#  Celestial Yang-Mills Amplitudes and
$D\\!=\\!4$ Conformal Blocks

Wei Fan1, Angelos Fotopoulos2, Stephan Stieberger3, Tomasz R. Taylor2, Bin
Zhu2
1Department of Physics, School of Science, Jiangsu University of Science and
Technology, Zhenjiang, 212003, China
2Department of Physics
Northeastern University, Boston, MA 02115, USA
3Max–Planck–Institut für Physik, Werner–Heisenberg–Institut,
80805 München, Germany<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We discuss the properties of recently constructed “single-valued” celestial
four-gluon amplitudes. We show that the amplitude factorizes into the
“current” part and the “scalar” part. The current factor is given by the
group-dependent part of the Wess-Zumino-Witten correlator of four holomorphic
currents with a non-vanishing level of Kač-Moody algebra. The scalar factor
can be expressed in terms of a complex integral of the Koba-Nielsen form,
similar to the integrals describing four-point correlators in Coulomb gas
models and, more generally, in the infinite central charge limit of Liouville
theory. The scalar part can be also obtained by a dimensional reduction of a
single $D=4$ conformal block and the shadow block from Minkowski space to the
celestial sphere.

###### Keywords:

conformal field theory, holography, scattering amplitudes

##  1 Introduction

Classical Yang-Mills (YM) theory is invariant under $O(4,2)$ conformal
transformations. In quantum theory, at the tree level, gluon scattering
amplitudes are conformally invariant, but this invariance is broken by loop
corrections. Furthermore, there is an overwhelming evidence for the existence
of a mass gap in YM theory. Hence studying conformal properties of gluon
amplitudes should lead to a better understanding of conformal symmetry
breaking in gauge theories.

There had not been much attention given to the conformal properties of YM
scattering amplitudes until Witten formulated perturbative YM as a string
theory in conformally-invariant twistor space in 2003 Witten:2003nn . More
recently, two of us Stieberger:2018onx  examined the symmetries of celestial
amplitudes Pasterski:2017ylz ; Pasterski:2021rjz  obtained by applying Mellin
transforms to the conventional (momentum space) amplitudes that convert them
into the correlators of a putative two-dimensional conformal theory on the
celestial sphere (CCFT). In CCFT, the dimensions of primary field operators
associated to gluons are $\Delta_{n}=1+i\lambda_{n}$, with real $\lambda_{n}$
Pasterski1705 . The dilatation generator Stieberger:2018onx  acts on the
celestial amplitudes as

${\cal D}=-i\sum_{n}(\Delta_{n}-1)=\sum_{n}\lambda_{n}\ ,$ (1.1)

therefore scale-invariant tree-level gluon amplitudes are non-zero only if
$\sum_{n}\lambda_{n}=0$. This constraint is enforced in all celestial gluon
amplitudes by the universal delta-function factor
$\delta(\sum_{n}\lambda_{n})$. Furthermore, the invariance under special
conformal transformations implies a set of second order differential equations
for celestial amplitudes.

In celestial holography, the directions of four-dimensional momenta are mapped
into the positions of primary field operators. As a result, the momentum
conservation law imposes constraints Mizera:2022sln  on these positions, what,
from the point of view of two-dimensional CFT, is an unwanted feature of the
correlation functions. In a recent work Fan:2022vbz , we circumvented these
constraints by constructing the “single-valued” CCFT correlation functions.
They are obtained from the Mellin transforms of the scattering amplitudes
evaluated in the presence of a dilaton background with the source located on
the celestial sphere. While this background violates translational symmetry,
it preserves scale invariance. In this work, we discuss the properties of
“single-valued” CCFT correlators. We find new representations of the celestial
four-gluon correlator and discuss its relation to the limits of $D=4$
conformal blocks at null infinity.

##  2 Integral representation of the single-valued four-gluon amplitude

The four-gluon celestial MHV amplitude is expressed in terms of two partial
amplitudes, $M(z_{i},\bar{z}_{i})$ and $\widetilde{M}(z_{i},\bar{z}_{i})$ in
the following way Fan:2022vbz :

$\displaystyle\Big{\langle}\phi_{\Delta_{1},-}^{a_{1},-\epsilon}(z_{1},\bar{z}_{1})\,\phi_{\Delta_{2},-}^{a_{2},-\epsilon}$
$\displaystyle(z_{2},\bar{z}_{2})\,\phi_{\Delta_{3},+}^{a_{3},+\epsilon}(z_{3},\bar{z}_{3})\,\phi_{\Delta_{4},+}^{a_{4},+\epsilon}(z_{4},\bar{z}_{4})\Big{\rangle}=$
$\displaystyle~{}~{}=f^{a_{1}a_{2}b}f^{a_{3}a_{4}b}M(z_{i},\bar{z}_{i})+f^{a_{1}a_{3}b}f^{a_{2}a_{4}b}\widetilde{M}(z_{i},\bar{z}_{i}).$
(2.1)

Here, the subscripts $\Delta_{i},J_{i}=\pm 1$ refer to the dimensions and
spins (gluon helicities) of the primary fields, respectively, group indices
are labeled by $a_{i}$, while the superscripts $\pm\epsilon$ indicate incoming
$(-\epsilon)$ and outgoing $(+\epsilon)$ particles. Taking advantage of
conformal symmetry, one defines

$\displaystyle G(x,\bar{x})$
$\displaystyle=\lim_{z_{1},\bar{z}_{1}\rightarrow\infty}z_{1}^{2h_{1}}\bar{z}_{1}^{2\bar{h}_{1}}M(z_{1};z_{2}=1;z_{3}=x;z_{4}=0)\,,$
(2.2) $\displaystyle\widetilde{G}(x,\bar{x})$
$\displaystyle=\lim_{z_{1},\bar{z}_{1}\rightarrow\infty}z_{1}^{2h_{1}}\bar{z}_{1}^{2\bar{h}_{1}}\widetilde{M}(z_{1};z_{2}=1;z_{3}=x;z_{4}=0)\,,$
(2.3)

with the conformal weights
$h_{1}=(\Delta_{1}+J_{1})/2,\bar{h}_{1}=(\Delta_{1}-J_{1})/2$. The complex
variable $x$ is identified with the complex cross ratio

$x=\frac{z_{12}z_{34}}{z_{13}z_{24}}\ .$ (2.4)

For the future use, it is convenient to define another cross ratio,

$z={\frac{z_{12}z_{43}}{z_{14}z_{23}}}={\frac{x}{x-1}}$ (2.5)

and real conformal invariants

$u=\left|{\frac{x}{x-1}}\right|^{2}=|z|^{2}\ ,\qquad
v={\frac{1}{|1-x|^{2}}}=|1-z|^{2}\ .$ (2.6)

Note that upon interchanging $3\leftrightarrow 4$, $x\leftrightarrow z$.

The partial amplitudes can be written as

$M(z_{i},\bar{z}_{i})=\Pi(z_{i},\bar{z}_{i})\,G(x,\bar{x})\
,\qquad\widetilde{M}(z_{i},\bar{z}_{i})=\Pi(z_{i},\bar{z}_{i})\,\widetilde{G}(x,\bar{x})\
,$ (2.7)

where

$\displaystyle\Pi(z_{i},\bar{z}_{i})$
$\displaystyle=~{}z_{12}^{~{}-h_{1}-h_{2}+h_{3}+h_{4}}z_{13}^{~{}-2h_{3}}z_{14}^{h_{2}+h_{3}-h_{1}-h_{4}}z_{24}^{h_{1}-h_{2}-h_{3}-h_{4}}$
$\displaystyle\quad~{}\times\bar{z}_{12}^{~{}-\bar{h}_{1}-\bar{h}_{2}+\bar{h}_{3}+\bar{h}_{4}}\bar{z}_{13}^{~{}-2\bar{h}_{3}}\bar{z}_{14}^{\bar{h}_{2}+\bar{h}_{3}-\bar{h}_{1}-\bar{h}_{4}}\bar{z}_{24}^{\bar{h}_{1}-\bar{h}_{2}-\bar{h}_{3}-\bar{h}_{4}}.$
(2.8)

With the dimensions parameterized as $\Delta_{n}=1+i\lambda_{n}$,
$\lambda_{n}\in{\bf R}$, the conformal prefactor (2.8) becomes

$\Pi(z_{i},\bar{z}_{i})=z_{12}^{~{}-i\lambda_{1}-i\lambda_{2}+2}z_{13}^{~{}-2-i\lambda_{3}}z_{14}^{~{}-i\lambda_{1}-i\lambda_{4}}z_{24}^{i\lambda_{1}-2}\bar{z}_{12}^{~{}-i\lambda_{1}-i\lambda_{2}-2}\bar{z}_{13}^{~{}-i\lambda_{3}}\bar{z}_{14}^{~{}-i\lambda_{1}-i\lambda_{4}}\bar{z}_{24}^{i\lambda_{1}}\
.$ (2.9)

There are three ways of constructing the “single-valued” four-gluon
correlators, all of them leading to the same result.

In Refs. Fan:2021isc ; Fan:2021pbp , we started from the celestial amplitude
of Pasterski, Shao and Strominger Pasterski:2017ylz  and performed the shadow
transformation on one of gluons. The shadow transform relaxes the kinematic
constraints due to momentum conservation, but it leads to a correlator that is
a multi-valued function of complex coordinates. It can be used, however, to
construct a single-valued correlator by applying the same procedure as in
minimal models DiF , by adding a contribution of additional conformal blocks.
At the end, we inverted the shadow transformation and obtained a well-defined
“unshadowed” four-gluon correlator. In Fan:2021pbp , this procedure was
performed explicitly only in the “soft” limit $\Delta\to 1$ of the shadowed
gluon, but it can be easily extended to arbitrary $\Delta$.

More recently Fan:2022vbz , we constructed this correlator by solving the
differential equations written by Banerjee and Ghosh (BG) in Ref.
Banerjee:2020vnt  and further developed in Ref.Hu:2021lrx . BG equations
follow from the consistency of soft theorems with the operator product
expansions of CCFT. They are similar to the equations describing the
decoupling of null states in minimal models. In Ref. Fan:2022vbz , we also
showed that the same correlator describes four-gluon scattering in the
presence of a dilaton background. This (third) representation of the single-
valued celestial gluon amplitude is given by the following integral:

$\displaystyle~{}~{}G(x,$
$\displaystyle\bar{x})=\delta(\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4})\
\frac{1}{x(1-x)}$ (2.10)
$\displaystyle\times\Bigg{\\{}\int_{0}^{1}dt\,B(-i\lambda_{1},-i\lambda_{2})(1+i\lambda_{1}+i\lambda_{2})\,t^{i\lambda_{3}-1}(1-t)^{i\lambda_{4}-1}$
$\displaystyle\times\big{(}t(1-x)(1-\bar{x})+1-t\big{)}^{i\lambda_{1}}\,_{2}F_{1}\left({-i\lambda_{1},-i\lambda_{2}\atop-1-i\lambda_{1}-i\lambda_{2}};\frac{t(1-t)\,x\,\bar{x}}{t(1-x)(1-\bar{x})+1-t}\right)$
$\displaystyle+\int_{0}^{1}dt\,B(2+i\lambda_{1},2+i\lambda_{2})(3+i\lambda_{1}+i\lambda_{2})\,t^{1-i\lambda_{4}}(1-t)^{1-i\lambda_{3}}(x\bar{x})^{2+i\lambda_{1}+i\lambda_{2}}$
$\displaystyle\times\big{(}t(1-x)(1-\bar{x})+1-t\big{)}^{-2-i\lambda_{2}}\,_{2}F_{1}\left({2+i\lambda_{1},2+i\lambda_{2}\atop
3+i\lambda_{1}+i\lambda_{2}};\frac{t(1-t)\,x\,\bar{x}}{t(1-x)(1-\bar{x})+1-t}\right)\Bigg{\\}}\,,$

where ${}_{2}F_{1}$ denotes Gauß hypergeometric function. Similarly,

$\widetilde{G}(x,\bar{x})=-xG(x,\bar{x})$ (2.11)

