# Silo NLP’s Participation at WAT2022

Shantipriya Parida*, Subhadarshi Panda†, Stig-Arne Grönroos *‡,
Mark Granroth-Wilding *, Mika Koistinen *
*Silo AI, Helsinki, Finland
<EMAIL_ADDRESS>
†Graduate Center, City University of New York, USA
<EMAIL_ADDRESS>
‡University of Helsinki, Finland

###### Abstract

This paper provides the system description of “Silo NLP’s" submission to the
Workshop on Asian Translation (WAT2022). We have participated in the Indic
Multimodal tasks (English$\rightarrow$Hindi, English$\rightarrow$Malayalam,
and English$\rightarrow$Bengali, Multimodal Translation). For text-only
translation, we trained Transformers from scratch and fine-tuned mBART-50
models. For multimodal translation, we used the same mBART architecture, and
extracted object tags from the images to use as visual features concatenated
with the text sequence.

Our submission tops many tasks including English$\rightarrow$Hindi multimodal
translation (evaluation test), English$\rightarrow$Malayalam text-only and
multimodal translation (evaluation test), English$\rightarrow$Bengali
multimodal translation (challenge test), and English$\rightarrow$Bengali text-
only translation (evaluation test).

##  1 Introduction

Machine translation (MT) is a classic sub-field in NLP which investigates the
usage of computer software to translate text or speech from one language to
another without human involvement Yang et al. (2020). Although MT performance
has reached near the level of human translators for many high-resource
languages, it remains challenging for many low-resource languages Popel et al.
(2020); Costa-jussà et al. (2022). Also, effective usage of other modalities
(e.g. image) in MT is an important research area in the past few years
Sulubacak et al. (2020); Parida et al. (2021b, a).

The WAT is an open evaluation campaign focusing on Asian languages since 2013
Nakazawa et al. (2020). In the WAT2022 Multimodal track, a new Indian language
Bengali was introduced for English$\rightarrow$Bengali text, multimodal
translation, and Bengali image captioning
task.111https://ufal.mff.cuni.cz/bengali-visual-genome/wat-2022-english-
bengali-multim

The multimodal translation tasks in WAT2022 consist of image caption
translation, in which the input is a descriptive source language caption
together with the image it describes, while the output is a target language
caption. The multimodal input enables the use of image context to disambiguate
source words with multiple senses.

In this system description paper, we explain our approach for the tasks
(including the sub-tasks) we participated in:

Task 1:

English$\rightarrow$Hindi (EN-HI) Multimodal Translation

  * •

EN-HI text-only translation

  * •

EN-HI multimodal translation

Task 2:

English$\rightarrow$Malayalam (EN-ML) Multimodal Translation

  * •

EN-ML text-only translation

  * •

EN-ML multimodal translation

Task 3:

English$\rightarrow$Bengali (EN-BN) Multimodal Translation

  * •

EN-BN text-only translation

  * •

EN-BN multimodal translation

##  2 Data sets

We used the data sets specified by the organizer for the related tasks along
with additional synthetic data for performance improvement. The use of
additional data places some222All except the EN–HI and EN–ML text-only
systems. of our submissions in the unconstrained track.

#### Task 1: English$\rightarrow$Hindi Multimodal Translation

For this task, the organizers provided HindiVisualGenome 1.1 Parida et al.
(2019)333https://lindat.mff.cuni.cz/repository/xmlui/handle/11234/1-3267
dataset (HVG for short). The training part consists of 29k English and Hindi
short captions of rectangular areas in photos of various scenes and it is
complemented by three test sets: development (D-Test), evaluation (E-Test) and
challenge test set (C-Test). Our WAT submissions were for E-Test (denoted “EV”
in WAT official tables) and C-Test (denoted “CH” in WAT tables).

For the synthetic image features, we use the Flickr8k data set (Hodosh et al.,
2013). Even though it is an image captioning data set, we discard the images,
treating the data set as in-domain monolingual data. We use a machine
translation into Hindi (Rathi, 2020) as the target side, and generate image
features using the procedure described in Section 3.4.

The statistics of the datasets are shown in Table 1.

|  | Tokens
---|---|---
Set | Sentences | English | Hindi | Malayalam | Bengali
Train | 28930 | 143164 | 145448 | 107126 | 113978
D-Test | 998 | 4922 | 4978 | 3619 | 3936
E-Test | 1595 | 7853 | 7852 | 5689 | 6408
C-Test | 1400 | 8186 | 8639 | 6044 | 6657
Table 1: Statistics of our data used in the English$\rightarrow$Hindi,
English$\rightarrow$Malayalam, and English$\rightarrow$Bengali Multimodal
task: the number of sentences and tokens.

#### Task 2: English$\rightarrow$Malayalam Multimodal Translation

For this task, the organizers provided MalayalamVisualGenome 1.0
dataset444https://lindat.mff.cuni.cz/repository/xmlui/handle/11234/1-3533 (MVG
for short). MVG is an extension of the HVG dataset for supporting Malayalam,
which belongs to the Dravidian language family Kumar et al. (2017). The
dataset size and images are the same as HVG. While HVG contains bilingual
English–Hindi segments, MVG contains bilingual English–Malayalam segments,
with the English, shared across HVG and MVG, see Table 1.

#### Task 3: English$\rightarrow$Bengali Multimodal Translation

For this task, the organizers provided BengaliVisualGenome 1.0
dataset555http://hdl.handle.net/11234/1-3722 (BVG for short). BVG is an
extension of the HVG dataset for supporting Bengali. The dataset size and
images are the same as HVG, and MVG, see Table 1.

##  3 Experimental Details

This section describes the experimental details of the tasks we participated
in.

###  3.1 EN-HI, EN-ML, EN-BN text-only translation

For EN–HI and EN–ML text-only (E-Test and C-Test) translation, we fine-tuned a
pre-trained mBART-50 model Tang et al. (2020) without using any additional
resources.

For EN-BN text-only (E-Test) translation, we used the Transformer base model
as implemented in Open-NMT-Py666https://opennmt.net/OpenNMT-py/ using Bangla
Natural Language Image to Text (BNLIT) Jishan et al. (2019) as an additional
dataset. The BNLIT is an image-to-text dataset containing 8743 images and
their corresponding text in Bengali. For our experiment, we used Bengali text
and translated it into English. The training steps are defined for 300K with
5K as validation steps and checkpoint steps.

For C-Test we fine-tuned a pre-trained m-BART-50 model without any additional
resources.

Figure 1: Multimodal translation pipeline.

###  3.2 EN-HI, EN-ML, EN-BN Multimodal translation

Our multimodal translation pipeline is shown in Figure 1. For EN-HI multimodal
(E-Test and C-Test) translation, we used the object tags extracted from the
HVG dataset images (see Section 3.3) for image features and concatenated them
with the text. Additionally, we used synthetic image features (see Section
3.4. The combined data set was used to fine-tune a pre-trained mBART-50 model.

For EN-ML multimodal (E-Test and C-Test) translation, we used object tags
extracted from the MVG dataset images and concatenated with the text and fine-
tuned on the mBART-50 model.

For EN-BN (E-Test and C-Test) translation, we used object tags extracted from
the BVG dataset images and concatenated with the text and fine-tuned on the
mBART-50 model.

###  3.3 Extracted image features

We derive the list of object tags for a given image using the pretrained
Faster R-CNN with ResNet-101-C4 backbone. It can recognize 80 object types
from the COCO data set (Lin et al., 2014). Based on their confidence scores,
we pick the top 10 object tags. In cases where less than 10 object tags are
detected, we consider all the detected tags. Figure 2 shows examples of object
tags detected for images from the challenge test set. The detected object tags
are then concatenated to the English sentence which needs to be translated to
Hindi, Malayalam, and Bengali. The concatenation is done using the special
token ‘##’ as the separator. The separator is followed by comma-separated
object tags. Adding objects enables the model to utilize visual concepts which
may not be readily available in the original sentence. The English sentences
along with the object tags are fed to the encoder of the mBART model.

Figure 2: Sample translations for the challenge test set. Both the text-only
and multimodal translations are shown. The object tags detected and used in
the multimodal translation setup are also shown. Hindi translations are shown
for the top two images, Bengali translations are shown for the middle two
images, and Malayalam translations are shown for the bottom two images.

###  3.4 Synthetic image features

Type of data Data set Images Source text Target text Sentences MMT Hindi
Visual Genome ✓ ✓ ✓ 29k Monolingual Flickr8k $\boldsymbol{+}$ ✓
$\boldsymbol{+}$ 40k CaptioningTranslationMTfeat. synth.

Figure 3: Process for generating synthetic image features. Green arrows
indicate processes to synthesize data, and green plus signs indicate the
resulting synthetic data. MT is short for machine translation, feat. synth.
for image feature synthesis.

We generate synthetic training data for the multimodal translation task by
enriching text-only data using synthetic image features (Grönroos et al.,
2018). Grönroos et al. (2018) use continuous image features and generate the
synthetic dummy features by taking the average vector of the features in the
training data. We improve on this procedure by generating discrete features
individually for each enriched training example by decoding from a sequence-
to-sequence (s2s) model. The s2s model is trained using the multimodal
training data (HVG), but instead of training the normal way, we use both
source language and target language text as input (with a separator token
between), and our object tags as the output. The model is a Transformer-base
(Vaswani et al., 2017) model trained using the Marian-NMT (Junczys-Dowmunt et
al., 2018) framework. The trained model is then used to enrich text-only
parallel data with synthetic features.

Our method applies to any text-only parallel data, even though in this
experiment we use it to enrich the text from an image captioning data set. Due
to the relatively small size of the multimodal training data sets, domain
match of additional training data has great importance for multimodal
translation (Grönroos et al., 2018). It is therefore important to apply domain
adaptation techniques when using general-domain text-only parallel data.

##  4 Results

We report the official automatic evaluation results of our models for all the
participated tasks in Table 2.

| WAT BLEU
---|---
System and WAT Task Label | Silo NLP | Best Comp
English$\rightarrow$Hindi MM Task |  |
MMEVTEXT21en-hi | 36.2 | 42.9
MMEVMM22en-hi | 42 | 39.4
MMCHTEXT22en-hi | 29.6 | 41.8
MMCHMM22en-hi | 39.1 | 39.3
English$\rightarrow$Malayalam MM Task |  |
MMEVTEXT21en-ml | 30.8 | 30.6
MMEVMM22en-ml | 41 | –
MMCHTEXT22en-ml | 14.6 | 19.5
MMCHMM22en-ml | 20.4 | –
English$\rightarrow$Bengali MM Task |  |
MMEVTEXT22en-bn | 41 | 41
MMEVMM22en-bn | 42.1 | 43.9
MMCHTEXT22en-bn | 22.6 | 32.9
MMCHMM22en-bn | 28.7 | 28.7

Table 2: WAT2022 Automatic Evaluation Results for English$\rightarrow$Hindi,
English$\rightarrow$Malayalam, and English$\rightarrow$Bengali. Rows
containing “TEXT" in the task label name denote text-only translation track,
and the rows representing “MM" denote multimodal translation including text
and images. “–" indicates single submission for the task. For each task, we
show the score of our system (Silo NLP) and the score of the best competitor
in the respective task.

