# A new pairwise boost quantum number from celestial states

Francesco Alessio<EMAIL_ADDRESS>NORDITA, KTH Royal Institute of
Technology and Stockholm University,
Hannes Alfvéns väg 12, SE-11419 Stockholm, Sweden
Department of Physics and Astronomy, Uppsala University,
Box 516, SE-75120 Uppsala, Sweden Michele Arzano<EMAIL_ADDRESS>Dipartimento di Fisica “E. Pancini”, Università di Napoli Federico II, I-80125
Napoli, Italy
INFN, Sezione di Napoli,
Complesso Universitario di Monte S. Angelo,
Via Cintia Edificio 6, 80126 Napoli, Italy

###### Abstract

Infrared effects in the scattering of particles in gravity and electrodynamics
entail an exchange of relativistic angular momentum between pairs of particles
and the gauge field. Due to this exchange particles can carry an
asymptotically non-vanishing “pairwise” boost-like angular momentum
proportional to the product of their couplings to the field. At the quantum
level this asymptotic angular momentum suggests the existence of a new quantum
number carried by multi-particle states. We argue that such quantum number is
related to a modification of the action of the generators of Lorentz
transformations on multi-particle states. We derive such a modification using
a group-theoretic argument based on the little group of the conformal primary
basis for asymptotic states. The corresponding representation is an extension
of the ordinary multi-particle Fock representation of the Poincaré group. The
new multi-particle states belonging to such representation no longer factorize
into tensor products of one-particle states. Viewed from a gravitational point
of view, our results provide evidence for a universal breakdown of the
description of multi-particle sates in terms of Fock space due to infrared
back-reaction.

##  I Introduction

More than fifty years ago, in a work Zwanziger (1972) that remained mostly
unnoticed until recent (see e.g. Csáki et al. (2021, 2022); Csaki et al.
(2021)), Zwanziger showed that in the scattering of electrically and
magnetically charged particles the electromagnetic field carries,
asymptotically, a time independent angular momentum proportional to the
following combination of the product of two different particles electric
$e_{i}$ and magnetic $g_{j}$ charges: $\mu_{ij}\sim e_{i}g_{j}-g_{i}e_{j}$. In
that work it was observed that the existence of such additional angular
momentum requires a non-trivial action of the generators of Lorentz
transformations on multiparticle states. This, in turn, leads to asymptotic
scattering states no longer transforming like tensor products of free one-
particle states i.e. according to the ordinary Fock representation. This fact
was picked up recently in Csáki et al. (2021) where the authors argued that
the non-trivial transformation of asymptotic states can be described in terms
of an additional quantum number associated to pairs of particles dubbed
pairwise helicity.

Interestingly, as already noticed by Zwanziger in Zwanziger (1972), there are
potential additional Coulombic contributions to the asymptotic angular
momentum of the electromagnetic field proportional to the product of electric
and magnetic charges of pairs of particles. Zwanziger dismissed these
contributions arguing that they should vanish. Recent works Gralla and Lobo
(2022a); Bhardwaj and Lippstreu (2022), however, suggest that these
contributions should be present even in the case of electromagnetism without
magnetic charges. In Gralla and Lobo (2022a) the authors calculated the
balance of four-momentum and relativistic angular momentum for the scattering
of charged particles at leading order in the interaction. Their result shows
that there is a shift of the net boost-like angular momentum or mass moment of
the particles which is balanced by an equal and opposite change of the boost-
like angular momentum carried by the field. They dubbed such effect
electromagnetic scoot. The same contributions to the boost-like angular
momentum of the particles and electromagnetic field were later re-derived in
Bhardwaj and Lippstreu (2022) in a covariant fashion along the lines of the
orginal calculation of Zwanziger in Zwanziger (1972).

The authors of Gralla and Lobo (2022a) found also111Actually the authors
discovered the gravitational scoot before the electromagnetic one. that a
similar effect arises in the gravitational scattering of massive particles
Gralla and Lobo (2022b). In this case the gravitational scoot is proportional
to the Newton’s constant and the product of the two particles masses. These
observations suggest that the original argument of Zwanziger in Zwanziger
(1972) (and its modern reinterpretation given in Csáki et al. (2021))
concerning the inadequacy of a Fock representation for free particle states
can be extended to ordinary electromagnetism (i.e. without resorting to the
existence of magnetic charges) and, most importantly, to gravity. Given the
universal coupling of the latter to all forms of matter and enegy this
strongly hints at an inadequacy of the Fock representation for free particles
particles when gravity is switched-on. The root of such inadequacy lies in the
presence of an additional quantum number, a pairwise boost-like angular
momentum, associated to an infrared back-reaction effect sourced by soft
gravitons. Indeed, as we discuss, the presence of a non-vanishing scoot can be
related to logarithmic terms in the subleading soft graviton and
electromagnetic theorems, as extesively discussed in Laddha and Sen (2018);
Sahoo and Sen (2019); Saha et al. (2020); Sahoo (2020); Krishna and Sahoo
(2023).

In this work we provide group-theoretic evidence for the existence of this new
pairwise quantum number. Our argument relies on the conformal primary basis
for asymptotic scattering states introduced in Pasterski et al. (2017);
Pasterski and Shao (2017) which play a pivotal role in celelestial holography
Pasterski (2021); Pasterski et al. (2021); Arkani-Hamed et al. (2021);
Raclariu (2021); Iacobacci et al. (2024); Iacobacci and Mück (2020); Sleight
and Taronna (2023); Ball et al. (2019); Pate et al. (2021); Fotopoulos and
Taylor (2019); Law and Zlotnikov (2020a); Fotopoulos et al. (2020). Such
states are labelled by coordinates on the celestial sphere and a conformal
dimension which is the eigenvalue of the generator of boosts in a given
direction. When considering a two-particle system we point out that, besides
rotations along the axis of motion of the two-particles already discussed by
Zwanziger in Zwanziger (1972), the little group can be consistently enhanced
in order to include an additional Lorentz transformation preserving the two-
particle configuration in the conformal primary representation: boosts along
the direction of motion of the two particles. It is precisely the eigenvalue
associated with such transoformation which will account for the pairwise
boost-like angular momentum discovered in Gralla and Lobo (2022b, a).

The paper is organized as follows: in the next section we briefly recall
recent results on the existence of an symptotic boost-like angular momentum in
particle scattering in gravity and electromagnetism. Section III is devoted to
an introduction and review of massive conformal primary states. In particular,
in III.1 we introduce the basis in terms of the standard on-shell momentum
basis and fix our conventions. In III.2 we discuss the action of Lorentz
transformations on such basis and in III.3 we derive the form of their
generators both in the on-shell momentum and conformal primary basis. In
Section IV we discuss an enhancement of the standard Wigner little group for
massive multiparticle states to include boosts in the direction of motion of
then particles. We first discuss how such the pairwise boost quantum number is
related to the little group of the conformal primary states and then extend
the derivation to multiparticles states described in the ordinary on-shell
four-momentum basis. This allows us to relate the new pairwise boost quantum
number to the scoot effect discussed in Section II. We conclude with a summary
of the results presented and a brief discussion of future perspectives.

Throughout the paper we use mostly plus signature.

##  II IR scoot in gravity and electrodynamics

Motivated by the study of gravitational self-force effects in the post-
Minkowskian (PM) regime, the authors of Gralla and Lobo (2022b) derived a new
formula for the shift of the total mass moment (or boost charge) $N^{i}$ of a
system of two point particles involved in a gravitational scattering process.
As remarked in the introduction, such quantity is defined as the boost-like
component of the total mechanical angular momentum of the system,

$\displaystyle
M^{\mu\nu}=\sum_{a=1,2}M^{\mu\nu}_{a}=\sum_{a=1,2}p_{a}^{\mu}x^{\nu}_{a}-p_{a}^{\nu}x^{\mu}_{a},\qquad
N^{i}\equiv M^{0i}=\sum_{a=1,2}E_{a}x_{a}^{i}-t\hskip 1.42271ptp_{a}^{i},$ (1)

where $p_{a}^{\mu}=(E_{a},p_{a}^{i})$ and $x_{a}^{\mu}=(t,x^{i})$ are the
momenta and the positions of the particles involved in the scattering process.
In particular, in the rest frame of particle 2 the position four-vectors are
$\bar{x}^{\mu}_{1}=(t,b,0,vt)$ and $\bar{x}^{\mu}_{2}=(t,0,0,0)$, whereas
their initial momenta are,

$\displaystyle\bar{p}_{1}^{\mu}=m_{1}\gamma(1,0,0,v),\qquad\bar{p}_{2}^{\mu}=(m_{2},0,0,0),\qquad\gamma=-\frac{\bar{p}_{1}\cdot\bar{p}_{2}}{m_{1}m_{2}}=\frac{1}{\sqrt{1-v^{2}}},$
(2)

where $v$ and $b$ are their relative velocity and position, namely the impact
parameter, and where $\gamma$ is the usual Lorentz factor. It can be
immediately checked that in this frame $\bar{N}^{i}_{\mathrm{mec}}$ does not
vanish. However, in the center of energy-momentum (CEM) frame, defined by the
Poincaré transformation,

$\displaystyle
x_{i}^{\mu}=\Lambda^{\mu}{}_{\nu}\bar{x}_{i}^{\nu}+a^{\mu}\qquad
p_{i}^{\mu}=\Lambda^{\mu}{}_{\nu}\bar{p}_{i}^{\nu},$ (3)

where,

$\displaystyle\Lambda^{\mu}{}_{\nu}=\left(\begin{matrix}\frac{m_{2}+\gamma
m_{1}}{E_{0}}&0&0&-\frac{\gamma m_{1}v}{E_{0}}\\\ 0&1&0&0\\\ 0&0&1&0\\\
-\frac{\gamma m_{1}v}{E_{0}}&0&0&\frac{m_{2}+\gamma
m_{1}}{E_{0}}\end{matrix}\right),\qquad a^{\mu}=-b\frac{m_{1}(\gamma
m_{2}+m_{1})}{E_{0}^{2}}\delta^{\mu}_{1},$ (4)

and,

$\displaystyle
E_{0}=\sqrt{-(p_{1}+p_{2})^{2}}=\sqrt{m_{1}^{2}+m_{2}^{2}+2m_{1}m_{2}\gamma},$
(5)

is the total energy, we have that the total spatial momentum is zero,
$P^{i}_{\mathrm{mech}}=p_{1}^{i}+p_{2}^{i}=0$ and also that
$N^{i}_{\mathrm{mech}}=p_{1}^{i}+p_{2}^{i}=0$. Explicitly for the momenta we
have,