By a series of algebraic manipulations and changes of integration variables
described in Appendix A, Eq.(2.10) can be recast as

$\displaystyle G(x,$
$\displaystyle\bar{x})=\delta(\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4})\
\frac{|1-x|^{2i\lambda_{1}}}{x(1-x)}\ $ (2.12) $\displaystyle\times{\cal
D}_{z}\;\Bigg{\\{}B(-1+i\lambda_{4},i\lambda_{3})\;B(-i\lambda_{2},-1-i\lambda_{1})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times{}_{2}F_{1}\left({-1+i\lambda_{4},-1-i\lambda_{1}\atop-1+i\lambda_{3}+i\lambda_{4}};z\right)\;_{2}F_{1}\left({-1+i\lambda_{4},-1-i\lambda_{1}\atop-1+i\lambda_{3}+i\lambda_{4}};\bar{z}\right)$
$\displaystyle-B(1-i\lambda_{3},2-i\lambda_{4})\;B(2+i\lambda_{1},1+i\lambda_{2})$
$\displaystyle~{}~{}~{}\left.\times\,|z|^{2(2+i\lambda_{1}+i\lambda_{2})}\;_{2}F_{1}\left({1-i\lambda_{3},1+i\lambda_{2}\atop
3+i\lambda_{1}+i\lambda_{2}};z\right)\;_{2}F_{1}\left({1-i\lambda_{3},1+i\lambda_{2}\atop
3+i\lambda_{1}+i\lambda_{2}};\bar{z}\right)\Bigg{\\}}\right|_{z={\frac{x}{x-1}}},$

with the differential operator:

${\cal D}_{z}={\frac{1}{z-\bar{z}}}\
\left(z\;\partial_{z}-\bar{z}\;\partial_{\bar{z}}\right)=-{\frac{\partial}{\partial
v}}\ .$ (2.13)

The expression in the curly bracket of (2.12) can be related to the complex
integrals considered by Dotsenko and Fateev Dotsenko:1984ad ; Dotsenko:1984nm
in the context of Coulomb gas models. To that end, we define

${\cal I}(z,\bar{z})=\int d^{2}w\ w^{\hat{a}+a}\ \bar{w}^{\hat{a}+\bar{a}}\
(w-1)^{\hat{b}+b}\ (\bar{w}-1)^{\hat{b}+\bar{b}}\ (w-z)^{\hat{c}+c}\
(\bar{w}-\bar{z})^{\hat{c}+\bar{c}}\ ,$ (2.14)

with the parameters $\hat{a},\hat{b},\hat{c}\notin{\bf Z}$ and
$a,b,c,\bar{a},\bar{b},\bar{c}\in{\bf Z}$. The integral (2.14) can be
evaluated through analytic continuation, by disentangling the holomorphic and
antiholomorphic parts Dotsenko . This procedure, which is also known as the
Kawai-Lewellen-Tye (KLT) method, yields

$\displaystyle{\cal I}(z,\bar{z})$
$\displaystyle=\frac{s(\hat{b})s(\hat{a}+\hat{b}+\hat{c})}{s(\hat{a}+\hat{c})}\
I_{1}(\hat{a}+a,\hat{b}+b,\hat{c}+c;z)\
I_{1}(\hat{a}+\bar{a},\hat{b}+\bar{b},\hat{c}+\bar{c};\bar{z})$
$\displaystyle+\ (-1)^{b+\bar{b}+c+\bar{c}}\
\frac{s(\hat{a})s(\hat{c})}{s(\hat{a}+\hat{c})}\
I_{2}(\hat{a}+a,\hat{b}+b,\hat{c}+c;z)\
I_{2}(\hat{a}+\bar{a},\hat{b}+\bar{b},\hat{c}+\bar{c};\bar{z})\ ,$ (2.15)

with $s(\alpha):=\sin(\pi\alpha)$ and

$\displaystyle I_{1}(a,b,c;z)$ $\displaystyle=\int_{1}^{\infty}dw\ w^{a}\
(w-1)^{b}\ (w-z)^{c}$ $\displaystyle=B(-a-b-c-1,b+1)\
{}_{2}F_{1}\left({-c,-a-b-c-1\atop-a-c};z\right)\ ,$ (2.16) $\displaystyle
I_{2}(a,b,c;z)$ $\displaystyle=\int_{0}^{z}dw\ w^{a}\ (1-w)^{b}\ (z-w)^{c}$
$\displaystyle=z^{1+a+c}\ B(a+1,c+1)\ {}_{2}F_{1}\left({-b,1+a\atop
a+c+2};z\right)\ .$ (2.17)

By comparing the above equations with (2.12), we obtain

$\displaystyle G(x,\bar{x})$
$\displaystyle={\frac{1}{\pi}}\;\delta(\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4})\
\frac{|1-x|^{2i\lambda_{1}}}{x(1-x)}$
$\displaystyle\times\left.(2+i\lambda_{1}+i\lambda_{2})\;B(2+i\lambda_{1},1+i\lambda_{2})\;B(i\lambda_{3},-1+i\lambda_{4})\
{\cal D}_{z}\;{\cal I}(z,\bar{z})\right|_{z={\frac{x}{x-1}}}\ ,$ (2.18)

with the assignments:

${\cal I}(z,\bar{z})\ \
\left\\{\begin{array}[]{ccccc}\hat{a}=-i\lambda_{3}&,&\ a=\bar{a}=0\ ,\\\
\hat{b}=-i\lambda_{2}&,&\ b=\bar{b}=-1\ ,\\\ \hat{c}=-i\lambda_{4}&,&\
c=\bar{c}=1\ .\\\ \end{array}\right.$ (2.19)

To be more explicit, we can replace the cross ratio $z$ by $x$ and express
(2.18) in terms of

$\left.{\cal
I}(z,\bar{z})\right|_{z={\frac{x}{x-1}}}=|1-x|^{-2(i\lambda_{1}+1)}\;\int
d^{2}w\ |w|^{-2i\lambda_{4}+2}\ |w-1|^{-2i\lambda_{2}-2}\
|w-x|^{-2i\lambda_{3}}\ ,$ (2.20)

and

${\cal
D}_{z}={\frac{|1-x|^{2}}{x-\bar{x}}}\;\left\\{x(1-x)\;\partial_{x}-\bar{x}(1-\bar{x})\;\partial_{\bar{x}}\right\\}\
.$ (2.21)

The fact that the four-gluon celestial amplitude can be expressed in terms of
the integrals encountered in Coulomb gas models points to a deeper relation
between CCFT and the Coulomb gas models and more generally, to a relation
between CCFT and the complexified Liouville theory.

##  3 Relation to Aomoto–Gelfand hypergeometric functions

In this section, we wish to discuss an interesting relation of the four-gluon
celestial amplitude to the Aomoto–Gelfand hypergeometric functions. This
relation remains somehow outside the main scope of this paper, therefore it
can be skipped at the first reading.

The Aomoto–Gelfand hypergeometric functions are constructed in the framework
of Grassmannians Aomoto . Among some well-known examples, there are four
hypergeometric functions of two variables, introduced by Appell. Here, we
focus on Appell’s $F_{4}$, which is given by the following integral Aomoto

$\displaystyle F_{4}\left[\alpha,\beta,\gamma,\gamma^{\prime};u,v\right]$
$\displaystyle={\frac{\Gamma(1-\alpha)}{\Gamma(1-\gamma)\Gamma(1-\gamma^{\prime})\Gamma(\gamma+\gamma^{\prime}-\alpha-1)}}$
$\displaystyle\times\int_{\Delta_{2}}dx_{1}\;dx_{2}\;x_{1}^{-\gamma}\;x_{2}^{-\gamma^{\prime}}\;(1-x_{1}-x_{2})^{\gamma+\gamma^{\prime}-\alpha-2}\;\left(1-{\frac{u}{x_{1}}}-{\frac{v}{x_{2}}}\right)^{-\beta}$
(3.1)

over the two-dimensional simplex:

$\Delta_{2}=\\{(x_{1},x_{2})\in{\bf R}^{2}\ |\ |x_{1}|+|x_{2}|\leq 1\\}\ .$
(3.2)

Appell’s $F_{4}$ has the following power series expansion around $u,v=0$:

$F_{4}\left[\alpha,\beta,\gamma,\gamma^{\prime};u,v\right]=\sum_{m,n=0}^{\infty}{\frac{(\alpha)_{m+n}\;(\beta)_{m+n}}{(\gamma)_{m}(\gamma^{\prime})_{n}}}\
{\frac{u^{m}v^{n}}{m!n!}}\ ,\ u,v\in{\bf C}\ ,\ |u|^{1/2}+|v|^{1/2}<1\ .$
(3.3)

The above series solves a system of two partial (hypergeometric) differential
equations called the $F_{4}$ equations. The general solution of the $F_{4}$
system can be written as a linear combination of four independent functions
Goto :

$\displaystyle f_{1}$
$\displaystyle=F_{4}\left[\alpha,\beta,\gamma,\gamma^{\prime};u,v\right]\ ,$
$\displaystyle f_{2}$
$\displaystyle=v^{1-\gamma^{\prime}}\;F_{4}\left[\alpha+1-\gamma^{\prime},\beta+1-\gamma^{\prime},\gamma,2-\gamma^{\prime};u,v\right]\
,$ $\displaystyle f_{3}$
$\displaystyle=u^{1-\gamma}\;F_{4}\left[\alpha+1-\gamma,\beta+1-\gamma,2-\gamma,\gamma^{\prime};u,v\right]\
,$ $\displaystyle f_{4}$
$\displaystyle=u^{1-\gamma}\;v^{1-\gamma^{\prime}}\;F_{4}\left[\alpha+2-\gamma-\gamma^{\prime},\beta+2-\gamma-\gamma^{\prime},2-\gamma,2-\gamma^{\prime};u,v\right]\
.$ (3.4)

The relation between the four-gluon celestial amplitude and the $F_{4}$ system
can be obtained by starting from (2.12) and applying various hypergeometric
function relations. The variables $u$ and $v$ are related to the cross ratios
$x$ and $z$ according to Eq.(2.6). The details of the derivation are given in
Appendix B. At the end, the amplitude can be expressed in the basis (3.4) as

$\displaystyle G(x,\bar{x})$
$\displaystyle=-\delta(\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4})\
{\frac{1}{x(1-x)}}\ v^{\beta}\
\Bigg{\\{}{\frac{\Gamma(1-\gamma^{\prime})\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\gamma)}}\;f_{1}$
$\displaystyle+{\frac{\Gamma(\gamma^{\prime}-1)\Gamma(\alpha+1-\gamma^{\prime})\Gamma(\beta+1-\gamma^{\prime})}{\Gamma(\gamma)}}\;f_{2}+{\frac{\Gamma(1-\gamma^{\prime})\Gamma(\alpha+1-\gamma)\Gamma(\beta+1-\gamma)}{\Gamma(2-\gamma)}}\;f_{3}$
$\displaystyle+{\frac{\Gamma(\gamma^{\prime}-1)\Gamma(\alpha+2-\gamma-\gamma^{\prime})\Gamma(\beta+2-\gamma-\gamma^{\prime})}{\Gamma(2-\gamma)}}\;f_{4}\Bigg{\\}}\
,$ (3.5)

with

$\displaystyle\alpha=i\lambda_{4}\ \ \ $ $\displaystyle,\ \ \
\beta=-i\lambda_{1}\ ,$ $\displaystyle\gamma=-1+i\lambda_{3}+i\lambda_{4}\ \ \
$ $\displaystyle,\ \ \ \gamma^{\prime}=1+i\lambda_{2}+i\lambda_{4}\ ,$ (3.6)

satisfying

$\alpha+\beta=\gamma+\gamma^{\prime}\ .$ (3.7)

The integral representation (2.18), written as in Eq.(3.5) and expanded by
using the series (3.3), agrees with the series expansion solution of BG
equations written explicitly in Section 2.1 of Ref.Fan:2022vbz .

