# Viscous and resistive accretion flows with radially self-similar and
outflows

Kazem Faghei and Azam Mollatayefeh
School of Physics, Damghan University, Damghan, Iran
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The existence of outflow in accretion flows is confirmed by observations and
magnetohydrodynamics (MHD) simulations. In this paper, we study outflows of
accretion flows in the presence of resistivity and toroidal magnetic field.
The mechanism of energy dissipation in the flow is assumed to be the viscosity
and the magnetic diffusivity due to turbulence in the accretion flow. It is
also assumed that the magnetic diffusivity and the kinematic viscosity are not
constant and vary by position and $\alpha$-prescription is used for them. The
influence of outflow emanating from accretion disc is considered as a sink for
mass, angular momentum and energy. The self-similar method is used to solve
the integrated equations that govern the behavior of the accretion flow in the
presence of outflow. The solutions represent the disc which rotates faster and
becomes cooler for stronger outflows. Moreover, by adding the magnetic
diffusivity, the surface density and rotational velocity decrease, while the
radial velocity and temperature increase. The study of present model with the
magnitude of magnetic field implies that the disc rotates and accretes faster
and becomes hotter, while the surface density decreases. The disc thickness
increases by adding the magnetic field or resistivity, while it becomes
thinner for more losses of mass and energy due to the outflows.

###### keywords:

accretion, accretion discs, magnetohydrodynamics: MHD

##  1 Introduction

Accretion disc is an important physical object in astrophysics. The standard
model of thin accretion disc is widely regarded as a successful model for
explaining the observational features in active galactic nuclei (AGN) and
X-ray binaries (Shakura & Sunyaev 1973). However, the standard disk model
cannot explain the spectral energy distributions (SED) of many sources, such
as Sgr A∗. To understand such systems, the model of advection dominated
accretion flow (ADAF) is introduced (Ichimaru 1977; Narayan & Yi 1994). In
this accretion flows, the energy released due to dissipation processes is
retained in the fluid rather than being radiated away. The models of ADAF
place an intermediate position between the spherically symmetric accretion
flow of non-rotating fluid (Bondi 1952) and the cool, thin disc of classical
accretion disc theory (e. g. Pringle 1981).

The observational evidence of accretion flows imply that outflow is an
important property in these systems. For example, comparison of the accretion
luminosities of neutron stars and white dwarfs in quiescence with similar
binary companions implies that the accretion rate onto white dwarfs is larger
by three orders of magnitude than that on the surface of neutron star (Loeb et
al. 2001). This indicates significant outflows in accretion flows. Moreover,
outflows seem to be common in the nuclei of galaxies. Marrone et al. (2006)
suggested the accretion rate of Sgr A∗ at small radii, much smaller than the
Bondi radius, must be low, below $10^{-7}M_{\odot}yr^{-1}$. While, Baganoff et
al. (2003) estimated the hot plasma surrounding Sgr A∗ should supply the
accretion rate of $10^{-6}M_{\odot}yr^{-1}$ at the Bondi radius. This
significant difference between the inner and outer mass accretion rates
indicates mass loss from the accretion flow due to the outflow (Kawabata &
Mineshige 2009). Thus, the accretion discs in the presence of outflow have
been studied by several authors (Knigge 1999; Fukue 2002, 2004; Shadmehri
2008; Xie & Yuan 2008; Kawabata & Mineshige 2009; Bu et al. 2009; Li & Cao
2009; Abbassi et al. 2010).

Knigge (1999) derived the radial distribution of dissipation rate and
effective temperature across a Keplerian, steady-state, mass losing accretion
disc, using a simple, parametric approach that is sufficiently general to be
applicable to many types of dynamical disc-wind models. Fukue (2002) examined
a hydrodynamical wind, which emanates from an accretion disc and is driven by
thermal and radiation pressure, under an one-dimensional approximation along
supposed streamlines. In another study, Fukue (2004) studied a supercritical
accretion regime, where the mass accretion rate was regulated just at the
critical rate with the help of wind mass-loss. He derived a critical radius
that outside of it, the disc is in a radiation-pressure dominated standard
state, while inside this radius the disc is in a critical state, where the
excess mass is expelled by wind, and the accretion rate is kept being just at
the critical rate at any radius inside of critical radius. Shadmehri (2008)
studied the effects of thermal conduction and outflows on ADAFs. He found that
in comparison to accretion flows without winds, the disc which rotates faster
and becomes cooler because of the angular momentum and energy flux which are
taking away by the winds. Xie & Yuan (2008), based on one-and-a-half-
dimensional description of the accretion flow, considered the interchange of
mass, radial and azimuthal momentum, and the energy between the outflow and
inflow. Bu et al. (2009) presented the self-similar solutions for ADAFs with
outflows and ordered magnetic fields. They assumed the magnetic field has a
strong toroidal component and a vertical component in addition to a stochastic
component. They found that the dynamical properties of ADAFs can be
significantly changed in the presence of ordered magnetic fields and outflows.
Abbassi et al. (2010) examined the effects of a hydrodynamical wind on ADAFs
in the presence of a toroidal magnetic field under a self-similar treatment.
Their results implied that in the presence of the wind, the disc temperature
decreases due to energy flux, which is taken away by winds and the accretion
velocity enhances.

The importance of magnetic diffusivity has been studied in several accreting
systems, such as the protostellar discs (Stone et al. 2000; Fleming & Stone
2003), discs in dwarf nova systems (Gammie & Menou 1998), the discs around
black holes (Kudoh & Kaburaki 1996), and Galactic centre (Melia & Kowalenkov
2001; Kaburaki et al. 2010). Moreover, two and three-dimensional MHD
simulations have shown that resistive dissipation is one of the crucial
processes that determines the saturation amplitude of the magnetorotational
instability (MRI). As, the linear growth rate of MRI can be reduced
significantly due to the suppression by ohmic dissipation (Sano et al. 1998;
Fleming et al. 2000; Masada & Sano 2008). Moreover, from comparison of ideal
and resistive MHD simulations, it seems the magnetic diffusivity may play an
important role in astrophysical outflows (Fendt & Čemeljić 2002; Čemeljić et
al. 2008).

As mentioned, semi-analytical studies of ADAFs with outflow are typically
related to systems without magnetic diffusivity. While, non-ideal MHD
simulations imply that the resistivity plays an important role on outflows (e.
g. Fendt & Čemeljić 2002; Čemeljić et al. 2008). Akizuki & Fukue (2006)
proposed a self-similar solution for ADAFs with a highly ionized gas. Thus,
they assumed that the plasma resistivity is zero, and only viscosity is due to
turbulence and dissipation in the disc. Also, they ignored the effects of
outflow/wind in their model. In this paper, we want to explore the effects of
the resistivity on magnetized ADAFs in the presence of outflows. Thus, we
adopt the presented solutions by Knigge (1999), Akizuki & Fukue (2006), and
Shadmehri (2008). The paper is organized as follow. In section 2, the basic
equations of constructing a model for ADAF in the presence of toroidal
magnetic field, resistivity and outflows will be defined. In section 3, self-
similar method for solving equations, which govern the behavior of the
accreting gas will be used. Results of the present model are brought in
sections 4 and 5. The summary of the model will appear in section 6.

##  2 Basic Equations

We consider a non-ideal magnetohydrodynamics of steady, axisymmetric, viscous,
accreting and rotating flow in presence of a purely toroidal magnetic field.
We use a cylindrical coordinate ($r$, $\phi$, $z$) centred on the accreting
object. We ignore the general-relativistic effects and use Newtonian gravity.
Under these assumptions, the continuity equation is

$\frac{\partial}{\partial r}(r\Sigma
v_{r})+\frac{1}{2\pi}\frac{\partial\dot{M}_{w}}{\partial r}=0,$ (1)

where $v_{r}$ is the radial velocity, $\Sigma=2\rho H$ is the surface density
of the disc, $\rho$ and $H$ being the mid-plane density and half-thickness of
the disc, respectively, and $\dot{M}_{w}$ is the mass loss rate by
outflow/wind. The half-thickness of the disc is given by
$H=c_{s}\sqrt{1+\Pi}/\Omega_{K}$, where $\Omega_{K}$ is the Keplerian angular
velocity, $c_{s}$ is the sound speed, $\Pi$ is defined below in equation (9).
It will be reduced to its traditional form of $H=c_{s}/\Omega_{K}$ in absence
of the toroidal component of magnetic field ($B_{\varphi}$; equation 8). The
sound speed is defined as $c_{s}=(p_{gas}/\rho)^{1/2}$, where $p_{gas}$ is the
gas pressure. The cumulative mass-loss rate from the disc can be written as

$\dot{M}_{w}(r)=\int^{r}_{{r_{in}}}4\,\pi\,r^{\prime}\,\dot{m}_{w}(r^{\prime})\,dr^{\prime},$
(2)

where $r_{in}$ denotes the radius at the inner edge of the disc and
$\dot{m}_{w}(r)$ is the mass-loss rate per unit area from each disc face.
Since the mass accretion rate is $\dot{M}=-2\pi r\Sigma v_{r}$, from the
equations (1) and (2), we can write

$\frac{\partial\dot{M}}{\partial r}=\frac{\partial\dot{M}_{w}}{\partial r}.$
(3)