$\displaystyle
p_{1}^{\mu}=\frac{m_{1}\gamma}{E_{0}}(m_{2}-m_{1}\gamma(v^{2}-1),0,0,m_{2}v),\qquad
p_{2}^{\mu}=\frac{m_{2}}{E_{0}}(m_{2}+\gamma m_{1},0,0,-m_{1}v\gamma).$ (6)

In this frame, it was found that the aforementioned shift is explicitly given
by the following formula,

$\displaystyle\Delta
N^{z}=\frac{2Gm_{1}m_{2}\gamma(3-2\gamma^{2})}{\gamma^{2}-1}\log\left(\frac{m_{2}+\gamma
m_{1}}{m_{1}+\gamma m_{2}}\right)\,.$ (7)

Note that the numerator in this formula is the same of the classical
subleading soft factor discussed e.g. in the last line of (3.6) of Sahoo and
Sen (2019), where $G$ is the gravitational constant. This effect has been
referred to as gravitational scoot. Being linear in $G$, from the PM expansion
point of view Kosower et al. (2019); Cristofoli et al. (2022); Di Vecchia et
al. (2023); Kosower et al. (2022); Buonanno et al. (2022); Travaglini et al.
(2022), the above formula can be interpreted as a 1PM contribution to the the
radiated angular momentum during the scattering process. The 2PM (one-loop)
order Manohar et al. (2022); Di Vecchia et al. (2022); Alessio and Di Vecchia
(2022) has been shown to be entirely due to the radiation of soft (i.e. zero
frequency) gravitons. Note that while the PM expansion assumes a small
dimensionless quantity $Gm/b$, where $m$ is one of the two particles masses,
equation (7) is actually independent of the impact parameter $b$ of the
process, which drops in the final result. Therefore, the gravitational scoot
is universal in the sense that it does not depend on the details of the
scattering process and it is a consequence of the long-range nature of the
gravitational interaction that influences the asymptotic behaviour of the
particle trajectories, which is not the same as that of free particles.
Indeed, as already pointed out in Zwanziger (1973) in the context of
electromagnetism, two free particle trajectories due to similar infrared
effects acquire a correction which depends logarithmically on their proper
time. On these lines of thought, the authors of Di Vecchia et al. (2022)
conjectured that this contribution could be captured by the subleading soft
graviton (or photon) theorem Cachazo and Strominger (2014); Bern et al.
(2014a, b). The latter indeed plays an important role in celestial holography
and its connection to asymptotic symmetries Compère et al. (2018); Donnay et
al. (2020, 2022); Agrawal et al. (2024); Strominger (2014); He et al. (2015);
Strominger (2017); Cachazo and Strominger (2014); Alessio and Arzano (2019) as
well as in the context of amplitudes for classical gravitational scattering Di
Vecchia et al. (2022, 2021); Alessio and Di Vecchia (2022, 2024); Lippstreu
(2023); McLoughlin et al. (2022).

In the easier case of electromagnetism, which already captures the relevant
structures of the problem, a similar formula was found in Gralla and Lobo
(2022a) for the shift in the total boost charge of a two-particle system:

$\displaystyle\Delta
N^{z}=\frac{2e_{1}e_{2}}{\gamma^{2}-1}\log\left(\frac{m_{2}+\gamma
m_{1}}{m_{1}+\gamma m_{2}}\right),$ (8)

where now $e_{1}$ and $e_{2}$ are the electric charges of the particles. In
this context, it has been demonstrated Gralla and Lobo (2022a) that the
mechanical mass moment in (8) is exactly compensated by an equal and opposite
mass moment shift of the electromagnetic field $\Delta N^{z}_{\mathrm{em}}$:

$\displaystyle\Delta N^{z}+\Delta N^{z}_{\mathrm{em}}=0,\qquad
N^{i}_{\mathrm{em}}=\int d^{3}x\hskip 1.42271pt\mathcal{E}x^{i}-t\int
d^{3}x\mathcal{P}^{i},$ (9)

where $\mathcal{E}$ and $\mathcal{P}^{i}$ are the field energy and momentum
densities. This shows that there is an exchange between mechanical and field
degrees of freedom. One concludes that even if it is possible to set $N^{z}=0$
at early times by the coordinate transformation in (3), the total boost charge
at late times, after the scattering process, will be non-vanishing.

The shift in the boost charge can be easily derived from the corrected
equations of the worldlines describing the asymptotic motion for large values
of the proper time of two classical relativistic particles under the mutual
influence of their Coulomb fields Zwanziger (1972); Sahoo and Sen (2019)

$\displaystyle
x^{\mu}_{a}(\tau_{a})=\frac{p^{\mu}_{a}}{m_{a}}\tau_{a}+\eta_{a}\log|\tau_{a}|\sum_{b\neq
a}e_{a}e_{b}\frac{\gamma
u^{\mu}_{a}-u_{b}^{\mu}}{m_{a}(\gamma^{2}-1)^{\frac{3}{2}}},$ (10)
$\displaystyle
p^{\mu}_{a}(\tau_{a})=p^{\mu}_{a}+\eta_{a}\frac{1}{\tau_{a}}\sum_{b\neq
a}e_{a}e_{b}\frac{\gamma
u^{\mu}_{a}-u_{b}^{\mu}}{(\gamma^{2}-1)^{\frac{3}{2}}},$ (11)

where $\eta_{a}=1$ $(-1)$ for outgoing (incoming) particles and $\tau_{a}$ is
the proper time of the $a$-th particle. The $\log$ dependence in the position
four-vectors is an unavoidable consequence of the long-range nature of the
Coulomb force, for an inverse-square force integrates up to a logarithmically
divergent position. Similarly, such logarithms appear already in the tree-
level elastic eikonal Di Vecchia et al. (2023); Alessio and Di Vecchia (2024)
in four-dimensions.

We denote by $\Delta M^{\mu\nu}$ the net difference between the asymptotic
angular momentum at late and early times and we assume that the late and early
time cutoffs for the particles are $\tau_{a}^{+}$ and $\tau_{a}^{-}$,
respectively. The asymptotic angular momentum of the $a$-th particle is

$\displaystyle M^{\mu\nu}_{a}(\tau^{\eta_{a}}_{a})=-2\eta_{a}\sum_{b\neq
a}\frac{e_{a}e_{b}}{(\gamma^{2}-1)^{\frac{3}{2}}}u^{[\mu}_{a}u^{\nu]}_{b}\log|\tau_{a}^{\eta_{a}}|$
(12)

and hence

$\displaystyle\Delta M^{\mu\nu}$
$\displaystyle=\sum_{a}M^{\mu\nu}(\tau_{a}^{+})-M^{\mu\nu}(\tau_{a}^{-})=-2\sum_{a}\sum_{b\neq
a}\frac{e_{a}e_{b}u_{a}^{[\mu}u_{b}^{\nu]}}{(\gamma^{2}-1)^{\frac{3}{2}}}\log|\tau_{a}^{+}\tau_{a}^{-}|.$
(13)

If we further assume that $\tau_{a}^{+}=\tau_{a}^{-}\equiv\tau_{a}$, we have

$\displaystyle\Delta M^{\mu\nu}=-4\sum_{a}\sum_{b\neq
a}\frac{e_{a}e_{b}u_{a}^{[\mu}u_{b}^{\nu]}}{(\gamma^{2}-1)^{\frac{3}{2}}}\log|\tau_{a}|.$
(14)

The above equation could also be inferred from the analysis in Bhardwaj and
Lippstreu (2022). When specifying to the case of a two-particle scattering,
plugging in the previous formula the explicit parametrization of the momenta
(6) in the CEM frame we get

$\displaystyle\Delta
N^{z}=\frac{2e_{1}e_{2}}{\gamma^{2}-1}\log\bigg{|}\frac{\tau_{1}}{\tau_{2}}\bigg{|},$
(15)

matching equation (8) upon identification of the arguments of the logarithms.
The actual value of the latter is not important in the present analysis and it
depends on the initial conditions of the scattering problem which we did not
include in the trajectories displayed in (10) and (11).

Similarly, the asymptotic trajectories of particles subjected to their
reciprocal gravitational interaction are given by Sahoo and Sen (2019)

$\displaystyle
x_{a}^{\mu}(\tau_{a})=\frac{p_{a}^{\mu}}{m_{a}}\tau_{a}+\eta_{a}\log|\tau_{a}|\sum_{b\neq
a}Gm_{b}\frac{u_{a}^{\mu}+u_{b}^{\mu}\gamma(2\gamma^{2}-3)}{(\gamma^{2}-1)^{\frac{3}{2}}},$
(16) $\displaystyle
p_{a}^{\mu}(\tau_{a})=p_{a}^{\mu}+\eta_{a}\frac{1}{\tau_{a}}\sum_{b\neq
a}Gm_{b}\frac{u_{a}^{\mu}+u_{b}^{\mu}\gamma(2\gamma^{2}-3)}{(\gamma^{2}-1)^{\frac{3}{2}}}.$
(17)

Proceeding as before, we get for the asymptotic angular momentum of
outgoing/incoming particles

$\displaystyle M^{\mu\nu}_{a}(\tau_{a})=2\eta_{a}G\sum_{b\neq
a}\frac{m_{a}m_{b}u_{a}^{[\mu}u_{b}^{\nu]}\gamma(2\gamma^{2}-3)}{(\gamma^{2}-1)^{\frac{3}{2}}}\log|\tau_{a}|,$
(18)

and

$\displaystyle\Delta M^{\mu\nu}=4G\sum_{a}\sum_{b\neq
a}\frac{m_{a}m_{b}u_{a}^{[\mu}u_{b}^{\nu]}\gamma(2\gamma^{2}-3)}{(\gamma^{2}-1)^{\frac{3}{2}}}\log|\tau_{a}|.$
(19)

for the shift. In the CEM frame and for two particles we recover (7), as we
should:

$\displaystyle\Delta
N^{z}=\frac{2Gm_{1}m_{2}\gamma(3-2\gamma^{2})}{\gamma^{2}-1}\log\left|\frac{\tau_{1}}{\tau_{2}}\right|.$
(20)

In the rest of this work we argue that the existence of this asymptotic boost
charge is reflecte into a modification of the generator of boosts on multi-
particle states described in the conformal primary representation which we
introduce in the next section.