##  4 Relation to $D=4$ conformal blocks

In this section, we establish a connection between celestial Yang-Mills
amplitudes and conformal blocks in $D=4$ CFT. The relevant blocks appear in
the four-point correlation functions of scalar (spin zero) $D=4$ fields with
dimensions $d_{n}=i\lambda_{n}$. They depend on the spacetime coordinates
$x_{n}^{\mu}$ of scalar fields. Due to $O(4,2)$ invariance, apart from a
standard conformal prefactor, they depend on only two cross ratios,

$U=\frac{x_{12}^{2}x_{34}^{2}}{x_{13}^{2}x_{24}^{2}}\,,\quad
V=\frac{x_{14}^{2}x_{23}^{2}}{x_{13}^{2}x_{24}^{2}}\,.$ (4.1)

It is often convenient to use another set of variables, $X$ and $\bar{X}$,
defined by

$U=X\bar{X}\,,\quad V=(1-X)(1-\bar{X})\,.$ (4.2)

The four-scalar correlator can be decomposed into conformal blocks associated
to various representations of the conformal group. A single conformal block
associated to a scalar field of dimension $\Delta$ contributes Dolan:2000ut ;
Dolan:2003hv ; Simmons-Duffin:2012juh

$W_{\Delta}(x_{n})=\frac{1}{(x_{12}^{2})^{\frac{d_{1}+d_{2}}{2}}(x_{34}^{2})^{\frac{d_{3}+d_{4}}{2}}}\Big{(}\frac{x_{14}^{2}}{x_{13}^{2}}\Big{)}^{\frac{d_{34}}{2}}\Big{(}\frac{x_{24}^{2}}{x_{14}^{2}}\Big{)}^{\frac{d_{12}}{2}}g_{\Delta}(U,V)\,,$
(4.3)

where $d_{ij}=d_{i}-d_{j}$ and, with the cross ratios parameterized as in
(4.2),

$\displaystyle g_{\Delta}(X,\bar{X})$
$\displaystyle={\frac{1}{X-\bar{X}}}\;|X|^{\Delta}$ (4.4)
$\displaystyle\times\left\\{X\,_{2}F_{1}\left({{\frac{1}{2}}(\Delta-
d_{12}),{\frac{1}{2}}(\Delta+d_{34})\atop\Delta};X\right){}_{2}F_{1}\left({{\frac{1}{2}}(\Delta-
d_{12})-1,{\frac{1}{2}}(\Delta+d_{34})-1\atop\Delta-2};\bar{X}\right)\right.$
$\displaystyle\left.-\bar{X}\,_{2}F_{1}\left({{\frac{1}{2}}(\Delta-\Delta_{12})-1,{\frac{1}{2}}(\Delta+d_{34})-1\atop\Delta-2};X\right){}_{2}F_{1}\left({{\frac{1}{2}}(\Delta-
d_{12}),{\frac{1}{2}}(\Delta+d_{34})\atop\Delta};\bar{X}\right)\right\\}.$

We are interested in the “celestial limit” of the conformal block (4.3),
leading from $D=4$ Minkowski space to $D=2$ celestial sphere. To that end, we
use Milne coordinates deBoer:2003vf ; Cheung:2016iub , with

$x^{\mu}=e^{\tau}\Big{(}\frac{k^{\mu}}{\rho}+\rho q^{\mu}\Big{)}\,,$ (4.5)

where $k^{\mu}$ and $q^{\mu}$ are null vectors

$k^{\mu}=\frac{1}{2}(1+z\bar{z},z+\bar{z},-iz+i\bar{z},-1+z\bar{z})\quad\text{and}\quad
q^{\mu}=\frac{1}{2}(1,0,0,1)\,.$ (4.6)

These coordinates describe a foliation of $D=4$ Minkowski space, divided into
the Milne and Rindler regions. Milne coordinates are used in the Milne region,
while Rindler coordinates are used in the Rindler region deBoer:2003vf ;
Cheung:2016iub . In the limit of $\rho\rightarrow 0$,

$x^{\mu}\to\frac{e^{\tau}k^{\mu}}{\rho}\ ,\qquad
x_{ij}^{2}\to\frac{e^{\tau_{i}+\tau_{j}}}{\rho^{2}}\,z_{ij}\bar{z}_{ij}\,.$
(4.7)

Furthermore, we choose common $\tau$ for all four $x^{\mu}$, and take
$\tau\rightarrow+\infty$. This limit sends the spacetime coordinates of scalar
fields to null infinity.111A similar limit can be taken in Rindler
coordinates. Such a limit can be also applied to (2,2) Klein space, with the
foliation described in Ref.Casali:2022fro . Then

$\lim_{\rho\rightarrow
0,\,\tau\rightarrow+\infty}U=x\bar{x}\,,\quad\lim_{\rho\rightarrow
0,\,\tau\rightarrow+\infty}V=(1-x)(1-\bar{x})\,,$ (4.8)

where $x$ and $\bar{x}$ are the usual cross ratios on the celestial sphere:

$x=\frac{z_{12}z_{34}}{z_{13}z_{24}}\,,\quad\bar{x}=\frac{\bar{z}_{12}\bar{z}_{34}}{\bar{z}_{13}\bar{z}_{24}}\
.$ (4.9)

In this limit,

$\lim_{\rho\rightarrow
0,\,\tau\rightarrow+\infty}g_{\Delta}(U,V)=g_{\Delta}(x,\bar{x})\,,$ (4.10)

with $g_{\Delta}(x,\bar{x})$ given in Eq.(4.4). Moreover, with
$d_{n}=i\lambda_{n}$, the prefactor of Eq.(4.3) becomes

$\displaystyle\lim_{\rho\rightarrow 0,\,\tau\rightarrow+\infty}$
$\displaystyle\frac{1}{(x_{12}^{2})^{\frac{d_{1}+d_{2}}{2}}(x_{34}^{2})^{\frac{d_{3}+d_{4}}{2}}}\Big{(}\frac{x_{14}^{2}}{x_{13}^{2}}\Big{)}^{\frac{d_{34}}{2}}\Big{(}\frac{x_{24}^{2}}{x_{14}^{2}}\Big{)}^{\frac{d_{12}}{2}}$
$\displaystyle=$
$\displaystyle\Big{(}\frac{\rho}{e^{\tau}}\Big{)}^{i\lambda_{1}+i\lambda_{2}+i\lambda_{3}+i\lambda_{4}}|z_{12}|^{-i\lambda_{1}-i\lambda_{2}}|z_{34}|^{-i\lambda_{3}-i\lambda_{4}}\Big{|}\frac{z_{14}}{z_{13}}\Big{|}^{i\lambda_{3}-i\lambda_{4}}\Big{|}\frac{z_{24}}{z_{14}}\Big{|}^{i\lambda_{1}-i\lambda_{2}}$
$\displaystyle=$
$\displaystyle|z_{12}|^{-i\lambda_{1}-i\lambda_{2}}|z_{34}|^{-i\lambda_{3}-i\lambda_{4}}\Big{|}\frac{z_{14}}{z_{13}}\Big{|}^{i\lambda_{3}-i\lambda_{4}}\Big{|}\frac{z_{24}}{z_{14}}\Big{|}^{i\lambda_{1}-i\lambda_{2}}\,,$
(4.11)

where we used $\sum_{k}i\lambda_{k}=0$ in the last step. As a result, the
celestial limit of $D=4$ conformal block reads

$\lim_{\rho\rightarrow
0,\,\tau\rightarrow+\infty}W_{\Delta}(x_{n})=|z_{12}|^{-i\lambda_{1}-i\lambda_{2}}|z_{34}|^{-i\lambda_{3}-i\lambda_{4}}\Big{|}\frac{z_{14}}{z_{13}}\Big{|}^{i\lambda_{3}-i\lambda_{4}}\Big{|}\frac{z_{24}}{z_{14}}\Big{|}^{i\lambda_{1}-i\lambda_{2}}g_{\Delta}(x,\bar{x})\,.$
(4.12)

In order to express the celestial amplitude in terms of celestial limits of
conformal blocks, we start from Eq.(2.12) and apply the following identity
derived in Appendix B:

$\displaystyle{\cal
D}_{z}\;{}_{2}F_{1}\left({\alpha,\beta\atop\gamma};z\right){}_{2}F_{1}\left({\alpha,\beta\atop\gamma};\bar{z}\right)$
$\displaystyle=-{\frac{\alpha\beta}{\gamma}}\;{\frac{1}{z-\bar{z}}}\left\\{\bar{z}\;{}_{2}F_{1}\left({\alpha,\beta\atop\gamma-1};z\right){}_{2}F_{1}\left({\alpha+1,\beta+1\atop\gamma+1};\bar{z}\right)\right.$
$\displaystyle\qquad\qquad\qquad\left.-z\;{}_{2}F_{1}\left({\alpha+1,\beta+1\atop\gamma+1};z\right){}_{2}F_{1}\left({\alpha,\beta\atop\gamma-1};\bar{z}\right)\right\\}.$
(4.13)

In this way, we obtain

$\displaystyle G(x,\bar{x})$
$\displaystyle=\delta(\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4})\
\frac{|x|^{i\lambda_{1}+i\lambda_{2}}|1-x|^{i\lambda_{1}-i\lambda_{2}}}{x(1-x)}$
(4.14)
$\displaystyle\times\sum_{\Delta=i\lambda_{3}+i\lambda_{4}\atop\Delta=4+i\lambda_{1}+i\lambda_{2}}^{\prime}(1-\Delta)\;B\left({\frac{\Delta-
d_{12}}{2}},{\frac{\Delta+d_{12}}{2}}\right)B\left({\frac{\Delta+d_{34}}{2}},{\frac{\Delta-
d_{34}}{2}}\right)\;f_{\Delta}(z,\bar{z})\ ,$

where

$\displaystyle f_{\Delta}(z,\bar{z})$
$\displaystyle={\frac{1}{z-\bar{z}}}\;|z|^{\Delta}$ (4.15)
$\displaystyle\times\left\\{z\;_{2}F_{1}\left({{\frac{1}{2}}(\Delta-
d_{12}),{\frac{1}{2}}(\Delta+d_{43})\atop\Delta};z\right){}_{2}F_{1}\left({{\frac{1}{2}}(\Delta-
d_{12})-1,{\frac{1}{2}}(\Delta+d_{43})-1\atop\Delta-2};\bar{z}\right)\right.$
$\displaystyle\left.-\bar{z}\;{}_{2}F_{1}\left({{\frac{1}{2}}(\Delta-
d_{12})-1,{\frac{1}{2}}(\Delta+d_{43})-1\atop\Delta-2};z\right){}_{2}F_{1}\left({{\frac{1}{2}}(\Delta-
d_{12}),{\frac{1}{2}}(\Delta+d_{43})\atop\Delta};\bar{z}\right)\right\\}\ .$

In Eq.(4.14), the prime over the sum accounts for a relative minus sign
between the two terms. Next, we use the well-known hypergeometric identity

${}_{2}F_{1}\left({a,b\atop c};{\frac{x}{x-1}}\right)=(1-x)^{a}\
{}_{2}F_{1}\left({a,c-b\atop c};x\right)\ $ (4.16)

to show that

$f_{\Delta}(z,\bar{z})=|x-1|^{-d_{12}}\ g_{\Delta}(x,\bar{x})\ ,$ (4.17)

