MS-TP-23-47

TUM-HEP-1486-23

Soft-photon spectra and the LBK theorem

R. Balsach♭,(a), D. Bonocore♯,(b), A. Kulesza♭,(c)

♭ Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster,
Wilhelm-Klemm-Straße 9, D-48149 Münster, Germany

♯ Technical University of Munich, TUM School of Natural Sciences,

Physics Department T31, James-Franck-Straße 1, D-85748, Garching, Germany

The study of next-to-leading-power (NLP) corrections in soft emissions
continues to attract interest both in QCD and in QED. Soft-photon spectra in
particular provide a clean case-study for the experimental verification of the
Low-Burnett-Kroll (LBK) theorem. In this paper we study the consistency of the
LBK theorem in the context of an ambiguity arising from momentum-conservation
constraints in the computation of non-radiative amplitudes. We clarify that
this ambiguity leads to various possible formulations of the LBK theorem,
which are all equivalent up to power-suppressed effects (i.e. beyond the
formal accuracy of the LBK theorem). We also propose a new formulation of the
LBK theorem with a modified shifted kinematics which facilitates the numerical
computation of non-radiative amplitudes with publicly available tools.
Furthermore, we present numerical results for soft-photon spectra in the
associated production of a muon pair with a photon, both in $e^{+}e^{-}$
annihilation and proton-proton collisions.

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###### Contents

  1. 1 Introduction
  2. 2 LBK theorem and shifted kinematics
    1. 2.1 Traditional LBK formulation
    2. 2.2 Non-radiative amplitude and unphysical momenta
    3. 2.3 From derivatives to shifts
    4. 2.4 Modified shifted kinematics
  3. 3 Numerical predictions for $e^{+}e^{-}\to\mu^{+}\mu^{-}\gamma$ and $pp\to\mu^{+}\mu^{-}\gamma$
  4. 4 Conclusions
  5. A Case study: $2\to 2$ scattering
    1. A.1 LBK theorem for $\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}\gamma$
    2. A.2 Comparison between LNS and the original work of Low
    3. A.3 Comparison between LNS and other literature
  6. B LBK invariance under momenta transformation
  7. C Calculation of the modified shifts

##  1 Introduction

Perturbative calculations are the cornerstone of theoretical predictions for
high energy physics experiments. The expansion in the coupling constant is
arguably the most important example, with the expansion terms denoted as
leading order (LO), next-to-leading order (NLO), and so forth. For processes
involving several scales, a larger number of dimensionless parameters can be
small in particular kinematic limits, hence other expansions are possible. A
case that has received substantial attention is the study of power corrections
to the strict soft and/or collinear limit, whose expansion terms are
conventionally denoted as leading power (LP), next-to-leading power (NLP),
etc.

The theoretical foundations of NLP emissions date back to the theorems of Low,
Burnett and Kroll (LBK) [1, 2] (see also [3, 4]), which continue to be
reformulated and generalized also in the recent years111Soft theorems are an
active field of research also at LP, see e.g. [5, 6, 7, 8, 9, 10, 11]. [12,
13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31,
32, 33, 34, 35, 36, 37]. In particle phenomenology, these subleading effects
have mainly attracted attention due to their potential relevance for QCD
resummation222 NLP effects are also relevant for the numerical stability of
differential NNLO calculations, both in QCD [38, 39] and in QED [40, 41]..
Indeed, it is well-known that infrared divergences due to unresolved soft and
collinear radiation yield logarithms in the cross-section that become large
when approaching some kinematic threshold, thus spoiling the predictive power
of finite-order perturbation theory. The goal of the traditional (i.e. LP)
resummation program is to reorganize the towers of these logarithms at a given
logarithmic accuracy to all-orders in perturbation theory. In this context,
subleading corrections due to emissions of gluons (and quarks) give rise to
NLP logarithms which, although power-suppressed in the threshold limit, could
give significant contribution to the cross-section. In the last decade, a
considerable effort has been invested in this direction [42, 43, 44, 45, 46,
47, 48, 49, 50].

The soft limit in the photon bremsstrahlung [51, 52, 53, 54, 55] provides
another probe of NLP effects. In this case, the study of the photon spectrum
gives direct access to the individual terms of NLP soft theorems, unlike the
QCD resummation case, where one is blind to the energy of the undetected gluon
since its momentum is integrated over the whole phase space. In fact, although
the LBK theorem is very old and the conditions that ensure the soft limit for
a given process are known in terms of a well-defined hierarchy of scales, to
the best of our knowledge, no study in the literature has studied numerically
what is the resolution in energy and momentum of a soft photon that one has to
reach in order for NLP effects to be measurable.

This question is not a mere theoretical exercise. Soft-photon spectra in
hadronic decays have been puzzling physicists for years. The discrepancy
between the LP prediction and the observed yields of photons produced together
with hadrons is outstanding and the results of the measurements remain not
understood at present [56, 57, 58, 59, 60, 61, 62, 63, 64]. Moreover, there
are plans for an upgrade of the ALICE detector at the Large Hadron Collider
that would enable the possibility of scrutinizing photons at ultra-soft
energies [65, 66]. In order to shed light on these discrepancies and correctly
interpret data from future measurements, it is therefore of the utmost
importance to have reliable theoretical predictions, including also NLP
corrections as first proposed in this context in [28]. With this long-term
goal in mind, in this paper we study the tree-level form of the LBK theorem
for the production of a photon in association with a $\mu^{+}\mu^{-}$ pair in
$e^{+}e^{-}$ and $pp$ collisions.

To analyse the soft-photon spectrum, one has to evaluate the expressions given
by the LBK theorem. In fact, several issues must be addressed, both
analytically and numerically. Most notably, as it has been already pointed out
in [28], the traditional form of the theorem expressed through derivatives of
the non-radiative amplitude is not optimal. Indeed, the non-radiative
amplitude depends on a set of unphysical momenta that violates momentum
conservation when the soft-photon momentum $k\neq 0$. This is problematic
since, by definition, photon spectra are calculated for a non-vanishing
momentum $k$. To overcome this issue, the strategy proposed first for two
massless legs in [27] 333See also [67] and the recent [68]. and then
generalized in [28] for an arbitrary number of (massless or massive) legs,
appears promising. The strategy relies on removing the derivatives of the non-
radiative amplitude by computing such amplitude on momenta which are slightly
shifted in value. Remarkably, the sum of these momenta shifts is equal to the
soft-photon momentum, so that momentum conservation is restored. The price to
pay for this trick is that the shifted momenta do not fulfill the on-shell
conditions. This issue prevents the calculation of the non-radiative amplitude
with most of the available public tools which can be used for the numerical
evaluation of matrix elements. It is one of the goals of this paper to
explicitly show how the momenta of the external particle can be kept on-shell
by proposing a modified version of the shifts that are equivalent to the ones
discussed in [28] up to NNLP corrections.

The observation of the dependence of the amplitude on non-physical momenta at
NLP is an old one, and it was first discussed by Burnett and Kroll [2]. More
recent and detailed discussions on this aspect can be found in [24] (see also
[31]). Despite the long history and the large body of papers which studied,
reformulated and generalized the LBK theorem, the issue of having non-physical
momenta in the non-radiative amplitude led some authors [35, 36, 37] to
question the validity of all known formulations of the theorem and to propose
a modified version. In this paper, we argue that such criticism has no valid
foundation by explicitly showing that the formulation in [35] is equivalent at
NLP to the ones previously derived in the literature. More generally, we prove
the invariance of the LBK theorem at NLP under a specific transformation of
the non-radiative amplitude, which leads to many equivalent formulations that
differ by NNLP corrections. As a consequence, the ambiguities contained in the
LBK theorem due to violation of momentum conservation in the non-radiative
amplitude are power-suppressed beyond the formal validity of the theorem.

Besides the issue of evaluating the non-radiative processes using physical on-
shell momenta, other technical aspects must be addressed in a numerical
implementation. In fact, the integration over phase space becomes unstable in
the soft limit. To overcome these instabilities, the numerical results of this
work have been generated with a program specifically targeted to treat these
extreme phase-space configurations. In addition, to obtain NLP predictions for
an arbitrary process that can be compared with experimental data, one wishes
to calculate the non-radiative amplitude using general-purpose event
generators. Thus, in this work, we demonstrate that with our modified shifted
momenta it is possible to obtain predictions for the radiative amplitude in
the soft-photon limit, using non-radiative amplitudes that are automatically
generated.

The structure of this paper is as follows. In Section 2 we discuss the LBK
theorem in all formulations that will be relevant for the numerical
implementation: the one with derivatives, the one with unmodified shifts and
the one with modified shifts. In doing so, we thoroughly analyse the
ambiguities in the computation of the non-radiative process when the theorem
is expressed through derivatives of the non-radiative amplitude. Section 3
contains numerical results for the $e^{+}e^{-}\to\mu^{+}\mu^{-}\gamma$ and
$pp\to\mu^{+}\mu^{-}\gamma$ processes. Specifically, after comparing numerical
results based on the aforementioned three versions of the LBK theorem, we
study the predictive power of the soft approximation at LP and NLP in various
kinematic ranges. We conclude in Section 4 with a brief discussion.

##  2 LBK theorem and shifted kinematics

We begin this section with a compact review of known results on the LBK
theorem. More specifically, in Section 2.1 we recall the traditional form of
the theorem in terms of derivatives of the non-radiative amplitude, while in
Section 2.3 we recall the equivalent form of the theorem with shifted
kinematics, recently introduced in [27] and [28]. The reason for reviewing
these known forms of the theorem (apart from the sake of comprehensibility and
the need to fix the notation) is twofold. On the one hand, we discuss how an
intrinsic ambiguity in the computation of non-radiative processes does not
invalidate the traditional formulation of the theorem, which has been recently
questioned [35, 36, 37]. On the other hand, we want to stress the virtue of
shifting the kinematics, which removes such ambiguity by restoring momentum
conservation. We then present a new formulation of the theorem in Section 2.4,
where the shifts are modified in order to keep the external lines on the mass
shell.

###  2.1 Traditional LBK formulation

$H$$k$$N$$M$ (a)

$H$$k$$N$$M$ (b)

$H$$k$$N$$M$ (c)

Figure 1: Diagram (a) corresponds to the non-radiative amplitude $\cal H$,
where the hard blob $H$ represent an unknown hard interaction. Diagrams (b)
and (c) represent respectively the external and internal contribution to the
radiative amplitude $\cal A$.

We consider a generic scattering amplitude ${\cal H}(p_{1},\dots,p_{n})$ where
$N$ particles of hard momenta $p_{1},\dots,p_{N}$ scatter into $M$ particles
of hard momenta $p_{N+1},\dots,p_{N+M}$, with $n=N+M$. The particles interact
via an unspecified hard dynamics which can be represented diagrammatically by
the dashed blob ${H}(p_{1},\dots,p_{n})$, as in fig. 1. For spinning particles
$H$ is equal to the full scattering amplitude ${\cal H}$ stripped off of the
external-state wave functions, while for scalars one trivially has ${\cal
H}=H$.

In the radiative process $N\to M+\gamma$, the bremsstrahlung amplitude ${\cal
A}(p_{1},\dots,p_{n},k)$ includes a photon of momentum $k$ in the final state.
For reasons that will become clearer in the next sections, it is convenient to
introduce a parameter $\eta=1$ for initial particles ($1\leq i\leq N$) and
$\eta=-1$ for final particles ($N<i\leq n$), so that momentum conservation
reads

$\sum_{i=1}^{N}p_{i}-\sum_{i=N+1}^{N+M}p_{i}=\sum_{i=1}^{n}\eta_{i}p_{i}=k~{}.$
(2.1)

In this way, momenta are incoming for particles in the initial states and
outgoing for particles in the final states. We also denote with $Q_{i}$ the
charge of the $i$-th particle. In the following, we assume that the momenta
$p_{i}$ appearing both in the non-radiative (i.e. elastic) amplitude ${\cal
H}(p_{1},\dots,p_{n})$ and in the radiative (i.e. inelastic) ${\cal
A}(p_{1},\dots,p_{n},k)$ fulfill momentum conservation in the radiative
configuration, as in eq. 2.1. We will discuss this aspect in detail in Section
2.2.

The radiative process can be represented diagrammatically by two classes of
diagrams, as depicted in fig. 1. In the first one, the emitted photon is
attached to one of the external lines. In the second one, the photon couples
directly to some internal line of the unspecified hard subdiagram ${H}$. We
denote the two corresponding radiative amplitudes (stripped off of the photon
polarization vector $\epsilon_{\mu}(k)$) as ${\cal A}_{\text{ext}}^{\mu}$ and
${\cal A}_{\text{int}}^{\mu}$, respectively. We begin with the former.