Above equation implies that the mass accretion rate varies by radius due to
outflow. Thus, we exploit a power-law dependence for mass accretion rate as
follows (e.g. Blandford & Begelman 1999)

$\dot{M}(r)=\dot{M}(R)\left(\frac{r}{R}\right)^{s},$ (4)

where $R$ is the radius at the outer edge of the disc, $\dot{M}(R)$ is the
mass accretion rate at $R$, and $s$ is a free parameter, which for a disc
without outflow/wind, $s=0$ and in the presence of the outflow/wind, $s>0$
(e.g. Fukue 2004). The observed broad-band spectra of Sgr A∗ and soft X-ray
transients can also be fitted by ADAF models with moderate outflows, $s\sim
0.3$ – $0.4$, if the direct heating of electrons in ADAFs is efficient
(Quataert & Narayan 1999; Yuan et al. 2003). Equations (1)-(4) imply that

$\dot{m}_{w}(r)=s\,\frac{\dot{M}(R)}{4\pi
R^{2}}\left(\frac{r}{R}\right)^{s-2}.$ (5)

The radial equation of momentum is

$v_{r}\frac{dv_{r}}{dr}=r\,(\Omega^{2}-\Omega_{K}^{2})-\frac{1}{\Sigma}\frac{d}{dr}(\Sigma
c_{s}^{2})-\frac{c_{A}^{2}}{r}-\frac{1}{2\Sigma}\frac{d}{dr}(\Sigma
c_{A}^{2})$ (6)

where $\Omega$ is the angular velocity of the flow and $c_{A}$ is Alfven
speed, which is defined as $c_{A}^{2}\equiv
B_{\varphi}^{2}/(4\pi\rho)=2p_{mag}/\rho$, $p_{mag}$ being the magnetic
pressure.

The angular momentum transfer equation with consideration of the outflow/wind
can be written as (e.g. Knigge 1999)

$\Sigma
v_{r}\frac{d}{dr}(r^{2}\Omega)=\frac{1}{r}\frac{d}{dr}\left(r^{3}\nu\Sigma\frac{d\Omega}{dr}\right)-\frac{(lr)^{2}\Omega}{2\pi
r}\frac{d\dot{M}_{w}}{dr}$ (7)

where the two terms on the right-hand side of above equation describe the
effects of viscous torques due to shear ($\nu$, which is the effective
kinematic viscosity) and the angular momentum sink provided by the outflow.
Here, it will be assumed that matter outflowed at radius $r$ on the disc
carries away specific angular momentum $(lr)^{2}\Omega$. Thus, $l=0$
corresponds to a non-rotating disc wind and $l=1$ to outflowing material that
carries away the specific angular momentum it had at the point of outflow
(Knigge 1999).

The hydrostatic balance in the vertical direction is integrated to

$\frac{GM}{r^{3}}H^{2}=c_{s}^{2}\left[1+\frac{1}{2}\left(\frac{c_{A}}{c_{s}}\right)^{2}\right]=(1+\Pi)c_{s}^{2}.$
(8)

Here, we introduce the parameter $\Pi$ by

$\Pi=\frac{p_{mag}}{p_{gas}}=\frac{1}{2}\left(\frac{c_{A}}{c_{s}}\right)^{2},$
(9)

which is the degree of magnetic pressure to the gas pressure. Since we will
apply a steady self-similar method to solve system equation, this parameter
will be constant throughout the disc. Really, this parameter is a function of
position and time (Machida et al. 2006; Oda et al. 2007; Khesali & Faghei
2008, 2009). Studies of hot accretion flows represent the typical value of
$\Pi$ lies in the range $0.01$-$1$ (De Villiers et al. 2003; Beckwith et al.
2008), but here we also consider the magnetically dominated case ($\Pi>1$).
Because, MHD simulation by Machida et al. (2006) shows as thermal instability
grows in an accretion flow, the magnetic pressure exceeds the gas pressure due
to the disc shrink in the vertical direction and conservation of the toroidal
magnetic flux. This will result in large $\Pi$ and forms a magnetically
dominated accretion flow (Oda et al. 2007).

We assume both of the viscosity and the diffusivity are due to turbulence in
the disc, so that it is reasonable to use these parameters in analogy to the
$\alpha$-prescription of Shakura & Sunyaev (1973) for the turbulent

$\nu=P_{m}\eta=\alpha c_{s}H,$ (10)

where $P_{m}$ is the magnetic Prandtl number of the turbulence assumed a
constant of order of unity, $\eta$ is the magnetic diffusivity, and $\alpha$
is a free parameter less than unity.

Here, we can write the energy equation considering energy balance in the
system. We will assume the energy released due to viscous and resistive
dissipations can be balanced by the advection cooling and energy loss of
outflow (e.g. Shadmehri 2008). Thus,

$\displaystyle\frac{\Sigma
v_{r}}{\gamma-1}\frac{dc_{s}^{2}}{dr}-2Hv_{r}c_{s}^{2}\frac{d\rho}{dr}=$
$\displaystyle\frac{\alpha\sqrt{1+\Pi}fc_{s}^{2}}{\Omega_{K}}\left[\Sigma
r^{2}\left(\frac{d\Omega}{dr}\right)^{2}+\frac{H}{2\pi
P_{m}}\left(\frac{1}{r}\frac{d}{dr}(rB_{\varphi})\right)^{2}\right]$
$\displaystyle-\frac{1}{2}\xi\dot{m}_{w}(r)v_{K}^{2}(r),$ (11)

where $f$ is a constant less than unity and is called advection parameter. The
parameter $f$ measures how much the flow is advection-dominated (Narayan & Yi
1994). The first two terms on the right-hand side of above equation represent
the energy generated due to viscous and resistive dissipation, respectively.
The resistive dissipation is derived by $(4\pi/c^{2})\eta\textbf{J}^{2}$,
where $\textbf{J}=(c/4\pi)\nabla\times\textbf{B}$ is the current density.
Moreover, the last term on the right-hand side of energy equation is the
energy loss due to wind or outflow. Depending on the energy loss mechanism,
dimensionless parameter $\xi$ may change. We consider it as a free parameter
of our model so that larger $\xi$ corresponds to more energy extraction from
the disc due to the outflows (Knigge 1999; Shadmehri 2008).

The creation/escape rate of the magnetic field can be described by dynamo and
diffusion. We define the advection rate of the toroidal magnetic field as (Oda
et al. 2007)

$\dot{\Phi}=\int v_{r}B_{\varphi}dz,$ (12)

which is used instead of the induction equation. Since we study a steady-state
accreting system, the above quantity will be constant in absence of the dynamo
and the diffusion effects. This quantity can vary with radius due to the
presence of the dynamo/diffusion effect (Machida et al. 2006; Oda et al.
2007). In the present model, we expect the magnetic flux advection rate varies
with radius due to the presence of resistivity. We will consider this property
in next section.

##  3 Self-Similar Solutions

We seek self-similar solutions in the following form (e.g. Akizuki & Fukue
2006; Shadmehri 2008):

$\Sigma(r)=c_{0}\Sigma_{0}\left(\frac{r}{R}\right)^{s-\frac{1}{2}}$ (13)
$v_{r}(r)=-c_{1}\sqrt{\frac{GM}{R}}\left(\frac{r}{R}\right)^{-\frac{1}{2}}$
(14)
$\Omega(r)=c_{2}\sqrt{\frac{GM}{R^{3}}}\left(\frac{r}{R}\right)^{-\frac{3}{2}}$
(15)
$c_{s}^{2}(r)=c_{3}\left(\frac{GM}{R}\right)\left(\frac{r}{R}\right)^{-1}$
(16) $c_{A}^{2}(r)=\frac{B_{\varphi}^{2}}{4\pi\rho}=2\Pi
c_{3}\left(\frac{GM}{R}\right)\left(\frac{r}{R}\right)^{-1}$ (17)

where $\Sigma_{0}$ and $R$ are exploited to write the equations in the non-
dimensional forms. Substituting the above solutions in the continuity, radial
momentum, angular momentum, hydrostatic, and energy equations [(1),(6)-(8),
and (11)], we can obtain the following relations:

$c_{0}c_{1}=\dot{m}$ (18)
$-\frac{1}{2}c_{1}^{2}+\left[(s+\frac{1}{2})\Pi+s-\frac{3}{2}\right]c_{3}-c_{2}^{2}+1=0$
(19) $-\frac{1}{2}c_{0}\left[c_{1}-3\alpha
c_{3}(s+\frac{1}{2})\sqrt{1+\Pi}\right]+l^{2}s\dot{m}=0$ (20)
$\frac{H}{r}=\sqrt{(1+\Pi)c_{3}}$ (21) $\displaystyle\alpha\Pi
f(\gamma-1)\left(s-\frac{1}{2}\right)^{2}c_{0}c_{3}^{2}+P_{m}c_{0}c_{3}\times$
$\displaystyle\left\\{\frac{9}{2}\alpha
f(\gamma-1)c_{2}^{2}-\frac{2c_{1}}{\sqrt{1+\Pi}}\left[\left(s-\frac{3}{2}\right)\gamma-\left(s-\frac{5}{2}\right)\right]\right\\}-$
$\displaystyle\frac{2\,s\,\dot{m}\,\xi\,P_{m}(\gamma-1)}{\sqrt{1+\Pi}}=0$ (22)

where $\dot{m}$ is non-dimensional mass accretion rate and is defined

$\dot{m}=\frac{\dot{M}(R)}{2\pi\Sigma_{0}\sqrt{GMR}}.$ (23)