Finally, we combine all these expressions with Eqs.(2.7) and (2.1), to obtain

$\displaystyle\Big{\langle}\phi_{\Delta_{1},-}^{a_{1},-\epsilon}($
$\displaystyle
z_{1},\bar{z}_{1})\,\phi_{\Delta_{2},-}^{a_{2},-\epsilon}(z_{2},\bar{z}_{2})\,\phi_{\Delta_{3},+}^{a_{3},+\epsilon}(z_{3},\bar{z}_{3})\,\phi_{\Delta_{4},+}^{a_{4},+\epsilon}(z_{4},\bar{z}_{4})\Big{\rangle}=$
(4.18)
$\displaystyle=\Big{(}f^{a_{1}a_{2}b}f^{a_{3}a_{4}b}\,\frac{1}{z_{12}z_{23}z_{34}z_{41}}+f^{a_{1}a_{3}b}f^{a_{2}a_{4}b}\,\frac{1}{z_{13}z_{32}z_{24}z_{41}}\Big{)}\frac{z_{12}^{2}}{\bar{z}_{12}^{2}}\,{\cal
S}(z_{n},\bar{z}_{n})\ ,$

where

$\displaystyle{\cal S}(z_{n}$
$\displaystyle,\bar{z}_{n})=|z_{12}|^{-i\lambda_{1}-i\lambda_{2}}|z_{34}|^{-i\lambda_{3}-i\lambda_{4}}\Big{|}\frac{z_{14}}{z_{13}}\Big{|}^{i\lambda_{3}-i\lambda_{4}}\Big{|}\frac{z_{24}}{z_{14}}\Big{|}^{i\lambda_{1}-i\lambda_{2}}$
$\displaystyle~{}~{}~{}\times\sum^{\prime}_{\Delta=i\lambda_{3}+i\lambda_{4}\atop\Delta=4+i\lambda_{1}+i\lambda_{2}}\\!\\!\\!(1-\Delta)\;B\left({\frac{\Delta-
d_{12}}{2}},{\frac{\Delta+d_{12}}{2}}\right)B\left({\frac{\Delta-
d_{34}}{2}},{\frac{\Delta+d_{34}}{2}}\right)\;g_{\Delta}(x,\bar{x})\,.$ (4.19)

By comparing with Eq.(4.12), we see that ${\cal S}(z_{n},\bar{z}_{n})$
represents the celestial limit of $D=4$ CFT correlator of four gauge singlet
scalars with dimensions $d_{n}=i\lambda_{n}$, consisting of two $D=4$ blocks:
one with dimension $\Delta=i\lambda_{3}+i\lambda_{4}$ and one with the ($D=4$)
shadow dimension
$\Delta=4+i\lambda_{1}+i\lambda_{2}=4-i\lambda_{3}-i\lambda_{4}$.

In the correlator (4.18), the first factor on the r.h.s. (enclosed in the
bracket) represents the group-dependent part of the correlator of four
holomorphic (positive helicity) Wess-Zumino-Witten currents DiF ,

$\Big{\langle}J^{a_{1}}(z_{1})J^{a_{2}}(z_{2})J^{a_{3}}(z_{3})J^{a_{4}}(z_{4})\Big{\rangle}=k^{2}\frac{\delta^{a_{1}a_{2}}\delta^{a_{3}a_{4}}}{z_{12}^{2}z_{34}^{2}}-k\frac{f^{a_{1}a_{2}b}f^{ba_{3}a_{4}}}{z_{12}z_{23}z_{34}z_{41}}+(2\leftrightarrow
3)+(2\leftrightarrow 4).$ (4.20)

where $k$ denotes the level of Kač-Moody algebra. The group-independent part
is absent in our celestial amplitude because we did not include the effects of
gravitational interactions. As in the heterotic superstring theory, the level
$k$ could be determined by computing the relative weight of such
contributions. The subsequent factor $z_{12}^{2}/\bar{z}_{12}^{2}$, present in
Eq.(4.18), “flips” the helicity of gluons number 1 and 2 from $+$ to $-$. Both
factors do not depend on $\lambda_{n}$. The $\lambda_{n}$-dependence is
contained entirely in the scalar factor ${\cal S}(z_{n},\bar{z}_{n})$. This
correlator depends neither on helicities nor on gauge charges of external
gluons. Besides Eq.(4.19), the scalar part of the correlator can be also
represented in a Coulomb gas form as

$\displaystyle{\cal S}(z_{n},\bar{z}_{n})=$
$\displaystyle~{}|z_{12}|^{-2i\lambda_{1}-2i\lambda_{2}}|z_{13}|^{-2i\lambda_{1}-2i\lambda_{3}}|z_{14}|^{-2i\lambda_{4}}|z_{23}|^{2i\lambda_{1}}$
$\displaystyle\times(2+i\lambda_{1}+i\lambda_{2})B(2+i\lambda_{1},1+i\lambda_{2})B(i\lambda_{3},-1+i\lambda_{4})$
(4.21) $\displaystyle\times{\cal D}_{z}\Big{(}|1-x|^{-2(i\lambda_{1}+1)}\;\int
d^{2}w\ |w|^{-2i\lambda_{4}+2}\ |w-1|^{-2i\lambda_{2}-2}\
|w-x|^{-2i\lambda_{3}}\Big{)}\,$

where ${\cal D}_{z}$ is given in Eq.(2.21). This suggest that the CCFT
operators associated to gluons factorize into the current parts and certain
operators similar to those encountered in Coulomb gas models and in the
infinite central charge limit of Liouville theory Dorn:1994xn . In string
theory, the integrals of the form (4.21), with the kinematic invariants
(instead of dimensions $i\lambda_{n}$) in the exponents, are also known as the
Koba-Nielsen integrals.

##  5 $D=2$ conformal block decomposition of $D=4$ blocks

As concluded in the previous section, we expect $D=4$ conformal blocks (4.19)
to describe an autonomous sector of CCFT. This motivates us to perform $D=2$
conformal decomposition of this correlator. The $D=2$ chiral weights of
external scalars are $h_{k}=\bar{h}_{k}=i\lambda_{k}/2$. We consider the
conformal block decomposition of222Here, we ignore the
$\delta(\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4})$ prefactor.

$G_{\cal
S}(x,\bar{x})~{}=~{}\lim_{z_{1},\bar{z}_{1}\rightarrow\infty}z_{1}^{i\lambda_{1}}\bar{z}_{1}^{i\lambda_{1}}{\cal
S}(z_{1};z_{2}=1;z_{3}=x;z_{4}=0)~{}=~{}x(1-x)\,G(x,\bar{x})\ .$ (5.1)

In Ref.Fan:2022vbz , we performed the conformal block decomposition of the
four-gluon correlator $G(x,\bar{x})$ by using the power series solution of BG
equations. The $s$-channel ($x\approx 0$) conformal blocks associated to
primary fields with chiral weights $(h,\bar{h})$ were denoted by
$K_{34}^{21}\left[h,\bar{h}\right](x,\bar{x})$ and their explicit form was
given in Ref.Fan:2022vbz . The conformal block decomposition of $G_{\cal
S}(x,\bar{x})$ can be performed in a similar way, with the following result:

$\displaystyle G_{\cal S}(x,\bar{x})=\sum_{n=0}^{\infty}$
$\displaystyle\left\\{a_{n}\,K_{34}^{21}\left[n+\frac{i\lambda_{3}+i\lambda_{4}}{2},n+\frac{i\lambda_{3}+i\lambda_{4}}{2}\right](x,\bar{x})\right.$
(5.2)
$\displaystyle+\left.b_{n}\,K_{34}^{21}\left[n+2-\frac{i\lambda_{3}+i\lambda_{4}}{2},n+2-\frac{i\lambda_{3}+i\lambda_{4}}{2}\right](x,\bar{x})\right\\}\,.$
(5.3)

with the coefficients

$\displaystyle a_{n}$
$\displaystyle=B(n-i\lambda_{1},n-i\lambda_{2})B(n+i\lambda_{3},n+i\lambda_{4})\left(1-2n+i\lambda_{1}+i\lambda_{2}\right),$
(5.4) $\displaystyle b_{n}$
$\displaystyle=-B(n+2-i\lambda_{3},n+2-i\lambda_{4})B(n+2+i\lambda_{1},n+2+i\lambda_{2})\left(-3-2n+i\lambda_{3}+i\lambda_{4}\right).$

There are two sets of conformal blocks propagating in the $s$-channel: one
with the chiral weights

$h_{n}=\bar{h}_{n}=n+\frac{i\lambda_{3}+i\lambda_{4}}{2}$ (5.5)

and one with

$h_{n}={\bar{h}}_{n}=n+2-\frac{i\lambda_{3}+i\lambda_{4}}{2}\ .$ (5.6)

Note that all blocks have zero spin. We stress that they are gauge singlets
because the group-dependence is contained in the separate, Wess-Zumino-Witten
part of the four-gluon correlator.333The block decomposition presented here
can be viewed as a dimensional reduction of conformal blocks from $D$ to
$D{-}2$ dimensions, see Ref.Hogervorst:2016hal . For the reverse case, see
Ref.Kaviraj:2019tbg , where $(D{-}2)$-dimensional blocks are written as finite
linear combinations of $D$-dimensional blocks.

##  6 Conclusions

In this work, we studied the properties of four-gluon celestial amplitudes
constructed in Ref.Fan:2022vbz . We focused on the integral representation
describing four-gluon scattering in the presence of a dilaton background. We
expressed it in terms of Appell’s $F_{4}$ hypergeometric function and showed
that it agrees with the series expansion obtained by solving Banejree-Ghosh
equations.

Our main result is that the four-gluon celestial amplitude factorizes into the
“current” part and the “scalar” part, as written in Eq.(4.18). The current
factor is given by the group-dependent part of the Wess-Zumino-Witten
correlator of four holomorphic currents with a non-vanishing level of Kač-
Moody algebra. The scalar factor can be expressed in terms of a complex
integral of the Koba-Nielsen form, similar to the integrals describing four-
point correlators in Coulomb gas models and, more generally, in the infinite
central charge limit of Liouville theory. The connection to Liouville theory
is not precise at this point, but we believe that once it is put on solid
grounds, it will establish a close connection between CCFT and Liouville
theory. CCFT is similar to the heterotic superstring theory, which also
contains an “internal” holomorphic WZW current sector, but the “spacetime”
sector of heterotic strings is replaced in CCFT by Liouville theory.

A very interesting feature of the scalar part of the correlator is that it can
be obtained by a dimensional reduction of a single $D=4$ conformal block and
the shadow block from Minkowski space to the celestial sphere. This is
certainly related to $D=4$ conformal symmetry of Yang-Mills theory at the tree
level, but the fact that a “minimal” number of one block and its shadow appear
in the amplitude must have a deeper explanation. Understanding the role of
conformal symmetry and its breaking by quantum loops and by nonperturbative
effects in CCFT will help in studying Yang-Mills dynamics from a novel angle.
Acknowledgments
This material is based in part upon work supported by the National Science
Foundation under Grant Number PHY–1913328. Any opinions, findings, and
conclusions or recommendations expressed in this material are those of the
authors and do not necessarily reflect the views of the National Science
Foundation. Wei Fan is supported in part by the National Natural Science
Foundation of China under Grant No. 12105121.

