# Soft theorems in de Sitter spacetime

Pujian Mao and Kai-Yu Zhang

Soft theorems in de Sitter spacetime

Pujian Mao and Kai-Yu Zhang

Center for Joint Quantum Studies and Department of Physics,

School of Science, Tianjin University, 135 Yaguan Road, Tianjin 300350, China

Emails<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract: In this paper, we derive a soft photon theorem and a soft gluon
theorem in the de Sitter spacetime from the Ward identity of the near
cosmological horizon large gauge transformation. Taking the flat limit of the
de Sitter spacetime, the soft theorems naturally recover the corresponding
flat spacetime soft theorems.

###### Contents

  1. 1 Introduction
  2. 2 Soft photon theorem in de Sitter spacetime
    1. 2.1 Gauge fixing, boundary condition and matching
    2. 2.2 Solution space
    3. 2.3 Surface charge of large gauge symmetry
    4. 2.4 Mode expansion near the horizon
    5. 2.5 Soft photon theorem in coordinates space
    6. 2.6 Soft theorem in momentum space
    7. 2.7 Flat limit
  3. 3 Soft gluon theorem in de Sitter spacetime
    1. 3.1 Gauge fixing and boundary Condition
    2. 3.2 Solution space
    3. 3.3 Surface charge of large gauge symmetry
    4. 3.4 Mode expansion near the horizon
    5. 3.5 Soft gluon theorem and flat limit
  4. 4 Conclusion

##  1 Introduction

In the last few years there have been renewed interests on soft theorems. The
new enthusiasm comes from a purely theoretical side which resides in the
connection between soft theorems and asymptotic symmetries [1, 2, 3, 4, 5].
Soft theorems are nothing but the Ward identities of asymptotic symmetries.
Since asymptotic symmetries do not preserve the vacuum of the theory, they are
spontaneously broken. Consequently, soft particles are the Goldstone bosons
associated to the (broken) asymptotic symmetries. Because any asymptotically
flat spacetime shares the same boundary structure, Hawking, Perry, and
Strominger (HPS) argued that black holes will carry soft hairs when acting
asymptotic symmetry transformations on stationary black hole solutions [6].
The soft hairs are soft particles on the black hole horizon which are just the
Goldstone bosons of asymptotic symmetries. The newly discovered black hole
soft degree of freedom provides a possible resolution of the black hole
information paradox. While, in the original proposal of HPS, the horizon soft
particles are created from the point of view of a broken symmetry, the
relevance of the soft particles to soft theorems in curved spacetime was
examined in [7, 8]. The soft theorems in curved spacetime [7, 8] are derived
from the Ward identity of the near black hole horizon symmetry. In this paper,
we will extend the work of [7, 8] to the cosmological horizon in de Sitter
(dS) spacetime.

There are many reasons to consider soft theorems or soft limits in dS
spacetime. For instance, they provide certain relations or constraints on
different point correlation functions in dS spacetime, see, e.g., the
investigations in [9, 10, 11, 12, 13, 14, 15, 16, 17]. In this paper, we will
compute soft photon and gluon theorems in dS spacetime from the Ward identity
of the near cosmological horizon symmetry. Apart from providing a
complementary to the previous studies, such as [15, 18, 19, 20], there are
several reasons to study the soft theorems from the near horizon symmetry
point of view. The cosmological horizon, as a null hypersurface, locally is
very similar to the black hole horizon. Naively, the derivation should be just
repeating the black hole horizon cases, e.g., in [7, 8]. However, the presence
of a cosmological horizon is a consequence of the fact that the dS
cosmological universe is expanding so fast that there are events which will
never be seen by an observer inside. In this sense, the cosmological horizon
behaves more like a physical boundary.111See, for instance, [21, 22] for
considering the cosmological horizon as physical boundary to obtain the
thermodynamics of a rotating black hole by the Euclidean Schwarzschild-de
Sitter solution.  So the near cosmological horizon analysis should be
intuitively connected to the asymptotic analysis near null infinity of the
asymptotically flat spacetime. More precisely, there is well defined flat
limit from the cosmological solution in the coordinates system that we have
adopted [23]. In the flat limit, the cosmological horizon somehow becomes the
null infinity of the asymptotically flat spacetime. Hence, the soft theorems
derived in dS spacetime must recover the flat spacetime ones in the flat
limit. This is precisely we will verify in this work.222In the black hole
cases, flat spacetime soft theorems can be recovered from the curved space
ones. But the conditions to achieve that are somewhat only with mathematical
consistency. Physically, those conditions are not reasonable.

There are two main parts in this paper. In section 2, we derive a soft theorem
photon theorem from the Ward identity with respect to the near cosmological
horizon symmetry in the static patch of dS spacetime in four dimensions. We
consider only interaction of massless particles in the present work. To our
interest, the in-state $\ket{\mathrm{in}}$ is defined on $\mathcal{H}^{-}$ and
the out-state $\ket{\mathrm{out}}$ is defined on $\mathcal{H}^{+}$, see Fig. 1
for the notations. We introduce a retarded coordinates system to cover near
horizon region for $\mathcal{H}^{-}$ part of the horizon. The near horizon
symmetry and the solution space of the Maxwell theory coupled with a generic
background source is derived under properly selected gauge and boundary
conditions which are adapted from the null infinity case in the flat
spacetime. For the $\mathcal{H}^{+}$ part, one simply needs to adopt an
advanced coordinate which can be considered as a time-reverse transformation
from the retarded coordinate. Since the computations on the $\mathcal{H}^{+}$
part is exactly the same, we will not present the full details. By introduing
an antipodal matching on the bifurcation sphere $B$, one can obtain the full
near horizon symmetry and solution space on the horizon. Then we derive a soft
theorem from the Ward identity,

$\displaystyle\matrixelement{\mathrm{out}}{Q_{A}^{+}\mathcal{S}-\mathcal{S}Q_{A}^{-}}{\mathrm{in}}=0,$
(1)

where $Q^{+}_{A}$ and $Q^{-}_{A}$ are charges generating the transformation of
near horizon symmetry defined on $\mathcal{H}^{+}$ and on $\mathcal{H}^{-}$
respectively. In section 3, we derive a soft gluon theorem for a scalar matter
field coupled Yang-Mills theory, again from the Ward identity of the near
horizon symmetry. For the soft photon and gluon theorems in dS spacetime, the
dS radius has the same effect in the soft factor. Both soft theorems have a
clear flat limit, in which case, the flat spacetime soft theorems are
naturally recovered. In the last section, we give some concluding remarks.

##  2 Soft photon theorem in de Sitter spacetime

In this section, we study the near horizon behavior of a Maxwell theory
coupled with a background source in four dimensional dS spacetime, which is
described by the action

$\displaystyle
S=\int\mathrm{d}^{4}x\,\sqrt{-g}\left(\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+J_{\mu}A^{\mu}\right)\,,$
(2)

and derive the soft theorem both in coordinates space and momentum space.