On the E-Test sets, our multimodal systems receive the highest BLEU scores for
English$\rightarrow$Hindi and English$\rightarrow$Malayalam, and our text-only
systems outperform other text-only systems for English$\rightarrow$Malayalam
and English$\rightarrow$Bengali. Our multimodal systems consistently
outperform our text-only systems, with increases between +1.1 and +10.2 BLEU.

On the C-Test challenge sets, we have the highest BLEU score for
English$\rightarrow$Bengali multimodal translation. Again, our multimodal
systems outperform our text-only systems, with increases between +5.7 and +9.5
BLEU.

It should be noted that our English$\rightarrow$Hindi multimodal system and
English$\rightarrow$Bengali text-only system are unconstrained, making use of
substantial additions of in-domain data. For these language pairs, the
increase in translation quality can not be attributed entirely to
multimodality. However, for English$\rightarrow$Malayalam both systems use the
same training sentences, making these systems comparable. The
English$\rightarrow$Malayalam multimodal system outperforms the text-only
system by +10.2 BLEU for E-test and +5.7 BLEU for C-test. As the C-test
challenge set is constructed to contain translational ambiguity, the
improvement is an indication that the image features are useful for
disambiguation.

We demonstrate examples of translations obtained for the challenge test set in
Figure 2. The extracted image features in the form of object tags are also
shown for each image in the figure. We observe that the multimodal
translations are notably better than the text-only translations. This is
consistent with the pattern of the BLEU scores in Table 2.

##  5 Conclusions

In this system description paper, we presented our system for three tasks in
WAT2022: (a) English$\rightarrow$Hindi, (b) English$\rightarrow$Malayalam, and
(c) English$\rightarrow$Bengali Multimodal Translation. We released the code
through Github for research777https://github.com/shantipriyap/SiloNLP_WAT2022.

In the next step, we will explore the usage of synthetic features in
multimodal translation for the Malayalam and Bengali languages, to make use of
available text-only corpora for these language pairs.

## Acknowledgements

We are thankful to Silo AI888https://silo.ai, Helsinki, Finland for the
computation resources and the necessary support for participating in WAT2022.

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# YFS Resummation for Future Lepton-Lepton Colliders in S  HERPA

F. Krauss  Institute for Particle Physics Phenomenology, Durham University,
United Kingdom  A. Price  University of Siegen, Department of Physics, 57068
Siegen, Germany  M. Schönherr  Institute for Particle Physics Phenomenology,
Durham University, United Kingdom

###### Abstract

We present an implementation of the Yennie–Frautschi–Suura (YFS) scheme for
the all-orders resummation of logarithms from the emission of soft real and
virtual photons in processes that are critical for future lepton colliders.
They include, in particular, $e^{-}e^{+}\to f\bar{f}$ and $e^{-}e^{+}\to
W^{-}W^{+}$, where we validate the results of our implementation, improved
with fixed-order corrections, with those obtained from the most precise
calculations. We also show, for the first time, results for the Higgs-
Strahlungs process, $e^{-}e^{+}\to ZH$, in YFS resummation including fixed-
order improvements up to order $\alpha^{3}L^{3}$.

###### Contents

  1. 1 Introduction
  2. 2 Theory
  3. 3 Validation: $e^{-}e^{+}\to f\bar{f}$ and $e^{-}e^{+}\to W^{-}W^{+}$
  4. 4 YFS vs Collinear Resummation
  5. 5 Higgs Production
  6. 6 Summary and Outlook
  7. A YFS Infrared Functions
  8. B Photon Generation

##  1 Introduction

With an unprecedented amount of data gathered, the Large Hadron Collider (LHC)
continues to provide a rich environment, that allows us to further test and
scrutinise our knowledge of fundamental particles and their interactions with
matter. The discovery of the Higgs boson by the ATLAS[1] and CMS[2]
collaborations provided conclusive validation of spontaneously broken gauge
theories as the construction principle underpinning our understanding of
Nature, with the Standard Model (SM) of Particle Physics as its manifest
realization. To date, there has not been any direct discovery of physics
beyond the SM at the LHC, but there are observations that cannot be fully
explained within its framework: for example, the observation of non-zero
neutrino masses, dark matter and energy, and the anti-matter matter asymmetry.
Being intimately tied to the generation of fundamental masses and, through
them, CP violation as encoded in the CKM matrix, the Higgs-Boson may provide
key inputs to answer some of these questions.

While the LHC, and potential hadron collider successors, may yet provide a
deeper understanding to the nature of the Higgs-Boson, a future lepton-lepton
collider or ”Higgs-Factory” could provide unprecedented measurements of the
electroweak nature of the SM and thus provide an alternative experimental
avenue for the community to explore [3]. This is also stated in the European
Strategy update from 2020 [4]: _”An electron-positron Higgs-factory is the
highest-priority next collider”_. Due to their beams composed of elementary
particles, unlike a hadron machine, a lepton-lepton collider has a very clean
initial state, which facilitates measurements of electroweak pseudo-
observables (EWPO) [5] with unprecedented precision. One of these collider
designs is the Future Circular Collider, a post-LHC particle accelerator, that
initially would be operate in an $e^{+}e^{-}$ mode (FCC-ee) [6, 7] before
being repurposed to run as a high energy hadron-hadron collider (FCC-hh). For
the initial leptonic operations four centre-of-mass energies have been
proposed: After a first four-year stage, where the collider would run at the
Z-pole and collect a projected 150 $\mathrm{ab}^{-1}$ of data, corresponding
to the production of a staggering $10^{12}$ $Z$ bosons [6], the energy would
be increased for two years to $\sqrt{s}=161\,\mathrm{GeV}$, the $W^{+}W^{-}$
production threshold, with a projected integrated luminosity of 12
$\mathrm{ab}^{-1}$ or $10^{8}$ $W^{+}W^{-}$ pairs. After these two first
phases, a three-year run at $\sqrt{s}=240\,\mathrm{GeV}$ would turn the
collider into a Higgs factory, with copious production of the Higgs boson
through the Higgsstrahlung process $e^{+}e^{-}\rightarrow ZH$ and 1 million
$ZH$ events at an integrated luminosity of 5 $\mathrm{ab}^{-1}$. While this
energy does not correspond to the maximum cross section for Higgsstrahlung, it
would maximize the event per unit time due to the colliders luminosity
profile. This run will produce 1 million ZH events. The last phase of
operations would be at the $t\bar{t}$ production threshold, with a multipoint
scan around the threshold range $\sqrt{s}=345-365$ $\mathrm{GeV}$ and an
integrated luminosity of 1.5 $\mathrm{ab}^{-1}$, resulting in a million top-
pair production events. As an alternative option for an $e^{+}e^{-}$ collider,
the Circular Electron Positron Collider (CEPC) [8, 9], is projected as a
China-based Higgs-factory with a circumference of $80\,\mathrm{km}$. Similar
to the FCC-ee it is designed to operate at different centre-of-mass energies
during subsequent operation phases, namely at $91.2\,\mathrm{GeV}$ as a
$Z$-factory, producing close to 1012 $Z$ bosons, at $160\,\mathrm{GeV}$ for of
the production of 108 $W$ bosons at threshold, and at $240\,\mathrm{GeV}$ as a
Higgs-factory, resulting in 106 Higgs bosons. Like the FCC-ee, there is also
the possibility for the CEPC to be transformed into a hadron-hadron collider
at later stages.

While circular colliders can reach large integrated luminosities, they are
rather restricted in their energy reach due to QED Bremsstrahlung effects.
This is not a primary concern for linear colliders which can thus reach
energies in the multi-TeV range and provide polarised beams further amplifying
their physics potential, but they will operate at significantly reduced
luminosities. The Compact Linear Collider (CLIC) [10, 11, 12] has been
proposed as a multi-TeV linear $e^{+}e^{-}$ machine, to be built at CERN. It
would operate in three stages, with centre-of-mass energies at
$380\,\mathrm{GeV}$, $1.5\,\mathrm{TeV}$, and $3\,\mathrm{TeV}$. In the first
stage, about 160,000 Higgs bosons will be produced through Higgsstrahlung and
$WW$ fusion while during operations in the TeV range, it is expected to
produce millions of Higgs-bosons. The final $e^{+}e^{-}$ collider being under
current consideration is the International Linear Collider (ILC) [13, 14, 15,
16] in Japan. Being smaller than CLIC, it is expected to run at multiple
energies from the Z-pole up to $500\,\mathrm{GeV}$, with a possible later
upgrade to $1\,\mathrm{TeV}$. Dedicated runs at $91\,\mathrm{GeV}$ and
$160\,\mathrm{GeV}$ will focus on the $Z$ boson in $e^{+}e^{-}\rightarrow
f\bar{f}$ and the production of $W$ pairs at threshold, $e^{+}e^{-}\rightarrow
W^{+}W^{-}$, respectively. The run at $250\,\mathrm{GeV}$ will allow precision
studies of the Higgs-boson through $e^{+}e^{-}\rightarrow ZH$ with a projected
integrated luminosity of $250\,\mathrm{fb}^{-1}$. Final runs at the $t\bar{t}$
threshold and at $500\,\mathrm{GeV}$ could be further supplemented through
machine upgrades that would allow to probe couplings of the Higgs boson to the
top quark and to itself, with runs at $1\,\mathrm{TeV}$.

To take full advantage of these highly precise machines, the corresponding
theory uncertainties have to be smaller or, at least, match the size of the
experimental ones, cf. Table 1 for some examples. There are two main types of
theory uncertainties that need to be addressed by the community for the
successful analysis of the experimental results: parametric uncertainties,
which reflect our limited knowledge of the fundamental SM input parameters,
and uncertainties due to missing higher-order terms in our perturbative
calculations.

