular species and star formation activity. We find that
many molecules widely assumed to specifically trace the dense, star-forming
component of molecular clouds (e.g. HCN, HCO+, CS) actually also exist in
substantial quantities in material only transiently enhanced in density, which
will eventually return to a more diffuse state without forming any stars. By
contrast, N2H+ only exists in detectable quantities above a volume density of
$10^{4}\,{\rm cm}^{-3}$, the point at which CO, which reacts destructively
with N2H+, begins to deplete out of the gas phase onto grain surfaces. This
density threshold for detectable quantities of N2H+ corresponds very closely
to the volume density at which gas becomes irreversibly gravitationally bound
in the simulations: the material traced by N2H+ never reverts to lower
densities, and quiescent regions of molecular clouds with visible N2H+
emission are destined to eventually form stars. The N2H+ line intensity is
likely to directly correlate with the star formation rate averaged over
timescales of around a Myr.

###### keywords:

astrochemistry – stars: formation – ISM: molecules – ISM: clouds

††pubyear: 2023††pagerange: NEATH II: N2H+ as a tracer of imminent star
formation in quiescent high-density gas–NEATH II: N2H+ as a tracer of imminent
star formation in quiescent high-density gas

##  1 Introduction

Stars form in molecular clouds, specifically in the densest regions (often
referred to as cores or clumps; Bergin & Tafalla 2007) where self-gravity is
able to overcome the various forces opposing collapse. Lada et al. (2010)
proposed that star formation only occurs in regions with K-band extinctions
$A_{\rm K}\gtrsim 0.8\,{\rm mag}$, based on the correlation between the mass
of material above this threshold and the star formation rate (as traced by the
number of protostars in the cloud). Studies using entirely independent methods
(e.g. Könyves et al., 2015) have found evidence of similar thresholds for star
formation activity, corresponding to a visual extinction of $A_{\rm V}\sim
8\,{\rm mag}$ or a column density of $N_{\rm H}\sim 10^{22}\,{\rm cm}^{-2}$,
but the significance of this value and the reasons for its existence remain
unclear.

One limitation of observational studies is that they are restricted to probing
the line-of-sight column density, while the volume density of the gas is
presumably more relevant to its evolution; the Jeans mass, for example, scales
with local gas properties ($M_{J}\propto\rho^{-1/2}\,T^{3/2}$; Klessen &
Glover 2016) rather than the column of material. Lada et al. (2010) argue that
their column density threshold corresponds to a volume density of $n_{\rm
H}\sim 10^{4}\,{\rm cm}^{-3}$, but this relies on the assumption of a specific
cloud geometry, and in reality one would expect a distribution of volume
densities along the line of sight, rather than a single value. Establishing
line-of-sight structure in molecular clouds is fraught with difficulties; see,
for example, the ongoing debate about whether the Musca cloud is a filament or
an edge-on sheet (Tritsis et al., 2022; Kaminsky et al., 2023).

A common approach to mitigate this issue is to assume that the high volume-
density gas associated with star formation is specifically traced by molecular
lines with high critical densities, most commonly the $J=1-0$ HCN transition.
However, the emission in these lines is typically dominated by material with
significantly lower density ($\sim 10^{3}\,{\rm cm}^{-3}$; Pety et al. 2017;
Kauffmann et al. 2017; Evans et al. 2020; Jones et al. 2023; Neumann et al.
2023) than the assumed threshold for star formation, making them unsuitable
for this purpose. The one exception appears to be the N2H+ $J=1-0$ line, which
is almost exclusively detected in regions with column densities above
$10^{22}\,{\rm cm}^{-2}$ (Kauffmann et al., 2017; Tafalla et al., 2021),
coinciding quite closely with the Lada et al. (2010) column density threshold
for star formation.

Another limitation is that observations can only capture a single moment of a
molecular cloud’s $\,{\rm Myr}$-scale lifetime. While previous levels of star
formation activity can be measured, such as by counting the number of
protostellar objects as in Lada et al. (2010), the potential of a cloud for
future star formation is not easily accessible. Many studies divide objects
between those which are gravitationally bound, and so presumably destined to
collapse, and unbound (and therefore transient) objects, but this distinction
relies on a number of quantities which are either difficult or impossible to
measure accurately (Enoch et al., 2008), and neglects the possibility of
continuing accretion of material (Rigby et al., 2021; Anderson et al., 2021).

In this paper, we post-process hydrodynamical simulations of molecular clouds
with a time-dependent chemical network, and investigate molecular tracers of
material which will form stars. We find that N2H+ only exists in substantial
quantities in gas which is irreversibly undergoing gravitational collapse,
making it a unique signature of future star formation activity in molecular
clouds.

##  2 Method

Table 1: Simulation collision velocities, total durations, durations of star formation (time after formation of the first sink particle), final mass ($M_{\rm sink}$) and number ($N_{\rm sink}$) of sink particles, the final mass of gas with density above $10^{3}$ ($M_{\rm mid}$) and $10^{4}\,{\rm cm}^{-3}$ ($M_{\rm dense}$), and the average star formation rate (defined as $M_{\rm sink}/t_{\rm SF}$). Model |  $v_{\rm col}$/$\,{\rm km\,s^{-1}}$ |  $t_{\rm end}$/$\,{\rm Myr}$ |  $t_{\rm SF}$/$\,{\rm Myr}$ |  $M_{\rm sink}$/$\,{\rm M}_{\odot}$ | $N_{\rm sink}$ |  $M_{\rm mid}$/$\,{\rm M}_{\odot}$ |  $M_{\rm dense}$/$\,{\rm M}_{\odot}$ |  $\left<{\rm SFR}\right>$/$\,{\rm M}_{\odot}\,{\rm Myr}^{-1}$
---|---|---|---|---|---|---|---|---
V3 | $3$ | $6.79$ | $1.22$ | $155$ | $86$ | $3341$ | $263$ | $127$
V7 | $7$ | $5.53$ | $1.70$ | $102$ | $105$ | $2986$ | $211$ | $60$
V10 | $10$ | $7.66$ | $1.13$ | $74$ | $35$ | $1610$ | $265$ | $65$
V14 | $14$ | $8.70$ | $-$ | $0$ | $0$ | $4$ | $0$ | $-$
Table 2: Elemental abundances used in the chemical modelling. Element | Abundance |  | Element | Abundance
---|---|---|---|---
C | $1.4\times 10^{-4}$ |  | S | $1.2\times 10^{-5}$
N | $7.6\times 10^{-5}$ |  | Si | $1.5\times 10^{-7}$
O | $3.2\times 10^{-4}$ |  | Mg | $1.4\times 10^{-7}$

Figure 1: Column densities of hydrogen nuclei (top row), CO (second row), HCN
(third row) and N2H+ (bottom row) for the four simulations. The initial bulk
velocities of the clouds are oriented along the horizontal axis. Note that all
post-processed tracer particles are located within a sphere of radius
$16.2\,{\rm pc}$, so the molecular column densities outside the projected
boundaries of this sphere are set to zero. Figure 2: Time evolution of the
mass of intermediate ($n_{\rm H}>10^{3}\,{\rm cm}^{-3}$; dotted lines) and
high ($n_{\rm H}>10^{4}\,{\rm cm}^{-3}$; solid lines) density gas, and
accreted sink mass (dashed lines), for the four simulations. Figure 3:
Distribution of tracer particles by peak density reached during the duration
of the simulation, and the density at the simulation end, for the V7 model.
The dashed black line shows the one-to-one relationship for tracer particles
which only reach their peak density at the end of the simulation.

Figure 4: Average peak gas-phase abundances of CO (left), HCN (centre) and
N2H+ (right), as a function of the peak density reached during the duration of
the simulation, and the density at the simulation end, for the V7 model. The
dashed black lines show the one-to-one relationship for tracer particles which
only reach their peak density at the end of the simulation.

Figure 5: Distribution of tracer particles by peak density reached during the
duration of the simulation, and the density at the simulation end, for the V3
(left), V10 (centre) and V14 (right) models. The dashed black lines show the
one-to-one relationship for tracer particles which only reach their peak
density at the end of the simulation.

Figure 6: Average peak gas-phase abundances of N2H+ as a function of the peak
density reached during the duration of the simulation, and the density at the
simulation end, for the V3 (left), V10 (centre) and V14 (right) models. The
dashed black lines show the one-to-one relationship for tracer particles which
only reach their peak density at the end of the simulation.

Figure 7: Average peak gas-phase abundances of NH3 (left), HCO+ (centre) and
CS (right), as a function of the peak density reached during the duration of
the simulation, and the density at the simulation end, for the V7 model. The
dashed black lines show the one-to-one relationship for tracer particles which
only reach their peak density at the end of the simulation.