##  III Conformal primary basis

The Wigner little group of massive particles, i.e. the subset of the Lorentz
group that leaves a reference four-momentum invariant is that of three-
dimensional spatial rotations, $\mathrm{SO}(3)$, whose associated quantum
number $\sigma$ is the spin labelling massive irreducible representations of
the Lorentz group. Correspondingly, single-particle states are represented by
kets $|p,\sigma\rangle$, where $p$ is the on-shell four-momentum of the
particle. Multiparticle states are customarily constructed as “bare” tensor
products of single-particle states such as
$|p_{1},p_{2},\sigma_{1},\sigma_{2}\rangle=|p_{1},\sigma_{1}\rangle\otimes|p_{2}.\sigma_{2}\rangle$.
However, as it is clear from the discussion in the previous section, when
considering the interaction of massive particles mediated by electromagnetic
or gravitational interaction, the presence of a asymptotically non-vanishing
boost-like angular momentum coupling pairs of particles makes this simple
construction inadequate. In order to address such shortcoming, as a first
step, we look for a basis of asymptotic multiparticle states characterized by
a boost quantum number i.e. which diagonalizes the action of boosts. This is
achieved by the conformal primary basis Pasterski et al. (2017); Pasterski and
Shao (2017) which we introduce here in the case of a scalar massive particle.
A generalization for particles with spin can be found e.g. in Pasterski and
Puhm (2021).

###  III.1 The basis

We start by considering the Minkowski metric in momentum space

$\displaystyle
ds^{2}=-dE^{2}+d\vec{p}^{2}=-dE^{2}+dp^{2}+p^{2}(d\theta^{2}+\sin\theta^{2}d\phi^{2}).$
(21)

The induced metric on the mass-shell hyperboloid $H_{3}$ defined by
$E^{2}=p^{2}+m^{2}$ for a fixed value of the mass $m$ is

$\displaystyle ds^{2}=m^{2}ds^{2}_{H_{3}},\qquad
ds^{2}_{H_{3}}=\frac{dy^{2}+dzd\bar{z}}{y^{2}},$ (22)

where $ds^{2}_{H_{3}}$ is the metric on the unit mass hyperboloid and the
Poincaré coordintes $(y,z,\bar{z})$ are defined as

$\displaystyle y=\frac{m}{\sqrt{m^{2}+p^{2}}+p\cos\theta},\qquad
z=\frac{p\sin\theta}{\sqrt{p^{2}+m^{2}}+p\cos\theta}e^{i\phi},\qquad\bar{z}=z^{*}\,.$
(23)

Notice that the conformal boundary is located at $y=0$ and it is conformally a
2-sphere $S^{2}$.

The on-shell four-momentum of a massive particle can be thus parametrized
Raclariu (2021); Pasterski (2021) in terms of such coordinates as222We will
use the shorthand notation $f(y,z,\bar{z})\equiv f(y,z)$.

$p^{\mu}(y,z)=\eta\frac{m}{2y}\left(1+y^{2}+z\bar{z},z+\bar{z},i(\bar{z}-z),1-y^{2}-z\bar{z}\right)\equiv\eta
m\,\hat{p}^{\mu}(y,z),$ (24)

with $\hat{p}^{2}=-1$ and $\eta=\pm 1$ for outgoing/incoming particles as in
the previous section. One-particle states of scalar particles in the
$(y,z,\bar{z})$ momentum basis are taken to be $|p\,\rangle\equiv|y,z\rangle$
and they are associated to plane wave solutions $\langle x|y,z\rangle$ of the
Klein-Gordon equation. Completeness of the $|y,z\rangle$ basis implies that

$\displaystyle|y,z\rangle=\int d\mu(p^{\prime})\langle
y^{\prime},z^{\prime}|y,z\rangle|y^{\prime},z^{\prime}\rangle=\int_{0}^{\infty}\frac{dy^{\prime}}{y^{\prime
3}}\int dz^{\prime}d\bar{z}^{\prime}{}^{\eta}\langle
y^{\prime},z^{\prime}|y,z\rangle|y^{\prime},z^{\prime}\rangle,$ (25)

where $d\mu(p)$ is the Lorentz invariant integration measure on $H_{3}$,
$d\mu(p)=\frac{d^{3}\hat{p}}{\hat{p}^{0}}$, from which we read the
orthonormality condition

$\displaystyle\langle
y^{\prime},z^{\prime}|y,z\rangle=y^{3}\delta(y-y^{\prime})\delta^{(2)}(z-z^{\prime}).$
(26)

The conformal primary basis $|\Delta,w,\bar{w}\rangle\equiv|\Delta,w\rangle$
is defined as Lam and Shao (2018)

$\displaystyle|\Delta,w\rangle=\int d\mu(p)\langle
y,z|w,\Delta\rangle|y,z\rangle\equiv\int_{0}^{\infty}\frac{dy}{y^{3}}\int
dzd\bar{z}G_{\Delta}(\hat{p},w)|y,z\rangle,$ (27)

where the coefficients of the integral map are given by the scalar bulk-to-
boundary propagator on $H_{3}$ Witten (1998); Pasterski and Shao (2017)

$G_{\Delta}(\hat{p},w)\equiv\langle
y,z|\Delta,w\rangle=\frac{1}{\sqrt{4\pi^{3}}}\frac{\Gamma(1+i\lambda)}{\Gamma(i\lambda)}\bigg{(}\frac{y}{y^{2}+|w-z|^{2}}\bigg{)}^{\Delta},\qquad\Delta=1+i\lambda.$
(28)

$\Delta$ is the conformal dimension. The states are orthonormal

$\displaystyle\langle\Delta^{\prime},w^{\prime}|\Delta,w\rangle=\int_{0}^{\infty}\frac{dy}{y^{3}}\int
dzd\bar{z}G^{*}_{\Delta^{\prime}}(\hat{p},w^{\prime})G_{\Delta}(\hat{p},w)=\delta(\lambda-\lambda^{\prime})\delta^{(2)}(w-w^{\prime}),$
(29)

where we used (26) and the value of the integral333We thank the authors of
Sleight and Taronna (2023) for bringing Costa et al. (2014) to our
attention.Costa et al. (2014)

$\int\frac{dy}{y^{d+1}}\int
d^{d}\vec{z}\frac{y^{\Delta+\Delta^{\prime}+\epsilon}}{[y^{2}+(\vec{z}-\vec{w}^{\prime})^{2}]^{\Delta^{\prime}}[y^{2}+(\vec{z}-\vec{w})^{2}]^{\Delta}}=\frac{2\pi^{d+1}|\Gamma(i\lambda)|^{2}}{|\Gamma(\frac{d}{2}+i\lambda)|^{2}}\delta(\lambda+\lambda^{\prime})\delta^{(d)}(\vec{w}-\vec{w}^{\prime}).$
(30)

where
$(\Delta,\Delta^{\prime})=(\frac{d}{2}+i\lambda,\frac{d}{2}+i\lambda^{\prime})$.
The integral transform in (27) can also be inverted and states $|y,z\rangle$
in the on-shell momentum basis can be obtained from the conformal primary ones
$|\Delta,w\rangle$ as

$\displaystyle|y,z\rangle=\int
d\mu(\lambda,w)\langle\Delta,w|y,z\rangle|\Delta,w\rangle\equiv\int_{0}^{\infty}d\lambda\int
dwd\bar{w}G^{*}_{\Delta}(\hat{p};w)|\Delta,w\rangle,$ (31)

and the orthonormality condition in (26) implies that

$\displaystyle\int_{0}^{\infty}d\lambda\int
dwd\bar{w}G^{*}_{\Delta}(\hat{p},w)G_{\Delta}(\hat{p}^{\prime},w)=y^{3}\delta(y-y^{\prime})\delta^{(2)}(w-w^{\prime}).$
(32)

In the momentum basis the eigenfunctions are simply given by standard plane
waves,

$\displaystyle f(x;y,z)\equiv\langle
x|y,z\rangle=\frac{1}{(2\pi)^{\frac{3}{2}}}e^{i\eta m\hat{p}(y,z)\cdot x}.$
(33)

In the conformal primary basis they can be explicitly constructed by inserting
the identity as

$\displaystyle\phi_{\Delta}(x;w)$ $\displaystyle\equiv\langle
x|\Delta,w\rangle=\int d^{3}\mu(p)\langle y,z|w,\Delta\rangle\langle
x|y,z\rangle$
$\displaystyle=\frac{1}{(2\pi)^{\frac{3}{2}}}\int_{0}^{\infty}\frac{dy}{y^{3}}\int
dzd\bar{z}G_{\Delta}(\hat{p},w)e^{i\eta m\hat{p}(y,z)\cdot x}.$ (34)

For the sake of completeness the explicit form of the conformal primary
wavefunctions in (III.1) is given by Pasterski and Shao (2017); Pasterski and
Puhm (2021); Raclariu (2021); Iacobacci and Mück (2020)

$\displaystyle\phi_{\Delta}(x;w)=\frac{\Gamma(1+i\lambda)}{\Gamma(i\lambda)}\frac{1}{\sqrt{2}\pi^{2}}\frac{1}{im}\frac{(\sqrt{-x^{2}})^{\Delta-1}}{(-q(w)\cdot
x-i\eta\epsilon)^{\Delta}}K_{\Delta-1}(im\sqrt{-x^{2}}),$ (35)

where $q^{\mu}(w)$ is the null vector

$\displaystyle q^{\mu}(w)=(1+|w|^{2},w+\bar{w},i(\bar{w}-w),1-|w|^{2}),\qquad
q^{2}=0,$ (36)

and where $K_{\nu}(x)$ are modified Bessel function of the second kind. The
wave functions in (35) comprise a complete set of orthonormal solutions to the
massive Klein-Gordon equation provided that $\lambda\in\mathbb{R}^{+}$ so that
$\Delta$ belongs to principal continuous series of the irreducible
representations of the Lorentz group Pasterski and Shao (2017). Finally, we
also notice that, in terms of $q^{\mu}(w)$, the bulk-to-boundary propagator
admits a simple representation,

$\displaystyle
G_{\Delta}(\hat{p},w)=\frac{1}{\sqrt{4\pi^{3}}}\frac{\Gamma(1+i\lambda)}{\Gamma(i\lambda)}\frac{1}{[-\hat{p}(y,z)\cdot
q(w)]^{\Delta}}.$ (37)

###  III.2 Action of Lorentz transformations

We are now interested in the transformation properties of the new basis under
the action of SL$(2,\mathbb{C})/\mathbb{Z}_{2}$. As shown in appendix A, on-
shell momenta (24) transform under the action of
SL$(2,\mathbb{C})/\mathbb{Z}^{2}$ through the coordinate map