Using the equations (18)-(22), we obtain a quadratic equation for the
coefficient of $c_{3}$:

$\displaystyle-\frac{81}{16}\alpha^{2}(1+\Pi)\left[\frac{1+2s}{2l^{2}s-1}\right]^{2}c_{3}^{2}+$
$\displaystyle\Big{\\{}\frac{3}{(\gamma-1)f}\left[\frac{1+2s}{2l^{2}s-1}\right]\left[\left(s-\frac{3}{2}\right)\gamma-\left(s-\frac{5}{2}\right)\right]+$
$\displaystyle\frac{9}{2}\left[\left(s+\frac{1}{2}\right)\Pi+\left(s-\frac{3}{2}\right)\right]+\frac{\Pi}{P_{m}}\left(s-\frac{1}{2}\right)^{2}\Big{\\}}c_{3}+$
$\displaystyle\frac{9}{2}\left[1+\frac{2s\xi}{3f}\left(\frac{1+2s}{2l^{2}s-1}\right)\right]=0,$
(24)

and the rest of the coefficients are

$c_{0}=-\frac{2}{3}\frac{\dot{m}}{\alpha\sqrt{1+\Pi}}\left[\frac{2l^{2}s-1}{1+2s}\right]c_{3}^{-1},$
(25)
$c_{1}=-\frac{3}{2}\alpha\sqrt{1+\Pi}\left[\frac{1+2s}{2l^{2}s-1}\right]c_{3},$
(26)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}c_{2}^{2}=1-\frac{9}{8}\alpha^{2}(1+\Pi)\left[\frac{1+2s}{2l^{2}s-1}\right]^{2}c_{3}^{2}+$
$\displaystyle\left[(s+\frac{1}{2})\Pi+s-\frac{3}{2}\right]c_{3}.$ (27)

Without mass outflows, resistivity, and toroidal magnetic field, i.e.
$s=l=\xi=0$, $P_{m}=\infty$, and $\Pi=0$, above relations reduce to the
standard ADAF solutions (Narayan & Yi 1994). Moreover, in the absence of wind
and resistivity but with toroidal magnetic field, above relations reduce to
result of Akizuki & Fukue (2006). However the present model includes toroidal
magnetic field, outflows, and resistivity.

The studies of resistive and magnetized ADAFs (e. g. Faghei 2011) imply that
the solution for a set of the input parameters reaches to a non-rotating limit
at a specific of $\Pi$ which we call it by $\Pi_{c}$. Assuming $c_{2}=0$ and
$s=l=\xi=0$ (no wind case) in equations of (18)-(22), $\Pi_{c}$ can be written
as

$\Pi_{c}=\frac{18P_{m}}{f}\left[\frac{5/3-\gamma}{\gamma-1}\right].$ (28)

For typical values of adiabatic index and advection parameter in ADAFs,
$\gamma=4/3$ and $f=1$, we can write $\Pi_{c}=18P_{m}$. We cannot extend the
solutions for larger values of $\Pi_{c}$, because the right-hand side of
equation (27) becomes negative and a negative $c_{2}$ is clearly unphysical.
Moreover, $\Pi_{c}=18P_{m}$ implies that the flow can be magnetically
dominated for $P_{m}>1/18\sim 0.056$. It means the flow can be had a strong
magnetic field in the presence of resistivity. This property is in accord with
resistive MHD simulations of Machida et al. (2006).

Now, we can investigate the radial dependence of the magnetic flux advection
rate (equation 12). The self-similar solution for this quantity implies that

$\dot{\Phi}(r)=\dot{\Phi}_{out}~{}(\frac{r}{R})^{(s-3/2)/2},$ (29)

where $\dot{\Phi}_{out}$ is the magnetic flux advection rate at outer edge of
the disc, $R$. Since $s<3/2$, the magnetic flux increases with approaching to
central object and the stronger wind/outflow reduces this increasing. The
radial dependence of $\dot{\Phi}$ is qualitatively consistent with results of
Machida et al. (2006) and Oda et al. (2007).

##  4 Results

Now we can investigate the behavior of the solutions in the presence of the
outflows and resistivity. The effects of outflows and resistivity are studied
via parameters $s$, $l$, $\xi$, and $P_{m}^{-1}$. Here, the inverse of Prandtl
number specifies the resistivity of the fluid. Because,
$\eta\propto\alpha/P_{m}$ and $\alpha$ parameter is $0.1$ in all Figures. The
behavior of physical variables as a function of $P_{m}^{-1}$ are shown in
Figures 1 and 2. The solutions in Figures 1 and 2 represent the radial inflow
speed, $c_{1}$, and sound speed, $c_{3}$, both increase with the magnitude of
resistivity ($P_{m}^{-1}$). These properties are qualitatively consistent with
Faghei (2011). The density profiles, $c_{0}$, show that it decreases by adding
resistivity. It can be due to temperature raise of the flow. The rotational
velocity decreases, $c_{2}$, with the magnitude of resistivity. Because, the
viscous torque increases with the temperature ($\nu\propto c_{s}^{2}\propto
T$). These properties are also consistent with results of Faghei (2011).

Figure 1: Physical quantities of the flow as a function of $P_{m}^{-1}$ for
$\gamma=4/3$, $\alpha=0.1$, $\Pi=0.5$, and $f=l=\xi=1$. The solid, dotted,
dashed, and dot-dashed lines represent $s=0,0.01,0.02$, and $0.03$.

In Figure 1, we also studied the effect of parameter $s$ on physical
variables. The value of $s$ measures the strength of outflow and a larger $s$
denotes a stronger outflow. Figure 1 shows that for non-zero $s$, surface
density is lower than the standard ADAF solution and for stronger outflows
this reduction of surface density is more evidence. We can see that ADAF with
wind rotates more quickly than those without winds and lead to enhance
accretion velocity. The solution shows that temperature decreases for stronger
outflows. On the other hand, outflows play as a cooling agent. These
properties are in accord with results of Shadmehri (2008) and Abbassi et al.
(2010).

In Figure 2, the effect of energy loss due to outflows is studied by $\xi$
parameter. As with the magnitude of $\xi$ parameter, the more energy will
carry by outflows. Due to this energy loss, we expect the temperature should
decrease by adding the $\xi$ parameter. Temperature profiles confirmed this
property. Since the turbulence viscosity is proportional to temperature
($\nu\propto T$), the efficiency of angular momentum transport decreases with
the $\xi$ parameter. Decreasing viscous torque increases the rotational
velocity and decreases the radial infall velocity. These properties are
consistent with previous works (e. g. Shadmehri 2008).

Figure 2: Same as Figure 1, but $s=0.1$, and the solid, dotted, dashed, and
dot-dashed lines represent $\xi=0,0.1,0.2$, and $0.3$.

The physical variables as a function of parameter $\Pi$ and several values of
mass and energy losses are shown in Figures 3 and 4. By adding $\Pi$ which
indicates the role of magnetic filed on the dynamics of accretion discs, we
see the sound speed becomes larger, while surface density decreases. Moreover,
Figure 3 represents that the radial and rotational velocities increase with
the magnitude of $\Pi$. Increase in the radial velocity is due to the magnetic
tension term dominates the magnetic pressure term in the radial momentum
equation, which assists the radial velocity of accretion flows. Moreover,
increase in the rotational velocity is because of that the disc should rotate
faster than the case without the magnetic field which results the magnetic
tension. These properties are qualitatively consistent with the previous works
on magnetized ADAFs (e.g. Akizuki & Fukue 2006; Khesali & Faghei 2009; Abbassi
et al. 2010). Figures 3 and 4 imply that the mass and energy losses due to
wind give the same results by Figures 1 and 2. On the other hand, the effects
of mass and energy losses on the physical variables do not change in low and
high values of magnetic field.

Figures 5 and 6 represent that the disc thickness increases with the magnitude
of the resistivity or the magnetic field. Because the temperature increases by
adding the resistivity or the magnetic field, and from equation (21) we can
see the disc thickness increases with temperature. In Figures 5 and 6, the
disc thickness is also studied for several values of parameters $s$ and $\xi$.
The disc thickness profiles imply that it becomes thinner for the stronger
mass or energy losses due to outflows. Because these parameters reduces the
temperature of the flow.

##  5 The Bernoulli parameter

Here, we exploit Bernoulli parameter ($Be$) to consider the effects of the
resistivity to generate/enhance outflows in magnetized ADAFs. Because, this
parameter measures the likelihood that outflow or wind may originate
spontaneously (Narayan & Yi 1994). An adiabatic flow has a constant $Be$ along
streamlines. If $Be$ is positive for any of accreting gas, then this gas can
potentially reach infinity with a net positive kinetic energy. The Bernoulli
parameter, defined as the sum of the kinetic energy, the enthalpy and the
potential energy of the accretion flow,

$Be=\frac{1}{2}(v^{2}+r^{2}\Omega^{2})+\frac{\gamma}{\gamma-1}c_{s}^{2}-r^{2}\Omega_{K}^{2}.$
(30)

Figure 3: Physical quantities of the flow as a function of $\Pi$ for
$\gamma=4/3$, $\alpha=0.1$, $P_{m}=1$, $f=1$, $l=1$ and $\xi=1$. The solid,
dotted, dashed, and dot-dashed lines represent $s=0,0.01,0.03$, and $0.05$.