Observable | Where from | Current (LEP) | FCC (stat.) | FCC (syst.) | $\frac{\mathrm{N}ow}{\mathrm{F}CC}$
---|---|---|---|---|---
$M_{Z}$ [MeV] |  $Z$ linesh.[17] | $91187.5\pm 2.1\\{0.3\\}$ | $0.005$ | $0.1$ | 3
$\Gamma_{Z}$ [MeV] |  $Z$ linesh.[17] | $2495.2\pm 2.1\\{0.2\\}$ | $0.008$ | $0.1$ | 2
$R^{Z}_{l}=\Gamma_{h}/\Gamma_{l}$ |  $\sigma(M_{Z})$[18] | $20.767\pm 0.025\\{0.012\\}$ | $6\cdot 10^{-5}$ | $1\cdot 10^{-3}$ | 12
$\sigma^{0}_{\mathrm{h}ad}$[nb] |  $\sigma^{0}_{\mathrm{h}ad}$ [17] | $41.541\pm 0.037\\{0.025\\}$ | $0.1\cdot 10^{-3}$ | $4\cdot 10^{-3}$ | 6
$N_{\nu}$ |  $\sigma(M_{Z})$[17] | $2.984\pm 0.008\\{0.006\\}$ | $5\cdot 10^{-6}$ | $1\cdot 10^{-3}$ | 6
$N_{\nu}$ |  $Z\gamma$ [19] | $2.69\pm 0.15\\{0.06\\}$ | $0.8\cdot 10^{-3}$ | $<10^{-3}$ | 60
$\sin^{2}\theta_{W}^{eff}\\!\times 10^{5}$ |  $A_{FB}^{lept.}$[18] | $23099\pm 53\\{28\\}$ | $0.3$ | $0.5$ | 55
$\sin^{2}\theta_{W}^{eff}\\!\times 10^{5}$ |  $\langle{\cal P}_{\tau}\rangle$,$A_{\mathrm{F}B}^{pol,\tau}$[17] | $23159\pm 41\\{12\\}$ | $0.6$ | $<0.6$ | 20
$M_{W}$ [MeV] | ADLO[20] | $80376\pm 33\\{6\\}$ | 0.5 | 0.3 | 12
$A_{FB,\mu}^{M_{Z}\pm 3.5{\mathrm{G}eV}}$ |  $\frac{d\sigma}{d\cos\theta}$[17] | $\pm 0.020\\{0.001\\}$ | $1.0\cdot 10^{-5}$ | $0.3\cdot 10^{-5}$ | 100

Table 1:  The current systematic and statistical uncertainties on QED
sensitive observables, with terms in {…} denoting the contributions to QED
alone. The FCC-ee uncertainty estimates have been taken from [21], the overall
table has been reproduced from [22]

A large source of intrinsic theoretical uncertainties are corrections due to
QED radiation, where improvements by factors of 2-100, depending on the
observable, are mandatory to reduce the QED uncertainties to an acceptable
level. This demand reflects the simple fact that QED effects of the order
$0.1\%$, that could be safely ignored ignored at LEP, turn into limiting
factors in the full analysis of experimental results at future Higgs-
Factories. For example, from a precise measurement of the total hadronic cross
section of $e^{+}e^{-}\rightarrow f\bar{f}$ near the Z-pole, the mass and
width of the Z boson can measured with an uncertainty of the order of
$0.1\,\mathrm{MeV}$. This translates into the corresponding QED uncertainty
that will have to be reduced to $\delta_{QED}\leq 0.03\,\mathrm{MeV}$ [23],
mandating the inclusion of corrections up to
$\mathcal{O}\left(\alpha^{4}L^{4}\right)$ with $L=\log(s/m_{f}^{2})$.

This paper reports first steps towards achieving the necessary theoretical
precision within the framework of the S  HERPA event generator [24, 25, 26],
with a particular focus on the description of photon radiation off the
incoming leptons. In S  HERPA , as in many other Monte Carlo event generators
[27, 28, 29], this has so far been included through the structure function
approach [30] which resums the large logarithms associated to multiple
(collinear) photon emission through the
Dokshitser–Gribov–Lipatov–Altarelli–Parisi (DGLAP) equations [31, 32, 33, 34].
While S  HERPA implements the well-known leading-order accuracy [35, 36, 37],
in recent years the structure functions have been extended to next-to-leading
logarithmic accuracy [38, 39]. At fixed order, sub-leading logarithmic
corrections to $e^{+}e^{-}\rightarrow\gamma^{*}/Z$ have been calculated up to
$\mathcal{O}\left(\alpha^{6}L^{6}\right)$ [40, 41, 42].

However, instead of further improving the structure function approach
implementation in S  HERPA , we will here calculate the emission of soft
photons and resum the associated logarithms in the Yennie–Frautschi–Suura
formulation (YFS) [43]. In YFS, and in contrast to the structure function
approach, photon emissions are treated in a fully differential form,
explicitly creating the photons in an exact treatment of the emission phase
space and the recoil. We will built on previous experiences with the YFS
method, applied to final state radiation in decays of unstable particles in
the S  HERPA ’s P  HOTONS module [44], which has recently been supplemented
with the inclusion of weak corrections and the exact treatment of second-order
QED effects in the decays of $Z$ and Higgs bosons [45]. We will present a
process-independent implementation of the YFS method to initial state
radiation (ISR), resumming the associated soft logarithms to all orders, and
we will add the interplay with QED emissions in final-state radiation (FSR).
We will further add explicit higher-order QED and weak corrections for certain
processes, in particular for $e^{+}e^{-}\to f\bar{f}$, $e^{+}e^{-}\to
W^{+}W^{-}$, and for $e^{+}e^{-}\rightarrow HZ\rightarrow H\ell^{+}\ell^{-}$
which will pave the way for the systematic inclusion of such matrix element
corrections into the resummation framework.

In Section 2 we will summarise the YFS approach to the description of photon
emissions to all orders and the corresponding resummation of the associated
logarithms. This will be followed in Section 3 by the careful validation of
our results, comparing them with equivalent results obtained from K  KMC [46,
47] for $e^{+}e^{-}\to f\bar{f}$ and K  ORAL W/Y  FS WW [48] for
$e^{+}e^{-}\to W^{+}W^{-}$. We will quickly turn to a comparison with results
obtained from a structure function approach for the description of QED initial
state radiation in Section 4. In Section 5 we will show and discuss our
results for the Higgs–Strahlungs process $e^{+}e^{-}\rightarrow HZ\rightarrow
H\ell^{+}\ell^{-}$ in YFS resummation, which is of course of paramount
interest for the success of experiments at future $e^{+}e^{-}$ colliders. We
will summarise our work in Section 6, where we will also detail some future
steps.

##  2 Theory

The Yennie-Frautschi-Suura (YFS) formalism [43] provides a robust method for
resumming, to all orders, the potentially large logarithms that are associated
with the emission of real and virtual photons in the soft limit. YFS
systematically incorporates further improvements of the theoretical accuracy
of the resummation through the inclusion of exact fixed-order expressions. In
contrast to the structure function method, the YFS approach explicitly
generates resolved photons, with a resolution criterion given by an energy and
angle cut-off. In so doing the full kinematic structure of scattering events
is reconstructed which leads to a straightforward implementation of the YFS
method as both cross section calculator and event generator. We will detail
the relevant algorithms in App. B and focus here on a discussion of the theory
background only.

Summing over all real and virtual photon emissions, the total cross section
for a $2\to N$ scattering process with kinematics $p_{1}+p_{2}\to
p_{3}+p_{4}\dots+p_{N+2}$ is given by,

${\mathrm{d}}\sigma=\sum_{n_{\gamma}=0}^{\infty}\frac{1}{n_{\gamma}!}\,{\mathrm{d}}\Phi_{Q}\left[\prod_{i=1}^{n_{\gamma}}{\mathrm{d}}{\Phi_{i}^{\gamma}}\right]\left(2\pi\right)^{4}\delta^{4}\left(\sum_{i=1}^{2}p_{i}-\sum_{j=3}^{N+2}q_{j}-\sum_{k=1}^{n_{\gamma}}k_{k}\right)\left\lvert\sum_{\bar{n}_{\gamma}=0}^{\infty}\mathcal{M}^{\bar{n}_{\gamma}+\frac{1}{2}n_{\gamma}}_{n_{\gamma}}\right\rvert^{2}\\!\\!\,,$
(2.1)

where the outgoing momenta $q_{j}$ emerge from the original $p_{j}$ after the
effect of the real photon emissions has been taken into account. Here,
${\mathrm{d}}\Phi_{Q}$ denotes the modified final state phase space element,
the ${\mathrm{d}}\Phi_{i}^{\gamma}$ are the phase space elements spanned by
the $n_{\gamma}$ real photon momenta $k_{i}$ emitted off the leading order
configuration. Similarly, $\bar{n}_{\gamma}$ counts the number of virtual
photons added to it. The indices $i$ and $j$ matrix elements
$\mathcal{M}_{i}^{j}$ indicate the number of real photons, $i$, and the
overall additional orders in $\alpha$, $j$, relative to the Born
configuration. Consequently the Born matrix element for the core process
without any additional real or virtual photons is designated as
$\mathcal{M}_{0}^{0}$. This expression includes photon emissions to all
orders, but it is obviously an unrealistic ambition to be able to calculate
all its contributing terms in Eq. 2.1; instead we may only calculate the first
few terms in the perturbative series, until a target accuracy is reached.

### Exponentiation of virtual photon contributions

To analyse the structure of the resummation and its fixed-order improvements,
consider the case of a single virtual photon. In the soft limit, the matrix
element factorises as

$\displaystyle\mathcal{M}_{0}^{1}=\alpha BM_{0}^{0}+M_{0}^{1}\,.$ (2.2)

Therein $\alpha$ is the QED coupling constant, $B$ is an integrated off-shell
eikonal encoding the universal soft-photon limit, see App. A, and
$\mathcal{M}_{0}^{0}=M_{0}^{0}$ is the leading order matrix element. Then, the
finite remainder $M_{0}^{1}$ is the infrared-subtracted matrix element
including one virtual photon. YFS showed that the insertion of further virtual
photons leads to a power series; for up to two virtual photons we have

$\displaystyle\mathcal{M}_{0}^{0}=$ $\displaystyle M_{0}^{0},$
$\displaystyle\mathcal{M}_{0}^{1}=$ $\displaystyle M_{0}^{1}+\alpha
BM_{0}^{0},$ $\displaystyle\mathcal{M}_{0}^{2}=$ $\displaystyle
M_{0}^{2}+\alpha BM_{0}^{1}+\frac{(\alpha B)^{2}}{2!}M_{0}^{0}\,,$ (2.3)

where $M_{0}^{2}$ is now the infrared finite remainder of the matrix element
containing two virtual photons. This generalises to any number
$\bar{n}_{\gamma}$ of virtual photons,

$\displaystyle\mathcal{M}_{0}^{\bar{n}_{\gamma}}=\sum_{r=0}^{\bar{n}_{\gamma}}M_{0}^{\bar{n}_{\gamma}-r}\frac{(\alpha
B)^{r}}{r!}\;,$ (2.4)

and resumming all virtual photon emissions therefore gives

$\displaystyle\sum_{\bar{n}_{\gamma}=0}^{\infty}\mathcal{M}_{0}^{\bar{n}_{\gamma}}=\exp(\alpha
B)\sum_{\bar{n}_{\gamma}=0}^{\infty}M_{0}^{\bar{n}_{\gamma}}\,.$ (2.5)

Due to the abelian nature of QED and the absence of collinear singularities in
the soft-photon limit, this can be further generalised to squared matrix
elements that include any number of additional real photon emissions, such
that

$\displaystyle\left|\sum_{\bar{n}_{\gamma}=0}^{\infty}\mathcal{M}_{n_{\gamma}}^{\bar{n}_{\gamma}+\frac{1}{2}n_{\gamma}}\right|^{2}=$
$\displaystyle\exp(2\alpha
B)\left|\sum_{\bar{n}_{\gamma}}^{\infty}M_{n_{\gamma}}^{\bar{n}_{\gamma}+\frac{1}{2}n_{\gamma}}\right|^{2}\,.$
(2.6)

By construction, $M_{n_{\gamma}}^{\bar{n}_{\gamma}+\frac{1}{2}n_{\gamma}}$ is
completely free of soft singularities due to virtual photons but it will still
contain those due to real photons.