Figure 1: Penrose diagram of de Sitter spacetime

We focus on the static patch in dS spacetime, see Fig. 1 for the Penrose
diagram. The line element in static coordinates $(t,r,z,\bar{z})$ is given by

$\displaystyle\mathrm{d}s^{2}=-f(r)\mathrm{d}t^{2}+f(r)^{-1}\mathrm{d}r^{2}+\Omega(r)^{2}\gamma_{z\bar{z}}\mathrm{d}z\mathrm{d}\bar{z}\,,$
(3) $\displaystyle
f(r)=\frac{r(2\ell-r)}{\ell^{2}}\,,\qquad\Omega=\ell-r\,,\qquad\gamma_{z\bar{z}}=\frac{2}{(1+z\bar{z})^{2}}\,,$
(4)

where $\ell$ is the dS radius which is related to the positive cosmological
constant by $\Lambda=3\ell^{-2}$ and we have shifted the cosmological horizon
at $r=0$. To study the theory in the near horizon region, we introduce the
retarded time coordinate
$u=t-\frac{1}{2}\ell\log\left(\frac{r}{2\ell-r}\right)$. In the
$(u,r,z,\bar{z})$ coordinates, the line element becomes

$\mathrm{d}s^{2}=-f(r)\mathrm{d}u^{2}-2\mathrm{d}r\mathrm{d}u+\Omega(r)^{2}\gamma_{z\bar{z}}\mathrm{d}z\mathrm{d}\bar{z}\,,$
(5)

which covers the $\mathcal{H}^{-}$ part of the horizon.

A similar analysis for $\mathcal{H}^{+}$ could be generated by the time-
reverse transformation of de Sitter spacetime near the bifurcation sphere $B$,
and thus in the rest of this paper, we just show the detail on
$\mathcal{H}^{-}$ case. In the retarded coordinates all non-zero components of
Christoffel are

$\displaystyle\Gamma_{uu}^{u}$
$\displaystyle=-\Gamma_{ur}^{r}=-\frac{\Omega}{\ell^{2}}=-\frac{1}{2}\partial_{r}f$
$\displaystyle\Gamma_{z\bar{z}}^{u}$
$\displaystyle=-\gamma_{z\bar{z}}\Omega=\gamma_{z\bar{z}}\Omega\partial_{r}\Omega$
(6) $\displaystyle\Gamma_{uu}^{r}$
$\displaystyle=\frac{f\Omega}{\ell}=\frac{1}{2}f\partial_{r}f$
$\displaystyle\Gamma_{z\bar{z}}^{r}$
$\displaystyle=\gamma_{z\bar{z}}f\Omega=-\gamma_{z\bar{z}}\Omega\partial_{r}\Omega$
(7) $\displaystyle\Gamma_{rz}^{z}$
$\displaystyle=\Gamma_{r\bar{z}}^{\bar{z}}=-\Omega^{-1}=\Omega^{-1}\partial_{r}\Omega$
$\displaystyle\Gamma_{zz}^{z}$
$\displaystyle=\left(\Gamma_{\bar{z}\bar{z}}^{\bar{z}}\right)^{*}=\gamma_{z\bar{z}}^{-1}\partial_{z}\gamma_{z\bar{z}}\,.$
(8)

###  2.1 Gauge fixing, boundary condition and matching

In this section, we take the radial gauge and the near horizon fall-off
conditions for the gauge fields and the conserved current are

$\displaystyle A_{r}$ $\displaystyle=0$ $\displaystyle A_{u}$
$\displaystyle=\mathcal{O}(r)$ $\displaystyle A_{z}$
$\displaystyle=\mathcal{O}(1)$ (9) $\displaystyle J_{r}$ $\displaystyle=0$
$\displaystyle J_{u}$ $\displaystyle=\mathcal{O}(1)$ $\displaystyle J_{z}$
$\displaystyle=\mathcal{O}(1).$

It is easy to see that the only residual gauge transformation is $\delta
A_{\mu}=\partial_{\mu}\epsilon(z,\bar{z})$, where $\epsilon(z,\bar{z})$ is an
arbitrary differentiable function defined on the horizon. Because of the non-
vanishing volume in phase space for large gauge symmetry[24], this symmetry is
non-trivial and leads to the infrared relation between near horizon physics
and soft theorem.

The surface charge of the asymptotic symmetry could be constructed by the
lower form through the generalized Noether’s second theorem [25]. We define
the surface charge on the bifurcation sphere of the horizon

$Q_{\epsilon}=\int_{B}\epsilon\star F,$ (10)

which generates the large gauge transformation in the solution space. In the
null infinity case [4], the antipodal matching condition is imposed to
preserve the conservation law between two null infinity, because the spatial
infinity $i^{0}$ is not a common boundary of $\mathcal{I}^{+}$ and
$\mathcal{I}^{-}$. But in this work, horizons live in a finite region, and the
bifurcation sphere $B$ is well defined in spacetime without any singularity
nor ambiguity. One can consistently study the symmetries near the bifurcation
sphere [26]. The conservation law of the gauge theories should hold
automatically without any additional matching condition when $\mathcal{H}^{-}$
deforms to $\mathcal{H}^{+}$ as a Cauchy slice. However, we work in the
retarded coordinates $(u,r,z,\bar{z})$ or the advanced coordinates
$(v,r,z,\bar{z})$ on $\mathcal{H}^{+}$. Neither of them can really cover the
bifurcation point $B$. Thus, the charges computed in the two coordinates are
not connected to each other automatically. To reveal the conservation law on
the horizon, we introduce a horizon antipodal matching

$F_{ru}^{0}(+\infty,0,z,\bar{z})=F_{rv}^{0}(-\infty,0,z,\bar{z})\,,$ (11)

following the standard treatment at null infinity. As one can see from the
solution space in the next section, the initial data of the gauge field is
completely free. Thus, the matching condition can always be consistently
imposed. Then the charge can be defined on the bifurcation sphere $B$ as a
limiting case

$\displaystyle Q_{\epsilon}=$
$\displaystyle\lim_{u\rightarrow+\infty}\int_{B}\mathrm{d}^{2}z\ell^{2}\gamma_{z\bar{z}}\epsilon(z,\bar{z})F_{ru}(u,0,z,\bar{z})$
(12) $\displaystyle=$
$\displaystyle\lim_{v\rightarrow-\infty}\int_{B}\mathrm{d}^{2}z\ell^{2}\gamma_{z\bar{z}}\epsilon(z,\bar{z})F_{rv}(v,0,z,\bar{z})\,.$

###  2.2 Solution space

Under the previously introduced gauge and near horizon fall-off conditions, we
can arrange all equations of motion and constraints into several parts.

  * •

Continuity equation. We treat the matter current with no dynamics, as a
consequence this current must be conserved.

$\displaystyle\nabla^{\mu}J_{\mu}=0\,.$ (13)

  * •

Hyper-surface equation

$\displaystyle\nabla_{\mu}F^{\mu u}=0\,.$ (14)

  * •

Standard equations

$\displaystyle\nabla_{\mu}F^{\mu z}=J^{z}\,,\
\nabla_{\mu}F^{\mu\bar{z}}=J^{\bar{z}}\,,$ (15)

  * •

Supplementary equation

$\displaystyle\nabla_{\mu}F^{\mu r}=J^{r}\,.$ (16)

The previous gauge fixing does not eliminate all the gauge freedom, the
remaining gauge symmetry also connects the equations of motion by the Noether
identity

$\displaystyle\nabla_{\nu}(J^{\nu}-\nabla_{\mu}F^{\mu\nu})=0\,,$ (17)

which indicates if the equations (13)–(15) are fulfilled, one have

$\displaystyle\partial_{r}\left[\sqrt{-g}\left(J^{r}-\nabla_{\mu}F^{\mu
r}\right)\right]=0\,.$ (18)

The $r$-dependence parts of the supplementary equation are satisfied
automatically and the only indeterminate equation is at $\mathcal{O}(1)$. All
independent equations of motion yield the solutions as