Narayan & Yi (1994) showed that Bernoulli parameter is positive in height-
integrated advection dominated flows, and suggested this may explain the
frequency occurrence of outflows and wind in many accretion systems. Using the
self-similar transformations of (13)-(17), the Bernoulli parameter can be
written as

$b=\frac{Be}{v_{K}^{2}}=\frac{1}{2}(c_{1}^{2}+c_{2}^{2})+\frac{\gamma}{\gamma-1}c_{3}-1,$
(31)

where $b$ is the normalized Bernoulli parameter. In Figure 7, the behavior of
this parameter as a function of $P_{m}^{-1}$ is studied for different values
of magnetic field. The $Be$ profiles represent that it is positive and
increases with the magnitude of resistivity or magnetic field. It can be due
to increase of the flow temperature by adding the resistivity or magnetic
field. Moreover, the $Be$ profiles show the magnetic field is more important
in high resistivity. Thus, the outflows can be enhanced in resistive and
magnetized ADAFs.

Figure 4: Same as Figure 3, but $s=0.1$, and the solid, dotted, dashed, and
dot-dashed lines represent $\xi=0,1,2$, and $3$.

##  6 Summary and Discussion

Mass loss appears to be a common phenomenon among accretion systems. The
observational evidences of ADAFs imply that outflow is important in such
system. Moreover, the non-ideal MHD simulation results represent that
resistivity can play an importance role in outflow of accretion discs.

In this paper, the structure of a magnetized ADAF in the presence of
resistivity and outflow is investigated. We assumed that the magnetic field
has a purely toroidal component. The outflow emanating affects on the
equations of continuity, angular momentum and energy, and can therefore act as
a sink for mass, angular momentum and energy. We adopted the presented
solutions by Knigge (1999), Akizuki & Fukue (2006), and Shadmehri (2008).
Thus, we assumed that angular momentum transport is due to viscous turbulence
and the $\alpha$-prescription is used for the kinematic coefficient of
viscosity. We also assumed that the flow does not have a good cooling
efficiency and so a fraction of energy accretes along with matter onto the
central object. To solve the equations that govern the structure behavior of
magnetized ADAF with outflow, we have used a steady self-similar solution.

Figure 5: The ratio of height thickness to radius as a function of
$P_{m}^{-1}$ for different $s$ (left panel), and $\xi$ (right panel). The
input parameters in left and right panels are same as Figures 1 and 2,
respectively.

The present model represented the outflows in accretion flows can improve by
the resistivity and magnetic field. These properties are in accord with
resistive MHD simulations of Fendt & Čemeljić (2002) and Čemeljić et al.
(2008). For example, Fendt & Čemeljić (2002) showed that resistivity can
affect the outflows structure in accretion discs. Moreover, they found that
the outflow velocity increases with the magnitude of the resistivity and
toroidal magnetic field.

In this paper, we assumed a purely toroidal magnetic field that provides
restrictions in the present model. For example, a purely toroidal magnetic
field is not enough to have magnetically driven outflows. Thus, the present
model is unsuitable to use in disc with magnetically driven outflows.
Moreover, we considered the presented model in a height-integrated approach
and applied it to calculate Bernoulli parameter. Narayan & Yi (1995) showed
that Bernoulli parameter in ADAF varies by latitude. As Bernoulli parameter in
equatorial is negative and becomes positive for high latitude. One can
investigate latitudinal behavior of the present model.

## Acknowledgments

We would like to thank referee for his/her invaluable comments and very
careful reading of the manuscript that helped us to improve the initial
version of the paper.

Figure 6: The ratio of height thickness to radius as a function of $\Pi$ for
different values of $s$ (left panel), and $\xi$ (right panel). The input
parameters in left and right panels are same as Figures 3 and 4, respectively.

Figure 7: The dimensionless of Bernoulli constant as a function of
$P_{m}^{-1}$ for different values of $\Pi$. The input parameters are same as
Figure 1, but $s=\xi=l=0$, and the solid, dotted, dashed, and dot-dashed lines
represent $\Pi=0.1,0.3,0.5$, and $0.7$, respectively.

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[a,b]Martin Beneke

# Endpoint factorization and next-to-leading power resummation of gluon thrust

Mathias Garny  Sebastian Jaskiewicz  Julian Strohm  Robert Szafron  Leonardo
Vernazza  Jian Wang

###### Abstract

Endpoint divergences in the convolution integrals appearing in next-to-
leading-power factorization theorems prevent a straightforward application of
standard methods to resum large logarithmic power-suppressed corrections in
collider physics. We study the power-suppressed configuration of the thrust
distribution in the two-jet region, where a gluon-initiated jet recoils
against a quark-antiquark pair. With the aid of operatorial endpoint
factorization conditions, we derive a factorization formula where the
individual terms are free from endpoint divergences and can be written in
terms of renormalized hard, (anti) collinear, and soft functions in four
dimensions. This framework enables us to perform the first resummation of the
endpoint-divergent SCETI observables at the leading logarithmic accuracy using
exclusively renormalization-group methods.

##  1 Introduction

The endpoint factorization and next-to-leading power (NLP) resummation of
gluon thrust employing renormalization group (RG) methods has recently been
presented in [1]. This proceeding highlights the key insights alongside
showcasing the main results. For precise definitions and technicalities of the
derivation, we direct the interested reader to the original publication.

In the past, hadronic event shape variables in the two-jet region were of
great importance in the development of diagrammatic resummation methods for
QCD [2, 3]. Later, the advent of soft-collinear effective theory (SCET)
enabled the accuracy of resummation to be further improved. This was first
shown for the thrust variable $T$ [4], defined as

$\displaystyle
T=\mbox{max}_{\vec{n}}\,\frac{\sum_{i}\left|\vec{p_{i}}\cdot\vec{n}\right|}{\sum_{i}\left|\vec{p_{i}}\right|},$
(1)

where the index $i$ sums over all final state hadrons (partons). As
$\tau=1-T\to 0$, back-to-back jets are formed by the particles, and large
logarithms $\ln\tau$ appear at every order in $\alpha_{s}$, signalling a
breakdown in perturbation theory. This phenomenon has been scrutinized in
great detail for the quark-antiquark two-jet process that contributes at
leading power (LP) in the $\tau$ expansion.

In the work presented here, we instead focus on the “gluon thrust” phase-space
region where at leading order (LO) the gluon recoils a quark-antiquark pair

$e^{+}e^{-}\to\gamma^{*}\to[g]_{c}+[q\bar{q}]_{\bar{c}}\,.$ (2)

The gluon and the quark-antiquark jets are chosen to be in the collinear and
anticollinear directions, respectively. As shown in figure 1, this process
begins at $\mathcal{O}(\alpha_{s})$ and it is of particular interest in our
investigations as in the limit of $\tau\to 0$ the LP $[\ln\tau/\tau]_{+}$
soft-gluon behaviour is absent. Instead, the process begins at NLP in the
$\tau$ expansion, with the leading term being $\alpha_{s}\ln\tau$.

The study of power corrections in a multitude of contexts has recently
gathered considerable attention. However, as we discuss below, further rapid
progress has thus far been hindered by the ubiquitous appearance of endpoint
divergences in convolution integrals between the hard, (anti-) collinear, and
soft functions in the NLP factorization theorems, such as for the Drell-Yan
threshold [5]. We address this key conceptual issue in the context of gluon
thrust, as it allows us to focus on the problem of endpoint divergences
without needing to address additional difficulties such as factorization of
parton distribution functions, which would affect Drell-Yan and DIS processes.

The result for the all-order logarithmic structure of “gluon thrust” at the
double logarithmic (DL) accuracy was first written down in [6] and later
derived from $d$-dimensional consistency relations in [7]. An interesting
feature of this result is the unconventional “quark” Sudakov form factor,
which arises due to the colour mismatch between the back-to-back energetic
particles when either the quark or anti-quark in the $q\bar{q}$-jet becomes
soft, causing the double logarithms to be proportional to the difference of
the colour charges, $C_{A}-C_{F}$. Despite the progress in the description of
this process within SCET [6, 8, 7], it has not yet been possible to improve
the resummation accuracy beyond DL order due to the presence of endpoint-
divergences in the relevant convolution integrals.

In this work, we develop the refactorization ideas employed for the DIS
process [7], and Higgs decay to two photons through light-quark loops [9, 10]
in order to derive a SCETI NLP factorization theorem valid in $d=4$
dimensions. To arrive at this result, we make use of standard factorization
within SCET combined with endpoint factorization.

Figure 1:  Diagrams contributing to gluon thrust at leading $\alpha_{s}$ order
in QCD.

##  2 Heuristic discussion

In this section, we present a sketch of the factorization formula in the EFT
framework, which aids in motivating the endpoint rearrangement and subtraction
terms introduced below. At $\mathcal{O}(\alpha_{s})$, there are two ways to
induce the gluon jet:

  1. I

Both the quark and the anti-quark can carry large anti-collinear momentum and
create a single jet recoiling against the collinear gluon.