### Exponentiation of real photon contributions

For real photon emissions, the factorization occurs at the level of squared
matrix elements. For the case of a single real photon this can be expressed
as,

$\displaystyle\frac{1}{2(2\pi)^{3}}\left|\sum_{\bar{n}_{\gamma}=0}^{\infty}M_{1}^{\bar{n}_{\gamma}+\frac{1}{2}}\right|^{2}=\tilde{S}\left(k\right)\left|\sum_{\bar{n}_{\gamma}=0}^{\infty}M_{0}^{\bar{n}_{\gamma}}\right|^{2}+\sum_{\bar{n}_{\gamma}=0}^{\infty}\tilde{\beta}_{1}^{\bar{n}_{\gamma}+1}(k)\,.$
(2.7)

In this expression, all the (residual real emission) singularities are
contained within the eikonal, $\tilde{S}\left(k\right)$, defined in Eq. B.4.
The $\tilde{\beta}_{n_{\gamma}}^{\bar{n}_{\gamma}+n_{\gamma}}$ are the
infrared-finite squared matrix elements. They correspond to the Born level
process plus emissions of $n_{\gamma}$ real and $\bar{n}_{\gamma}$ virtual
photons at order $n_{\gamma}+\bar{n}_{\gamma}$ in the QED coupling $\alpha$.
For convenience we introduce the following notation,

$\displaystyle\tilde{\beta}_{n_{\gamma}}=\sum_{\bar{n}_{\gamma}=0}^{\infty}\tilde{\beta}_{n_{\gamma}}^{\bar{n}_{\gamma}+n_{\gamma}}\,.$
(2.8)

Extracting all real-emission soft photon divergences through eikonal factors,
the squared matrix element for any $n_{\gamma}$ real emissions, summed over
all possible virtual photon corrections, can be written as

$\displaystyle\left(\frac{1}{2(2\pi)^{3}}\right)^{n_{\gamma}}\left|\sum_{\bar{n}_{\gamma}=0}^{\infty}M_{n_{\gamma}}^{\bar{n}_{\gamma}+\frac{1}{2}n_{\gamma}}\right|^{2}$
$\displaystyle=$
$\displaystyle\;\;\;\tilde{\beta}_{0}\prod_{i=1}^{n_{\gamma}}\left[\tilde{S}(k_{i})\right]+\sum_{i=1}^{n_{\gamma}}\left[\frac{\tilde{\beta}_{1}(k_{i})}{\tilde{S}(k_{i})}\right]\prod_{j=1}^{n_{\gamma}}\left[\tilde{S}(k_{j})\right]+\sum_{\genfrac{}{}{0.0pt}{}{i,j=1}{i<j}}^{n_{\gamma}}\left[\frac{\tilde{\beta}_{2}(k_{i},k_{j})}{\tilde{S}(k_{i})\tilde{S}(k_{j})}\right]\prod_{l=1}^{n_{\gamma}}\left[\tilde{S}(k_{l})\right]+\dots$
$\displaystyle{}+\tilde{\beta}_{n_{\gamma}-1}(k_{1},\ldots,k_{i-1},k_{i+1},\ldots,k_{n_{\gamma}})\sum_{i=1}^{n_{\gamma}}\tilde{S}(k_{i})+\tilde{\beta}_{n_{\gamma}}(k_{1},\ldots,k_{n_{\gamma}})\,.$
(2.9)

Within this expression, all $\tilde{\beta}_{i}$ are free from all infrared
divergences due to either real or virtual photon emissions.

The first term in Section 2, $\tilde{\beta_{0}}$, contains all virtual photon
corrections to the Born matrix element and approximates the emission of
$n_{\gamma}$ real photons through the eikonals $\tilde{S}\left(k\right)$. The
second term corrects the eikonal approximation for one photon at a time to the
exact single photon emission matrix element, including all virtual corrections
to it. Similarly, the next term corrects the coherent emission of two real
photons to the exact expression including all virtual corrections, and so on.

In order to recombine all terms into an expression for the inclusive cross
section and facilitate the cancellation of all infrared singularities it is
useful to define an unresolved region $\Omega$ in which the kinematic impact
of any real photon emission is unimportant. Integrating over this unresolved
real emission phase space gives the integrated on-shell eikonal $\tilde{B}$,

$\displaystyle 2\alpha\tilde{B}(\Omega)$ $\displaystyle=$
$\displaystyle\int\frac{{\mathrm{d}}^{3}{k}}{k^{0}}\,\tilde{S}\left(k\right)\big{[}1-\Theta(k,\Omega)\big{]}\,,$
(2.10)

which contains all infrared poles due to real soft photon emission.111
$\Theta(k,\Omega)=1$ if the photon $k$ does not reside in $\Omega$ and zero
otherwise.  Substituting this expression back into Eq. (2.1), the
contributions originating from $\tilde{B}$ for all $n_{\gamma}$ photons again
exponentiate. This gives

$\displaystyle{\mathrm{d}}\sigma=\sum_{n_{\gamma}=0}^{\infty}\frac{e^{Y(\Omega)}}{n_{\gamma}!}\,{\mathrm{d}}{\Phi_{Q}}\left[\prod_{i=1}^{n_{\gamma}}{\mathrm{d}}{\Phi_{i}^{\gamma}}\,\tilde{S}\left(k_{i}\right)\,\Theta(k_{i},\Omega)\right]\left(\tilde{\beta}_{0}+\sum_{j=1}^{n_{\gamma}}\frac{\tilde{\beta}_{1}(k_{j})}{\tilde{S}\left(k_{j}\right)}+\sum_{{j,k=1}\atop{j<k}}^{n_{\gamma}}\frac{\tilde{\beta}_{2}(k_{j},k_{k})}{\tilde{S}\left(k_{j}\right)\tilde{S}\left(k_{k}\right)}+\cdots\right)\,,$
(2.11)

with the YFS form-factor

$Y(\Omega)=2\alpha\left[B+\tilde{B}(\Omega)\right]\;.$ (2.12)

Therein, all infrared singularities originating from real and virtual soft
photon emission, contained in $\tilde{B}$ and $B$ respectively, cancel,
leaving a finite remainder. An explicit expression for the form-factor can be
found in App. A.

Finally, let us comment on a technical complication in Eq. 2.11, related to
the question on how to evaluate matrix elements for the emission of a fixed
number of photons when the event itself contains many more soft photons. For
example, the $\tilde{\beta}_{1}$ terms are defined in the $(N+1)$-particle
phase space while the full event populates an $(N+n_{\gamma})$-particle phase
space. This necessitates the projection of the momenta of the latter onto a
phase space with lower dimension [46] and reflects the fact that the
subtraction, and, consequently, the calculation of the $\tilde{\beta}_{i}$,
proceeds at the end-point where the momenta of the emitted photons vanish. In
our example of the evaluation of the matrix correction for one real photon,
$\tilde{\beta}_{1}$, the phase space projection must satisfy four-momentum
conservation:

$\displaystyle\sum\limits_{i=1}^{2}p_{i}=\sum_{j=3}^{N+2}q_{j}+\sum_{k=1}^{n_{\gamma}}k_{k}\;\;\longrightarrow\;\;\sum_{i=1}^{2}\mathcal{R}p_{i}=\sum_{j=3}^{N+2}\mathcal{R}q_{j}+k_{1}\,.$
(2.13)

In this reduction step, there is some freedom in how the projection
$\mathcal{R}$ is chosen, exploiting the Lorenz invariance of the matrix
elements and the phase space. Different choices will lead to different
resulting, additional Jacobeans, and it is advantageous to employ such
mappings $q\to\mathcal{R}q$, where the Jacobean us unitary. We detail how we
generate the photon momentum in App. B.

### Collinear logarithms: EEX vs. CEEX exponentiation

While fixed-order perturbative calculations truncate the expansion at some
predefined order, for example at $\mathcal{O}\left(\alpha^{n}\right)$,
resummed calculations include certain classes of logarithms times coupling
constants to all orders. The YFS scheme resums, to all orders, the logarithms
emerging from soft real and virtual photon emissions, but misses the collinear
logarithms in the exponentiation, which, formally speaking are of the same
size. Instead, the inclusion of these logarithms is delegated to the infrared,
or, better, soft–finite hard remainders $\tilde{\beta}$. In principle, of
course, these corrections could be included to all orders, but in practice
this series is truncated at a given order
$\mathcal{O}\left(\alpha^{n}L^{m}\right)$, with $L$ the collinear logarithm
$L=\log(s/m_{i}^{2})$.

There are two approaches for inclusion of these higher order effects. The
first, known as exclusive exponentiation (EEX), closely follows the logic in
the original YFS paper by constructing the $\tilde{\beta}_{i}^{j}$ using
analytical differential distributions from the corresponding Feynman diagrams.
These expressions are given in terms of products of fours vectors – at low
orders usually Mandelstam variables or similar quantities. They have the
advantage that they are relatively straightforward to implement and their
behaviour in certain limits, like the soft or collinear limits, can be easily
checked for the correct behaviour. Independently constructing terms
corresponding to initial and final state radiation (ISR and FSR) contributions
facilitates the simple automation and implementation of the ISR contributions
for lepton ($e^{+}e^{-}$ or $\mu^{+}\mu^{-}$) colliders. Due to the possibly
complex final states such a simple automation is not feasible for the FSR
parts, and the corresponding contributions have to be included on a
case–by–case basis. The EEX corrections that we have included have been
calculated in [49, 50, 30, 51, 52, 53, 54] and the explicit expressions for
the IR subtracted terms can be found in [55].