Without loss of generality, we restrict the analysis to the case of an
external emission from an initial-state fermion-antifermion pair of charge
$Q_{1}=-Q_{2}\equiv Q$ and momenta $p_{1}$ and $p_{2}$, respectively,
satisfying $p_{1}+p_{2}=k$. The sum of the diagrams corresponding to the two
emissions reads

$\displaystyle{\cal A}_{\text{ext}}^{\mu}(p_{1},p_{2},k)=$
$\displaystyle\bar{v}(p_{2}){H}(p_{1}-k,p_{2})\frac{i(\not{p}_{1}-\not{k}+m)}{(p_{1}-k)^{2}-m^{2}}(-iQ\gamma^{\mu})u(p_{1})$
$\displaystyle+\bar{v}(p_{2})(-iQ\gamma^{\mu})\frac{i(-\not{p}_{2}+\not{k}+m)}{(p_{2}-k)^{2}-m^{2}}{H}(p_{1},p_{2}-k)u(p_{1})~{}.$
(2.2)

In the limit where the photon momentum $k$ is small compared to the hard
momenta $p_{1}$ and $p_{2}$, we can expand444The expansion in the four-
momentum $k$ is equivalent to the expansion in the photon energy $\omega$
since all components of $k$ scale homogeneously in the soft limit. in $k$ both
the fermion propagators and the hard subdiagram ${H}$. After using the Dirac
equation, enforcing the on-shell condition $k^{2}=0$ and neglecting terms
proportional to $k^{\mu}$ that vanish by gauge invariance, we get

$\displaystyle{\cal A}_{\text{ext}}^{\mu}(p_{1},p_{2},k)$
$\displaystyle=Q\bar{v}(p_{2}){H}(p_{1},p_{2})\left(-\frac{p_{1}^{\mu}}{p_{1}\cdot
k}+\frac{p_{2}^{\mu}}{p_{2}\cdot k}+\frac{ik_{\nu}S^{\mu\nu}}{p_{1}\cdot
k}\right)u(p_{1})$
$\displaystyle\quad+Q\bar{v}(p_{2})\frac{ik_{\nu}S^{\mu\nu}}{p_{2}\cdot
k}{H}(p_{1},p_{2})u(p_{1})+Q\frac{p^{\mu}_{1}}{p_{1}\cdot
k}k^{\nu}\bar{v}(p_{2})\frac{\partial{H}(p_{1},p_{2})}{\partial
p_{1}^{\nu}}u(p_{1})$ $\displaystyle\quad-Q\frac{p^{\mu}_{2}}{p_{2}\cdot
k}k^{\nu}\bar{v}(p_{2})\frac{\partial{H}(p_{1},p_{2})}{\partial
p_{2}^{\nu}}u(p_{1})\,\,+\,\,{\cal O}(k)~{},$ (2.3)

where we defined
$S^{\mu\nu}=\frac{i}{4}\commutator{\gamma^{\mu}}{\gamma^{\nu}}$ and we
exploited the functional dependence of ${H}$ by setting

$\displaystyle\frac{\partial{H}(p_{1}-k,p_{2})}{\partial
k^{\nu}}\Big{|}_{k=0}=-\frac{\partial{H}(p_{1},p_{2})}{\partial
p^{\nu}_{1}}~{}.$ (2.4)

At this point we should note that in order to derive eq. 2.3 we have expanded
eq. 2.2 in $k$ while keeping all the other momenta fixed, in analogy with
Low’s derivation in [1]. Mathematically, one can regard the r.h.s. of eq. 2.2
as a function defined on the entire space spanned by the vectors
$\\{p_{1},p_{2},k\\}$, where the vectors $p_{1}$ and $p_{2}$ are not
restricted to the surface $p_{1}+p_{2}=k$. Although the result of such
expansion is then defined for arbitrary momenta $p_{1}$ and $p_{2}$,
eventually we are of course only interested in the physical value of the
expanded function on the momentum-conservation surface. An alternative
approach, followed e.g. by Burnett and Kroll [2] and more recently by555We
thank O. Nachtmann for clarifying that in [35, 36, 37] this is how the
expansion is performed. [35, 36, 37], consists of expanding eq. 2.2 on the
momentum-conservation surface by inserting a dependence over $k$ in the
momenta $p_{1}(k)$ and $p_{2}(k)$. The parametrization of the momenta
$p_{1}(k)$ and $p_{2}(k)$, which are not fixed, is then only constrained by
$p_{1}(k)+p_{2}(k)=k$. The two approaches yield equivalent expressions on the
momentum-conservation constraint up to power-corrections in the expansion.

To proceed further, one has to compute the internal emission contribution
${\cal A}_{\text{int}}$. However, since in general one cannot know how the
photon couples to the internal hard subdiagrams, one is seemingly prevented
from an explicit calculation of ${\cal A}_{\text{int}}$. However, gauge
invariance comes to the rescue, since

$\displaystyle k_{\mu}\left({\cal A}_{\text{ext}}^{\mu}+{\cal
A}_{\text{int}}^{\mu}\right)=0~{}.$ (2.5)

From this, we deduce that

$\displaystyle{\cal A}_{\text{int}}^{\mu}$
$\displaystyle=-\sum_{i=1}^{2}Q_{i}\bar{v}(p_{2})\frac{\partial{H}(p_{1},p_{2})}{\partial
p_{i\,\mu}}u(p_{1})+K^{\mu}~{},$ (2.6)

where $K^{\mu}$ is a gauge invariant term ($k\cdot K=0$). A power counting
analysis reveals that at the tree level $K^{\mu}$ is power-suppressed at NLP
and can be set to zero.666See [24] for a more detailed analysis.

Therefore, combining eq. 2.3 and eq. 2.6 we get the final form for the LBK
theorem for the radiative amplitude ${\cal A}(p_{1},p_{2},k)$, which reads

$\displaystyle{\cal A}(p_{1},p_{2},k)=$
$\displaystyle-\epsilon_{\mu}(k)\sum_{i=1}^{2}Q_{i}\,\frac{p_{i}^{\mu}}{p_{i}\cdot
k}\,{\cal
H}(p_{1},p_{2})-\epsilon_{\mu}(k)\sum_{i=1}^{2}Q_{i}\,\bar{v}(p_{2})G^{\mu\nu}_{i}\,\frac{\partial{H}(p_{1},p_{2})}{\partial
p_{i}^{\nu}}u(p_{1})$
$\displaystyle+\epsilon_{\mu}(k)Q\,\bar{v}(p_{2})\left[{H}(p_{1},p_{2})\frac{ik_{\nu}S^{\mu\nu}}{p_{1}\cdot
k}+\frac{ik_{\nu}S^{\mu\nu}}{p_{2}\cdot k}{H}(p_{1},p_{2})\right]u(p_{1})~{},$
(2.7)

where we have introduced the following tensor

$\displaystyle
G^{\mu\nu}_{i}=g^{\mu\nu}-\frac{(2p_{i}-k)^{\mu}k^{\nu}}{2p_{i}\cdot
k}=g^{\mu\nu}-\frac{p_{i}^{\mu}k^{\nu}}{p_{i}\cdot k}+{\cal O}(k)~{}.$ (2.8)

The first term in eq. 2.7 represents the well-known LP factorization in terms
of the eikonal factor $p\cdot\epsilon/(p\cdot k)$. The remaining terms
correspond to NLP corrections.

A generalization of the calculation above to an arbitrary number of initial or
final state particles is straightforward, although the final result is not
quite compact since one has to distinguish the four cases where the
(anti-)fermion is in the initial or final state. A short-hand notation that is
quite common in the literature on scattering amplitudes [15, 16, 17, 18, 19,
20, 21] consists of factoring out the spin generator and the derivatives from
the non-radiative amplitude, yielding

$\displaystyle{\cal A}(p_{1},\dots,p_{n},k)$ $\displaystyle=\left({\cal
S}_{\text{LP}}+{\cal S}_{\text{NLP}}\right){\cal H}(p_{1},\dots,p_{n})~{},$
(2.9)

where

$\displaystyle{\cal
S}_{\text{LP}}=-\sum_{i=1}^{n}\eta_{i}\,Q_{i}\,\frac{p_{i}\cdot\epsilon(k)}{p_{i}\cdot
k}~{},\qquad{\cal
S}_{\text{NLP}}=-\sum_{i=1}^{n}\eta_{i}\,Q_{i}\,\frac{ik_{\nu}J_{i}^{\mu\nu}\epsilon_{\mu}(k)}{p_{i}\cdot
k}~{}.$ (2.10)

Here, $J^{\mu\nu}_{i}=S^{\mu\nu}_{i}+L^{\mu\nu}_{i}$ is the total angular
momentum, while $L^{\mu\nu}_{i}=i\left(p_{i}^{\mu}\frac{\partial}{\partial
p_{i\nu}}-p_{i}^{\nu}\frac{\partial}{\partial p_{i\mu}}\right)$ is the orbital
angular momentum which is related to the tensor $G^{\mu\nu}$ via
$G^{\mu\nu}_{i}\,\frac{\partial}{\partial
p_{i}^{\nu}}=i\frac{k_{\nu}}{p_{i}\cdot k}L^{\mu\nu}_{i}$. However, one should
not be fooled by the simplicity of eq. 2.9, since $J^{\mu\nu}$ is not a simple
multiplicative factor but rather an operator that contains derivatives and
gamma matrices. The derivatives act on the hard coefficient ${H}$ only (not
the full amplitude $\cal H$), while the spin generator must be inserted in the
correct order within the spinors, as shown in eq. 2.7 for the simple case of
an initial state fermion-antifermion pair.

Things become much simpler for the squared unpolarized amplitude
$\overline{|{\cal A}|}^{2}$, since all NLP corrections can be recast in terms
of derivatives of the squared non-radiative amplitude, as first shown in [2].
This can be seen again by considering the simple case of two charged incoming
particles as in eq. 2.7, where NLP corrections correspond to a derivative
contribution (second term) and a spin contribution (third and fourth term).
When squaring and averaging over the polarizations, the non-radiative
amplitude reads simply

$\displaystyle\overline{|{\cal H}(p_{1},p_{2})|}^{2}$
$\displaystyle=\Tr\left[{(\not{p}_{2}-m)}{H}(p_{1},p_{2}){(\not{p}_{1}+m)}\overline{H}(p_{1},p_{2}))\right]~{},$
(2.11)

where we defined $\overline{H}=\gamma^{0}{H}^{{\dagger}}\gamma^{0}$. For the
radiative amplitude instead one has the following schematic structure at NLP

$\displaystyle{|{\cal A}(p_{1},p_{2},k)|}^{2}$ $\displaystyle=|{\cal
S}_{\text{LP}}|^{2}{|{\cal H}(p_{1},p_{2})|}^{2}+2\real\left({\cal
S}_{\text{LP}}{\cal H}(p_{1},p_{2}){\cal S}_{\text{NLP}}^{\dagger}{\cal
H}^{\dagger}(p_{1},p_{2})\right)~{},$ (2.12)

where ${\cal S}_{\text{LP}}$ and ${\cal S}_{\text{NLP}}$ have been defined in
eq. 2.9. The second term in eq. 2.12 corresponds to the interference between
the LP factor and either the derivative or the spin contribution as in eq.
2.7. For an emission from the leg with momentum $p_{1}$, the spin term becomes

$\displaystyle\Tr\left[(\not{p}_{2}-m){H}(p_{1},p_{2})\left(\frac{\not{k}\gamma_{\mu}}{{2}p_{1}\cdot
k}(\not{p}_{1}+m)+(\not{p}_{1}+m)\frac{\gamma_{\mu}\not{k}}{{2}p_{1}\cdot
k}\right)\overline{H}(p_{1},p_{2}))\right]\sum_{i}\eta_{i}{Q_{i}}\frac{p_{i}^{\mu}}{p_{i}\cdot
k}~{}.$ (2.13)

Up to terms proportional to $k^{\mu}$ one then has

$\displaystyle\frac{\not{k}\gamma^{\mu}}{{2}p_{1}\cdot
k}(\not{p}_{1}+m)+(\not{p}_{1}+m)\frac{\gamma^{\mu}\not{k}}{{2}p_{1}\cdot
k}\,\,=\,\,-\gamma^{\mu}+\frac{p_{1}^{\mu}}{p_{1}\cdot
k}\not{k}\,\,=\,\,-G^{\mu\nu}_{1}\frac{\partial}{\partial
p^{\nu}_{1}}(\not{p}_{1}+m)~{}.$ (2.14)

Recalling that derivatives in eq. 2.7 act on the hard function only, we
conclude that both the spin and the orbital contribution combine into
derivatives of the full squared non-radiative amplitude $\overline{|{\cal
H}(p_{1},p_{2})|}^{2}$. Hence we obtain

$\displaystyle\overline{\left|{\cal A}(p_{1},p_{2},k)\right|}^{2}$
$\displaystyle=Q^{2}\sum_{ij=1}^{2}\frac{p_{i}\cdot p_{j}}{(p_{i}\cdot
k)(p_{j}\cdot k)}\overline{\left|{\cal
H}(p_{1},p_{2})\right|}^{2}+Q^{2}\sum_{ij=1}^{2}\frac{p_{i\,\mu}}{p_{i}\cdot
k}G_{j}^{\mu\nu}\frac{\partial}{\partial p_{j}^{\nu}}\overline{\left|{\cal
H}(p_{1},p_{2})\right|}^{2}~{}.$ (2.15)

Although so far we have considered fermions, the result above holds also in
the case of spin $0$ and spin $1$ charged particles. In the former case, the
spin generator vanishes, hence it is obvious that NLP terms include only the
derivative contribution. For spin $1$ one can exploit the gauge invariance of
the amplitude to set
$\sum_{\lambda}\epsilon_{\mu}^{(\lambda)}(k)\epsilon_{\nu}^{(\lambda)}(k)=-g_{\mu\nu}$,
which does not depend on any momenta and therefore $\frac{\partial}{\partial
p}|{\cal H}|=\frac{\partial}{\partial p}|{H}|$, leaving again only a
derivative contribution.