##  Appendix A Intermediate steps from Eq.(2.10) to Eq.(2.12)

By using the integral representation of the hypergeometric function
${}_{2}F_{1}$,

${}_{2}F_{1}\left({a,b\atop c};x\right)=B(b,c-b)^{-1}\
\int_{0}^{1}d\eta\;\eta^{b-1}\;(1-\eta)^{c-b-1}\;(1-\eta x)^{-a}\ ,$

we can express the integral (2.10) as the following double integral

$\displaystyle
G(x,\bar{x})=\delta(\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4})\
\frac{1}{x(1-x)}$ (A.1)
$\displaystyle{\textstyle\times\Bigg{\\{}(1+i\lambda_{1})\int_{0}^{1}d\xi\;\xi^{i\lambda_{3}-1}\;(1-\xi)^{i\lambda_{1}+i\lambda_{4}-1}\int_{0}^{1}d\eta\;\eta^{-i\lambda_{2}-1}\;(1-\eta)^{-2-i\lambda_{1}}\;\left(1+{\frac{\xi
u_{2}}{1-\xi}}-\xi\eta u_{1}\right)^{i\lambda_{1}}}$
$\displaystyle{\textstyle+(1+i\lambda_{2})\;u_{1}^{2+i\lambda_{1}+i\lambda_{2}}\;\int_{0}^{1}d\xi\;\xi^{1-i\lambda_{4}}\;(1-\xi)^{-1-i\lambda_{3}-i\lambda_{2}}\int_{0}^{1}d\eta\;\eta^{1+i\lambda_{1}}\;(1-\eta)^{i\lambda_{2}}\;\left(1+{\frac{\xi
u_{2}}{1-\xi}}-\xi\eta u_{1}\right)^{-2-i\lambda_{2}}\Bigg{\\}},}$

with:

$u_{1}=|x|^{2}\ \ \ ,\ \ \ u_{2}=|1-x|^{2}\ .$ (A.2)

After introducing the new integration variables,

$\xi={\frac{1}{1+\beta}}\ \ \ ,\ \ \ \eta=\frac{1+\beta}{\gamma}\ ,$ (A.3)

with $d\xi d\eta=\tfrac{d\beta d\gamma}{(1+\beta)\gamma^{2}}$, we obtain

$\displaystyle G(x,\bar{x})$
$\displaystyle=\delta(\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4})\
\frac{1}{x(1-x)}$ (A.4)
$\displaystyle\times\Bigg{\\{}(1+i\lambda_{1})\int_{0}^{\infty}{\frac{d\beta}{\beta}}\int_{1+\beta}^{\infty}{\frac{d\gamma}{\gamma}}\
{\frac{\beta^{i\lambda_{4}}\;\gamma^{2+i\lambda_{2}}}{(\beta\gamma+\gamma
u_{2}-\beta u_{1})^{-i\lambda_{1}}\;(\gamma-\beta-1)^{2+i\lambda_{1}}}}$
$\displaystyle+(1+i\lambda_{2})\left({\frac{u}{v}}\right)^{2+i\lambda_{1}+i\lambda_{2}}\int_{0}^{\infty}{\frac{d\beta}{\beta}}\int_{1+\beta}^{\infty}{\frac{d\gamma}{\gamma}}\
{\frac{\beta^{2-i\lambda_{3}}\;\gamma^{-i\lambda_{1}}}{(\beta\gamma+\gamma
u_{2}-\beta
u_{1})^{2+i\lambda_{2}}\;(\gamma-\beta-1)^{-i\lambda_{2}}}}\Bigg{\\}},$

with:

$\displaystyle v$
$\displaystyle={\frac{1}{u_{2}}}={\frac{1}{|1-x|^{2}}}=|z-1|^{2}\ ,$ (A.5)
$\displaystyle u$
$\displaystyle={\frac{u_{1}}{u_{2}}}=\left|{\frac{x}{x-1}}\right|^{2}=|z|^{2}\
.$ (A.6)

Furthermore, (A.4) can be expressed as:

$\displaystyle G$
$\displaystyle(x,\bar{x})=-\delta(\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4})\
\frac{v^{-i\lambda_{1}}}{x(1-x)}\ $ (A.7)
$\displaystyle\times{\frac{\partial}{\partial v}}\
\Bigg{\\{}\int_{0}^{\infty}{\frac{d\beta}{\beta}}\int_{1+\beta}^{\infty}{\frac{d\gamma}{\gamma}}\;{\frac{\beta^{i\lambda_{4}-1}\;\gamma^{1+i\lambda_{2}}}{(\gamma+\beta\gamma
v-\beta u)^{-1-i\lambda_{1}}\;(\gamma-\beta-1)^{2+i\lambda_{1}}}}$
$\displaystyle\qquad\qquad-u^{2+i\lambda_{1}+i\lambda_{2}}\int_{0}^{\infty}{\frac{d\beta}{\beta}}\int_{1+\beta}^{\infty}{\frac{d\gamma}{\gamma}}\;{\frac{\beta^{1-i\lambda_{3}}\;\gamma^{-1-i\lambda_{1}}}{(\gamma+\beta\gamma
v-\beta u)^{1+i\lambda_{2}}\;(\gamma-\beta-1)^{-i\lambda_{2}}}}\Bigg{\\}}\ .$

Now we can borrow the following integral Simmons-Duffin:2012juh

$\displaystyle\int_{0}^{\infty}{\frac{d\beta}{\beta}}\int_{1+\beta}^{\infty}{\frac{d\gamma}{\gamma}}$
$\displaystyle\;{\frac{\beta^{b-1}\;\gamma^{e-1}}{(\gamma+v\beta\gamma-u\beta)^{1-f}\;(\gamma-\beta-1)^{f}}}$
$\displaystyle=B(-1+b,4-b-e-f)\;B(2-e,1-f)\ $ (A.8)
$\displaystyle\times{}_{2}F_{1}\left({-1+b,1-f\atop
3-e-f};z\right)\;_{2}F_{1}\left({-1+b,1-f\atop 3-e-f};\bar{z}\right)\ ,$

to cast (A.7) into Eq.(2.12).

##  Appendix B Hypergeometric function relations

In this appendix we detail the steps from the integral representation
Eq.(2.12) to Eq.(3.5).

We first notice the duplication formula Slater

${}_{2}F_{1}\left({\alpha,\beta\atop\gamma};z\right){}_{2}F_{1}\left({\alpha,\beta\atop\gamma};\bar{z}\right)={\cal{G}}\left[\alpha,\beta,\gamma,\gamma;u,1-v\right]\
,$ (B.1)

with the variables related as in Eq.(A.6), and

$\displaystyle{\cal{G}}\left[\alpha,\beta,\gamma,\delta;u,1-v\right]$
$\displaystyle={\frac{\Gamma(\delta)\Gamma(\delta-\alpha-\beta)}{\Gamma(\delta-\alpha)\Gamma(\delta-\beta)}}\
F_{4}\left[\alpha,\beta,\gamma,\alpha+\beta+1-\delta;u,v\right]$
$\displaystyle+{\frac{\Gamma(\delta)\Gamma(\alpha+\beta-\delta)}{\Gamma(\alpha)\Gamma(\beta)}}\;v^{\delta-\alpha-\beta}\;F_{4}\left[\delta-\alpha,\delta-\beta,\gamma,\delta-\alpha-\beta+1;u,v\right]$
$\displaystyle=\sum_{m,n=0}^{\infty}{\frac{u^{m}(1-v)^{n}}{m!n!}}\;{\frac{(\delta-\alpha)_{m}(\delta-\beta)_{m}}{(\gamma)_{m}}}{\frac{(\alpha)_{m+n}(\beta)_{m+n}}{(\delta)_{2m+n}}}\
.$ (B.2)

The function (B.2) enjoys the transformation property Dolan:2000uw :

${\cal{G}}\left[\alpha,\beta,\gamma,\delta;u,1-v\right]=v^{-\alpha}\
{\cal{G}}\left[\alpha,\delta-\beta,\gamma,\delta;{\frac{u}{v}},1-{\frac{1}{v}}\right]\
.$ (B.3)

Next, we consider the relation

$\partial_{v}\;{\cal{G}}\left[\alpha,\beta,\gamma,\delta;u,1-v\right]=-{\frac{\alpha\beta}{\delta}}\
{\cal{G}}\left[\alpha+1,\beta+1,\gamma,\delta+1;u,1-v\right]\ ,$ (B.4)

following from

$\partial_{v}F_{4}\left[\alpha,\beta,\gamma,\delta;u,v\right]={\frac{\alpha\beta}{\delta}}\
F_{4}\left[1+\alpha,1+\beta,\gamma,1+\delta;u,v\right]\ ,$ (B.5)

which together with (B.1) gives rise to:

$\displaystyle{\cal
D}_{z}\;{}_{2}F_{1}\left({\alpha,\beta\atop\gamma};z\right){}_{2}F_{1}\left({\alpha,\beta\atop\gamma};\bar{z}\right)$
$\displaystyle={\frac{\alpha\beta}{\gamma}}\
{\cal{G}}\left[\alpha+1,\beta+1,\gamma,\gamma+1;u,1-v\right]$ (B.6)
$\displaystyle=-{\frac{\alpha\beta}{\gamma}}\;{\frac{1}{z-\bar{z}}}\left\\{\bar{z}\;{}_{2}F_{1}\left({\alpha,\beta\atop\gamma-1};z\right){}_{2}F_{1}\left({\alpha+1,\beta+1\atop\gamma+1};\bar{z}\right)\right.$
$\displaystyle\qquad\qquad\qquad\left.-z\;{}_{2}F_{1}\left({\alpha+1,\beta+1\atop\gamma+1};z\right){}_{2}F_{1}\left({\alpha,\beta\atop\gamma-1};\bar{z}\right)\right\\}$

After inserting Eqs.(B.6) and (B.2) into Eq.(2.12), we obtain Eq.(3.5).

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# A maximum magnetic moment to angular momentum conjecture

John D. Barrow1 and G. W. Gibbons1,2,3
1DAMTP, Centre for Mathematical Sciences,
University of Cambridge,
Wilberforce Rd., Cambridge CB3 0WA, UK
2LE STUDIUM, Loire Valley Institute for Advanced Studies,
Tours and Orleans, France
3Laboratoire de Mathématiques et de Physique
Théorique, Université de Tours, France

###### Abstract

Conjectures play a central role in theoretical physics, especially those that
assert an upper bound to some dimensionless ratio of physical quantities. In
this paper we introduce a new such conjecture bounding the ratio of the
magnetic moment to angular momentum in nature. We also discuss the current
status of some old bounds on dimensionless and dimensional quantities in
arbitrary spatial dimension. Our new conjecture is that the dimensionless
Schuster-Wilson-Blackett number, $c\mu/JG^{\frac{1}{2}}$, where $\mu$ is the
magnetic moment and $J$ is the angular momentum, is bounded above by a number
of order unity. We verify that such a bound holds for charged rotating black
holes in those theories for which exact solutions are available, including the
Einstein-Maxwell theory, Kaluza-Klein theory, the Kerr-Sen black hole, and the
so-called STU family of charged rotating supergravity black holes. We also
discuss the current status of the Maximum Tension Conjecture, the Dyson
Luminosity Bound, and Thorne’s Hoop Conjecture.

##  1 Introduction

Regardless of what one thinks of the debate concerning the relative merits of
the traditional Baconian or inductionist, versus Bayesian or Popperian,
viewpoints about the nature of science, few would disagree that making
precisely stated conjectures or exhibiting counter-examples has an important
place in theoretical physics. In making such conjectures it is important to
bare in mind that although it is frequently convenient to adopt units well
suited to practical aspects of the subject being discussed, any physically
meaningful statement must be independent of an arbitrary choice of units. In
fact, adopting an appropriate set of “natural units” can afford insights which
may be otherwise obscured. In this paper we are led in section 2, by our
consideration of natural units for physical quantities which are independent
of Planck’s constant, to conjecture new fundamental bounds on dimensionless
quantities in classical gravitation, in particular that there is an upper
bound on the magnetic moment to angular momentum ratio. In section 3, we
verify that such a bound holds for charged rotating black holes in those
theories for which exact solutions are available, including the Einstein-
Maxwell theory, Kaluza-Klein theory, the Kerr-Sen black hole, and the so-
called STU family of charged rotating supergravity black holes. We discuss the
current status of the Maximum Tension Conjecture in section 4, the Dyson
luminosity bound in section 5, and new approaches to Thorne’s Hoop Conjecture
in section 6.

##  2 Units and dimensional analyses

Natural units were first introduced into physics and metrology by George
Johnstone Stoney at the British Association Meeting in 1874, in an attempt to
cut through the proliferation of parochial units of measurement spawned by the
industrial revolution and the expansion of Victorian engineering and commerce
[1, 3]. He sought to devise units that, unlike feet and horsepower, avoided
any anthropomorphic benchmark, and made no use of changing parochial
standards, like days or standard weights. A similar universal approach had
also been advocated by Maxwell in 1870, who suggested that constants be
founded on atomic or optical standards [2, 3]. He also saw a new opportunity
to promote his prediction of a new elementary particle, which he first dubbed
the ’electrolion’ in 1881 and then renamed the ’electron’ in 1894, carrying a
basic unit of electric charge, $e$, whose numerical value he predicted using
Faraday’s Law and Avogadro’s Number. The electron was subsequently discovered
by Thomson in 1897, and Stoney remains the only person to have successfully
predicted the numerical value of a new fundamental constant of physics.