Order | ISR Corrections | FSR Corrections | Reference
---|---|---|---
$\tilde{\beta}_{0}^{0}$ | Born | Born |
$\tilde{\beta}_{1}^{1}$ \+ $\tilde{\beta}_{0}^{1}$ | $\mathcal{O}\left(\alpha,\alpha L\right)$ | $\mathcal{O}\left(\alpha,\alpha L\right)$ | Eq. (13)
$\tilde{\beta}_{2}^{2}$ \+ $\tilde{\beta}_{1}^{2}$ \+ $\tilde{\beta}_{0}^{2}$ | $\mathcal{O}\left(\alpha^{2}L^{2}\right)$ | $\mathcal{O}\left(\alpha^{2}L\right)$ | Eq. (18,19)
$\tilde{\beta}_{3}^{3}$ \+ $\tilde{\beta}_{2}^{3}$ \+ $\tilde{\beta}_{1}^{3}$ | $\mathcal{O}\left(\alpha^{3}L^{3}\right)$ | — | Eq. (26)

Table 2:  The explicit beta terms that have been implemented in S

HERPA

’s EEX. The ISR corrections are universal and are applied to any lepton-
collider process. For the FSR the corrections are only applied to the
$f\bar{f}$ dipoles from decays of neutral bosons. The corrections to the decay
of charged bosons have been implemented in the P

HOTONS

module [44, 45] and the two methods will be combined in future work. The
reference column provides the equation number from [55] where the explicit
form of the corrections are given.

In the second approach, known as coherent exclusive exponentiation (CEEX)
[55], the soft–finite $\tilde{\beta}_{i}^{j}$ are constructed at the amplitude
level and therefore the subtraction is performed before squaring the
amplitudes. In turn, they will automatically include all ISR, FSR and ISR-FSR
interference effects as well as a consistent treatment of any spin effects.
The downside of this theoretically superior approach is the drastically
increased difficulty compared to EEX, which presents a veritable obstacle to
its straightforward and transparent automation. As a consequence, CEEX so far
has only been implemented for $e^{+}e^{-}\rightarrow f\bar{f}$, and an
algorithm underpinning an $e^{+}e^{-}\rightarrow W^{+}W^{-}$ implementation
has been described in [56].

### Infrared Boundary for FSR and ISR

To combine the effects of soft photon emissions from ISR and FSR, it is
important to guarantee the absence of double-counting, either positive or
negative. Following the literature and other implementations, e.g. in H  ERWIG
[57], in the EEX approach implemented in S  HERPA this is achieved by
analysing the singular domains in each case. They are given by rejecting
photons with an energy of $k^{0}$ in the c.m. frame of the emitting dipole
creating the eikonal as

$\displaystyle\Omega=\left\\{\begin{array}[]{lclcl}k^{0}_{I}&<&\displaystyle{\epsilon_{I}\,\cdot\,\frac{\sqrt{s_{Q}}}{2}}&\mbox{for}&\Omega_{I}\\\\[14.22636pt]
k^{0}_{F}&<&\displaystyle{\epsilon_{F}\,\cdot\,\frac{\sqrt{s_{Q}}}{2}\left(1+\frac{2\sum_{i}k_{i}\cdot
Q}{s_{Q}}\right)}&\mbox{for}&\Omega_{F}\,,\end{array}\right.$ (2.16)

where $s_{Q}=(q_{1}+q_{2})^{2}$, the c.m.-energy squared after initial photons
have been radiated off. These two regions need to be consistently combined,
for example [47] by choosing $\epsilon_{F}$ small enough such that the FSR
domain lies with the ISR domain. This requires to also remove any FSR photon
that resides in the overlap region
$\delta\Omega=\Omega_{I}\setminus\Omega_{F}$. This removal process does not
come for free. As we are in essence redefining the IR domain for the FSR
photons, the corresponding phase space integration has changed and the
contribution of the double-counted region to the overall exponentiated form
factor has to be removed. For each dipole constituting an FSR eikonal with
charges $Q_{f}$, this removal amounts to multiplication with a factor

$\displaystyle W_{\text{FSR/ISR removal}}$ $\displaystyle=$
$\displaystyle\exp\bigg{[}-2\alpha
Q_{f}^{2}\left[\tilde{B}(\Omega_{I};\bar{q}_{1},\bar{q}_{2})-\tilde{B}(\Omega_{F};\bar{q}_{1},\bar{q}_{2})\right]$
(2.17) $\displaystyle\hskip 25.60747pt+2\alpha
Q_{f}^{2}\left[\tilde{B}(\Omega_{I};q_{1},q_{2})-\tilde{B}(\Omega_{F};q_{1},q_{2})\right]\bigg{]}$

where $q_{1,2}$ are the final state momenta after the photon emission and
$\bar{q}_{1,2}$ are scaled momenta such that
$\bar{q}_{i}^{2}=m_{i}^{2}\frac{s_{Q}}{s}$. The second term in the exponential
is used to remove $\tilde{B}(\Omega_{F})$ from the original YFS form factor as
$\Omega_{I}$ now includes $\Omega_{F}$.

##  3 Validation: $e^{-}e^{+}\to f\bar{f}$ and $e^{-}e^{+}\to W^{-}W^{+}$

In this section we will compare our results with other implementations of the
YFS method. In particular, we will focus on the $e^{+}e^{-}$ generators K  KMC
[46, 47] and K  ORAL W/Y  FS WW [48]. As we are examining the effects of
multiple soft photon emissions, their QED coupling should be evaluated in the
Thompson limit, making $\alpha(0)$ the natural choice. We decouple this from
scheme choices for electroweak parameters which we keep process-dependent,
allowing in particular to use $\alpha$ at more suitable scales for the hard
scatter.

Figure 1:  Total cross-section for $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$ for
80$\mathrm{GeV}$ to 500$\mathrm{GeV}$ including ISR+FSR up to
$\mathcal{O}\left(\alpha^{3}L^{3}\right)$. In the first subplot we show the
deviation from the K

KMC

generator at the same accuracy. In the second subplot, we compare the
$\mathcal{O}\left(\alpha^{3}L^{3}\right)$ against the pure resummed
prediction. For the resummation only result all higher–order $\tilde{\beta}$
have been set to zero and only $\tilde{\beta}_{0}^{0}$
$=\left|\mathcal{M}_{0}^{0}\right|^{2}$ has been kept.

### $e^{-}e^{+}\to f\bar{f}$

For this process we employ the $\alpha(0)$ electroweak scheme, with values for
its input parameters listed in Table 3. In this scheme the widths of the gauge
bosons are taken to be fixed, $\sin\theta_{W}$ is calculated from the W and Z
masses as $\sin\theta_{W}$$=\sqrt{1-M_{W}^{2}/M_{Z}^{2}}$. In general, all
couplings associated with the emissions of soft photons are evaluated as
$\alpha(0)$, while the couplings associated with hard process can be chosen to
be different, for example $\alpha(M_{Z})$ for fermion-pair production on the
$Z$-pole.

| Mass [ $\mathrm{GeV}$ ] | Width [ $\mathrm{GeV}$ ]
---|---|---
Z | 91.1876 | 2.4952
W | 80.385 | 2.09692
H | 125 | 0.00407
e | 0.000511 | -
$\mu$ | 0.105 | -
$\alpha_{\text{QED}}^{-1}\left(0\right)$ | 137.03599976 |

Table 3:  Electroweak input parameters in the $\alpha(0)$ scheme.

In the comparison between S  HERPA and K  KMC , we first compared the total
$e^{-}e^{+}\to\mu^{-}\mu^{+}$ cross sections from both codes at a wide range
of energies, with results presented in Fig. 1. In these plots both ISR and FSR
have been included as well as perturbative fixed–order corrections up to
$\mathcal{O}\left(\alpha^{3}L^{3}\right)$. We used a minimal phasespace cut of
$M_{\mu^{+}\mu^{-}}>20\mathrm{GeV}$, and we took the infrared cut-off for
photon energies to be $E^{\gamma}_{cut}=0.1\mathrm{MeV}$. Both electron and
muon masses are included, as required by the YFS formalism. To facilitate a
like–for–like comparison between our implementation and K  KMC , we use the
same input parameters and cuts, and turn off any initial-final interference
(IFI) effects in K  KMC ; consequently, the K  KMC matrix elements are taken
in the exclusive exponentiation (EEX).

Figure 2:  Plot of the invariant mass distribution of the final state
particles. The nominal plots from S

HERPA

are displayed in the main frame. The sub-plots are the ratios with respect to
K

KMC

.

Figure 3:  The total number of photons for both ISR and FSR (left), and the
total sumeed photon energy (right), comparing S

HERPA

with K

KMC

.

In Figs. 2 and 3 we plot some differential distributions that further
elucidate the dynamics of soft photon emissions and verify that results
obtained from our implementation agree with the corresponding K  KMC results.
In Fig. 2 we exhibit the invariant mass and $p_{T}$ distributions of the final
state leptons, and we again find agreement at the permil level or below. At
large values of the transverse momentum we notice small deviations between the
two generators attributable to a lack of statistics in this highly suppressed
phase space region, which requires multiple non collinear photon emissions. In
particular the $p_{T}$ plot shows the fully exclusive nature of the YFS
scheme, in contrast to the more inclusive structure function approach, where
the lepton pair transverse momentum is integrated out and could be recovered
only a posteriori, for example, by combining the calculation with a parton
shower.

The left plot in Fig. 3 compares the number of ISR+FSR photons, generated in S
HERPA and in K  KMC , and we again find excellent agreement at the sub-permil
level. In the right part of the same figure, we show the summed photon
energies $\sum_{n}E_{\gamma}$, which exhibits a distinct shape at
$\frac{\sqrt{s}}{2}$. This shape results from the fact that the energy of a
single photon is constrained to be less than $\frac{\sqrt{s}}{2}$, to exceed
this kinematic limit the emission of at least two hard photons is needed. As
before, we notice a slight statistical fluctuation in the tail of the photon-
energy distribution.

### $e^{-}e^{+}\to W^{-}W^{+}$

Dedicated tools for the calculation of $W^{+}W^{-}$ production cross sections
fall into two categories: The first, including for example R  acoonWW [58, 59,
60] treat ISR effects through the structure function approach, while in the
second, exemplified by K  ORAL W/Y  FS WW [48], ISR is treated using YFS
resummation. Both of these tools include the full matrix elements for
four–fermion production, $e^{-}e^{+}\to 4f$, and K  ORAL W/Y  FS WW also
incorporates diagrams where the four fermions are produced through pairs of
$Z$ bosons.