$\displaystyle J_{u}$
$\displaystyle=\Omega^{-2}\int_{0}^{r}\mathrm{d}r^{\prime}\gamma_{z\bar{z}}^{-1}(\partial_{z}J_{\bar{z}}+\partial_{\bar{z}}J_{z})+\Omega^{-2}J_{u}^{0}\,,$
(19) $\displaystyle A_{u}$
$\displaystyle=\int_{0}^{r}\mathrm{d}r^{\prime}\Omega^{-2}\int_{0}^{r^{\prime}}\mathrm{d}r^{\prime\prime}\gamma_{z\bar{z}}^{-1}\partial_{r^{\prime}}(\partial_{z}A_{\bar{z}}+\partial_{\bar{z}}A_{z})+\left(\Omega^{-1}-\ell^{-1}\right)A_{u}^{0}\,,$
(20) $\displaystyle J_{u}^{0}$
$\displaystyle=-\gamma_{z\bar{z}}^{-1}\Omega^{-2}\partial_{u}\left(\partial_{z}A_{\bar{z}}^{0}+\partial_{\bar{z}}A_{z}^{0}\right)-\partial_{u}A_{u}^{0}\,,$
(21) $\displaystyle J_{z}$
$\displaystyle=-2\partial_{u}\partial_{r}A_{z}+\partial_{r}(f\partial_{r}A_{z})+\partial_{r}\partial_{z}A_{u}+\Omega^{-2}\partial_{z}\left[\gamma_{z\bar{z}}^{-1}(\partial_{\bar{z}}A_{z}-\partial_{z}A_{\bar{z}})\right]\,,$
(22) $\displaystyle J_{\bar{z}}$
$\displaystyle=-2\partial_{u}\partial_{r}A_{\bar{z}}+\partial_{r}(f\partial_{r}A_{\bar{z}})+\partial_{r}\partial_{\bar{z}}A_{u}+\Omega^{-2}\partial_{\bar{z}}\left[\gamma_{z\bar{z}}^{-1}(\partial_{z}A_{\bar{z}}-\partial_{\bar{z}}A_{z})\right]\,,$
(23)

where $A^{0}_{u}$ is an integral constant, defined by
$A^{0}_{u}=\ell^{2}\partial_{r}A_{u}|_{r=0}$, and $A_{z}^{0}=A_{z}|_{r=0}$,
$A_{\bar{z}}^{0}=A_{\bar{z}}|_{r=0}$, $J_{u}^{0}=\ell^{2}J_{u}|_{r=0}$. One
can see from the standard equations that $A_{\bar{z}}^{0}$ and $A_{z}^{0}$ do
not have constraints on their time evolution. They are the news functions of
this system that indicating the local propagating degree of freedom. It is
obvious that $A_{u}$ is totally determined by the integration constant and
$A_{z}$, $A_{\bar{z}}$ which generate the whole solution space of the near
horizon Maxwell theory. Any differentiable scalar function $\epsilon$ defined
on a $2$-sphere can be used to transform $A_{\mu}$ through a large gauge
transformation of $\delta A_{\mu}=\partial_{\mu}\epsilon(z,\bar{z})$,
resulting in an isomorphism of the solution space.

###  2.3 Surface charge of large gauge symmetry

We express the surface charge as

$\displaystyle
Q^{-}_{\epsilon}=\int_{\mathcal{H}^{-}}\mathrm{d}^{2}z\mathrm{d}u\
\gamma_{z\bar{z}}\epsilon\left[\gamma_{z\bar{z}}^{-1}\partial_{u}\left(\partial_{z}A_{\bar{z}}^{0}+\partial_{\bar{z}}A_{z}^{0}\right)+J_{u}^{0}\right]\,,$
(24)

where the solutions of the equations of motion are inserted. One can split
this charge into two parts

  * •

Soft Charge.

$\displaystyle Q_{S}^{-}=\int_{\mathcal{H}^{-}}\mathrm{d}^{2}z\mathrm{d}u\
\epsilon\partial_{u}\left(\partial_{z}A_{\bar{z}}^{0}+\partial_{\bar{z}}A_{z}^{0}\right)\,.$
(25)

If the symmetry parameter $\epsilon$ is a constant at the boundary, the
integrand is exactly a total derivative for some functions, and thus this
charge is trivial. But for the nontrivial case where $\epsilon$ is a function
defined on the horizon, this charge serves a key role in the infrared relation
between asymptotic symmetry and the soft theorem.

  * •

Hard Charge.

$\displaystyle Q^{-}_{H}=\int_{\mathcal{H}^{-}}\mathrm{d}^{2}z\mathrm{d}u\
\gamma_{z\bar{z}}\epsilon J_{u}^{0}\,.$ (26)

This part acts on $\ket{\mathrm{in}}$ linearly, and the state of incoming
particles defined on Hilbert space are eigenstates of the hard charge.

The action of the large gauge transformation on the gauge field can be
recovered from the canonical commutation relation. The symplectic form of the
free Maxwell field, defined on the $\mathcal{H}^{-}$ as a Cauchy surface, is

$\displaystyle\Omega_{\mathcal{H}^{-}}$
$\displaystyle=\int_{\mathcal{H}^{-}}\delta A\wedge\delta(\star F)$ (27)
$\displaystyle=\int_{\mathcal{H}^{-}}\mathrm{d}u\mathrm{d}^{2}z\delta\partial_{u}A_{\bar{z}}^{0}\wedge\delta
A_{z}^{0}+\delta\partial_{u}A_{z}^{0}\wedge\delta A_{\bar{z}}^{0}\,,$

which yields the canonical commutation relation

$\displaystyle\left[\partial_{u}A_{\bar{z}}^{0}(u,z,\bar{z}),A_{z}^{0}(g,w,\bar{w})\right]=-i\delta(u-g)\delta^{2}(z-w)\,.$
(28)

Thus

$\displaystyle\left[Q_{\epsilon}^{-},A_{z}^{0}(u,z,\bar{z})\right]=i\partial_{z}\epsilon(z,\bar{z})\,,$
(29)

which reveals that $Q_{\epsilon}^{-}$ generates the gauge transformation of
near horizon symmetry, as anticipated.

If we assume no magnetic monopole and long-range magnetic force [4],
$\mathrm{d}F=0$, $F_{z\bar{z}}|_{u\rightarrow-\infty}=0$, one can find a
scalar field $N$ such that

$\displaystyle\partial_{z}N=\int\mathrm{d}u\partial_{u}A_{z}^{0}\,,\qquad\partial_{\bar{z}}N=\int\mathrm{d}u\partial_{u}A_{\bar{z}}^{0}\,,$
(30)

and thus a useful form of soft charge is

$\displaystyle
Q_{S}^{-}=-2\int\mathrm{d}^{2}z\partial_{\bar{z}}\epsilon\partial_{z}N\,.$
(31)

###  2.4 Mode expansion near the horizon

Now we derive the mode expansions of the gauge field which are the main
ingredients for deriving a soft theorem. It is convenient to write down the
mode expansion of the free field operator under the isotropic coordinates
$(t,x_{1},x_{2},x_{3})$, in which the line element of the dS spacetime is

$\displaystyle\mathrm{d}s^{2}=-\left(\frac{\ell^{2}-\rho^{2}}{\ell^{2}+\rho^{2}}\right)^{2}\mathrm{d}t^{2}+\mathrm{\left(\frac{2\ell^{2}}{\ell^{2}+\rho^{2}}\right)^{2}d}\vec{x}\cdot\mathrm{d}\vec{x}\,,$
(32) $\displaystyle t=u+\frac{1}{2}\ell\log$
$\displaystyle\left(\frac{r}{2\ell-r}\right)\,,\quad
x_{1}=\rho\frac{z+\bar{z}}{z\bar{z}+1}\,,\quad
x_{2}=\rho\frac{z-\bar{z}}{i(z\bar{z}+1)}\,,\quad
x_{3}=\rho\frac{1-z\bar{z}}{z\bar{z}+1}\,,$ (33)

where

$\displaystyle r=\ell\frac{(\ell-\rho)^{2}}{\ell^{2}+\rho^{2}}\,.$ (34)