$y\rightarrow y^{\prime}=\frac{y}{|cz+d|^{2}+|c|^{2}y^{2}}\equiv\Lambda
y,\qquad z\rightarrow
z^{\prime}=\frac{(az+b)(\bar{c}\bar{z}+\bar{d})+a\bar{c}y^{2}}{|cz+d|^{2}+|c|^{2}y^{2}}\equiv\Lambda
z,\qquad,$ (38)

where $ad-bc=\bar{a}\bar{d}-\bar{b}\bar{c}=1$. The $|\Delta,w\rangle$ basis is
defined through the integral transform (27) involving the bulk-to-boundary
propagator. In the massless case, due to the nature of null hypersurfaces,
$(w,\bar{w})$ and $(z,\bar{z})$ are the same angular coordinates on both bulk
and asymptotic spheres, defined through the standard stereographic map.
However, in the massive case $(w,\bar{w})$ have not to be interpreted as the
same as $(z,\bar{z})$. Rather, the full set of variables $(\Delta,w,\bar{w})$
has to be regarded as variables “dual” to $(y,z,\bar{z})$. As discussed in
Pasterski et al. (2017) , trajectories of free massive particles
$x^{\mu}(s)=\hat{p}^{\mu}s$ at early and late times can be written as
$x^{\mu}(s)/\sqrt{-x^{2}(s)}\rightarrow\hat{p}^{\mu}$. Worldlines of massive
particles thus asymptote to a fixed position on the hyperbolic slices of
Minkowski space-time determined by their four-momenta. Hence $(w,\bar{w})$ can
be interpreted as a boundary coordinate of the early- and late-time asymptotic
$H_{3}$ slice and can be considered as coordinates on past and future timelike
infinity $i^{\mp}$, which are indeed conformal spheres. Hence, their
transformation follows from evaluating (38) at $y=0$ and it reads

$\displaystyle w\rightarrow w^{\prime}=\frac{aw+b}{cw+d}\equiv\Lambda w,\qquad
ad-bc=1,$ (39)

i.e. they undergo standard Möbius transformations. By explicit computation, we
can show the bulk-to-boundary propagator transforms with conformal weight
$\Delta$

$\displaystyle G_{\Delta}(\Lambda\hat{p},\Lambda
w)=|cw+d|^{2\Delta}G_{\Delta}(\hat{p},w).$ (40)

We can easily derive the transformation law $U(\Lambda)|\Delta,w\rangle$. We
assume that

$\displaystyle U(\Lambda)|\Delta,w\rangle\equiv f(\Lambda;w)|\Delta,\Lambda
w\rangle,$ (41)

with $f(\Lambda;w)$ to be determined. Then we have,

$\displaystyle\langle y,z|\Delta,w\rangle$
$\displaystyle=G_{\Delta}(\hat{p},w)\stackrel{{\scriptstyle\eqref{F20}}}{{=}}G_{\Delta}(\Lambda\hat{p},\Lambda
w)|cw+d|^{-2\Delta}=\langle\Lambda y,\Lambda z|\Delta,\Lambda
w\rangle|cw+d|^{-2\Delta}$
$\displaystyle\stackrel{{\scriptstyle\eqref{tr7}}}{{=}}\langle
y,z|\underbrace{U^{\dagger}(\Lambda)U(\Lambda)}_{\mathbf{1}}|\Delta,w\rangle\frac{|cw+d|^{-2\Delta}}{f(\Lambda;w)}=G_{\Delta}(\hat{p},w)\frac{|cw+d|^{-2\Delta}}{f(\Lambda;w)}.$
(42)

Therefore, comparing the last to the first line we must have
$f(\Lambda;w)=|cw+d|^{-2\Delta}$ and thus

$\displaystyle U(\Lambda)|\Delta,w\rangle=|cw+d|^{-2\Delta}|\Delta,\Lambda
w\rangle\,,$ (43)

as also discussed in Banerjee (2019); Lippstreu (2021) in the massless case.
Our equation (43) shows that the same considerations of these works can be
extended to the massive case. In particular, if we take $|\Delta,0\rangle$ as
the reference state associated to a particle moving along the $x^{3}$ axis,
its little group, together with rotations about the $x^{3}$ axis, is enhanced
to include boosts along the $x^{3}$ direction. Indeed, it is easy to prove
that rotations and boosts along $x^{3}$ map $w=0$ into itself through (39).

The transformation of the conformal primary wavefunctions (35) follows easily
from (43). We have

$\displaystyle\phi_{\Delta}(\Lambda\cdot x;\Lambda
w)=\int\frac{d^{3}\hat{p}}{\hat{p}^{0}}G_{\Delta}(\hat{p},\Lambda w)e^{i\eta
m\hat{p}\cdot\Lambda\cdot x}=\int\frac{d^{3}\hat{p}^{\prime}}{\hat{p}^{\prime
0}}G_{\Delta}(\hat{p}^{\prime},w^{\prime})e^{i\eta m\hat{p}^{\prime}\cdot
x^{\prime}}.$ (44)

where we defined $x^{\prime}\equiv\Lambda\cdot x$ and we changed the
integration variable to $p^{\prime}=\Lambda\cdot p$ and used Lorentz
invariance of the measure $d\mu(p)$. Using now (40) we obtain,

$\displaystyle\phi_{\Delta}(\Lambda\cdot x;\Lambda w)$
$\displaystyle=|cw+d|^{2\Delta}\int\frac{d^{3}\hat{p}^{\prime}}{\hat{p}^{\prime
0}}G_{\Delta}(\hat{p},w)e^{i\eta m\hat{p}^{\prime}\cdot
x^{\prime}}=|cw+d|^{2\Delta}\phi_{\Delta}(x,w),$ (45)

where in the last step we used again Lorentz invariance of the measure and
that $p^{\prime}\cdot x^{\prime}=p\cdot x$. In the case of wavefunctions with
spin, i.e. for fields $\phi_{h,\bar{h}}(x,w)=\langle x|h,\bar{h},w\rangle$
with $(h,\bar{h})\equiv\frac{1}{2}(\Delta+\sigma,\Delta-\sigma)$ with
$h\neq\bar{h}$ and $\sigma$ is the spin, the transformations for the states
and the wavefunctions are Pasterski and Puhm (2021),

$\displaystyle
U(\Lambda)|h,\bar{h},w\rangle=(cw+d)^{-2h}(\bar{c}\bar{w}+\bar{d})^{-2\bar{h}}|h,\bar{h},\Lambda
w\rangle,$ (46) $\displaystyle\phi_{h,\bar{h}}(\Lambda\cdot x;\Lambda
w)=(cw+d)^{2h}(\bar{c}\bar{w}+\bar{d})^{2\bar{h}}D_{h,\bar{h}}(\Lambda)\phi_{h,\bar{h}}(x;w),$
(47)

where $D_{h,\bar{h}}(\Lambda)$ is a representation of the Lorentz
transformation acting on the internal indices of $\phi_{h,\bar{h}}(x,w)$.

###  III.3 Generators

In this last subsection we derive an explicit form of the generators in the
two bases discussed above. We assume that a Lorentz transformation is
parametrized by the variable $\alpha$, that can be either an angle $\varphi$
in the case of spatial rotations or a rapidity $\chi$ in the case of boosts. A
spatial rotation about an axis
$\hat{n}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ of an angle
$\varphi$ is given by the SL(2,$\mathbb{C}$) matrix $R_{\hat{n}}(\varphi)$

$\displaystyle
R_{\hat{n}}(\varphi)=\left(\begin{matrix}\cos\frac{\varphi}{2}-i\cos\theta\sin\frac{\varphi}{2}&-i\sin\theta\sin\frac{\varphi}{2}e^{-i\phi}\\\
\\\
-i\sin\theta\sin\frac{\varphi}{2}e^{i\phi}&\cos\frac{\varphi}{2}+i\cos\theta\sin\frac{\varphi}{2}\end{matrix}\right),$
(48)

whereas a boost about $\hat{n}$ of rapidity $\chi$ by $B_{\hat{n}}(\chi)$

$\displaystyle
B_{\hat{n}}(\chi)=\left(\begin{matrix}\cosh\frac{\chi}{2}-\cos\theta\sinh\frac{\chi}{2}&-\sin\theta\sinh\frac{\chi}{2}e^{-i\phi}\\\
\\\
-\sin\theta\sinh\frac{\chi}{2}e^{i\phi}&\cosh\frac{\chi}{2}+\cos\theta\sinh\frac{\chi}{2}\end{matrix}\right).$
(49)

The generators in the $(y,z,\bar{z})$ basis can be easily computed from (38)
as Alessio and Esposito (2018),

$\displaystyle
G_{z}(\theta,\phi)=\frac{\partial}{\partial\alpha}\bigg{(}\frac{(a(\alpha)z+b(\alpha))(\bar{c}(\alpha)\bar{z}+\bar{d}(\alpha))+a(\alpha)\bar{c}(\alpha)y^{2}}{(c(\alpha)z+d(\alpha))(\bar{c}(\alpha)\bar{z}+\bar{d}(\alpha))+c(\alpha)\bar{c}(\alpha)y^{2}}\bigg{)}\bigg{|}_{\alpha=0},\qquad
G_{\bar{z}}(\theta,\phi)=G^{*}_{z}(\theta,\phi),$ (50) $\displaystyle
G_{y}(\theta,\phi)=\frac{\partial}{\partial\alpha}\bigg{(}\frac{y}{(c(\alpha)z+d(\alpha))(\bar{c}(\alpha)\bar{z}+\bar{d}(\alpha))+c(\alpha)\bar{c}(\alpha)y^{2}}\bigg{)}\bigg{|}_{\alpha=0}.$
(51)

and they are

$\displaystyle
G(\theta,\phi)=G_{z}(\theta,\phi)\partial_{z}+G_{\bar{z}}(\theta,\phi)\partial_{\bar{z}}+G_{y}(\theta,\phi)\partial_{y},$
(52)

for each direction specified by the angles $(\theta,\phi)$. Explicitly they
are444Note that in our conventions we use the inverse of the matrices in (48)
and (49) corresponding to the inverse Möbius transformation
$w=\frac{dw^{\prime}-b}{-cw^{\prime}+a}$.

