# Anomalous soft photons: status and perspectives

R. Bailhache  0000-0001-7987-4592 D. Bonocore  0000-0002-8672-5584 P. Braun-
Munzinger  0000-0003-2527-0720 X. Feal  0000-0002-3258-2020 S. Floerchinger
0000-0002-3428-4625 J. Klein  0000-0002-1301-1636 K. Köhler  P. Lebiedowicz
0000-0003-1963-6263 C.M. Peter  0009-0008-9207-9911 R. Rapp
0000-0003-3020-4402 K. Reygers  0000-0001-9808-1811 W. Schäfer
0000-0002-0764-0372 H.S. Scheid  0000-0003-1184-9627 K. Schweda
0000-0001-9935-6995 J. Stachel  0000-0003-0750-6664 H. van Hees
0000-0003-0729-2117 C.A. van Veen  0000-0003-1199-4445 M. Völkl
0000-0002-3478-4259 Institut für Kernphysik, Johann Wolfgang Goethe-
Universität Frankfurt, Frankfurt, Germany  Technical University of Munich, TUM
School of Natural Sciences, Physics Department T31, James-Franck-Straße 1,
D-85748, Garching, Germany  Research Division and ExtreMe Matter Institute
EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany
IGFAE, Universidade de Santiago de Compostela, Campus Vida, 15705 Santiago de
Compostela, Spain  Theoretisch-Physikalisches Institut, Friedrich-Schiller-
Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany  European Organization
for Nuclear Research (CERN), Geneva, Switzerland  Physikalisches Institut,
Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany  Institute of
Nuclear Physics Polish Academy of Sciences, Radzikowskiego 152, PL-31342
Kraków, Poland  Cyclotron Institute and Department of Physics and Astronomy,
Texas A&M University, College Station, TX 77843-3366, USA  Institut für
Theoretische Physik, Goethe-Universität Frankfurt am Main, Max-von-Laue-
Strasse 1, D-60438 Frankfurt am Main, Germany and Helmholtz Research Academy
Hesse for FAIR, Campus Frankfurt, D-60438 Frankfurt, Germany

(May 2024)

###### Abstract

This report summarizes the work of the EMMI Rapid Reaction Task Force on “Real
and Virtual Photon Production at Ultra-Low Transverse Momentum and Low Mass at
the LHC”. We provide an overview of the soft-photon puzzle, i.e., of the long-
standing discrepancy between experimental data and predictions based on Low’s
soft-photon theorem, also referred to as “anomalous” soft photon production,
and we review the current theoretical understanding of soft radiation and soft
theorems. We also focus on low-mass dileptons as a tool for determining the
electrical conductivity of the medium produced in high-energy nucleus-nucleus
collisions. We discuss how both topics can be addressed with the planned ALICE
3 detector at the LHC.

###### keywords:

EMMI-RRTF-ER20-01 , TUM-HEP-1496-24

††journal: Physics Reports

###### Contents

  1. 1 Introduction
  2. 2 The soft-photon puzzle
    1. 2.1 Introduction
      1. 2.1.1 Overview of soft-photon measurements
      2. 2.1.2 Attempts from theory
    2. 2.2 Soft radiation and soft theorems
      1. 2.2.1 Low’s leading soft theorem: classical bremsstrahlung
      2. 2.2.2 Low’s leading soft theorem: symmetry considerations
      3. 2.2.3 Expansion scheme: Leading order, Derivative operators versus momentum shifts in next-to-leading order
      4. 2.2.4 Loop corrections
    3. 2.3 Experimental consequences and opportunities at the LHC
      1. 2.3.1 Measuring the forward soft-photon spectrum with the Forward Conversion Tracker of ALICE 3
    4. 2.4 Specific examples: Internal bremsstrahlung in decays of $\text{J}/\psi$
      1. 2.4.1 Estimates of rates and spectra for $\text{J}/\psi$ production in ultraperipheral Pb–Pb collisions with STARlight
      2. 2.4.2 Results for the $\text{p}\text{p}\to\text{p}\text{p}(\text{J}/\psi\to\mu^{+}\mu^{-}\gamma)$ reaction in a Regge inspired toy model
  3. 3 Electrical conductivity via soft photons and dielectrons from heavy-ion collisions
    1. 3.1 Electric conductivity
    2. 3.2 Thermal radiation signal for two different scenarios
    3. 3.3 Physical backgrounds
      1. 3.3.1 Hadronic decay background
      2. 3.3.2 Electromagnetic background
      3. 3.3.3 Normalization of the background sources
    4. 3.4 Total dielectron yield
  4. 4 Conclusions and Outlook

##  1 Introduction

The measurement of low-energy real and virtual photons in high-energy
collisions of hadrons and nuclei provides access to fundamental physics
questions. In this report we focus on soft-photon production and tests of the
range of validity of Low’s soft photon theorem as well as on the extraction of
the electrical conductivity of the medium created in nucleus-nucleus
collisions with the aid of soft virtual photons. In experimental tests of
Low’s theorem, striking discrepancies between predictions and data were found.
No agreement exists on their possible origin, despite more than 40 years of
research. On the theory side, there is a considerable interest in soft
theorems as they are related to symmetries reflecting the infrared structure
of gravity and gauge theory. The electrical conductivity is a key property
that characterizes the produced quark-gluon plasma and the hadron gas. The
production rate of low-energy virtual photons, measurable as $e^{+}e^{-}$
pairs in the experiment, is sensitive to the electrical conductivity. However,
extracting this electrical conductivity is challenging and so far few
experimental constraints exist. The planned ALICE 3 detector at the LHC
provides the opportunity to quantitatively address both questions. We
summarize the current theoretical understanding of the two topics and discuss
experimental opportunities and challenges.

##  2 The soft-photon puzzle

###  2.1 Introduction

Bremsstrahlung produced in scattering processes can be attributed to
accelerated charges between the incoming initial-state and outgoing final-
state charged particles, as well as to accelerations happening in the short-
lived intermediate state. However, bremsstrahlung photons with a sufficiently
long wavelength cannot resolve the intermediate state and consequently their
production is solely determined by the overall change of currents from the
initial to the final state. Photons which satisfy this long-wavelength
criterion are called soft photons. For inelastic hadronic interactions, the
order of magnitude for the spatial extent of the intermediate state and its
lifetime are usually estimated to be $\Delta x\approx 1\,\mathrm{fm}$ and
$\Delta t\approx 1\,\mathrm{fm}/c$, respectively. The wavelength of a 100 MeV
photon is about $12\,\mathrm{fm}$. Thus, photons with energies less than
$10\text{--}100\,\mathrm{MeV}$ are expected to qualify as soft photons for
inelastic hadronic interactions.

Particle production in hadronic collisions at high energies is often modeled
as the fragmentation of a QCD string spanned along the beam axis. The various
string fragments can be viewed as particle sources that move along the beam
direction with different velocities. Following an old argument by Gribov [1],
for such systems the transverse momentum with respect to the beam axis has
been traditionally used to characterize the softness of a produced photon. As
the mean transverse momentum of pions is in the order of $p_{T}^{\pi}\approx
300\text{--}500\,\mathrm{MeV}/c$, soft photons are often defined as photons
whose transverse momentum $k_{\text{T}}$ satisfies $k_{\text{T}}\ll
p_{T}^{\pi}$.

In 1958 Francis Low wrote a seminal paper on how to relate the production of
soft photons with the production of charged particles involved in a scattering
process [2]. Low considered 2-to-2 scattering processes plus an additional
photon in the final state and based on a quantum mechanical treatment, he
provided expressions for $\sigma_{0}$ and $\sigma_{1}$ in the power expansion

$\sigma=\frac{\sigma_{0}}{\omega_{k}}+\sigma_{1}+\omega_{k}\sigma_{2}+\ldots$
(1)

of the cross section where $\omega_{k}$ is the photon energy corresponding to
a four-momentum $k$. A generalized expression for the leading-power term for
$n$-to-$m$ scattering processes is tested in various hadronic scattering
experiments. Surprisingly, some of the experiments found a significant excess
of low-energy photons compared to the expectation from the leading term
$\sigma_{0}$ in Low’s theorem. The ratio of measured-over-expected photons is
in the range between 4–8, see Tab. 1. Moreover, an excess by a factor of four
is also found by the DELPHI experiment in $\text{e}^{+}\text{e}^{-}\to
n\,\mathrm{jets}$ collisions. This is now referred to as the soft-photon
puzzle, and one refers to the excess as anomalous soft-photon production. The
experimental situation, however, is unsatisfactory, as some experiments did
not observe an excess above the expectation from Low’s theorem. Previous
reviews of this subject and the existing measurements can be found in [3, 4,
5, 6, 7, 8, 9].

####  2.1.1 Overview of soft-photon measurements

Here we give a brief overview of data on soft-photon production in high-energy
collisions, and we highlight selected results. Important results are
summarized in Tab. 1. To study the soft-photon signal, all experiments
subtract the background photons coming from the decay of hadrons, which is
dominated by the decay $\pi^{0}\to\gamma\gamma$. A remaining serious
experimental background for the inner-bremsstrahlung signal as calculated with
Low’s formula then is external bremsstrahlung created by electrons and
positrons in the detector material.

Early bubble-chamber experiments gave conflicting results on soft-photon
production. While no excess above the expected inner-bremsstrahlung signal is
found in $\pi^{-}\mathrm{p}$ collisions with a beam momentum of 10.5 GeV/$c$
at the Stanford Linear Accelerator Center (SLAC) [10], a strong enhancement is
found by the WA27 collaboration in $70\,\mathrm{GeV}/c$
$\mathrm{K}^{+}\mathrm{p}$ collisions using the Big European Bubble Chamber
(BEBC) at the CERN Super Proton Synchrotron (SPS) [11]. A few years later,
anomalous soft-photon production was reported at higher energies in
$\pi^{+}\mathrm{p}$ and $\mathrm{K}^{+}\mathrm{p}$ collisions at 250 GeV/$c$
by the EHS-NA22 collaboration using the Rapid Cycling Bubble Chamber (RCBC)
[12].

A first confirmation of a photon excess from an experiment not using a bubble
chamber came from the SOPHIE/WA83 experiment at CERN’s Omega spectrometer.
Employing electromagnetic calorimeters, this experiment found a significant
low-$p_{T}$ excess in $\pi^{-}\mathrm{p}$ collisions at 280 GeV/$c$ beam
momentum [13]. However, a few years later, Antos et al. did not find a
significant excess above the expected hadronic inner bremsstrahlung in p–Be
collisions at 450 GeV/$c$ in two independent measurements, one based on the
photon conversion method and the other based on a barium-fluoride calorimeter
[14]. Using also barium-fluoride calorimeters, the same conclusion was reached
by Tincknell et al. for p–Be and p–W collisions at 18 GeV/$c$ [15].

Figure 1: Left panel: Setup of the WA102 experiment, located at CERN’s Omega
spectrometer. Photons from p–p collisions in a liquid hydrogen target are
measured by reconstructing $\text{e}^{+}\text{e}^{-}$ tracks resulting from
photon conversions in a 1 mm thick lead converter [6]. Right panel: The left-
hand figure shows the measured photon $p_{T}$ spectrum along with the expected
background photons (BG, open circles) and inner-bremsstrahlung signal (Brems,
open triangles). The background photons comprise the photons from hadron
decays and secondary photons. Secondary photons are external bremsstrahlung
photons, resulting from electrons and positrons produced in a photon
conversion. The right-hand figure shows the measured photon yield after the
subtraction of the background. The excess above the inner bremsstrahlung
signal has approximately the same shape as the inner bremsstrahlung spectrum.

In a series of measurements at CERN’s Omega spectrometer, again, a significant
photon excess was observed [16, 17, 6]. These measurements were based on the
photon-conversion method, allowing the reconstruction of the photons’ line of
flight. This made it possible to suppress background of photons not coming
from the target region. An excess above the expected inner-bremsstrahlung
signal of a factor of about 4–5 was observed in these experiments in
$\pi^{-}\mathrm{p}$ collisions at 280 GeV/$c$ and pp collisions at 450
GeV/$c$. A first publication quoted an even larger excess, however, later a
problem in the calculation of the bremsstrahlung signal was identified [17].

Figure 2: DELPHI results on soft-photon production in
$\text{e}^{+}\text{e}^{-}\to n\,\mathrm{jets}$ (upper panels) and
$\text{e}^{+}\text{e}^{-}\to\mu^{+}\mu^{-}$ (lower panels) at
$\sqrt{s}=91\,\mathrm{GeV}$ as a function of the transverse momentum $p_{T}$
and the angle $\theta$ relative to the reconstructed jet axis or muon
direction, respectively [7, 18]. The measured photon yield shows an excess
above the inner-bremsstrahlung signal, as calculated based on Low’s theorem in
$\text{e}^{+}\text{e}^{-}\to n\,\mathrm{jets}$. In
$\text{e}^{+}\text{e}^{-}\to\mu^{+}\mu^{-}$, a suppression of photons with
angles $\theta\lesssim m_{\mu}/E_{\mu}\approx 0.0023$ due to the dead-cone
effect is visible.

An important addition to the set of soft-photon measurements was made by the
DELPHI experiment in $\text{e}^{+}\text{e}^{-}\to\mathrm{jets}$ reactions at
the Z pole ($\sqrt{s}=91\,\mathrm{GeV}$) [7]. At relatively small transverse
momenta with respect to the jet axis ($p_{T}<80\,\mathrm{MeV}/c$), an excess
of photons above the inner-bremsstrahlung signal of about a factor 4 was
observed. In $\text{e}^{+}\text{e}^{-}\to\mu^{+}\mu^{-}$ collisions, however,
the measured photon signal agreed with the calculated inner-bremsstrahlung
signal [7], see Fig. 2. In particular, the dependence of the photon yield on
the angle $\theta$ between the radiating muon and the emitted photon exhibited
an expected depletion at small angles $\theta\lesssim m_{\mu}/E_{\mu}$ known
as dead-cone effect, see Sec. 2.2.1. These findings suggest that the anomalous
soft-photon production is related to the production of hadrons. Studying the
characteristics of the photon excess in
$\text{e}^{+}\text{e}^{-}\to\mathrm{jets}$, DELPHI found that the ratio
$\gamma_{\mathrm{meas}}/\gamma_{\mathrm{brems}}$ of the measured signal over
the expected inner-bremsstrahlungs signal was constant as a function of the
charged-particle multiplicity within a jet. The neutral-particle multiplicity
of a jet (dominated by the number $\pi^{0}$’s) was estimated based on the
number of clusters reconstructed in the electromagnetic calorimeter and the
number of reconstructed conversions of photons with energies above 1 GeV.
Somewhat surprisingly, the ratio
$\gamma_{\mathrm{meas}}/\gamma_{\mathrm{brems}}$ was found to strongly
increase with the multiplicity of neutral particles of a jet, reaching an
excess up to about a factor 15 or larger for large neutral-particle
multiplicities [19]. This is illustrated in Fig. 3.

Figure 3: Photon yield per jet and the ratio
$\gamma_{\mathrm{meas}}/\gamma_{\mathrm{brems}}$ as a function of the charged
($N_{\mathrm{ch}}$) and neutral ($N_{\mathrm{neu}}$) multiplicity of the jets
in $e^{+}e^{-}\to n\,\mathrm{jets}$ events, as measured by DELPHI [19].
$N_{\mathrm{neu}}$ is calculated based on the number of showers in the
electromagnetic calorimeter and the number of converted photons, and is
defined in a way that it approximates the number of $\pi^{0}$’s in the jet.

In Ref. 3, the DELPHI collaboration stresses that the dependence of the photon
excess on the multiplicity of neutral particles of a jet is a particular
challenge for most model approaches. As explained in this paper, this feature
of the data might point to a general picture, where the extra radiation
results from dipole radiation of $q\bar{q}$ quark pairs formed in the
fragmentation process. The dipole moment of a neutral $q\bar{q}$ pair is
larger than the one of a $q\bar{q}$ pair with electric charge of $\pm 1$. As
dipole radiation scales with the square of the dipole moment, the radiation
strength of a neutral $q\bar{q}$ pair can be up to almost an order of
magnitude higher than for a charged $q\bar{q}$ pair. DELPHI also points out
that soft-photon production associated with $q\bar{q}$ production built upon
the space-time structure of the Lund string fragmentation model is not strong
enough to describe the data [20].

We close this section by mentioning further measurements that are interesting
with regard to Low’s theorem. Yu. M. Antipov et. al. studied pion radiative
scattering ($\pi^{-}\mathrm{p}\to\pi^{-}\mathrm{p}\gamma$) at the Serpukhov
accelerator [21]. In this experiment, $43\,\mathrm{GeV}/c$ $\pi^{-}$ impinged
on a liquid-hydrogen target. The scattered pion was deflected by a magnetic
field, and the photon energy was measured in the forward direction with a lead
glass calorimeter. Calculations based on the Low theorem were found to agree
with the data up to photon energies of $40\,\mathrm{GeV}$. The results were
interpreted as a confirmation of Gribov’s prediction of an extended range of
applicability of Low’s theorem [1]. The WA93 collaboration measured inclusive
photons in $200\,\mathrm{GeV}$/nucleon ${}^{32}\mathrm{S}+\mathrm{Au}$
collisions with a small-acceptance BGO detector [22]. In the first bin of the
$k_{T}$ spectrum ($k_{T}\lesssim 68\,\mathrm{MeV}/c$), an excess above the
expected decay-photon yield was observed. Experimental background and hadronic
bremsstrahlung were not sufficient to explain the excess [23]. Finally, we
note that inner bremsstrahlung was also studied in radiative decays of the
neutron ($\mathrm{n}\to\mathrm{p}e^{-}\bar{\nu}_{e}\gamma$). The sudden
acceleration of the electron in the electromagnetic field of the proton gives
rise to bremsstrahlung which can be calculated within quantum electrodynamics
(QED). The energy spectrum of the electron has a maximum at around
$200\,\mathrm{keV}$ and an endpoint energy of $782\,\mathrm{keV}$ [24].
Bremsstrahlung photons were measured from $0.4\,\mathrm{keV}$ up to the
endpoint energy [25]. No deviations from a QED calculation were found [26].
The rather low lower-energy limit of the measurement of $0.4\,\mathrm{keV}$
appears to suggest that Low’s theorem holds for radiative neutron decays.