Figure 8: Evolution of the gas density (left) and the CO (centre) and N2H+
(right) abundances, for three tracer particles with peak density
$>10^{4}\,{\rm cm}^{-3}$ and final density $<10^{3}\,{\rm cm}^{-3}$ from the
V10 model. Figure 9: Distribution of time spent above $10^{4}\,{\rm cm}^{-3}$
($t_{\rm dense}$) versus final density, for all tracer particles with a peak
density above $10^{4}\,{\rm cm}^{-3}$ in the V10 model. Boxes show the median
and 25th/75th percentiles, whiskers the 10th and 90th percentiles. The dashed
red line shows the freeze-out timescale at a density of $10^{4}\,{\rm
cm}^{-3}$ ($0.34\,{\rm Myr}$).

We perform simulations of colliding, initially-atomic clouds using arepo
(Springel, 2010), a moving-mesh magnetohydrodynamics (MHD) code. The
simulations include an extensive suite of processes affecting the
thermodynamics of the gas and dust (Glover & Mac Low, 2007; Glover & Clark,
2012), including an on-the-fly chemical network (a modified version of the
Gong et al. 2017 network, as described in Hunter et al. 2023) for the key
atomic and molecular coolants, with self-shielding from the external
ultraviolet (UV) radiation field calculated self-consistently (Clark et al.,
2012).

We initialise two spherical clouds of mass $10^{4}\,{\rm M}_{\odot}$ and
radius $19\,{\rm pc}$, for an initial volume density of $10\,{\rm cm}^{-3}$
and a gas (dust) temperature of $300\,{\rm K}$ ($15\,{\rm K}$), displaced
along the $x$-axis so that the clouds are initially just touching. The clouds
have a virialised turbulent velocity field with a root-mean-square value of
$0.95\,{\rm km\,s^{-1}}$, and a bulk motion along the $x$-axis $v_{x}=\pm
v_{\rm col}$ for the cloud displaced in the $\mp x$ direction; we investigate
four values of $v_{\rm col}$ between $3$ and $14\,{\rm km\,s^{-1}}$, giving
net collision velocities $2v_{\rm col}$ between $6$ and $28\,{\rm
km\,s^{-1}}$. Model properties are listed in Table 1. A $3\,{\rm\mu G}$
magnetic field is included parallel to the collision axis. Sink particles are
introduced following Tress et al. (2020) (see also Prole et al. 2022), with a
threshold density of $2\times 10^{-16}\,{\rm g\,cm}^{-3}$ and formation radius
of $9\times 10^{-4}\,{\rm pc}$.

We use Monte Carlo tracer particles (Genel et al., 2013) to record the
evolution of the physical properties of individual parcels of gas. We then
post-process the time-dependent chemical evolution of the tracers with the
NEATH framework111https://fpriestley.github.io/neath/ (Priestley et al.,
2023), using a modified version of the uclchem code (Holdship et al., 2017)
with the UMIST12 reaction network (McElroy et al., 2013). We assume elemental
abundances from Sembach et al. (2000) with the refractory elements depleted by
an additional factor of 100 (as is appropriate for the dense interstellar
medium; Lee et al. 1998; Jenkins 2009), listed in Table 2. The external UV
field is set to $1.7$ Habing units (Habing, 1968), and the cosmic ray
ionisation rate to $10^{-16}\,{\rm s^{-1}}$. These chemical parameters are
shared with the underlying arepo simulations.

We select $10^{5}$ tracer particles from each simulation to model chemically,
chosen from particles with a final position within $16.2\,{\rm pc}$ of the
simulation centre; this region includes virtually all of the dense gas in the
simulations. Particles are chosen to evenly sample final densities in the
range $10-10^{6}\,{\rm cm}^{-3}$, with the exception of the V14 simulation,
where the upper boundary is the maximum density present at the simulation end
point ($2000\,{\rm cm}^{-3}$). The limit of $10^{6}\,{\rm cm}^{-3}$ is
motivated by the $44\,{\rm kyr}$ time resolution of the simulations (i.e. the
interval between updates of tracer properties), which is insufficient to fully
resolve the freefall time of gas at higher densities. This leads to the
rapidly-evolving physical properties in collapsing regions not being captured
by the chemical modelling, making the resulting molecular abundances
unreliable. A time resolution of $44\,{\rm kyr}$ produces abundances which are
converged up to gas densities of $10^{5}\,{\rm cm}^{-3}$, and accurate to
within a factor of two up to $10^{6}\,{\rm cm}^{-3}$ (Priestley et al., 2023),
which is more than sufficient for our purposes.

##  3 Results

###  3.1 Physical properties

Table 1 lists the mass and number of sinks formed in each of the four
simulations, and the mass of gas above volume density thresholds of $10^{3}$
and $10^{4}\,{\rm cm}^{-3}$ at the final time $t_{\rm end}$. We choose $t_{\rm
end}$ so that the V3, V7 and V10 simulations have all formed a similar
quantity of dense ($n_{\rm H}>10^{4}\,{\rm cm}^{-3}$) gas; the V14 simulation
is evolved to the point where it appears to be dispersing, as it never forms
any substantial amount of dense gas.

Figure 1 shows the simulation column densities, viewed perpendicular to the
collision axis at $t_{\rm end}$. The V3 and V7 models have both formed a
compressed layer at the interface between the two clouds, which contains the
vast majority of the dense material. In the V10 simulation, the higher kinetic
energy of the initial cloud motions has disrupted this layer, and the dense
material is more spread out along the collision axis. In the V14 simulation,
the collision velocity is so high that no layer forms, and there is very
little visible dense substructure. Some of the overdense regions that do form
may eventually collapse under their own self-gravity (as in Jones et al.
2023), but there is no sign of this happening at the point we terminate the
simulation.

Figure 2 shows the time evolution of the masses of intermediate density
($M_{\rm mid}$; $n_{\rm H}>10^{3}\,{\rm cm}^{-3}$) and genuinely dense gas
($M_{\rm dense}$, defined using a volume density threshold of $10^{4}\,{\rm
cm}^{-3}$ as suggested by Lada et al. 2010), and the total mass accreted onto
sink particles ($M_{\rm sink}$). Intermediate density gas forms earlier for
faster collision velocities, but the two higher velocity models subsequently
show a decline in the mass of this material, which for the V14 model appears
to be irreversible. The onset of dense gas and sink formation has a more
complex relationship with $v_{\rm col}$; model V7 forms dense gas and sinks
before V3, but has a slower rate of star formation, as indicated by the lower
average values222We note that the star formation rates in our simulations are
comparable to those estimated observationally by Lada et al. (2010). in Table
1. Model V10 first forms dense gas at a comparable time to V7, but this gas
then disperses before a second round of dense gas formation several $\,{\rm
Myr}$ later, this time accompanied by sink formation. Model V14 never forms
any significant quantity of dense gas, nor any sinks, over the runtime of the
simulation. Star formation activity (as represented by sink particles) only
seems to occur after the formation of a substantial ($>10\,{\rm M}_{\odot}$)
mass of dense gas. The mass of stars formed approximately tracks the mass of
this dense gas with a delay of $\sim 1-2\,{\rm Myr}$.

###  3.2 Chemical properties

Figure 1 shows the column densities of CO, which is typically thought to trace
gas with densities above $\sim 100\,{\rm cm}^{-3}$ (e.g. Clark et al., 2019),
and HCN and N2H+, both commonly used as tracers of denser gas. Regions of
significant CO column density exist above a threshold total column of $\sim
10^{21}\,{\rm cm}^{-2}$ (Priestley et al., 2023), but much of the substructure
in regions above this threshold is not visible in the CO column; there is
essentially no distinction between moderate- and high-density sightlines. HCN
fares slightly better, but substantial columns are still present in fairly
low-density sightlines, in agreement with its widespread detection in regions
of molecular clouds with $A_{\rm V}<8\,{\rm mag}$ (e.g. Evans et al., 2020).

As found by previous observational studies (Kauffmann et al., 2017; Tafalla et
al., 2021), N2H+ only exists in significant quantities for $N_{\rm H}\gtrsim
10^{22}\,{\rm cm}^{-2}$ (Priestley et al., 2023). This is apparent in Figure
1, where the morphology of the clouds seen in N2H+ is very different to that
in CO or HCN, with only the highest-density substructures being visible. The
V14 model, which forms no sink particles, also has no substantial N2H+ column
density along any sightline. This suggests a possible role for the molecule as
a direct tracer of material with the potential to form stars.

Figure 3 shows the distribution of tracer particles333As the tracer particles
do not correspond to physical gas parcels, this is not a direct measure of the
mass distribution, but is a reasonable proxy quantity. in the post-processed
sample, according to the peak density reached at any point during their
evolution, and the final density at $t_{\rm end}$, for the V7 model (the
intermediate case out of the three models which form sink particles). Tracer
particles which fall below the one-to-one relationship (represented by the
dashed black line) were, at some point, at a higher density than their final
value; they have undergone a transient enhancement in density, which has not
resulted in runaway gravitational collapse.