###  2.1 Stoney Units

In response to a challenge from the British Association to reduce or organise
the plethora of special units that had sprung up to service the industrial
revolution and Britain’s trading empire, in 1874 Johnstone Stoney first
introduced a system of ”natural units” of mass, length and time using the
speed of light, $c$, the Newtonian gravitational constant, $G,$and his
proposed electron charge, $e$, [4, 5, 6]. Stoney’s natural units were

$M_{S}=\left(\frac{e^{2}}{G}\right)^{1/2},\text{
}L_{S}=\left(\frac{Ge^{2}}{c^{4}}\right)^{1/2},\text{
}T_{S}=\left(\frac{Ge^{2}}{c^{6}}\right)^{1/2}.$ (1)

These were the first natural units. However, we should note that in those days
before the theory of special relativity, the speed of light, $c$, did not
possess the absolute status that it would later assume and $e$ was still just
a hobby-horse of Stoney’s (for some context see the history ref. [7]).

###  2.2 Planck Units

In 1899, a similar idea was introduced by Max Planck [8] to create another set
of natural units based on $c,G,$ and $h,$ the quantum constant of action that
bears his name. They differ from Stoney’s units by a factor
$\frac{1}{2\pi}(\frac{e^{2}}{\mathchar 1406\relax c})^{1/2}$ – the square root
of the fine structure constant divided by $2\pi$. These units are now
commonplace in physics and cosmology and they define units of mass, length and
time that combine relativistic, gravitational and quantum aspects of physics:

$M_{Pl}=\left(\frac{hc}{G}\right)^{1/2},\text{
}L_{Pl}=\left(\frac{Gh}{c^{3}}\right)^{1/2},\text{
}T_{Pl}=\left(\frac{Gh}{c^{5}}\right)^{1/2}.$ (2)

However, innumerable related Planck units may be constructed for other
physical quantities in any number of space dimensions by dimensional analysis.
Those involving thermal physics can be included by adding the Boltzmann
constant, $k_{B}$, to $G,c$ and $h$. Some of the Planck units are especially
interesting for classical physics if they do not contain Planck’s constant.
This signals that they are purely classical in origin and may highlight a
limiting physical principle. This is trivially so for the Planck unit of
velocity, $V_{Pl}=c$, but less obvious for the Planck units of force
$F_{Pl}=c^{4}/G$ and power $P_{Pl}=c^{5}/G$ which are strongly suspected to be
maximal quantities in classical physics. It has been conjectured [9, 11, 12,
13, 14] 111for an earlier anticipation of this idea but based on a different
physical motivation see [16]that in general relativity (with and without a
cosmological constant) there should be a maximum value to any physically
attainable force (or tension) given by

$F_{\max}=\frac{c^{4}}{4G}\,,$ (3)

where $c$ is the velocity of light and $G$ is the Newtonian gravitational
constant. For possible relations to the holographic principle and to quantum
clocks, see [17, 18, 19].

###  2.3 De Sitter units

If one believes that the observed acceleration of the scale factor of the
universe [20] is due to a classical cosmological constant $\Lambda$ rather
than some form of slowly-evolving ’dark energy’, with time-dependent density,
then a set of absolute de Sitter units of mass, length, and time can be
introduced:

$M_{dS}=c^{2}G^{-1}\Lambda^{-\frac{1}{2}}\,,L_{ds}=\Lambda^{-\frac{1}{2}}\,,T_{ds}=c^{-1}\Lambda^{-\frac{1}{2}}\,.$
(4)

In these units $c^{4}/G$ is still the unit of force and the upper bound (3)
still appears to hold [14].

###  2.4 Fundamental principles and dimensions

We referred above to ’limiting principles’, or what are sometimes called
’impotence principles’. In [9] the phrase ’maximum tension principle’ was used
in the usual sense of ’fundamental principles’, that is general statements
expected to be true of all viable theories and which may follow as a valid
consequences of a precisely formulated mathematical statement within any well-
defined mathematically theory. Such principles may have heuristic value in
motivating and formulating a theory, but cannot be used in themselves to
define a theory. For example ’Heisenberg’s Uncertainty Principle’ is an
elementary theorem in wave mechanics but is insufficient in itself to define
wave mechanics. Moreover, it not only rests heavily on translation invariance,
but may not hold in more general quantum mechanical theories, such as
relativistic quantum field theory, in which the notion of a position
observable is problematic. Other examples in general relativity include
’Mach’s principle’ 222For an incisive account of many inequivalent
formulations this can be given see [10]. ’equivalence principles. and Thorne’s
’hoop conjecture’ (to which we return below). Other ’principles’, like the
’cosmological principle’ may be simplifying symmetry assumptions, or
approximations, that cannot be precisely true in reality, or straightforward
methodological principles, like the ’weak anthropic principle’, or various
variational principles.

The maximum force conjecture gives rise to the closely related conjecture [21]
that there is a maximum power defined by

$P_{\max}=cF_{\max}=\frac{c^{5}}{4G},$ (5)

the so-called Dyson luminosity [22], or some multiple of it (to account for
geometrical factors that are $O(1)$). This will be treated in detail in
section 4.

We note that some of the non-quantum Planck units, like the velocity,
$V_{Pl}=c$, are independent of the dimension of space but others, like
$F_{Pl}$, are not, because in $N$-dimensional space the dimensions of $G$ are
$M^{-1}L^{N}T^{-2}$. Thus, in $N$ dimensions the non-quantum Planck unit is
mass $\times$ (acceleration)N-2, which is only a force when $N=3,$ as shown in
ref. [14].

In this paper, we display another physically interesting non-quantum Planck
unit formed by the ratio of the magnetic moment of a body, $\mu$, to its total
angular momentum, $J$, and conjecture that classically all bodies satisfy an
inequality

$\frac{\mu}{J}<\beta\frac{G^{1/2}}{c},$ (6)

where $\beta$ is a numerical factor $O(1),$ and we explore the evidence for
this maximum bound. Unlike the Planck units of force and power, the Planck
unit for the ratio $\mu/J$ is independent of spatial dimension.

To show this, if we use unrationalised units the dimensions $[.]$ of magnetic
$Q$ and electric charge $\bar{Q}$ are the same and are given by the inverse-
square laws discovered by Michell and Priestley, [23, 24], respectively, with

$[Q]=[\bar{Q}]=M^{\frac{1}{2}}L^{\frac{3}{2}}T^{-1}\,.$ (7)

The dimensions of a magnetic moment $\mu$ are therefore

$\bigl{[}\mu\bigr{]}=M^{\frac{1}{2}}L^{\frac{5}{2}}T^{-1}\,.$ (8)

Thus, the ratio of magnetic moment to angular momentum $J$ has dimensions

$\bigl{[}\frac{\mu}{J}\bigr{]}=\bigl{[}\frac{G^{\frac{1}{2}}}{c}\bigr{]}=\bigr{[}\frac{Q}{Mc}\bigr{]}\,.$
(9)

which is independent of Planck’s constant $\mathchar 1406\relax$. This
property continues to hold in $N$-dimensional space because there we have
$[Q]=M^{1/2}L^{N/2}T^{-1}$, $[\mu]=M^{1/2}L^{1+N/2}T^{-1}$ and
$[J]=ML^{2}T^{-1}$.

The ratio

$Z\equiv\frac{Q^{2}}{GM^{2}}$ (10)

may be regarded as the separation-independent ratio of the electrostatic
repulsion to the gravitational attraction between two identical bodies of mass
$M$ and charge $Q$. It has been claimed [25] that Zöllner was the first person
to recognise its significance and so one might call it the Zöllner number. A
famous, but now discredited, theory of Dirac’s predicting the time variation
of the gravitation ’constant’ $G\propto 1/t$, with the age of the universe
$t$, [26] was partly motivated by the very small value of $Z$ when the mass
$M=m_{e}$ and charge $Q=e$ of the electron (or even the proton mass $m_{pr}$)
are substituted. Thus, giving

$N=\frac{e^{2}}{Gm_{e}^{2}}\approx 3\times 10^{42},$ (11)

which suggested to Dirac its possible equality (in some yet to be found
theory) with the square root of the total number of protons or electrons in
the visible universe, $c^{3}t/Gm_{e}\sim 10^{83}$, up to a factor $O(1)$. In
fact, the value of $N$ and its numerical proximity to the ratio of the
classical electron radius to the Hubble radius was first noticed by Weyl in
1919 [27, 5] and the numerical ’coincidence’ is anthropic because it is
equivalent to the statement that the present age of the universe is of order
the main sequence lifetime of a star [28, 5].

Classically, we have the Larmor relation

$\frac{\mu}{J}=\frac{Q}{2Mc}$ (12)

where $M$ is the mass of a system with charge $Q$. More generally, we have

$\frac{\mu}{J}=g\frac{Q}{2Mc}$ (13)

where $g$ is the gyromagnetic ratio. Famously, Dirac showed that for electrons
$g=2$, at least at lowest order in the fine structure constant
$e^{2}/\mathchar 1406\relax c,$ [29], and this value has some significance in
supersymmetric theories [30].

After earlier suggestions made by Schuster [31] and Wilson [32], Blackett [33]
conjectured that all rotating bodies should acquire a magnetic moment given by

$\frac{\mu}{J}=\beta\frac{G^{\frac{1}{2}}}{c}\,,$ (14)

where the dimensionless Schuster-Wilson-Blackett number has $\beta\approx
O(1),$ and was once regarded as a possible universal constant. Although
$\beta$ is found to be of order unity for a variety of rotating astronomical
bodies ranging from the earth, the sun, and a variety of stars, as a general
statement for macroscopic bodies, the Schuster-Wilson-Blackett conjecture has
fallen foul of astronomical data. Yet it remains of interest to enquire
whether it provides a natural upper bound for bodies with significant
gravitational self-energy.

Since, for electrons

$\beta=N^{\frac{1}{2}}\,,$ (15)

no interesting bound holds for the elementary particles. However, it is of
interest to ask what is known about $\beta$ in Einstein-Maxwell and
supergravity theories, since for black holes there is typically an upper to
$|Q|/G^{\frac{1}{2}}M$ of order unity. For Planck mass particles with charges
of order $e$, we find $\beta$ is not far from unity. Such objects can arise in
string theory, whose low-energy limit is supergravity theory, so this further
motivates the investigation that follows.

##  3 The Schuster-Wilson-Blackett Number for electrically charged rotating
black holes

Brandon Carter first discovered [34] that Kerr-Newman black holes in Einstein-
Maxwell theory have a gyromagnetic ratio equal to $2$:

$\frac{\mu}{J}=\frac{Q}{Mc}\,.$ (16)

Now, to avoid naked singularities, we require (if we assume the black hole has
no magnetic charge)

$GM^{2}\geq Q^{2}+\frac{J^{2}}{M^{2}}.$ (17)

Thus,

$1\leq\frac{c^{2}\mu^{2}}{GJ^{2}}+\frac{J^{2}}{GM^{4}},$ (18)

and so we have the required bound:

$\big{|}\frac{\mu}{J}\big{|}\leq\frac{G^{\frac{1}{2}}}{c}\,.$ (19)

Hence, we have $\beta<1$ for Kerr-Newman black holes. The literature on
extensions of Carter’s result is quite large. A notable example [35] is a
detailed analysis of a current loop surrounding a static black hole. As the
loop moves towards the horizon the gyromagnetic ratio smoothly interpolates
between the classical value $g=1$ and the Carter-Dirac value $g=2$.