  2. II

Either the quark or the anti-quark is anti-collinear and balances the
collinear gluon momentum, in which case the other fermion is soft.

Both situations provide identical power suppression in $\tau$, and the further
evolution of the process is governed by LP interactions. However, the
separation into possibilities I and II introduces an ambiguity in what is
precisely meant by the “soft” and “anti-collinear” modes. If we begin with
configuration I and lower the large anti-collinear momentum of either the
quark or the anti-quark, eventually it will become soft, at which point it
should in fact be counted as a contribution to situation II rather than I. As
long as the definition of the jet is infrared safe, this separation is not an
issue for fixed-order computations. It is a problem, however, if the goal is
to perform resummation since, in this case, a clear separation of soft and
(anti) collinear modes is required in order to disentangle large logarithms
into single-scale pieces. This tension between the mathematical description of
the modes and the physical picture is the origin of the endpoint-divergences
which appear in convolution integrals of the NLP factorization theorems.

We now focus on the Feynman diagrams in figure 1 and analyze possibilities I
and II from the point of view of SCET. Starting with I, later referred to as
“B-type”, we see that the internal propagator in the left diagram carries hard
momentum $q_{a}=p_{g}+p_{a}$ (since $p_{g}$ is collinear and $p_{a}$ anti-
collinear). As the intermediate propagator is hard, $q_{a}^{2}\sim Q^{2}$, it
is integrated out, and the hard scattering vertex in SCET directly produces a
$q\bar{q}g$ state. This is shown in the left-most diagram of figure 2. Only
the total momentum of the $q\bar{q}$ anti-collinear pair is fixed by momentum
conservation, while the amplitude depends on the fraction of the momentum
carried by each particle. In the case where one of these fractions becomes
vanishingly small, the corresponding parton is effectively soft and should be
counted as possibility II, which we later refer to as the “A-type”
contribution. Here, the $q_{a}$-intermediate propagator, or $q_{b}$ for the
soft anti-quark case, ceases to be hard. Therefore, the hard scattering vertex
is the LP $\gamma^{*}\to q\bar{q}$ process and the entire momentum of the
energetic quark (or anti-quark) is subsequently transferred to the gluon,
rendering the daughter fermion soft. In SCET, such a situation is described by
an insertion of a power-suppressed SCET Lagrangian interaction term
$\mathcal{L}^{(1)}_{\xi q}$, which constitutes soft (anti-) quark emission, as
depicted in the middle and right-most diagrams of figure 2.

Figure 2:  Representation of the gluon-thrust amplitude in the two-jet region
in SCET.

The fact that there exists an overlap region where the limits of two different
expressions can describe the same physical process is central to the idea of
endpoint factorization. Before we make these concepts concrete, let us start
with the following schematic discussion. In Laplace space, the factorization
theorem for two-hemisphere invariant mass distribution of gluon thrust takes
the following form:

$\displaystyle\frac{1}{\sigma_{0}}\frac{\widetilde{d\sigma}}{ds_{R}ds_{L}}$
$\displaystyle=\int_{0}^{\Lambda}d\omega
d\omega^{\prime}\,\left|C^{A0}\right|^{2}\times\mathcal{J}^{(\bar{q})}_{\bar{c}}\times\mathcal{J}_{c}\left(\omega,\omega^{\prime}\right)\otimes
S_{\rm NLP}\left(\omega,\omega^{\prime}\right)$
$\displaystyle+\int_{\Lambda/Q}^{1-\Lambda/Q}drdr^{\prime}\,C^{B1}(r)C^{B1}(r^{\prime})^{*}\otimes\mathcal{J}^{q\bar{q}}_{\bar{c}}\left(r,r^{\prime}\right)\times\mathcal{J}_{c}^{(g)}\times
S^{(g)}\,.$ (3)

In this formula, the hard matching coefficients are denoted by $C$, the jet
functions by $\mathcal{J}$ and the soft functions by $S$. In the arguments of
functions, we only retain the dependence on convolution variables which
contain divergent integrals. The $\omega$ and $\omega^{\prime}$ soft-collinear
convolutions diverge logarithmically for $\omega,\omega^{\prime}\to\infty$,
and the $r$, $r^{\prime}$ hard-anti-collinear convolution integrals are
logarithmically divergent for $r,r^{\prime}\to 0,1$.

The soft function $S_{\rm NLP}$ contains the soft quark from situation II. In
the overlap region, this soft quark carries a large soft momentum $\omega$ and
could be thought of as being a part of the $\mathcal{J}^{q\bar{q}}_{\bar{c}}$
anti-collinear function appearing in the bottom line of (3), with a small
anti-collinear momentum fraction $r$. Taking away this quark from $S_{\rm
NLP}$ leaves behind only $S^{(g)}$, therefore in total $S_{\rm NLP}\to
S^{(g)}$,
$\mathcal{J}^{(\bar{q})}_{\bar{c}}\to\mathcal{J}^{q\bar{q}}_{\bar{c}}$, and
the hard process changes from A0-type to B1-type. Hence, in these limits, the
integrands of the two terms in (3) should be identical. This fact allows us,
in the singular limits, to perform a rearrangement at the integrand level,
such that both terms are individually finite. At this stage, we can employ
standard RG techniques to resum the logarithms in the hard, jet, and soft
functions, as we show in more detail in the following sections.

##  3 Bare factorization theorem

Before focusing on endpoint factorization, we state the results of the
derivation of the $d$-dimensional SCET factorization formula for gluon thrust
[1]. To start, we integrate out the hard modes and match the electromagnetic
current to

$\displaystyle\bar{\psi}\gamma_{\perp}^{\mu}\psi(0)$ $\displaystyle=$
$\displaystyle\int dtd\bar{t}\,\widetilde{C}^{\rm
A0}(t,\bar{t})\,\bar{\chi}_{c}(tn_{+})\gamma_{\perp}^{\mu}\chi_{\bar{c}}(\bar{t}n_{-})+(c\leftrightarrow\bar{c})$
(4) $\displaystyle+\,\sum_{i=1,2}\int
dtd\bar{t}_{1}d\bar{t}_{2}\,\widetilde{C}^{\rm
B1}_{i}(t,\bar{t}_{1},\bar{t}_{2})\,\bar{\chi}_{\bar{c}}(\bar{t}_{1}n_{-})\Gamma_{i}^{\mu\nu}\mathcal{A}_{c\perp\nu}(tn_{+})\chi_{\bar{c}}(\bar{t}_{2}n_{-})+\ldots\qquad$

In the first line, we have the LP hard scattering vertex, which produces a
back-to-back quark-antiquark pair. It contributes to gluon thrust through a
time-ordered product with the $\mathcal{O}(\lambda)$ suppressed SCET
Lagrangian [11] term

$\displaystyle\mathcal{L}_{\xi
q}(x)=\bar{q}_{s}(x_{-})\not{\mathcal{A}}_{c\perp}(x)\chi_{c}(x)+\rm{h.c.}\,,$
(5)

which transforms a collinear quark into a collinear gluon and a soft quark.
The soft quark argument $x_{\mp}^{\mu}$ is defined as
$x_{\mp}^{\mu}=(n_{\pm}\cdot x)\frac{n_{\mp}^{\mu}}{2}$. In the second line of
(4), we have the $\mathcal{O}(\lambda)$ suppressed “B-type” SCET operator,
which produces a collinear gluon and an anti-collinear quark-antiquark pair
directly, see the left-most diagram in figure 2. The relevant Dirac structures
in the B-type operator are

$\Gamma_{1}^{\mu\nu}=\frac{\not{n}_{-}}{2}\gamma_{\perp}^{\nu}\gamma_{\perp}^{\mu}\,,\qquad\quad\Gamma_{2}^{\mu\nu}=\frac{\not{n}_{-}}{2}\gamma_{\perp}^{\mu}\gamma_{\perp}^{\nu}\,.$
(6)

The remaining steps of the derivation are fairly standard. The collinear,
anti-collinear and soft fields are decoupled at LP by the redefinition of the
collinear field with the soft Wilson line [12]
$Y_{n_{-}}(x)=\mathcal{P}\exp\left[ig_{s}\int_{0}^{\infty}dsn_{-}A_{s}(x+sn_{-})\right]$.
We then square the matrix elements, sum and integrate over all possible final
states. Since the final state
$|X\rangle=|X_{c}\rangle|X_{\bar{c}}\rangle|X_{s}\rangle$ is made up of
corresponding modes, the matrix element factorizes into collinear, anti-
collinear and soft functions. Following this prescription for the two terms in
(4) gives rise to the A-type and B-type contributions to the factorization
formula, for which we present the results separately. We give results for the
two-hemisphere invariant-mass distribution, which is related to the thrust
distribution by

$\frac{d\sigma}{d\tau}=\int
dM_{R}^{2}dM_{L}^{2}\,\delta\\!\left(\tau-\frac{M_{R}^{2}+M_{L}^{2}}{Q^{2}}\right)\frac{d\sigma}{dM_{R}^{2}dM_{L}^{2}}\,.$
(7)