Finally, we note that eq. 2.15 can be trivially generalized to an arbitrary
number $n$ of external (charged or neutral) particles. One simply has to
repeat the derivation above for each particle-antiparicle pair, paying special
care to whether the particles are in the initial or final states. Thus, the
general form for the LBK theorem in the traditional formulation reads

$\displaystyle\overline{\left|{\cal A}(p_{1},\dots,p_{n},k)\right|}^{2}=$
$\displaystyle-\sum_{ij=1}^{n}\eta_{i}\eta_{j}Q_{i}Q_{j}\frac{p_{i}\cdot
p_{j}}{(p_{i}\cdot k)(p_{j}\cdot k)}\overline{\left|{\cal
H}(p_{1},\dots,p_{n})\right|}^{2}$
$\displaystyle-\sum_{ij=1}^{n}\eta_{i}Q_{i}Q_{j}\frac{p_{i\,\mu}}{p_{i}\cdot
k}G_{j}^{\mu\nu}\frac{\partial}{\partial p_{j}^{\nu}}\overline{\left|{\cal
H}(p_{1},\dots,p_{n})\right|}^{2}~{},$ (2.16)

where we used $\eta_{j}^{2}=1$. In Section 2.3 and Section 2.4 we will discuss
two alternative forms of the theorem that do not involve derivatives. Before
doing so, in the next section we analyse an important property of eq. 2.16.

###  2.2 Non-radiative amplitude and unphysical momenta

In the traditional form of the theorem of eq. 2.16, the non-radiative
amplitude
${\cal H}(p_{1},\dots,p_{n})$ is affected by an ambiguity related to the fact
that it must be evaluated outside the physical region. Indeed, in order for
${\cal H}(p_{1},\dots,p_{n})$ to represent a physical process with no photon
radiation, the momenta $p_{i}$ should fulfill $\sum_{i}\eta_{i}p_{i}=0$.
However, the momenta $p_{i}$ that have been introduced in eq. 2.1 fulfill
momentum conservation in the radiative amplitude ${\cal
A}(p_{1},\dots,p_{n},k)$, i.e. $\sum_{i}\eta_{i}p_{i}=k$. Therefore, in eq.
2.16 we are in fact evaluating $\mathcal{H}$ using radiative momenta, which
for $k\neq 0$ are unphysical for the non-radiative process and thus induce an
unphysical ambiguity in the final result. It is the aim of this section to
demonstrate that the use of unphysical momenta in the non-radiative amplitude
does not invalidate the consistency of eq. 2.16 at LP and NLP.

We start with the observation that every amplitude, and in particular the non-
radiative amplitude of the LBK theorem $\mathcal{H}(p_{1},\ldots,p_{n})$, is
intrinsically ambiguous if momentum conservation is not imposed. In fact, one
can always find a function $\Delta$ such that the transformation

${\mathcal{H}}(p_{1},\ldots,p_{n})\to\mathcal{H}(p_{1},\ldots,p_{n})+\Delta(p_{1},\ldots,p_{n})$
(2.17)

leads to the exact same physics, as long as $\Delta$ fulfills

$\Delta(p_{1},\ldots,p_{n})\,\delta\left(\sum_{i}\eta_{i}p_{i}\right)=0~{}.$
(2.18)

We would like to exploit this property to show that eq. 2.16 does not depend
on $\Delta$, up to NNLP corrections. To do so, we have to assign a scaling in
$k$ to $\Delta$ in eq. 2.17. In this regard, we note that momentum
conservation for the radiative amplitude must be fulfilled in order for eq.
2.16 to give physical results. Therefore, what matters for the invariance of
eq. 2.16 under eq. 2.17 is the value of $\Delta$ on the momentum-conservation
surface $\sum_{i}\eta_{i}p_{i}=k$. By imposing this constraint, the momenta
$p_{i}$ can be effectively interpreted as functions $p_{i}(k)$, with an
arbitrary functional dependence over $k$, constrained only by total momentum
conservation. This induces an implicit dependence of
$\Delta(p_{1},\ldots,p_{n})$ over $k$ through the momenta $p_{i}(k)$, such
that $\Delta$ can be expanded in $k$. Therefore, we can now check whether the
r.h.s. of eq. 2.16 is invariant at NLP under the transformation of eq. 2.17 on
the momentum-conserving surface $\sum_{i}\eta_{i}p_{i}=k$. Let us consider the
LP and NLP cases separately. To simplify the discussion, we will first
consider the form of the LBK theorem at the amplitude level as in eq. 2.9 and
eq. 2.10 in the scalar case only. We will then discuss how the generalization
for the squared amplitude (which is valid also in the spinning case) follows
analogously.

To check whether the LBK theorem in the form of eq. 2.9 and eq. 2.10 is
invariant under eq. 2.17 at LP, one has to verify that

$\left(\sum_{i}\eta_{i}Q_{i}\frac{p^{\mu}_{i}(k)}{p_{i}(k)\cdot
k}\right)\Delta(p_{1}(k),\ldots,p_{n}(k))=\order{1},$ (2.19)

or alternatively

$\displaystyle\Delta(p_{1}(k),\ldots,p_{n}(k))=O(k)~{}.$ (2.20)

The key point here is to notice that the limit $k\to 0$ implies the expression
$\sum_{i}\eta_{i}p_{i}=0$. Then, from eq. 2.18 one concludes that $\Delta\to
0$ for $k\to 0$. Since in this paper we are restricting the scope of our
analysis to a tree-level calculation, where the absence of non-analytic terms
allows a Laurent expansion in $k$, we conclude that $\Delta$ is at worst of
$\mathcal{O}(k)$ and hence eq. 2.20 is fulfilled, thus validating the theorem
at LP.

At NLP we have to modify the consistency condition of eq. 2.19 as follows

$\displaystyle\sum_{i}\eta_{i}Q_{i}\left[\frac{p_{i}^{\mu}(k)}{k\cdot
p_{i}(k)}\Delta(p_{1}(k),\ldots,p_{n}(k))+\eta_{i}G_{i}^{\mu\nu}\partialderivative{\Delta(p_{1}(k),\ldots,p_{n}(k))}{p_{i}^{\nu}}\right]=\order{k}~{},$
(2.21)

where $\mathcal{O}(k)$ in the r.h.s. represent NNLP corrections. In order to
verify this condition, once again we introduce a $k$ dependence in
$\Delta(p_{1},\ldots,p_{n})$ via $p_{i}(k)$. Given that we have to deal with
derivatives, it is convenient to make the dependence on $k$ explicit by
defining a new function $\tilde{\Delta}_{\mu}(p_{1},\ldots,p_{n},k)$ which is
constrained on the momentum-conservation surface by

$\displaystyle
k^{\mu}\tilde{\Delta}_{\mu}(p_{1},\ldots,p_{n},k)\,\,\delta\left(\sum_{i}\eta_{i}p_{i}-k\right)={\Delta}(p_{1}(k),\ldots,p_{n}(k))\,\,\delta\left(\sum_{i}\eta_{i}p_{i}-k\right)~{}.$
(2.22)

By enforcing the delta constraints of eq. 2.22, one can effectively substitute
$k=k(p)=\sum_{i}\eta_{i}p_{i}$ in $k^{\mu}\tilde{\Delta}_{\mu}$. Therefore,
the following relation between the derivatives of $\Delta$ and
$\tilde{\Delta}_{\mu}$ can be found

$\displaystyle\partialderivative{\Delta(p_{1},\ldots,p_{n})}{p_{j}^{\mu}}$
$\displaystyle=\frac{d}{dp_{j}^{\mu}}\left(k^{\nu}(p)\tilde{\Delta}_{\nu}(p_{1},\ldots,p_{n},k(p))\right)$
$\displaystyle=\eta_{j}\tilde{\Delta}_{\mu}(p_{1},\ldots,p_{n},k(p))+k^{\nu}(p)\partialderivative{\tilde{\Delta}_{\nu}(p_{1},\ldots,p_{n},k(p))}{p_{j}^{\mu}}$
$\displaystyle\quad+\eta_{j}k^{\nu}(p)\partialderivative{\tilde{\Delta}_{\nu}(p_{1},\ldots,p_{n},k(p))}{k^{\mu}}$
$\displaystyle=\eta_{j}\tilde{\Delta}_{\mu}(p_{1},\ldots,p_{n},k(p))+\order{k}~{},$
(2.23)

where in the last equality we dropped terms of order $\order{k}$, using the
fact that $\tilde{\Delta}_{\mu}=\order{1}$. At this point eq. 2.21 follows
straightforwardly. Indeed, the l.h.s. of eq. 2.21 becomes

$\displaystyle\sum_{i}\eta_{i}Q_{i}\left[\frac{p_{i}^{\mu}}{k\cdot
p_{i}}k^{\nu}\tilde{\Delta}_{\nu}(p_{1},\ldots,p_{n},k)+\eta_{i}\left(g^{\mu\nu}-\frac{p_{i}^{\mu}k^{\nu}}{p_{i}\cdot
k}\right)\eta_{i}\tilde{\Delta}_{\nu}(p_{1},\ldots,p_{n},k)\right]+\order{k}~{}.$
(2.24)

For a given $i$, thanks to eq. 2.19 and eq. 2.22, all terms in eq. 2.24 are
NLP. However, because $\eta_{i}^{2}=1$, there is a cancellation between the
terms in eq. 2.24, yielding

$\displaystyle\left(\sum_{i}\eta_{i}Q_{i}\right)\tilde{\Delta}^{\mu}(p_{1},\ldots,p_{n},k)+\order{k}~{}.$
(2.25)

Finally, given that $\sum_{i}\eta_{i}Q_{i}=0$ by charge conservation, only a
residual $\mathcal{O}(k)$ (i.e. NNLP) term remains, as required by eq. 2.21.
We then conclude that the NLP theorem at the amplitude level as in eq. 2.9 and
eq. 2.10 is invariant under eq. 2.17 on the surface $\sum_{i}\eta_{i}p_{i}=k$
and thus it is consistent also when the corresponding non-radiative amplitude
is evaluated with unphysical momenta.

A crucial step in the derivation above is the cancellation of NLP ambiguities
between the LP term and the derivative term. To make the general arguments
discussed above more concrete and see this cancellation explicitly, in
Appendix A we consider the soft bremsstrahlung in a simple case of a $2\to 2$
non-radiative process involving only scalar particles. This discussion is also
meant to clarify the relation with the work of [35, 37, 36] where the validity
of the traditional form of the LBK theorem has been questioned.

The generalization of the previous arguments to the squared-matrix elements of
eq. 2.16 straightforwardly carries over, by simply adjusting the correct power
counting in $k$. More specifically, one has to check that eq. 2.16 remains
invariant under777Note that although the definition of $\Delta$ here is
different from the one in eq. 2.17, it obeys eq. 2.18.
$\overline{|{\cal H}|}^{2}\to\overline{|{\cal H}|}^{2}+\Delta$. At LP this is
equivalent to showing that

$\left(\sum_{ij}\eta_{i}\eta_{j}Q_{i}Q_{j}\frac{p_{i}(k)\cdot
p_{j}(k)}{p_{i}(k)\cdot k\,\,p_{j}(k)\cdot
k}\right)\Delta(p_{1}(k),\ldots,p_{n}(k))=\order{k^{-1}},$ (2.26)

while at NLP the consistency condition reads

$\displaystyle\sum_{ij}\eta_{i}\eta_{j}Q_{i}Q_{j}\Bigg{(}\frac{p_{i}(k)\cdot
p_{j}(k)}{p_{i}(k)\cdot k\,\,p_{j}(k)\cdot k}\Delta(p_{1}(k),\ldots,p_{n}(k))$
$\displaystyle\quad+\frac{p_{j\,\mu}(k)}{p_{j}(k)\cdot
k}\eta_{i}G_{i}^{\mu\nu}\partialderivative{\Delta(p_{1}(k),\ldots,p_{n}(k))}{p_{i}^{\nu}}\Bigg{)}=\order{1}~{}.$
(2.27)

Both conditions can be verified with the same arguments as outlined above,
thus showing that the r.h.s. of eq. 2.16 does not depend on $\Delta$ at LP and
NLP. Therefore, even though the non-radiative amplitude is evaluated with
unphysical momenta, the formulation of the theorem as in eq. 2.16 is
consistent at NLP.