Up to a peak density of about $10^{4}\,{\rm cm}^{-3}$, there is a significant
population of tracers below the one-to-one line. These tracers reached
relatively high densities due to compression by supersonic gas motions, but
subsequently returned to much lower values444This re-expansion is also due to
the turbulent gas motions, as the magnetic field strength is - by design - too
weak to prevent the collapse of gravitationally-bound structures., extending
down to the initial cloud density of $10\,{\rm cm}^{-3}$. By contrast, this
behaviour is almost non-existent above a peak density of $10^{4}\,{\rm
cm}^{-3}$. Material which reaches this density never becomes strongly rarefied
in its subsequent evolution, and is effectively destined to form (or accrete
onto) a sink particle. This corresponds very well with the volume density
threshold proposed by Lada et al. (2010), and appears to be a natural outcome
of the combined hydrodynamical and thermal evolution of molecular gas,
although the exact reasons for the existence and value of this threshold are
currently unclear.

Figure 4 shows how the peak molecular abundances of CO, HCN and N2H+ vary with
the peak and final densities reached by the tracer particles, again for the V7
model. In addition to their limited ability to discern high-density
substructures, CO and HCN both reach high peak abundances (relative to their
maximum gas-phase abundance at any density) in material which reaches
densities of a few thousand $\,{\rm cm}^{-3}$, but subsequently returns to
$\sim 10\,{\rm cm}^{-3}$ and thus plays no role in star formation. By
contrast, N2H+ has an abundance of $\lesssim 10^{-12}$ in this material, which
we demonstrate in Section 3.4 would be observationally challenging to detect.
Significant quantities of this molecule are only produced in gas which
reaches, and then stays above, the $10^{4}\,{\rm cm}^{-3}$ density threshold.

Figures 5 and 6 show that this behaviour - for both the physical and chemical
evolution - is independent of the simulation collision velocity. For all
models, we find substantial quantities of gas which reaches a density above
$10^{3}\,{\rm cm}^{-3}$ and subsequently returns to much lower values, but
this behaviour is vanishingly rare for material which exceeds a volume density
of $10^{4}\,{\rm cm}^{-3}$. This volume density threshold for eventual
gravitational collapse corresponds very closely to the density at which
significant quantities of N2H+ are produced. We therefore suggest that
detectable N2H+ emission in a molecular cloud directly traces material which
is destined to eventually form stars, rather than transiently-enhanced,
unbound material, which may eventually disperse.

This behaviour is not shared by any other molecular species we investigate.
Figure 7 shows the behaviour of NH3, HCO+ and CS for the V7 model, all of
which are often assumed to trace high-density, star-forming material. While
NH3 and HCO+ reach their peak abundances only in material which exceeds the
$10^{4}\,{\rm cm}^{-3}$ threshold for subsequent collapse, both molecules also
exist in appreciable quantities below this boundary. CS behaves more like HCN
and CO, and is even more likely to be readily detectable in material which is
only transiently enhanced in density. We address the detectability issue
quantitatively in Section 3.4 below.

###  3.3 Transient high-density gas

The two faster collisions, models V10 and V14, do appear to produce some dense
gas which is not directly associated with star formation, reflected in the
temporary peaks in dense gas mass between $\sim 2-3\,{\rm Myr}$ in Figure 2.
Figure 8 shows the time evolution of three tracer particles from the V10
model, which have peak densities above $10^{4}\,{\rm cm}^{-3}$ but final
densities significantly below this. The enhancement to high density is
extremely brief ($\ll\,{\rm Myr}$) in all cases, and the tracer particles
quickly return to lower densities of $\sim 100\,{\rm cm}^{-3}$.

The increase in density initially results in a rapid increase in the
abundances of both CO and N2H+. However, once the CO abundance reaches $\sim
10^{-5}$, its ability to efficiently destroy N2H+ terminates the rise in the
abundance of the latter species, and two of the three tracer particles never
exceed an N2H+ abundance of $10^{-11}$. One tracer particle does reach a high
enough density to begin depleting CO from the gas phase, allowing a peak N2H+
abundance of $10^{-10}$ which we show in Section 3.4 may be observationally
relevant, but this only lasts for the brief period of time when the gas is at
high density. Once the density begins to decline again, CO is rapidly desorbed
back off grain surfaces, as the rates of the main desorption processes (via
cosmic rays and UV photons) decline more slowly (proportional to $n_{\rm H}$)
than freeze-out (proportional to $n_{\rm H}^{2}$). This causes a steep
reduction in the abundance of N2H+.

Forming and maintaining a high N2H+ abundance requires depletion of CO, which
itself requires high densities. The rate of depletion onto grain surfaces is
given by

$\dot{n}_{\rm freeze}=4.57\times
10^{4}\,\left<d_{g}a^{2}\right>\,\left(T/m\right)^{0.5}n_{\rm H}\,n\;\;{\rm
cm^{-3}\,s^{-1}},$ (1)

where $n$ is the gas-phase abundance of the molecule,
$\pi\left<d_{g}a^{2}\right>$ is the grain surface area per hydrogen nucleus,
$T$ is the gas temperature, $m$ is the atomic mass of the molecule, and we
have assumed a sticking coefficient of unity (Rawlings et al., 1992). Taking
$n_{\rm H}=10^{4}\,{\rm cm}^{-3}$, $T=20\,{\rm K}$, $m=28$ for CO, and
$\left<d_{g}a^{2}\right>=2.2\times 10^{-22}\,{\rm cm^{2}}$ from Rawlings et
al. (1992), the freeze-out timescale is $t_{\rm freeze}=n/\dot{n}_{\rm
freeze}=0.34\,{\rm Myr}$.

In Figure 9, we compare this value with the amount of time spent by tracer
particles above a density of $10^{4}\,{\rm cm}^{-3}$, $t_{\rm dense}$ (CO
freeze-out is inefficient at lower densities; Priestley et al. 2023), for the
V10 model (we exclude tracer particles for which $t_{\rm dense}=0$). Virtually
all tracer particles with a final density below $10^{4}\,{\rm cm}^{-3}$ have
$t_{\rm dense}<t_{\rm freeze}$; material which reaches high densities only
transiently does not spend enough time under these conditions to deplete CO,
and so does not form significant quantities of N2H+. For final densities above
$10^{4}\,{\rm cm}^{-3}$, $t_{\rm dense}\sim t_{\rm freeze}$, and CO is
sufficiently depleted for substantial quantities of N2H+ to survive. As
$t_{\rm freeze}$ scales as $n_{\rm H}^{-1}$, even tracer particles with
$t_{\rm dense}<0.34\,{\rm Myr}$ can deplete most of their CO if their final
density is above $10^{4}\,{\rm cm}^{-3}$, as they only need to spend a small
fraction of time at the higher density in order for freeze-out to proceed to
completion.

###  3.4 Detectability of line emission

Figure 10: Main beam temperature versus density for the $J=1-0$ lines of N2H+ (blue), HCO+ (orange) and HCN (green), the $J=2-1$ line of CS (red), and the $(1,1)$ inversion transition of NH3 (purple). Table 3: Input parameters used for the radiative transfer modelling. $n_{\rm H}$/$\,{\rm cm}^{-3}$ |  $T$/$\,{\rm K}$ |  $N_{\rm H}$/$\,{\rm cm}^{-2}$ |  $\sigma_{v}$/$\,{\rm km\,s^{-1}}$ |  $X($N2H+$)$ |  $X($HCO${}^{+})$ |  $X($HCN$)$ |  $X($CS$)$ |  $X($NH${}_{3})$
---|---|---|---|---|---|---|---|---
$10^{2}$ | $47$ | $1.2\times 10^{21}$ | $1.0$ | $10^{-14}$ | $10^{-12}$ | $10^{-12}$ | $10^{-12}$ | $10^{-10}$
$10^{3}$ | $23$ | $2.4\times 10^{21}$ | $1.0$ | $10^{-13}$ | $10^{-10}$ | $10^{-9}$ | $10^{-9}$ | $10^{-8}$
$10^{4}$ | $15$ | $4.8\times 10^{21}$ | $1.0$ | $10^{-11}$ | $10^{-9}$ | $10^{-8}$ | $10^{-8}$ | $10^{-7}$
$10^{5}$ | $10$ | $9.5\times 10^{21}$ | $1.0$ | $10^{-10}$ | $10^{-8}$ | $10^{-8}$ | $10^{-7}$ | $10^{-6}$
$10^{6}$ | $7$ | $1.9\times 10^{22}$ | $1.0$ | $10^{-10}$ | $10^{-8}$ | $10^{-8}$ | $10^{-7}$ | $10^{-6}$

We estimate the intensity of molecular emission lines as a function of volume
density using radex (van der Tak et al., 2007), assuming a uniform slab
geometry. The required input parameters are the volume density and temperature
of the gas, the column density of the molecule in question, and the velocity
dispersion in the slab. Both the gas temperature and the total column density
are quite tightly correlated with volume density in our simulations, so we
adopt the power-law fits to the average relationships between these quantities
from Priestley et al. (2023) to obtain temperature and total column555Strictly
speaking, this is the effective shielding column, not the line-of-sight value
required by the radiative transfer model. However, the difference between
these two quantities is typically small enough to make no qualitative
difference to our results (Clark & Glover, 2014). as a function of density. We
then convert the total column to the column density of the relevant molecule
using its typical abundance in gas of the required volume density, estimated
from Figures 4 and 7. As there as no equivalent relationship between velocity
dispersion and gas density, we assume a value of $1.0\,{\rm km\,s^{-1}}$
typical of molecular clouds, and note that observed line widths do not appear
to vary strongly with density (Tafalla et al., 2021). Input parameters used in
the modelling are given in Table 3.