Exp. |  year |  $p_{\mathrm{beam}}$ or $\sqrt{s}$ |  photon $k_{T}$ |  $\gamma_{\mathrm{meas}}/\gamma_{\mathrm{brems}}$ |  method |  Ref.
---|---|---|---|---|---|---
$\pi^{+}\text{p}$ |  1979 |  10.5 GeV/$c$ |  $p_{T}<20\,\mathrm{MeV}/c$ |  $1.25\pm 0.25$ |  bubble chamber |  Goshaw et al. [10]
$\text{K}^{+}\text{p}$ WA27, CERN |  1984 |  70 GeV/$c$ |  $k_{T}<60\,\mathrm{MeV}/c$ |  $4.0\pm 0.8$ |  bubble chamber (BEBC) |  Chliapnikov et al. [11]
$\pi^{+}\text{p}$ CERN, EHS, NA22 |  1991 |  250 GeV/$c$ |  $k_{T}<40\,\mathrm{MeV}/c$ |  $6.4\pm 1.6$ |  bubble chamber (RCBC) |  Botterweck et al. [12]
$\text{K}^{+}\text{p}$ CERN, EHS, NA22 |  1991 |  250 GeV/$c$ |  $k_{T}<40\,\mathrm{MeV}/c$ |  $6.9\pm 1.3$ |  bubble chamber (RCBC) |  Botterweck et al. [12]
$\pi^{-}\text{p}$ CERN, WA83, OMEGA |  1993 |  280 GeV/$c$ |  $k_{T}<10\,\mathrm{MeV}/c$ |  |  |
($0.2<E_{\gamma}<1\,\mathrm{GeV}$) |  $7.9\pm 1.4$ |  calorimeter |  Banerjee et al. [13] |  |  |
p–Be |  1993 |  450 GeV/$c$ |  $k_{T}<20\,\mathrm{MeV}/c$ |  $<2$ |  pair conversion, calorimeter |  Antos et al. [14]
p–Be, p–W |  1996 |  18 GeV/$c$ |  $k_{T}<50\,\mathrm{MeV}/c$ |  $<2.65$ |  calorimeter |  Tincknell et al. [15]
$\pi^{-}\text{p}$ CERN, WA91, OMEGA |  1997 |  280 GeV/$c$ |  $k_{T}<20\,\mathrm{MeV}/c$ |  |  |
($0.2<E_{\gamma}<1\,\mathrm{GeV}$) |  $7.8\pm 1.5$ |  pair conversion |  Belogianni at al. [16] |  |  |
$\pi^{-}\text{p}$ CERN, WA91, OMEGA |  2002 |  280 GeV/$c$ |  $k_{T}<20\,\mathrm{MeV}/c$ |  |  |
($0.2<E_{\gamma}<1\,\mathrm{GeV}$) |  $5.3\pm 1.0$ |  pair conversion |  Belogianni at al. [17] |  |  |
pp CERN, WA102, OMEGA |  2002 |  450 GeV/$c$ |  $k_{T}<20\,\mathrm{MeV}/c$ |  |  |
($0.2<E_{\gamma}<1\,\mathrm{GeV}$) |  $4.1\pm 0.8$ |  pair conversion |  Belogianni at al. [6] |  |  |
$\text{e}^{+}\text{e}^{-}\to n\,\mathrm{jets}$ CERN, DELPHI |  2006 |  91 GeV ($\sqrt{s}$) |  $k_{T}<80\,\mathrm{MeV}/c$ |  |  |
($0.2<E_{\gamma}<1\,\mathrm{GeV}$) |  $4.0\pm 0.3\pm 1.0$ |  pair conversion |  DELPHI [7, 19] |  |  |
$\text{e}^{+}\text{e}^{-}\to\mu^{+}\mu^{-}$ CERN, DELPHI |  2008 |  91 GeV ($\sqrt{s}$) |  $k_{T}<80\,\mathrm{MeV}/c$ |  |  |
($0.2<E_{\gamma}<1\,\mathrm{GeV}$) |  $\sim 1$ |  pair conversion |  DELPHI [18] |  |  |

Table 1:  Overview of soft-photon measurements. The column
“$\gamma_{\mathrm{meas}}/\gamma_{\mathrm{brems}}$” shows the ratio of the
measured photon yield over the inner-bremsstrahlung yield calculated based on
Low’s theorem. The column “$p_{\mathrm{beam}}$ or $\sqrt{s}$” lists the
momentum of the beam for fixed-targets experiments and the center-of-mass
energy for the DELPHI measurements.

####  2.1.2 Attempts from theory

Various attempts were made to explain the observed excess of soft photons,
see, e.g., [27, 28, 29, 3, 30, 31, 32, 33, 34]. Most models introduce an
additional source of soft-photon production. The authors of the latter
references point out that the DELPHI results suggest an additional mechanism
for soft-photon production due to non-perturbative QCD evolution and the
dynamics of hadronization. However, while the proposed processes may be
relevant when approaching the Low region they generally vanish as
$k_{\text{T}}\rightarrow 0$.

Presently, no agreement exists on the possible origin of the observed excess
in soft-photon-production. Clearly, this calls for a new effort in measuring
ultra-soft-photon production in hadronic collisions in a systematic way.

##### Paper by Barshay

In Ref. [27], S. Barshay argues that the approach to a transient phase of
coherent pion condensation could, through the modification of the dispersion
relation for pions in such a phase, induce an enhanced production of photons
in the low-momentum regime.

##### Papers by Shuryak

In refs. [28, 35], E. Shuryak discusses a model for a “pion liquid” that could
form in hadronic collisions. Here, charged pions undergo several random
collisions inside the medium. For photon frequencies that are small compared
to the lifetime of this liquid the usual Low-theorem result is recovered while
an enhancement is observed at somewhat larger frequencies where the in-medium
collisions can be resolved.

##### Papers by Lichard and Van Hove

In Ref. [29], P. Lichard and L. Van Hove discuss a model that goes back to
earlier work of Van Hove [36] where photons are emitted from supposedly rather
long-lived “globs” of very soft partons. As they argue, such blobs of “cold
quark-gluon plasma” could need much time to hadronize, and, during this
hadronization process, could produce photons at intermediate frequencies. A
more detailed analysis of this model is presented together with a discussion
of experimental data in a later paper [3].

##### Paper by Czyż and Florkowski

In Ref. [30], the authors investigate photon production from a boost-invariant
model for fragmenting color-flux tubes described as classical color strings
between quarks that are themselves created by Schwinger pair production from
the color fields. Since this fragmentation dynamics extends in time and space,
one finds an enhancement over the leading Low formula at intermediate photon
frequencies.

##### Paper by Botz, Haberl and Nachtmann

In Ref. [31], the authors propose synchroton radiation of quarks in hadron-
hadron collisions. When quarks and antiquarks in a hadron move through the QCD
vacuum, they are under the influence of chromoelectric and chromomagnetic
vacuum fields. The chromomagnetic Lorentz force causes deflections of the
electrically charged quarks and antiquarks leading to the emission of
synchroton radiation with photon energies below 300 MeV in the center-of-mass
system of the collision in addition of hadron bremsstrahlung.

##### Paper by Hatta and Ueda

In Ref. [32], the authors determine an analog of the inclusive cross section
for photon production in electron-positron annihilation in $\mathcal{N}=4$
supersymmetric Yang-Mills theory. The calculation is done both perturbatively
and, at leading order in the large-$N$ limit, from holography. They find a
contribution with similar kinematic dependence as bremsstrahlung photons, and
speculate that this might help to resolve the soft-photon puzzle. Since the
particle content of $\mathcal{N}=4$ supersymmetric Yang-Mills theory differs
substantially from the real world (it has many massless charged particles and
is non-confining), it is difficult to judge whether the found mechanism is
universal enough or not.

##### Paper by Wong

In Ref. [33], Wong constructs a two-dimensional model with the gauge group
$\text{SU}(3)\times\text{U}(1)$, and finds a way to map it to a scalar quantum
field theory which contains mesons, but also a massive photon. He then argues
that the non-equilibrium dynamics of this model, which features hadronization
through meson production in its strong sector, can describe anomalous soft-
photon-production.

##### Paper by Kharzeev and Loshaj

In Ref. [34], D. E. Kharzeev and F. Loshaj describe a mechanism for photon
production from the oscillating electromagnetic currents that arise during the
process of string fragmentation. When a pair of highly energetic quarks
separates, the QCD string between them pulls additional quark-antiquark pairs
from the vacuum which screen the original color fields but also generate
additional ones, leading to a multistaged process of particle production. This
results in oscillating electromagnetic currents, which in turn act as sources
for electromagnetic fields or photons. The constructed model leads to an
enhanced production of photons at intermediate frequencies with the energy
scale set by the QCD string tension, and the width of a neutral isoscalar
resonance. The latter is taken as a fit parameter by Kharzeev and Loshaj, and
they arrive thus at a model which describes the photon yields found by the
DELPHI Collaboration in hadronic events as a function of the jet momentum,
reasonably well.

##### Attempts from perturbative QCD

Useful information in the calculation of soft-photon spectra has been brought
by analyses in perturbative QCD with massless quarks. Specifically, Ma,
Sterman, and Venkata [37] computed a three-loop correction of order
$\alpha_{s}^{3}/\omega$ to the leading power form of Low’s theorem. Similarly,
other authors [38, 39] pointed out that the next-to-leading power term in the
soft expansion receives a one-loop correction which is logarithmic in the soft
momentum (i.e., schematically of order $\alpha_{s}\log(\omega)$), and thus
might be enhanced for small momenta. However, it still remains an open
question how these perturbative results can be linked to the non-perturbative
regime.

##### Conclusion from different theory attempts

Overall, none of the proposed approaches can convincingly explain the
experimentally observed excess of soft photon production. However, this soft-
photon puzzle might hint at non-perturbative QCD physics, specifically
hadronization. When the duration $\Delta\tau$ of the collision, i.e., the time
between the moment of collision and the formation of final-state momenta is
prolonged by the hadronization dynamics, the effective expansion parameter of
the soft expansion $\omega\Delta\tau$ increases which implies that the strict
regime of validity of the soft expansion is shifted to much smaller photon
momenta. Accordingly, if the considered momentum range is kept fixed,
subleading terms in the expansion might become sizeable, which would lead to
an apparent deviation from the soft theorem. Although the contribution of
these subleading terms near the Low divergence should be negligible, it would
be interesting to measure them. Such an estimate would give access to effects
that so far have not been quantified in soft photon spectra, such as the spin
and the recoil of the charge emitter, the role of collinear effects and the
dependence of the soft spectrum on the strong coupling constant.

###  2.2 Soft radiation and soft theorems

####  2.2.1 Low’s leading soft theorem: classical bremsstrahlung

The phenomenology of soft-photon radiation is closely related to the problem
of infrared divergences in QED and to the long-range nature of the
electromagnetic interaction, i.e., the vanishing mass of the photon as
dictated by the underlying Abelian gauge symmetry. In contradistinction to the
problem of ultraviolet (UV) divergences in the perturbative evaluation of
higher-order corrections, the IR problem has been addressed as early as in
1937 by Bloch and Nordsieck [40] on the level of cross sections for
transitions including the emission of arbitrary numbers of low-energy photons.
They showed that in the soft photon limit the total probability for an
electron transition is unaffected by the emission of an infinite number of
low-energy photons while the total energy emitted remains finite, thus
recovering the well-known result from classical electrodynamics. The modern
interpretation and solution to this IR problem can be attributed to Yennie,
Frautschi and Suura [41] and to Weinberg [42] who showed that by considering
the finite energy resolution of detectors — where an arbitrary number of “soft
photons” with energies below this resolution can escape undetected — the cross
sections for e.g., elastic scattering, in addition to radiative corrections to
all orders of perturbation theory, are IR finite.

From a more physical point of view, the infrared (IR) problems are due to the
long-range nature of the electromagnetic interaction and thus to the
inadequacy of naive Fock charged-particle states as the asymptotic states of
scattering theory. Rather, the true asymptotic state is a charged particle
surrounded by its own electromagnetic field. Formally, these asymptotic states
are given by so-called “infra-particle states”, i.e., states describing a
charged particle together with a photon-coherent state. As shown in [40, 43,
44, 45, 46], indeed when using the correct infra-particle states with the
usual QED scattering operator, all IR divergences cancel, and the
corresponding cross sections coincide with those obtained in the more
conventional Bloch-Nordsieck approach. For a recent review, including the
relation to asymptotic or large gauge symmetries, see [47].

Low’s soft theorem [2] states that the amplitude of emission or absorption of
very low-energy quanta in any given transition with charged particles can
always be expressed as a soft factor times the amplitude of the transition
without the emission or absorption of the very low-energy quanta. Low obtained
this result for spin-0 particles using the Ward identity, giving expressions
for the leading and next-to-leading terms in the multipole or soft expansion
of cross sections in Eq. (1). The extension for particles of spin 1/2 is due
to Burnett and Kroll [48]. The generalization for particles of arbitrary spin
has been given by Bell and Van Royen [49]. The modern treatment of Low’s
theorem is due to Weinberg [42], who demonstrated the universal behavior of
this soft factorization for all Abelian theories as well as gravity, and
showed the exact cancellation of IR divergences of cross sections between
unobserved real and virtual low-energy bosons in these theories to all orders
in perturbation theory. This section presents a comprehensive introduction to
Low’s theorem and the general features of classical soft-photon radiation.

Figure 4: A classical depiction of a single-particle transition experiment
where a charge $Q_{i}$ is prepared with initial momentum $p_{i}$ and goes out
with charge $Q_{f}=Q_{i}$ and final momentum $p_{f}$ after undergoing an
unspecified interaction in a localized region of space-time, illustrated by a
blob.

Before introducing Low’s theorem, let us consider the classic radiation
scenario of a single charged-particle transition like the one depicted in Fig.
4. There, a particle carrying charge $Q_{i}$ in units of the electron charge
$e$ is prepared with initial momentum $p_{i}$. After an interaction, which
does not need to be further specified, it goes out with final momentum $p_{f}$
and charge $Q_{f}=Q_{i}=Q$. The classical photon field induced by this charge
during the process is easily obtained from the solution to the Maxwell
equations, namely

$A_{\mu}(x)=\int
d^{4}yD_{\mu\nu}(x-y)J^{\nu}(y)=\int\frac{d^{4}k}{(2\pi)^{4}}e^{-ik\cdot
x}\bigg{\\{}-\frac{g_{\mu\nu}}{k^{2}+i0^{+}}+(1-\xi)\frac{k_{\mu}k_{\nu}}{(k^{2}+i0^{+})^{2}}\bigg{\\}}\tilde{J}^{\nu}(k)$
(2)

where $D_{\mu\nu}(x-y)$ is the photon propagator and
$J^{\mu}(x)\equiv\big{(}\rho(x),\bm{j}(x)\big{)}$ the classical current
density induced by the charged particle along its path $x_{\mu}(\tau)$ in the
transition

$J^{\mu}(x)=Qe\int^{\infty}_{-\infty}d\tau\dot{x}^{\mu}(\tau)\delta^{4}(x-x(\tau))\,.$
(3)

Here $\tau$ is the particle’s proper time, and $Q$ its electric charge in
units of $e$. In the last step in Eq. (2) we Fourier transformed to momentum
space, where $k^{\mu}$ is the momentum of the photon, $\mu=0,1,2,3$,
$k^{2}=k_{\mu}k^{\mu}$, $\xi$ the gauge fixing parameter, and the small
imaginary part $i0^{+}$ ensures that positive and negative energy modes of the
gauge field propagate forward and backward in time, respectively. We are
interested in computing the form of this photon field at low momentum $k$ or,
equivalently, at large distances $x$. At large $x$, the particle follows the
classical straight-line trajectory of Fig. 4 given by

$x^{\mu}(\tau)=\frac{p^{\mu}_{i}}{m}\tau\,\,\,\,\,\text{for}\,\,\,\,\tau<0\,,\,\,\,\,\,\,\,\,x^{\mu}(\tau)=\frac{p^{\mu}_{f}}{m}\tau\,\,\,\,\,\text{for}\,\,\,\,\tau>0\,,$
(4)

where $m$ is the mass of the charged particle. Using Eq. (4), the current at
large $x$ can now be easily obtained as

