# Latent Variable Models for Visual Question Answering

Zixu Wang1, Yishu Miao1, Lucia Specia1,2
1Department of Computing, Imperial College London, UK
2Department of Computer Science, University of Sheffield, UK
{zixu.wang, y.miao20<EMAIL_ADDRESS>
###### Abstract

Current work on Visual Question Answering (VQA) explore deterministic
approaches conditioned on various types of image and question features. We
posit that, in addition to image and question pairs, other modalities are
useful for teaching machine to carry out question answering. Hence in this
paper, we propose latent variable models for VQA where extra information
(_e.g_. captions and answer categories) are incorporated as latent variables,
which are observed during training but in turn benefit question-answering
performance at test time. Experiments on the VQA v2.0 benchmarking dataset
demonstrate the effectiveness of our proposed models: they improve over strong
baselines, especially those that do not rely on extensive language-vision pre-
training.

##  1 Introduction

As a classic multi-modal machine learning problem, Visual Question Answering
(VQA) [2] systems are tasked with providing a correct textual answer given an
image and a textual question. Current VQA models [14, 8, 1] are trained to
learn the relationship between areas in an image and the question, and to
choose the correct answer from a vocabulary of answer candidates, _i.e_., they
are modelled as a classification problem. The majorities of popular VQA [7, 9,
18] models are created in a deterministic manner and explore solely
information from the given image-question pair. There are other approaches
attempting to incorporate extra information, such as image captions [17] and
mutated inputs [6, 3]. However, it in turn restricts the practical
applications as the extra information is required to be explicitly available
during testing.

In this paper, we propose an approach to explore additional information as
latent variables in VQA: we employ latent variables for VQA to exploit extra
information (_i.e_. image captions and answer categories) to complement
limited textual information from image and question pairs. We assume a
realistic setting where this information – esp. captions – may only be
available during the training phase. To that end, we introduce a continuous
latent variable as the caption representation to capture the essential
information from this modality. Moreover, the answer category is modelled as a
discrete latent variable, which acts as an inductive bias to benefit the
learning of answer prediction, and can be integrated out during testing. The
motivation is that the generative framework is able to incorporate many other
types of information as continuous or discrete latent variables, and as such
it effectively leverages additional resources to constrain the original image-
question distribution while omitting them in testing. This grants the models
with stronger generalisation ability compared to its deterministic
counterparts, which generally require off-the-shelf pipelines to model the
information from external modalities.

Intuitively, image captions describe diverse aspects of an image and include
attributes and relations of objects in a more informative way. In our work, a
continuous latent variable is employed for capturing the caption distributions
and constraining the generative distribution conditioned on image and question
pairs. In this way, the joint multimodal representations from images and
question can benefit from the caption modality during training, and it
requires no explicit caption inputs in testing. Similarly, there exists a
strong connection between a question and answer pair when the question
provides informative signals on its type or the category of possible answers.
For example, “How many”, “Where is” and “what is” normally connect to numbers,
locations, and objects respectively. We propose a discrete latent variable is
employed for modelling answer categories and providing better inductive bias
from the question and answer pairs.

In summary, our main contributions are:

  * •

A novel generative VQA framework combining the modularity of latent variables
with the flexibility to introduce extra information as continuous and/or
discrete latent variables.

  * •

A method to incorporate additional information which does not rely on building
multiple deterministic pipelines, aiming at learning the underlying
compositional, relational, and hierarchical structures of multiple modalities.
The models benefit from the extra information during training without
providing explicit inputs in testing.

  * •

The improvements over deterministic baseline models (_e.g_. UpDn [1] and VL-
BERT [15]) in experiments with the VQA v2.0 dataset demonstrate the
effectiveness of our proposed latent variable models. Our qualitative analysis
also indicates that using extra resources (_i.e_. captions and answer
categories) as latent variables captures complementary information during
training and benefits the VQA performance in testing.

Figure 1: Architecture of our latent variable model for VQA. We use dotted
lines to denote the process of proposed latent variables.

##  2 Model

We first present an overview of our general model structure, followed by the
encoders for different modalities, and the proposed corresponding latent
variables.