It was shown by Reina and Treves [36] that any asymptotically-flat solution of
the Einstein-Maxwell equations obtained by performing a Harrison
transformation on a neutral solution must also have $g=2$. Furthermore, it has
been shown [37, 38] that, provided any sources obey the constraint that $G$
times the energy density bounds the charge density, then all asymptotically-
flat solutions of the Einstein Maxwell equations, possibly with sources of the
kind specified which are regular outside a regular event horizon, obey the
following Bogomolnyi bound on the Zöllner number:

$Z=\frac{Q^{2}}{GM^{2}}\leq 1.$ (20)

Combining this with Reina and Treves’ result, implies

$\beta<1.$ (21)

###  3.1 Kerr-Newman AdS black holes

Using the notation of [39], and temporarily setting $G=c=1$, we must
distinguish the parameters $M,Q,J,$ in the spacetime metric from the physical
quantities. The latter are denoted by primes. From [40], we introduce

$M^{\prime}\equiv\frac{M}{\Xi^{2}}\,,\qquad
J^{\prime}\equiv\frac{aM}{\Xi^{2}}$ (22)

where $\Xi\equiv 1-\frac{a^{2}}{l^{2}}$.

Aliev gives the physical charge as

$Q^{\prime}=\frac{Q}{\Xi}\,,$ (23)

He finds

$\mu^{\prime}=\frac{Qa}{\Xi}\,,$ (24)

so we have

$\frac{|\mu^{\prime}|}{|J^{\prime}|}=\frac{|Q|}{M}(1-\frac{a^{2}}{l^{2}})\,.$
(25)

Now, for a horizon to exist, we require

$\Delta_{r}=(1+\frac{a^{2}}{l^{2}})\Bigl{(}r^{2}-\frac{2Mr}{1+\frac{a^{2}}{l^{2}}}+\frac{Q^{2}+a^{2}}{1+\frac{a^{2}}{l^{2}}}\Bigr{)}+\frac{r^{4}}{a^{2}}$
(26)

to have at least one real root. A _necessary_ condition for this is that the
quadratic in the first term be negative. This requires

$\frac{|Q|}{M}<\frac{1}{\sqrt{1+\frac{a^{2}}{l^{2}}}}.$ (27)

Thus, we also require

$\frac{|\mu^{\prime}|}{|J^{\prime}|}<\frac{1-\frac{a^{2}}{l^{2}}}{\sqrt{1+\frac{a^{2}}{l^{2}}}}.$
(28)

Now,

$(1-x)(1+x)=1-x^{2}\leq
1\,,\quad\Rightarrow\quad\frac{1-x}{\sqrt{1+x}}\leq\frac{1}{(1+x)^{\frac{3}{2}}}\leq
1\,,$ (29)

so

$\frac{|\mu^{\prime}|}{|J^{\prime}|}<1\,.$ (30)

Therefore, we have shown that $\beta<1$ for Kerr-Newman-AdS black holes.
333Note that since Harrison transformations are not available when the
cosmological constant is non-vanishing, there is no analogue of the Reina-
Treves result with which to combine the Bogomolnyi bound of. [41] in this
case.

###  3.2 Einstein-Maxwell-Dilaton black holes

These have only been discussed for general dilaton-photon coupling constant
$\alpha$ for the case of slow rotation [47, 48]. One has a uniqueness theorem
for general $\alpha$, angular momentum, electric and magnetic charges [49]
provided that $\alpha^{2}\leq 3$.

In the general slow-rotation case one finds that [47]

$J=\frac{a}{2}\bigl{(}r_{+}+\frac{3-\alpha^{2}}{3(1+\alpha^{2})}r_{-}\bigr{)}\,,\qquad\mu=aQ\,.$
(31)

If $a$ is small, then the mass $M$ and charge $Q$ are given by

$M=\frac{1}{2}\Bigl{(}r_{+}+\frac{1-\alpha^{2}}{1+\alpha^{2}}r_{-}\Bigr{)}\,,\qquad|Q|=\sqrt{\frac{r_{+}r_{-}}{1+\alpha^{2}}}\,.$
(32)

Since $r_{+}\geq r_{-}\geq 0,$ we have

$\frac{|Q|}{M}\leq\sqrt{1+\alpha^{2}},$ (33)

so that in accordance with the Bogolmolnyi bound of [50], this gives

$M\geq\frac{|Q|}{\sqrt{1+\alpha^{2}}}\,.$ (34)

We have

$\frac{|J|}{|\mu|}=\frac{1}{2}\sqrt{1+\alpha^{2}}\Bigl{(}\sqrt{\frac{r_{+}}{r_{-}}}+\frac{3-\alpha^{2}}{3(1+\alpha^{2}}\sqrt{\frac{r_{-}}{r_{+}}}\Bigr{)}\,,$
(35)

so, provided $\alpha^{2}\leq 3$, this gives

$\frac{|\mu|}{|J|}\leq\frac{1}{2}\sqrt{1+\alpha^{2}}\leq 1\,.$ (36)

As pointed out in [47], we can then obtain a gyro-magnetic ratio:

$g=2-\frac{4\alpha^{2}r_{-}}{(3-\alpha^{2})r_{-}+3(1+\alpha^{2})r_{+}}\,.$
(37)

###  3.3 Kerr-Kaluza-Klein black holes

The observational and theoretical failures of the old Schuster-Blackett
conjecture (14) led some to resort to Kaluza-Klein theory (see [42]). Rotating
charged black holes in this theory may be obtained by boosting the neutral
Kerr solution (sometimes referred to in this context as a rotating ’black
string’) along the fifth dimension. If $v$ is the velocity parameterizing the
boost, and $a$ and $M_{s}$ the parameters of the original Kerr solution, then
in units in which $G=c=1$ [43],[44], we have

$\displaystyle
M=M_{s}\bigl{(}1+\frac{1}{2}\frac{v^{2}}{1-v^{2}}\bigr{)}\,,\qquad J$
$\displaystyle=$ $\displaystyle\frac{M_{s}a}{\sqrt{1-v^{2}}}$ $\displaystyle
Q=M_{s}\frac{v}{1-v^{2}}\,,\qquad\mu$ $\displaystyle=$
$\displaystyle\frac{M_{s}av}{\sqrt{1-v^{2}}}$ (38)

and the gyromagnetic ratio is $g=2-v^{2}$. Restoring units, we have

$\frac{|\mu|}{|J|}=\frac{G^{\frac{1}{2}}}{c}v,$ (39)

and, remarkably, we see that $\beta=v/c\leq 1$.

We may also regard Kaluza-Klein black holes as Einstein-Maxwell-Dilaton black
holes with $\alpha=\sqrt{3}$. For the gyromagnetic ratios of elementary
particles in Kaluza-Klein theory and their comparison with black holes, the
reader may consult [46, 42, 43].

###  3.4 Kerr-Sen electrically-charged black holes

These black holes satisfy the low-energy equations of motion of heterotic
string theory [51] and may be regarded as an Einstein-Maxwell-Dilaton black
hole with coupling constant $\alpha=1$. According to [51], the mass $M$,
charge $Q$, angular momentum $J$ and magnetic dipole moment, $\mu,$ are given
by

$\displaystyle M$ $\displaystyle=$
$\displaystyle\frac{m}{2}(1+\cosh\theta)\,,\qquad J=Ma$ $\displaystyle Q$
$\displaystyle=$ $\displaystyle\frac{m}{\sqrt{2}}\sinh\theta\,,\quad\mu=Qa\,.$
(40)

where $m,a,\theta$ are parameters 444Our $\theta$ is Sen’s $\alpha$.. Thus, we
find

$\beta=\frac{|\mu|}{|J|}=\sqrt{2}\frac{\sinh\theta}{1+\cosh\theta}=\sqrt{2}\tanh\frac{\theta}{2}\leq\sqrt{2}\,.$
(41)

and

$g=2\,.$ (42)

We also find a Bogomolnyi inequality (34) with $\alpha^{2}=1$ is satisfied,
that is,

$\frac{|Q|}{M}\leq\sqrt{2}.$ (43)

###  3.5  STU electrically charged black holes

The electromagnetic properties of a more general family of 4-charged black
holes, which are solutions of the so-called $STU$ supergravity theory
(characterised by $S$, $T$, and $U$ dualities) are reviewed in [45]. These
solutions depend upon 4 boost parameters, $\delta_{i}$. If
$c_{i}=\cosh\delta_{i}$, $s_{i}=\sinh\delta_{i}$,
$\Pi_{c}=c_{1}c_{2}c_{3}c_{4}$, $\Pi_{s}=s_{1}s_{2}s_{3}s_{4}$,
$\Pi_{c}^{1}=c_{2}c_{3}c_{4}\,\mathrm{etc}$,
$\Pi_{s}^{1}=s_{2}s_{3}s_{4}\,\mathrm{etc}$, then according to [45]

$\displaystyle\qquad
M=\frac{m}{4}\sum_{i}\bigl{(}c_{i}^{2}+s_{i}^{2}\bigr{)}\,,\qquad J$
$\displaystyle=$ $\displaystyle ma\big{(}\Pi_{c}-\Pi_{s}\bigr{)}$ (44)
$\displaystyle Q_{i}=2ms_{i}c_{i}\,,\qquad\mu_{i}$ $\displaystyle=$
$\displaystyle 2ma\bigl{(}s_{i}\Pi_{c}^{i}-c_{i}\Pi_{s}^{i}\bigr{)}\,.$ (45)

Evidently

$4M\geq\sum_{i}|Q_{i}|\,.$ (46)

We also have

$\frac{1}{2}\frac{\mu_{i}}{J}=\frac{s_{i}\Pi_{c}^{i}-c_{i}\Pi_{s}^{i}}{\Pi_{c}-\Pi_{s}}\,.$
(47)

If we assume that $s_{i}>0$, $\forall\,i$, then

$\frac{1}{2}\frac{\mu_{i}}{J}\leq\frac{s_{i}(\Pi_{c}^{i}-\Pi_{s}^{i})}{c_{i}(\Pi_{c}^{i}-\Pi_{s}^{i})}\leq\tanh\delta_{i}\leq
1\,.$ (48)

There are some special cases which coincide with solutions of the Einstein-
Maxwell-Dilaton theory.

  * •

Einstein-Maxwell Black Holes: $\delta_{1}=\delta_{2}=\delta_{3}=\delta_{4}$,
$Q_{i}=Q$, $\mu_{1}=\mu$.

Thus, from (45), we have

$\displaystyle M=m\cosh{2\delta}\,,\qquad J$ $\displaystyle=$ $\displaystyle
ma\cosh 2\delta,$ (49) $\displaystyle Q_{i}=m\sinh 2\delta\,,\quad\mu_{i}$
$\displaystyle=$ $\displaystyle ma\sinh 2\delta,$ (50)

and if we set $Q=Q_{i}$ and $\mu=\mu_{i}$ so that

$Q^{2}=\frac{1}{4}\bigl{(}Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}+Q_{4}^{2}\bigr{)},$
(51)

then we find that $g=2$ and

$\frac{|\mu|}{|J|}=\tanh 2\delta=\frac{|Q|}{M}\leq 1\,.$ (52)

Evidently both the charge inequality (46) and the dipole inequality ( 48) are
satisfied, the latter by a comfortable margin since for $x>0$,

$\tanh 2x\leq 2\tanh x\,.$ (53)

.

  * •

Kerr-Kaluza-Klein electrically charged black holes:
$\delta_{2}=\delta_{3}=\delta_{4}=0$, $\delta_{1}=\delta$.

$\displaystyle M=\frac{m}{4}(3+\cosh{2\delta}),\qquad J$ $\displaystyle=$
$\displaystyle ma\cosh\delta$ (54) $\displaystyle Q_{1}=m\sinh
2\delta\,,\qquad\mu_{1}$ $\displaystyle=$ $\displaystyle 2ma\sinh\delta$ (55)

Now, if $v=\tanh\delta$ then from (40) we have

$\displaystyle M$ $\displaystyle=$
$\displaystyle\frac{M_{s}}{4}\bigl{(}3+\cosh 2\delta\bigr{)}\,,\qquad
J=M_{s}a\cosh\delta$ $\displaystyle Q$ $\displaystyle=$
$\displaystyle\frac{M_{s}}{2}\sinh 2\delta\,,\qquad\mu=M_{s}a\sinh\delta$ (56)

Thus, $M_{s}=m$, $Q=\frac{1}{2}Q_{1}$ and $\mu=\frac{1}{2}\mu_{1},$ so that we
have

$Q^{2}=\frac{1}{4}Q_{1}^{2},$ (57)

and we find that

$\beta=\frac{|\mu|}{|J|}=\tanh\delta\leq 1\,.$ (58)

We also have

$M\geq\frac{1}{2}|Q|,$ (59)

which is consistent with (34) provided that $\alpha^{2}=3$.