Figure 10 shows the main beam temperatures as a function of density for the
$J=1-0$ lines of N2H+, HCO+ and HCN, and the $J=2-1$ CS line, all of which are
commonly observed in $3\,{\rm mm}$ line surveys (e.g. Pety et al., 2017;
Kauffmann et al., 2017; Tafalla et al., 2021) and assumed to originate from
the higher-density regions of molecular clouds. We also include the $(1,1)$
NH3 transition, another line thought to trace higher-density gas than CO, and
for which observational data is similarly plentiful (Ragan et al., 2011; Ragan
et al., 2012; Friesen et al., 2017; Fehér et al., 2022). The N2H+ and HCN
intensities correspond to the $(1,1,2)-(0,1,0)$ and $(1,0)-(0,1)$ hyperfine
components, respectively; other hyperfine components typically have
intensities within a factor of a few of each other, so the exact choice of
transition does not affect our conclusions.

At a density of $100\,{\rm cm}^{-3}$, all lines other than NH3 are extremely
weak ($<10^{-5}\,{\rm K}$), but by $10^{3}\,{\rm cm}^{-3}$ the HCN, HCO+ and
CS lines have strengthened to $\sim 10^{-2}\,{\rm K}$ and NH3 is above
$0.1\,{\rm K}$. These intensities are well within reach of modern telescopes
with a modest666Using the IRAM $30\,{\rm m}$ sensitivity calculator
(https://oms.iram.fr/tse/), we estimate that a $3\,{\rm mm}$ line reaching an
intensity of a few $10^{-2}\,{\rm K}$ would be detectable (exceeding the
sensitivity limit by a factor of a few) in $\lesssim 10\,{\rm hr}$ of
telescope time. investment of observing time. N2H+ remains several orders of
magnitude fainter than this at $10^{3}\,{\rm cm}^{-3}$, and does not even
begin to approach comparable line brightnesses until a density of
$10^{4}\,{\rm cm}^{-3}$. As noted above, this corresponds almost exactly to
the density at which material appears to become gravitationally bound. By a
quirk of its chemistry, and unlike any other molecular species we have
investigated, N2H+ is only detectable in line emission in gas which will
inevitably form stars.

##  4 Discussion

The fact that N2H+ specifically traces gas which is in the process of forming
stars, unlike most of the other species commonly used in this role, is due to
its chemical behaviour. N2H+ reacts rapidly with CO to form HCO+ and N2, and
due to the high abundance of CO at even moderate densities, this is typically
the main route of destruction. The N2H+ abundance is therefore kept low until
CO begins to deplete onto grain surfaces in significant quantities, which only
begins to occur above a density of $10^{4}\,{\rm cm}^{-3}$ (Tafalla et al.
2002, 2004; Caselli et al. 2002; see also Priestley et al. 2023), and requires
that the gas remains at or above this density for a significant period of time
(Section 3.3). No equivalent reaction with CO exists for the other species,
meaning that they exist in detectable quantities below what appears to be the
star formation threshold, at least in the simulations presented here.

One implication of our results is that although molecules such as HCN and HCO+
do trace the material which will eventually go on to form stars, they also
trace gas that is only transiently shocked to moderately high density and so
plays no part in the star formation process. This makes them unreliable as
tracers of star-forming gas. At first glance, this seems incompatible with the
tight correlation between HCN luminosity and the star formation rate (Gao &
Solomon, 2004; Neumann et al., 2023), which has been interpreted as HCN
directly tracing the gas involved with star formation. However, the HCN
$J=1-0$ line intensity is known to be almost-linearly correlated with the
total column of gas (Tafalla et al., 2021), and therefore the mass. The
correlation with star formation rate may therefore just be a correlation with
the amount of molecular gas, rather than any specifically star-forming
component of clouds. This would imply that the mass fraction of genuinely
dense material is more-or-less constant, at least when averaged over large
enough scales, which appears to be borne out by recent observational work
(García-Rodríguez et al., 2023; Jiménez-Donaire et al., 2023).

On smaller scales, when the structure of individual molecular clouds can be
resolved, the distinction between material traced by HCN and by N2H+ is likely
to become important, as the ratio between their lines is clearly not constant
(Pety et al., 2017; Tafalla et al., 2021). If HCN emission is assumed to trace
star-forming gas, one might infer that the V14 model has a small but non-zero
potential for star formation, despite the fact that it is actually in the
process of dispersing. Tafalla et al. (2021) find a near-linear correlation
between the strength of the N2H+ $J=1-0$ line and the total column density of
material when $N_{\rm H}\gtrsim 3\times 10^{22}\,{\rm cm}^{-2}$. We suggest
that this can be used to estimate the mass of material corresponding to the
immediate reservoir for star formation, $M_{\rm SF}$, via

$M_{\rm SF}=310\,\left(L_{\rm N_{2}H^{+}}/\,{\rm K}\,{\rm km\,s^{-1}}\,{\rm
pc}^{2}\right)\,{\rm M}_{\odot},$ (2)

where $L_{\rm N_{2}H^{+}}$ is the luminosity of the N2H+ $J=1-0$ line
integrated over all hyperfine components, and the proportionality constant is
converted from the column density-line intensity scaling factor $X_{\rm
N_{2}H^{+}}=1.4\times 10^{22}\,{\rm cm}^{-2}\,(\,{\rm K}\,{\rm
km\,s^{-1}})^{-1}$ found by Tafalla et al. (2021). If this material is
collapsing on approximately the freefall time for gas at a density of
$10^{4}\,{\rm cm}^{-3}$, $t_{\rm ff}\sim 0.4\,{\rm Myr}$, the star formation
rate can be estimated as $M_{\rm SF}/t_{\rm ff}$, giving

${\rm SFR}_{\rm N_{2}H^{+}}=775\,\left(L_{\rm N_{2}H^{+}}/\,{\rm K}\,{\rm
km\,s^{-1}}\,{\rm pc}^{2}\right)\,{\rm M}_{\odot}\,{\rm Myr}^{-1}.$ (3)

Based on the delay between the formation of dense gas and sink particles in
Figure 2, this quantity can be thought of as representing the expected star
formation rate over the next $\sim\,{\rm Myr}$ of evolution, rather than an
estimate of previous activity as provided by methods such as protostar counts.

Although our simulations do not include protostellar feedback, heating by
newly-formed protostars is expected to rapidly desorb molecules from grain
surfaces in their vicinity (e.g. Viti et al., 2004), in particular CO. This
increases the destruction rate of N2H+, and should eventually reduce its
abundance back to undetectable levels. In reality, N2H+ and even its
deuterated isotopologues are commonly detected towards regions of active star
formation (Fontani et al., 2011; Friesen et al., 2013; Cosentino et al.,
2023), so its presence is not uniquely associated with material that is about
to form stars, but also with material that already has. We do not consider
this distinction hugely important: N2H+ still remains the best indicator of
the potential for star formation of a molecular cloud, regardless of whether
that potential has already been realised or not.

Finally, we note that our results are based on cloud properties, a radiation
field and an ionisation rate appropriate for the Solar neighbourhood, and may
not apply in more extreme environments. Unlike most other observational
studies (Pety et al., 2017; Kauffmann et al., 2017; Tafalla et al., 2021),
Barnes et al. (2020) find that N2H+ behaves quite similarly to HCN in the W49
complex, a region of intense star formation activity. It is possible that an
enhanced formation rate due to more highly ionised gas, or a reduced
destruction rate due to greater photodissociation of CO, could change the
chemical behaviour of N2H+ to something closer to that of HCN, making it more
readily detectable at moderate densities. However, it also seems possible that
these same effects would change the threshold density for star formation, so
it is unclear what impact this would have on the properties of various
molecules as tracers. While this is beyond the scope of the present paper (the
chemical network is only currently calibrated for Solar neighbourhood
conditions), we plan to investigate this topic more extensively in future
work.