Finally, we note that the arbitrariness in the evaluation of the non-radiative
function with non-physical momenta was already observed by Burnett and Kroll
in their original work [2]. In fact, Burnett and Kroll proposed a prescription
to evaluate the non-radiative ampitude by shifting the unphysical momenta by
an arbitrary quantity that restore momentum conservation in the elastic
amplitude. The argument we have presented here, instead, is more general. By
exploiting the invariance at NLP of the non-radiative amplitude under eq. 2.17
we have proven that eq. 2.16 is consistent without the need to restore
momentum conservation. In fact, one could restrict the transformations of eq.
2.17 to the special case of linear shifts on the external momenta. Then, the
proposal of Burnett and Kroll would correspond to the specific case of shifts
that fulfill momentum conservation in the elastic configuration. In order to
shed light on the relation between the general argument of this section and
the strategy of Burnett and Kroll, in Appendix B we discuss the invariance of
eq. 2.16 in the special case where eq. 2.17 can be represented by linear
transformations of the momenta.

###  2.3 From derivatives to shifts

In the previous section we have verified that the traditional form of the LBK
theorem with derivatives of the non-radiative process is consistent at NLP,
since non-physical ambiguities arising in the computation of the non-radiative
process are NNLP. Still, the dependence of eq. 2.16 on an unphysical non-
radiative amplitude seems unsatisfactory. In particular, if one intends to
automatically generate the amplitude of the non-radiative process using
publicly available tools, having a form of the theorem that is defined from
scratch for physical amplitudes with momenta that fulfill momentum
conservation is desirable. The non-radiative process is then computed for
physical momenta and is thus unambiguous. Hence, it is natural to ask whether
it is possible to find a simpler formulation of the theorem which is
particularly suitable for numerical implementations.

The answer is yes, as proposed in [28], building on previous work in QCD [27].
It stems from the fact that since derivatives are the generators of
translations, one can convert the term with derivatives in the LBK theorem
into momentum shifts in the non-radiative amplitude. In fact, one can write
eq. 2.16 as

$\displaystyle\overline{\left|\mathcal{A}(p_{1},\dots,p_{n},k)\right|}^{2}$
$\displaystyle=\overline{\left|{\cal
S}_{\text{LP}}\right|}^{2}\left[1+\sum_{j}\delta
p_{j}^{\nu}\partialderivative{p_{j}^{\nu}}\right]\overline{\left|\mathcal{H}(p_{1},\dots,p_{n})\right|}^{2}~{},$
(2.28)

where the shifts $\delta p_{i}$ are to be determined, while from eq. 2.10

$\overline{\left|{\cal
S}_{\text{LP}}\right|}^{2}=-\sum_{ij=1}^{n}\eta_{i}\eta_{j}Q_{i}Q_{j}\frac{p_{i}\cdot
p_{j}}{(p_{i}\cdot k)(p_{j}\cdot k)}~{}.$ (2.29)

By comparison with eq. 2.16, we deduce

$\displaystyle\delta p_{j}^{\nu}$
$\displaystyle=Q_{j}\left(\sum_{k,l}\eta_{k}\eta_{l}Q_{k}Q_{l}\frac{p_{k}\cdot
p_{l}}{(p_{k}\cdot k)(p_{l}\cdot
k)}\right)^{-1}\sum_{i}\left(\frac{\eta_{i}Q_{i}p_{i\mu}}{k\cdot
p_{i}}\right)G_{j}^{\mu\nu}~{}.$ (2.30)

Thus, one obtains

$\displaystyle\overline{\left|\mathcal{A}(p_{1},\dots,p_{n},k)\right|}^{2}$
$\displaystyle=\overline{\left|{\cal
S}_{\text{LP}}\right|}^{2}\overline{\left|\mathcal{H}(p_{1}+\delta
p_{1},\dots,p_{n}+\delta p_{n})\right|}^{2}~{},$ (2.31)

i.e. a form of the LBK theorem without derivatives and with a single LP soft
factor.

We note immediately that $\delta p_{j}^{\nu}=\order{k}$, hence the shifts
vanish at LP, as expected. Another crucial property that can be readily
verified is that

$\displaystyle\sum_{j}\eta_{j}\delta p_{j}^{\mu}=-k^{\mu}~{}.$ (2.32)

Therefore, recalling that $\sum_{j}\eta_{j}p_{j}^{\mu}=k^{\mu}$, we deduce
that momentum conservation is restored in the non-radiative amplitude of eq.
2.31, which can be then computed without the ambiguities discussed in the
previous section.

Note also that by getting rid of the derivatives, we obtained a form of the
theorem with just a single positive-defined term. Naturally, as long as the
soft expansion is meaningful, we expect the derivative term in eq. 2.16 to be
small w.r.t. the LP term. Thus, for soft-photon momenta, also eq. 2.16 remains
positive, as expected for a cross-section. Still, for a theorem whose scope is
to extend the range of validity of the soft approximation to larger soft
momenta, the formulation in eq. 2.31 seems more elegant, since it ensures that
the cross-section remains positive. We will come back to this point in Section
3.

Finally, one can easily verify that the momenta shifts are orthogonal to each
momentum, i.e.

$\displaystyle\delta p_{j}\cdot p_{j}=0~{}.$ (2.33)

This implies that

$\displaystyle(p_{j}+\delta p_{j})^{2}=m_{j}^{2}+\order{k^{2}}~{},$ (2.34)

thus fulfilling the on-shell condition up NLP. We notice however that the
condition is violated already at NNLP. More precisely, one can verify that

$\displaystyle(\delta p_{j})^{2}$
$\displaystyle=Q^{2}_{j}\left(\sum_{k,l}\eta_{k}\eta_{l}Q_{k}Q_{l}\frac{p_{k}\cdot
p_{l}}{(p_{k}\cdot k)(p_{l}\cdot k)}\right)^{-1}\neq 0~{},$ (2.35)

hence masses do get shifted by a non-zero NNLP amount for non-vanishing $k$.
This feature might be problematic when using automatically generated
amplitudes, since most of the public tools typically require momenta to be
exactly on-shell. In the next section we discuss how to overcome this problem.

###  2.4 Modified shifted kinematics

We seek another expression for $\delta p_{i}$ that ensures that masses are not
shifted, without spoiling the NLP terms of the LBK theorem. Hence we require
the new definition for $\delta p_{i}$ to fulfill the following conditions:

  1. (i)

it conserves momentum to all orders in $k$, i.e.

$\displaystyle\sum_{i}\eta_{i}\delta p_{i}+k=0~{},$ (2.36)

  2. (ii)

it fulfills the on-shell condition to all orders in $k$, i.e.

$\displaystyle(p_{i}+\delta p_{i})^{2}=m_{i}^{2}~{},$ (2.37)

  3. (iii)

it reduces to eq. 2.30 up to NNLP corrections, i.e.

$\displaystyle\delta p_{j}^{\nu}=-Q_{j}\left(\overline{\left|{\cal
S}_{\text{LP}}\right|}^{2}\right)^{-1}\sum_{i}\left(\frac{\eta_{i}Q_{i}}{k\cdot
p_{i}}\right)\left(p_{i}^{\nu}-\frac{p_{i}\cdot p_{j}}{p_{j}\cdot
k}k^{\nu}\right)+\order{k^{2}}~{}.$ (2.38)

We can find such definition by considering the following ansatz,

$\displaystyle\delta p_{i}^{\mu}=\sum_{j}A_{ij}p_{j}^{\mu}+B_{i}k^{\mu}~{},$
(2.39)

and, by imposing the constraints (i)-(iii), subsequently determine the unknown
coefficients $A_{ij}$ and $B_{i}$ . It turns out that these conditions are not
too restrictive and one is free to select a single solution. Details of this
calculation can be found in Appendix C. The final result reads

$\displaystyle\delta p_{i}^{\mu}=AQ_{i}\sum_{j}\frac{\eta_{j}Q_{j}}{k\cdot
p_{j}}p_{j\nu}G^{\nu\mu}_{i}+\frac{1}{2}\frac{A^{2}Q^{2}_{i}{\overline{\left|{\cal
S}_{\text{LP}}\right|}^{2}}}{p_{i}\cdot k}k^{\mu}~{},$ (2.40)

with

$\displaystyle
A=\frac{1}{\chi}\left(\sqrt{1-\frac{2\chi}{\overline{\left|{\cal
S}_{\text{LP}}\right|}^{2}}}-1\right)~{},\qquad\chi=\sum_{i}\frac{\eta_{i}Q^{2}_{i}}{p_{i}\cdot
k}~{},$ (2.41)

and $\overline{\left|{\cal S}_{\text{LP}}\right|}^{2}$ defined in eq. 2.29. It
is straightforward to check that the conditions of eq. 2.36, eq. 2.37 and eq.
2.38 are satisfied by this solution. Indeed, eq. 2.40 reduces to eq. 2.30 up
to NNLP corrections and therefore it still correctly reproduces the LBK
theorem at NLP. Moreover, both momentum conservation and the on-shell
condition hold to all-orders in the soft expansion.

The price to pay is that we need to introduce spurious NNLP terms in the hard
momenta, which unavoidably affect the numerical evaluation of the non-
radiative amplitude $\cal H$, hence the prediction for the photon spectra.
However, one should bear in mind that the sensitivity of $\cal H$ to NNLP
effects is not a feature that belongs only to the modified kinematics. We
encountered it also in the other two versions of the LBK theorem.
Specifically, in the traditional form with derivatives, $\cal H$ is not
uniquely defined. Thus, even though the NLP ambiguities cancel, as we showed
in Section 2.2, NNLP spurious terms do survive. In the formulation with
unmodified shifted kinematics, although there are no unphysical ambiguities
due to violation of momentum conservation, when going from eq. 2.28 to eq.
2.31 we are implicitly adding spurious NNLP terms. Hence eq. 2.31 is also
valid only up to NLP.

More generally, we note that a residual arbitrariness in the final result due
to missing higher-order terms is a feature common to all perturbative
expansions. In fact, choosing a specific definition for the modified shifts in
eq. 2.31 corresponds to the choice of a “scheme”, which is specified by the
inclusion of power-suppressed (i.e. beyond NLP) terms. In this regard, we note
that the modified shifts make this scheme-dependence transparent, since the
choice is process-independent. Instead, in the traditional formulation of the
LBK theorem of eq. 2.16, $\cal H$ is not univocally determined and thus the
(hidden) scheme-dependence corresponds to the choice of a specific functional
form for the amplitude. This choice is obviously process-dependent.

The question that remains is what is the role of these NNLP effects in a
numerical computation of photon spectra, i.e. what is the version of the LBK
theorem that gives the best approximation of the exact radiative process.
Among other things, we investigate these aspects in the next section.

##  3 Numerical predictions for $e^{+}e^{-}\to\mu^{+}\mu^{-}\gamma$ and
$pp\to\mu^{+}\mu^{-}\gamma$

In the following, we present a numerical study of the spectra of soft photons
produced in association with a muon pair in $e^{+}e^{-}$ and $pp$ collisions.
The cross-sections for the $ij\to\mu^{+}\mu^{-}(+\gamma)$ processes
($ij=e^{+}e^{-},\,q\bar{q}$) are calculated at the tree level, including both
$Z$ and $\gamma$ exchange. We consider the $e^{+}e^{-}$ collisions at the
center-of-mass (c.m.) energy of 91 GeV, i.e. the LEP1 collision energy, at
which the measurements of photon spectra were carried out by the DELPHI
collaboration[57]. The $pp$ collisions are considered at 14 TeV c.m. energy.
To ensure that we are not sensitive to any infrared effects other than that
related to the soft photon, we impose kinematical cuts on the transverse
momentum of the muons, $p_{T,\mu}>10$ GeV, pseudorapidity of all final state
particles, $|\eta_{i}|<2.5$, as well as the photon-muon separation, $\Delta
R>0.4$. The photon distributions are computed with an in-house code using the
VEGAS+ algorithm [69] for performing the phase-space integration. In the case
of $pp$ collisions, we make use of LHAPDF6[70] and choose to evaluate the
cross sections with NNPDF4.0[71] LO set of parton distribution functions. The
tree-level amplitudes for the non-radiative and (exact) radiative amplitudes
are either generated by MadGraph5@NLO[72] or calculated analytically.888 The
analytical expression for the non-radiative amplitude used in this section is
specified by
$\mathcal{H}(p_{1},p_{2},p_{3},p_{4})=\mathcal{H}(s(p_{1},p_{2}),t(p_{1},p_{3})))$.
All exact (i.e. obtained without imposing the soft-photon approximation)
results have been cross-checked against numerical results generated using the
SHERPA event generator.999Numerical checks with MadGraph5 were also performed.
The results of MadGraph5 for the
$e^{+}e^{-}\to\mu^{+}\mu^{-}\gamma$ process appear to depend on the chosen
integration strategy and the way of grouping the Feynman diagrams for
calculations. We thank the MadGraph team for clarifying that point.