Figure 4: Total cross-section for hadronic and leptonic decay channel of
$e^{+}e^{-}\rightarrow W^{+}W^{-}$.

The infrared factorization of soft-photon emissions can be easily extended to
include photon radiation off on-shell $W$ bosons. This is achieved by simply
replacing the mass $m_{f}$ and charge $Q_{f}$ of spin-$\frac{1}{2}$ fermions
in the amplitude and, correspondingly, the YFS form factor App. A, with the
mass $m_{W}$ and the charge $Q_{W}$ of the $W$ bosons [61]. In the case of
$e^{-}e^{+}\to W^{-}W^{+}$ these corrections induce an additional term in the
YFS form factor, namely

$Y(p_{e^{+}},p_{e^{-}})\rightarrow
Y(p_{e^{+}},p_{e^{-}})+Y(p_{W^{+}},p_{W^{-}}).$ (3.1)

As we focus on the EEX version of the soft photon resummation, we neglect any
coherence effects. One such effect occurs when a charged final state particle
becomes close to a beam particle with the same, or very similar, charge such
that the sum of initial and final state charges approaches 0. In these
configurations the charges screen each other, and the pattern of photon
emissions becomes increasingly similar to that expected from electric dipoles.
As a result the emitted photon spectrum becomes softer, an effect pointed out
e.g. in [62]. To correctly capture such effects a fully coherent description
of $e^{+}e^{-}\rightarrow 4f$ is necessary, which is beyond the scope of this
paper. However, this effect can be safely neglected in cases where the
particles are well separated, for example in the case of narrow resonant
production.

Figure 5:  $\cos\Theta_{W^{-}}$ of the $W^{-}$ boson with respect to the
incoming $e^{-}$ beam (left), and the corresponding transverse momentum
(right), comparing S

HERPA

with Y

FS

WW and the Born-level calculation.

Another important correction to $W^{+}W^{-}$ pair production cross sections
stems from electromagnetic Coulomb interactions [63, 64, 65] at the production
threshold, which leads to large corrections of the form
$\approx\frac{\alpha\pi}{\beta_{W}}$, i.e. an enhancement by the inverse
$W$-boson velocity $\beta_{W}$, which of course tends to zero at threshold.
The analytic result for the all order Coulomb correction was presented in [66]
and the effects of the second order contribution was investigated and shown to
yield a positive correction to the production cross section in the threshold
region. Being loop-induced this correction however is already included in the
virtual part of the YFS form-factor once QED emissions from the $W$ bosons are
included. Since the Coulomb effect can be treated to even higher orders we
keep it as a separate contribution in S  HERPA and therefore subtract it
consistently from the YFS form factor. Following [61] this results in a
subtracted virtual contribution, namely

$\displaystyle B_{\text{Subtracted}}\left(p_{W^{+}},p_{W^{-}}\right)$
$\displaystyle=$ $\displaystyle
B_{\text{Full}}\left(p_{W^{+}},p_{W^{-}}\right)-\frac{\theta(\beta_{\text{cut}}-\beta_{W})}{\beta_{W}}\lim_{\beta_{W}\rightarrow
0}\beta B_{\text{Full}}\left(p_{W^{+}},p_{W^{-}}\right)$ (3.2)
$\displaystyle=$ $\displaystyle
B_{\text{Full}}\left(p_{W^{+}},p_{W^{-}}\right)-\frac{\pi}{4\beta_{W}}\theta(\beta_{\text{cut}}-\beta_{W})\,.$

As made explicit in the equation above, the Coulomb correction can be
extracted from the full virtual YFS form factor by taking a suitable limit,
which is then subtracted in a region of small
$\beta_{W}<\beta_{\mathrm{c}ut}$, the phase space regime where the Coulomb
correction dominates. $\beta_{\mathrm{cut}}$ can be taken as a user input and
for the default value we follow the choice of K  ORAL W/Y  FS WW with
$\beta_{\mathrm{cut}}=0.382$.

Figure 6:  The effect of ISR and the Coulomb exchange at the $WW$ threshold,
with the ratio taken with respect to Y

FS

WW. The ISR reduces the total cross-section by about 30% and the Coulomb
correction enhancement is of order 5%.

For the validation of the S  HERPA implementation we employed the $G\mu$
scheme including all particle masses, Table 4 for the specific values we used.

| Mass [ $\mathrm{GeV}$ ] | Width [ $\mathrm{GeV}$ ]
---|---|---
Z | 91.1876 | 2.4952
W | 80.385 | 2.09692
H | 125 | 0.00407
e | 0.000511 | -
$\mu$ | 0.105 | -
$\alpha^{-1}_{G_{\mu}}$ | 132.231948 |
$G_{F}$ | $1.16637\times 10^{-5}\mathrm{GeV}^{-2}$ |

Table 4: List of electroweak parameters in the $G_{\mu}$ scheme used for the
validation of S

HERPA

with K

ORAL

W in $e^{-}e^{+}\to W^{-}W^{+}$ production processes.

For the comparison with K  ORAL W we only consider the contribution of three
diagrams, the $Z/\gamma$ s-channel and the neutrino mediated t-channel, to the
four-fermion final state [67] 222 Of course the full set of diagrams can be
easily included within the S  HERPA framework, and first order real and
virtual corrections can be extracted from S  HERPA ’s in-built matrix element
generators or from automated NLO tools such as O  PEN L  OOPS [68] or R  ECOLA
[69].. In Fig. 4 we exhibit the total cross-sections for hadronic and leptonic
final states at different c.m. energies, obtained by multiplying the $WW$-pair
production cross section with corresponding branching ratios. Again we see
excellent agreement between S  HERPA ’s YFS implementation and the K  ORAL W
benchmark. In terms of differential distributions, we see from Fig. 5 that
both the $\cos\Theta_{W^{-}}$ and the transverse momentum distribution agree
between both calculations. We also include the transverse momenta distribution
associated with born calculation. This emphasises the effect of the ISR has on
the $p_{T}$ which can be as large as 10%. It should be emphasised that for the
ISR treatment in $e^{+}e^{-}\rightarrow W^{+}W^{-}$ nothing has changed in the
resummation algorithm, demonstrating that the S  HERPA implementation is
independent of the final state. In Fig. 6 we demonstrate the impact of
including the Coulomb correction around the threshold region. As expected, it
provides a small, but non-negligible correction to the cross section in the
threshold region which must be taken into account at future $e^{+}e^{-}$
colliders. It is also noteworthy that our current implementation does not take
into account the effect of the sizeable $W$ width, or, conversely, its limited
lifetime. This will be left to a future publication.

##  4 YFS vs Collinear Resummation

In this section we will present a comparison of the YFS treatment of QED
initial-state radiation with its description in the collinear resummation or
structure function approach [30], that has been traditionally used in MC tools
such as S  HERPA or W  HIZARD . Instead of an expansion of the radiation
pattern around the soft region in the YFS approach, the stucture function
method focuses on large collinear logarithms and resums the corresponding
leading-log (LL) terms using the DGLAP evolution equations [31, 32, 33, 34].
Combining these evolution equations with leading-order (LO) initial
conditions, trivially the Dirac-$\delta$ function, yields the structure
functions. Going beyond LO, the trivial $\delta$ function is replaced with a
more complex mixed electron-photon state, giving rise to the recently derived
NLO structure functions [70, 71, 72]. These new PDFs are poised to become the
new standard in MC tools, which will systematically include NLO corrections,
for example in future versions of M  adGraph5_aMC@NLO [73].

In any case, at the currently mainly used LL accuracy, the structure functions
are convoluted with the ”partonic” cross section as

${\mathrm{d}}\sigma(s)=\int{\mathrm{d}}x_{1}{\mathrm{d}}x_{2}\;f_{e^{+}}(x_{1},Q^{2})\,f_{e^{-}}(x_{2},Q^{2})\;{\mathrm{d}}\hat{\sigma}(x_{1}x_{2}s),$
(4.1)

where ${\mathrm{d}}\hat{\sigma}$ is the differential partonic cross section.
At this accuracy level, $f_{e^{+}}(x)=f_{e^{-}}(x)$, and it is given by

$f_{e^{\pm}}(x,Q^{2})=\beta\,\frac{\exp\left(-\gamma_{E}\beta+\frac{3}{4}\beta_{S}\right)}{\Gamma\\!\left(1+\beta\right)}(1-x)^{\beta-1}\,+\beta_{H}\sum_{n=0}^{\infty}\beta_{H}^{n}\,\mathcal{H}_{n}(x)\;,$
(4.2)

with $\beta=\tfrac{\alpha}{\pi}\left(\ln(s/m_{e}^{2})-1\right)$. This form of
$\beta$ is a direct result of the analytical integration over the entire soft-
photon phasespace and any change to its definition will cause the IR
subtraction to fail. There is however some residual freedom in how non-leading
terms are taken into account, reflected in the treatment of the soft photon
residue $\beta_{S}$ as either $\ln(s/m_{e}^{2})$ or $\ln(s/m_{e}^{2})-1$. This
contrasts with the treatment in YFS resummation, where this freedom is absent.
The hard coefficients $\mathcal{H}_{n}(x)$ are given in [74, 75, 76, 77]. By
default, S  HERPA includes the hard coefficients up to second order ($n=2$).
For some popular choices of the various $\beta$’s, cf. Table 5.

Scheme | $\beta_{S}$ | $\beta_{H}$ | Refs
---|---|---|---
Beta | $\beta$ | $\beta$ |  [74]
Eta | $\eta$ | $\eta$ |  [75]
Mixed | $\beta$ | $\eta$ |  [76]

Table 5:  The different scheme choices available in S

HERPA

for the electron structure function, where
$\beta=\frac{\alpha}{\pi}\left(\ln(s/m_{e}^{2})-1\right)$ and
$\eta=\frac{\alpha}{\pi}\ln(s/m_{e}^{2})$.

Figure 7:  Total cross-section for $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$ for
80$\mathrm{GeV}$ to 500$\mathrm{GeV}$ where the ISR is modelled using a
collinear (blue) resummation and compared against the soft resummation up to
$\mathcal{O}\left(\alpha^{3}L^{3}\right)$. The red line represents the
strictly leading logarithmic YFS resummation while the green represents the
full YFS calculation.