###  2.1 General Model Structure

In a VQA task, images and questions are normally used to learn a joint
multimodal distribution for answer predictions. We postulate that the joint
representation can be improved by other multimodal information. Hence, we
introduce captions and answer categories to our VQA model as continuous and
discrete latent variables respectively to encourage a better learning in the
joint distribution of image and question pairs during training. A notable
advantage of the latent variable models is that they do not explicitly require
captions or answer categories during testing, and therefore can be easily
extended to condition on any other useful information.

Firstly we introduce the notations used in the general VQA model. $V$, $Q$,
$A$ are used to denote the input image, question, and answer instances
respectively. The image feature $v$, question representation $q$, and answer
representation $a$ are extracted from the image encoder, question encoder, and
answer encoder. The VQA task is constructed as a classification problem to
output the most likely answer $\hat{a}$ from a fixed set of answers based on
the content of the image $v$ and question $q$:

$\hat{a}=\textrm{argmax}\ p(a|v,q)$ (1)

In our latent variable model, we introduce image captions $C$ to the training
phase. Similarly, we extract the caption features $c$ by a caption encoder.
However, instead of directly feeding in the caption features $c$ into to the
model, we employ a continuous latent distribution $z$ to be the caption
representations. Here $z\sim q(z|c)$ is modelled as variational distribution.
We then build a generative distribution $z\sim p(z|v,q)$ to infer the caption
information by conditioning on image and question pairs, which is optimised
during training via neural variational inference. We originally experimented
using $q(z|v,q,c)$ as the vairational distribution. However, this distribution
is quite close (_i.e_. small KL divergence) to the generative distribution
$p(z|v,q)$, which weakens the learning signal from KL divergence.

In addition, we introduce a discrete latent variable $d$ for modelling answer
category inferred via $d\sim p(d|v,q)$, which is also conditioned on image and
question pairs.

Hence, the training of the latent variable model is carried out by the samples
$(v,q,a,c,d)$. During testing, the answer $a$ is predicted from the image and
question pair $(v,q)$:

$\hat{a}\\!=\\!\textrm{argmax}\\!\sum_{d,z}p(a|v,q,d,z)p(d|v,q)p(z|v,q)$ (2)

where the discrete latent variable $d$ is directly integrated out, and the $z$
is the Monte-Carlo sample from $p(z|v,q)$.

###  2.2 Continuous Latent Variable: Caption

As captions are modelled by a continuous latent variable, we only have
explicit captions during training. Here we present the generative distribution
that is conditioned on images and questions during testing, and the
variational distribution that is conditioned on explicit captions during
training. Therefore, the caption encoder is only used in the training phase.

Generative Distribution - $p_{\theta}(z|v,q)$. We use a latent distribution
$p_{\theta}(z|v,q)$ to model the joint multimodal distributions of images and
questions. Compared to its deterministic counterpart using concatenated
multimodal features, we parameterise the stochastic distribution with
$\mathcal{N}(z|\mu_{\theta}(v,q),\sigma^{2}_{\theta}(v,q))$.

Variational Distribution - $q_{\phi}(z|c)$. We first apply a RNN model to
embed the caption inputs $C$ and a latent variable $q_{\phi}(z|c)$ to model
the caption semantics and distributions, where
$z\sim\mathcal{N}(z|\mu_{\phi}(c),\sigma^{2}_{\phi}(c))$.

###  2.3 Discrete Latent Variable: Answer Category

Assuming that each image and question pair $(v,q)$ can be projected to an
answer category to help find a correct answer, we are able to encourage the
model to distinguishing candidates across answer categories instead of only
the spurious relationships between questions and answers via simple linguistic
features. Therefore, in order to leverage this useful inductive bias, we
propose a discrete latent variable to model the answer category given an image
and question pair $(v,q)$. In particular, for each answer category $d$, we
have a conditional independent distribution $p(a|v,q,d)$ over the answers in
the certain answer category.

$p(a|v,q)=\sum_{d}p(a|v,q,d)\cdot p(d|v,q)$ (3)

We trained an answer category classifier using joint image-question pairs as
the input, given the true labels as shown at left bottom in Figure 1. We then
use the category distribution to modify the answer distribution through
element-wise production to get more precise answer distribution.