##  5 Conclusions

We have post-processed hydrodynamical simulations of molecular clouds with a
full time-dependent chemical network, in order to investigate the ability of
various molecular species to trace the reservoirs of material for star
formation. We find that many molecules commonly used for this purpose, such as
HCN and HCO+, exist in substantial quantities in gas which is presently at
high density ($n_{\rm H}\gtrsim 10^{3}\,{\rm cm}^{-3}$), but only transiently,
and which will eventually return to lower densities without undergoing any
star formation. N2H+, by contrast, is only present in detectable amounts when
the volume density of the gas exceeds $10^{4}\,{\rm cm}^{-3}$, due to the
onset of CO freeze-out. This coincides closely with the threshold density at
which cloud material typically becomes gravitationally bound: the material
traced by N2H+ will inevitably form stars (if it is not doing so already),
unless disrupted by some outside process. Regions of clouds detected
observationally in N2H+ therefore correspond directly to sites of current and
future star formation, whereas other molecular tracers will give a misleading
impression of a cloud’s potential to form stars.

## Acknowledgements

FDP, PCC, SER and OF acknowledge the support of a consolidated grant
(ST/K00926/1) from the UK Science and Technology Facilities Council (STFC).
SCOG and RSK acknowledge funding from the European Research Council (ERC) via
the ERC Synergy Grant “ECOGAL-Understanding our Galactic ecosystem: From the
disk of the Milky Way to the formation sites of stars and planets” (project ID
855130, from the Heidelberg Cluster of Excellence (EXC 2181 - 390900948)
“STRUCTURES: A unifying approach to emergent phenomena in the physical world,
mathematics, and complex data”, funded by the German Excellence Strategy, and
from the German Ministry for Economic Affairs and Climate Action in project
“MAINN” (funding ID 50OO2206). The team in Heidelberg also thanks for
computing resources provided by The Länd through bwHPC and DFG through grant
INST 35/1134-1 FUGG and for data storage at SDS@hd through grant INST
35/1314-1 FUGG. This research was undertaken using the supercomputing
facilities at Cardiff University operated by Advanced Research Computing at
Cardiff (ARCCA) on behalf of the Cardiff Supercomputing Facility and the
Supercomputing Wales (SCW) project. We acknowledge the support of the latter,
which is part-funded by the European Regional Development Fund (ERDF) via the
Welsh Government.

## Data Availability

The data underlying this article will be shared on request.

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A strategy for a simultaneous measurement of $C\\!P$ violation parameters
related to the $C\\!K\\!M$ angle $\gamma$ in multiple $B$ meson decay channels

Jordi Garra Ticó1

1Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom

Several methods exist [1, 2, 3, 4, 5] to measure $C\\!P$ violation observables
related to the $C\\!K\\!M$ angle $\gamma$. These observables are different for
every $B$ meson decay channel. However, the information they contain on
$\gamma$ is encoded in a similar way for all of them. This paper describes a
strategy for a simultaneous measurement including several $B$ meson decay
channels, while taking into account any possible correlations between them.

##  1 Introduction

The CKM angle $\gamma$ is currently the least constrained of the angles of the
unitarity triangle. It is defined as

$\gamma=\arg\left(-\frac{V_{ud}V^{\star}_{ub}}{V_{cd}V^{\star}_{cb}}\right).$

Because $V_{ub}$ is the largest contribution to this phase, $\gamma$ is
measured in decays that involve an interference between the transitions
$b\rightarrow u$ and $b\rightarrow c$, such as $B^{\pm}\rightarrow
D^{(*)}K^{(*)\pm}$, $B^{0}\rightarrow DK^{*0}$ or, eventually in the future,
$B^{\pm}\rightarrow D^{(*)}\pi^{\pm}$, where $D$ represents some admixture of
the $D$ meson flavor eigenstates $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$, and the notation $(*)$ indicates
a regular expression representing either one $*$ or none.

Several methods exist [1, 2, 3, 4, 5] to measure different $C\\!P$ violation
observables that constrain the $C\\!K\\!M$ angle $\gamma$. This paper presents
an approach to simultaneously measure a reduced set of observables in multiple
$B$ decay channels, independently of which of these methods is used for any
$B$ decay considered.

##  2 Unified approach

###  2.1 The admixture coefficients $z_{\pm}^{c}$

Several $B$ meson decay channels produce admixtures of neutral $D$ mesons that
involve $\gamma$. The notation for the $D$ meson states is
$\left|\vphantom{\bar{D}^{0}}{D}^{0}\right\rangle$ and
$\left|\bar{D}^{0}\right\rangle$ for the flavor eigenstates and
$\left|\vphantom{\bar{D}^{0}}{D}_{\pm}\right\rangle$ for the $D$ meson
produced in a $B$ decay. In this paper,
$\left|\vphantom{\bar{D}^{0}}{D}_{+}\right\rangle$ denotes the $D$ meson
produced in a $B^{+}$ or $B^{0}$ decay, and
$\left|\vphantom{\bar{D}^{0}}{D}_{-}\right\rangle$ denotes the $D$ meson
produced in a $B^{-}$ or $\bar{B}^{0}$ decay. A superscript $c$ is used to
denote the $B$ decay channel. In general, one can write

$\begin{aligned}
\left|D_{-}^{c}\right\rangle&\sim\left|{D}^{0}\right\rangle+z_{-}^{c}\left|\bar{D}^{0}\right\rangle,\\\
\left|D_{+}^{c}\right\rangle&\sim\left|\bar{D}^{0}\right\rangle+z_{+}^{c}\left|{D}^{0}\right\rangle,\end{aligned}\Rightarrow\left\\{\begin{aligned}
A_{-}^{c}&\sim\vphantom{A_{\bar{D}}}A_{D}+z_{-}^{c}A_{\bar{D}},\\\
A_{+}^{c}&\sim
A_{\bar{D}}+z_{+}^{c}\vphantom{A_{\bar{D}}}A_{D},\end{aligned}\right.$ (1)

where $\vphantom{A_{\bar{D}}}A_{D}=\left\langle
f\left|\mathcal{H}\vphantom{D^{0}}\right|\\!\vphantom{\bar{D}^{0}}{D}^{0}\right\rangle$,
$A_{\bar{D}}=\left\langle
f\left|\mathcal{H}\vphantom{D^{0}}\right|\\!\bar{D}^{0}\right\rangle$ and
$A_{\pm}^{c}=\left\langle
f\left|\mathcal{H}\vphantom{D^{0}}\right|\\!\vphantom{\bar{D}^{0}}{D}_{\pm}^{c}\right\rangle$.
The complex coefficients $z_{\pm}^{c}$ are generally specific for every $B$
decay channel $c$, and are typically expressed in either Cartesian or polar
coordinates as

$z_{\pm}^{c}=x_{\pm}^{c}+i\,y_{\pm}^{c}=r_{c}\,e^{i\delta}\,e^{\pm i\gamma},$
(2)

where all the parameters with a subscript or superscript $c$ are specific for
every $B$ decay channel. The total number of parameters in Cartesian
coordinates for $N$ different $B$ decay channels is $4\,N$.

It should be noted that, by defining

$z_{c}=r_{c}\,e^{i\delta_{c}},$ (3)

we have

$z_{\pm}^{c}=z_{c}\,e^{\pm i\gamma}.$ (4)

This reveals a clear channel invariant,

$\frac{z_{+}^{c}}{z_{-}^{c}}=e^{2i\gamma}\quad\Rightarrow\quad\gamma=\frac{1}{2}\arg\left(\frac{z_{+}^{c}}{z_{-}^{c}}\right).$
(5)

If the $z_{\pm}^{c}$ coefficients are both multiplied by any complex
coefficient $\xi$, the result will contain the exact same information on
$\gamma$. In particular, it is always possible to express the $z_{\pm}^{c}$
coefficients for channel $c$ as

$z_{\pm}^{c}=\xi_{c}\,z_{\pm}^{\mathrm{DK}},$ (6)

where

$\xi_{c}=\frac{z_{c}}{z_{\mathrm{DK}}}.$ (7)

Notice that, by definition, from expressions (3) and (7), the $\xi_{c}$
coefficients do not depend on $\gamma$ so they are formally nuisance
parameters.

In order to simplify the notation, this paper uses
$z_{\pm}=z_{\pm}^{\mathrm{DK}}$, where DK refers to the $B^{\pm}\rightarrow
DK^{\pm}$ decay mode, so that $z_{\pm}^{c}=\xi_{c}\,z_{\pm}$.

Performing a simultaneous fit for the Cartesian parameters using these
$\xi_{c}$ coefficients reduces the number of independent parameters in the fit
from $4\,N$ to only $2\,(N+1)$ ($4$ for the $DK$ channel, and then only $2$
for each of the rest of the channels). This is only one more parameter than a
simultaneous fit for the polar coordinates, but it has the advantage that,
with a Cartesian fit, the real and imaginary components of $z_{\pm}$ and
$\xi_{c}$ are expected to exhibit Gaussian behavior.