For the A-type contribution we find

$\displaystyle\frac{1}{\sigma_{0}}\frac{d\sigma}{dM_{R}^{2}dM_{L}^{2}}|_{\rm
A-type}$ $\displaystyle=$ $\displaystyle\frac{2C_{F}}{Q}\,f(\epsilon)\,|C^{\rm
A0}(Q^{2})|^{2}\int_{0}^{\infty}dl_{+}dl_{-}\,\int d\omega
d\omega^{\prime}\,\mathcal{J}_{\bar{c}}^{(\bar{q})}(M_{R}^{2}-Ql_{+})$ (8)
$\displaystyle\times\,\bigg{\\{}\,\mathcal{J}_{c}(M_{L}^{2}-Ql_{-},\omega,\omega^{\prime})\,S_{\rm
NLP}(l_{+},l_{-},\omega,\omega^{\prime})+\,\widehat{\mathcal{J}}_{c}(M_{L}^{2}-Ql_{-},\omega,\omega^{\prime})\,\widehat{S}_{\rm
NLP}(l_{+},l_{-},\omega,\omega^{\prime})\,\bigg{\\}}\,,$

and for the B-type

$\displaystyle\frac{1}{\sigma_{0}}\frac{d\sigma}{dM_{R}^{2}dM_{L}^{2}}|_{\rm
B-type}$ $\displaystyle=$
$\displaystyle\frac{2C_{F}}{Q^{2}}\,f(\epsilon)\int_{0}^{\infty}dl_{+}dl_{-}\,\sum_{i,i^{\prime}=1,2}\int
drdr^{\prime}\,C^{\rm B1*}_{i^{\prime}}(Q^{2},r^{\prime})C^{\rm
B1}_{i}(Q^{2},r)$
$\displaystyle\times\bigg{\\{}\delta_{ii^{\prime}}\mathcal{J}_{\bar{c}}^{q\bar{q}(8)}(M_{R}^{2}-Ql_{+},r,r^{\prime})+(1-\delta_{ii^{\prime}})\mathcal{\widehat{J}}_{\bar{c}}^{q\bar{q}(8)}(M_{R}^{2}-Ql_{+},r,r^{\prime})\bigg{\\}}\mathcal{J}_{c}^{(g)}(M_{L}^{2}-Ql_{-})\,S^{(g)}(l_{+},l_{-})\,.$

In the above equations, $f(\epsilon)$ is a $d$-dimensional factor with
$f(0)=1$ in four dimensions. The precise operator definitions for all the
functions, along with the lowest-order results, can be found in sections 3.1
and 3.2 of [1] for the A and B-type parts, respectively.

###  3.1 Tree-level evaluation

It is instructive to investigate the structure of the expressions already at
the lowest order. We take the tree-level results for all required functions
from [1] and insert them into equations (8) and (3) which yields the following
expressions. The A-type soft-quark term is

$\displaystyle\frac{1}{\sigma_{0}}\frac{d^{2}\sigma}{dM^{2}_{R}\,dM^{2}_{L}}|_{\scriptsize{\begin{array}[]{l}$\rm
A--type$,\\\\[-2.84544pt] \rm
tree\end{array}}}\\!=\frac{\alpha_{s}}{4\pi}\frac{2C_{F}}{Q^{2}}\frac{f(\epsilon)e^{\epsilon\gamma_{E}}}{\Gamma(1-\epsilon)}\bigg{[}\frac{1}{\epsilon}\delta^{+}(M^{2}_{L})\left(\frac{({M^{2}_{R}})^{2}}{{Q}^{2}\mu^{2}}\right)^{\\!-\epsilon}\\!-\frac{\delta^{+}(M^{2}_{R})}{1-\epsilon}\left(\frac{(M^{2}_{L})^{2}}{Q^{2}\mu^{2}}\right)^{\\!-\epsilon}\bigg{]},$
(12)

where the $\epsilon$-pole originates from the logarithmic divergence in the
$\int_{l_{+}}^{\infty}d\omega/\omega$ integral for large $\omega$ values. The
soft-antiquark term is identical to the above. Turning to the B-type term, we
find

$\displaystyle\frac{1}{\sigma_{0}}\frac{d^{2}\sigma}{dM^{2}_{L}\,dM^{2}_{R}}|_{\scriptsize{\begin{array}[]{l}$\rm
B--type$,\\\\[-2.84544pt] \rm
tree\end{array}}}\\!=\frac{\alpha_{s}}{4\pi}\,\frac{4C_{F}}{Q^{2}}\,f(\epsilon)\,\bigg{\\{}-\frac{1}{\epsilon}+\frac{\epsilon}{(1-\epsilon)^{2}}\,\bigg{\\}}\,\delta^{+}(M^{2}_{L})\bigg{(}\frac{M^{2}_{R}}{\mu^{2}e^{\gamma_{E}}}\bigg{)}^{\\!-\epsilon}\frac{\Gamma(2-\epsilon)}{\Gamma(2-2\epsilon)}\,.$
(15)

Here, the $\epsilon$-pole is due to a logarithmic divergence in the integral
over the momentum fraction $r$ as $r\to 0$ and $r\to 1$. Summing both
contributions, and taking the limit $\epsilon\to 0$, we arrive at

$\displaystyle\frac{1}{\sigma_{0}}\frac{d^{2}\sigma}{dM^{2}_{L}\,dM^{2}_{R}}$
$\displaystyle=$
$\displaystyle\frac{\alpha_{s}C_{F}}{\pi}\,\frac{1}{Q^{2}}\,\bigg{\\{}\delta^{+}(M^{2}_{L})\left[\ln\frac{Q^{2}}{M_{R}^{2}}-1\right]-\delta^{+}(M^{2}_{R})\bigg{\\}}\,.$
(16)

After accounting for the soft-antiquark contribution to the A-type term, the
$\epsilon$-poles cancel in the sum of A-type and B-type expressions, as it is
needed since gluon-thrust is an infrared safe observable. We also reproduce
the coefficient $-\alpha_{s}C_{F}/\pi$ of the $\ln\tau$ term in [13] after
converting to thrust.

The single logarithm in (16) comes from dimensionally regulated convolution
integrals that are divergent in $d=4$. However, in order to perform
resummation, we must define renormalized hard, (anti) collinear, and soft
functions and set $\epsilon\to 0$ before the convolution integrals are
performed. This order of proceeding is at this point ill-defined, and in what
follows, we explain how this problem can be solved. Before proceeding, we make
a helpful tree-level observation. Considering (8) with tree-level expressions
for the appearing functions, we can expand the integrand in the limit of large
$\omega,\omega^{\prime}$ before performing the convolutions and arrive at the
expression

$\displaystyle\frac{\alpha_{s}}{4\pi}\,\frac{2C_{F}}{Q^{2}}\,\delta^{+}(M^{2}_{L})\,\frac{\delta(\omega-\omega^{\prime})}{\omega\omega^{\prime}}\,\frac{f(\epsilon)}{\Gamma(1-\epsilon)}\,\omega\left(\frac{M_{R}^{2}\omega}{Q\mu^{2}e^{\gamma_{E}}}\right)^{\\!-\epsilon}$
(17)

for the integrand of the $\omega,\omega^{\prime}$ integral. Similarly for the
B-type term, inserting tree-level results into (3), expanding the integrand in
the small-$r$ limit, and only then performing the $dl_{+}dl_{-}$ integrals
yields

$\displaystyle\frac{\alpha_{s}}{4\pi}\,\frac{2C_{F}}{Q^{2}}\,\delta^{+}(M^{2}_{L})\,\frac{\delta(r-r^{\prime})}{rr^{\prime}}\frac{f(\epsilon)}{\Gamma(1-\epsilon)}\,r\,\bigg{(}\frac{M^{2}_{R}r}{\mu^{2}e^{\gamma_{E}}}\bigg{)}^{\\!-\epsilon}$
(18)

for the $r,r^{\prime}$ integrand. The expression in (18) is identical to one
in (17) if we identify $r=\omega/Q$, $r^{\prime}=\omega^{\prime}/Q$. From the
heuristic discussion given in section 2, it is apparent that the agreement is
not a coincidence, but rather it must hold to all orders in $\alpha_{s}$
expansion. We formalize these statements next.

##  4 Endpoint factorization

In this section, we use the coincidence of integrands of the A-type and B-type
terms in the asymptotic limits discussed above to rearrange and factorize the
endpoint contributions so as to render the convolution integrals finite.
Similar techniques were previously used in amplitude-factorization problems to
derive endpoint factorization for exclusive $B$ decays to P-wave charmonia
[14], which uses both SCET and non-relativistic QCD, and Higgs decay to two
photons [10], which is a SCETII process. Gluon thrust, on the other hand, is a
cross-section level SCETI problem, but the main mechanism that achieves
endpoint factorization resembles the above-mentioned cases.

###  4.1 B1 matching coefficients in the soft-collinear limit

A key ingredient in the endpoint factorization discussion is the factorization
property of the $C^{{\rm{B}}1}_{i}$ coefficient function of the $q\bar{q}g$
SCET B1 operators in the $r\to 0,1$ limits. The endpoint divergence in this
contribution arises because the intermediate quark or anti-quark goes on-
shell, see figure 1. The matching coefficient in the original definition is a
single-scale, hard function. However, for $r\to 0$ (soft quark) and $r\to 1$
(soft anti-quark) cases it becomes a two-scale object and can be itself
factorized as follows [7]:

$\displaystyle\hskip 14.22636pt\displaystyle C_{1}^{\rm B1}(Q^{2},r)=C^{\rm
A0}(Q^{2})\times\frac{D^{\rm B1}(rQ^{2})}{r}+\mathcal{O}(r^{0})\,.$ (19)

Now, the two scales $Q^{2}$ and $rQ^{2}$ are separated into the LP hard
matching coefficient, and a new coefficient $D^{\rm B1}(rQ^{2})$, which
depends on the endpoint-scale $\sqrt{r}Q$.