In the static patch of dS spacetime, $\partial_{t}$ is a timelike Killing
vector which suggests to define a positive energy of a particle as
$\omega=-p_{0}$. The dispersion relation for massless particles in the
isotropic coordinates is

$\displaystyle-\left(\frac{\rho^{2}+\ell^{2}}{\rho^{2}-\ell^{2}}\right)^{2}\omega^{2}+\left(\mathrm{\frac{\ell^{2}+\rho^{2}}{2\ell^{2}}}\right)^{2}\vec{p}^{2}=0\,.$
(35)

One can find a covariant measure of one-particle phase space

$\displaystyle\int\frac{\mathrm{d}\omega\mathrm{d}^{3}\vec{p}}{(2\pi)^{3}}\delta(p^{\mu}p_{\mu})\Theta(\omega)=\left(\frac{\ell^{2}-\rho^{2}}{\ell^{2}+\rho^{2}}\right)^{2}\int\frac{\mathrm{d}^{3}\vec{p}}{(2\pi)^{3}2\omega}\,.$
(36)

For the free massless scalar field $\phi(x)$, we can write this operator as

$\displaystyle\phi(x^{\mu})=\left(\frac{\ell^{2}-\rho^{2}}{\ell^{2}+\rho^{2}}\right)^{2}\int\frac{\mathrm{d}^{3}\vec{p}}{(2\pi)^{3}2\omega}\left[\mathfrak{a}(p)e^{ip\cdot
x}+\mathfrak{a}(p)^{\dagger}e^{-ip\cdot x}\right]\,.$ (37)

The extra factor
$\left(\frac{\ell^{2}-\rho^{2}}{\ell^{2}+\rho^{2}}\right)^{2}$ comes from the
gravitational effect through the dispersion relation. Literally, this mode
expansion is the flat spacetime one with the correction from the dispersion
relation of the null momentum. The full mode expansion of scalar field in dS
spacetime can be performed perturbatively, see, e.g., in [20]. However, for
the gauge theories, they are the exact mode expansions since the dS spacetime
is conformally flat and the gauge theories are conformal invariance in four
dimensions at the classical level. The mode expansion for the Maxwell field
$A_{\mu}$ with two polarization vectors that are perpendicular to the
direction of propagation is

$\displaystyle
A_{\mu}(x^{\mu})=\left(\frac{\ell^{2}-\rho^{2}}{\ell^{2}+\rho^{2}}\right)^{2}\sum_{\alpha=\pm}\int\frac{\mathrm{d}^{3}\vec{p}}{(2\pi)^{3}2\omega}\left[\epsilon^{*\alpha}_{\mu}a_{\alpha}(p)e^{ip\cdot
x}+\epsilon^{\alpha}_{\mu}a_{\alpha}^{\dagger}(p)e^{-ip\cdot x}\right]\,.$
(38)

One can parameterize the photon momentum as

$\displaystyle
p_{\mu}=\frac{\omega}{1+\zeta\bar{\zeta}}\frac{2\ell^{2}}{\rho^{2}-\ell^{2}}\left(\frac{\rho^{2}-\ell^{2}}{2\ell^{2}}(1+\zeta\bar{\zeta}),(\zeta+\bar{\zeta}),-i(\zeta-\bar{\zeta}),(1-\zeta\bar{\zeta})\right)\,,$
(39)

and similarly construct the polarization vectors as

$\displaystyle\epsilon^{+\mu}$
$\displaystyle=\frac{1}{\sqrt{2}}\frac{\ell^{2}+\rho^{2}}{2\ell^{2}}\left(\frac{2\ell^{2}}{\rho^{2}-\ell^{2}}\bar{\zeta},1,-i,-\bar{\zeta}\right)\,,$
(40) $\displaystyle\epsilon^{-\mu}$
$\displaystyle=\frac{1}{\sqrt{2}}\frac{\ell^{2}+\rho^{2}}{2\ell^{2}}\left(\frac{2\ell^{2}}{\rho^{2}-\ell^{2}}\zeta,1,i,-\zeta\right)\,,$
(41)

which satisfies

$\displaystyle\epsilon^{\alpha\mu}\epsilon^{\beta*}_{\mu}=\delta^{\alpha\beta}\,,\qquad
p_{\mu}\epsilon^{\alpha\mu}=0\,,\qquad\epsilon^{\alpha}_{r}=0\,.$ (42)

Projecting those two vectors to the sphere, we have

$\displaystyle\epsilon_{\bar{z}}^{+}$
$\displaystyle=\frac{2\sqrt{2}\ell^{2}\rho}{(1+\zeta\bar{\zeta})\left(\ell^{2}+\rho^{2}\right)}\,,$
$\displaystyle\epsilon_{z}^{-}$
$\displaystyle=\frac{2\sqrt{2}\ell^{2}\rho}{(1+\zeta\bar{\zeta})\left(\ell^{2}+\rho^{2}\right)}\,.$
(43)

To work on the energy representation, we should perform an integral to remove
the degree of freedom with respect to the space angle

$\displaystyle\int\frac{\mathrm{d}^{3}p}{2\omega}\epsilon^{*\alpha}_{\mu}a_{\alpha}(p)e^{ip\cdot
x}=$
$\displaystyle\left(\frac{2\ell^{2}}{\rho^{2}-\ell^{2}}\right)^{3}\int\pi\omega\mathrm{d}\omega\epsilon^{*\alpha}_{\mu}(\omega\hat{x})a_{\alpha}(\omega\hat{x})$
(44) $\displaystyle\times\int\sin\mathrm{\theta
d}\theta\exp\left[i\omega\left(-t+\frac{2\ell^{2}}{\rho^{2}-\ell^{2}}\rho\cos\theta\right)\right]\,.$

This allows for an exact computation of the integral,

$\displaystyle\int\sin\mathrm{\theta d}\theta\exp$
$\displaystyle\left[i\omega\left(-t+\frac{2\ell^{2}}{\rho^{2}-\ell^{2}}\rho\cos\theta\right)\right]$
(45) $\displaystyle=$
$\displaystyle\frac{1}{\rho\omega}\left(\frac{r}{2\ell-r}\right)^{-i\ell\omega/2}\left(\frac{\rho^{2}-\ell^{2}}{2\ell^{2}}\right)\times
e^{-i\omega
u}\sin\left[\frac{2\ell^{2}}{\rho^{2}-\ell^{2}}\rho\omega\right]\,.$

Finally, the near horizon transverse field operator $A^{0}_{z}$ is expressed
in terms of plane modes

$\displaystyle A_{z}^{0}(x)=\frac{2\sqrt{2}}{\pi^{2}(1+z\bar{z})}$
$\displaystyle\frac{\ell^{6}}{\left[\ell^{2}+\left(R+\ell\right)^{2}\right]^{3}}\int\mathrm{d}\omega\left(\frac{r}{2\ell-r}\right)^{-i\ell\omega/2}$
(46)
$\displaystyle\times\sin\left[\frac{2\ell^{2}(R+\ell)\omega}{R(R+2\ell)}\right]\left[e^{\mathrm{-i\omega}u}a_{+}(\omega\hat{x})+e^{\mathrm{i\omega}u}a_{-}^{\dagger}(\omega\hat{x})\right]\,,$

where $R=\rho-\ell$, and $R=0$ at the horizon. Following the treatment in [7,
8], we introduce a near horizon regularization $R\rightarrow R+i\mathcal{R}$
to deal with the divergence from the near horizon limit.