###  2.2 The $\eta$ function

It is useful to define the $\eta$ function as

$\eta\left(a,b,\kappa\right)=\left|a\right|^{2}+\left|b\right|^{2}+2\kappa{{\rm\,Re}\left(a^{\star}b\right)},$
(8)

where $a,b\in\mathbb{C}$ and $\kappa\in\mathbb{R}$. The $\kappa$ coefficient
is called the coherence factor because it gives an idea of the fraction of
coherent sum that contributes to $\eta$,

$\eta\left(a,b,\kappa\right)=\kappa\left|a+b\right|^{2}+(1-\kappa)\left(\left|a\right|^{2}+\left|b\right|^{2}\right).$
(9)

In this paper, when the coherence factor argument is omitted it should be
assumed that it is implicit. If one of the complex arguments is omitted, it
should be assumed that it is $1$. So,

$\eta\left(a\right)=\eta\left(a,1,\kappa\right).$ (10)

The $\eta$ function has the following properties:

$\displaystyle\eta(a,b,\kappa)$ $\displaystyle=\eta(b,a,\kappa),$ Symmetric
(11) $\displaystyle\eta(a,b,\kappa)$
$\displaystyle=\left|a\right|^{2}\eta\left(1,\frac{b}{a},\kappa\right),$
Scaling (12)

###  2.3 Signal amplitude

The signal probability distribution function is proportional to the squared
amplitude integrated over a certain region of the $B$ phase space,

$p_{\pm}^{c}\sim\int d\mathcal{P}_{B}\left|A_{\pm}^{c}\right|^{2}.$ (13)

If $A_{c}$ is the decay amplitude corresponding to a $b\rightarrow c$
transition and $A_{u}\,e^{\pm i\gamma}$ is the decay amplitude corresponding
to a $b\rightarrow u$ transition, then

$\displaystyle A_{-}$ $\displaystyle\sim
A_{c}\vphantom{A_{\bar{D}}}A_{D}+A_{u}e^{-i\gamma}A_{\bar{D}},$ (14)
$\displaystyle A_{+}$ $\displaystyle\sim
A_{c}A_{\bar{D}}+A_{u}e^{+i\gamma}\vphantom{A_{\bar{D}}}A_{D}.$ (15)

It is important not to forget that $A_{c}$ and $A_{u}$ are different for each
$B$ decay channel, but to avoid cluttering the following notation with too
many indices, I skip them for just a moment.

In the case of a 2-body $B$ decay, such as $B^{\pm}\rightarrow DK^{\pm}$, the
$B$ decay amplitudes $A_{c}$ and $A_{u}$ are constants, and one can write

$\displaystyle A_{-}$
$\displaystyle\sim\vphantom{A_{\bar{D}}}A_{D}+z_{-}A_{\bar{D}},$ (16)
$\displaystyle A_{+}$ $\displaystyle\sim
A_{\bar{D}}+z_{+}\vphantom{A_{\bar{D}}}A_{D},$ (17)

where

$z_{\pm}=\frac{A_{u}}{A_{c}}e^{\pm i\gamma}.$ (18)

This implies that, for 2-body decays,

$\displaystyle p_{-}$
$\displaystyle\sim\eta(\vphantom{A_{\bar{D}}}A_{D},z_{-}\,A_{\bar{D}}),$ (19)
$\displaystyle p_{+}$
$\displaystyle\sim\eta(A_{\bar{D}},z_{+}\,\vphantom{A_{\bar{D}}}A_{D}).$ (20)

In the case of a multibody $B$ decay with $3$ or more particles in the final
state, such as $B^{0}\rightarrow DK\pi$, the amplitude
$\left|A_{\pm}^{c}\right|^{2}$ is usually integrated over some region in the
$B$ decay phase space (around the $K^{*0}$ resonance for $B^{0}\rightarrow
DK\pi$).

By squaring the modulus of expressions (16) and (17) and by defining

$\displaystyle N_{\alpha\beta}$ $\displaystyle=\int
d\mathcal{P}_{B}A^{\star}_{\alpha}A_{\beta},$ (21) $\displaystyle
X_{\alpha\beta}$
$\displaystyle=\frac{N_{\alpha\beta}}{\sqrt{N_{\alpha\alpha}N_{\beta\beta}}},$
(22)

one can write

$\displaystyle p_{-}$
$\displaystyle\sim\left|\vphantom{A_{\bar{D}}}A_{D}\right|^{2}+\frac{N_{uu}}{N_{cc}}\left|A_{\bar{D}}\right|^{2}+2\left|X_{cu}\right|{{\rm\,Re}\left(\sqrt{\frac{N_{uu}}{N_{cc}}}\,\frac{X_{cu}}{\left|X_{cu}\right|}e^{-i\gamma}\vphantom{A_{\bar{D}}}A_{D}^{\star}A_{\bar{D}}\right)},$
(23) $\displaystyle p_{+}$
$\displaystyle\sim\left|A_{\bar{D}}\right|^{2}+\frac{N_{uu}}{N_{cc}}\left|\vphantom{A_{\bar{D}}}A_{D}\right|^{2}+2\left|X_{cu}\right|{{\rm\,Re}\left(\sqrt{\frac{N_{uu}}{N_{cc}}}\,\frac{X_{cu}}{\left|X_{cu}\right|}e^{+i\gamma}A_{\bar{D}}^{\star}\vphantom{A_{\bar{D}}}A_{D}\right)}.$
(24)

Notice that, because of the Cauchy-Schwarz inequality,
$\left|X_{\alpha\beta}\right|\leq 1$. Also, one can define

$\displaystyle\kappa$ $\displaystyle=\left|X_{cu}\right|,$ (25) $\displaystyle
r$ $\displaystyle=\sqrt{\frac{N_{uu}}{N_{cc}}},$ (26) $\displaystyle
e^{i\delta}$ $\displaystyle=\frac{X_{cu}}{\left|X_{cu}\right|},$ (27)
$\displaystyle z$ $\displaystyle=r\,e^{i\delta},$ (28) $\displaystyle z_{\pm}$
$\displaystyle=z\,e^{\pm i\gamma},$ (29)

so, adding back the channel index and using the formalism described in §2.1,
the signal amplitude probability distribution can always be expressed as

$\displaystyle p^{c}_{-}$
$\displaystyle\sim\eta(\vphantom{A_{\bar{D}}}A_{D},\xi_{c}\,z_{-}\,A_{\bar{D}},\kappa_{c})=\left|\vphantom{A_{\bar{D}}}A_{D}\right|^{2}+\left|\xi_{c}\,z_{-}\right|^{2}\left|A_{\bar{D}}\right|^{2}+2\,\kappa_{c}\,{{\rm\,Re}\left(\xi_{c}\,z_{-}\vphantom{A_{\bar{D}}}A_{D}^{\star}A_{\bar{D}}\right)},$
(30) $\displaystyle p^{c}_{+}$
$\displaystyle\sim\eta(A_{\bar{D}},\xi_{c}\,z_{+}\,\vphantom{A_{\bar{D}}}A_{D},\kappa_{c})=\left|A_{\bar{D}}\right|^{2}+\left|\xi_{c}\,z_{+}\right|^{2}\left|\vphantom{A_{\bar{D}}}A_{D}\right|^{2}+2\,\kappa_{c}\,{{\rm\,Re}\left(\xi_{c}\,z_{+}A_{\bar{D}}^{\star}\vphantom{A_{\bar{D}}}A_{D}\right)}.$
(31)

It should be noted that these expressions describe the physics of the
admixture that leads to the different $C\\!P$ observables used to measure
$\gamma$, but they are not specific to any method.

##  3 Specific equations for established methodologies

###  3.1 GLW equations

The GLW method uses, on one hand, states that are accessible from only one of
the flavor eigenstates, either
$\left|\vphantom{\bar{D}^{0}}{D}^{0}\right\rangle$ or
$\left|\bar{D}^{0}\right\rangle$, such that $A^{{D}}_{{f}}=\left\langle
f_{{D}}|\mathcal{H}|{D}^{0}\right\rangle$, $A^{\bar{D}}_{\bar{f}}=\left\langle
f_{\bar{D}}|\mathcal{H}|\bar{D}^{0}\right\rangle$, but $\left\langle
f_{\bar{D}}|\mathcal{H}|\vphantom{\bar{D}^{0}}{D}^{0}\right\rangle=\left\langle\vphantom{f_{\bar{D}}}f_{{D}}|\mathcal{H}|\bar{D}^{0}\right\rangle=0$,
and, on the other hand, states that are accessible from one of the $CP$
eigenstates $\left|{D}_{\pm}^{\mathrm{CP}}\right\rangle$, such that
$A^{\mathrm{CP}}_{\pm}=\left\langle
f_{\pm}|\mathcal{H}|D^{\mathrm{CP}}_{\pm}\right\rangle$, but $\left\langle
f_{\mp}|\mathcal{H}|D^{\mathrm{CP}}_{\pm}\right\rangle=0$.