The exact predictions and the predictions to which we refer to as NLP are
obtained by integrating the exact matrix elements or their particular NLP
approximation over the full 3-particle phase space. At this point, we note
that one could also consider the expansion of the phase-space factor in powers
of the soft momentum and truncate it at LP or NLP depending on whether the
matrix elements are evaluated at NLP or LP, respectively. However, as
discussed in the last section, the soft approximation of the matrix elements
based on the LBK theorem in all its forms receives NNLP contributions.
Therefore, integrating over the full phase space leads to the same level of
accuracy. On the other hand, our LP predictions are obtained by imposing
momentum conservation on all external particles other than the photon, which
effectively corresponds to truncating the expansion of the phase-space factor
at LP and calculating the LP term of the non-radiative amplitude on such
external momenta.

We begin with a comparison of numerical predictions for the
$e^{+}e^{-}\to\mu^{+}\mu^{-}\gamma$ process obtained using the NLP
approximations of the amplitude derived in the previous chapter, i.e. the
traditional form with derivatives of eq. 2.16, the form of eq. 2.31 with the
off-shell momenta shifts defined in eq. 2.30, and the form of eq. 2.31 with
on-shell shifts defined in eq. 2.40. To this aim, we use the analytic result
for the non-radiative amplitude to calculate the derivatives in eq. 2.16
analytically, as well as to compute eq. 2.31 involving the off-shell momenta
shifts. We also compare the NLP predictions to the full result where no soft
approximation has been applied. The corresponding differential distributions
in photon energy $\omega$ are shown in fig. 2, in the range of 1-500 MeV (left
plot) and 0.1-10 GeV (right plot). As expected, we observe that all three
approaches converge to the exact result in the limit of small $\omega$, and
depart from it with growing photon energy. However, the approximation of the
exact result provided by the formulation of the LBK theorem involving
derivatives, eq. 2.16, is distinctively worse than those based on formulations
involving shifting of momenta. While the latter agree with the exact result
within 1-2% for $\omega\lesssim 500\text{MeV}$, where one naively expects the
soft approximation to work, the former differs from the exact predictions
within the same range by up to 6%. Besides, its behaviour is qualitatively
different: at $\omega\gtrsim 6\text{GeV}$, the derivative approach gives a
non-physical negative result, in contrast to the other predictions which stay
positive. This can be understood from eq. 2.16 which is a sum of two not-
positive-defined terms and as such can get negative when the expansion breaks
down. As the difference between various NLP approximations is due to the NNLP
terms, these results clearly show the relevance of the subleading terms beyond
the formal accuracy of the LBK theorem.

Figure 2: Comparison of the soft-photon energy spectra calculated using three
formulations of the LBK theorem discussed in this paper with the exact (i.e.
no soft expansion) result for the process $e^{-}e^{+}\to\mu^{-}\mu^{+}\gamma$
at $\sqrt{s}=91$ GeV.

We also see that the NLP approximations of the photon spectrum calculated
using the non-radiative amplitude with momenta shifted on-shell or off-shell
perform equally well for the photon energies considered here, indicating that
the NNLP effects introduced in the on-shell shifts are not significant. Since
the formulation with momenta shifted on-shell enables sourcing the amplitude
subroutines from a wide range of public tools, we employ this formulation in
further studies.

The NLP distributions in photon energy $\omega_{\gamma}$ and transverse
momentum $p_{T,\gamma}$, obtained using the radiative amplitude with on-shell
momenta eq. 2.40, are then compared to the LP and exact predictions in fig. 3
and fig. 4, respectively. In particular, we show distributions for very soft
photons with 1 MeV$<\omega_{\gamma},\,p_{T,\gamma}<100$ MeV. As discussed
above, the NLP formula relying on shifting momenta on-shell returns
predictions which provide a very good approximation of the exact result. Up to
the scale of 100 MeV, the difference between the two predictions is at a few
per mille level and grows to a 1-2% level for $\omega_{\gamma}$ or
$p_{T,\gamma}$ of up to ca. 1 GeV. In contrast, the LP approximation differs
from the exact result by up to ca. 2% (6%) and up to ca. 40% (70%) in these
two ranges of $\omega_{\gamma}$ ($p_{T,\gamma}$), correspondingly.

Figure 3: The $\mathrm{LP}$ and $\mathrm{NLP}$ approximations and the exact
result for the energy spectrum of the photon in the
$e^{-}e^{+}\to\mu^{-}\mu^{+}\gamma$ process at $\sqrt{s}=91$ GeV.

Figure 4: The $\mathrm{LP}$ and $\mathrm{NLP}$ approximations and the exact
result for the transverse momentum spectrum of the photon in the
$e^{-}e^{+}\to\mu^{-}\mu^{+}\gamma$ process at $\sqrt{s}=91$ GeV.

Next, we study the soft-photon spectra in the process
$pp\to\mu^{+}\mu^{-}\gamma$. The differential distributions in photon energy
and transverse momentum are shown in fig. 5 and fig. 6, respectively. The
photon energy spectrum in fig. 5 is presented in both the partonic c.m. frame
and the laboratory frame. No perceptible difference is observed between the
results in the two frames for our choice of kinematical cuts. Perhaps not
surprisingly, the behaviour of the LP and NLP approximations is very similar
to the one found for the $e^{+}e^{-}$ collisions. Quantitatively, however, in
the ranges of $\omega_{\gamma}$ and $p_{T,\gamma}$ studied here, the LP and
NLP predictions appear to be relatively closer to the exact results than in
the $e^{+}e^{-}$ case. To be more precise, within an accuracy of roughly
$10\%$, the LP spectrum deviates from the exact result for
$\omega_{\gamma}\gtrsim 400$MeV, $p_{T,\gamma}\gtrsim 150$MeV ($e^{+}e^{-}$)
and $\omega_{\gamma}\gtrsim 1$GeV, $p_{T,\gamma}\gtrsim 500$MeV ($pp$). The
NLP predictions reach this level of deviation only at $p_{T,\gamma}\gtrsim 1$
GeV in the $e^{+}e^{-}$ case, and at $p_{T,\gamma}\gtrsim 10$ GeV in the $pp$
case, i.e. outside of the soft regime.

Figure 5: The $\mathrm{LP}$ and $\mathrm{NLP}$ approximations and the exact
result for the energy spectrum of the photon in the partonic c.m. frame (top
row) and the laboratory frame (bottom row) in the $pp\to\mu^{-}\mu^{+}\gamma$
process at $\sqrt{s}=14$ TeV.

Figure 6: The $\mathrm{LP}$ and $\mathrm{NLP}$ approximations and the exact
result for the transverse momentum spectrum of the photon in the
$pp\to\mu^{-}\mu^{+}\gamma$ process at $\sqrt{s}=14$ TeV.

##  4 Conclusions

In this paper, we have presented an extensive study of the LBK theorem and its
implications on numerical predictions for the soft-photon spectra.

First, motivated by recent discussions in the literature[35, 36, 37], we have
addressed the consistency of the LBK theorem. As known for a long time, in the
original formulation of the theorem[1] the non-radiative amplitude is
calculated on a set of unphysical momenta. Seen in the most general way, this
violation of momentum conservation leads to an ambiguity in the functional
form of the non-radiative amplitude. We have provided a proof that the
aforementioned ambiguity can only affect the expansion of the radiative
amplitude starting from NNLP in soft-photon energy, i.e. beyond the formal
accuracy of the LBK theorem, thus proving the validity of the LBK theorem at
NLP. In doing so, we have generalized the remark by Burnett and Kroll by
observing an invariance of the theorem at NLP under a specific transformation
of the non-radiative amplitude. The consequence of this invariance is the
presence of many equivalent forms of the LBK theorem, which include the
original formulation by Low as one possibility.

Among the different versions of the theorem, for practical reasons, it is
particularly attractive to consider those that restore momentum conservation
in the calculation of the non-radiative amplitude. Such restoration can be
achieved by reformulating the LBK theorem in terms of the non-radiative
amplitude calculated on momenta which values are modified wrt. the momenta of
the radiative amplitude. The modification that has been put forward in the
literature [27, 67, 28] relies on adding small shifts of the order of the
soft-photon momentum $k$. Apart from reviewing the derivation of the
reformulated LBK theorem in terms of shifted momenta, we have proposed
expressions for the momenta shifts which not only restore momentum
conservation, but also ensure that each of the shifted momenta is on-shell. In
this way, we facilitate the generation and numerical calculation of the non-
radiative amplitude with a wide range of publicly available tools.

We have also studied the quality of the soft approximation provided by the LBK
theorem in its various formulations on the example of the
$e^{+}e^{-}\to\mu^{+}\mu^{-}\gamma$ process, analysed at LEP1 energies. We
have found a remarkable improvement in the quality of the approximation for
the formulations involving shifts w.r.t. the formulation of the theorem with
derivatives, although we have considered the latter only for a specific form
of the non-radiative amplitude. Depending on the required quality of the
approximation, NNLP effects can be thus numerically relevant and the form of
the LBK theorem used to calculate the NLP approximation of the soft-photon
spectra needs to be chosen carefully. In this regard, our studies indicate
that the formulations involving shifts should be preferred. Notably, for the
implementation involving on-shell shifts, we have used the amplitudes
generated by the MadGraph5 code, in this way demonstrating the feasibility of
the calculations for a wide range of processes, given a corresponding phase-
space integrating code is available.

Despite the long history of the LBK theorem, to the best of our knowledge, no
study in the literature has explicitly identified when power-suppressed
effects become visible in the soft-photon spectra. In order to address this
question, we compared the exact, LP and NLP results for the
$e^{+}e^{-}\to\mu^{+}\mu^{-}\gamma$ process at the LEP1 energy of $91$ GeV and
the $pp\to\mu^{+}\mu^{-}\gamma$ process at the LHC energy of $14$ TeV. We have
studied various ranges of photon energy and transverse momentum. For the cases
studied here, the quality of the NLP approximation in a reasonably soft regime
is of the order of, or better than, one percent, even though the specific
results depend on the process, observable and the analysis set-up. The quality
of the LP approximation is significantly worse, meaning that for measurements
where a precision of a few percent can be reached in the soft regime, the LP
might not provide a good enough approximation. Correspondingly, our results
suggest that in order to access the power-suppressed terms, a percent-level
precision is needed, especially in the case of the $pp\to\mu^{+}\mu^{-}\gamma$
process at the LHC. Obviously, this is only a crude estimate. A more precise
statement would require a careful analysis of all theoretical and experimental
uncertainties.

The results of this work open up the possibility of many follow-up analyses.
The two processes we studied here involve simple leptonic final states and
central rapidity photons. Given that the excess in the soft-photon spectrum
was observed for $e^{+}e^{-}$ collisions in hadronic final states, it would be
interesting to investigate the impact of NLP corrections on the predictions
for the associated photon production with jets. In this regard, it would also
be interesting to extend the analysis by including QCD corrections at
1-loop[28], which will be relevant for both initial and final state hadrons.
In the long run, the studies need to be extended to $pp$ collisions resulting
in various hadronic final states with photons at forward rapidities, as
planned to be investigated with the ALICE 3 detector[66].

## Acknowledgments

We would like to thank A. Andronic, V. Hirschi, F. Kling and S. Weinzierl for
useful discussions. We would also like to thank the MadGraph, Sherpa and
Vegas+ authors for helping us with their software. DB would like to thank the
participants to the EMMI workshop “Real and virtual photon production at
ultra-low transverse momentum and low mass at LHC” and in particular P. Braun-
Munziger, X. Feal, S. Flörchinger, H. van Hees, P. Lebiedowicz, O. Nachtmann,
K. Reygers, K. Schweda, J. Stachel and M. Völkl for stimulating discussions on
the topic of soft photons. DB and AK are grateful to the Galileo Galilei
Institute for hospitality and support during the scientific program “Theory
Challenges in the Precision Era of the Large Hadron Collider”, where part of
this work was done. AK thanks the CERN Theory Division for the hospitality and
support during the early stages of this work. The work of RB was partially
supported by the Deutsche Forschungsgemeinschaft (DFG) through the Research
Training Group “GRK 2149: Strong and Weak Interactions: from Hadrons to Dark
Matter”. The work of DB was supported by the Excellence Cluster ORIGINS funded
by the DFG under Grant No. EXC- 2094-390783311.

##  Appendix A Case study: $2\to 2$ scattering

In this section we apply the general arguments of Section 2.2 concerning the
traditional form of the LBK theorem with derivatives to the simple case of
photon bremsstrahlung in pion scattering. In doing so, we compare with the
analysis of Lebiedowicz, Nachmann, Szczurek (LNS) [35], showing that their
result is equivalent to the traditional form of the LBK theorem and with
previous studies in the literature.