###  2.5 Soft photon theorem in coordinates space

Following previous treatment [4, 7], we choose $\epsilon=\frac{1}{z-w}$, which
leads to

$\displaystyle\partial_{\bar{z}}\epsilon(z,\bar{z})=2\pi\delta^{2}(z-w)\,.$
(47)

Hence, the soft charge (31) has the form in terms of annihilation and creation
operators

$\displaystyle Q_{S}^{-}$ $\displaystyle=-8\pi^{2}i\lim_{\omega\rightarrow
0}\left[\omega\widetilde{A_{z}^{0}}\right]\,,$ (48)

where $\widetilde{A_{z}^{0}}$ is defined by Fourier relation
$A_{z}^{0}=\int_{-\infty}^{+\infty}\mathrm{d}\omega
e^{iu\omega}\widetilde{A_{z}^{0}}$, and thus

$\displaystyle
Q_{S}^{-}=-\frac{i2\sqrt{2}}{1+w\bar{w}}\frac{\ell^{2}}{R}\lim_{\omega\rightarrow
0}\left(\omega^{2}a_{+}(\omega\hat{x})+\omega^{2}a_{-}^{\dagger}(\omega\hat{x})\right)\,.$
(49)

The soft charge acts on the in-state as

$Q_{S}^{-}\ket{\mathrm{in}}=-\frac{i2\sqrt{2}}{1+w\bar{w}}\frac{\ell^{2}}{R}\lim_{\omega\rightarrow
0}\omega^{2}a_{-}^{\dagger}(\omega\hat{x})\ket{\mathrm{in}}\,.$ (50)

Regarding to the action of the hard charge on the state, for simplicity, we
consider a complex scalar field source. The conserved current in terms of
complex scalar fields is

$J^{0}_{u}=iQ\left(\phi^{0}\partial_{u}\bar{\phi}^{0}-\bar{\phi}^{0}\partial_{u}\phi^{0}\right)\,.$
(51)

The symplectic form of a free scalar theory on the horizon is

$\displaystyle\Omega_{\mathcal{H}^{-}}=\int_{\mathcal{H}^{-}}\mathrm{d}u\mathrm{d}^{2}z\gamma_{z\bar{z}}\ell^{2}\left(\delta\partial_{u}\bar{\phi}^{0}_{k}\wedge\delta\phi^{0}_{k}+\delta\partial_{u}\phi^{0}\wedge\delta\bar{\phi}^{0}_{k}\right)\,,$
(52)

which yields

$\displaystyle\left[\partial_{u}\bar{\phi}^{0}_{k}(u,z,\bar{z}),\phi^{0}_{k}(g,w,\bar{w})\right]=-i\gamma^{-1}_{z\bar{z}}\ell^{-2}\delta^{2}(z-w)\delta(u-g)\,.$
(53)

Thus the commutation relation between the hard charge and the scalar fields
defined on $\mathcal{H}^{-}$ is

$\displaystyle\left[Q_{H}^{-},\phi^{0}_{k}(u_{k},z_{k},\bar{z}_{k})\right]=Q_{k}\epsilon(z_{k},\bar{z}_{k})\phi^{0}_{k}(u_{k},z_{k},\bar{z}_{k})\,.$
(54)

Hence

$\displaystyle
Q_{H}^{-}\ket{\mathrm{in}}=\sum_{k=1}^{n}Q_{k}\epsilon(z_{k},\bar{z}_{k})\ket{\mathrm{in}}\,.$
(55)

Then the action of the hard charge on the in-state is

$Q_{H}^{-}\ket{\mathrm{in}}=\sum_{k=1}^{n}\frac{Q_{k}^{\mathrm{in}}}{z_{k}^{\mathrm{in}}-w}\ket{\mathrm{in}}\,,$
(56)

Finally, combining those terms and considering also the out-state part, the
Ward identity in the coordinates space yields the soft theorem as

$\displaystyle\lim_{\omega\rightarrow
0}\matrixelement{\mathrm{out}}{a_{+}\mathcal{S}-\mathcal{S}a_{-}^{\dagger}}{\mathrm{in}}$
$\displaystyle=i\frac{R}{2\ell^{2}\omega}\frac{1+w\bar{w}}{\sqrt{2}\omega}\left[\sum_{l=1}^{m}\frac{Q_{l}^{\mathrm{out}}}{w-z_{l}^{\mathrm{out}}}-\sum_{k=1}^{n}\frac{Q_{k}^{\mathrm{in}}}{w-z_{k}^{\mathrm{in}}}\right]\matrixelement{\mathrm{out}}{\mathcal{S}}{\mathrm{in}}\,.$
(57)

###  2.6 Soft theorem in momentum space

From the null parameterization of the incoming hard momenta in the near
horizon region,

$\displaystyle
p_{k\mu}^{\mathrm{in}}=\frac{E_{k}^{\mathrm{in}}}{1+z_{k}^{\mathrm{in}}\bar{z}_{k}^{\mathrm{in}}}\frac{2\ell^{2}}{\rho^{2}-\ell^{2}}\left(\frac{\ell^{2}-\rho^{2}}{2\ell^{2}}(1+z_{k}^{\mathrm{in}}\bar{z}_{k}^{\mathrm{in}}),(z_{k}^{\mathrm{in}}+\bar{z}_{k}^{\mathrm{in}}),-i(z_{k}^{\mathrm{in}}-\bar{z}_{k}^{\mathrm{in}}),(1-z_{k}^{\mathrm{in}}\bar{z}_{k}^{\mathrm{in}})\right)\,.$
(58)

it is easy to show that

$\displaystyle\sum_{k=1}^{n}Q_{k}\frac{p_{k}^{\mathrm{in}}\cdot\epsilon}{p_{k}^{\mathrm{in}}\cdot
p^{\mathrm{in}}}=\sum_{k=1}^{n}\frac{R}{\ell}\frac{(1+w\bar{w})}{\sqrt{2}\omega}\frac{Q_{k}^{\mathrm{in}}}{w-z_{k}^{\mathrm{in}}}\,.$
(59)

Similar relation can be derived for the outgoing hard momenta. Those relations
allow us to write the soft photon theorem in momentum space as

$\displaystyle\lim_{\omega\rightarrow
0}\matrixelement{\mathrm{out}}{a_{+}\mathcal{S}-\mathcal{S}a_{-}^{\dagger}}{\mathrm{in}}$
$\displaystyle=\frac{i}{2\omega\ell}\left[\sum_{l=1}^{m}Q_{l}^{\mathrm{out}}\frac{p_{l}^{\mathrm{out}}\cdot\epsilon}{p_{l}^{\mathrm{out}}\cdot
p}-\sum_{k=1}^{n}Q_{k}^{\mathrm{in}}\frac{p_{k}^{\mathrm{in}}\cdot\epsilon}{p_{k}^{\mathrm{in}}\cdot
p}\right]\matrixelement{\mathrm{out}}{\mathcal{S}}{\mathrm{in}}\,.$ (60)

###  2.7 Flat limit

A limit of the spacetime geometries is in general a subtle issue [27]. For
instance, the flat limit $\ell\to\infty$ is not well defined for the line
element (5). While, the flat limit is well defined for the case using $\Omega$
as the radial coordinate, where the line element is

$\mathrm{d}s^{2}=(\frac{\Omega^{2}}{\ell^{2}}-1)\mathrm{d}u^{2}-2\mathrm{d}\Omega\mathrm{d}u+\Omega^{2}\gamma_{z\bar{z}}\mathrm{d}z\mathrm{d}\bar{z}\,,\quad\Omega=\ell-r\,.$
(61)

The location of the cosmological horizon in term of $\Omega$ is simply
$\Omega=\ell$. Hence, the flat limit yields that the cosmological horizon
becomes the null infinity of the flat spacetime.