##  3 Datasets & Setup

###  3.1 Datasets

We use the VQA v2.0 dataset [2] for our proposed latent variable model. The
answers are balanced in order to minimise the effectiveness of dataset priors.
We report the results on validation set and test-standard set through the
official evaluation server. The source of image captions in our work is the
MSCOCO dataset [12].

We use answer categories from the annotations of [10]. The answers in the VQA
v2.0 dataset are annotated with a set of 15 categories for the top 500 answers
that makes up the 82% 111Although the category definitions cannot cover all
types of answers, and false prediction during testing might be observed, the
latent variable can still maintain the robustness in predicting correct
answers by summing over all the probabilities of predicted categories. of the
VQA v2.0 dataset; and the other answers are treated as an additional category.

##  4 Experiments

In this section, we first describe the experimental results of our latent
variable model, compared with both a UpDn (Bottom-up Top-down) baseline model
and a state-of-the-art pre-trained visual-linguistic model (VL-BERT); then we
conduct qualitative analysis to validate the effectiveness of proposed
components.

| VQA v2.0 test-dev (%) | test-std (%)
---|---|---
| All | Yes/No | Num | Other | All
Caption [17] | - | - | - | - | 68.37
DFAF [13] | 70.22 | 86.09 | 53.32 | 60.49 | 70.34
MLIN [5] | 71.09 | 87.07 | 53.39 | 60.49 | 71.27
UpDn [1] | 65.32 | 81.82 | 44.21 | 56.05 | 65.67
\+ latent (ours) | 66.01 | 82.96 | 44.58 | 55.94 | 66.29
VL-BERT${}_{\textrm{large}}$ [15] | 71.79 | - | - | - | 72.22
\+ latent (ours) | 72.03 | 88.03 | 54.16 | 62.42 | 72.37

Table 1: Experimental results on VQA v2.0 test-dev and test-standard (test-
std) set. Accuracies are reported in percentage (%) terms. The state-of-the-
art scores are in bold; underlined scores are best among (baseline _vs_.
latent variable extension); and both underlines and bold scores are the
overall best results.

###  4.1 Quantitative Analysis

We compare the results of our latent variable model with the baseline model
(UpDn), a state-of-the-art visual-linguistic pre-training model (VL-BERT), and
three other related VQA models; where [17] uses generated captions to assist
answer predictions and [13, 5] explore the interactions between visual and
linguistic inputs.

As demonstrated in Table 1, our latent variable model outperforms when acting
as an extension. In particular, our latent variable model outperforms UpDn by
0.69% accuracy on test-dev set and by 0.62% accuracy on test-standard set. In
addition, our model improves the performance by 0.24% accuracy than its VL-
BERT counterpart on test-dev set and by 0.15% accuracy on test-standard set.
These results indicate the effectiveness of including captions and answer
categories as latent variables, to promote the distribution of image and
caption pairs to be closer to the captions’ space, and to learn a better
distinction among different kinds of answers, or different answers within the
same answer category.

The result of [17] (68.37%) is a very strong baseline, which follows a
traditional deterministic approach. However, their model is trained to
generate captions that can be used at test time, while in our case only image
and question pairs are required for answer prediction. [13] and [5] both
achieve comparable performance (70.34% and 71.27% on test-standard set,
respectively) to VL-BERT (72.22%) without pre-training, by dynamically
modulating the intra-modality information and exploring the latent interaction
between modalities. Our latent variable model has the overall best result when
combined with the strong pre-training VL-BERT, which indicates both the
effectiveness of the visual-linguistic pre-training framework, and the
incorporation of continuous (captions) and discrete (answer categories) latent
variables.

Compared to the results on the standard baseline (UpDn), the improvements
achieved by our proposed model on the VL-BERT framework is smaller. This is
because VL-BERT has been pre-trained on massive image captioning data, where
the learning of visual features have largely benefited from the modality of
captions already. Nevertheless, based upon the strong baseline model, our
proposed model can still improve performance slightly, which further indicates
the effectiveness of the latent variable framework.