###  A.1 LBK theorem for $\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}\gamma$

Following [1] and [35], and using the notation established in [35], we
consider the following process:

$\pi^{-}(p_{a})\pi^{0}(p_{b})\to\pi^{-}(p^{\prime}_{1})\pi^{0}(p^{\prime}_{2})\gamma(k)~{},$

with

$p_{a}+p_{b}=p^{\prime}_{1}+p^{\prime}_{2}+k~{}.$

For this process, the LBK theorem in the form of eq. 2.7, adapted to the case
of scalar particles, reads

$\displaystyle\mathcal{A}^{\mu}(p_{a},p_{b},p^{\prime}_{1},p^{\prime}_{2},k)$
$\displaystyle=e\left[\frac{p_{a}^{\mu}}{k\cdot
p_{a}}\mathcal{H}(p_{a},p_{b},p^{\prime}_{1},p^{\prime}_{2})+G_{a}^{\mu\nu}\partialderivative{\mathcal{H}(p_{a},p_{b},p^{\prime}_{1},p^{\prime}_{2})}{p_{a}^{\nu}}\right]$
$\displaystyle\phantom{=}-e\left[\frac{p_{1}^{\prime\mu}}{k\cdot
p^{\prime}_{1}}\mathcal{H}(p_{a},p_{b},p^{\prime}_{1},p^{\prime}_{2})-G_{1}^{\mu\nu}\partialderivative{\mathcal{H}(p_{a},p_{b},p^{\prime}_{1},p^{\prime}_{2})}{p_{1}^{\prime\nu}}\right]+\order{k}$
$\displaystyle=e\left[\frac{p_{a}^{\mu}}{k\cdot
p_{a}}-\frac{p_{1}^{\prime\mu}}{k\cdot
p^{\prime}_{1}}+G_{a}^{\mu\nu}\partialderivative{p_{a}^{\nu}}+G_{1}^{\mu\nu}\partialderivative{p^{\prime\nu}_{1}}\right]\mathcal{H}(p_{a},p_{b},p^{\prime}_{1},p^{\prime}_{2})+\order{k}~{}.$
(A.1)

To show the ambiguity in the calculation of the non-radiative amplitude with
radiative kinematics, we consider the following two choices for $\mathcal{H}$:

$\mathcal{H}_{1}=\mathcal{H}(s^{\prime}_{L},t_{1}),\qquad\mathcal{H}_{2}=\mathcal{H}(s^{\prime}_{L},t_{2})~{},$
(A.2)

where we defined

$s^{\prime}_{L}=p_{a}\cdot p_{b}+p^{\prime}_{1}\cdot p^{\prime}_{2},\qquad
t_{1}=(p_{a}-p^{\prime}_{1})^{2},\qquad t_{2}=(p_{b}-p^{\prime}_{2})^{2}.$
(A.3)

Here, $\mathcal{H}(s_{L},t)$ is the amplitude for the non-radiative process
$\pi^{-}(p_{a})\pi^{0}(p_{b})\to\pi^{-}(p_{1})\pi^{0}(p_{2})$. Obviously,
$p_{a}+p_{b}=p_{1}+p_{2}$, so in the elastic limit $t_{1}=(p_{a}-p_{1})^{2}$
and $t_{2}=(p_{b}-p_{2})^{2}$, hence we have $t_{1}=t_{2}$, and thus
$\mathcal{H}_{1}=\mathcal{H}_{2}$.

Now we consider

$\displaystyle\partialderivative{\mathcal{H}_{i}}{p_{a}^{\nu}}$
$\displaystyle=\partialderivative{\mathcal{H}_{i}}{s^{\prime}_{L}}\partialderivative{s^{\prime}_{L}}{p_{a}^{\nu}}+\partialderivative{\mathcal{H}_{i}}{t_{i}}\partialderivative{t_{i}}{p_{a}^{\nu}},$
$\displaystyle\partialderivative{\mathcal{H}_{i}}{p^{\prime\nu}_{1}}$
$\displaystyle=\partialderivative{\mathcal{H}_{i}}{s^{\prime}_{L}}\partialderivative{s^{\prime}_{L}}{p^{\prime\nu}_{1}}+\partialderivative{\mathcal{H}_{i}}{t_{i}}\partialderivative{t_{i}}{p^{\prime\nu}_{1}}~{},$
(A.4)

where no sum over the index $i=1,2$ has been assumed. The derivatives of
$s^{\prime}_{L}$ in eq. A.4 read

$\partialderivative{s^{\prime}_{L}}{p_{a}^{\nu}}=p_{b\nu},\qquad\partialderivative{s^{\prime}_{L}}{p^{\prime\nu}_{1}}=p^{\prime}_{2\nu}~{}.$

The terms with derivatives of $t_{i}$ are different for $i=1,2$, and are given
by

$\partialderivative{t_{1}}{p^{\nu}_{a}}=2(p_{a}-p^{\prime}_{1})_{\nu},\qquad\partialderivative{t_{1}}{p^{\prime\nu}_{1}}=-2(p_{a}-p^{\prime}_{1})_{\nu},\qquad\partialderivative{t_{2}}{p^{\nu}_{a}}=\partialderivative{t_{2}}{p^{\prime\nu}_{1}}=0~{}.$
(A.5)

Putting the above equations together we obtain two seemingly distinct forms
for the LBK theorem, depending on whether we take ${\cal H}={\cal H}_{1}$ or
${\cal H}={\cal H}_{2}$ in eq. A.1. They read

$\displaystyle\mathcal{A}^{\mu}(p_{a},p_{b},p^{\prime}_{1},p^{\prime}_{2},k)=e\Bigg{[}$
$\displaystyle\left(\frac{p_{a}^{\mu}}{k\cdot
p_{a}}-\frac{p_{1}^{\prime\mu}}{k\cdot
p^{\prime}_{1}}\right)+\left(p_{b}^{\mu}-\frac{p_{b}\cdot k}{p_{a}\cdot
k}p_{a}^{\mu}+p^{\prime\mu}_{2}-\frac{p^{\prime}_{2}\cdot
k}{p^{\prime}_{1}\cdot
k}p^{\prime\mu}_{1}\right)\partialderivative{s^{\prime}_{L}}$
$\displaystyle-2(p_{a}-p^{\prime}_{1})\cdot k\left(\frac{p_{a}^{\mu}}{k\cdot
p_{a}}-\frac{p_{1}^{\prime\mu}}{k\cdot
p^{\prime}_{1}}\right)\partialderivative{t_{1}}\Bigg{]}\mathcal{H}_{1}+\order{k}~{},$
$\displaystyle\mathcal{A}^{\mu}(p_{a},p_{b},p^{\prime}_{1},p^{\prime}_{2},k)=e\Bigg{[}$
$\displaystyle\left(\frac{p_{a}^{\mu}}{k\cdot
p_{a}}-\frac{p_{1}^{\prime\mu}}{k\cdot p^{\prime}_{1}}\right)$
$\displaystyle+\left(p_{b}^{\mu}-\frac{p_{b}\cdot k}{p_{a}\cdot
k}p_{a}^{\mu}+p^{\prime\mu}_{2}-\frac{p^{\prime}_{2}\cdot
k}{p^{\prime}_{1}\cdot
k}p^{\prime\mu}_{1}\right)\partialderivative{s^{\prime}_{L}}\Bigg{]}\mathcal{H}_{2}+\order{k}~{}.$
(A.6)

If one ignores the fact that $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ are not
the same, the two expressions indeed yield different results, thus seemingly
invalidating the LBK theorem. However, by relating $\mathcal{H}_{1}$ and
$\mathcal{H}_{2}$, the difference disappears at NLP. In fact, one has

$\mathcal{H}_{2}=\mathcal{H}(s^{\prime}_{L},t_{2})=\mathcal{H}(s^{\prime}_{L},t_{1})+(t_{2}-t_{1})\partialderivative{\mathcal{H}(s^{\prime}_{L},t_{1})}{t_{1}}+\order{k^{2}}~{}.$
(A.7)

Using then the expression

$t_{2}-t_{1}=(p_{b}-p^{\prime}_{2})^{2}-(p_{a}-p^{\prime}_{1})^{2}=(p_{a}-p^{\prime}_{1}-k)^{2}-(p_{a}-p^{\prime}_{1})^{2}=-2k\cdot(p_{a}-p^{\prime}_{1})+\order{k^{2}}~{},$
(A.8)

we get

$\mathcal{H}_{2}=\mathcal{H}_{1}-2k\cdot(p_{a}-p^{\prime}_{1})\partialderivative{\mathcal{H}_{1}}{t_{1}}+\order{k^{2}}~{},$
(A.9)

which accounts for the difference between the two previous expressions. This
is the cancellation we saw in Section 2.2, where we proved it in the general
case.

###  A.2 Comparison between LNS and the original work of Low

The soft bremsstrahlung in pion scattering has been computed also by the
authors of ref. [35]. Their final result, which is given by eq. (3.27) of
their paper, reads

$\displaystyle\mathcal{A}^{\mu}=$ $\displaystyle
e\mathcal{H}(s_{L},t)\left[\frac{p_{a}^{\mu}}{p_{a}\cdot
k}-\frac{p^{\prime\mu}_{1}}{p^{\prime}_{1}\cdot
k}\right]+2e\partialderivative{\mathcal{H}(s_{L},t)}{s_{L}}\left[p_{b}^{\mu}-\frac{p_{b}\cdot
k}{p_{a}\cdot k}p_{a}^{\mu}\right]$
$\displaystyle-2e\partialderivative{\mathcal{H}(s_{L},t)}{t}\left[(p_{a}-p_{1})\cdot
k-p_{a}\cdot l_{1}\right]\left[\frac{p_{a}^{\mu}}{p_{a}\cdot
k}-\frac{p^{\mu}_{1}}{p_{1}\cdot k}\right]+\order{k}~{},$ (A.10)

where the non-radiative amplitude $\cal H$ is written as a function of the two
variables $s_{L}$ and $t$

$s_{L}=p_{a}\cdot p_{b}+p_{1}\cdot p_{2},\qquad
t=(p_{a}-p_{1})^{2}=(p_{b}-p_{2})^{2}.$ (A.11)

The momenta fulfill the relations

$p_{a}+p_{b}=p_{1}+p_{2}=p^{\prime}_{1}+p^{\prime}_{2}+k$ (A.12)

and $l_{i}$ are defined as a shift between $p$ and $p^{\prime}$ as follows:

$l_{i}=p_{i}-p^{\prime}_{i}.$ (A.13)

As done in [1], here we set $k^{2}=0$ and drop all terms proportional to
$k^{\mu}$ since we assume all final states to be on-shell.

The authors of [35] compare then Section A.2 to the one derived by Low in ref.
[1], which they report in eq. (3.29) of their paper. It reads

$\tilde{\mathcal{A}}^{\mu}=e\mathcal{H}(s_{L},t)\left[\frac{p_{a}^{\mu}}{p_{a}\cdot
k}-\frac{p^{\mu}_{1}}{p_{1}\cdot
k}\right]+e\partialderivative{\mathcal{H}(s_{L},t)}{s_{L}}\left[p_{b}^{\mu}-\frac{p_{b}\cdot
k}{p_{a}\cdot k}p_{a}^{\mu}+p_{2}^{\mu}-\frac{p_{2}\cdot k}{p_{1}\cdot
k}p_{1}^{\mu}\right]+\order{k}~{}.$ (A.14)

They conclude that, while the LP terms agree, there is a discrepancy in the
NLP terms. We first point out that (A.14) does not precisely coincide with the
result given by Low in equation (2.16) of [1]. In fact, by looking at eq.
(2.1) and eq. (2.16) in [1], one concludes that the correct expression should
be101010In the final stages of writing this paper, we have been informed that
the authors are aware of this mistake.

$\mathcal{A}^{\mu}=e\mathcal{H}(s^{\prime}_{L},t_{2})\left[\frac{p_{a}^{\mu}}{p_{a}\cdot
k}-\frac{p^{\prime\mu}_{1}}{p^{\prime}_{1}\cdot
k}\right]+e\partialderivative{\mathcal{H}(s^{\prime}_{L},t_{2})}{s^{\prime}_{L}}\left[p_{b}^{\mu}-\frac{p_{b}\cdot
k}{p_{a}\cdot k}p_{a}^{\mu}+p^{\prime\mu}_{2}-\frac{p^{\prime}_{2}\cdot
k}{p^{\prime}_{1}\cdot k}p^{\prime\mu}_{1}\right]+\order{k}~{},$ (A.15)

with $s^{\prime}_{L}$ and $t_{2}$ defined as in eq. A.3. This form of the LBK
theorem is indeed what we obtained in eq. A.6.