Regarding to the flat limit of the soft theorem in dS spacetime, it is more
reasonable to consider the momentum space one due to the subtleties from the
coordinates in the spacetime limit. It is obvious from (60) that the soft
theorem in the flat limit recovers the flat spacetime soft photon theorem. The
only condition is that the dS radius multiplied by the soft energy is a finite
constant in the flat limit, see also [28] for relevant discussions about the
flat limit of AdS spacetime and soft limit. This constraint can be understood
as follows. In terms of the cosmological constant $\Lambda$, the flat limit is
only physically reasonable when $\Lambda$ approaching zero as fast as the soft
energy $\omega$. Note that any constant prefactor can be absorbed in the null
parametrization (39) and the imaginary unit $i$ would not change the physical
observable cross section. The imaginary unit $i$ can also be understood as a
regularization in the flat limit in terms of $\Lambda$. Similar to the black
hole cases [7, 8], $\Lambda=0$ means there is no horizon in the spacetime at
all. Since our derivation is based on the near horizon symmetry, the flat
limit should not be taken directly, but by introducing a regularization
$\Lambda\to\Lambda+i\lambda$. Then the condition for $\lambda$ in the flat
limit is that it should be proportional to the soft energy $\omega$.

##  3 Soft gluon theorem in de Sitter spacetime

In this section, we explore the soft gluon theorem from the Ward identity with
respect to the near horizon symmetry of a scalar matter field coupled Yang-
Mills theory in dS spacetime. For generality, we do not specify the gauge
group. The theory is described by the action

$\displaystyle
S=\int\mathrm{d}^{4}x\,\sqrt{-g}\left(\frac{1}{4}F^{a\mu\nu}F^{a}_{\mu\nu}+D_{\mu}\phi^{a}D^{\mu}\bar{\phi}^{a}\right)\,.$
(62)

The scalar fields $\phi^{a}$ are in the adjoint representation of the gauge
group,

$\displaystyle\left[T^{a},T^{b}\right]=C^{abc}T^{c}\,,\qquad\left(T^{a}\right)_{bc}=C^{abc}\,,\qquad\
(D_{\mu}\phi)^{a}=\nabla_{\mu}\phi^{a}+ig_{YM}QC^{abc}A_{\mu}^{b}\phi^{c}\,$
(63)

the gauge field strength is defined by

$\displaystyle
F^{a}_{\mu\nu}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+ig_{YM}C^{abc}A_{\mu}^{a}A_{\nu}^{b}\,.$
(64)

The gauge transformation satisfies the symmetry algebra

$\displaystyle\left[\delta_{\epsilon_{1}},\delta_{\epsilon_{2}}\right]A^{a}_{\mu}=\delta_{\left[{\epsilon_{1}},{\epsilon_{2}}\right]}A^{a}_{\mu}\,,\qquad\left[\delta_{\epsilon_{1}},\delta_{\epsilon_{2}}\right]\phi^{a}=\delta_{\left[{\epsilon_{1}},{\epsilon_{2}}\right]}\phi^{a}\,.$
(65)

###  3.1 Gauge fixing and boundary Condition

We choose the radial gauge and the following near horizon fall-off conditions,

$\displaystyle A_{r}^{a}=0\,,\quad A_{u}^{a}=\mathcal{O}(r)\,,\quad
A_{z}^{a}=\mathcal{O}(1)\,,\quad\phi^{a}=\mathcal{O}(1)\,,\quad\bar{\phi}^{a}=\mathcal{O}(1)\,.$
(66)

The only residual gauge transformation is $\delta
A_{\mu}^{a}=D_{\mu}\epsilon^{a}(z,\bar{z})$ where $\epsilon^{a}(z,\bar{z})$
are arbitrary differentiable functions defined on $S^{2}$. The surface charge
is constructed by [29]

$\displaystyle Q_{\epsilon}=$ $\displaystyle 2\int_{B}\Tr[\epsilon\star F]$
(67) $\displaystyle=$
$\displaystyle\int_{B}\mathrm{d}^{2}z\Omega^{2}\gamma_{z\bar{z}}\epsilon^{a}F^{a}_{ru}\,,$

which generates the nontrivial large gauge transformation in the solution
space of the near horizon scalar fields coupled Yang-Mills theory.

###  3.2 Solution space

Following the analysis of the Abelian case, we divide equations of motion into
three parts

  * •

Hyper-surface equation

$\displaystyle D_{\mu}F^{a\mu u}=0\,.$ (68)

  * •

Standard equations

$\displaystyle D_{\mu}F^{a\mu z}=J^{az}\,,\
D_{\mu}F^{a\mu\bar{z}}=J^{a\bar{z}}\,,\
D^{\mu}D_{\mu}\phi^{a}=D^{\mu}D_{\mu}\bar{\phi}^{a}=0\,.$ (69)

  * •

Supplementary equation

$\displaystyle D_{\mu}F^{\mu r}=J^{r}\,.$ (70)

The conserved current is given by

$\displaystyle
J_{\mu}^{a}=iQC^{abc}\left(\phi^{b}D_{\mu}\bar{\phi}^{c}-\bar{\phi}^{b}D_{\mu}\phi^{c}\right)\,.$
(71)

The Noether identity

$\displaystyle D_{\mu}\left(g_{YM}J^{a\mu}-D_{\nu}F^{a\nu\mu}\right)$
$\displaystyle=ig_{YM}QC^{abc}\left[\phi^{b}D_{\mu}D^{\mu}\bar{\phi}^{c}-\bar{\phi}^{b}D_{\mu}D^{\mu}\phi^{c}\right]\,,$
(72)

yields that, once equations (68) and (69) are satisfied, we have

$\displaystyle\partial_{r}\left[\sqrt{-g}\left(D_{\mu}F^{a\mu
r}-g_{YM}J^{ar}\right)\right]$ $\displaystyle=0\,,$ (73)

which means only the $\mathcal{O}(1)$ equation is indeterminate. We present
all necessary equations