The state-of-the-art performance on VQA v2.0 among pre-training frameworks is
achieved by LXMERT, Oscar and Uniter [16, 11, 4]. They have been extensively
pre-trained using massive datasets on languages and vision tasks (including
VQA) in a multi-task learning fashion. Our work is not directly comparable,
and is not aimed at improving and beating the state-of-the-art performance.
Instead, it is focused on exploring the potential of latent variable models to
represent additional useful information in multimodal learning and to
contribute to pre-trained vision-language frameworks. We draw attention to the
advantage of using generative framework on the VQA task. In this case, we can
employ more information during training (which is omitted in testing) to
regularise the original multimodal distribution. This can be demonstrated by
the improvements on VL-BERT brought by the latent variables.

| VQA v2.0 val
---|---
| All | Yes/No | Num | Other
UpDn | 63.15 | 80.38 | 42.84 | 55.86
UpDn + caption | 63.85 | 81.10 | 43.63 | 55.90
UpDn + category | 63.51 | 81.62 | 42.17 | 55.38
Ours w/o caption | 64.09 | 81.82 | 44.37 | 55.74
Ours w/ caption | 64.24 | 82.36 | 44.52 | 56.02

Table 2: Ablation study to investigate the effect of each component: caption,
and answer category. “Ours w/o caption” indicates our final model in which
only image and question pairs are needed at test time; while “Ours w/ caption”
represents the model using caption during evaluation. The result of our best
model are in bold; while the best performance with captions as inputs during
testing is underlined for comparison.

###  4.2 Qualitative Analysis

We perform an ablation study to qualitatively analyse the effect of the
components introduced in our work brought by the continuous (image caption)
and discrete (answer category) latent variables, as shown in Table 2.

####  4.2.1 Effect of Captions

The introduction of captions as a continuous latent variable improves the
classification performance, with an additional modality as input to benefit
the learning of multimodal representations. According to the breakdown numbers
in 2, the improvements brought by the latent variables of captions and answer
categories are 0.70 and 0.36 respectively for All questions altogether. The
combined strategy reaches 0.94 which indicates that the benefits from the two
latent variables are complementary. Note that neither the captions nor the
answer categories is available during testing; we only make use of these
modalities during training.

To further investigate potential benefit of the captions, we design an
experiment that feed in ground truth captions via variational distribution for
caption representations instead of inferring them from question and answer
pairs (_i.e_. use $q_{\phi}(z|c)$ to replace $p_{\theta}(z|v,q)$). We test
this out in the validation dataset and obtain 64.24 (‘Ours w/ caption’)
compared to 64.09 (‘Ours w/o caption’). It shows that having explicit captions
as input gives slightly better performance. However, the captions in these
experiments are ground truth, which means that if we were to use instead
automatically generated captions from an image captioning pipeline, the
numbers might drop due to the possible captioning errors. Primarily, our
proposed model (‘Ours w/o caption’) achieves the performance on par with with
the model with ground truth captions, which demonstrates the effectiveness of
the strategy that incorporates extra modality by latent variables.

Figure 2: Examples of our latent variable model outperforming the baseline
UpDn model from the introduction of answer category as a discrete latent
variable. The answer predicted by UpDn is highlighted in red and the answer
from our model is in blue. We also show other sample answer candidates within
each category.

####  4.2.2 Effect of Answer Category

As it can be observed from Table 2, after introducing answer category as an
additional discrete latent variable, our proposed model can also be improved
over the UpDn baseline, where the largest improvement can be observed for the
“Yes/No” type. For the question types “Num” and “Other”, the results of
UpDn+category are lower than the baseline. This may be due to the multiple
answer candidates under the two categories. For example, although the category
classier can accurately predict the answer category (_e.g_., “count”, “color”,
_etc_.), it can be still difficult to distinguish among the answers - {“9”,
“20”, …, “many” for “count”; “black”, “brown”, …, “black and white” for
“color”}. We highlight that the contribution of answer categories as a
discrete latent variable is to introduce an inductive bias which helps predict
the correct answer categories and answers given a specific image and question.