A careful comparison of expressions Section A.2 and eq. A.15 shows that they
are in perfect agreement with each other, up to $\order{k}$. To simplify the
comparison, it is worth noticing that $p$ and $p^{\prime}$ are equal up to
$\order{k}$ corrections. Therefore, to order $\order{k}$, the difference
between $p$ and $p^{\prime}$ is only relevant for the first term, and eq. A.15
can be rewritten as

$\mathcal{A}^{\mu}=e\mathcal{H}(s^{\prime}_{L},t_{2})\left[\frac{p_{a}^{\mu}}{p_{a}\cdot
k}-\frac{p^{\prime\mu}_{1}}{p^{\prime}_{1}\cdot
k}\right]+e\partialderivative{\mathcal{H}(s_{L},t)}{s_{L}}\left[p_{b}^{\mu}-\frac{p_{b}\cdot
k}{p_{a}\cdot k}p_{a}^{\mu}+p^{\mu}_{2}-\frac{p_{2}\cdot k}{p_{1}\cdot
k}p^{\mu}_{1}\right]+\order{k}~{}.$ (A.16)

To compare Section A.2 with eq. A.16 it is necessary to use the formula

$\mathcal{H}(s^{\prime}_{L},t_{2})=\mathcal{H}(s_{L},t)+\delta
s^{\prime}_{L}\partialderivative{\mathcal{H}}{s_{L}}+\delta
t_{2}\partialderivative{\mathcal{H}}{t}~{},$ (A.17)

where we defined

$\displaystyle\delta s^{\prime}_{L}$
$\displaystyle=s^{\prime}_{L}-s_{L}=p^{\prime}_{1}\cdot
p^{\prime}_{2}-p_{1}\cdot p_{2}=(p_{1}-l_{1})\cdot(p_{2}-l_{2})-p_{1}\cdot
p_{2}$ $\displaystyle=-(l_{1}\cdot p_{2}+l_{2}\cdot
p_{1})+\order{k^{2}}=-(p_{1}+p_{2})\cdot k+\order{k^{2}}$
$\displaystyle=-(p_{a}+p_{b})\cdot k+\order{k^{2}}~{},$ $\displaystyle\delta
t_{2}$
$\displaystyle=t_{2}-t=(p_{b}-p^{\prime}_{2})^{2}-(p_{b}-p_{2})^{2}=(p_{b}-p_{2}+l_{2})^{2}-(p_{b}-p_{2})^{2}$
$\displaystyle=2l_{2}\cdot(p_{b}-p_{2})+\order{k^{2}}=-2(k-l_{1})\cdot(p_{a}-p_{1})+\order{k^{2}}$
$\displaystyle=-2\left[(p_{a}-p_{1})\cdot k-p_{a}\cdot
l_{1}\right]+\order{k^{2}}~{}.$ (A.18)

Here we used the relations $p_{1}\cdot l_{2}=p_{1}\cdot k$ and $p_{2}\cdot
l_{1}=p_{2}\cdot k$, which can be derived from the two relations
$l_{1}+l_{2}=k$ and $p_{i}\cdot l_{i}=0$ (i.e. eq. (3.17) and eq. (3.22) in
[35]).

Inserting now eq. A.17 into eq. A.16, we see that the term proportional to
$\mathcal{H}$ is identical to the one in Section A.2. Additionally, the term
proportional to $\partialderivative{\mathcal{H}}{t}$ is given by

$e\delta
t_{2}\partialderivative{\mathcal{H}}{t}\left[\frac{p_{a}^{\mu}}{p_{a}\cdot
k}-\frac{p^{\prime\mu}_{1}}{p^{\prime}_{1}\cdot
k}\right]=-2e\partialderivative{\mathcal{H}}{t}\left[(p_{a}-p_{1})\cdot
k-p_{a}\cdot l_{1}\right]\left[\frac{p_{a}^{\mu}}{p_{a}\cdot
k}-\frac{p^{\prime\mu}_{1}}{p^{\prime}_{1}\cdot k}\right]~{}.$ (A.19)

The r.h.s. of eq. A.19 coincides with the third term in Section A.2 after
dropping the prime in the last parenthesis (this is possible because this term
is already a NLP term, and thus the difference due to replacing $p$ with
$p^{\prime}$ is a NNLP effect). Finally, the term proportional to
$\partialderivative{\mathcal{H}}{s_{L}}$ has now the following expression

$e\delta
s^{\prime}_{L}\partialderivative{\mathcal{H}}{s_{L}}\left[\frac{p_{a}^{\mu}}{p_{a}\cdot
k}-\frac{p^{\mu}_{1}}{p_{1}\cdot
k}\right]+e\partialderivative{\mathcal{H}}{s_{L}}\left[p_{b}^{\mu}-\frac{p_{b}\cdot
k}{p_{a}\cdot k}p_{a}^{\mu}+p^{\mu}_{2}-\frac{p_{2}\cdot k}{p_{1}\cdot
k}p^{\mu}_{1}\right]~{},$ (A.20)

where again the prime in the first term can dropped because $\delta
s^{\prime}_{L}$ is already of order $\order{k}$. After some algebra, it is
easy to show that this term reads

$\displaystyle
2e\partialderivative{\mathcal{H}}{s_{L}}\left[p_{b}^{\mu}-k\cdot
p_{b}\frac{p_{a}^{\mu}}{p_{a}\cdot k}\right]~{},$ (A.21)

in perfect agreement with the second term in Section A.2. Therefore, the LNS
result (Section A.2) and Low’s original result (eq. A.15) are completely
equivalent at NLP.

We note also that although here we focused on pion scattering, we expect
analogous arguments to hold for the proton scattering discussed in [37, 36].

###  A.3 Comparison between LNS and other literature

A comparison between the result of LNS, i.e. Section A.2, and the previous
literature has been carried out in the appendix of [35]. It is claimed there
that all previous known forms of the LBK theorem are problematic. In
particular, LNS claim that the result of reference [24] is not consistent,
since it depends on an arbitrary quantity, as discussed below. However, a
careful analysis shows that this claim has no valid foundation, since the
dependence on such quantity vanishes.

The argument in [35] is the following. In [24] Low’s theorem is written in a
form that depends on the following four quantities:

$\displaystyle I_{1}$
$\displaystyle=\left(-l_{1}\cdot\partialderivative{p_{1}}-l_{2}\cdot\partialderivative{p_{2}}-k\cdot\partialderivative{p_{a}}\right)\mathcal{H}(p_{a},p_{b},p_{1},p_{2})~{},$
$\displaystyle I_{2}$
$\displaystyle=\left(-l_{1}\cdot\partialderivative{p_{1}}-l_{2}\cdot\partialderivative{p_{2}}+k\cdot\partialderivative{p_{1}}\right)\mathcal{H}(p_{a},p_{b},p_{1},p_{2})~{},$
$\displaystyle I_{3}^{\mu}$
$\displaystyle=\partialderivative{\mathcal{H}(p_{a},p_{b},p_{1},p_{2})}{p_{a\mu}},\qquad
I_{4}^{\mu}=\partialderivative{\mathcal{H}(p_{a},p_{b},p_{1},p_{2})}{p_{1\mu}}~{}.$
(A.22)

If one then restricts the analysis to an amplitude $\mathcal{H}$ that depends
solely on the quantity $p_{a}^{2}+p_{1}^{2}-p_{b}^{2}-p_{2}^{2}$, as done in
[35], the elastic amplitude is given by a constant, since 111111 Note that the
definition of $\eta$ and $\xi$ given by eq. (14) in [24] implies that the
elastic momenta are not on-shell. However, this detail is irrelevant for our
discussion in this section.

$\displaystyle\mathcal{H}(p_{a},p_{b},p_{1},p_{2})=f(p_{a}^{2}+p_{1}^{2}-p_{b}^{2}-p_{2}^{2})=f(m_{a}^{2}+m_{1}^{2}-m_{b}^{2}-m_{2}^{2})\equiv
f_{0}.$ (A.23)

Thus, the derivatives of the elastic amplitude vanish and the final result
only depends on the value of $f_{0}$. However, if one considers the
corresponding expressions in eq. A.22, they become

$I_{1}=-2(k\cdot p_{a})f^{\prime}_{0},\qquad I_{2}=2(k\cdot
p_{1})f^{\prime}_{0},\qquad I_{3}^{\mu}=2p_{a}^{\mu}f^{\prime}_{0},\qquad
I_{4}^{\mu}=2p_{1}^{\mu}f^{\prime}_{0}~{}.$ (A.24)

Therefore, in eq. A.24 there is a dependence on $f^{\prime}_{0}$ (the
derivative of $f$ evaluated on non-radiative momenta). $f^{\prime}_{0}$ is an
arbitrary quantity, which seems to invalidate the consistency of the result in
[24].

To see why this argument does not imply an inconsistency of the LBK theorem in
the form given in ref. [24], a more detailed analysis of the complete
expressions is needed. A direct comparison between [24] and [35] is
unfortunately not possible, since the processes under consideration are
different ($e^{-}\pi^{0}\to e^{-}\pi^{0}$ for ref. [24], and
$\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}$ for LNS). The comparison thus requires a
dictionary to relate the theorems with fermions and scalar fields,
respectively, which is summarized in Table 1.

Gervais | Lebiedowicz, Nachtmann, Szczurek
---|---
$\frac{1}{\not{p}-m}$ | $\frac{1}{p^{2}-m^{2}}$
$\gamma_{\mu}$ | $p_{\mu}+p^{\prime}_{\mu}$
$u,v,\bar{u},\bar{v}$ | $1$
$p_{1}$, $p^{\prime}_{1}$ | $p_{a}$
$k_{1}$, $k^{\prime}_{1}$ | $p_{b}$
$p_{2}$ | $p^{\prime}_{1}$
$k_{2}$ | $p^{\prime}_{2}$
$p^{\prime}_{2}$ | $p_{1}$
$k^{\prime}_{2}$ | $p_{2}$
$q$ | $k$
$\eta_{1}$, $\xi_{1}$ | 0
$\xi_{2}$ | $-l_{1}$
$\eta_{2}$ | $-l_{2}$
Table 1: Relation between the notations used in [24] and [35].

After carefully taking this translation into account, Low’s theorem in the
notation of ref. [24] (see equation (20) there) reads

$\mathcal{A}^{\mu}=e\left(\frac{p_{1}^{\prime\mu}}{p^{\prime}_{1}\cdot
k}-\frac{p_{a}^{\mu}}{p_{a}\cdot
k}\right)\mathcal{H}(p_{a},p_{b},p_{1},p_{2})-e\left(\frac{p_{a}^{\mu}}{p_{a}\cdot
k}I_{1}-\frac{p_{1}^{\mu}}{p_{1}\cdot
k}I_{2}+I_{3}^{\mu}+I_{4}^{\mu}\right)~{}.$ (A.25)

In particular, it is worth noting that the expression for $I_{i}$ in eq. A.25
enter via the following combinations

$p_{a}^{\mu}I_{1}+(p_{a}\cdot k)I^{\mu}_{3},\qquad
p_{1}^{\mu}I_{2}-(p_{1}\cdot k)I_{4}^{\mu}~{}.$ (A.26)

Therefore, after inserting the expressions of eq. A.24 in eq. A.25, the
dependence on $f_{0}^{\prime}$ vanishes. Hence, for the case of eq. A.23
studied in this section, eq. A.25 is consistent with other formulations of the
LBK theorem, such as LNS (Section A.2) and Low’s (eq. A.15).