  * •

Standard equations

$\displaystyle\mathcal{D}^{A}\mathcal{D}_{A}\bar{\phi}^{a}+\Omega^{-1}$
$\displaystyle\left[\partial_{r}\left(\Omega
f\partial_{r}\bar{\phi}^{a}\right)+f\partial_{r}\Omega\partial_{r}\bar{\phi}^{a}\right]-2\Omega^{-1}\partial_{r}\partial_{u}\left(\Omega\bar{\phi}^{a}\right)$
(74) $\displaystyle-
ig_{YM}C^{abc}\left(h^{AB}A_{A}^{b}D_{B}\bar{\phi}^{c}+fA_{r}^{b}\partial_{r}\bar{\phi}^{c}-A_{u}^{b}\partial_{r}\bar{\phi}^{c}\right)$
$\displaystyle\quad\quad\quad+ig_{YM}C^{abc}\left[\mathcal{D}_{A}\left(A^{bA}\bar{\phi}^{c}\right)-\Omega^{-2}\partial_{r}\left(\Omega^{2}A_{u}^{b}\bar{\phi}^{c}\right)\right]=0\,,$
$\displaystyle\mathcal{D}^{A}\mathcal{D}_{A}\phi^{a}+\Omega^{-1}$
$\displaystyle\left[\partial_{r}\left(\Omega
f\partial_{r}\phi^{a}\right)+f\partial_{r}\Omega\partial_{r}\phi^{a}\right]-2\Omega^{-1}\partial_{r}\partial_{u}\left(\Omega\phi^{a}\right)$
(75)
$\displaystyle+ig_{YM}C^{abc}\left(h^{AB}A_{A}^{b}D_{B}\phi^{c}+fA_{r}^{b}\partial_{r}\phi^{c}-A_{u}^{b}\partial_{r}\phi^{c}\right)$
$\displaystyle\quad\quad\quad-
ig_{YM}C^{abc}\left[\mathcal{D}_{A}\left(A^{bA}\phi^{c}\right)-\Omega^{-2}\partial_{r}\left(\Omega^{2}A_{u}^{b}\phi^{c}\right)\right]=0\,,$
$\displaystyle\partial_{r}\partial_{z}A_{u}^{a}-2\partial_{u}\partial_{r}A_{z}^{a}+\partial_{r}\left(f\partial_{r}A_{z}^{a}\right)$
$\displaystyle-\partial_{z}\left(F_{\ \ \
z}^{a\bar{z}}\right)-g_{YM}J_{z}^{a}$ (76) $\displaystyle=ig_{YM}C^{abc}$
$\displaystyle\left[A_{\bar{z}}^{b}\partial_{r}A_{z}^{c}+A_{z}^{b}F_{\ \ \
z}^{c\bar{z}}+A_{z}^{b}\partial_{r}A_{\bar{z}}^{c}-2A_{u}^{b}\partial_{r}A_{z}^{c}\right]\,,$
$\displaystyle\partial_{r}\partial_{\bar{z}}A_{u}^{a}-2\partial_{u}\partial_{r}A_{\bar{z}}^{a}+\partial_{r}\left(f\partial_{r}A_{\bar{z}}^{a}\right)$
$\displaystyle-\partial_{\bar{z}}\left(F_{\ \ \
\bar{z}}^{az}\right)-g_{YM}J_{\bar{z}}^{a}$ (77)
$\displaystyle=ig_{YM}C^{abc}$
$\displaystyle\left[A_{z}^{b}\partial_{r}A_{\bar{z}}^{c}+A_{\bar{z}}^{b}F_{\ \
\
\bar{z}}^{cz}+A_{\bar{z}}^{b}\partial_{r}A_{z}^{c}-2A_{u}^{b}\partial_{r}A_{\bar{z}}^{c}\right]\,,$

where $\mathcal{D}^{A}$ and $h_{AB}$ are the covariant derivative and metric
defined on the horizon respectively. It is clear that
$A^{a0}_{z}=A^{a}_{z}|_{r=0}$, $A^{a0}_{\bar{z}}=A^{a}_{\bar{z}}|_{r=0}$,
$\phi^{a0}=\phi^{a}|_{r=0}$ do not have constraints for their time evolution.
They are the news functions of the system that characterize the local
propagating degree of freedom.

  * •

Hyper-surface equation

$\displaystyle A_{u}^{a}$
$\displaystyle=\int_{0}^{r}\mathrm{d}r^{\prime}\Omega^{-2}\int_{0}^{r^{\prime}}\mathrm{d}r^{\prime\prime}\gamma_{z\bar{z}}^{-1}\partial_{r^{\prime\prime}}\left(\partial_{\bar{z}}A_{z}^{a}+\partial_{z}A_{\bar{z}}^{a}\right)+\left(\ell^{-1}-\Omega^{-1}\right)A_{u}^{a0}$
(78) $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+\int_{0}^{r}\mathrm{d}r^{\prime}\Omega^{-2}\int_{0}^{r^{\prime}}\mathrm{d}r^{\prime\prime}\gamma_{z\bar{z}}^{-1}ig_{YM}C^{abc}\left(A_{z}^{b}\partial_{r^{\prime\prime}}A_{\bar{z}}^{c}+A_{\bar{z}}^{b}\partial_{r^{\prime\prime}}A_{z}^{c}\right)\,,$

where $A^{a0}_{u}$ is an integral constant and defined by
$A^{a0}_{u}=\ell^{2}\partial_{r}A^{a}_{u}|_{r=0}$.

  * •

Supplementary equation

$\displaystyle\partial_{u}A_{u}^{a0}=-\gamma_{z\bar{z}}^{-1}\partial_{u}$
$\displaystyle\left(\partial_{\bar{z}}A_{z}^{a0}+\partial_{z}A_{\bar{z}}^{a0}\right)$
(79) $\displaystyle-
ig_{YM}C^{abc}\gamma_{z\bar{z}}^{-1}\left(A_{z}^{0b}\partial_{u}A_{\bar{z}}^{0c}+A_{\bar{z}}^{0b}\partial_{u}A_{z}^{0c}\right)$
$\displaystyle\quad\quad\quad\quad\quad\quad-
iQg_{YM}C^{abc}\Omega^{2}\left(\phi^{0b}\partial_{u}\bar{\phi}^{0c}-\bar{\phi}^{0b}\partial_{u}\phi^{0c}\right)\,.$

The full solution space of the non-Abelian theory is generated by the
integration constant $A^{a0}_{u}$ and the initial data
$\\{A_{z}^{a},A_{\bar{z}}^{a},\phi^{a},\bar{\phi}^{a}\\}$ and the news
functions at any time.

###  3.3 Surface charge of large gauge symmetry

Inserting the near horizon solution, the boundary charge is reduced to

$\displaystyle Q_{\epsilon}=$
$\displaystyle\int_{\mathcal{H}^{-}}\mathrm{d}^{2}z\mathrm{d}u\epsilon^{a}\partial_{u}\left(\partial_{\bar{z}}A_{z}^{a0}+\partial_{z}A_{\bar{z}}^{a0}\right)$
(80)
$\displaystyle+ig_{YM}C^{abc}\int_{\mathcal{H}^{-}}\mathrm{d}^{2}z\mathrm{d}u\epsilon^{a}\left[\left(A_{z}^{a0}\partial_{u}A_{\bar{z}}^{0b}+A_{\bar{z}}^{a0}\partial_{u}A_{z}^{0b}\right)\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\left.+Q\gamma_{z\bar{z}}\Omega^{2}\left(\phi^{0b}\partial_{u}\bar{\phi}^{0c}-\bar{\phi}^{0b}\partial_{u}\phi^{0c}\right)\right]\,.$

We decompose this charge into two parts

  * •

Soft Charge

$\displaystyle
Q_{S}^{-}=-2\int_{B}\mathrm{d}^{2}z\partial_{\bar{z}}\epsilon^{a}\partial_{z}N^{a}\,,$
(81)

where

$\displaystyle\partial_{z}N^{a}=\int\mathrm{d}u\partial_{u}A_{z}^{a0}\,,\
\partial_{\bar{z}}N^{a}=\int\mathrm{d}u\partial_{u}A_{\bar{z}}^{a0}\,.$ (82)

The soft charge is very similar to the case of the Maxwell theory. The only
adaption is the group index on the gauge field.

  * •

Hard Charge

$\displaystyle Q_{H}^{-}=$ $\displaystyle
ig_{YM}C^{abc}\int_{\mathcal{H}^{-}}\mathrm{d}^{2}z\mathrm{d}u\epsilon^{a}\left[\left(A_{z}^{0b}\partial_{u}A_{\bar{z}}^{0c}+A_{\bar{z}}^{b0}\partial_{u}A_{z}^{0c}\right)\right.$
(83)
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\left.+Q\gamma_{z\bar{z}}\Omega^{2}\left(\phi^{0b}\partial_{u}\bar{\phi}^{0c}-\bar{\phi}^{0b}\partial_{u}\phi^{0c}\right)\right]$

The first line is zero in the Abelian case since a linearization will turn off
the self-interacting pieces.