$J^{\mu}(x)\simeq\int^{\infty}_{0}d\tau\bigg{[}Q_{f}e\frac{p^{\mu}_{f}}{m}\bigg{]}\delta^{4}\Big{(}x-\frac{p_{f}}{m}\tau\Big{)}+\int^{0}_{-\infty}d\tau\bigg{[}Q_{i}e\frac{p^{\mu}_{i}}{m}\bigg{]}\delta^{4}\Big{(}x-\frac{p_{i}}{m}\tau\Big{)}\,,$
(5)

and its Fourier transform then reads

$\begin{split}\tilde{J}^{\mu}(k)&=\int d^{4}xe^{ik\cdot
x}J^{\mu}(x)=\int^{\infty}_{0}d\tau\bigg{[}Q_{f}e\frac{p^{\mu}_{f}}{m}\bigg{]}e^{i\frac{k\cdot
p_{f}}{m}\tau}+\int^{0}_{-\infty}d\tau\bigg{[}Q_{i}e\frac{p^{\mu}_{i}}{m}\bigg{]}e^{i\frac{k\cdot
p_{i}}{m}\tau}\\\ &=ie\bigg{\\{}Q_{f}\frac{p^{\mu}_{f}}{k\cdot
p_{f}}-Q_{i}\frac{p_{i}^{\mu}}{k\cdot p_{i}}\bigg{\\}}\,,\end{split}$ (6)

where in the last step we integrated in $\tau$ and neglected the boundary
terms at plus and minus infinity. When substituting now Eq. (6) into Eq. (2)
we see that the gauge-dependent terms — those proportional to $k_{\mu}k_{\nu}$
within the photon propagator in Eq. (2) — will vanish due to charge
conservation $\partial_{\mu}J^{\mu}(x)=0$, namely they will give rise to terms

$k_{\mu}\tilde{J}^{\mu}(k)=ie(Q_{f}-Q_{i})=0\,.$ (7)

Thus, plugging Eq. (6) into Eq. (2) and setting $Q_{f}=Q_{i}=Q$ yields

$A_{\mu}(x)=iQe\int\frac{d^{4}k}{(2\pi)^{4}}e^{-ik\cdot
x}\frac{-g_{\mu\nu}}{k^{2}+i0^{+}}\bigg{\\{}\frac{p^{\nu}_{f}}{k\cdot
p_{f}}-\frac{p_{i}^{\nu}}{k\cdot p_{i}}\bigg{\\}}\,.$ (8)

The integral in $k_{0}$ in this expression can easily be performed by using
complex analysis. Defining the $k_{0}$ integration contour for $(t>0)$ in the
lower half complex plane, enclosing the $k_{0}=\omega_{{k}}-i0^{+}$ pole,
where $\omega_{{k}}=|\bm{k}|$ is the frequency of an on-shell photon, simply
gives111We neglected the absorptive mode of the photon field for $t<0$
corresponding to the $k_{0}=-\omega_{{k}}+i\varepsilon$ pole, and the two
Lienard-Wiechert fields at large distances, corresponding to the Coulomb field
always present in the presence of the charge, as we are interested in
computing the probability of emitting an on-shell photon.

$A_{\mu}(x)=Qe\int\frac{d^{3}\bm{k}}{(2\pi)^{3}}e^{-i\omega_{k}t+i\bm{k}\cdot\bm{x}}\frac{-g_{\mu\nu}}{2\omega_{k}}\bigg{\\{}\frac{p_{f}^{\nu}}{k\cdot
p_{f}}-\frac{p_{i}^{\nu}}{k\cdot
p_{i}}\bigg{\\}}\,,\,\,\,\,\text{for}\,\,\,\,t>0\,.$ (9)

It remains to determine the amplitude of finding within this field an on-shell
photon mode of momentum $k=(\omega_{k},\bm{k})$ and polarization $\lambda$.
This can easily be done by noticing that the sum over physical photon
polarizations can be expressed as

$\sum_{\lambda=1,2}\varepsilon_{\mu}(\bm{k},\lambda)\varepsilon_{\nu}^{*}(\bm{k},\lambda)=-g_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{(k\cdot
n)^{2}}+\frac{k_{\mu}n_{\nu}+k_{\nu}n_{\mu}}{(k\cdot n)}\,,$ (10)

where $n_{\mu}=(1,\bm{0})$ is a normalized time-like vector. Substituting then
$-g_{\mu\nu}$ in Eq. (9) with Eq. (10) and noticing once again that the
$k_{\mu}k_{\nu}$, $k_{\mu}\eta_{\nu}$ and $k_{\nu}\eta_{\mu}$ terms can be
neglected because they vanish due to charge conservation, one can rewrite Eq.
(9) as

$A_{\mu}(x)=\int\frac{d^{3}\bm{k}}{(2\pi)^{3}}\frac{1}{\sqrt{2\omega_{k}}}\epsilon_{\mu}(\bm{k},\lambda)e^{-i\omega_{k}t+i\bm{k}\cdot\bm{x}}M(p_{f};k,\lambda\,|\,p_{i})\,,$
(11)

which is an infinite superposition of normalized free plane waves of a photon
with on-shell momentum $k=(\omega_{k},\bm{k})$ and polarization $\lambda$,

$A_{\mu}^{\text{free}}(x)=\frac{1}{\sqrt{2\omega_{k}}}\epsilon_{\mu}(\bm{k},\lambda)e^{-i\omega_{k}t+i\bm{k}\cdot\bm{x}}\,,$
(12)

with amplitude denoted $M(p_{f};k,\lambda\,|\,p_{i})$ given by

$M(p_{f};k,\lambda\,|\,p_{i})=\frac{Qe}{\sqrt{2\omega_{k}}}\epsilon_{\mu}^{*}(\bm{k},\lambda)\bigg{\\{}\frac{p_{f}^{\mu}}{k\cdot
p_{f}}-\frac{p_{i}^{\mu}}{k\cdot p_{i}}\bigg{\\}}\,.$ (13)

This is the well-known result from classical electrodynamics for the amplitude
of emitting a soft photon with momentum $k=(\omega_{k},\bm{k})$ and
polarization $\lambda$ from an accelerated charge that goes from initial
momentum $p_{i}$ to final momentum $p_{f}$ due to some interaction.

Figure 5: Feynman diagrams for the emission of a photon from a single charged-
particle transition. The initial state is prepared with a particle of charge
$Q_{i}$ with momentum $p_{i}$ and spin $s_{i}$. An arbitrary interaction is
denoted with a blob with amplitude $M_{fi}^{0}$ in the absence of emission of
a soft-photon. The experiment measures the final momentum $p_{f}$, spin
$s_{f}$ and charge $Q_{f}$ of the particle after the interaction, and the
momentum and polarization of the soft-photon emitted.

Let us compare now this classical result with the full prediction from quantum
electrodynamics, and introduce to this end Low’s theorem. In the following, we
summarize the treatment by Weinberg [50]. The Feynman diagrams for the soft-
photon emission in a single charged-particle transition are depicted in Fig.
5. For concreteness, we analyze the first diagram (a) in Fig. 5, where the
photon is emitted after the interaction, assuming the charged particle to be a
spin-1/2 particle with final charge $Q_{f}$, momentum $p_{f}$, spin $s_{f}$
and mass $m$.

According to the usual Feynman rules, the diagram (a) of Fig. 5 requires
changing the spinor $\bar{u}(p_{f},s_{f})$ for the outgoing charged fermion to

$Q_{f}e\bar{u}(p_{f},s_{f})(-i)\gamma^{\mu}\frac{i(\not{p}_{f}+\not{k}+m)}{(p_{f}+k)^{2}-m^{2}+i0^{+}}\frac{\epsilon_{\mu}^{*}(k,\lambda)}{\sqrt{2\omega_{k}}}.$
(14)

For $k=(\omega_{k},\bm{k})$ with $\omega_{k}=|\bm{k}|\rightarrow 0$ we can
neglect $\not{k}$ in the numerator and use

$\bar{u}(p_{f},s_{f})\gamma^{\mu}(\not{p}_{f}+m)=\bar{u}(p_{f},s_{f})\big{[}\\{\gamma^{\mu},\not{p}_{f}\\}-(\not{p}_{f}-m)\gamma^{\mu}\big{]}=\bar{u}(p_{f},s_{f})2p_{f}^{\mu}.$
(15)

Further, using the on-shell conditions $p_{f}^{2}=m^{2}$ and $k^{2}=0$, we
finally get

$M_{a}(p_{f},s_{f};k,\lambda|p_{i},s_{i})=\frac{Q_{f}e}{\sqrt{2\omega_{k}}}\frac{\epsilon^{*}(\bm{k},\lambda)\cdot
p_{f}}{p_{f}\cdot
k}M_{fi}^{0}(p_{f},s_{f};p_{i},s_{i})+\mathcal{O}(\omega_{k}^{0}).$ (16)

This shows that in the limit $|\bm{k}|\rightarrow 0$ for the soft photon
emitted from the _outgoing_ charged-particle line, the amplitude is given by
the product of the amplitude for the process without the emission of this soft
photon multiplied with a factor which is of order $\mathcal{O}(1/\omega_{k})$.
If the soft photon is emitted from the _incoming_ charged particle line
instead like in (b) of Fig. 5, in the corresponding fermion propagator we will
have $p_{i}-k$ and thus the sign changes compared to Eq. (16) which results in

$M_{b}(p_{f},s_{f};k,\lambda|p_{i},s_{i})=-\frac{Q_{i}e}{\sqrt{2\omega_{k}}}\frac{\epsilon^{*}(\bm{k},\lambda)\cdot
p_{i}}{p_{i}\cdot
k}M_{fi}^{0}(p_{f},s_{f};p_{i},s_{i})+\mathcal{O}(\omega_{k}^{0}).$ (17)

If the soft photon is radiated from an inner charged-particle line of a
diagram, i.e., from any internal line within the blob in Fig. 5, there is no
divergence for $\omega_{k}\rightarrow 0$, and thus for the _leading-order
soft-photon amplitude_ $\mathcal{O}(1/\omega_{k})$ the amplitude is given by
the sum of diagrams (a) and (b) of Fig. 5 in Eqs. (16) and (17),

$M(p_{f},s_{f};k,\lambda\,|\,p_{i},s_{i})=M_{a}+M_{b}=\frac{Qe}{\sqrt{2\omega_{k}}}\epsilon_{\mu}^{*}(\bm{k},\lambda)\bigg{\\{}\frac{p_{f}^{\mu}}{k\cdot
p_{f}}-\frac{p_{i}^{\mu}}{k\cdot
p_{i}}\bigg{\\}}M_{fi}^{0}(p_{f},s_{f}\,|\,p_{i},s_{i})+\mathcal{O}(\omega_{k}^{0}),$
(18)

where we used $Q_{f}=Q_{i}=Q$.

Equation (18) is Low’s theorem for the special case of the emission of a
single soft photon in a $1\to 1$ transition of a spin-1/2 particle. We see
that the soft factor does neither depend on the particular features of the
transition nor on the spin dimensions of the charged particle, and it
completely agrees with the classical amplitude of soft-photon emission we
obtained in Eq. (13). The only difference that appears is the presence of the
non-radiative transition amplitude $M_{fi}^{0}$, i.e., without the emission of
any photon. Indeed, in a classical calculation there is not such an amplitude
for a charged particle to go from $p_{i}$ to $p_{f}$, the final momentum
$p_{f}$ is completely determined by the knowledge of the initial condition
$p_{i}$ and the classical equations of motion.

Figure 6: Set of real soft-photon diagrams - one for each external charged
particle line of the transition - contributing to the total amplitude
$M_{fi}^{1}$ of emitting a soft real photon of momentum $k$ and polarization
$\lambda$ in any given $i\to f$ transition.

The previous calculation can easily be extended to the case in which the
transition involves not one but many charged particles, like the one depicted
in Fig. 6. Using the shorthand notation

$\begin{split}&M^{1}_{fi}\equiv
M(p_{1}^{f},s_{1}^{f},\ldots,p_{N_{f}}^{f},s_{N_{f}}^{f};k,\lambda\,|\,p_{1}^{i},s_{1}^{i},\ldots,p_{N_{i}}^{i},s_{N_{i}}^{i})\,,\\\
&M^{0}_{fi}\equiv
M(p_{1}^{f},s_{1}^{f},\ldots,p_{N_{f}}^{f},s_{N_{f}}^{f}\,|\,p_{1}^{i},s_{1}^{i},\ldots,p_{N_{i}}^{i},s_{N_{i}}^{i})\,,\end{split}$
(19)

for the amplitude of the $i\to f$ transition with and without the emission of
the real soft photon, Low’s theorem for arbitrary transitions reads

$M_{fi}^{1}=\frac{e}{\sqrt{2\omega_{k}}}\sum_{n=1}^{N_{i}+N_{f}}\eta_{n}Q_{n}\frac{\epsilon^{*}(\bm{k},\lambda)\cdot
p_{n}}{k\cdot p_{n}}M_{fi}^{0}+\mathcal{O}(\omega_{k}^{0})\,,$ (20)

where the sum over $n$ runs over initial and final particles and $\eta_{n}=+1$
or $\eta_{n}=-1$ if the charged particle is outgoing or incoming,
respectively.

The amplitude $M_{fi}^{1}$ for the production of a soft photon then factorizes
into $M_{fi}^{0}$, the Feynman amplitude for the production of charged
particles only (non-radiative) and the soft-photon production factor,

$\frac{e}{\sqrt{2\omega_{k}}}\sum_{n=1}^{N_{i}+N_{f}}\eta_{n}Q_{n}\frac{\epsilon^{*}(\bm{k},\lambda)\cdot
p_{n}}{k\cdot p_{n}}.$ (21)

It should be noted that the process can of course be accompanied by the
production of neutral particles which are in general not detected.

We remark that, from Eq. (20), the soft-photon production factor arises from
the interference of photon production from incoming and outgoing charged
particles. Importantly, one should recognize that in this soft-photon limit
all Feynman diagrams where the soft-photon line is connected to an internal
line corresponding to a virtual charged particle yield a non-diverging and,
therefore, negligible contribution to the soft-photon production cross
section. It is then not necessary to evaluate the contribution of all possible
internal loops to the cross section: Low’s leading term is “tree-level
correct”.

The photon production amplitude in Eq. (21) has a pole whenever $p\cdot k$
vanishes (in any reference frame). In the rest frame of the photon emitting
charged particle, this implies that the photon energy must be much smaller
than the mass of the charged particle to approach the Low limit.

Generally we have

$k\cdot p=\omega(E-|\bm{p}|\cos\theta),$ (22)

where $E$ is the energy of the charged particle, $|\bm{p}|$ the magnitude of
its momentum, and $\theta$ the opening angle between the photon and the
charged particle. Here, all quantities are evaluated either in the frame of
the radiating particle (particle frame) or in a frame with z-direction along
the direction of the colliding beams (beam frame).

To start with, we consider in the particle frame ultra-relativistic particles
($E\approx|\bm{p}|$) and not too large angles $\theta$. This results in

$k\cdot p\approx\omega E(1-\cos\theta)\approx\omega
E\frac{\theta^{2}}{2}=k_{T}E\frac{\theta}{2},$ (23)

explicitly exhibiting the connection between $k\cdot p$ and the photon
transverse momentum $k_{T}$. More generally, the photon energy $\omega$ can,
in the particle frame or in the beam frame, always be expressed as
$\omega=k_{T}\cosh y_{k}$ with photon rapidity $y_{k}$ and transverse momentum
$k_{T}$, both in the respective frame. Clearly, $k\cdot p$ vanishes for
$\omega\rightarrow 0$, leading to the “Low” pole. Note that strictly speaking
the expression $(E-|\bm{p}|\cos\theta)$ is always positive, since there exist
no massless charged particles. The above considerations demonstrate that the
divergence in the photon energy $\omega$ manifests itself as a divergence in
the transverse momentum $k_{T}$. This was first considered in [1] with the
conclusion that the applicability of the Low theorem might reach well beyond
the region originally investigated by Low.

In collider experiments, the beam frame is used. Then the Low divergence can
be measured as a divergence in the $k_{T}$ of the photons with respect to the
beam axis. This is the approach that will be taken in the ALICE 3 project. The
first steps in ALICE 3 will be to test for inclusive and exclusive reactions
the leading order predictions based on Low’s original result with precision
measurements of the associated charged particle distributions. The obvious
next steps will be to extend with ALICE 3 the measurements into higher photon
$k_{T}$ regions until $\pi^{0}$ decay background becomes prohibitive.

The generalization of Low’s theorem in Eq. (20) to the emission of many very
low-energy photons of momentum $\\{k_{r}\\}$ and polarization
$\\{\lambda_{r}\\}$, with $r=1,\ldots,N_{\gamma}$, is given by multiplying the
amplitude of the transition $i\to f$ by $N_{\gamma}$ soft factors like the one
in Eq. (20), one for each low-energy photon accompanying the transition,

$M^{N_{\gamma}}_{fi}=\prod_{r=1}^{N_{\gamma}}\bigg{\\{}\frac{\epsilon_{\mu}^{*}(\bm{k}_{r},\lambda_{r})}{\sqrt{2\omega_{k}}}e\sum_{n=1}^{N_{i}+N_{f}}\eta_{n}Q_{n}\frac{p_{n}^{\mu}}{k_{r}\cdot
p_{n}}\bigg{\\}}M^{0}_{fi}+\mathcal{O}(\omega_{k}^{0})\,,$ (24)

where we analogously abbreviated the radiative amplitude as

$M^{N_{\gamma}}_{fi}\equiv
M(p_{1}^{f},s_{1}^{f},\ldots,p_{N_{f}}^{f},s_{N_{f}}^{f},k_{1},\lambda_{1},\ldots,k_{N_{\gamma}},\lambda_{N_{\gamma}}\,|\,p_{1}^{i},s_{1}^{i},\ldots,p_{N_{i}}^{i},s_{N_{i}}^{i})\,.$
(25)

Equation (24) describes the emission of $N_{\gamma}$ very low-energy photons
as mutually independent processes and also insensitive to the particular
details of the interactions described by the non-radiative transition
amplitude $M^{0}_{fi}$. The soft factor in Eq. (20) continues to exhibit the
same features as the classical radiation induced by a system of $N_{i}$
charges coming in from the far past in uniform motion with momenta
$\\{p^{i}_{n}\\}$, interacting then in some complicated way that does not need
to be specified to go into a system of $N_{f}$ charges going out to the far
future in uniform motion as well with shifted momenta $\\{p^{f}_{n}\\}$. Note
that charge is conserved, but $N_{f}$ can be different from $N_{i}$. The
leading term in Low’s soft theorem then just reflects the fact that photons
with very low momenta $k\to 0$ radiated during the transition are sensitive
only to very large regions of the space-time $x^{\mu}$, $k_{\mu}x^{\mu}\sim
1$, such that the transition from the initial to the final state becomes for
them effectively instantaneous.