Figure 3:  Representation of the B1 matching coefficient factorization in the
soft-quark limit given in (19).

The matching coefficient $D^{\rm B1}(p^{2})$ is a universal function that will
appear in different processes involving soft quark emission. The DL
resummation of $D^{\rm B1}(p^{2})$ has been derived in [7]. It comprises the
all-order $(C_{A}-C_{F})^{n}$ colour coefficient that appears to be
characteristic in soft quark emissions [15, 6, 7]. The identical coefficient
also enters the $ggH$ amplitude with a bottom-quark loop and was computed
recently at two-loops [16]. The $D^{\rm B1}(p^{2})$ coefficient and its
evolution equation can be obtained from the corresponding B1 operator
coefficients and anomalous dimension [17, 18] by taking the limit $r\to 0$.
The extraction of the anomalous dimension resembles to the derivation of the
asymptotic kernel for the QED light-meson light-cone distribution amplitude
[19], see also [10]. For the derivation and results, we refer to appendix A of
[1]. When the gluon is replaced by a photon, the abelian version of $D^{\rm
B1}(p^{2})$ corresponds to the jet function in the LP factorization of
$B\to\gamma\ell\nu$ [20], and in the NLP endpoint factorization of the
$H\to\gamma\gamma$ amplitude [10]. In the abelian case, the evolution up to
two loops was inferred from renormalization-group consistency of the
$B\to\gamma\ell\nu$ observable [21]. The one-loop evolution kernel of the jet
function has also been obtained by direct computation in [22].

###  4.2 Endpoint factorization consistency conditions

As motivated in section 2 and confirmed explicitly above at
$\mathcal{O}(\alpha_{s})$, we expect the integrands of the A- and B-type terms
to have the same asymptotic limits to all orders, which is a prerequisite for
endpoint factorization. Concretely, the limit of the anti-collinear momentum
$n_{-}\cdot p_{\bar{c}}=rQ,r^{\prime}Q\to 0$ in the B-type term should match
the limit $n_{-}\cdot k=\omega,\omega^{\prime}\to\infty$ of the corresponding
soft momentum component in the A-type term. The above picture is formalized by
the two refactorization conditions [1]:

(I) $\displaystyle\hskip
14.22636pt\mathcal{J}_{c}\left(p^{2},\omega,\omega^{\prime}\right)=\mathcal{J}_{c}^{(g)}(p^{2})\,\displaystyle\frac{D^{\rm
B1}(\omega Q)}{\omega}\frac{{D^{\rm
B1}}^{*}(\omega^{\prime}Q)}{\omega^{\prime}}+\mathcal{O}\\!\left(\frac{1}{\omega^{(\prime)}}\right)\,,$
(20)

and

(II)
$\displaystyle\hskip-42.67912ptQ\,\widetilde{\mathcal{J}}^{(\bar{q})}_{\bar{c}}(s_{R})\,\widetilde{S}_{\rm
NLP}\left(s_{R},s_{L},\omega,\omega^{\prime}\right)\Big{|}_{\omega^{(\prime)}\to\infty}$
(21) $\displaystyle\hskip
71.13188pt\displaystyle=\widetilde{\mathcal{J}}^{q\bar{q}(8)}_{\bar{c}}\left(s_{R},r,r^{\prime}\right)\widetilde{S}^{(g)}(s_{R},s_{L})\Big{|}_{r^{(\prime)}=\omega^{(\prime)}/Q\to
0}\,.$

For large $\omega$, the soft quark field in $S_{\rm NLP}$ turns anti-
collinear, $\bar{q}_{s}Y_{n_{-}}\to\bar{\chi}_{\bar{c}}$, so it moves from
$S_{\rm NLP}$ to $\mathcal{J}^{q\bar{q}(8)}_{\bar{c}}$. Removing
$\bar{q}_{s}Y_{n_{-}}$ from $S_{\rm NLP}$ leaves the LP soft function
$S^{(g)}$, and adding it as $\bar{\chi}_{\bar{c}}$ to
$\mathcal{J}^{(\bar{q})}_{\bar{c}}$ changes the anti-quark jet function into
$\mathcal{J}^{q\bar{q}(8)}_{\bar{c}}$. In total, this results in $S_{\rm
NLP}\mathcal{J}^{(\bar{q})}_{\bar{c}}\to
S^{(g)}\mathcal{J}^{q\bar{q}(8)}_{\bar{c}}$, which is relation (II). At the
same time, the quark fields in the A-type collinear function become highly
off-shell, which removes them from
$\mathcal{J}_{c}(p^{2},\omega,\omega^{\prime})$. This leaves behind only the
collinear gluon, so $C^{\rm A0}$ turns into $C^{\rm B1}$. Therefore, $|C^{\rm
A0}|^{2}\mathcal{J}_{c}\to|C^{\rm B1}|^{2}\mathcal{J}_{c}^{(g)}$, which is
relation (I).

###  4.3 Endpoint factorization formula

We can now state the endpoint-finite factorization formula. To do so
concisely, we use the double-bracket notation introduced in [9] to denote the
asymptotic behaviours of the various functions. In functions of
$\omega,\omega^{\prime}$, rescale $\omega\to\kappa\omega$,
$\omega^{\prime}\to\kappa\omega^{\prime}$ and take $\kappa\to\infty$. Then

$\displaystyle\llbracket S_{\rm
NLP}\left(l_{+},l_{-},\omega,\omega^{\prime}\right)\rrbracket\equiv S_{\rm
NLP}\left(l_{+},l_{-},\omega,\omega^{\prime}\right)|_{\mathcal{O}(\kappa^{0})},$
(22)
$\displaystyle\llbracket\mathcal{J}_{c}(p^{2},\omega,\omega^{\prime})\rrbracket\equiv\mathcal{J}_{c}(p^{2},\omega,\omega^{\prime})|_{\mathcal{O}(\kappa^{-2})}\,.$
(23)

The right-hand side of the above equation equals the right-hand side of the
consistency relation (I). Similarly, in functions of $r,r^{\prime}$, rescale
$r\to r\kappa$, $r^{\prime}\to r^{\prime}\kappa$ and take $\kappa\to 0$, or
the corresponding rescaling is applied to $\bar{r}$, $\bar{r}^{\prime}$. Which
of the two is meant, will be indicated by the subscript 0 or 1 on the double
bracket. Then $\llbracket C^{\rm B1}_{1}(Q^{2},r)\rrbracket_{0}\equiv C^{\rm
B1}_{1}(Q^{2},r)|_{\mathcal{O}(\kappa^{-1})}\,$, and $\llbracket C^{\rm
B1}_{2}(Q^{2},r)\rrbracket_{1}\equiv C^{\rm
B1}_{2}(Q^{2},r)|_{\mathcal{O}(\kappa^{-1})}\,$, while $\llbracket C^{\rm
B1}_{1}(Q^{2},r)\rrbracket_{1}=\llbracket C^{\rm
B1}_{2}(Q^{2},r)\rrbracket_{0}=0$. To implement the rearrangement of endpoint-
singular terms we introduce the scaleless integral

$\displaystyle\frac{2C_{F}}{Q}\,f(\epsilon)\,|C^{\rm
A0}|^{2}\widetilde{\mathcal{J}}^{(\bar{q})}_{\bar{c}}\widetilde{\mathcal{J}}_{c}^{(g)}\int_{0}^{\infty}d\omega
d\omega^{\prime}\,\frac{D^{\rm B1}(\omega Q)}{\omega}\frac{D^{\rm
B1*}(\omega^{\prime}Q)}{\omega^{\prime}}\left\llbracket\widetilde{S}_{\rm
NLP}(s_{R},s_{L},\omega,\omega^{\prime})\right\rrbracket,$ (24)

which vanishes in $d$-dimensions. We split this integral in two terms
$I_{1,2}$, $I_{1}+I_{2}=0$, with $I_{1}$ being defined by $\omega$ or
$\omega^{\prime}$ smaller than a parameter $\Lambda$ and $I_{2}$ the
complement region, as shown on the left-hand side of figure 4. In the
complement region $\omega,\omega^{\prime}>\Lambda$, and the double-bracket
asymptotic behaviour can be used for functions of $\omega,\omega^{\prime}$ in
the A-type term. The endpoint rearrangement consists of subtracting $I_{1}$
from the B-type term and $I_{2}$ from the A-type term. The subtracted
expressions are now separately endpoint-finite, but depend on $\Lambda$ which
cancels exactly between the two terms as long as we do not make further
approximations.

Figure 4:  On the left we show the split of (24) into $I_{1}+I_{2}$ according
to the correspondingly indicated regions in the $\omega-\omega^{\prime}$ plane
as described in the text below (24). In the overlap region in green the
asymptotic behaviour of the A- and B-type term must agree. On the right, we
display the Laplace-space LL gluon thrust distribution. Variation of the
initial scales as described in the text yields the light-red band.