In order to further elaborate the effectiveness of answer categories, we
extract examples where our model predicted the correct answers while the UpDn
baseline failed to do so, as shown in Figure 2. For the two cases in the top
row, both models predict answers under the same and correct answer categories,
hence the answer space is similar; however, our latent variable model can
effectively distinguish and learn the difference among the answers which fall
within the same category. The bottom row of Figure 2 shows two cases where the
two models predict answers in different answer categories, and therefore they
are also very different in meaning. Our model not only outputs the highest
probability for the correct answer category, but also makes the correct final
prediction.

##  5 Conclusions

In this paper, we propose to tackle VQA under the framework of latent variable
models, employing captions and answer categories as the continuous and the
discrete latent variables respectively to constrain the original image-
question distribution while omitting the extra information during the test
phase. Our experimental results and qualitative analysis show the
effectiveness of the latent variables in boosting answering performance at
test time when only image and question pairs are available. This framework
could be easily generalised to incorporate other types of information or
modalities to enhance VQA and other tasks.

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NNLO Photon Fragmentation within Antenna Subtraction

Thomas Gehrmann and Robin Schürmann$\star$

Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich,
Switzerland

*<EMAIL_ADDRESS>
15th International Symposium on Radiative Corrections:
Applications of Quantum Field Theory to Phenomenology,
FSU, Tallahasse, FL, USA, 17-21 May 2021 10.21468/SciPostPhysProc.?

## Abstract

We report on our recent progress towards including the photon fragmentation
contribution in next-to-next-to-leading order (NNLO) QCD predictions for
photon production cross sections. This extension to previous NNLO calculations
requires the identification of the photon in singular parton-photon collinear
limits. We discuss how these limits can be subtracted within antenna
subtraction using fragmentation antenna functions and we outline their
integration.

###### Contents

  1. 1 Introduction
  2. 2 NLO Photon Fragmentation in Antenna Subtraction
  3. 3 Ingredients at NNLO
    1. 3.1 Integration of the $X^{0}_{4}$ Fragmentation Antenna Functions
    2. 3.2 Integration of the $X^{1}_{3}$ Fragmentation Antenna Functions
    3. 3.3 Coefficient Functions for Semi-Inclusive Deep Inelastic Scattering
  4. 4 Conclusion

##  1 Introduction

The production of a photon ($\gamma$) in hadronic collisions can proceed
through different mechanisms. Besides direct photons produced in the hard
underlying scattering process, photons can also be produced in jet events
where the jet radiates a photon in the process of hadronisation. The latter is
called fragmentation contribution and is described by non-perturbative parton-
to-photon fragmentation functions $D_{p\to\gamma}$ [1, 2]. To reduce the
contribution from fragmentation and eliminate the large background of
secondary photons from hadronic decays, a photon isolation is imposed in the
experimental analysis. The fixed-cone isolation limits the hadronic energy
inside a fixed cone around the photon and is used in all experimental
measurements to date. As it allows a limited amount of hadronic energy in the
photon direction, measurements using this isolation also contain a
fragmentation contribution (besides the contribution from direct photons).
Alternative isolation prescriptions use a dynamical cone [3] in which the
allowed hadronic energy is decreasing towards the center of the cone. These
idealised isolation prescriptions fully eliminate the fragmentation
contribution.
The calculation of isolated photon production and photon-plus-jet production
cross sections at hadron colliders using a fixed-cone isolation procedure and
including both, the direct and the fragmentation contribution has been
achieved to next-to-leading order (NLO) QCD accuracy [4, 5, 6, 7, 8, 9, 10].
Available next-to-next-to-leading order (NNLO) QCD predictions [11, 12, 13]
for this process suppress the fragmentation contribution using an idealised
isolation. Consequently, they rely on an empirical tuning of isolation
parameters to mimic the experimental isolation.
To overcome this drawback of available NNLO QCD predictions, a calculation at
this level of accuracy with a fixed-cone isolation is needed. Predictions with
a fixed-cone isolation have to handle additional photon-parton collinear
singularities which have to be extracted using a subtraction procedure, which
was accomplished up to now only for $e^{+}e^{-}$ collisions [14]. In contrast
to singular limits in QCD, in which no kinematic information on individual
partons has to be retained, in photonic singular limits the information on the
photon momentum must not be lost. We describe how these photonic limits can be
subtracted in the antenna subtraction formalism [15] with a new class of
fragmentation antenna functions and outline their integration which remains
differential in the final-state photon momentum fraction.