The relevant parameter for the soft approximation is the dimensionless
quantity $\omega_{T}/\Lambda$ where $\omega_{T}$ is the total energy radiated
in the form of very low-energy photons during the transition and $\Lambda$ the
infrared (IR) region of soft photons in which the leading term of the soft
expansion in Eq. (24) is strictly valid. To estimate $\Lambda$ we note that in
deriving Eq. (24) we assumed that the total energy $\omega_{T}$ emitted
through the $N_{\gamma}$ soft photons shall be negligible compared with the
energy of the charged particles in the initial and final states,

$\omega_{T}=\omega_{1}+\ldots+\omega_{N_{\gamma}}\ll E_{n}^{f,i}\,.$ (26)

According to Weinberg [42], a good strategy is to set $\Lambda$ equal to $m$,
the typical mass of the external charged particles and define the IR region of
soft photons as those satisfying

$\frac{\omega_{T}}{\Lambda}<1\,,\,\,\,\,\,\text{with}\,\,\,\,\,\Lambda\simeq
m\,.$ (27)

However, while the condition in Eq. (27) is necessary to take the soft-photon
limit in any of the external charged-particle lines of the transition it might
not be sufficient as the previous calculation ignored soft photons emitted
from the inner charged particle lines. Eventually if the typical time extent
$\Delta\tau$ of the interactions is sufficiently large — like in processes
involving high-multiplicity events long-lived virtual charged particles and/or
interactions with extended matter222The limit of applicability of Low’s
theorem in the Bethe-Heitler cross section in the presence of matter is the
well-known Landau-Pomeranchuk-Migdal (LPM) effect [51, 52]. The IR region
$\Lambda\simeq 1/\Delta\tau$ in this case can be of the order of a few MeV, as
measured by the SLAC E-146 collaboration [53]. The QCD analog of the LPM
effect plays a fundamental role in describing the physics of jets in ultra-
relativistic heavy-ion collisions. — the corresponding IR scale $\Lambda\simeq
1/\Delta\tau$ for which internal line emissions can be safely ignored can be
in fact much smaller than the condition in Eq. (27). As a consequence, the
choice of $\Lambda$ will generally be process dependent. For instance, if the
transition involves the production of an intermediate 10 GeV $\mathrm{J}/\psi$
— with typical decay width of $90~{}\mathrm{keV}$ and mean life-time of
$7\,\mathrm{pm}/c$ in the center of mass frame — the IR region of soft photons
emitted by the charged pair decays will be of the order
$\omega\lesssim\Lambda=1/\Delta\tau\sim 30$ keV.

We turn now to analyze the general features of the observed soft-photon
radiation, considering for simplicity the emission of a single soft photon in
Eq. (20). The generalization to many soft photons is straightforward. The
probability density $dI$ of finding one very-low energy photon of frequency
$\omega<\Lambda$ and momentum $k$ in a volume $d^{3}\bm{k}$ around $\bm{k}$
accompanying the transition in Fig. 6 is given by taking the absolute square
of the leading term in the soft-photon approximation in Eq. (20) and summing
over the two possible photon polarizations

$dI=\frac{d^{3}\bm{k}}{(2\pi)^{3}}\sum_{\lambda=\pm
1}|M_{fi}^{1}|^{2}=\frac{d^{3}\bm{k}}{(2\pi)^{3}}\frac{e^{2}}{2\omega_{k}}\sum_{\lambda=\pm
1}\epsilon_{\mu}^{*}(\bm{k},\lambda)\epsilon_{\nu}(\bm{k},\lambda)\sum_{n,m=1}^{N_{i}+N_{f}}\eta_{n}Q_{n}\eta_{m}Q_{m}\frac{p_{n}^{\mu}p_{m}^{\nu}}{k\cdot
p_{n}k\cdot p_{m}}|M_{fi}^{0}|^{2}\,,$ (28)

where $|M_{fi}^{0}|^{2}$ is the differential cross section for the transition
without the emission of the very low-energy photon. To discuss the angular
distribution of soft radiation we write
$k=(\omega_{k},\bm{k})=\omega_{k}(1,\hat{\bm{k}})$ and
$p_{n}=(E_{n},\bm{p}_{n})=E_{n}(1,\bm{\beta}_{n})$ and therefore

$k\cdot
p_{n}=\omega_{k}E_{n}(1-\bm{\beta}_{n}\cdot\hat{\bm{k}})\,,\,\,\,\,k\cdot
p_{m}=\omega_{k}E_{m}(1-\bm{\beta}_{m}\cdot\hat{\bm{k}})\,,$ (29)

where $\beta_{n}=(1,\bm{\beta}_{n})$ and $\beta_{m}=(1,\bm{\beta}_{m})$. We
can also re-write the polarization sum in Eq. (30) more conveniently as

$\sum_{\lambda=\pm
1}\epsilon^{*}_{\mu}(\bm{k},\lambda)\epsilon_{\nu}(\bm{k},\lambda)p^{\mu}_{n}p^{\nu}_{m}=\bm{p}_{n}\cdot\bm{p}_{m}-\frac{(\bm{k}\cdot\bm{p}_{n})(\bm{k}\cdot\bm{p}_{m})}{\omega^{2}}=E_{n}E_{m}\big{(}\bm{\beta}_{n}\times\hat{\bm{k}}\big{)}\cdot\big{(}\bm{\beta}_{m}\times\hat{\bm{k}}\big{)}\,.$
(30)

Noticing $d^{3}\bm{k}=\omega_{k}^{2}d\omega_{k}d\Omega_{\bm{k}}$, where
$\Omega_{\bm{k}}$ is the soft-photon solid angle, this leads to

$dI=\alpha\frac{d\omega_{k}}{\omega_{k}}\frac{d\Omega_{\bm{k}}}{(2\pi)^{2}}\sum_{n,m=1}^{N_{i}+N_{f}}\eta_{n}Q_{n}\eta_{m}Q_{m}\frac{\big{(}\bm{\beta}_{n}\times\hat{\bm{k}}\big{)}\cdot\big{(}\bm{\beta}_{m}\times\hat{\bm{k}}\big{)}}{(1-\bm{\beta}_{n}\cdot\hat{\bm{k}})(1-\bm{\beta}_{m}\cdot\hat{\bm{k}})}|M_{fi}^{0}|^{2}\,,$
(31)

where $\alpha=e^{2}/4\pi$. Due to the $d\omega_{k}/\omega_{k}$ term, the
differential spectrum is logarithmically divergent in the infrared limit when
$\omega_{k}\to 0$. In addition, the angular distribution of radiation has
peaks orthogonal to the direction of the charged particles in the far past and
far future if these are non-relativistic $|\bm{\beta}_{n}|\to 0$, or mostly
forward along a small cone of angle $\theta_{n}\sim 1/\gamma_{n}$, where
$\theta_{n}$ is the angle between $\bm{k}$ and $\bm{\beta}_{n}$, if these are
relativistic $|\bm{\beta}_{n}|\to 1$. A direct consequence of the particular
form of the angular distribution in Eq. (31) is the emergence of collinear
divergences when the emitting particle is ultrarelativistic and the dead cone
of soft-photon radiation when the emitting particle is massive. In order to
see these two related phenomena, we assume for simplicity that all charged
particles in the transition go in different directions, and therefore their
radiation is resolved angularly for each particle. We focus on the angular
distribution of soft photon radiation around a single charged particle in Eq.
(31), with charge $Q$, mass $m$ and energy $E$ and then

$dI=\alpha
Q^{2}\frac{d\omega_{k}}{\omega_{k}}\frac{d\Omega_{k}}{(2\pi)^{2}}\frac{|\bm{\beta}|^{2}\sin^{2}\theta}{(1-|\bm{\beta}|\cos\theta)^{2}}|M_{fi}^{0}|^{2}\,,$
(32)

where $\theta$ is the angle between the charged particle and the soft photon.
If a charged particle would be propagating along the light cone, i.e., with
$m=0$ and $|\bm{\beta}|=1$, in addition to the IR divergence $1/\omega_{k}$,
another divergence from the collinear emission at $\theta=0$ would occur, in
which soft photons are emitted in the same exact direction as the massless
charged particle. In contrast, if the charged particle is massive, there is
instead a suppression of soft-photon radiation around $\theta=0$, commonly
called the “dead cone”. The width of this dead cone can be obtained from the
denominator in Eq. (32) that can be rewritten as

$1-|\bm{\beta}|\cos\theta=(1-\cos\theta)+(1-|\bm{\beta}|)\cos\theta\,,$ (33)

with the first term corresponding to the massless charged-particle case, with
$|\bm{\beta}|=1$, and the second to a mass dependent correction. Assuming the
massive charged particle is relativistic, then $\gamma\gg 1$ and $\theta\ll
1$, and we can expand

$|\bm{\beta}|=\sqrt{1-\frac{1}{\gamma^{2}}}=1-\frac{1}{2\gamma^{2}}+\cdots\,,\,\,\,\cos\theta=1-\theta^{2}/2+\cdots\,,\,\,\,\sin\theta=\theta+\cdots,$
(34)

so that Eq. (32) becomes

$dI=\alpha
Q^{2}\frac{d\omega_{k}}{\omega_{k}}\frac{d\theta}{2\pi}\frac{4\theta^{3}}{\big{[}\theta^{2}+\theta_{0}^{2}\big{]}^{2}}|M_{fi}^{0}|^{2},$
(35)

with $\theta_{0}=1/\gamma$. This well-known suppression of small-angle soft
radiation in the dead cone, $\theta\lesssim\theta_{0}$, has a number of
phenomenological implications in heavy-ion collisions and finds immediate
application in the perturbative calculations of heavy-quark fragmentation
functions [54]. From the previous equation, small-angle soft radiation reaches
a maximum when

$\frac{d}{d\theta}\frac{\theta^{3}}{(\theta^{2}+\theta_{0}^{2})^{2}}\bigg{|}_{\theta=\theta_{\text{max}}}=\frac{\theta^{2}_{\text{max}}(3\theta_{0}^{2}-\theta^{2}_{\text{max}})}{(\theta^{2}_{\text{max}}+\theta_{0}^{2})^{3}}=0\,,$
(36)

this is, at an angle $\theta_{\text{max}}=\sqrt{3}\theta_{0}=\sqrt{3}/\gamma$
with respect to the emitting ultra-relativistic charged particle. The heavy-
to-light ratio of soft radiation off an ultra-relativistic heavy particle (HP)
and off a massless particle (LP) is often expressed as

$\frac{dI_{\text{HP}}}{dI_{\text{LP}}}=\frac{\bm{\beta}^{2}\sin^{2}\theta}{\big{[}\big{(}1-\cos\theta)+(1-|\bm{\beta}|)\cos\theta\big{]}^{2}}\frac{(1-\cos\theta)^{2}}{\sin^{2}\theta}\simeq\bigg{\\{}1+\frac{1}{\gamma^{2}\theta^{2}}\bigg{\\}}^{-2}=\bigg{\\{}1+\frac{\theta_{0}^{2}}{\theta^{2}}\bigg{\\}}^{-2}.$
(37)

Coming back to the differential spectrum in Eq. (31), it can be integrated
over all emission angles to obtain the differential rate of emission of a soft
photon with a frequency between $\omega_{k}$ and $\omega_{k}+d\omega_{k}$. To
perform this integration it is however convenient to go one step back and use
the form in Eq. (28) and the polarization sum in Eq. (10) to find

$dI=\frac{\alpha}{(2\pi)^{2}}\frac{d^{3}\bm{k}}{\omega_{k}}\sum_{n,m=1}^{N_{i}+N_{f}}\eta_{n}Q_{n}\eta_{m}Q_{m}\frac{-p_{n}\cdot
p_{m}}{(k\cdot p_{n})(k\cdot p_{m})}|M_{fi}^{0}|^{2}.$ (38)

We note that Eq. (38) constitutes a coherent sum over all emitting charged
particles and thus interference is implicitly included. This is demonstrated
in Fig. 7 that shows the intensity of photon emission of four charged
particles in pseudorapidity $\eta$ and azimuth $\phi$. Suppression of photon
emission along the direction of motion of the emitting particle is visible
(dead cone) as well as an interference pattern bridging the region between
charged particles.

Figure 7: Intensity of photon emission of four charged particles in
pseudorapidity $\eta$ and azimuth $\phi$ according to Eq. (38). The direction
of the particles is given by the dots, with blue and red signifying opposite
values of $\eta_{n}Q_{n}$. This example shows an increased emission between
particles of equal charge, and suppressed emission between particles of
opposite charge.

Integrating Eq. (38) in the soft-photon solid angle one then gets

$\frac{dI}{d\omega_{k}}=|M_{fi}^{0}|^{2}\frac{1}{\omega_{k}}\frac{\alpha}{2\pi}\sum_{n,m=1}^{N_{i}+N_{f}}\eta_{n}Q_{n}\eta_{m}Q_{m}\frac{1}{v_{nm}}\log\frac{1+v_{nm}}{1-v_{nm}}\,,\,\,\,\,\,\,\,\,{\rm
with}\ v_{nm}=\sqrt{1-\frac{m_{n}^{2}m_{m}^{2}}{(p_{n}\cdot p_{m})^{2}}},$
(39)

where $v_{nm}$ is the relative velocity of each charge $n$ in the rest frame
of another charge $m$. The above result is sometimes expressed in terms of
four-dimensional pairwise cusp angles $\gamma_{nm}$ between charged particles
as

$\frac{dI}{d\omega_{k}}=|M_{fi}^{0}|^{2}\frac{1}{\omega_{k}}\frac{\alpha}{\pi}\sum_{n,m=1}^{N_{i}+N_{f}}\eta_{n}Q_{n}\eta_{m}Q_{m}\gamma_{nm}\coth\gamma_{nm}\,,\,\,\,\,\,\,{\rm
with}\,\,\,\,\,\,\cosh\gamma_{nm}=\frac{p_{n}\cdot p_{m}}{m_{n}m_{m}}.$ (40)

The total probability for emission of a soft photon in the IR region
$\omega_{k}<\Lambda$ logarithmically diverges in the low-frequency limit as

$I=|M_{fi}^{0}|^{2}\frac{\alpha}{\pi}\sum_{n,m=1}^{N_{i}+N_{f}}\eta_{n}Q_{n}\eta_{m}Q_{m}\gamma_{nm}\coth\gamma_{nm}\int^{\Lambda}_{\lambda}\frac{d\omega_{k}}{\omega_{k}}\,,$
(41)

where $\lambda$ is the IR cut-off and hence the number of emitted photons in
the IR region is effectively infinite. However, the total emitted energy is
still finite since the quantity $\omega_{k}dI/d\omega_{k}$ is well-behaved in
the $\lambda\to 0$ limit. As it is well known these IR divergences always
appear in cross sections due to the emission of arbitrary numbers of soft real
photons and cancel exactly to all-orders in perturbation theory with the IR
divergences introduced by the existence also of arbitrary numbers of virtual
soft photons accompanying any transition.

####  2.2.2 Low’s leading soft theorem: symmetry considerations

As shown in the previous section using either classical or quantum mechanical
arguments, invariant S-matrix elements for the amplitude of emission of soft
photons during a transition relate to leading power in the $1/\omega$
expansion where $\omega$ is the emitted photon energy to S-matrix elements for
the transitions to take place without the emission of the soft photon. In this
section, we will show how this very general result can be traced back to very
fundamental aspects of the gauge interactions, ultimately related to Lorentz
and gauge invariance and the massless nature of gauge bosons.

Gauge interactions are typically described on a very fundamental level with
Quantum Field Theories (QFTs). In these formulations, both particles and
forces are described on equal footing through _local_ field operators in
space-time. The analogous mechanism of the classical “action at a distance” or
force between any two “particles” is hence described by the propagation of a
gauge field from one space-time point to another. In order to make this
propagation and the underlying theory fully compatible with the (special)
relativistic structure of space-time, i.e., consistent with the limiting speed
of light one introduces a microcausality principle according to which any two
local observables in nature must necessarily commute for space-like separated
coordinates,

$[\hat{O}_{1}(x),\hat{O}_{2}(y)]=0\quad\text{for}\quad(x-y)^{2}<0.$ (42)

These local observables can always be expressed in terms of local quantized
field operators. The latter are constructed by finding irreducible
representations of the proper orthochronous Poincaré group, i.e., the group
generated by space-time translations, Lorentz boosts, and spatial rotations
[55, 50]. Any irreducible representation can be constructed by first
considering the momentum eigenstates. Since $p^{2}=m^{2}$ is a Casimir
operator of the Poincaré group, it is one of the parameters characterizing any
irreducible representation. Accordingly, only representations with $m^{2}>0$
(leading to the description of massive particles) or $m^{2}=0$ (leading to
massless particles) are consistent with the microcausality principle for
interacting fields in Eq. (42).