$\displaystyle
R_{1}(y,z)=\frac{i}{2}\bigg{[}(\bar{z}^{2}+y^{2}-1)\partial_{\bar{z}}-(z^{2}+y^{2}-1)\partial_{z}+y(\bar{z}-z)\partial_{y}\bigg{]},$
(53) $\displaystyle
R_{2}(y,z)=\frac{1}{2}\bigg{[}(\bar{z}^{2}-y^{2}+1)\partial_{\bar{z}}+(z^{2}-y^{2}+1)\partial_{z}+y(\bar{z}+z)\partial_{y}\bigg{]}$
(54) $\displaystyle R_{3}(y,z)=-i(\bar{z}\partial_{\bar{z}}-z\partial_{z}),$
(55) $\displaystyle
B_{1}(y,z)=-\frac{1}{2}\bigg{[}(\bar{z}^{2}-y^{2}-1)\partial_{\bar{z}}+(z^{2}-y^{2}-1)\partial_{z}+y(\bar{z}+z)\partial_{y}\bigg{]},$
(56) $\displaystyle
B_{2}(y,z)=\frac{i}{2}\bigg{[}(\bar{z}^{2}+y^{2}+1)\partial_{\bar{z}}-(z^{2}+y^{2}+1)\partial_{z}+y(\bar{z}-z)\partial_{y}\bigg{]},$
(57) $\displaystyle
B_{3}(y,z)=\bar{z}\partial_{\bar{z}}+z\partial_{z}+y\partial_{y}.$ (58)

They satisfy the Lorentz algebra
$[R_{i}(y,z),R_{j}(y,z)]=\epsilon^{ij}{}^{k}R_{k}(y,z)$,
$[B_{i}(y,z),R_{j}(y,z)]=\epsilon^{ij}{}^{k}B_{k}(y,z)$ and
$[B_{i}(y,z),B_{j}(y,z)]=-\epsilon^{ij}{}^{k}R_{k}(y,z)$.

Similarly, we can derive the generators in the basis $(\Delta,w,\bar{w})$.
Together with the coordinate part, that can be derived from (39), now the
generators comprise also an additional internal piece coming from (45). We get

$\displaystyle
G(\theta,\phi)=\frac{\partial}{\partial\alpha}|c(\alpha)w+d(\alpha)|^{2\Delta}\bigg{|}_{\alpha=0}+\frac{\partial
w}{\partial\alpha}\bigg{|}_{\alpha=0}\partial_{w}+\frac{\partial\bar{w}}{\partial\alpha}\bigg{|}_{\alpha=0}\partial_{\bar{w}},\qquad
w=\frac{dw^{\prime}-b}{-cw^{\prime}+a}.$ (59)

Explicitly Law and Zlotnikov (2020b); Stieberger and Taylor (2019),

$\displaystyle
R_{1}(\Delta,w)=\frac{i}{2}\bigg{[}(\bar{w}^{2}-1)\partial_{\bar{w}}-(w^{2}-1)\partial_{w}-\Delta(\bar{w}-w)\bigg{]},$
(60) $\displaystyle
R_{2}(\Delta,w)=\frac{1}{2}\bigg{[}(\bar{w}^{2}+1)\partial_{\bar{w}}+(w^{2}+1)\partial_{w}-\Delta(w+\bar{w})\bigg{]},$
(61) $\displaystyle
R_{3}(\Delta,w)=-i(\bar{w}\partial_{\bar{w}}-w\partial_{w}),$ (62)
$\displaystyle
B_{1}(\Delta,w)=-\frac{1}{2}\bigg{[}(\bar{w}^{2}-1)\partial_{\bar{w}}+(w^{2}-1)\partial_{w}-\Delta(w+\bar{w})\bigg{]},$
(63) $\displaystyle
B_{2}(\Delta,w)=\frac{i}{2}\bigg{[}(\bar{w}^{2}+1)\partial_{\bar{w}}-(w^{2}+1)\partial_{w}-\Delta(\bar{w}-w)\bigg{]},$
(64) $\displaystyle
B_{3}(\Delta,w)=\bar{w}\partial_{\bar{w}}+w\partial_{w}-\Delta.$ (65)

It is easy to show that the above generators satisfy the Lorentz algebra. We
notice that the differential part of the above generators can be obtained from
the ones in (53)-(58) by sending $y\rightarrow 0$. Notice that, as emphasised
at the beginning of this section, the conformal dimension $\Delta$ is the
eigenvalue of boosts along the $x^{3}$ direction. This shows that the
conformal primary basis just introduced diagonalizes the action of boosts
along a given direction.

In the more general case of spinning wavefunctions transforming as in (46) and
(47), the generators have to be modified in order to include the spin
contribution. One can conclude by a similar computation that

$\displaystyle
R_{1}(\Delta,w)=\frac{i}{2}\bigg{[}(\bar{w}^{2}-1)\partial_{\bar{w}}-(w^{2}-1)\partial_{w}-2(\bar{h}\bar{w}-hw)\bigg{]},$
(66) $\displaystyle
R_{2}(\Delta,w)=\frac{1}{2}\bigg{[}(\bar{w}^{2}+1)\partial_{\bar{w}}+(w^{2}+1)\partial_{w}-2(hw+\bar{h}\bar{w})\bigg{]},$
(67) $\displaystyle
R_{3}(\Delta,w)=-i\bigg{[}\bar{w}\partial_{\bar{w}}-w\partial_{w}-(\bar{h}-h)\bigg{]},$
(68) $\displaystyle
B_{1}(\Delta,w)=-\frac{1}{2}\bigg{[}(\bar{w}^{2}-1)\partial_{\bar{w}}+(w^{2}-1)\partial_{w}-2(hw+\bar{h}\bar{w})\bigg{]},$
(69) $\displaystyle
B_{2}(\Delta,w)=\frac{i}{2}\bigg{[}(\bar{w}^{2}+1)\partial_{\bar{w}}-(w^{2}+1)\partial_{w}-2(\bar{h}\bar{w}-hw)\bigg{]},$
(70) $\displaystyle
B_{3}(\Delta,w)=\bar{w}\partial_{\bar{w}}+w\partial_{w}-(h+\bar{h})\,.$ (71)

In this case it is clear that the generator of rotations along the $x^{3}$
direction contains an intrinsic extra-piece which is exactly the spin of the
representation, given by $\sigma=h-\bar{h}$.

##  IV Massive pairwise little group

So far we have focused on single-particle states introducing a basis which is
diagonal under the action of boosts along a given direction. Such boosts are
naturally a subgroup of the little group associated to this basis. The results
of Gralla and Lobo (2022a, b) suggest the existence of a new quantum number
associated to pairs of particles which, as discussed at the beginning of
Section 3, renders the description of multiparticle states as ordinary tensor
products of one-particle states inadequate. In this section we demonstrate
that the new pairwise boost quantum number can be obtained from the little
group of two-particle states in the conformal primary basis extended to
include boost transformations along the direction of motion.

As discussed in Lippstreu (2021) in the conformal primary basis the little
group of the state describing two massless particles in a frame where their
spatial momenta are aligned, besides rotations around the axis of motion555As
remarked in Zwanziger (1972) rotations around the common axis of motion
obviously preserve both particles momenta., contains also boosts along the
direction of motion, which we take again to be along the $x^{3}$ axis.
Therefore, the little group of the two-particle state is generated by the pair
$\\{R_{3},B_{3}\\}$ satisfying $[R_{3},B_{3}]=0$. Two massless momenta aligned
along $x^{3}$ are described by stereographic coordinates $w_{1}=0$ and
$w_{2}=\infty$ because, as remarked in Section III.2, the angular coordinates
of massless momenta are the same for bulk and boundary states. It follows that
such reference two-particle states must be eigenstates of these operators with
eigenvalues $\sigma_{12}$ and $\Delta_{12}$ and we denote them by
$|\Delta_{1},0;\Delta_{2},\infty;\Delta_{12},\sigma_{12}\rangle$. Hence we
have Banerjee (2019)

$\displaystyle
R_{3}(\Delta_{1},w_{1},\Delta_{2},w_{2},\Delta_{12})|\Delta_{1},0;\Delta_{2},\infty;\Delta_{12},\sigma_{12}\rangle=-i(\eta_{1}+\eta_{2})\frac{\sigma_{12}}{2}|\Delta_{1},0;\Delta_{2},\infty;\Delta_{12},\sigma_{12}\rangle,$
(72) $\displaystyle
B_{3}(\Delta_{1},w_{1},\Delta_{2},w_{2},\Delta_{12})|\Delta_{1},0;\Delta_{2},\infty;\Delta_{12},\sigma_{12}\rangle=-(\eta_{1}+\eta_{2})\frac{\Delta_{12}}{2}|\Delta_{1},0;\Delta_{2},\infty;\Delta_{12},\sigma_{12}\rangle,$
(73)

and finite pairwise little group transformations with Wigner parameters
$\alpha$ and $\beta$ read

$\displaystyle
U(R_{3})|\Delta_{1},0;\Delta_{2},\infty;\Delta_{12},\sigma_{12}\rangle=e^{\mp
i\alpha\sigma_{12}}|\Delta_{1},0;\Delta_{2},\infty;\Delta_{12},\sigma_{12}\rangle,$
(74) $\displaystyle
U(B_{3})|\Delta_{1},0;\Delta_{2},\infty;\Delta_{12},\sigma_{12}\rangle=e^{\mp\beta\Delta_{12}}|\Delta_{1},0;\Delta_{2},\infty;\Delta_{12},\sigma_{12}\rangle.$
(75)

The eigenvalues $\sigma_{12}$ and $\Delta_{12}$ are referred to as pairwise
helicity and pairwise boost. In the remainder of this paper we will only focus
on $\Delta_{12}$ and its connection to the asymptotic boost-like angular
momentum introduced in section II. The explicit dependence of the Wigner
parameter $\beta$ on the Lorentz transformation $\Lambda$ has been discussed
in Lippstreu (2021) and it is

$\displaystyle
U(\Lambda)|\Delta_{1},w_{1};\Delta_{2},w_{2};\Delta_{12}\rangle=\prod_{i=1,2}|cw_{i}+d_{i}|^{-2\Delta_{i}+\eta_{i}\Delta_{12}}|\Delta_{1},\Lambda
w_{1};\Delta_{2},\Lambda w_{2};\Delta_{12}\rangle.$ (76)

Notice that the above formula, although derived for massless particles, holds
also for massive particles. This is a simple consequence of the fact the in
both the massive and massless case, the variables $w$ transform as in (39)
under Lorentz transformations.