  * •

String Theory: $\delta_{1}=\delta_{2}=\delta$, $\delta_{3}=\delta_{4}=0.$

Now, we have

$\displaystyle M=\frac{1}{2}m(1+\cosh 2\delta)\,,\qquad J$ $\displaystyle=$
$\displaystyle\frac{1}{2}ma(1+\cosh 2\delta),$ (60) $\displaystyle
Q_{1}=Q_{2}=m\sinh 2\delta\,,\qquad\mu_{1}=\mu_{2}$ $\displaystyle=$
$\displaystyle 2ma\sinh\delta\cosh\delta\,,$ (61)

and if we set $Q_{1}=Q_{2}=\sqrt{2}Q$ and $\mu_{1}=\mu_{2}=\sqrt{2}\mu,$ so
that

$Q^{2}=\frac{1}{4}(Q_{1}^{2}+Q_{2}^{2}),$ (62)

we obtain

$\frac{|\mu|}{|J|}=\sqrt{2}\tanh\delta\,.$ (63)

and

$\frac{|Q|}{M}=\sqrt{2}\tanh\delta\leq\sqrt{2}\,.$ (64)

We also find that $g=2$ and obtain consistency with (34) and agreement with
(41) provided $\theta=2\delta$.

Note that for all these special cases, the conversion from the conventions of
[45] and standard (Gaussian) units is

$Q^{2}=\frac{1}{4}\sum_{i}Q_{i}^{2}\,.$ (65)

##  4 The Dyson bound

The importance of some multiple of $c^{5}/G$ in studies of gravitational
radiation appears to have first been noticed in a paper of Dyson [22]. He
observed, by a scaling argument, that the luminosity in gravitational
radiation of an orbiting binary star system according to Einstein’s linearised
theory of gravitational radiation, must be a dimensionless multiple of
$c^{5}/G$, (see below for a more precise statement). Subsequently, Thorne [54]
introduced it into modern studies of possible sources of gravitational
radiation, linear or non-linear, detectable on earth using current technology.
Thorne’s paper seems to have introduced the idea of a Dyson bound [55, 56]: a
maximum possible luminosity in gravitational waves 555In reply to an enquiry
by Christoph Schiller, Dyson replied on 14th Feb 2011: ’It is not true that I
proposed the formula $c^{5}/G$ as a luminosity limit for anything. I make no
such claim. Perhaps this notion arose from a paper that I wrote in 1962 with
the title, “Gravitational Machines”, published as Chapter 12 in the book,
“Interstellar Communication” edited by Alastair Cameron, [New York, Benjamin,
1963]. Equation (11) in that paper is the well-known formula
$128V^{10}/5Gc^{5}$ for the power in gravitational waves emitted by a binary
star with two equal masses moving in a circular orbit with velocity $V$. As
$V$ approaches its upper limit $c$, this gravitational power approaches the
upper limit $128c^{5}/5G$. The remarkable thing about this upper limit is that
it is independent of the masses of the stars. It may be of some relevance to
the theory of gamma-ray bursts.’.

Six years after [22], Dyson wrote a short note, [57], posing a question the
answer to which was supplied by Hawking’s famous area theorem [58]. It seems
reasonable therefore to suggest (see footnote 9 of [60]) that $c^{5}/G,$ or
some multiple of it, be called “one Dyson”. If one accepts this, the maximum
luminosity of GW150914 (or the orbital merger of any equal-mass non-spinning
black holes) is about $1$ milli-Dyson.

##  5 The Maximum Tension Principle

Independent of these considerations, in an article written for the Festschrift
celebrating the 60th birthday of the late Jacob Bekenstein [9], it was
conjectured that $c^{4}/4G$ was the maximum possible tension or force in
classical general relativity. Dimensionally, this makes sense. The Einstein
equations read

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}g^{\alpha\beta}R_{\alpha\beta}=\frac{8\pi
G}{c^{4}}T_{\mu\nu}\,.$ (66)

The Ricci tensor $R_{\mu\nu}$ has dimensions $\mathrm{L}^{-2}$ and every
component of the energy-momentum tensor, $T_{\mu\nu}$, has dimensions force
per unit area and the Einstein constant $8\pi G/c^{4}$ has units of an inverse
force.

Some heuristic arguments in favour of this maximum tension conjecture were
given in [9] and the factor of $\frac{1}{4}$ justified by reference to conical
singularities and the requirement that the deficit angles of cosmic strings do
not exceed $2\pi$ radians. In fact, the deficit angle is subject to a so-
called Bogomolnyi bound [61] in this case. The extensions in the presence of a
cosmological constant were given in [14] but as yet there exists no formal
proof, or indeed precise mathematical formulation. Further work on the maximum
tension (or force) principle may be found in [11, 12, 13, 15]. Earlier
suggestions regarding a maximum force then came to light. In [16], the authors
claimed that $c^{4}/4G$ is the maximal force allowed in general relativity and
in [21] made the obvious maximal power hypothesis that $c^{5}/4G$ is the
maximum power allowed in nature. Neither paper makes any reference to [22] or
[54]. There are also earlier (unseen) papers on this subject, [62, 63], whose
titles clearly indicates that the author had the same order of magnitude for
the maximal force and maximal power in mind [64].

##  6 Thorne’s Hoop Conjecture

The proposed Dyson bound and the maximum tension principle resemble another,
as yet unresolved but possibly related, issue: how does one formulate in a
precise way Thorne’s hoop conjecture? [65] Recently, there has been some
progress in this direction.

In [66], a precise candidate was proposed for the hoop radius of an apparent
horizon in terms of its Birkhoff invariant $\beta_{b}$. The conjecture was
that every apparent horizon should satisfy

$\beta_{b}\leq 4\pi GM_{ADM}/c^{2},$ (67)

where $M_{ADM}$ is the ADM mass of the system. In [67], considerable support
was marshalled for (67) using known exact solutions of supergravity theories.
However, more recently, a counterexample was constructed using time-symmetric
vacuum initial data in ref. [68]. Following suggestions in [70, 71], one may
then reformulate the hoop conjecture as

$\beta_{b}\leq 2\pi GM_{BY}/c^{2},$ (68)

where $M_{BY}$ is the Brown-York quasi-local mass [72, 73] of the apparent
horizon. Note that the the Brown-York quasi-local mass is only defined for
time-symmetric data. Using a result of Paiva [74] 666There is an earlier and
weaker result due to Croke [78] which may possibly prove to be of use in the
present context. One may check that (68) holds for all initial data sets
constructed in ref. [68]. For a proof in Robinson-Trautmann metrics, see ref
[69].

The Brown-York quasi-local mass [72, 73] of the apparent horizon, which is
assumed to have positive Gaussian curvature, therefore admits a unique (up to
rigid motions) isometric embedding into Euclidean space as a convex body. The
definition of the Brown-York mass inside any 2-surface, $S,$ lying in a Cauchy
surface $\\{\Sigma,\hat{g}\\}$ is

$M_{BY}=\frac{1}{8\pi}\int_{S}\Bigl{(}k_{0}-k\Bigr{)}dA(S,\hat{g}|_{S}),$ (69)

where $k$ is the trace of the fundamental form of $S$ considered as embedded
in $\\{\Sigma,\hat{g}\\}$ and $k_{0}$ is the trace of the fundamental form of
$\\{S,g\\}$ when isometrically embedded in Euclidean space
$\\{\mathbb{E}^{3},\delta_{ij}\\}$, for which we have simply
$dA(S,\hat{g}|_{S})=dA(S,{\delta_{ij}}_{S})=dA$. From a spacetime point of
view, the Brown-York mass depends on both how the spacelike surface $S$ sits
in spacetime $\\{M,g_{\mu\nu}\\}$ (it has two fundamental forms) and also the
spacelike hypersurface $\Sigma$ passing through it (which picks out a linear
combination of its two second fundamental forms). The Brown-York mass is
believed to be a “quasi-local” measure of the amount of “energy” on $\Sigma$
inside $S$ [76]. The York-Brown mass suffers from a number of shortcomings but
in the present context has been shown that it is positive [77].

Among the shortcomings of the Brown-York mass is that it requires that the
surface $S$ admit an isometric embedding into three-dimensional Euclidean
space. This is not possible for the horizon of all Kerr black holes.
Embeddings into four-dimensional Euclidean space are known but are believed
not to be unique. There exists a unique isometric embedding into hyperbolic
three-space [86] and hence a (presumably not unique) embedding into four-
dimensional Minkowski spacetime.

The converse of the hoop conjecture remains to be considered; that is, the
question if some surface $S$ satisfies

$\beta_{b}\leq\ \frac{2\pi GM_{BY}}{c^{2}}\,,$ (70)

then must $S$ be, or lie inside, an apparent horizon? The various papers of
Shi and Tam [77, 79, 80, 81, 82, 83, 84, 85] contain some relevant results
here.

###  6.1 Relation to work of Tod

Tod [75] has looked at the hoop conjecture from the point of view of a
collapsing shell construction for which an isometric embedding is possible,
and seeks to define the hoop radius in terms of a maximum shadow circumference
$C_{m}$. This is defined as the supremum of the circumference of all
orthogonal projections of the surface. He finds that

$\frac{\pi}{2}C\leq\frac{1}{2}\int_{\S}k_{0}dA\leq 2C_{m}\,,$ (71)

where the upper bound is attained for any surface of constant breadth.

Thus, in the context of the time-symmetric initial value problem, rather than
the collapsing shell calculation 777for which if the apparent horizon lies in
a spacelike hyperplane, one does have [66]
$\frac{\beta_{b}}{4\pi}\leq\frac{GM_{ADM}}{c^{2}}$ (72) ., an apparent horizon
must satisfy

$\frac{C_{m}}{8}\leq\frac{GM_{BY}}{c^{2}}\leq\frac{C_{m}}{2\pi}\,$ (73)

Since

$\beta_{b}\leq C_{m}\,,$ (74)

the lower bound yields

$\frac{\beta_{b}}{8}\leq\frac{GM_{BY}}{c^{2}}$ (75)

which, since $8>2\pi$, is a weaker statement than (68) .

##  7 _
_ Conclusions

We have explored the nature of a number of upper bounds on fundamental
quantities in nature. Some of this involves further elaboration and
generalisation to higher dimensions of earlier upper bounds on forces and
power in general relativity, but our discussion has focussed on a detailed
analysis of our conjecture that the ratio of the magnetic moment to angular
momentum is bounded above in nature. Suspicion falls on this combination for a
maximum principle because it has a natural Stoney-Planck unit that is
independent of the quantum of action, $h$, and so it entirely classical. We
find evidence for our conjecture that the ratio $c\mu/JG^{\frac{1}{2}}$ is
bounded by a quantity of order unity by investigating a wide range of testing
theoretical situations. In particular, we verified that such a conjecture
holds for charged rotating black holes in those theories for which exact
solutions are available, including the Einstein-Maxwell and dilaton theories,
Kaluza-Klein theory, the Kerr-Sen black hole, and the so-called STU family of
charged rotating supergravity black holes. We also discussed the current
status of the Maximum Tension Conjecture, the Dyson Luminosity Bound, and
Thorne’s Hoop Conjecture and saw the possible points of contact between them
and our conjecture bounding $\mu/J$.

Acknowledgements

The authors are supported by the STFC of the United Kingdom. We would like to
thank Mirjam Cvetic, Chris Pope, Bruce Allen, Jorge Rueda, Michael Good and
Kostas Skenderis for helpful conversations and communications.

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