Let us now discuss these two possible situations separately. For an
irreducible representation with $m^{2}>0$, any momentum eigenstate
$|\bm{p},\alpha\rangle$ can be obtained through a Lorentz boost of the zero-
momentum eigenstate $|\bm{p}=0,\sigma\rangle$. The possible degeneracy of this
state is then characterized by the action of the Poincaré subgroup, which
leaves $\bm{p}=0$ invariant. This is the spatial rotation group SO(3) or, in
quantum theory, its covering group SU(2). In other words,
$|\bm{p}=0,\sigma\rangle$ must span an irreducible representation of SU(2).
The corresponding Casimir operator $\bm{s}^{2}$ defines the spin $s$ of the
particle, with $s\in\\{0,1/2,1,3/2,\ldots\\}$, corresponding to the eigenvalue
$s(s+1)$ for $\bm{s}^{2}$ and $s_{z}\in\\{-s,-s+1,\ldots,s-1,s\\}$. This SU(2)
is known as the little group of the massive irreducible representation of the
Poincare group.

On the other hand, for $m=0$, one has instead
$p^{2}=(p^{0})^{2}-\bm{p}^{2}=0$. Here one can choose an arbitrary light-like
“standard momentum” vector $p=(\kappa,0,0,\kappa)$, and the little group in
this case is the Lorentz subgroup which leaves this light-like four-vector
invariant, i.e., either rotations around the $z$-axis or “null rotations”.
Altogether, one finds that the little group is isomorphic to the symmetry
group of the Euclidean plane, ISO(2). Now, given that the “translations” in
this group lead to continuous eigenvalues of the corresponding generators,
this would physically imply the existence of a continuous spin/polarization of
the massless particles described by the theory. Since this has never been
observed in nature, this implies that the null rotations must be represented
trivially so that only the rotations around the $z$ axis can lead to non-
trivial operations on the fields. Since for the $z$-rotations the possible
spin values are $\\{0,1/2,1,\ldots\\}$, then the representation of the
Poincaré group constructed in this way leads to the right representation of
the spatial rotations.

The crucial observation to be made is that the previously mentioned triviality
of the null rotations can only be achieved within a local QFT by assuming
gauge invariance in our theory. For instance, for Abelian gauge bosons with
$s=1$ in quantum electrodynamics (QED), the standard way to realize the
representation of the Poincaré group is to use a four-vector field
$A_{\mu}(x)$ representing the photon. The massless nature of this particle is
then expressed as $\Box A_{\mu}=0$, and the triviality of the null rotations
is then achieved by assuming that for any scalar field $\chi$, the transformed
field $A_{\mu}^{\prime}=A_{\mu}+\partial_{\mu}\chi$ describes the same
physical scenario and therefore the theory is gauge invariant. It immediately
follows that only two polarization states for the photon are physical
corresponding to the helicity values
$h=\bm{p}\cdot\bm{s}/|\bm{p}|\in\\{s,-s\\}$ (for photons $s=1$, i.e.,
$h\in\\{1,-1\\}$).

To describe the interactions of such a massless gauge field with the matter
fields, for instance, electrons and positrons in QED, the Lagrangian, and
equations of motion must be also gauge invariant. The local gauge
transformations for the matter fields read
$\psi\rightarrow\exp(-\mathrm{i}q\chi)\psi$,
$\bar{\psi}\rightarrow\exp(+\mathrm{i}q\chi)\bar{\psi}$. In QED the gauge
invariant field-strength tensor is given by

$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu},\quad\quad\quad\partial_{\mu}F^{\mu\nu}=J^{\nu},$
(43)

such that necessarily

$\partial_{\mu}J^{\mu}=\partial_{t}J^{0}+\bm{\nabla}\cdot\bm{J}=0,$ (44)

which implies the conservation of the electric charge,

$Q=\int_{\mathbb{R}^{3}}d^{3}xj^{0}(x).$ (45)

We already found how this classical charge conservation in Eq. (7) is
precisely a Ward identity ensuring the gauge invariance of all radiative
S-matrix elements and constraining their exact form in the soft-photon $k\to
0$ limit, namely Low’s soft theorem. The previous equivalence between soft
theorems and Ward identities was first noted by Weinberg in [56, 57] who
showed that conservation of charge in QED and Einstein’s equivalence principle
in gravity — the equality of gravitational and inertial mass — follow directly
from Lorentz invariance the existence of massless particles of spins $s=1$ and
$s=2$ called photons and gravitons and the pole structure of the scattering
amplitudes. In short, charge and gravitational mass are soft photon and
graviton coupling constants, hence if a process violates charge conservation
then the same process with inner-bremsstrahlung of an extra soft photon (or
graviton) would violate Low’s soft theorem and hence Lorentz invariance.
Indeed, as noted by Weinberg the S-matrix element for the emission of a photon
or graviton can be written as the product of a photon field polarization
“vector” $\epsilon^{\mu}$ or “tensor” $\epsilon^{\mu}\epsilon^{\nu}$,
respectively, with a Lorentz covariant vector or tensor amplitude. The
“vector” $\epsilon_{\mu}(\bm{q},\lambda)$ is not a four-vector but under a
Lorentz transformation goes to
$\epsilon_{\mu}(q,\lambda)\rightarrow\epsilon_{\mu}(q^{\prime},\lambda)+Cq^{\prime}_{\mu}$,
where the additional term, $\propto q^{\prime}$, corresponds to an additional
gauge transformation. Since observables must be Lorentz covariant and gauge
independent, the $S$-matrix element for the emission of a photon can only be
Lorentz invariant and gauge independent if this “gauge contribution” $\propto
q$ cancels. This means that the S-matrix element must vanish if any
$\epsilon_{\mu}$ gets replaced by the emitted photon momentum $q_{\mu}$. Using
Eq. (20), in the soft-photon limit this statement necessarily leads to Low’s
form of the soft theorem and to charge conservation in any given transition,

$\sum_{n=1}^{N_{i}+N_{f}}\eta_{n}Q_{n}=0.$ (46)

More generally, the deep implications of gauge invariance of S-matrix
amplitudes were first analyzed by Dirac in [58]. He showed around the same
time as Low that a manifestly invariant formulation of QED under general gauge
transformations prevents one from defining electron states as free field
solutions without including their interactions with the photon field. In this
treatment, fermions appear always accompanied by dynamical Coulomb fields
around them with free field solutions $\Psi(x)$ replaced then by dressed
states $\Psi(x)e^{C(x)}$, and each choice of $C(x)$ corresponding to a choice
of the electric field around the charged fermion. Building on previous work by
Dollard [59], Chung [60] and Kibble [46, 44, 45, 43], Faddeev and Kulish [61]
realized that when these Coulomb fields — encoded as coherent states of
infinitely many soft photons dressing the incoming and outgoing vacuum states
and encoding all pairwise interactions at asymptotic space-time distances in
the transition — are considered in scattering problems then real and virtual
IR divergences systematically cancel from all QED amplitudes. Conversely, in
the conventional treatment of scattering problems in field perturbation theory
it is assumed that the gauge interactions vanish at very large space-time
distances so that Fock vacuum states can be effectively represented by free
field solutions. As a result, real and virtual IR divergences in Low’s theorem
cancel only at the cross section level [40, 62, 63, 41, 42]. A comprehensive
introduction to the Faddeev-Kulish S-matrix can be found in the Supplement S4
of the classic textbook of Jauch and Rohrlich [63]. More modern discussions of
the Faddeev-Kulish S-matrix and soft theorems in the context of Soft Collinear
Effective Field Theory (SCET) and worldline formulations can be found in [64,
39, 65, 66].

Besides the intrinsic interest of the Faddeev-Kulish S-matrix formulation in
practical calculations for the treatment of the IR divergences in Low’s
theorem the Faddeev-Kulish picture of dressed vacuum coherent states is of
fundamental importance for the discussion of asymptotic symmetries and general
gauge invariance. In the context of gravity, Bondi, Van der Burg, Metzner, and
Sachs [67, 68] showed a long time ago that degenerate vacua having different
numbers of soft gravitons are connected by supertranslations of the infinite-
dimensional BMS group of asymptotic space-time symmetries. Armed with this
insight, Strominger and collaborators [69] showed that in QED and likely QCD
one can in principle construct the analog of the BMS group as the group of
large gauge transformations at every point on the celestial sphere at null
infinity. Large gauge transformations are those satisfying $A_{\mu}\rightarrow
A_{\mu}+\partial_{\mu}\chi$, with $\chi$ a function which does not die down at
spatial infinity. They showed how soft theorems in QED and gravity can be
directly related to the Ward identities for the infinite dimensional symmetry
group of large U(1) gauge transformations [70] and the BMS group [71],
respectively, approaching angle dependent constants defined through the
corresponding cusp anomalous dimensions in the celestial sphere [72]. We refer
the reader to [73] for a comprehensive introduction. The spontaneous breaking
of this asymptotic symmetry leads to an infinite number of degenerate vacua
characterized by different numbers of soft photons and gravitons allowing thus
a suggestive interpretation of soft photons and gravitons as Goldstone modes
of the theory for the transition between such different vacua. The charges of
the broken generators satisfy a Kac-Moody algebra on the two-dimensional
celestial sphere at null infinity.

A direct phenomenological implication of the existence of the Faddeev-Kulish
soft-photon clouds can be found in the equivalence between soft theorems and
gravitational [74] and electromagnetic [75] displacement (or memory) effects.
In gravity, the clock times of an array of detectors will differ before and
after the passage of a gravitational-wave pulse by a BMS supertranslation
[74]. An analogous experiment in QED to measure this electromagnetic memory
effect has been proposed by Susskind in [76]. Likewise, the striking
similarities between the spontaneous breaking of Poincaré invariance in highly
occupied systems of soft gravitons and gluons has been proposed as a possible
correspondence between the IR sector of quantum chromodynamics and gravity and
hence of the dynamics of gluon condensates in relativistic nucleus-nucleus
collisions and that of black holes [77].

The discussion above focused on computing and describing the main features of
the soft radiation encoded in the leading power term
$\mathcal{O}(\omega^{-1}_{k})$ of the multipole or soft expansion of cross
sections in Eq. (1). As previously emphasized, this term agrees with the
classical power spectrum of the transition and thus far has provided the main
theoretical reference for experimental analysis [7, 18, 19]. In the next
section we will address the problem of computing the sub-leading power
corrections of $\mathcal{O}(\omega^{0}_{k})$ in this soft expansion.

####  2.2.3 Expansion scheme: Leading order, Derivative operators versus
momentum shifts in next-to-leading order

The extension of the factorization picture described above at subleading power
in the soft expansion has been investigated since the time of Low in the 1950s
and the 1960s [78, 2, 48, 49] and has continued to attract interest until
recent years both in QCD phenomenology [79, 80, 81, 82, 83, 84, 85, 86, 87,
88, 89] and in more formal investigations on the structure of scattering
amplitudes [90, 69]. More specifically, at Next-to-Leading-Power (NLP) in the
soft expansion, Low, Burnett and Kroll (LBK) derived a tree-level theorem
which yields an $\omega_{k}^{0}$ correction to the power spectrum. In the
language used in modern literature, these theorems at the amplitude level are
dubbed _next-to-soft_ theorems. Let us consider first for simplicity the
emission of a single real soft photon during a transition $i\to f$ involving
two charged particles of momenta $p_{1}$ and $p_{2}$ either on the initial or
final state. The next-to-soft theorem generalizes Eq. (20) to

${M}^{1}_{fi}(p_{1},p_{2},k)=\left({\cal S}_{\text{LP}}+{\cal
S}_{\text{NLP}}\right){M}_{fi}^{0}(p_{1},p_{2}),$ (47)

where the leading power (LP) and next-to-leading power (NLP) soft factors are
given by

${\cal
S}_{\text{LP}}=e\sum_{n=1}^{2}Q_{n}\eta_{n}\,\frac{p_{n}\cdot\epsilon(k)}{p_{n}\cdot
k}~{},\qquad{\cal
S}_{\text{NLP}}=e\sum_{n=1}^{2}Q_{n}\eta_{n}\,\frac{ik_{\nu}J^{\mu\nu}\epsilon_{\mu}(k)}{p_{n}\cdot
k}~{}.$ (48)

Here, $J^{\mu\nu}$ represents the total angular momentum operator of the hard
emitting particle. More precisely, it is the sum of the orbital angular
momentum generator

$L^{\mu\nu}_{n}=i\left(p_{n}^{\mu}\frac{\partial}{\partial
p^{n}_{\nu}}-p_{n}^{\nu}\frac{\partial}{\partial p^{n}_{\mu}}\right)$ (49)

and the spin generator $S^{\mu\nu}_{n}$ of the Lorentz group. It is clear that
${\cal S}_{\text{NLP}}$ in Eq. (47) is not a multiplicative factor but an
operator acting on the non-radiative amplitude. For instance, in the case of
an amplitude with only two charged particles of spin $1/2$ in the initial
state, the spin generator is
$S^{\mu\nu}=\frac{i}{4}[\gamma^{\mu},\gamma^{\nu}]$ and Eq. (47) can be
written explicitly as

$\begin{split}& {M}_{fi}^{1}(p_{1},p_{2},k)\\\
&=e\sum_{n=1}^{2}\eta_{n}Q_{n}\,\frac{p_{n}\cdot\epsilon(k)}{p_{n}\cdot
k}\,\bar{v}(p_{2}){\cal
H}(p_{1},p_{2})u(p_{1})-\epsilon_{\mu}(k)e\sum_{n=1}^{2}\eta_{n}Q_{n}\,\bar{v}(p_{2})G^{\mu\nu}_{n}\,\frac{\partial{\cal
H}(p_{1},p_{2})}{\partial p_{n}^{\nu}}u(p_{1})\\\
&-e\eta_{1}Q_{1}\,\bar{v}(p_{2})\frac{ik_{\nu}S^{\mu\nu}\epsilon_{\mu}(k)}{p_{1}\cdot
k}{\cal H}(p_{1},p_{2})u(p_{1})-e\eta_{2}Q_{2}\,\bar{v}(p_{2}){\cal
H}(p_{1},p_{2})\frac{ik_{\nu}S^{\mu\nu}\epsilon_{\mu}(k)}{p_{2}\cdot
k}u(p_{1})~{},\end{split}$ (50)

where we defined

$G^{\mu\nu}_{n}\equiv g^{\mu\nu}-\frac{(2p_{n}-k)^{\mu}k^{\nu}}{2p_{n}\cdot
k}=g^{\mu\nu}-\frac{p_{n}^{\mu}k^{\nu}}{p_{n}\cdot k}+{\cal O}(k)~{}.$ (51)

Note in particular that derivatives contained in Eq. (49) act on the hard
subdiagram ${\cal H}(p_{1},p_{2})$ only, i.e., the non-radiative amplitude
${M}_{fi}^{0}(p_{1},p_{2})$ stripped off the external spinors. This is a
consequence of the way the soft expansion is performed. Specifically, Eq. (50)
can be derived by assuming that momentum conservation in the form
$p_{1}+p_{2}+k=0$ is imposed after expanding the full radiative process
${M}_{fi}^{1}(p_{1},p_{2},k)$ in powers of the soft momentum $k$. This is in
analogy with the original work of Low [2] for $2\to 2$ scattering of scalar
particles.

An alternative approach that has been recently proposed in the literature [91,
92, 93, 94] consists of expanding the radiative amplitude on the momentum
conservation surface by implicitly inserting a dependence over $k$ in the
momenta $p_{1}$ and $p_{2}$ (and consequently also in the spinors, if
necessary). Even in the scalar case, the resulting expression for the expanded
radiative amplitude is then different from the original result by Low (see
[94] for a detailed discussion on this point). However, one should note that
once momentum conservation is enforced, the difference between the two
expressions is only sub-subleading (i.e. NNLP) and thus beyond the accuracy of
Low’s theorem [95, 96]. More generally, the freedom to choose the functional
dependence of the non-radiative amplitude leads to many forms of Low’s
theorem, which are all formally equivalent after momentum conservation is
imposed up to NNLP corrections. However, one should bear in mind that in
photon spectra where the soft momentum $k$ is necessarily non-vanishing these
NNLP effects do bring numerical differences that might be sizable for
sufficiently large values of $\omega_{k}$. Therefore, some formulation might
be more versatile and efficient than others.

Despite the slightly intricate structure at the amplitude level, soft
emissions at NLP acquire a remarkably simple form for unpolarized cross
sections. In fact, as originally observed by Burnett and Kroll, by summing
over the polarizations and neglecting next-to-next-to-leading power (NNLP)
terms the analogue of Eq. (24) for the bremsstrahlung cross section of a
single soft photon to NLP in a transition $i\to f$ involving $N$ particles of
arbitrary spin is given by

$|{M}_{fi}^{1}(p_{1},\ldots,p_{N},k)|^{2}=e^{2}\sum_{n,m=1}^{N}(-\eta_{n}Q_{n}\eta_{m}Q_{m})\bigg{\\{}\frac{p_{n}\cdot
p_{m}}{p_{n}\cdot k\,p_{m}\cdot k}+\frac{(p_{n})_{\mu}}{p_{n}\cdot
k}G_{m}^{\mu\nu}\frac{\partial}{\partial
p_{m}^{\nu}}\bigg{\\}}|M_{fi}^{0}(p_{1},\ldots,p_{N})|^{2}~{}.$ (52)

Note in particular that both the spin and the orbital contributions of eq. 48
are represented by the term proportional to the derivative of the non-
radiative process due to the unpolarized nature of the cross section.