The canonical commutation relations of the Abelian case on $\mathcal{H}^{-}$
can be generalized to the present case as

$\displaystyle\left[\partial_{u}A_{\bar{z}}^{0a}(u,z,\bar{z}),A_{z}^{0b}(g,w,\bar{w})\right]=-i\delta^{2}(z-w)\delta(u-g)\delta^{ab}\,,$
(84)

thus

$\displaystyle\left[Q^{-}_{\epsilon},A_{z}^{0d}(u,z,\bar{z})\right]=i\left[\partial_{\mu}\epsilon^{d}(z,\bar{z})+ig_{YM}C^{abd}A^{0a}_{z}\epsilon^{b}(z,\bar{z})\right]\,,$
(85)

which is the infinitesimal gauge transformation of the gauge field.

###  3.4 Mode expansion near the horizon

The mode expansion of the Yang-Mills field is almost the same as the Maxwell
field with the only adaption of a group index. We just give the final results
of the mode expansion here without repeating the same type of computation. The
mode expansion near the horizon field is given by

$\displaystyle A_{z}^{0a}(x)=\frac{2\sqrt{2}}{\pi^{2}(1+z\bar{z})}$
$\displaystyle\frac{\ell^{6}}{\left[\ell^{2}+\left(R+\ell\right)^{2}\right]^{3}}\int\mathrm{d}\omega\left(\frac{r}{2\ell-r}\right)^{-i\ell\omega/2}$
(86)
$\displaystyle\times\sin\left[\frac{2\ell^{2}(R+\ell)\omega}{R(R+2\ell)}\right]\left[e^{\mathrm{-i\omega}u}a_{+}^{a}(\omega\hat{x})+e^{\mathrm{i\omega}u}a_{-}^{a\dagger}(\omega\hat{x})\right]\,.$

###  3.5 Soft gluon theorem and flat limit

Following the choice in [8], we choose $\epsilon^{a}(z,\bar{z})=\frac{1}{z-w}$
where $a$ denotes a particular component of $\epsilon$ and the rest components
are set to zero. Then, the soft charge of the Yang-Mills theory in terms of
the annihilation and creation operators defined in the mode expansion is

$\displaystyle
Q_{S}^{-}=-\frac{i2\sqrt{2}}{1+w\bar{w}}\frac{\ell^{2}}{R}\lim_{\omega\rightarrow
0}\left(\omega^{2}a_{+}^{a}(\omega\hat{x})+\omega^{2}a_{-}^{a\dagger}(\omega\hat{x})\right)\,.$
(87)

Acting the soft charge on in-state yields

$\displaystyle Q_{S}^{-}\ket{\mathrm{in}}$
$\displaystyle=-\frac{i2\sqrt{2}}{1+w\bar{w}}\frac{\ell^{2}}{R}\lim_{\omega\rightarrow
0}\omega^{2}a_{-}^{a\dagger}(\omega\hat{x})\ket{\mathrm{in}}\,.$ (88)

Regarding to the hard part, one needs to compute the commutation relation from
the symplectic form of $\phi^{a}$, which is given by

$\displaystyle\Omega_{\mathcal{H}^{-}}=\int_{\mathcal{H}^{-}}\mathrm{d}u\mathrm{d}^{2}z\gamma_{z\bar{z}}\ell^{2}\left(\delta\partial_{u}\bar{\phi}^{0a}\wedge\delta\phi^{0a}+\delta\partial_{u}\phi^{0a}\wedge\delta\bar{\phi}^{0a}\right)\,.$
(89)

Hence,

$\displaystyle\left[\partial_{u}\bar{\phi}^{0a}(u,z,\bar{z}),\phi^{0b}(g,w,\bar{w})\right]=-i\gamma^{-1}_{z\bar{z}}\ell^{-2}\delta^{2}(z-w)\delta(u-g)\delta^{ab}\,,$
(90)

which leads to

$\displaystyle\left[Q^{-}_{H},\phi^{0d}_{k}(u,z_{k},\bar{z}_{k})\right]=Q_{k}g_{YM}C^{abd}\epsilon^{a}(z_{k},\bar{z}_{k})\phi^{0b}_{k}(u,z_{k},\bar{z}_{k})\,.$
(91)

One can find that the hard charge acts on in-state of the scalar particle
linearly

$\displaystyle
Q_{H}^{-}\ket{\mathrm{in}}=\sum_{k=1}^{n}Q^{\mathrm{in}}_{k}T^{a}_{k}\epsilon^{a}(z_{k}^{\mathrm{in}},\bar{z}_{k}^{\mathrm{in}})\ket{\mathrm{in}}\,.$
(92)

Note that the commutation relation in (84) will yield the same result for the
hard charge acting on in-state of the gluon. Finally, combining the actions of
the soft and hard charges, the Ward identity in the coordinates space leads to
a soft gluon theorem in dS spacetime

$\displaystyle\lim_{\omega\rightarrow
0}\omega^{2}\matrixelement{\mathrm{out}}{a_{+}^{b}\mathcal{S}-\mathcal{S}a_{-}^{b\dagger}}{\mathrm{in}}$
$\displaystyle=i\frac{1+w\bar{w}}{2\sqrt{2}}\frac{R}{\ell^{2}}\left[\sum_{l=1}^{m}\frac{Q_{l}^{\mathrm{out}}T^{b}_{l}}{w-z_{l}^{\mathrm{out}}}-\sum_{k=1}^{n}\frac{Q_{k}^{\mathrm{in}}T^{b}_{k}}{w-z_{k}^{\mathrm{in}}}\right]\matrixelement{\mathrm{out}}{\mathcal{S}}{\mathrm{in}}\,.$
(93)

Considering the null parametrization in (39), the soft theorem in momentum
space is

$\displaystyle\lim_{\omega\rightarrow
0}\matrixelement{\mathrm{out}}{a_{+}^{b}\mathcal{S}-\mathcal{S}a_{-}^{b\dagger}}{\mathrm{in}}$
$\displaystyle=\frac{i}{2\omega\ell}\left[\sum_{l=1}^{m}Q_{l}^{\mathrm{out}}\frac{T^{b}_{l}p_{l}^{\mathrm{out}}\cdot\epsilon}{p_{l}^{\mathrm{out}}\cdot
p}-\sum_{k=1}^{n}Q_{k}^{\mathrm{in}}\frac{T^{b}_{k}p_{k}^{\mathrm{in}}\cdot\epsilon}{p_{k}^{\mathrm{in}}\cdot
p}\right]\matrixelement{\mathrm{out}}{\mathcal{S}}{\mathrm{in}}\,.$ (94)

It is amusing to see that the dS radius has exactly the same type of
correction to soft photon and soft gluon theorem in dS spacetime. The flat
limit of the soft gluon theorem is just the same as the soft photon theorem.

##  4 Conclusion

In this paper, a soft photon theorem and a soft gluon theorem are derived in
dS spacetime from the Ward identity of the near horizon symmetry. Our results
indicate that the cosmological constant affects the soft theorems universally.
Both the soft photon and soft gluon theorem have a well defined flat limit.
The soft theorems after the flat limit recover exactly the flat spacetime soft
photon and soft gluon theorems.

As an ending remark, it is worth mentioning that we focus on massless hard
particles in the soft theorem. While, the soft photon theorem derived in [20]
involves only massive hard particles and the zero mass limit of the hard
particles seems not well defined. So it is not clear to us how to connect our
results in the present form to the ones in [20].

## Acknowledgments

The authors thank Sayali Bhatkar, Diksha Jain, and Adam Tropper for useful
correspondences. This work is supported in part by the National Natural
Science Foundation of China (NSFC) under Grants No. 11905156 and No. 11935009.
K.Y.Z. is also supported in part by NSFC Grant No. 11905158.

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