When undertaking a comparison between the YFS resummation and the collinear
resummation care should be taken about which terms are included. For example,
if we consider the form factor in Eq. 4.2 we see that it is LL in nature while
if we consider the YFS form factor it contains terms that are at NLL. This can
be easily seen if we take the high energy limit of the form factor,

$Y\left(\Omega_{I}\right)\underset{s\gg
m_{e}^{2}}{\Longrightarrow}\exp\left\\{\beta\left[\log(\frac{2k_{\text{min}}}{\sqrt{s}})+\frac{1}{4}\right]+\frac{\alpha}{\pi}\left(\frac{\pi^{2}}{3}-\frac{1}{2}\right)\right\\}.$
(4.3)

The numerical effect of this addition can be seen in Fig. 7 were we consider
the YFS cross-section with and without this NLL contribution. We see that the
agreement between the strictly LL YFS calculation and the collinear
resummation is around 0.1% when the full YFS calculation is used, including
the NLL terms of soft origin, the difference is around 0.6%.

##  5 Higgs Production

We finally turn to a first application of our implementation for a new case,
namely the production of Higgs bosons in the Higgs-Strahlungs process at a
future electron–positron collider, $e^{-}e^{+}\to ZH$. To frame the
discussion, we first depict in Fig. 8 the c.m.-energy dependent Higgs
production cross section in various channels, and compare the results at Born
level with those obtained by including QED ISR in the YFS scheme. We see, as
expected, that QED ISR tends to increase the cross sections at high
c.m.-energies in those processes that are driven by the exchange of an
$s$-channel propagator, such as $e^{+}e^{-}\rightarrow ZH$ and the top-
associated production, while it tends to decrease the cross section at their
peaks. The increase can be thought off as the effect of some form of a
”radiative return” to the peak, while the decrease at the peak can be
understood as a ”washing out” of the large peak cross section by reducing the
c.m.-energy due to the ISR. Conversely, QED ISR decreases the production cross
sections throughout in those processes that are $t$-channel dominated.

Figure 8: Total cross-section for $e^{+}e^{-}\rightarrow HX$ at the Born level
and with ISR corrections included. For the ISR we include the perturbative
corrections up to $\mathcal{O}\left(\alpha^{3}L^{3}\right)$

In Fig. 9 we validate, once more, the treatment of QED FSR in our
implementation, by comparing the invariant mass distribution of the muon pair
in $e^{-}e^{+}\to Z_{\to\mu^{-}\mu^{+}}H$ obtained at Born level and by adding
FSR in two independent implementations of the YFS scheme within S  HERPA . The
muons are defined at the dressed level, meaning that nearby photons are
recombined with the undressed momenta of the muons using a cone size of
$\Delta R=0.1$. We apply a phasespace cut on the final state muons satisfying
$80\leq M_{\mu^{-}\mu^{+}}<115$. The electroweak parameters are the same that
we used in the K  KMC validation and are summarized in Table 3. At the
perturbative level we note that there are slightly different approaches of how
the $\tilde{\beta}$ terms are implemented. In P  HOTONS these corrections are
calculated using matrix element corrections to the bosonic decays. These are
exact corrections and are constructed using spin amplitude methods. The
corrections we employ for the P  HOTONS calculation are $\mathrm{NLO}$
accurate in QED, which corresponds to the second row of Table 2. As a
validation of our new FSR implementation these are also the $\tilde{\beta}$
terms we keep for the YFS++ calculation. We see the by now familiar form of
the shift of the mass distribution once FSR is added, and in addition we
confirm that our two implementations in the established P  HOTONS module [44]
and in our new Y  FS++ module are in perfect agreement. It is worth noting
that we normalised the distributions on the production cross section, which is
necessary because in Y  FS++ both QED ISR and FSR are included, while P
HOTONS is specifically geared towards inclusion of QED radiation in decays
only.

Figure 9:  Photon multiplicity for P

HOTONS

module and the new implementation (top left), contrasting pure FSR (red and
blue) as well the combined ISR+FSR (Green), the reconstructed $Z$ resonance
mass (top right), and the Higgs transverse momentum (bottom left). The
reconstructed $Z$ mass, again, contrasting the impact of ISR as modelled in
collinear factorisation and YFS resummation (bottom right) as further
validation.

##  6 Summary and Outlook

In this publication we have described, in detail, the implementation of soft
photon resummation in the Yennie–Frautschi–Suura scheme for initial and final
state emissions in a new module Y  FS++ within the S  HERPA event generator.
To take maximal advantage of the calculation tools already existing in S
HERPA we have restricted our implementation to the EEX approach, and we
anticipate that we will fully combine our new framework with automated modern
spin-amplitude methods, for example Berends-Giele recursion [78], to calculate
the IR finite $\tilde{\beta}$ using S  HERPA ’s built-in ME generators, A
MEGIC [79] and C  OMIX [80].

We validated our new implementation by detailed comparison with established
tools. In particular we confirmed the correctness of our implementation in
$e^{-}e^{+}\to\mu^{-}\mu^{+}$ by comparison with K  KMC , where we found
perfect agreement between the two codes. We also checked our code in a second
process, $e^{-}e^{+}\to W^{-}W^{+}$. While we focused mainly on the effect of
QED initial state radiation, to confirm the process–independence of our
implementation, we also added the Coulomb correction to the $W$-pair
production at threshold. We confirmed the correctness of our results by
finding perfect agreement with corresponding results obtained from K  ORAL W.
In a second validation step we confirmed, again for
$e^{-}e^{+}\to\mu^{-}\mu^{+}$, that the treatment of QED ISR in the YFS
approach yields results that are comparable to those obtained by using the
structure function approach. This is an important finding, as it shows that
expansion of the radiation pattern around either soft or collinear regions of
emission phase space and the subsequent resummation of the emerging
corresponding large logarithms yield numerically consistent results.

We finally turned to produce a first new result, and analysed the impact of
QED radiation in the YFS scheme on Higgs production in the Higgs-Strahlung
process at electron–positron colliders. This further shows that our
implementation can be applied in a process independent way, rendering it an
interesting asset for future precision studies of important processes at the
planned lepton colliders of the next decades.

We intend to build on this successful implementation and extend it in various
ways. As already indicated, as a first step we plan to fully connect the YFS
resummation with automated fixed–order tools to automatically include at least
the first–order/one–loop corrections. We hope that we can further improve the
accuracy for important processes by adding even higher–order corrections where
possible. In addition, we realise that the treatment of unstable particles,
i.e. resonances with sizable widths, deserves special consideration. We plan
to build on the important work by Jadach et al. in [56] and extend their
treatment to other processes. It is anticipated that the features described in
this paper will be made available in the future S  HERPA 3 release.

## Acknowledgement

A.P would like to thank Stanislaw Jadach for his help throughout all stages of
this work. This work has received funding from the European Union’s Horizon
2020 research and innovation programme as part of the Marie Skłodowska-Curie
Innovative Training Network MCnetITN3 (grant agreement no. 722104) and the
Marie Skłodowska-Curie grant agreement No 945422. F.K. gratefully acknowledges
funding as Royal Society Wolfson Research fellow. M.S. is funded by the Royal
Society through a University Research Fellowship (URF\R1\180549) and an
Enhancement Award (RGF\EA\181033 and CEC19\100349).

##  Appendix A YFS Infrared Functions

In this appendix, we present some analytical representations of the YFS
infrared (IR) functions corresponding to the emission of virtual and real
photons from a pair of charged massive particles. The YFS-Form-Factor
$Y(\Omega)$ reads

$Y(\Omega)=2\alpha\sum_{i<j}\left(\mathcal{R}e\;B(p_{i},p_{j})+\tilde{B}(p_{i},p_{j},\Omega)\right)\,,$

where the virtual eikonal factor is given by

$\displaystyle B(p_{i},p_{j})$ $\displaystyle=$
$\displaystyle-\frac{i}{8\pi^{3}}Z_{i}Z_{j}\theta_{i}\theta_{j}\int\frac{{\mathrm{d}}^{4}k}{k^{2}}\left(\frac{2p_{i}\theta_{i}-k}{k^{2}-2(k\cdot
p_{i})\theta_{i}}+\frac{2p_{j}\theta_{j}+k}{k^{2}+2(k\cdot
p_{j})\theta_{j}}\right)^{2},$

and the real eikonal factor reads

$\displaystyle\tilde{B}(p_{i},p_{j},\Omega)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}}Z_{i}Z_{j}\theta_{i}\theta_{j}\int{\mathrm{d}}^{4}k\,\delta(k^{2})\left(1-\Theta(k,\Omega)\right)\left(\frac{p_{i}}{(p_{i}\cdot
k)}-\frac{p_{j}}{(p_{j}\cdot k)}\right)^{2}\,.$

Here $Z_{i}$ and $Z_{j}$ are the charges of particles $i$ and $j$ in units of
the positron charge, respectively, and $\theta_{i,j}=\pm 1$ for final
(initial) state particles. $\Omega$ is the region of the phase space for which
the soft photons cannot be resolved. The divergences present in this
expression need to be regularised, which can be achieved by either introducing
a fictitious small photon mass $m_{\gamma}$, as in the original YFS paper
[43], or through dimensional regularisation.

### Virtual IR Function

Here we present the expression for the virtual part of the YFS for any two
charged massive particles.

$\displaystyle 2\alpha\mathcal{R}B(p_{1},p_{2})$ $\displaystyle=$
$\displaystyle-
Z_{i}Z_{j}\theta_{i}\theta_{j}\frac{\alpha}{\pi}\bigg{[}\left(\frac{1}{\rho}\ln\frac{\mu(1+\rho)}{m_{1}m_{2}}-1\right)\ln\frac{m_{\gamma}^{2}}{m_{1}m_{2}}+\frac{\mu\rho}{s}\ln\frac{\mu(1+\rho)}{m_{1}m_{2}}+\frac{m_{1}^{2}-m_{2}^{2}}{2s}\ln\frac{m_{1}}{m_{2}}-1$
$\displaystyle\hskip
51.21504pt+\frac{1}{\rho}\left(\pi^{2}-\frac{1}{2}\ln\frac{\mu(1+\rho)}{m_{1}^{2}}\ln\frac{\mu(1+\rho)}{m_{2}^{2}}-\frac{1}{2}\ln^{2}\frac{m_{1}^{2}+\mu(1+\rho)}{m_{2}^{2}+\mu(1+\rho)}-\mathrm{Li}_{2}(\zeta_{1})-\mathrm{Li}_{2}(\zeta_{2})\right)\bigg{]},$

where,

$\mu=p_{1}p_{2},\quad
s=2\mu+m_{1}^{2}+m_{2}^{2},\quad\rho=\sqrt{1-\left(\frac{m_{1}m_{2}}{\mu}\right)^{2}},\quad\zeta_{i}=\frac{2\mu\rho}{m_{i}^{2}+\mu(1+\rho)}.$
(A.2)

### Real IR Function

Here we present an expression for the IR function $\tilde{B}$ which
corresponds to the emission of a real photon $k\in\Omega$ from a dipole
consisting of two charged massive particles $p_{1}$ and $p_{2}$.