Starting with the A-type contribution,subtracting from it the complement
region $I_{2}$ of the integral (24), and using the endpoint factorization
conditions results in

$\displaystyle\frac{1}{\sigma_{0}}\frac{\widetilde{d\sigma}}{ds_{R}ds_{L}}|_{\rm
A-type}\\!$ $\displaystyle=$ $\displaystyle\frac{2C_{F}}{Q}|C^{\rm
A0}|^{2}\,\widetilde{\mathcal{J}}_{\bar{c}}^{(\bar{q})}\,\int_{0}^{\infty}d\omega
d\omega^{\prime}\,\bigg{\\{}\,\widetilde{\mathcal{J}}_{c}(s_{L},\omega,\omega^{\prime})\,\widetilde{S}_{\rm
NLP}(s_{R},s_{L},\omega,\omega^{\prime})$ (25)
$\displaystyle-\,\theta(\omega-\Lambda)\theta(\omega^{\prime}-\Lambda)\,\llbracket\widetilde{\mathcal{J}}_{c}(s_{L},\omega,\omega^{\prime})\rrbracket\llbracket\widetilde{S}_{\rm
NLP}(s_{R},s_{L},\omega,\omega^{\prime})\rrbracket$
$\displaystyle+\;\,\widetilde{\\!\\!\widehat{\mathcal{J}}}_{\\!c}(s_{L},\omega,\omega^{\prime})\,\,\widetilde{\\!\widehat{S}}_{\rm
NLP}(s_{R},s_{L},\omega,\omega^{\prime})\,\bigg{\\}}\,,\quad$

where we set $f(\epsilon)=1$ as $\epsilon\to 0$. The remaining part $I_{1}$ of
the integral (24) is now combined with $i=i^{\prime}=1$ part of the B-type
term, and after using refactorization conditions we have

$\displaystyle\frac{1}{\sigma_{0}}\frac{\widetilde{d\sigma}}{ds_{R}ds_{L}}|_{\scriptsize{\begin{array}[]{l}$\rm
B--type$\\\\[-2.84544pt] $i=i'=1$\end{array}}}\\!$ $\displaystyle=$
$\displaystyle\frac{2C_{F}}{Q^{2}}\widetilde{\mathcal{J}}_{c}^{(g)}\,\widetilde{S}^{(g)}(s_{R},s_{L})\,\int_{0}^{\infty}drdr^{\prime}\,\bigg{[}$
(29) $\displaystyle\theta(1-r)\theta(1-r^{\prime})C^{\rm
B1*}_{1}(r^{\prime})C^{\rm
B1}_{1}(r)\,\widetilde{\mathcal{J}}_{\bar{c}}^{q\bar{q}(8)}(s_{R},r,r^{\prime})$
$\displaystyle-\big{[}1-\theta(r-\Lambda/Q)\theta(r^{\prime}-\Lambda/Q)\big{]}\,\llbracket
C^{\rm B1*}_{1}(r^{\prime})\rrbracket_{0}\,\llbracket C^{\rm
B1}_{1}(r)\rrbracket_{0}\,\llbracket\widetilde{\mathcal{J}}_{\bar{c}}^{q\bar{q}(8)}(s_{R},r,r^{\prime})\rrbracket_{0}\bigg{]}\,.$

The $i=i^{\prime}=2$ term with the anti-quark becoming soft takes a similar
form and is explicitly provided in [1].

##  5 Resummation

Solving of the necessary RG equations for the objects appearing in the above
factorization formulas is discussed in detail in sections 5.1 and 5.2 of [1].
The final result for the leading-logarithmic accurate resummed expression
after all the individual pieces combined together reads in Laplace space:

$\displaystyle\frac{1}{\sigma_{0}}\frac{\widetilde{d\sigma}}{ds_{R}ds_{L}}|_{\rm
LL}$ $\displaystyle=$ $\displaystyle
2\cdot\frac{2C_{F}}{Qs_{R}}\frac{\alpha_{s}(\mu_{c})}{4\pi}\,\,\exp\left[4C_{F}S\left(\mu_{h},\mu_{\bar{c}}\right)+4C_{A}S\left(\mu_{s},\mu_{c}\right)\right]\times\left(\frac{Q^{2}}{\mu_{h}^{2}}\right)^{\\!-2C_{F}A\left(\mu_{h},\mu_{\bar{c}}\right)}$
(30)
$\displaystyle\times\left(\frac{1}{s_{L}s_{R}e^{2\gamma_{E}}\mu_{s}^{2}}\right)^{\\!-2C_{A}A\left(\mu_{s},\mu_{c}\right)}\int_{\sigma}^{Q}\frac{d\omega}{\omega}\,\left(\frac{\omega}{s_{R}e^{\gamma_{E}}\mu_{s\Lambda}^{2}}\right)^{\\!-2\left(C_{F}-C_{A}\right)A\left(\mu_{s\Lambda},\mu_{h\Lambda}\right)}$
$\displaystyle\times\exp\left[4\left(C_{F}-C_{A}\right)S\left(\mu_{s\Lambda},\mu_{h\Lambda}\right)\right]\,\left(s_{R}e^{\gamma_{E}}Q\right)^{2C_{F}A\left(\mu_{h\Lambda},\mu_{\bar{c}}\right)+2C_{A}A\left(\mu_{c},\mu_{h\Lambda}\right)}\,,$

with the functions $S\left(\nu,\mu\right)$ and
$A_{\gamma_{i}}\left(\nu,\mu\right)$ defined as in [23]. We note that
$\mu^{2}_{h\Lambda}\sim\omega Q$ and $\mu^{2}_{s\Lambda}\sim\omega/s_{R}$ are
$\omega$-dependent scales that appear inside the integrand for reasons
explained in [1]. The importance of next-to-leading logarithms can be studied
by varying the various matching scales around the values adopted in (30). We
vary the three pairs of scales $(\mu_{h},\mu_{h\Lambda})$,
$(\mu_{c},\mu_{\bar{c}})$, $(\mu_{s},\mu_{s\Lambda})$ by a factor of $1/2$ and
2 around their default scales. Taking the minimum and maximum values to
compute the scale variation. We show this for the normalized Laplace-space
distribution $\frac{Qs}{\sigma_{0}}\frac{\widetilde{d\sigma}}{ds}$ in the
right-hand panel of figure 4 as the light-red band around the red curve (LL)
that represents (30). For comparison, the tree-level (LO) and
linear-$\beta_{0}$ truncation of the LL expression are displayed. The sizeable
scale variation in the figure emphasizes the need for NLL resummation. The
endpoint-rearranged factorization formula presented in this work provides the
starting point for this systematic improvement.

##  6 Conclusion

In this work, we summarized the derivation of a novel endpoint factorization
relation for the NLP gluon-thrust distribution in the two-jet region, as
achieved in [1]. This off-diagonal contribution contains a gluon-initiated jet
recoiling against a quark-antiquark pair, which involves subleading-power
cross-section level soft and jet functions. The framework shares many
similarities with the rearrangement employed for the resummation of the
$H\to\gamma\gamma$ bottom-loop amplitude [9, 10] and allows for the first time
to systematically remove endpoint divergences in the convolution integrals of
SCETI factorization theorems, opening the path to NLP resummation for collider
observables with soft quark emission. Employing the subtraction of endpoint
divergences with the aid of operatorial factorization conditions, we managed
to reshuffle the factorization theorem such that the individual terms are free
from endpoint divergences and can be written in terms of renormalized hard,
(anti) collinear, and soft functions in four dimensions. At this point,
standard RG techniques can be applied to obtain the resummed integrands. In
[1], we derived the anomalous dimensions of the NLP jet and soft functions
using RG consistency and endpoint factorization relations. We also calculated
the one-loop anomalous dimension for the hard matching coefficients, which
enabled us to perform the first resummation of the endpoint-divergent SCETI
observable at the LL accuracy using exclusively RG methods. We verified that
our results can recover the earlier results at DL accuracy [6, 7] and we
evaluated the numerical impact of the LL corrections. Our main result for
gluon-thrust is (30), which provides an expression for the two-hemisphere
invariant mass distribution in Laplace space. The next step in further
exploration of NLP resummation is the extension of the present work and
related soft-quark emission processes to NLL accuracy, which requires the
calculation of renormalization kernels for the NLP soft and jet functions at
order $\mathcal{O}(\alpha_{s})$, and application of the methods presented here
to address endpoint divergent issues appearing in diagonal channels of
relevant processes.

#### Acknowledgements

This work has been supported by the Excellence Cluster ORIGINS funded by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany’s Excellence Strategy –EXC-2094 –390783311. S.J. is supported by the
UK Science and Technology Facilities Council under grant ST/T001011/1. R.S. is
supported by the United States Department of Energy under Grant Contract DE-
SC0012704. L.V. is funded by Fellini Fellowship for Innovation at INFN, and by
the European Union’s Horizon 2020 research programme under the Marie
Skłodowska-Curie Cofund Action, grant agreement no. 754496\. J.W. is funded by
the National Natural Science Foundation of China (No. 12005117) and the
Taishan Scholar Foundation of Shandong province (tsqn201909011).

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