Equation (76) implies that the Lorentz generators acting on two-particle
incoming and outgoing states in the conformal primary basis derived in
(60)-(65), have to be modified according to the simple rule
$\Delta_{1}\rightarrow\Delta_{1}-\frac{1}{2}\eta_{2}\Delta_{12}$ and
$\Delta_{2}\rightarrow\Delta_{2}-\frac{1}{2}\eta_{1}\Delta_{12}$. In formulae,

$\displaystyle
R_{1}(\Delta_{1},w_{1};\Delta_{2},w_{2};\Delta_{12})=R_{1}(\Delta_{1},w_{1})+R_{1}(\Delta_{2},w_{2})+i\frac{\Delta_{12}}{4}[(\bar{w}_{1}-w_{1})\eta_{1}+(\bar{w}_{2}-w_{2})\eta_{2}],$
(77) $\displaystyle
R_{2}(\Delta_{1},w_{1};\Delta_{2},w_{2};\Delta_{12})=R_{2}(\Delta_{1},w_{1})+R_{2}(\Delta_{2},w_{2})+\frac{\Delta_{12}}{4}[(\bar{w}_{1}+w_{1})\eta_{2}+(\bar{w}_{2}+w_{2})\eta_{1}],$
(78) $\displaystyle
R_{3}(\Delta_{1},w_{1};\Delta_{2},w_{2};\Delta_{12})=R_{3}(\Delta_{1},w_{1})+R_{3}(\Delta_{2},w_{2}),$
(79) $\displaystyle
B_{1}(\Delta_{1},w_{1};\Delta_{2},w_{2};\Delta_{12})=B_{1}(\Delta_{1},w_{1})+B_{1}(\Delta_{2},w_{2})-\frac{\Delta_{12}}{4}[(w_{1}+\bar{w}_{1})\eta_{2}+(w_{2}+\bar{w}_{2})\eta_{1}],$
(80) $\displaystyle
B_{2}(\Delta_{1},w_{1};\Delta_{2},w_{2};\Delta_{12})=B_{2}(\Delta_{1},w_{1})+B_{2}(\Delta_{2},w_{2})+i\frac{\Delta_{12}}{4}[(\bar{w}_{1}-w_{1})\eta_{2}+(\bar{w}_{2}-w_{2})\eta_{1}],$
(81) $\displaystyle
B_{3}(\Delta_{1},w_{1};\Delta_{2},w_{2};\Delta_{12})=B_{3}(\Delta_{1},w_{1})+B_{3}(\Delta_{2},w_{2})+\frac{\Delta_{12}}{2}(\eta_{1}+\eta_{2}).$
(82)

The expression for $B_{3}(\Delta_{1},w_{1};\Delta_{2},w_{2};\Delta_{12})$
above shows that two-particle states in the conformal primary basis carry a
pairwise boost-like quantum number. From the signs of $\eta_{1}$ and
$\eta_{2}$ we see that in a scattering process the particles will exchange an
overall boost charge $\Delta B_{3}=2\Delta_{12}$. Such exchange is the
analogue of the scoot effect discussed in Section II with the crucial
difference that in this case the states of the particles are described in the
conformal primary basis while the scoot effect was derived using the
conventional momentum basis. In what follows we demonstrate that the effect we
just described is also present in the ordinary four-momentum parametrization
of two-particle states thus providing a group-theoretic derivation of the
scoot effect. Our analysis also extends the argument provided by Zwanziger
Zwanziger (1972) for the existence of a pairwise helicity appearing in the
scattering of electrically and magnetically charged particles to the boost-
like pairwise charge. Before proceeding further let us stress here an
important point. The generators in (77)-(82) satisfy the Lorentz algebra even
if $\Delta_{12}$ is not a constant, but a function of a Lorentz invariant
quantity such as $q_{1}(w_{1})\cdot q_{2}(w_{2})$, where $q_{i}(w_{i})$ are
null momenta as in (36). For simplicity, we will derive the algebra in the on-
shell momentum basis assuming that $\Delta_{12}$ is a constant.

Let us start by acting on (31) with a Lorentz transformation

$\displaystyle U(\Lambda)|y,z\rangle$
$\displaystyle=\int_{0}^{\infty}d\lambda\int\hskip 1.42271ptdwd\bar{w}\hskip
1.42271ptG^{*}_{\Delta}(\hat{p};w)U(\Lambda)|\Delta,w\rangle$
$\displaystyle\stackrel{{\scriptstyle\eqref{F21}}}{{=}}\int_{0}^{\infty}d\lambda\int\hskip
1.42271ptdwd\bar{w}\hskip
1.42271ptG^{*}_{\Delta}(\hat{p};w)|cw+d|^{-2\Delta}|\Delta,\Lambda w\rangle.$
(83)

Using now (27) one gets

$\displaystyle U(\Lambda)|y,z\rangle$
$\displaystyle=\int_{0}^{\infty}d\lambda\int\hskip 1.42271ptdwd\bar{w}\hskip
1.42271pt\int
d\mu(p^{\prime})G^{*}_{\Delta}(\hat{p};w)G_{\Delta}(\hat{p}^{\prime};\Lambda
w)|cw+d|^{-2\Delta}|y^{\prime},z^{\prime}\rangle,$
$\displaystyle=\int_{0}^{\infty}d\lambda\int\hskip 1.42271ptdwd\bar{w}\hskip
1.42271pt\int
d\mu(k)G^{*}_{\Delta}(\hat{p};w)G_{\Delta}(\hat{k};w)|cw+d|^{-2\Delta}|cw+d|^{2\Delta}|\Lambda
Y,\Lambda Z\rangle$ $\displaystyle=|\Lambda y,\Lambda z\rangle.$ (84)

In going from the first to the second line we defined
$\hat{p}^{\prime}=\Lambda\hat{k}$ with $\hat{k}=\hat{k}(Y,Z)$ and then used
Lorentz invariance of the measure. Equation (IV) is just the standard result
for the Lorentz transformations of scalar one-particle states.

In order to find the form of the generators acting on two-particle states
having a non-trivial pairwise boost charge $\Delta_{12}$ we need to compute
the action of $U(\Lambda)$ on them. We denote this basis by
$|p_{1};p_{2};\Delta_{12}\rangle\equiv|y_{1},z_{1};y_{2},z_{2};\Delta_{12}\rangle$.
Repeating the same steps illustrated for single-particle states but now using
(76) we get

$\displaystyle U(\Lambda)|y_{1},z_{1};y_{2},z_{2};\Delta_{12}\rangle=\int
d\mu(\lambda_{1},w_{1})d\mu(\lambda_{2},w_{2})\int
d\mu(k_{1})d\mu(k_{2})G_{\Delta_{1}}^{*}(\hat{p}_{1};w_{1})G_{\Delta_{1}}(\hat{k}_{1};w_{1})$
$\displaystyle\times
G_{\Delta_{2}}^{*}(\hat{p}_{2};w_{2})G_{\Delta_{2}}(\hat{k}_{2};w_{2})|cw_{1}+d|^{-\eta_{1}\frac{\Delta_{12}}{2}}|cw_{2}+d|^{-\eta_{2}\frac{\Delta_{12}}{2}}|\Lambda
Y_{1},\Lambda Z_{1};\Lambda Y_{2},\Lambda Z_{2};\Delta_{12}\rangle,$ (85)

where $\hat{k}_{i}=\hat{k}_{i}(Y_{i},Z_{i})$ that we rewrite as

$\displaystyle U(\Lambda)|y_{1},z_{1};y_{2},z_{2};\Delta_{12}\rangle=\int
d\mu(k_{1})d\mu(k_{2})I_{\Delta_{12}}(\hat{p}_{1};\hat{k}_{1})I_{\Delta_{12}}(\hat{p}_{2};\hat{k}_{2})|\Lambda
Y_{1},\Lambda Z_{1};\Lambda Y_{2},\Lambda Z_{2};\Delta_{12}\rangle,$ (86)

with,

$\displaystyle
I_{\Delta_{12}}(\hat{p}_{i};\hat{k}_{i})=\int_{0}^{\infty}d\lambda_{i}\int\hskip
1.42271ptdw_{i}d\bar{w}_{i}G_{\Delta_{i}}^{*}(\hat{p}_{i};w_{i})G_{\Delta_{i}}(\hat{k}_{i};w_{i})|cw_{i}+d|^{-\eta_{i}\frac{\Delta_{12}}{2}}\,.$
(87)

Clearly for $\Delta_{12}=0$, using (32) we have
$I_{0}(\hat{p}_{i},\hat{k}_{i})=\delta^{(3)}(\hat{p}_{i}-\hat{k}_{i})$. For
$\Delta_{12}\neq 0$, as shown in Appendix B, we have

$\displaystyle
I_{\Delta_{12}}(\hat{p}_{i},\hat{k}_{i})-I_{0}(\hat{p}_{i},\hat{k}_{i})\simeq-\eta_{i}\frac{\Delta_{12}}{2}(\alpha+\beta
z_{i}+\bar{\beta}\bar{z}_{i})I_{0}(\hat{p}_{i},\hat{k}_{i})$
$\displaystyle-\eta_{i}\frac{\Delta_{12}}{4\pi}\int_{0}^{\infty}d\lambda_{i}\frac{y_{i}^{\Delta_{i}}Y^{\Delta^{*}_{i}}_{i}}{|\Gamma(i\lambda_{i})|^{2}}\int_{0}^{\infty}dAA^{\Delta_{i}-3}e^{-Ay^{2}_{i}}\int_{0}^{\infty}dBB^{\Delta^{*}_{i}-2}e^{-BY^{2}_{i}}$
$\displaystyle\int\frac{d^{2}\vec{l}}{(2\pi)^{2}}e^{i\vec{l}\cdot(\vec{Z}_{i}-\vec{z}_{i})}e^{-\frac{l^{2}}{4A}}e^{-\frac{l^{2}}{4B}}[i(\beta+\bar{\beta})l^{1}-(\beta-\bar{\beta})l^{2}].$
(88)

with $\alpha=d^{\prime}(0)+\bar{d}^{\prime}(0)$ and $\beta=c^{\prime}(0).$ We
now show that the integral in the last two lines of the previous equation does
not give any contribution to (86). Indeed, we have to insert the previous
equation in (86) and integrate it over $\hat{k}_{i}(Y_{i},Z_{i})$ obtaining
the integral

$\displaystyle\int d^{2}\vec{Z}_{i}e^{i\vec{l}\cdot
Z_{i}}=(2\pi)^{2}\delta^{(2)}(\vec{l}_{i}).$ (89)

Hence we have to set everywhere in the integral in the second and third line
of (88) $\vec{l}=0$ and it gives exactly $0$ because it is linear in
$\vec{l}$. We have then