The observables of interest are

$\displaystyle R_{\mathrm{CP}}^{\pm\,c}$
$\displaystyle=\frac{\Gamma\left(B^{-}\rightarrow
D_{\pm}^{\mathrm{CP}}h^{-}\right)+\Gamma\left(B^{+}\rightarrow
D_{\pm}^{\mathrm{CP}}h^{+}\right)}{\Gamma\left(B^{-}\rightarrow{D}^{0}h^{-}\right)+\Gamma\left(B^{+}\rightarrow\bar{D}^{0}h^{+}\right)}=\frac{1}{2}\left|\frac{A^{\mathrm{CP}}_{\pm}}{A^{D}_{f}}\right|^{2}\frac{\eta\left(\pm\xi_{c}\,z_{-}\right)+\eta\left(\pm\xi_{c}\,z_{+}\right)}{2},$
(32) $\displaystyle A_{\mathrm{CP}}^{\pm\,c}$
$\displaystyle=\frac{\Gamma\left(B^{-}\rightarrow
D_{\pm}^{\mathrm{CP}}h^{-}\right)-\Gamma\left(B^{+}\rightarrow
D_{\pm}^{\mathrm{CP}}h^{+}\right)}{\Gamma\left(B^{-}\rightarrow
D_{\pm}^{\mathrm{CP}}h^{-}\right)+\Gamma\left(B^{+}\rightarrow
D_{\pm}^{\mathrm{CP}}h^{+}\right)}=\frac{\eta\left(\pm\xi_{c}\,z_{-}\right)-\eta\left(\pm\xi_{c}\,z_{+}\right)}{\eta\left(\pm\xi_{c}\,z_{-}\right)+\eta\left(\pm\xi_{c}\,z_{+}\right)}.$
(33)

###  3.2 ADS equations

The ADS method uses states that are accessible from both flavor eigenstates.
With the convention
$CP\left|\vphantom{\bar{D}^{0}}{D}^{0}\right\rangle=\left|\bar{D}^{0}\right\rangle$
and assuming no direct $CP$ violation in the $D$ decay, one can define the
$\rho$ ratio of amplitudes as

$\rho=\frac{A^{\bar{D}}_{{f}}}{A^{{D}}_{{f}}},$ (34)

and

$\displaystyle\Gamma^{\pm}_{\mathrm{fav}}$
$\displaystyle=\Gamma\left(B^{\pm}\rightarrow
D_{\mathrm{fav}}\,h^{\pm}\right)=\left|A^{\pm}_{\tilde{D}}A^{D}_{f}\right|^{2}\eta\left(1,\rho\,\xi_{c}\,z_{\pm}\right),$
(35) $\displaystyle\Gamma^{\pm}_{\mathrm{sup}}$
$\displaystyle=\Gamma\left(B^{\pm}\rightarrow
D_{\mathrm{sup}}\,h^{\pm}\right)=\left|A^{\pm}_{\tilde{D}}A^{D}_{f}\right|^{2}\eta\left(\rho,\xi_{c}\,z_{\pm}\right),$
(36)

where $D_{\mathrm{sup}}$ and $D_{\mathrm{fav}}$ refer to suppresed and favored
decay modes of the produced $D$ meson.

The observables of interest are

$R_{\mathrm{ADS}}^{\pm\,c}=\frac{\Gamma^{\pm}_{\mathrm{sup}}}{\Gamma^{\pm}_{\mathrm{fav}}}=\frac{\eta\left(\rho,\xi_{c}\,z_{\pm}\right)}{\eta\left(1,\rho\,\xi_{c}\,z_{\pm}\right)}.$
(37)

It is also frequent to use the observables

$\displaystyle R_{\mathrm{ADS}}^{c}$
$\displaystyle=\frac{\Gamma^{-}_{\mathrm{sup}}+\Gamma^{+}_{\mathrm{sup}}}{\Gamma^{-}_{\mathrm{fav}}+\Gamma^{+}_{\mathrm{fav}}}$
$\displaystyle=\frac{\eta\left(\rho,\xi_{c}\,z_{-}\right)+\eta\left(\rho,\xi_{c}\,z_{+}\right)}{\eta\left(1,\rho\,\xi_{c}\,z_{-}\right)+\eta\left(1,\rho\,\xi_{c}\,z_{+}\right)},$
(38) $\displaystyle A_{\mathrm{ADS}}^{\mathrm{sup}\,c}$
$\displaystyle=\frac{\Gamma^{-}_{\mathrm{sup}}-\Gamma^{+}_{\mathrm{sup}}}{\Gamma^{-}_{\mathrm{sup}}+\Gamma^{+}_{\mathrm{sup}}}$
$\displaystyle=\frac{\eta\left(\rho,\xi_{c}\,z_{-}\right)-\eta\left(\rho,\xi_{c}\,z_{+}\right)}{\eta\left(\rho,\xi_{c}\,z_{-}\right)+\eta\left(\rho,\xi_{c}\,z_{+}\right)},$
(39) $\displaystyle A_{\mathrm{ADS}}^{\mathrm{fav}\,c}$
$\displaystyle=\frac{\Gamma^{-}_{\mathrm{fav}}-\Gamma^{+}_{\mathrm{fav}}}{\Gamma^{-}_{\mathrm{fav}}+\Gamma^{+}_{\mathrm{fav}}}$
$\displaystyle=\frac{\eta\left(1,\rho\,\xi_{c}\,z_{-}\right)-\eta\left(1,\rho\,\xi_{c}\,z_{+}\right)}{\eta\left(1,\rho\,\xi_{c}\,z_{-}\right)+\eta\left(1,\rho\,\xi_{c}\,z_{+}\right)}.$
(40)

###  3.3 GGSZ equations

The GGSZ method uses $3$ or more body decays to final states that can be
accessed from any of $\left|\vphantom{\bar{D}^{0}}{D}^{0}\right\rangle$ or
$\left|\bar{D}^{0}\right\rangle$. As opposed to GLW or ADS, this method does
not involve intermediate observables, and the goal is to fit for the $C\\!P$
observables, using equations (30-31) directly.

###  3.4 Time dependent equations

The time evolution of the amplitude of the $B_{s}\rightarrow D_{s}K$ decay is
governed by the equation

$A^{B_{s}}_{D_{s}^{\pm}}(t)=A^{B_{s}}_{D_{s}^{\pm}}\left[g_{+}(t)+\lambda_{\pm}g_{-}(t)\right],$
(41)

where $t$ is the $B_{s}$ meson lifetime and $g_{\pm}(t)$ are the functions
that describe the mixing of the $B$ meson flavor eigenstates. The
$\lambda_{\pm}$ parameters can be expressed as

$\lambda_{\pm}=\xi_{c}\,z_{\pm}\,e^{\pm i\phi_{q}},$ (42)

where the channel $c$ is $B_{s}\rightarrow D_{s}K$ in this case.

Unfortunately, current measurements of $\gamma$ in decays with $B$ meson
mixing introduce intermediate parameters instead of targeting the
$z^{c}_{\pm}$ observables themselves. In these cases, the squared amplitude of
the time-dependent probability distribution function is expressed as

$\small e^{-\Gamma
t}\,\frac{1+\left|\lambda_{-}\right|^{2}}{2}\left[\vphantom{\frac{1-\left|\lambda_{-}\right|^{2}}{1+\left|\lambda_{-}\right|^{2}}}\right.\cosh(x\Gamma
t)+\underbrace{\frac{1-\left|\lambda_{-}\right|^{2}}{1+\left|\lambda_{-}\right|^{2}}}_{\displaystyle
C_{f}}\cos(y\Gamma
t)-\underbrace{\frac{-2{{\rm\,Re}\left(\lambda_{-}\right)}}{1+\left|\lambda_{-}\right|^{2}}}_{\displaystyle
A_{f}^{\Delta\Gamma}}\sinh(x\Gamma
t)-\underbrace{\frac{2{{\rm\,Im}\left(\lambda_{-}\right)}}{1+\left|\lambda_{-}\right|^{2}}}_{\displaystyle
S_{f}}\sin(y\Gamma
t)\left.\vphantom{\frac{1-\left|\lambda_{-}\right|^{2}}{1+\left|\lambda_{-}\right|^{2}}}\right],$
(43)

and similarly for the conjugated amplitude, where $x$ and $y$ are parameters
that describe the $B$ meson mixing and $\Gamma$ is its decay rate. This
approach makes this kind of analysis observable based, but even in this case,
it is possible to introduce the $C_{f}$, $A_{f}^{\Delta\Gamma}$ and $S_{f}$
parameters, and the similar parameters for the conjugate decay, in a
simultaneous fit that shares the $z_{\pm}$ parameters with other $B$ decay
channels and only introduces $\xi_{c}$ as a new coefficient.

##  4 Experimental considerations

There are several advantages in a simultaneous measurement, compared to a
combination of standalone measurements:

  * •

It allows to properly take into account the correlations introduced by
crossfeed components that are signal in one channel but background in another.