In analogy with Eq. (50), the NLP term in Eq. (52) contains derivatives acting
on a non-radiative amplitude $M_{fi}^{0}(p_{1},\ldots,p_{N})$ that is
evaluated for a configuration where momentum is not conserved since
$p_{1}+\ldots+p_{N}=-k\neq 0$. While this is not a problem in the strict $k\to
0$ limit, the finiteness of $k$ at NLP makes the amplitude on the r.h.s. of
Eq. (52) non-physical. Several possibilities to overcome this issue have been
recently put forward [91, 92, 93, 94, 39, 95, 97]. Reference [39] in
particular, building on previous studies in the context of gluon resummation
[98], proposed to rewrite the non-radiative amplitude in terms of shifted
kinematics as

$|M_{fi}^{1}(p_{1},\dots,p_{N},k)|^{2}=e^{2}\left(\sum_{n,m=1}^{N}-\eta_{n}Q_{n}\eta_{m}Q_{m}\frac{p_{n}\cdot
p_{m}}{p_{n}\cdot k\,p_{m}\cdot k}\right)|M_{fi}^{0}(p_{1}+\delta
p_{1},\dots,p_{N}+\delta p_{N})|^{2}~{},$ (53)

where the shift on the $\ell$-th momentum $p_{\ell}$ is defined at NLP as

$\delta
p_{\ell}^{\mu}=\left(\sum_{n,m=1}^{N}\eta_{n}Q_{n}\eta_{m}Q_{m}\frac{p_{n}\cdot
p_{m}}{p_{n}\cdot k\,p_{m}\cdot
k}\right)^{-1}\sum_{k=1}^{N}\left(\eta_{k}Q_{k}\eta_{\ell}Q_{\ell}\frac{(p_{k})_{\nu}G_{\ell}^{\nu\mu}}{p_{k}\cdot
k}\right)~{}.$ (54)

For the simple case of two external charged particle legs these become

$\delta p_{1}^{\mu}=\frac{1}{2}\left(-\frac{p_{2}\cdot k}{p_{1}\cdot
p_{2}}p_{1}^{\mu}+\frac{p_{1}\cdot k}{p_{1}\cdot
p_{2}}p_{2}^{\mu}-k^{\mu}\right)~{},\quad\delta
p_{2}^{\mu}=\frac{1}{2}\left(\frac{p_{2}\cdot k}{p_{1}\cdot
p_{2}}p_{1}^{\mu}-\frac{p_{1}\cdot k}{p_{1}\cdot
p_{2}}p_{2}^{\mu}-k^{\mu}\right)~{},$ (55)

from which one can readily see that $\delta p_{1}^{\mu}+\delta
p_{2}^{\mu}=-k^{\mu}$, thus restoring momentum conservation in the non-
radiative amplitude $M_{fi}^{0}(p_{1},\ldots,p_{N})$. Note that the cross
section in Eq. (53) contains the same soft factor as the LP approximation for
the cross section in Eq. (24), and therefore can be analogously implemented in
the photon spectrum.

At this point it should be noted that while the shifted kinematics restore
momentum conservation to all-order in the soft expansion, the on-shell
condition is fulfilled only at NLP, i.e. $(p_{i}+\delta
p_{i})^{2}=m^{2}_{i}+{\cal O}(k^{2})$, but it is violated at NNLP. Although
legitimate within the range of validity of Low’s theorem, this feature might
be problematic for numerical implementations. The solution to this problem
exploits the fact that it is possible to add spurious NNLP terms without
invalidating Low’s theorem. Specifically, one can modify the shifts $\delta
p_{i}$ so that they fulfil the on-shell condition and momentum conservation to
all orders in the soft expansion. A specific form for these modified shifts
has been recently provided [95] and reads

$\displaystyle\delta
p_{i}^{\mu}=AQ_{i}\sum_{j=1}^{N}\frac{\eta_{j}Q_{j}}{k\cdot
p_{j}}p_{j\nu}G^{\nu\mu}_{i}+\frac{1}{2}\frac{A^{2}Q^{2}_{i}{B}}{p_{i}\cdot
k}k^{\mu}~{},$ (56)

where

$\displaystyle
A=\frac{1}{\chi}\left(\sqrt{1-\frac{2\chi}{B}}-1\right)~{},\qquad
B=-\sum_{ij=1}^{N}\eta_{i}\eta_{j}Q_{i}Q_{j}\frac{p_{i}\cdot
p_{j}}{(p_{i}\cdot k)(p_{j}\cdot
k)}~{},\qquad\chi=\sum_{i=1}^{N}\frac{\eta_{i}Q^{2}_{i}}{p_{i}\cdot k}~{}.$
(57)

One can easily verify that $\delta p_{i}$ in Eq. (56) reduces to Eq. (54) by
expanding it in $k$ at NLP (thus fulfilling Eq. (53)) and that the momenta
$p_{i}+\delta p_{i}$ are both on-shell and fulfill momentum conservation to
all orders in $k$.

####  2.2.4 Loop corrections

The infrared structure of QED with massive particles to all orders in
perturbation theory is rather simple as can be seen by looking at the
structure of diagrams contributing to the exponent of the soft function [41].
One important consequence is that Low’s theorem at leading power (i.e.
$1/\omega$) remains unchanged at the loop level. Things are different for the
subleading term in the soft expansion and, when the massless limit is
considered also for the leading term. Let us consider the massive and massless
case separately, both in QED and QCD.

In the presence of massive particles, one expects the corresponding correction
to the NLP soft theorem to involve ratios of scales containing the masses of
the charged particles. In fact, these additional scales introduce an
additional quantity whose expression can be read in Eq. (4.41) of [99] (see
Appendix A of [97] for the results for the integrals). Remarkably, this
correction to the NLP soft theorem is just a multiplicative function (unlike
the factor ${\cal S}_{\text{NLP}}$ of Eq. (48)) which involves harmonic
polylogarithms. Most importantly, the aforementioned soft function is one-loop
exact and therefore the modified ${\cal S}_{\text{NLP}}$ holds to all orders
in the electromagnetic coupling constant.

The massless case is rather different. It is relevant both in QED when one can
approximate the charged particles to be massless and, perhaps more
importantly, in the partonic calculations of perturbative QCD. Specifically,
loop corrections affect the LP term ${\cal S}_{\text{LP}}$, both for soft
photons and soft gluons. However, while the LP soft-gluon current receives
corrections already at one-loop order in QCD [100], the LP factor for a soft-
photon emission is modified only starting from three loops [101, 102]. This
effect originates from the coupling of the soft photon to a loop of virtual
massless particles, which yields a correction to ${\cal S}_{\text{NLP}}$ of
order $\alpha_{s}^{3}/\omega$. This analysis has been recently generalized to
all orders for the simple case of quark-antiquark pair creation [37].

The NLP term ${\cal S}_{\text{NLP}}$ also receives loop corrections in the
massless case [103]. Specifically, at the one-loop level, corrections that are
non-analytic in the soft momentum modifies ${\cal S}_{\text{NLP}}$ by a term
of order $\alpha_{s}\log(\mu/p_{n}\cdot k)$, with $\mu$ the dimensional
regularization scale [39]. The origin of these corrections can be traced back
to the presence of other small scales, in this case the collinear one
$p_{n}\cdot k$ that prevents a naive Taylor expansion in the soft momentum
$k$. Diagrammatically, this effect is due to the soft photon coupling to a
subdiagram of collinear lines (hence called _radiative jet_) [79, 80, 104, 87,
105].

In conclusion, this recent body of work in perturbative calculations has
provided new information both in the infrared structure of quantum field
theories and in the phenomenological implications of power corrections to the
strict soft limit for hadronic physics whenever a perturbative treatment is
possible. However, it is neither clear how these results could provide a
solution to the soft-photon puzzle nor how they relate to the non-perturbative
regime. In fact, providing accurate theoretical predictions for photon spectra
in particular kinematical conditions remains an active field of research,
especially for processes with various hadronic final states at very high
rapidity. This aspect is of special relevance also for the planned future
measurements discussed in the next section.

###  2.3 Experimental consequences and opportunities at the LHC

The ALICE Collaboration intends to install a completely new silicon-based
detector (“ALICE 3”) during the fourth Long Shutdown of the Large Hadron
Collider that is presently scheduled for the years 2032–2033. ALICE 3 will
possess a unique pointing resolution over a large pseudorapidity range
$(-4<\eta<4)$, complemented by multiple subdetector systems for particle
identification including silicon time-of-flight layers, a ring-imaging
Cherenkov detector with high-resolution readout, a muon-identification system
and an electromagnetic calorimeter [106].

A dedicated soft-photon measurement at the LHC leveraging high collision rates
and improvements in detector technology could be key to resolving the soft-
photon puzzle. In particular, the energy dependence of the excess above the
inner bremsstrahlung expectation as calculated by Low’s leading-power
expression could point to the origin of the puzzle. The production of hadrons
in a non-diffractive proton-proton collision at high energies is sometimes
described as the fragmentation of two “beam jets” pointing in opposite
directions along the beam axis. The DELPHI experiment considered the
transverse momentum of soft photons with respect to the jet axis. This is
based on an argument by Gribov [1] about an extended range of applicability of
Low’s theorem. For non-diffractive proton-proton collisions at colliders, a
low transverse momentum relative to most produced particles would imply
measuring in the forward direction and at transverse momenta of $k_{T}\lesssim
m_{\pi}$ relative to the beam axis. The background of photons from decays of
neutral pions and $\eta$ mesons is a major experimental challenge. However,
for photon transverse momenta of $k_{T}\lesssim 5\,\mathrm{MeV}/c$ the
expected soft-photon signal becomes larger than the decay-photon background
owing to the rest mass of the neutral pion. Therefore, the photon transverse
momentum region $1\lesssim k_{T}\lesssim 5\,\mathrm{MeV}/c$ is a very
promising range for studying Low’s theorem at the LHC.

The clean photon identification as well as the good momentum and pointing
resolution at low momenta make the photon conversion method a natural choice
for the measurement of low $k_{T}$ photons. We consider this option in the
following. Reconstructing photons through conversion into $e^{+}e^{-}$ pairs
becomes possible for photon energies above $E_{\gamma,\,\mathrm{min}}\approx
50\,\mathrm{MeV}$. For lower energies, multiple scattering of the electrons
and positrons in the detector material makes the momentum reconstruction
difficult. In addition, Compton scattering becomes relevant for
$E_{\gamma}\lesssim 50\,\mathrm{MeV}/c$. The photon transverse momentum is
given by $k_{T,\gamma}=E_{\gamma}/\cosh\eta$. To measure low-$k_{T}$ photons
with $k_{T}\lesssim 5\,\mathrm{MeV}/c$ therefore requires measuring at large
pseudorapidities making use of the forward boost. The planned ALICE 3 detector
has a solenoid magnet creating a magnetic field in the direction of the beam
axis. To achieve reasonable momentum resolution for electrons and positrons
from photon conversions at forward pseudorapidities, these particles need to
be tracked in a dipole field which gives a much larger lever arm for the
momentum reconstruction. These basic considerations were followed in the
design of the Forward Conversion Tracker (FCT) of ALICE 3 as detailed in the
ALICE 3 letter of intent [106].

From the relation between Feynman amplitudes and cross sections and starting
from Eq. (38), the leading-power expression for the invariant yield of inner-
bremsstrahlung photons from, e.g., colliding $e^{+}e^{-}$ (initial state
radiation) and from final state $\mu^{+}\mu^{-}$ pairs can be calculated as

$\frac{dN_{\gamma}}{d^{2}\bm{k}_{\text{T}}\,d\eta\,d\phi}=\frac{\alpha}{(2\pi)^{2}}\int
d^{3}\bm{p}_{\mu^{+}}d^{3}\bm{p}_{\mu^{-}}\;\sum_{n,m=1}^{4}\eta_{n}Q_{n}\eta_{m}Q_{m}\frac{-(p_{n}\cdot
p_{m})}{(p_{n}\cdot k)(p_{m}\cdot
k)}\;\frac{dN_{\mu^{+}\mu^{-}}}{d^{3}\bm{p}_{\mu^{+}}d^{3}\bm{p}_{\mu^{-}}}\,.$
(58)

The $p_{n}$ and $\bm{p}_{n}$ denote the four and three momenta of the incoming
electron and positron, and the outgoing muons; $\bm{p}_{\mu^{+}}$ and
$\bm{p}_{\mu^{-}}$ are the three momentum of the muons, $k$ and $\bm{k}$
denote the four and three momentum of the photon
($E_{\gamma}=\omega_{k}=|\bm{k}|$). As defined previously (cf. Eq. (20)), the
factors $\eta_{n}$ are $\eta_{n}=1$ for the outgoing particles (the muons) and
$\eta_{n}=-1$ for the incoming particles (the $e^{+}$ and the $e^{-}$). Note
that this definition of $\eta$ differs from the one used by the DELPHI
collaboration, where the charge of the particles $Q_{n}$ in units of the
elementary charge $e$ is included in the definition of $\eta_{n}$ [7, 18]. The
total number of charged particles is $N$, i.e., the sum of the number of
charged particles in the initial state ($N_{i}$) and the final state
($N_{f}$). Here, $N_{i}=2$ (electrons) and $N_{f}=2$ (muons). The last factor
in the sum is the differential production rate of the produced muons. Eq. (58)
is expressed in terms of the photon production rate per muon.

The invariant yield of inner-bremsstrahlung photons for a reaction with two
incoming particles with fixed momenta and $N_{f}$ charged hadrons in the final
state, i.e., $N=N_{f}+2$, is given by

$\displaystyle\frac{dN_{\gamma}}{d^{2}\bm{k}_{\text{T}}\,d\eta\,d\phi}$
$\displaystyle=\frac{\alpha}{(2\pi)^{2}}\int d^{3}\bm{p}_{1}\ldots
d^{3}\bm{p}_{N_{f}}\;\sum_{n,m=1}^{N}\eta_{n}Q_{n}\eta_{m}Q_{m}\frac{-(p_{n}\cdot
p_{m})}{(p_{n}\cdot k)(p_{m}\cdot
k)}\;\frac{dN_{\mathrm{hadrons}}}{d^{3}\bm{p}_{1}\ldots d^{3}\bm{p}_{N_{f}}}$
(59) $\displaystyle\equiv\frac{-\alpha}{(2\pi)^{2}}\int d^{3}\bm{p}_{1}\ldots
d^{3}\bm{p}_{N_{f}}\;\left(\sum_{n=1}^{N}\eta_{n}Q_{n}\frac{p_{n}}{p_{n}\cdot
k}\right)^{2}\;\frac{dN_{\mathrm{hadrons}}}{d^{3}\bm{p}_{1}\ldots
d^{3}\bm{p}_{N_{f}}}\,,$ (60)

where, again, $k$ and $\bm{k}$ denote the four and three momentum of the
photon, $p_{n}$ and $\bm{p}_{n}$ denote the four and three momenta of the
incoming and outgoing charged particles; $d^{3}\bm{p}_{1}\ldots
d^{3}\bm{p}_{N_{f}}$ are the three momenta of the $N_{f}$ produced charged
hadrons; as above, $\eta=-1$ for the incoming particles and $\eta=+1$ for the
outgoing charged hadrons. Eqs. (58) and (59) were used by DELPHI [18] and
other experiments, and, in general, are evaluated on an event-by-event basis,
either from real data or event generators that were tuned to data. In terms of
computing time, Eq. (60) can be advantageous compared to Eq. (59). We note
that the expression of Eq. (60) given in parentheses squared is necessarily
negative. Through the factors $\eta_{n}$ and $Q_{n}$, the sum constitutes a
difference of four-momentum vectors, with its square always being negative.
This holds for an arbitrary number of charged particles due to the
conservation of charge and four-momentum.

Even with $e^{+},e^{-}$ beams, where the initial state radiation is much more
intense than with hadron beams, the contribution from initial state radiation
amounts to only about 1% in the kinematic range covered by the experiment [18]
and is thus negligible. In collider mode, photons from initial state radiation
are emitted at small polar angles with respect to the beam axis. The angular
distribution is suppressed in the regions $\theta<1/\gamma$ (dead cone) and
has a maximum around $\theta=\sqrt{3}/\gamma$, where $\gamma$ the Lorentz
factor of the beam, see Eq. (36). Thus, only a tiny fraction of the photons
are in the range of the central barrel.

Factoring out the energy of the charged particles in Eq. (60) replaces the
four-momentum $p_{i}$ by the vector $(1,\bm{\beta}_{i})$, i.e., the soft-
photon production depends explicitly on the velocities of the charged
particles (cf. Eq. (31)). A detector for a soft-photon measurement should
therefore be complemented with a detector for charged-particle identification,
e.g., a Ring Imaging Cherenkov Detector (RICH) extending the FCT design
described in the ALICE 3 letter of intent [106].