$\displaystyle[U(\Lambda)-\mathbf{1}]|y_{1},z_{1};y_{2},z_{2};\Delta_{12}\rangle\simeq$
(90)
$\displaystyle-\frac{\Delta_{12}}{2}[\alpha(\eta_{1}+\eta_{2})+\beta(z_{1}\eta_{1}+z_{2}\eta_{2})+\bar{\beta}(\bar{z}_{1}\eta_{1}+\bar{z}_{2}\eta_{2})]|y_{1},z_{1};y_{2},z_{2};\Delta_{12}\rangle,$
(91)

from which we can read the pairwise contribution to the generators in
(53)-(58). We have

$\displaystyle
R_{1}(y_{1},z_{1};y_{2},w_{2};\Delta_{12})=R_{1}(y_{1},z_{1})+R_{1}(y_{2},z_{2})+i\frac{\Delta_{12}}{4}[(\bar{z}_{1}-z_{1})\eta_{1}+(\bar{z}_{2}-z_{2})\eta_{2}],$
(92) $\displaystyle
R_{2}(y_{1},z_{1};y_{2},w_{2};\Delta_{12})=R_{2}(y_{1},z_{1})+R_{2}(y_{2},z_{2})+\frac{\Delta_{12}}{4}[(\bar{z}_{1}+z_{1})\eta_{2}+(\bar{z}_{2}+z_{2})\eta_{1}],$
(93) $\displaystyle
R_{3}(y_{1},z_{1};y_{2},w_{2};\Delta_{12})=R_{3}(y_{1},z_{1})+R_{3}(y_{2},z_{2}),$
(94) $\displaystyle
B_{1}(y_{1},z_{1};y_{2},w_{2};\Delta_{12})=B_{1}(y_{1},z_{1})+B_{1}(y_{2},z_{2})-\frac{\Delta_{12}}{4}[(z_{1}+\bar{z}_{1})\eta_{2}+(z_{2}+\bar{z}_{2})\eta_{1}],$
(95) $\displaystyle
B_{2}(y_{1},z_{1};y_{2},w_{2};\Delta_{12})=B_{2}(y_{1},z_{1})+B_{2}(y_{2},z_{2})+i\frac{\Delta_{12}}{4}[(\bar{z}_{1}-z_{1})\eta_{2}+(\bar{z}_{2}-z_{2})\eta_{1}],$
(96) $\displaystyle
B_{3}(y_{1},z_{1};y_{2},w_{2};\Delta_{12})=B_{3}(y_{1},z_{1})+B_{3}(y_{2},z_{2})+\frac{\Delta_{12}}{2}(\eta_{1}+\eta_{2}),$
(97)

In the center of mass frame, where $z_{i}=\bar{z}_{i}=0$, we have

$\displaystyle\Delta
B_{3}=B_{3}^{\mathrm{out}}-B_{3}^{\mathrm{in}}=2\Delta_{12}.$ (98)

Comparing with equation (15), we can make the identification

$\displaystyle\Delta_{12}=-\Delta_{21}=\frac{e_{1}e_{2}}{\gamma^{2}-1}\log\bigg{|}\frac{\tau_{1}}{\tau_{2}}\bigg{|},$
(99)

in electromagnetism and

$\displaystyle\Delta_{12}=-\Delta_{21}=\frac{2Gm_{1}m_{2}\gamma(3-2\gamma^{2})}{\gamma^{2}-1}\log\bigg{|}\frac{\tau_{1}}{\tau_{2}}\bigg{|},$
(100)

in gravity, where the Lorentz factor $\gamma$ is given by

$\displaystyle\gamma=-\hat{p}_{1}(y_{1},z_{1})\cdot\hat{p}_{2}(y_{2},z_{2})=\frac{y_{1}^{2}+y_{2}^{2}+|z_{12}|^{2}}{y_{1}y_{2}},\qquad
z_{12}=z_{1}-z_{2}\,.$ (101)

Notice that, with this identification666One can show that, keeping a non-
vanishing pairwise helicity $\sigma_{12}$ and repeating the matching procedure
with the asymptotic angular momentum found in Zwanziger (1972), it will imply
simply $\sigma_{12}\sim\mu_{12}=e_{1}g_{2}-e_{2}g_{1}$ without additional
dependence on the particles four-momenta., $\Delta_{12}$ depends on the two
on-shell momenta through the Lorentz factor $\gamma$. It is not immediately
obvious that in this case the modified generators in (92)-(97) will still
satisfy the Lorentz algebra. However, it can be show that this is the case
i.e.
$[R_{i}(y_{i},z_{i};\Delta_{12}),R_{j}(y_{i},z_{i};\Delta_{12})]=\epsilon^{ij}{}^{k}R_{k}(y_{i},z_{i};\Delta_{12})$,
$[B_{i}(y_{i},z_{i};\Delta_{12}),R_{j}(y_{i},z_{i};\Delta_{12})]=\epsilon^{ij}{}^{k}B_{k}(y_{i},z_{i};\Delta_{12})$
and
$[B_{i}(y_{i},z_{i};\Delta_{12}),B_{j}(y_{i},z_{i};\Delta_{12})]=-\epsilon^{ij}{}^{k}R_{k}(y_{i},z_{i};\Delta_{12})$
since $\gamma$ is a Lorentz invariant quantity.

This completes our proof that the asymptotic boost charge carried by multi-
particle states in scattering processes in electromagnetism and gravity can be
obtained in a group-theoretic fashion as a pairwise quantum number linked to
the little group of celestial multi-particle states.

##  V Conclusions

In his original 1972 paper Zwanziger (1972), Zwanziger observed that the
asymptotic electromagnetic field generated by pairs of particles with both
electric and magnetic charges carries a non-vanishing angular momentum due to
the charge-pole interaction. The group-theoretic origin of this new quantum
number was identified in a modification of the action of Lorentz generators on
scattering states associated to the little group of the four-momenta
associated to pairs of particles.

Zwanziger also identified potential pairwise Coulombic contributions
(depending only on the product of the particles electric and magnetic charges
separately) to the asymptotic relativistic angular momentum of the
electromagnetic field but dismissed them as vanishing or ill-defined. In
retrospect this conclusion appears also to be justified by the fact that, in
the ordinary four-momentum basis for scattering states, the only manifest
transformations which leave invariant the four-momentum configuration of pairs
of particles are rotations around a given spatial axis.

Recent results Gralla and Lobo (2022a) motivated by gravitational scattering
calculations suggest, however, that already in ordinary electromagnetism, in
the absence of magnetic charges, the electromagnetic field possesses an
asymptotic boost-like angular momentum proving that the Coulombic
contributions originally negelcted by Zwanziger are, in fact, real. Indeed
what has been dubbed as “scoot effect” consists in an exchange of boost
angular momentum between particles and field. A natural question then concerns
the group theoretic-origin of such boost-like angular momentum carried by
asymptotic states. In this work we provided an answer to this question showing
that the origin of the pairwise boost charge can be traced back to the little
group of a two-particle state in the conformal primary basis representation.
We explicitly showed that, using the integral transform connecting the
ordinary on-shell momentum representation to the conformal primary one, the
pairwise boost charge contribution to two-particle states in the conformal
primary basis translates into an analogous charge for two-particle states in
the ordinary momentum representation. Such charge appears in a modification of
the expression of the generators of Lorentz transformations acting on two-
particle states (92)-(97). In the case of the pairwise helicity discovered by
Zwanziger this modified action of Lorentz generators on multiparticle states
has been interpreted Csáki et al. (2021); Csaki et al. (2021); Lippstreu
(2021) in terms of an extension of the Fock space representation of the
Poincaré group in which two-particle states are no longer described by simple
tensor products of one-particle states but contain an additional factor
representing a “pairwise Hilbert space” in which the pairwise helicity is
stored. Alternatively one could keep the ordinary tensor product structure of
the two-particle states and interpret the modification of the action of
Lorentz generators as a modification of the ordinary Leibniz action on tensor
product states carrying an additional term accounting for the pairwise angular
momentum. Either way the existence of a pairwise relativistic angular momentum
carried by asymptotic states renders the conventional Fock space picture of
multiparticle states and their associated observables inadequate. In this work
we pointed out that, besides electromagnetic scattering, long range effects in
gravitational scattering require the existence of a pairwise boost-like
angular momentum and we evidenced the subtle connection between this quantum
number and the little group of of two-particle states in the conformal primary
basis. A remarkable upshot of our analysis is that, given the universal
character of the coupling of gravity to all forms of matter and energy, the
inadequacy of a Fock space description of multiparticle states appears to be a
general infrared consequence of the existence of gravitational interactions.
Given that the Fock space structure of multiparticle states in free quantum
field theory is a cornerstone of the description of particle creation effects
in the presence of causal horizons, it is tempting to speculate that an
infrared failure of such framework might have significant consequences and
challenge our current understanding of puzzling aspects like, for example, the
apparent failure of unitarity in black hole quantum evolution. We leave
further exploration of these implications to future studies.

### Acknowledgements

The research of FA is fully supported by the Knut and Alice Wallenberg
Foundation under grant KAW 2018.0116. MA acknowledges support from the INFN
Iniziativa Specifica QUAGRAP. The research of MA was also carried out in the
frame of Programme STAR Plus, financially supported by the University of
Napoli Federico II and Compagnia di San Paolo.