$\displaystyle 2\alpha\tilde{B}(p_{1},p_{2})$ $\displaystyle=$ $\displaystyle-
Z_{i}Z_{j}\theta_{i}\theta_{j}\frac{\alpha}{\pi}\bigg{[}\left(\frac{1}{\rho}\ln\frac{\mu(1+\rho)}{m_{1}m_{2}}-1\right)\ln\frac{\omega}{m_{\gamma}^{2}}+\frac{1}{2\beta_{1}}\ln\frac{1+\beta_{1}}{1-\beta_{1}}$
(A.3) $\displaystyle\hskip
51.21504pt+\frac{1}{2\beta_{2}}\ln\frac{1+\beta_{2}}{1-\beta_{2}}+\mu
G\left(p_{1},p_{2}\right)\bigg{]},$

where $\beta_{i}=\frac{|\vec{p}_{i}|}{E_{i}}$ and $\mu,\rho$ are defined in
Eq. A.2 and $\omega$ is the momentum cut-off specifying $\Omega$ in the frame
$\tilde{B}$ is to be evaluated in. $G\left(p_{1},p_{2}\right)$ is a
complicated function that can be expressed as a combination of logarithms and
dilogarithms,

$\displaystyle G\left(p_{1},p_{2}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\left(Q^{2}+M^{2}\right)\left(Q^{2}+\delta^{2}\right)}}\bigg{[}\ln\frac{\sqrt{\Delta^{2}+Q^{2}}-\Delta}{\sqrt{\Delta^{2}+Q^{2}}+\Delta}\left[\chi_{23}^{14}(\eta_{1})-\chi_{23}^{14}(\eta_{0})\right]+Y(\eta_{1})-Y(\eta_{0})\bigg{]},$

where,

$\displaystyle\chi_{kl}^{ij}$ $\displaystyle=$
$\displaystyle\ln\left|\frac{(\eta-y_{i})(\eta-y_{j})}{(\eta-y_{k})(\eta-
y_{l})}\right|$ $\displaystyle Y\left(\eta\right)$ $\displaystyle=$
$\displaystyle
Z_{14}\left(\eta\right)+Z_{21}\left(\eta\right)+Z_{32}\left(\eta\right)-Z_{34}\left(\eta\right)+\frac{1}{2}\chi_{34}^{12}\left(\eta\right)\chi_{14}^{23}\left(\eta\right)$
$\displaystyle Z_{ij}\left(\eta\right)$ $\displaystyle=$ $\displaystyle
2\mathrm{Li}_{2}\left(\frac{y_{j}-y_{i}}{\eta-
y_{i}}\right)+\frac{1}{2}\ln^{2}\left|\frac{\eta-y_{i}}{\eta-y_{j}}\right|,$
(A.5)

and

$\displaystyle\eta_{0}$ $\displaystyle=$
$\displaystyle\sqrt{E_{2}^{2}-m_{2}^{2}},\eta_{1}=\sqrt{E_{1}^{2}-m_{1}^{2}}+\sqrt{\Delta^{2}+Q^{2}}$
$\displaystyle y_{1,2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bigg{[}\sqrt{\Delta^{2}+Q^{2}}-E+\frac{M\delta\pm\sqrt{\left(Q^{2}+M^{2}\right)\left(Q^{2}+\delta^{2}\right)}}{\sqrt{\Delta^{2}+Q^{2}}+\Delta}\bigg{]}$
$\displaystyle y_{3,4}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bigg{[}\sqrt{\Delta^{2}+Q^{2}}+E+\frac{M\delta\pm\sqrt{\left(Q^{2}+M^{2}\right)\left(Q^{2}+\delta^{2}\right)}}{\sqrt{\Delta^{2}+Q^{2}}-\Delta}\bigg{]},$

where the following notation has been introduced,

$\displaystyle\Delta$ $\displaystyle=$ $\displaystyle
E_{1}-E_{2},E=E_{1}+E_{2},$ $\displaystyle\delta$ $\displaystyle=$
$\displaystyle m_{1}-m_{2},M=m_{1}+m_{2},$ $\displaystyle Q^{2}$
$\displaystyle=$ $\displaystyle-\left(p_{1}-p_{2}\right)^{2}.$ (A.7)

##  Appendix B Photon Generation

In this section we discuss the explicit construction of the photon momenta in
the Monte Carlo integration and event generation. In each event the overall
number of resolved photons is generated according to a Poissonian with mean

$\displaystyle\langle
n_{\gamma}\rangle=\,-\frac{\alpha}{\pi}\ln\left(\frac{E_{\text{max}}}{E_{\text{min}}}\right)\left(\frac{1+\beta_{1}\beta_{2}}{\beta_{1}+\beta_{2}}\ln\frac{\left(1+\beta_{1}\right)\left(1+\beta_{2}\right)}{\left(1-\beta_{1}\right)\left(1-\beta_{2}\right)}-2\right),$
(B.1)

where $\beta_{i}\,=\frac{|\vec{p_{i}}|}{p_{i}^{0}}$. We introduce an explicit
form for the photon momentum parameterizing the momentum using polar
coordinates.

$k_{i}=\frac{\sqrt{s}}{2}x_{i}(1,\sin\theta_{i}\cos\phi_{i},\sin\theta_{i}\sin\phi_{i},\cos\theta_{i}).$
(B.2)

In this parameterization the eikonal term is given by,

$\frac{d^{3}k_{i}}{k_{i}^{0}}\tilde{S}(k_{i})\,=\frac{dx_{i}}{x_{i}}\,d(\cos\theta_{i})\,d\phi_{i}\,g(\theta_{i}),$
(B.3)

where,

$\tilde{S}\left(k\right)=\sum_{i,j}\frac{\alpha}{4\pi^{2}}\,Z_{i}Z_{j}\theta_{i}\theta_{j}\left(\frac{p_{i}}{p_{i}\cdot
k}-\frac{p_{j}}{p_{j}\cdot k}\right)^{2}$ (B.4)

and $g(\theta)$ is given by,

$g(\theta)\,=\frac{\alpha}{\pi^{2}}\left(\frac{2(1+\beta_{1}\beta_{2})}{(1-\beta_{1}\cos\theta)(1+\beta_{2}\cos\theta)}-\frac{1-\beta_{1}^{2}}{(1-\beta_{1}\cos\theta)^{2}}-\frac{1-\beta_{2}^{2}}{(1+\beta_{2}\cos\theta)^{2}}\right),$
(B.5)

### Photon Angles

There are two angles used in the parametrisation of the photon momentum that
have to be generated. The first, $\phi$, is trivial and is given by,

$\phi=2\pi\\#,$ (B.6)

where # is a uniformly generated random number $\in\left(0,1\right)$. The
remaining angle $\theta$ is slightly more complex. It is generated by sampling
from the $\tilde{S}\left(k\right)$ distribution,

$\displaystyle\tilde{S}\left(k\right)$ $\displaystyle\propto$
$\displaystyle\left(\frac{p_{1}}{p_{1}k}-\frac{p_{2}}{p_{2}k}\right)^{2},$
(B.7) $\displaystyle=$
$\displaystyle\left(\frac{p_{1}^{2}}{\left(p_{1}k\right)^{2}}+\frac{p_{2}^{2}}{\left(p_{2}k\right)^{2}}-\frac{2p_{1}p_{2}}{\left(p_{1}k\right)\left(p_{2}k\right)}\right).$

By taking $p_{1}$ and $p_{2}$ to be the beam momenta, i.e.
$p_{1/2}\,=\left(\frac{\sqrt{s}}{2},0,0,\pm p_{z}\right)$, the eikonal is
given by

$\displaystyle\tilde{S}\left(k\right)\propto\left(\frac{1-\beta_{1}^{2}}{\left(1-\beta_{1}\cos\theta\right)^{2}}+\frac{1-\beta_{2}^{2}}{\left(1-\beta_{2}\cos\theta\right)^{2}}-\frac{2\left(1+\beta_{1}\beta_{2}\right)}{\left(1-\beta_{1}\cos\theta\right)\left(1+\beta_{2}\cos\theta\right)}\right)\,.$
(B.8)

The interference term can be rewritten as,

$\frac{1}{\left(1-\beta_{1}\cos\left(\theta_{i}\right)\right)\left(1+\beta_{2}\cos\left(\theta_{i}\right)\right)}=\frac{\beta_{1}\beta_{2}}{\beta_{1}+\beta_{2}}\left(\frac{1}{\beta_{2}\left(1-\beta_{1}\cos\left(\theta_{i}\right)\right)}+\frac{1}{\beta_{1}\left(1+\beta_{2}\cos\left(\theta_{i}\right)\right)}\right).$
(B.9)

Then $\cos\left(\theta_{i}\right)$ is generated according to either of the two
terms in the interference. Generating it according to
$\big{(}1\mp\beta_{i}\cos\left(\theta_{i}\right)\big{)}^{-1}$

$\displaystyle\int_{-1}^{y}d\cos\left(\theta_{i}\right)\frac{1}{1\mp\beta_{i}\cos\left(\theta_{i}\right)}$
$\displaystyle=$
$\displaystyle\\#\int_{-1}^{1}d\cos\left(\theta_{i}\right)\frac{1}{1\mp\beta_{i}\cos\left(\theta_{i}\right)},$
$\displaystyle\cos\left(\theta_{i}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\beta_{i}}\left(1-\left(1\pm\beta_{i}\right)\left(\frac{1\mp\beta_{i}}{1\pm\beta_{i}}\right)^{\\#}\right).$
(B.10)

The probability associated with either of these distributions is given by,

$P_{i}=\frac{\ln\left(\frac{1\pm\beta_{i}}{1\mp\beta_{i}}\right)}{\ln\left(\frac{1+\beta_{1}}{1-\beta_{1}}\right)+\ln\left(\frac{1+\beta_{2}}{1-\beta_{2}}\right)}$
(B.11)

The overall correction weight for this distribution is given by,

$W=\displaystyle\frac{\displaystyle\frac{2\left(1+\beta_{1}\beta_{2}\right)}{\left(1-\beta_{1}\cos\theta\right)\left(1+\beta_{2}\cos\theta\right)}-\displaystyle\frac{1-\beta_{1}^{2}}{\left(1-\beta_{1}\cos\theta\right)^{2}}-\displaystyle\frac{1-\beta_{2}^{2}}{\left(1-\beta_{2}\cos\theta\right)^{2}}}{\displaystyle\frac{2\left(1+\beta_{1}\beta_{2}\right)}{\left(1-\beta_{1}\cos\left(\theta_{i}\right)\right)\left(1+\beta_{2}\cos\left(\theta_{i}\right)\right)}}\,.$
(B.12)

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