Identifying all charged particles contributing to soft-photon production in a
given acceptance can be challenging. Many of the previous soft-photon
experiments listed in Tab. 1 therefore used event generators to determine the
input for Eq. (60). For instance, the DELPHI experiment used JETSET 7.3, and
WA83, WA91 and WA102 used tuned versions of the FRITIOF event generator that
were validated with data. In the measurements by Goshaw et al. and by WA27, on
the other hand, the input for the calculation of the inner bremsstrahlung was
derived from measurement. In the paper by WA27, it is noted that the
calculation of the inner bremsstrahlung was found to be rather insensitive to
the structure of the hadronic cross section.

Experimental background of bremsstrahlung photons produced in the detector
material, i.e., external bremsstrahlung, poses a formidable challenge to soft-
photon measurements. This is illustrated in Fig. 8. The background stems
mostly from secondary electrons which are produced through photon conversions
in the detector material. In addition, background bremsstrahlung photons can
also be created by electrons and positrons from $\pi^{0}$ Dalitz decays. The
bremsstrahlung background produced in the detector material follows the same
$1/k_{\text{T}}$ form as that from internal bremsstrahlung. Since particles at
forward rapidities cross material such as the beam pipe at shallow angles,
they are exposed to an increased effective material budget. For example, a
beryllium beam pipe of $500\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in
thickness corresponds to a radiation length of 0.14% and 10% at $\eta=0$ and
$\eta=5$, respectively.

Figure 8: Illustration of background photons for a measurement of the inner-
bremsstrahlung signal. Shown is the contribution from decay photons, from
external bremsstrahlung created by electrons and positrons, from photon
conversions, from external bremsstrahlung created by primary electrons and
positrons (e.g. from $\pi^{0}$ Dalitz decays), and from decays of $\pi^{0}$’s
produced in secondary interactions with the detector material.

The background from external bremsstrahlung can be estimated analytically. We
consider a photon detector at forward pseudorapidity with a passive material
of $x/X_{0}$ in units of radiations lengths in front of the detector. Due to
the forward boost, the electrons and positrons from photon conversions have
large enough energies so that we use the approximation

$\frac{dN_{\gamma}^{\mathrm{ext.\,brems}}}{d\omega_{k}}=\frac{x}{X_{0}}\left(\frac{4}{3}\frac{1}{\omega_{k}}-\frac{4}{3}\frac{1}{E_{e}}+\frac{k}{E_{e}^{2}}\right)\approx\frac{4}{3}\frac{x}{X_{0}}\frac{1}{\omega_{k}}$
(61)

for the energy spectrum of the bremsstrahlung photons per electron. Here,
$E_{e}$ is the energy of the electron or positron and $\omega_{k}$ is the
photon energy. The mean emission angle of the radiated photon is
$\theta\approx m_{e}/E_{e}=\sqrt{3}/\gamma$, i.e., the photons are emitted in
a narrow cone around the electron or positron. A photon, e.g., from a
$\pi^{0}$ decay, converts in the passive material with a probability of
$7/9\,x/X_{0}$. For a pseudorapidity density
$dN_{\gamma}^{\mathrm{decay}}/d\eta$ of the decay photons, the transverse
momentum spectrum of the external bremsstrahlung can be estimated as

$\frac{dN_{\gamma}^{\mathrm{ext.\,brems}}}{dk_{\text{T}}d\eta}=\frac{28}{27}\frac{dN_{\gamma}^{\mathrm{decay}}}{d\eta}\left(\frac{x}{X_{0}}\right)^{2}\frac{1}{k_{\text{T}}}\,.$
(62)

Here, it is assumed that the electrons and positrons from photon conversions
traverse on average half of the passive material in front of the photon
detector. For proton-proton collisions at $\sqrt{s}=13.6\,\mathrm{TeV}$ we
choose $dN_{\gamma}^{\mathrm{decay}}/d\eta\approx 4$ [107]. For a passive
material corresponding to 10% of a radiation length, we then obtain a
background-photon spectrum of $dN/dk_{\text{T}}d\eta\approx
0.04/k_{\text{T}}$. Evaluating Eq. (60) for proton-proton collisions simulated
with PYTHIA, we obtain for the signal of the inner bremsstrahlung
$dN_{\gamma}^{\mathrm{inner\,brems}}/dk_{\text{T}}d\eta\approx
0.017/k_{\text{T}}$. This simple estimate shows that for 10% of a radiation
length of material in front of the photon detector, the signal-to-background
ratio is already less than 1/2. We thus consider this an upper limit of the
maximum allowable material budget.

An elegant experimental technique for measuring bremsstrahlung photons is to
bend the charged particles by which the photons are emitted out of the
acceptance of the photon detector. This approach was used by the SLAC E-146
collaboration for the measurement of bremsstrahlung created by electrons in
various targets [53]. This experiment disentangled the soft photon from the
radiating charged particle by bending the scattered electron by a dipole
magnet and detecting it in a high-resolution spectrometer, while the soft
photon is detected in beam direction by a BGO crystal calorimeter. A similar
idea was used by the WA91 and the WA102 experiments. An isolation cut around
the detected photons was applied to suppress external bremsstrahlung produced
in the photon converter. This cut made use of the fact that the charged
particle that emitted an inner bremsstrahlung photon was bent away by the
magnet dipole field of the experiment. In the case of the ALICE 3 setup with a
solenoidal field in front of the photon detector, such an isolation cut is
less straightforward.

An essential aspect of the DELPHI measurement of soft photons in
$e^{+}e^{-}\to n\,\mathrm{jets}$ was the rejection of jets which contained an
identified electron or positron track. DELPHI systematically studied the
effect of changing the electron selection procedure resulting in an increased
electron identification efficiency. No effect on the signal rate was found.
The hypothesis that an extra amount of external bremsstrahlung produced an
apparent excess was therefore rejected. For a measurement of ultra-soft
photons in proton-proton collisions at the LHC, it will also be necessary to
reject events containing an electron or positron track in the acceptance of
the forward photon detector. Another essential ingredient is likely to be a
specially shaped beam pipe to minimize the crossing of electrons and positrons
at shallow angles. These two aspects are studied in detail with the aid of
GEANT simulations in the following section.

In order to address the soft-photon puzzle at the LHC, measurements in
different reaction systems are required. Exclusive proton-proton reactions,
e.g. $\mathrm{pp}\to\mathrm{pp}+\text{J}/\psi(\to\mu^{+}\mu^{-}\gamma)$ or
$\mathrm{pp}\to\mathrm{pp}\pi^{+}\pi^{-}\gamma$ serve as a baseline where no
deviations between the data and the expected inner bremsstrahlung signal are
likely to be found. In addition, ultra-peripheral Pb–Pb collisions offer a
unique opportunity. Jet production in ultra-peripheral Pb–Pb collisions might
be useful to study the production of ultra-soft photons in jets similar to the
DELPHI measurement with little background from an underlying event. The
central measurement, however, would be in non-diffractive proton-proton
collisions. Studying in particular the dependence of soft-photon radiation on
the multiplicity of the event can provide insights into the source of a
possible excess.

####  2.3.1 Measuring the forward soft-photon spectrum with the Forward
Conversion Tracker of ALICE 3

The Forward Conversion Tracker planned for ALICE 3 is a detector that focuses
on the measurement of soft-photons in the forward direction via the tracking
of conversion $e^{+}e^{-}$ pairs. The vacuum vessel around the collision point
of ALICE 3 will have a larger radius than the beam pipe to house its vertex
locator, as described in the ALICE 3 letter of intent [106]. A slight
modification of this design allows for a conical window in the beam pipe, such
that particles emitted at pseudorapidity $4<\eta<5$ will pass through less
effective material in comparison to a straight beam pipe. The design including
such a window together with the Forward Conversion Tracker are shown in Fig.
9.

Figure 9: A schematic layout of the Forward Conversion Tracker (in magenta), a
proposed soft-photon detector for ALICE 3, with a conical beam pipe (in blue)
to allow for a reduced effective material path length of particles emitted in
the pseudorapidity range $4<\eta<5$ (in red).

The FCT consists of 11 circular silicon disks which subtend the pseudorapidity
region $4<\eta<5$. The beam pipe and the window were constructed of beryllium
with a thickness of $800$ $\mu$m, with a radius of the beam pipe of $1.8$ cm
and a radius of $3.7$ cm for the vacuum vessel. The inner radius of the disks
of the FCT was kept the same at $5$ cm, and the outer radius varied from $17$
to $19$ cm in steps of $1$ cm. With this design, simulation studies to
investigate the magnitude of the signal photons in comparison to background
photons were carried out. Two options are considered: the inclusion of a
particle identification detector behind the FCT (not included in the figure)
to distinguish electrons and veto the events which contain an
electron/positron, and the inclusion of the conical window in the beam pipe.
All combinations of including and/or excluding these two options make up the
four scenarios examined.

For these simulation studies, PYTHIA 8.3 was used to generate proton-proton
collisions at $\sqrt{s}=14\,\text{TeV}$. The internal bremsstrahlung spectrum
at $4<\eta<5$ was then generated according to Eq. (60) using a dedicated
generator that takes the particle stack from PYTHIA. The generated particles
(including soft photons) are then propagated through the detector setup with
GEANT4 using the FTFP_BERT_EMV and G4OpticalPhoton physics lists. The
simulated photon spectrum, separated in four different channels, measured at
the first layer of the FCT, is illustrated in Fig. 10 for the four distinct
scenarios.

Figure 10: The photon transverse momentum spectrum for the scenarios: straight
beam pipe without electron ID (top left), straight beam pipe including
electron ID (top right), conical window in the beam pipe without electron ID
(bottom left) and conical window in the beam pipe including electron ID
(bottom right). The photon spectrum is divided in four channels: Hadronic
decays and direct $\gamma$’s (black), which contains direct photons as well as
photons from short-lived particle decays (e.g. $\pi^{0}\to\gamma\gamma$ and
$\eta\to\gamma\gamma$), Secondary interactions/decays (red), which contains
photons from the decay of secondary particles (e.g.
$K^{0}_{s}\to\pi^{0}\pi^{0}\to 4\gamma$) and secondary interactions (e.g.
$\pi^{+}+\,\text{material}\to X+\pi^{0}\to{2\gamma}$), External bremsstrahlung
(green), photons which find their origin in charged particle interactions with
material radiating a bremsstrahlung photon, and Internal bremsstrahlung
(magenta), which is the predicted soft-photon spectrum by Low’s formula. The
$k_{T}$ of the photons is expressed with respect to the beam axis, which is
used as a proxy for the direction of the charged particles that emitted the
photons.

For the ALICE 3 letter of intent [106], a simulation was done which showed
that the pointing angle resolution of the FCT for photons was good enough such
that those with a pointing angle above 25 mrad could be rejected. This
rejection substantially reduced the contribution from the decay of secondary
particles. The contribution from external bremsstrahlung was reduced as well
but remains very significant in comparison to the signal photons as can be
seen in the top left figure in Fig. 10. From these figures it becomes clear
that the inclusion of a conical window substantially reduces the external
bremsstrahlung contribution, but the signal (inner bremsstrahlung) to
background (all the other contributions) ratio in the region of $1<k_{T}<5$
MeV$/c$ is not yet satisfactory. The signal-to-background ratio in the region
of interest increases to about unity with the inclusion of a veto on events
containing electrons/positrons, and increases even further when a conical
window in the beam pipe is included.

Exp. |  year |  $p_{\mathrm{beam}}$ or $\sqrt{s}$ |  photon $k_{T}$ |  $\gamma_{\mathrm{meas}}/\gamma_{\mathrm{brems}}$ |  $\gamma_{\mathrm{brems}}/\gamma_{\mathrm{bkg}}$ |  Ref.
---|---|---|---|---|---|---
$\pi^{+}\text{p}$ |  1979 |  10.5 GeV/$c$ |  $k_{T}<20\,\mathrm{MeV}/c$ |  $1.25\pm 0.25$ |  0.67 |  Goshaw et al. [10]
$\pi^{-}\text{p}$ CERN, WA91, OMEGA |  2002 |  280 GeV/$c$ |  $k_{T}<20\,\mathrm{MeV}/c$ |  |  |
($0.2<E_{\gamma}<1\,\mathrm{GeV}$) |  $5.3\pm 1.0$ |  0.47 |  Belogianni at al. [17] |  |  |
pp CERN, WA102, OMEGA |  2002 |  450 GeV/$c$ |  $k_{T}<20\,\mathrm{MeV}/c$ |  |  |
($0.2<E_{\gamma}<1\,\mathrm{GeV}$) |  $4.1\pm 0.8$ |  0.38 |  Belogianni at al. [6] |  |  |
$\text{e}^{+}\text{e}^{-}\to n\,\mathrm{jets}$ CERN, DELPHI |  2006 |  91 GeV ($\sqrt{s}$) |  $k_{T}<80\,\mathrm{MeV}/c$ |  |  |
($0.2<E_{\gamma}<1\,\mathrm{GeV}$) |  $4.0\pm 0.3\pm 1.0$ |  0.036–0.013 |  DELPHI [7, 19] |  |  |
pp ALICE 3 no w., no $\text{e}^{\pm}$ veto |  $>$2034 |  14 TeV ($\sqrt{s}$) |  $1<k_{T}<5\,\mathrm{MeV}/c$ |  |  |
($0.1<E_{\gamma}<0.4\,\mathrm{GeV}$) |  ? |  0.07–0.12 |  ALICE 3 [106] |  |  |
pp ALICE 3 $\text{e}^{\pm}$ veto |  $>$2034 |  14 TeV ($\sqrt{s}$) |  $1<k_{T}<5\,\mathrm{MeV}/c$ |  |  |
($0.1<E_{\gamma}<0.4\,\mathrm{GeV}$) |  ? |  0.3–1.8 |  ALICE 3 [106] |  |  |
pp ALICE 3 window |  $>$2034 |  14 TeV ($\sqrt{s}$) |  $1<k_{T}<5\,\mathrm{MeV}/c$ |  |  |
($0.1<E_{\gamma}<0.4\,\mathrm{GeV}$) |  ? |  0.2–0.3 |  ALICE 3 [106] |  |  |
pp ALICE 3 window, $\text{e}^{\pm}$ veto |  $>$2034 |  14 TeV ($\sqrt{s}$) |  $1<k_{T}<5\,\mathrm{MeV}/c$ |  |  |
($0.1<E_{\gamma}<0.4\,\mathrm{GeV}$) |  ? |  0.33–2.5 |  ALICE 3 [106] |  |  |

Table 2:  Overview of the predicted signal ($\gamma_{\mathrm{brems}}$) over
background ($\gamma_{\mathrm{bkg}}$) at various soft-photon measurements done
in the past, as well as for a potential measurement at ALICE 3. In these past
experiments the variable that was used to express photon transverse momentum
is $p_{T}$, but in this paper $k_{T}$ is used for consistency.

It is interesting to compare ALICE 3’s signal-to-background ratio with
previous experiments. This is done in Tab. 2. It should be noted that the
calculation of the signal over background of the previous experiments is based
on the expected inner bremsstrahlung signal, not on the actually measured
soft-photon signal. In the following, the calculation for the predicted signal
over background as depicted in Table 2 will be explained. The kinematic range
in question can be inferred from the same table. For the first entry, the
paper [10] mentions that the predicted number of direct photons was 671. In
the same kinematic range, 1006 background photons were measured, which was
inferred by deduction from the reported measured amount of direct photons
which was $840\pm 166$ accounting for $(45.5\pm 8.9)\%$ of the total amount of
photons measured. This then gives the signal over background of 0.67. For the
second entry, the paper [17] lists the predicted yields for various soft
photons in its first table, which gives a signal over background of 0.47. For
the third entry, the paper [6] lists the predicted yields for various soft
photons in a similar table, its first table, which gives a predicted signal
over background of 0.38. For the fourth entry, it is a little more cumbersome
to calculate the predicted signal over background, as no direct values for the
yields were given. Instead, this paper [7] presents the Real Data (RD), Monte
Carlo data (MC) and expected inner bremsstrahlung (Brems) in graphs that
depict the ratio RD/MC, the difference $\text{RD}-\text{MC}$ and Brems. The MC
data do not include the inner bremsstrahlung photons. To get to the expected
signal over background (i.e. Brems/MC), the RD can be expressed as
$\text{RD}=(\text{MC}+A\cdot\text{Brems})$, where $A$ is the enhancement of
the expected signal which is 4.0 in this case. Combining this with
$(\text{RD}-\text{MC})/\text{Brems}=A$ gives
$(\text{RD}/\text{MC}-1)/A=\text{Brems}/\text{MC}$, which in the specified
kinematic region is 0.01 to 0.036.

As shown in Tab. 2, the signal-to-background ratio of the previous fixed-
target experiments is significantly smaller than that of the DELPHI
measurement in $\text{e}^{+}\text{e}^{-}\to n\,\mathrm{jets}$. In the fixed-
target experiments, the soft-photon excess is mostly within a 10–20 mrad cone
around the beam axis [7]. The background of photons from $\pi^{0}$ and $\eta$
decays at the correspondingly small $k_{T}$ with respect to the beam axis is
small. This follows from the decreasing yield of $\pi^{0}$ and $\eta$ decay
photons at low $k_{T}$. For a vanishing $p_{T}$ of a $\pi^{0}$ or $\eta$, the
decay-photon yield actually peaks at $k_{T}=m/2$ (“Jacobian peak”), where $m$
is the mass of the parent particle; for a finite $p_{T}$, the decay-photon
yield still drops rapidly to low $k_{T}$. For the jet production at the Z pole
studied by DELPHI, the initial direction of the quark and the antiquark is
determined from the reconstructed jet axis. This can be done with an accuracy
of 50–60 mrad [7]. As a result, the $k_{T}$ interval where the Low photons
dominate is blurred and the decay-photon background becomes larger.