##  Appendix B LBK invariance under momenta transformation

We consider here the invariance under eq. 2.17 in the case where $\Delta$
arises from linear transformations of the momenta in $\cal H$. Specifically,
we prove that, at NLP, eq. 2.16 is invariant under the following
transformation,

${\cal H}(p_{1},\dots,p_{n})\to{\cal
H}(\tilde{p}_{1},\dots,\tilde{p}_{n})~{},$ (B.1)

with

$\displaystyle\tilde{p}_{i}(k)=p_{i}+c_{i}k+\order{k^{2}},$ (B.2)

where the coefficients $c_{i}$ are arbitrary. To verify the invariance, let us
apply eq. B.1 to eq. 2.16. We get

$\displaystyle\overline{|{\cal A}(p_{1},\dots,p_{n},k)|}^{2}$
$\displaystyle=\sum_{ij=1}^{n}(-\eta_{i}\eta_{j}Q_{i}Q_{j})\frac{p_{i}\cdot
p_{j}}{p_{i}\cdot k\,p_{j}\cdot k}\overline{|{\cal
H}(\tilde{p}_{1},\dots,\tilde{p}_{n})|}^{2}$
$\displaystyle\quad+\sum_{ij=1}^{n}(-\eta_{i}\eta_{j}Q_{i}Q_{j})\frac{p_{i\,\mu}}{p_{i}\cdot
k}\eta_{j}\left(g^{\mu\nu}-\frac{p^{\mu}_{j}k^{\nu}}{p_{j}\cdot
k}\right)\frac{d}{dp_{j}^{\nu}}\overline{|{\cal
H}(\tilde{p}_{1},\dots,\tilde{p}_{n})|}^{2}~{},$ (B.3)

where partial derivatives have been replaced with total derivatives due to the
non-trivial functional dependence inside the non-radiative amplitude. The
function $\overline{|\cal H|}^{2}$ can be simply expanded in $k$ by using the
functional dependence of eq. B.2, to get

$\displaystyle\overline{|{\cal
H}(\tilde{p}_{1},\dots,\tilde{p}_{n})|}^{2}=\overline{|{\cal
H}(p_{1},\dots,p_{n})|}^{2}+k^{\mu}\sum_{i}c_{i}\frac{\partial}{\partial
p_{i}^{\mu}}\overline{|{\cal H}(p_{1},\dots,p_{n})|}^{2}+\order{k^{2}}~{}.$
(B.4)

To proceed further, we note that so far momentum conservation has not been
imposed. We can do so by making the dependence over the momenta $p_{i}$
explicit in the soft momentum, i.e. by enforcing $k^{\mu}\to
k^{\mu}(p_{1},\dots,p_{n})=\sum_{i}\eta_{i}p_{i}^{\mu}$. Subsequently, by
differentiating eq. B.4 we get

$\displaystyle\frac{d}{dp_{j}^{\nu}}\overline{|{\cal
H}(\tilde{p}_{1},\dots,\tilde{p}_{n})|}^{2}$
$\displaystyle=\frac{\partial}{\partial p_{j}^{\nu}}\overline{|{\cal
H}(p_{1},\dots,p_{n})|}^{2}+\delta_{\nu}^{\mu}\eta_{j}\sum_{i}c_{i}\frac{\partial}{\partial
p^{\mu}_{i}}\overline{|{\cal H}(p_{1},\dots,p_{n})|}^{2}+{\mathcal{O}}(k)~{}.$
(B.5)

Note that since eq. B.4 is expanded up to $\order{k^{2}}$, we have to truncate
eq. B.5 to $\order{k}$ because differentiating $k$ w.r.t the momenta $p_{i}$
reduces the order of the expansion. In particular,
$\frac{d\order{k^{2}}}{dp_{j}^{\nu}}=\order{k}$. This is not a problem, since
the l.h.s. of eq. B.5 is multiplied by an expression which is suppressed
w.r.t. the other term in eq. B.3 by one power of $k$. Therefore, plugging eq.
B.4 and eq. B.5 into eq. B.3 we get

$\displaystyle\overline{|{\cal A}(p_{1},\dots,p_{n},k)|}^{2}=$
$\displaystyle\sum_{ij=1}^{n}(-\eta_{i}\eta_{j}Q_{i}Q_{j})\frac{p_{i}\cdot
p_{j}}{p_{i}\cdot k\,p_{j}\cdot k}\overline{|{\cal
H}(p_{1},\dots,p_{n})|}^{2}$
$\displaystyle+\sum_{ij=1}^{n}(-\eta_{i}\eta_{j}Q_{i}Q_{j})\frac{p_{i\,\mu}}{p_{i}\cdot
k}\eta_{j}\left(g^{\mu\nu}-\frac{p^{\mu}_{j}k^{\nu}}{p_{j}\cdot
k}\right)\frac{\partial}{\partial p_{j}^{\nu}}\overline{|{\cal
H}(p_{1},\dots,p_{n})|}^{2}$ $\displaystyle+R(c_{i})~{},$ (B.6)

where

$\displaystyle R(c_{i})$
$\displaystyle=\sum_{ij=1}^{n}(-\eta_{i}\eta_{j}Q_{i}Q_{j})\frac{p_{i}\cdot
p_{j}}{p_{i}\cdot k\,p_{j}\cdot k}k^{\mu}\sum_{m}c_{m}\frac{\partial}{\partial
p^{\mu}_{m}}\overline{|{\cal H}(p_{1},\dots,p_{n})|}^{2}$
$\displaystyle\quad+\sum_{ij=1}^{n}(-\eta_{i}\eta_{j}Q_{i}Q_{j})\frac{p^{i}_{\mu}}{p_{i}\cdot
k}\eta_{j}\left(g^{\mu\nu}-\frac{p^{\mu}_{j}k^{\nu}}{p_{j}\cdot
k}\right)\eta_{j}\sum_{m}c_{m}\frac{\partial}{\partial
p^{\nu}_{m}}\overline{|{\cal H}(p_{1},\dots,p_{n})|}^{2}+{\cal O}(1)~{}.$
(B.7)

To prove the invariance of eq. 2.16 under eq. B.1 at NLP, we have to show that
the remainder term $R(c_{i})$, which depends on the arbitrary coefficients
$c_{i}$, is ${\cal O}(1)$ (i.e. NNLP). This follows straightforwardly by first
noting that, thanks to $\eta^{2}_{j}=1$, the term in the first line of eq. B.7
cancels with the analogous term in the second line. The term proportional to
$g^{\mu\nu}$ then vanishes since $\sum_{j}\eta_{j}Q_{j}=0$ by charge
conservation, thus leaving $R(c_{i})={\cal O}(1)$, as desired.

Having established that the LBK theorem in the form of eq. 2.16 is invariant
under eq. B.1 at NLP, the consistency of the theorem (i.e. the possibility to
evaluate $\cal H$ on a set of unphysical momenta as in eq. 2.16) follows as a
corollary. In fact, the invariance under eq. B.1 guarantees that at NLP there
is an infinite number of equivalent forms of the theorem, one for each choice
of the coefficients $c_{i}$. In particular, the unphysical momenta of eq. 2.16
corresponds to the trivial transformation with $c_{i}=0$. On the other hand,
we can choose $c_{i}$ so that momentum conservation for non-radiative
amplitude is restored (as in the strategy of Burnett and Kroll), i.e.

$\displaystyle\sum_{i}\eta_{i}c_{i}=-1~{},\qquad\sum_{i}\eta_{i}\tilde{p}_{i}=0~{}.$
(B.8)

Therefore, the form of the theorem in eq. 2.16 where $\cal H$ is evaluated on
a set of unphysical momenta is equivalent, up to NNLP corrections, to the form
where momentum conservation is restored (and thus no ambiguity is present).

Note that although the argument presented in this appendix is quite general,
it fails when the invariance of the amplitude cannot be represented by linear
shifts. A simple example is given by a constant amplitude that does not depend
on the external momenta. In that case, for the consistency of eq. 2.16 one has
to rely on the more general argument of Section 2.2.

##  Appendix C Calculation of the modified shifts

We show here the calculation that leads to the expression in eq. 2.40. We
consider the conditions (i)-(iii) of eq. 2.36, eq. 2.37 and eq. 2.38. The most
general form for $\delta p_{i}$ reads

$\displaystyle\delta
p_{i}^{\mu}=\sum_{j}A_{ij}^{\mu\nu}p_{j\nu}+B_{i}^{\mu\nu}k_{\nu}.$ (C.1)

However, it is enough for our purposes to consider

$\displaystyle\delta p_{i}^{\mu}=\sum_{j}A_{ij}p_{j}^{\mu}+B_{i}k^{\mu}.$
(C.2)

We can further restrict our ansatz by assuming the set
$\\{p_{i}^{\mu},k^{\mu}\\}$ to be linearly independent. Although clearly not
true in general, this is not a problem since we are only interested in finding
a single solution. With this assumption, we can now insert eq. C.2 into eq.
2.36, eq. 2.37 and eq. 2.38 to determine the coefficients $A_{ij}$ and
$B_{i}$. We get

  1. (i)

$\displaystyle\sum_{i,j}\eta_{i}A_{ij}p_{j}^{\mu}+\left(\sum_{i}\eta_{i}B_{i}+1\right)k^{\mu}=0~{},$
(C.3)

which gives

$\displaystyle\sum_{i}\eta_{i}A_{ij}=0,\quad\sum_{i}\eta_{i}B_{i}=-1~{}.$
(C.4)

  2. (ii)

$\displaystyle\left(\sum_{j}\left(\delta_{ij}+A_{ij}\right)p_{j}^{\mu}+B_{i}k^{\mu}\right)^{2}=m^{2}~{},$
(C.5)

which gives

$\displaystyle\sum_{j,k}\left(2\delta_{ij}A_{ik}+A_{ij}A_{ik}\right)(p_{j}\cdot
p_{k})+2\sum_{j}\left(\delta_{ij}B_{i}+A_{ij}B_{i}\right)(p_{j}\cdot k)=0~{}.$
(C.6)

  3. (iii)

$\displaystyle A_{ij}$ $\displaystyle=-Q_{i}\left(\overline{\left|{\cal
S}_{\text{LP}}\right|}^{2}\right)^{-1}\frac{\eta_{j}Q_{j}}{k\cdot
p_{j}}+\order{k^{2}}~{},$ $\displaystyle B_{i}$
$\displaystyle=Q_{i}\left(\overline{\left|{\cal
S}_{\text{LP}}\right|}^{2}\right)^{-1}\sum_{j}\left(\frac{\eta_{j}Q_{j}}{k\cdot
p_{j}}\right)\frac{p_{j}\cdot p_{i}}{p_{i}\cdot
k}+\order{k}=-\sum_{j}A_{ij}\frac{p_{i}\cdot p_{j}}{p_{i}\cdot
k}+\order{k}~{}.$ (C.7)

As we can see, the conditions given by eq. C.4, eq. C.6 and eq. C.7 are not
too restrictive. Thus, we still have the freedom to select a single solution
by introducing a scalar coefficient $A$ such that we have

$\displaystyle A_{ij}=AQ_{i}\frac{\eta_{j}Q_{j}}{k\cdot p_{j}}~{}.$ (C.8)

Then, the condition $\sum_{i}\eta_{i}A_{ij}=0$ is immediately satisfied by
charge conservation. The remaining conditions now yield

$\displaystyle\sum_{i}\eta_{i}B_{i}=-1~{},$ (C.9) $\displaystyle
2AQ_{i}\sum_{j}\eta_{j}Q_{j}\frac{p_{i}\cdot p_{j}}{k\cdot
p_{j}}-A^{2}Q^{2}_{i}\overline{\left|{\cal
S}_{\text{LP}}\right|}^{2}+2B_{i}(p_{i}\cdot k)=0~{},$ (C.10) $\displaystyle
A=\frac{-1}{\overline{\left|{\cal
S}_{\text{LP}}\right|}^{2}}+\order{k^{3}}\qquad
B_{i}=\frac{Q_{i}}{\overline{\left|{\cal
S}_{\text{LP}}\right|}^{2}}\sum_{j}\frac{\eta_{j}Q_{j}}{k\cdot
p_{j}}\frac{p_{i}\cdot p_{j}}{p_{i}\cdot k}+\order{k}~{}.$ (C.11)

We can finally determine the coefficients $A$ and $B_{i}$. Specifically, for
the coefficients $B_{i}$ we can use eq. C.10, which yields

$\displaystyle B_{i}=-AQ_{i}\sum_{j}\eta_{j}Q_{j}\frac{p_{i}\cdot
p_{j}}{(p_{i}\cdot k)(p_{j}\cdot
k)}+\frac{1}{2}\frac{A^{2}Q^{2}_{i}\overline{\left|{\cal
S}_{\text{LP}}\right|}^{2}}{p_{i}\cdot k}~{}.$ (C.12)

Assuming the behaviour of $A$ given in eq. C.11, the second term in eq. C.12
is $\order{k}$, so $B_{i}$ have the correct limit given in eq. C.11. To
determine $A$, we can use eq. C.9, which yields

$\displaystyle 1$
$\displaystyle=-\sum_{i}\eta_{i}B_{i}=A\sum_{i,j}\eta_{i}Q_{i}\eta_{j}Q_{j}\frac{p_{i}\cdot
p_{j}}{(p_{i}\cdot k)(p_{j}\cdot k)}-\frac{A^{2}\overline{\left|{\cal
S}_{\text{LP}}\right|}^{2}}{2}\sum_{i}\frac{\eta_{i}Q^{2}_{i}}{p_{i}\cdot k}$
$\displaystyle=-A\overline{\left|{\cal
S}_{\text{LP}}\right|}^{2}-\frac{A^{2}\overline{\left|{\cal
S}_{\text{LP}}\right|}^{2}}{2}\sum_{i}\frac{\eta_{i}Q^{2}_{i}}{p_{i}\cdot k},$
(C.13)

which is a quadratic equation. Defining
$\chi=\sum_{i}\frac{\eta_{i}Q^{2}_{i}}{p_{i}\cdot k}$, we find that

$\displaystyle
A=\frac{1}{\chi}\left(-1\pm\sqrt{1-\frac{2\chi}{\overline{\left|{\cal
S}_{\text{LP}}\right|}^{2}}}\right).$ (C.14)

Only the $+$ solution has the correct behaviour at low $k$. Combining thus eq.
C.2, eq. C.8, eq. C.12 and eq. C.14, we find the expression for the modified
shifts as given by eq. 2.40.

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