  * •

For $N$ $B$ decay channels, the number of parameters in the fit is reduced
from $4N$ to $2N+2$ in Cartesian coordinates or $2N+1$ in polar coordinates.
The minimum number of physical parameters is $2N+1$ and, introducing just one
additional redundant parameter, all the observables exhibit Gaussian behavior.

  * •

Systematic uncertainties can also be evaluated simultaneously, with $B$ decay
channel correlations properly taken into account.

##  5 Sensitivity study

As a proof of concept, I have used cfit [6] to generate $1000$ experiments of
toy Monte Carlo signal events for the $4$ $B$ decay channels
$B^{\pm}\\!\rightarrow DK^{\pm}$, $B^{0}\\!\rightarrow DK^{*0}$,
$B^{\pm}\\!\rightarrow D^{*0}K^{\pm}$ and $B^{\pm}\\!\rightarrow D\pi^{\pm}$,
and for the $2$ $D$ decay channels $K_{s}\pi^{+}\pi^{-}$ and
$K_{s}K^{+}K^{-}$, using world averages from UTfit [7]. Since there is no
information on the admixture coefficients for $B^{\pm}\\!\rightarrow
D\pi^{\pm}$, and considering that, for this channel, they are expected to be
one order of magnitude smaller than for $B^{\pm}\\!\rightarrow DK^{\pm}$, I
used $\xi_{D\pi}=0.08+0.06\,i$. The size of the toy experiments has been
estimated for $3\,\rm\,ifb$ of LHCb data, using world average branching
fractions and reasonable guesses for the efficiencies.

I have used the GGSZ method to perform a fit to the $C\\!P$ observables
$z_{\pm}$ and $\xi_{c}$ presented in §2.1. To ensure convergence, I have used
a triple fit strategy, with the following fit steps:

  1. 1.

Fit for $z_{\pm}$ to the $DK$ sample only.

  2. 2.

Fix $z_{\pm}$ from the previous step and fit for $\xi_{DK^{\star 0}}$,
$\xi_{D^{\star}K}$ and $\xi_{D\pi}$ to the whole sample.

  3. 3.

Float all parameters using the previous fit results as initial values.

All the fits for the $z_{\pm}$ and $\xi_{c}$ parameters have been obtained
using cfit [6].

A standalone fit to $B^{\pm}\\!\rightarrow DK^{\pm}$ is used as reference, and
several simultaneous fits are obtained by progressively adding more channels
to the approach. The obtained statistical uncertainties and biases are
summarised in table 1, where all collections include $B^{\pm}\\!\rightarrow
DK^{\pm}$ and the $K_{s}\pi^{+}\pi^{-}$ and $K_{s}K^{+}K^{-}$ $D$ decay
channels are considered.

collection | $\sigma_{x_{-}}$ | $\sigma_{y_{-}}$ | $\sigma_{x_{+}}$ | $\sigma_{y_{+}}$ | ${{\rm\,Re}\left(\sigma_{\xi_{D\pi}}\right)}$ | ${{\rm\,Im}\left(\sigma_{\xi_{D\pi}}\right)}$ | ${{\rm\,Re}\left(\sigma_{\xi_{DK^{\star 0}}}\right)}$ | ${{\rm\,Im}\left(\sigma_{\xi_{DK^{\star 0}}}\right)}$ | ${{\rm\,Re}\left(\sigma_{\xi_{D^{\star}K}}\right)}$ | ${{\rm\,Im}\left(\sigma_{\xi_{D^{\star}K}}\right)}$
---|---|---|---|---|---|---|---|---|---|---
$DK^{\pm}$ | $0.0172$ | $0.0186$ | $0.0210$ | $0.0240$ | — | — | — | — | — | —
$D\pi^{\pm}$ | $0.0175$ | $0.0185$ | $0.0212$ | $0.0245$ | $0.0673$ | $0.0794$ | — | — | — | —
$DK^{*0}$ | $0.0149$ | $0.0169$ | $0.0211$ | $0.0236$ | — | — | $2.11$ | $2.26$ | — | —
$D\pi^{\pm}$ | $0.0151$ | $0.0170$ | $0.0211$ | $0.0235$ | $0.0648$ | $0.0750$ | $2.09$ | $2.26$ | — | —
$DK^{*0}$
$D^{*0}K^{\pm}$ | $0.0146$ | $0.0169$ | $0.0211$ | $0.0239$ | — | — | — | — | $0.442$ | $0.426$
all | $0.0132$ | $0.0159$ | $0.0208$ | $0.0233$ | $0.0654$ | $0.0739$ | $2.02$ | $2.28$ | $0.429$ | $0.418$
collection | $\delta x_{-}$ | $\delta y_{-}$ | $\delta x_{+}$ | $\delta y_{+}$ | ${{\rm\,Re}\left(\delta\xi_{D\pi}\right)}$ | ${{\rm\,Im}\left(\delta\xi_{D\pi}\right)}$ | ${{\rm\,Re}\left(\delta\xi_{DK^{\star 0}}\right)}$ | ${{\rm\,Im}\left(\delta\xi_{DK^{\star 0}}\right)}$ | ${{\rm\,Re}\left(\delta\xi_{D^{\star}K}\right)}$ | ${{\rm\,Im}\left(\delta\xi_{D^{\star}K}\right)}$
$DK^{\pm}$ | $-0.00198$ | $-0.00209$ | $-0.00225$ | $-0.00178$ | — | — | — | — | — | —
$D\pi^{\pm}$ | $-0.00207$ | $-0.00077$ | $-0.00154$ | $-0.00109$ | $0.0133$ | $0.00917$ | — | — | — | —
$DK^{*0}$ | $-0.00184$ | $-0.00141$ | $-0.00185$ | $-0.00117$ | — | — | $-0.149$ | $-0.411$ | — | —
$D\pi^{\pm}$ | $-0.00162$ | $-0.00059$ | $-0.00154$ | $-0.00096$ | $0.0137$ | $0.00782$ | $-0.150$ | $-0.448$ | — | —
$DK^{*0}$
$D^{*0}K^{\pm}$ | $-0.00207$ | $0.00018$ | $-0.00138$ | $-0.00072$ | — | — | — | — | $-0.00980$ | $-0.0206$
all | $-0.00165$ | $0.00065$ | $-0.00143$ | $-0.00058$ | $0.0144$ | $0.00739$ | $-0.126$ | $-0.394$ | $0.00078$ | $-0.0091$
Table 1: Statistical uncertainties (upper part) and biases (lower part) of all
the parameters involved in each fit. All collections include
$B^{\pm}\\!\rightarrow DK^{\pm}$ plus the indicated $B$ decay channels.

It should be noted that including $B^{\pm}\\!\rightarrow D\pi^{\pm}$ does not
improve the sensitivity on $\gamma$ with the used sample size. However, each
other additional channel contributes significantly to a smaller uncertainty on
the $(x,y)_{\pm}$ parameters. All the biases obtained on any parameter are at
least an order of magnitude smaller than the statistical uncertainty on that
parameter.

##  6 Conclusions

I have presented an approach to measure the $C\\!P$ parameters $z_{\pm}$
simultaneously in different $B$ decay channels. This approach allows to
properly take into account the correlations introduced by crossfeed components
that are signal in one channel but background in another, it significantly
reduces the number of parameters in the fit and allows a common treatment of
the systematic uncertainties.

A more detailed study with realistic backgrounds expected at LHCb will follow.

## Acknowledgements

I wish to thank the Science and Technology Facilities Council (STFC, UK). I am
very thankful to my colleagues at Cambridge, Valerie Gibson, Susan Haines and
Chris Jones, for very interesting discussions.

## References

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  * [2] M. Gronau and D. Wyler, _On determining a weak phase from CP asymmetries in charged B decays_ , Phys. Lett. B 265 (1991) 172
  * [3] D. Atwood, I. Dunietz, and A. Soni, _Enhanced CP violation with $B\rightarrow KD^{0}(\overline{D}^{0})$ modes and extraction of the CKM angle $\gamma$_, Phys. Rev. Lett. 78 (1997) 3257, arXiv:hep-ph/9612433
  * [4] A. Giri, Y. Grossman, A. Soffer, and J. Zupan, _Determining $\gamma$ using $B^{\pm}\rightarrow DK^{\pm}$ with multibody $D$ decays_, Phys. Rev. D 68 (2003), no. 5 054018
  * [5] R. Fleischer, _New strategies to obtain insights into CP violation through $B_{(s)}\rightarrow D_{(s)}^{\pm}K^{\mp}$, $D_{(s)}^{*\pm}K^{\mp}$, … and $B_{(d)}\rightarrow D^{\pm}\pi^{\mp}$, $D^{*\pm}\pi^{\mp}$, … decays_, Nucl. Phys. B671 (2003) 459, arXiv:hep-ph/0304027
  * [6] J. Garra Ticó, _The cfit library_, https://github.com/cfit
  * [7] UTfit Collaboration, A. Bevan et al., Summer 2014 combination results. Updated results and plots available at https://www.utfit.org/foswiki/bin/view/UTfit.