At the end of this section, we summarize the main results. The range
$1\lesssim k_{T}\lesssim 5$ MeV/$c$ is an optimal region for testing Low’s
theorem, since the decay photon background is relatively small in this range.
According to the relation $k_{T}=E_{\gamma}/\cosh\eta$, photons with very low
transverse momentum can be measured at large pseudorapidities $\eta$ with the
photon conversion method using the forward boost. The suppression of external
bremsstrahlung from electrons and positrons is the main challenge for
measuring the soft inner bremsstrahlung spectrum. If the ALICE 3 FCT is
complemented by a detector that allows to veto events with an electron or
positron in the acceptance of the FCT, a soft photon measurement at the LHC to
test Low’s theorem becomes possible. In addition, a window in the beam pipe to
reduce the effective material in front of the FCT greatly simplifies the
measurement. In our view, the ALICE 3 FCT is the best option for resolving the
long-standing soft-photon puzzle in the foreseeable future.

###  2.4 Specific examples: Internal bremsstrahlung in decays of
$\text{J}/\psi$

####  2.4.1 Estimates of rates and spectra for $\text{J}/\psi$ production in
ultraperipheral Pb–Pb collisions with STARlight

Standard Model predictions for radiative decays of spin-one bosons such as
$Z$, $\Upsilon$, and $\mathrm{J}/\psi$ have been discussed in [108, 109].
Analytical expressions for single-photon emission that are valid down to
arbitrary photon energies were derived therein. It is photon emission from
inner bremsstrahlung that determines the line shape of these resonances when
experimentally reconstructing them from dileptons only. The first observation
of the radiative decay of $\text{J}/\psi\rightarrow e^{+}e^{-}+\gamma$ with a
photon energy $E_{\gamma}>100$ MeV was reported in [110]. The relative
branching ratio of $BR(\text{J}/\psi\rightarrow
e^{+}e^{-}+\gamma)/BR(\text{J}/\psi\rightarrow e^{+}e^{-})=(14.7\pm 2.8)\%$ is
in good agreement with expectations.

In ultraperipheral collisions, the two ions interact via their cloud of
virtual photons. At LHC energies, the radial electric field lines of a lead
nucleus are Lorentz contracted along the beam direction and add coherently. As
a consequence, cross sections for photonuclear processes are enhanced
quadratically by the atomic number $Z^{2}\approx 6700$ relative to the
inelastic hadronic cross section. Ultraperipheral collisions exhibit a
particularly clean event topology, e.g., when a vector meson such as a
$\text{J}/\psi$ meson is produced with typically rather low transverse
momentum, the two decay leptons (electrons or muons) are oriented almost back-
to-back and perpendicular to the beam axis with no other signal in the
detector. The clean separation between electrons and muons in such collisions
has already been achieved by ALICE in the central barrel [111].

We have used the event generator STARlight [112] to calculate $\text{J}/\psi$
meson production and their non-radiative decay to dileptons in ultraperipheral
Pb–Pb collisions as measured with the present ALICE apparatus which covers a
pseudorapidity range $(-1<\eta<1)$, and with the future ALICE 3 apparatus.
Further, we applied Low’s theorem, Eq. (58), to dilepton production from the
non-radiative $\text{J}/\psi$ decay. We note that any photon emission by the
$\text{J}/\psi$ meson is forbidden by charge-conjugation invariance. This
makes this channel unique and might be used as a benchmark.

Figure 11: Soft-photon spectrum as predicted by Low’s formula from
$\text{J}/\psi$ decays into pairs of muons (red dashed curve) and electrons
(blue dashed curve). The experimental background estimated by detailed Monte
Carlo simulations with the present ALICE apparatus is shown by histograms.

The soft-photon spectrum is shown in Fig. 11 for decays to a pair of muons
(red dashed curve) and electrons (blue dashed curve). Here, the scale for the
softness of photons is set by the two-body decay and thus
$m_{\text{J}/\psi}/2$. The $1/k_{\text{T}}$ pole is clearly visible. The
spectrum continues to arbitrarily large photon energies, where Low’s theorem
is not applicable anymore. Compared to the dielectron channel, the photon
yield in the dimuon channel is only smaller by a factor of about three. This
results from the velocity $\beta$ explicitly occurring in Eq. (32) with the
muon (electron) velocity $\beta_{\mu}=0.9977$ ($\beta_{e}\approx 1$) for a
$\text{J}/\psi$ decaying at rest.

Experimental background from external bremsstrahlung of the decay daughters in
the detector material has been estimated by detailed Monte Carlo simulations
and is shown in Fig. 11 by the red (blue) histograms for muons (electrons).
While the large background for electrons seems prohibitive for such a
measurement in the central barrel, the muon channel is essentially background
free due to the much larger mass of the muon, which suppresses the background
component by a factor $1/m$ to the power of four to six. The lower limit in
the photon energy is set by the capability of measuring the photon via
conversion to electron-positron pairs in the detector material in front of the
TPC.

We estimate the total number of detected decays of $\text{J}/\psi$ mesons to
$\mu^{+}\mu^{-}$ together with a bremsstrahlung photon of
$E_{\gamma}>100\,\text{MeV}$ to be about 1000, making such a measurement
possible. The estimated number of about 1000 detected inner-bremsstrahlung
photons might be reduced by a factor of up to two in case of strong shadowing
effects in the gluon distribution of the lead nuclei, which has not been
considered.

| Pb–Pb |  pp | Kr–Kr | O–O
---|---|---|---|---
|  $\sqrt{s_{NN}}=5.52$ TeV |  $\sqrt{s_{NN}}=14$ TeV |  $\sqrt{s_{NN}}=6.46$ TeV |  $\sqrt{s_{NN}}=7$ TeV
$\sigma$ | 41.761 mb | 82.293 nb | 2.974 mb |  $27.49\,\mu$b
$\sigma\cdot\text{BR}(\text{e}^{+}\text{e}^{-})$ | 2.494 mb | 4.914 nb | 0.1776 mb | 1.641$\mu$b
$\sigma\cdot\text{BR}(\mu^{+}\mu^{-})$ | 2.489 mb | 4.905 nb | 0.1773 mb | 1.639 $\mu$b
L${}_{\text{\rm month}}$ ($\text{nb}^{-1}$) | 5.6 | 5.1$\cdot 10^{5}$ | 8.4$\cdot 10^{1}$ | 1.6$\cdot 10^{3}$
$\sigma_{\mu^{\pm},\>|\eta|<4}$ | 2.044 mb | 2.836 nb | 0.142 mb | 1.260 $\mu$b
$N_{\mu^{\pm},\>|\eta|<4}$ | $11.5\cdot 10^{6}$ | $1.45\cdot 10^{6}$ | $11.9\cdot 10^{6}$ | $2.02\cdot 10^{6}$
$N_{\gamma,\>E_{\gamma}>100\,{\rm MeV}}$ | $99\cdot 10^{3}$ | $12\cdot 10^{3}$ | $102\cdot 10^{3}$ | $17\cdot 10^{3}$
$N_{\gamma,\>{\rm detected}}$ | 94 | 12 | 97 | 16
$\sigma_{\mu^{\pm},\>4<\eta<5}$ | 0.014 mb | 0.07 nb |  |
$N_{\mu^{\pm},\>4<\eta<5}$ | $76\cdot 10^{3}$ | $35.7\cdot 10^{3}$ |  |
$N_{\gamma,\>E_{\gamma}>100\,{\rm MeV}}$ | 4.1 $\cdot 10^{3}$ | $1.9\cdot 10^{3}$ |  |
$N_{\gamma,\>{\rm detected}}$ | 4 | 2 |  |
Table 3: The total cross section calculated with STARlight, the number of
lepton pairs decaying into the pseudorapidity range $|\eta|<4$ (central
barrel) and $4<\eta<5$ (Forward Conversion Tracker) per leptonic
$\text{J}/\psi$ decay with the projected integrated luminosities as taken from
[106].

Estimates for the cross sections and the number of emitted dileptons and
emitted and detected photons in the pseudorapidity range $|\eta|<4$ (ALICE 3
central barrel) and $4<\eta<5$ (ALICE 3, Forward Conversion Tracker) are
summarized in Table 3. The reported values for the collision energies and
luminosities are taken from [106]. The acceptance of $\text{J}/\psi$ detection
in the leptonic decay channels is increased by a factor of 7 compared to the
present ALICE setup. However, the fraction of photon conversions is reduced by
about an order of magnitude due to the substantially decreased material budget
when compared to the present ALICE setup. We assume a lepton tracking
efficiency of 90%, the effective material budget to be 1% in units of
radiation lengths resulting in a photon conversion probability of 0.78%, and a
reconstructing efficiency for the electron-positron pair from photon
conversions of 15%, thus an overall probability to detect a dilepton pair
together with an inner-bremsstrahlung photon with energy
$E_{\gamma}>100\,\text{MeV}$ to be approximately 0.095%.

The total number of photons generated per leptonic $\text{J}/\psi$ decay in
the pseudorapidity range of $|\eta|<4$ and an energy range of
$E_{\gamma}\in(100\,\text{MeV},1.5\,\text{GeV})$ is respectively

$\begin{split}N_{\text{J}/\psi\rightarrow e^{+}e^{-}\gamma}&\approx
0.2284\>\text{per}\>\text{J}/\psi\rightarrow\text{e}^{+}\text{e}^{-}\>\text{decay}\\\
N_{\text{J}/\psi\rightarrow\mu^{+}\mu^{-}\gamma}&\approx
0.0861\>\text{per}\>\text{J}/\psi\rightarrow\mu^{+}\mu^{-}\>\text{decay}.\end{split}$
(63)

This number is larger than for $|\eta|<1$ since, due to the Lorentz boost at
larger pseudorapidity, more photons appear above an energy threshold of
$E_{\gamma}>100\,\text{MeV}$. In the forward direction at $4<\eta<5$, the
boost is even larger. However, the limited angular coverage with respect to
the flight direction of the emitting muon overcompensates this increase and
the fraction of photon per muon pair is reduced to 5.39%.

The background signal occurring during the interaction, where the
$\text{J}/\psi$ is being produced, has not been taken into account. The
dominant contribution is the direct-dimuon production in ultraperipheral
collisions, discussed in further detail in [113] in general and in [111] with
respect to the production of the $\text{J}/\psi$-vector meson. This continuum
of non-resonant dimuon production has an even larger cross section than the
$\text{J}/\psi$-to-dimuon-production cross section and might serve as a
complementary measurement for inner bremsstrahlung.

####  2.4.2 Results for the
$\text{p}\text{p}\to\text{p}\text{p}(\text{J}/\psi\to\mu^{+}\mu^{-}\gamma)$
reaction in a Regge inspired toy model

An interesting process that would permit a rather clean identification of
photons is exclusive production of $\text{J}/\psi$ mesons in pp collisions,

$\text{p}\text{p}\to\text{p}\text{p}(\text{J}/\psi\to\mu^{+}\mu^{-}\gamma)\,.$
(64)

Due to the negative charge parity of the $\text{J}/\psi$ meson the photon-
Pomeron fusion shown in Fig. 12 is the dominant production mechanism. The full
amplitude of the reaction (Eq. (64)) is a sum of the $\gamma\mathbb{P}$ and
$\mathbb{P}\gamma$ exchanges.333A similar process for the
$\text{p}\text{p}\to\text{p}\text{p}\gamma$ reaction within the tensor-Pomeron
approach is discussed in [114]. It is shown there that the photoproduction
contribution wins over the diffractive bremsstrahlung one [92] for the photon
rapidity $|y|<4$ and the absolute value of the transverse momentum of the
photon $k_{\text{T}}\gtrsim 10$ MeV. Exclusive diffractive bremsstrahlung of
one and two photons at forward rapidities in proton-proton collisions at the
LHC was discussed in [115].

Figure 12:  Photon-Pomeron ($\gamma$-$\mathbb{P}$) fusion mechanism for
exclusive $\text{J}/\psi$ production in pp collisions with the decay
$\text{J}/\psi\to\ell^{+}\ell^{-}\gamma$. There is also the diagram
corresponding to the interchange of the two final leptons
$\ell^{+}(p_{+})\leftrightarrow\ell^{-}(p_{-})$.

The $\text{p}\text{p}\to\text{p}\text{p}(\text{J}/\psi\to\mu^{+}\mu^{-})$
reaction can be treated in analogy to the
$\text{p}\text{p}\to\text{p}\text{p}(\phi\to\mu^{+}\mu^{-})$ reaction
discussed in Sec. III of [116] within the tensor-Pomeron approach. For
simplicity, we have used the $b$-type coupling in the
$\mathbb{P}\text{J}/\psi\text{J}/\psi$ vertex and the transition
$\gamma$-$\text{J}/\psi$ is made here through the vector-meson-dominance
model. The values of the intercept and slope parameters of the Pomeron
trajectory, the coupling constants, and the form-factor parameters are
determined from a comparison of the model to the experimental data for the
$\gamma\text{p}\to\text{J}/\psi\text{p}$ reaction measured by ZEUS, H1, LHCb,
and ALICE Collaborations (see Fig. 3 of [117] for reference). It has been
checked that the model supplemented by the absorptive corrections (pp
rescattering) describes the LHCb data [118] on
$\text{p}\text{p}\to\text{p}\text{p}\text{J}/\psi$ at centre-of-mass energy
$\sqrt{s}=13$ TeV. The ratio of full and Born cross sections (the gap survival
factor) is about 0.85–0.9. In the case of Eq. (64) a similar effect can be
expected.

For the reaction
$\text{p}\text{p}\to\text{p}\text{p}(\text{J}/\psi\to\mu^{+}\mu^{-}\gamma)$,
we have ${\cal M}^{(2\to 5)}=(\epsilon^{\mu}(k))^{*}{\cal M}_{\mu}$, where
$\epsilon$ is the polarization vector of the photon and ${\cal M}_{\mu}={\cal
M}^{(\gamma\mathbb{P})}_{\mu}+{\cal M}^{(\mathbb{P}\gamma)}_{\mu}$. The
amplitude ${\cal M}^{(\gamma\mathbb{P})}_{\mu}$ is obtained as in Eq. (2.7) of
[116] but with $i\Gamma^{(\phi KK)}_{\kappa}(p_{3},p_{4})$ replaced by

$eg_{\text{J}/\psi\mu^{+}\mu^{-}}\bar{u}(p_{-})\left(\frac{2p_{-\mu}+\gamma_{\mu}\\!\\!\not\\!{k}}{2p_{-}\cdot
k}\gamma_{\kappa}-\gamma_{\kappa}\frac{2p_{+\mu}+\not\\!{k}\gamma_{\mu}}{2p_{+}\cdot
k}\right)v(p_{+})\,.$ (65)

The coupling parameter $g_{\text{J}/\psi\mu^{+}\mu^{-}}=8.197\times 10^{-3}$
is estimated from the decay rate $\text{J}/\psi\to\mu^{+}\mu^{-}$ (neglecting
radiative corrections). For the $\mathbb{P}\gamma$-exchange we have the same
structure with the replacements Eq. (2.23) of [116]. The
$\text{p}\text{p}\to\text{p}\text{p}(\text{J}/\psi\to\mu^{+}\mu^{-}\gamma)$
reaction is calculated within exact $2\to 5$ kinematics using the event
generator GenEx [119].

In the following we shall compare the full decay approach (Eq. (65)) to the
soft-photon approximation (SPA) where we keep only the pole terms
$\propto\omega^{-1}$ in the radiative amplitude. The ${\cal M}_{\mu}$
amplitude is then expressed by the amplitude without radiation ${\cal
M}^{(\text{p}\text{p}\to\text{p}\text{p}(\text{J}/\psi\to\mu^{+}\mu^{-}))}$
times the photon emission factor

${\cal M}_{\mu}\propto{\cal M}^{(2\to 4)}\;e\left(\frac{p_{-\mu}}{p_{-}\cdot
k}-\frac{p_{+\mu}}{p_{+}\cdot k}\right)\,.$ (66)

In Tab. 4 we have collected integrated cross sections for the exclusive
reactions $\text{p}\text{p}\to\text{p}\text{p}\mu^{+}\mu^{-}$ and
$\text{p}\text{p}\to\text{p}\text{p}\mu^{+}\mu^{-}\gamma$ calculated for
$\sqrt{s}=13$ TeV with different acceptance cuts. Both photons and muons cover
the same pseudorapidity range, $k_{\text{T}}$ and $E_{\gamma}$ are the
absolute value of the transverse momentum and the energy of the photon in the
overall c.m. system, respectively. The absorptive corrections were not
included in the calculations. We show the results for the SPA (Eq. (66)) and
complete decay (Eq. (65)). Note that the cut on $k_{\text{T}}$ significantly
reduces the cross section compared to that of the cut on $E_{\gamma}$,
especially at forward photon rapidities. In Fig. 13, we show the distributions
for the $\text{p}\text{p}\to\text{p}\text{p}\mu^{+}\mu^{-}\gamma$ reaction for
the complete result and SPA (leading term). One can see from the top panels
that both results are very close to each other. The deviation of the SPA from
the complete result are up to around 5% in the kinematic range considered
($3.5<\eta<5$, $k_{\text{T}}<0.06$ GeV, $E_{\gamma}<2$ GeV). For soft-photon
emission, the leading term dominates. The other terms have no singularity for