##  Appendix A Transformation of massive momenta

The isomorphism between the Lorentz group and
SL(2,$\mathbb{C}/\mathbb{Z}^{2}$) is realized as Oblak (2015)

$\displaystyle\Lambda^{\mu}{}_{\nu}(a,b,c,d)=$
$\displaystyle=\frac{1}{2}\begin{pmatrix}a\bar{a}+b\bar{b}+c\bar{c}+d\bar{d}&b\bar{a}+a\bar{b}+d\bar{c}+c\bar{d}&i(-b\bar{a}+a\bar{b}-d\bar{c}+c\bar{d})&-a\bar{a}+b\bar{b}-c\bar{c}+d\bar{d}\cr
c\bar{a}+a\bar{c}+d\bar{b}+b\bar{d}&d\bar{a}+a\bar{d}+c\bar{b}+b\bar{c}&i(-d\bar{a}+a\bar{d}+c\bar{b}-b\bar{c})&-c\bar{a}-a\bar{c}+d\bar{b}+b\bar{d}\cr
i(c\bar{a}-a\bar{c}+d\bar{b}-b\bar{d})&i(d\bar{a}-a\bar{d}+c\bar{b}-b\bar{c})&d\bar{a}+a\bar{d}-c\bar{b}-b\bar{c}&i(-c\bar{a}+a\bar{c}+d\bar{b}-b\bar{d})\cr-a\bar{a}-b\bar{b}+c\bar{c}+d\bar{d}&-b\bar{a}-a\bar{b}+d\bar{c}+c\bar{d}&i(b\bar{a}-a\bar{b}-d\bar{c}+c\bar{d})&a\bar{a}-b\bar{b}-c\bar{c}+d\bar{d}\end{pmatrix},$

where $ad-bc=\bar{a}\bar{d}-\bar{b}\bar{c}=1$. When acting on massive momenta
$\hat{p}^{\mu}$ in (24) we get

$\displaystyle\hat{p}^{\prime\mu}(y^{\prime},z^{\prime})=\Lambda^{\mu}{}_{\nu}(a,b,c,d)\hat{p}^{\nu}(y,z)$
$\displaystyle=\frac{1}{2y}(y^{2}(|a|^{2}+|c|^{2})+|az+b|^{2}+|cz+d|^{2},y^{2}(c\bar{a}+a\bar{c})+(cz+d)(\bar{a}\bar{z}+\bar{b})+(az+b)(\bar{c}\bar{z}+\bar{d}),$
$\displaystyle
i[y^{2}(\bar{a}c-a\bar{c})-(az+b)(\bar{c}\bar{z}+\bar{d})+(cz+d)(\bar{a}\bar{z}+\bar{b})],y^{2}(|c|^{2}-|a|^{2})-|az+b|^{2}+|cz+d|^{2})$
$\displaystyle=\frac{1}{2y^{\prime}}\left(1+y^{\prime
2}+z^{\prime}\bar{z}^{\prime},z^{\prime}+\bar{z}^{\prime},i(\bar{z}^{\prime}-z^{\prime}),1-y^{\prime
2}-z^{\prime}\bar{z}^{\prime}\right),$ (102)

with

$\displaystyle
z^{\prime}=\frac{(az+b)(\bar{c}\bar{z}+\bar{d})+a\bar{c}y^{2}}{(cz+d)(\bar{c}\bar{z}+\bar{d})+c\bar{c}y^{2}},\qquad
y^{\prime}=\frac{y}{(cz+d)(\bar{c}\bar{z}+\bar{d})+c\bar{c}y^{2}},$ (103)

as displayed in (38).

##  Appendix B Pairwise integral

In this Appendix we analyse the integral appearing in (87). Choosing the
parametrization $\hat{p}_{i}=\hat{p}_{i}(Y_{i},Z_{i})$ and
$\hat{k}_{i}=\hat{k}_{i}(y_{i},z_{i})$ we can re-write the integral as

$\displaystyle I_{\Delta_{12}}(\hat{p}_{i},\hat{k}_{i})=$
$\displaystyle\frac{1}{4\pi^{3}}\int_{0}^{\infty}d\lambda\int\hskip
1.42271ptdwd\bar{w}\frac{\Gamma(\Delta)\Gamma(\Delta^{*})}{\Gamma(i\lambda)\Gamma(-i\lambda)}\bigg{(}\frac{y_{i}}{y^{2}_{i}+|w-z_{i}|^{2}}\bigg{)}^{\Delta}\bigg{(}\frac{Y_{i}}{Y_{i}+|w-Z_{i}|^{2}}\bigg{)}^{\Delta^{*}}$
$\displaystyle\times|cw+d|^{-\eta_{i}\frac{\Delta_{12}}{2}}.$ (104)

We now consider the case with $\Delta_{12}\neq 0$. In order to find the
generator we need to compute the derivative

$\displaystyle\frac{d}{dx}(|c(x)w+d(x)|^{2})^{-\eta_{i}\frac{\Delta_{12}}{2}}|_{x=0}=-\eta_{i}\frac{\Delta_{12}}{2}(\alpha+\beta
w+\bar{\beta}\bar{w})=f(w,\bar{w}),$ (105)

where we used $d(0)=1$ and $c(0)=0$, and where $\alpha\equiv
d^{\prime}(0)+\bar{d}^{\prime}(0),\,\beta\equiv c^{\prime}(0)$. Therefore, we
see that the new integral we have to compute is

$\displaystyle I_{\Delta_{12}}(\hat{p}_{i},\hat{k}_{i})\simeq
I_{0}(\hat{p}_{i},\hat{k}_{i})$
$\displaystyle+\frac{1}{4\pi^{3}}\int_{0}^{\infty}d\lambda\int\hskip
1.42271ptdwd\bar{w}\frac{\Gamma(\Delta)\Gamma(\Delta^{*})}{\Gamma(i\lambda)\Gamma(-i\lambda)}\bigg{(}\frac{y_{i}}{y_{i}^{2}+|w-Z_{i}|^{2}}\bigg{)}^{\Delta}\bigg{(}\frac{Y_{i}}{Y^{2}_{i}+|w-Z_{i}|^{2}}\bigg{)}^{\Delta^{*}}f(w,\bar{w}).$
(106)

We have, using $w=x^{1}+ix^{2}$ and $dwd\bar{w}=2d^{2}\vec{x}$ we get

$\displaystyle I_{\Delta_{12}}(\hat{p}_{i},\hat{k}_{i})\simeq
I_{0}(\hat{p}_{i},\hat{k}_{i})$
$\displaystyle+\frac{1}{2\pi^{3}}\int_{0}^{\infty}d\lambda\frac{|\Gamma(\Delta)|^{2}y_{i}^{\Delta}Y_{i}^{\Delta^{*}}}{|\Gamma(i\lambda)|^{2}}\int
d^{2}\vec{x}\bigg{(}\frac{1}{y^{2}_{i}+|\vec{x}-\vec{z}_{i}|^{2}}\bigg{)}^{\Delta}\bigg{(}\frac{1}{Y^{2}_{i}+|\vec{x}-\vec{Z}_{i}|^{2}}\bigg{)}^{\Delta^{*}}f(\vec{x}),$
(107)

that, using Schwinger parametrization can be rewritten as

$\displaystyle
I_{\Delta_{12}}(\hat{p}_{i},\hat{k}_{i})-I_{0}(\hat{p}_{i},\hat{k}_{i})$
$\displaystyle\simeq\frac{1}{2\pi^{3}}\int_{0}^{\infty}d\lambda\frac{y_{i}^{\Delta}Y_{i}^{\Delta^{*}}}{|\Gamma(i\lambda)|^{2}}\int\frac{d^{2}\vec{l}}{(2\pi)^{2}}e^{i\vec{l}\cdot(\vec{x}-\vec{\rho})}$
$\displaystyle\times\int_{0}^{\infty}dAA^{\Delta-1}e^{-Ay^{2}_{i}}\int_{0}^{\infty}dBB^{\Delta^{*}-1}e^{-BY^{2}_{i}}\int
d^{2}\vec{x}e^{-A|\vec{x}-\vec{z}_{i}|^{2}}f(\vec{x})\int
d^{2}\vec{\rho}e^{-B|\vec{\rho}-\vec{Z}_{i}|^{2}}.$ (108)

We perform the shifts $\vec{x}^{\prime}=\vec{x}-\vec{z}_{i}$ and
$\vec{\rho}^{\prime}=\vec{\rho}-\vec{Z}_{i}$ and we get

$\displaystyle
I_{\Delta_{12}}(\hat{p}_{i},\hat{k}_{i})-I_{0}(\hat{p}_{i},\hat{k}_{i})$
$\displaystyle\simeq\frac{1}{2\pi^{3}}\int_{0}^{\infty}d\lambda\frac{y_{i}^{\Delta}Y_{i}^{\Delta^{*}}}{|\Gamma(i\lambda)|^{2}}\int\frac{d^{2}\vec{l}}{(2\pi)^{2}}e^{i\vec{l}\cdot(\vec{z}_{i}-\vec{Z}_{i})}\int_{0}^{\infty}dAA^{\Delta-1}e^{-Ay^{2}_{i}}e^{-\frac{l^{2}}{4A}}\int_{0}^{\infty}dBB^{\Delta^{*}-1}e^{-BY^{2}_{i}}e^{-\frac{l^{2}}{4B}}$
$\displaystyle\times\int
d^{2}\vec{x}e^{-A(\vec{x}-i\frac{\vec{l}}{2A})^{2}}f(\vec{x}+\vec{z}_{i})\int
d^{2}\vec{\rho}e^{-B(\vec{\rho}-i\frac{\vec{l}}{2B})^{2}}$
$\displaystyle=\frac{1}{2\pi^{3}}\int_{0}^{\infty}d\lambda\frac{y_{i}^{\Delta}Y_{i}^{\Delta^{*}}}{|\Gamma(i\lambda)|^{2}}\int\frac{d^{2}\vec{k}}{(2\pi)^{2}}e^{i\vec{l}\cdot(\vec{z}_{i}-\vec{Z}_{i})}\int_{0}^{\infty}dAA^{\Delta-1}e^{-Ay^{2}_{i}}e^{-\frac{l^{2}}{4A}}\int_{0}^{\infty}dBB^{\Delta^{*}-1}e^{-BY^{2}_{i}}e^{-\frac{l^{2}}{4B}}$
$\displaystyle\times\int
d^{2}\vec{x}e^{-A\vec{x}^{2}}f\bigg{(}\vec{x}+\vec{z}+i\frac{\vec{l}}{2A}\bigg{)}\int
d^{2}\vec{\rho}e^{-B\vec{\rho}^{2}}.$ (109)

We need to compute gaussian integral

$\displaystyle\int
d^{2}\vec{x}e^{-A\vec{x}^{2}}\bigg{[}\alpha+\beta\bigg{(}x^{1}+ix^{2}+z^{1}_{i}+iz^{2}_{i}+i\frac{l^{1}+il^{2}}{2A}\bigg{)}+\bar{\beta}\bigg{(}x^{1}-ix^{2}+z^{1}_{i}-iz^{2}_{i}+i\frac{l^{1}-il^{2}}{2A}\bigg{)}\bigg{]}$
$\displaystyle=$
$\displaystyle\frac{\pi}{A}\bigg{[}\alpha+(\beta+\bar{\beta})\bigg{(}z^{1}_{i}+i\frac{l^{1}}{2A}\bigg{)}+i(\beta-\bar{\beta})\bigg{(}z_{i}^{2}+i\frac{l^{2}}{2A}\bigg{)}\bigg{]},$
(110)

that yields equation (88).

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