##  2 NLO Photon Fragmentation in Antenna Subtraction

To accommodate photon fragmentation in calculations based on the antenna
subtraction formalism, we need to introduce a new type of fragmentation
antenna functions. Their unintegrated forms are identical to already known
antenna functions, but they come with novel types of phase space
factorisations, leading to different integrated antennae. At NLO, only the
simple-collinear quark-photon final-state singularity needs to be accounted
for (and the photon is required to be observed, thus preventing it from
becoming soft), which implies that $A_{3}^{0}(k_{q},k_{\gamma},k_{\bar{q}})$
in its final-final crossing and initial-final crossing with a quark in the
initial state are sufficient. In the following we will discuss the subtraction
of the quark-photon collinear limit in the initial-final configuration.
Using the notation established in [16], the real subtraction term for the
photon becoming collinear to a final-state quark in the initial-final
configuration reads

$\displaystyle{\rm d}\hat{\sigma}^{S}_{q(\gamma)}$
$\displaystyle=\mathcal{N}^{R}\sum_{{\rm perm.}}{\rm
d}\Phi_{n+1}(\dots,k_{q},k_{\gamma},\dots;p_{q},p_{2})\frac{1}{S_{n+1}}$ (1)
$\displaystyle\times Q_{q}^{2}\,A_{3}^{0}(k_{q},k^{\,{\rm
id.}}_{\gamma},\check{p}_{\check{q}})M_{n}^{0}(\dots,k_{(q\gamma)},,\dots;\bar{p}_{q},p_{2})\,J_{m}^{(n)}(\\{\tilde{k}\\}_{n};z)\,,$

where $Q_{q}$ is the charge of the final-state quark and $z$ is the photon
momentum fraction in the mapped momentum $k_{(q\gamma)}$. It is
$z=z_{3}(\check{p}_{\check{q}},k_{\gamma}^{{\rm id.}},k_{q})$ with the
definition of the NLO momentum fraction

$z_{3}\left(\check{k}_{a},k_{b}^{{\rm
id.}},k_{c}\right)=\frac{s_{ab}}{s_{ab}+s_{ac}}\,.$ (2)

The jet function $J_{m}^{(n)}$ applies the jet algorithm as well as any cuts
on the photon. Consequently, it retains an explicit functional dependence on
$z$. The initial-state quark acts as a reference parton, indicated by the
check-mark assigned to its momentum, $\check{p}_{\check{q}}$. In the initial-
final configuration the reference parton is always the initial-state parton.
Since the photon is identified within the cluster $k_{(q\gamma)}$, it carries
the superscript ’id’.
To integrate the subtraction term we have to rewrite the three-body phase
space appearing in the initial-final phase-space factorisation [17] to make
the integration over $z$ explicit,

$\displaystyle{\rm d}\Phi_{n+1}(\dots,k_{q},k_{\gamma},\dots;p_{q},p_{2})$
$\displaystyle={\rm
d}\Phi_{n}(\dots,k_{(q\gamma)},\dots;\bar{p}_{q},p_{2})\frac{{\rm
d}x}{x}\frac{Q^{2}}{2\pi}{\rm
d}\Phi_{2}\delta\left(z-\frac{s_{\check{q}\gamma}}{s_{\check{q}\gamma}+s_{\check{q}q}}\right){\rm
d}z\,,$ (3)

where $q^{2}=(p_{q}-k_{\gamma}-k_{q})^{2}=-Q^{2}$ and ${\rm d}\Phi_{2}={\rm
d}\Phi_{2}(q,p_{q};k_{\gamma},k_{q})$. We can integrate the initial-final real
subtraction term over the unresolved phase space while staying differential in
$z$. We obtain

$\displaystyle\int{\rm
d}\Phi_{2}(q,p_{q};k_{\gamma},k_{q})\frac{Q^{2}}{2\pi}{\rm
d}\hat{\sigma}^{S}_{q(\gamma)}$
$\displaystyle=\mathcal{N}^{V}\frac{1}{S_{n}}\sum_{{\rm perm.}}{\rm
d}\Phi_{n}(\dots,k_{(q\gamma)},\dots;\bar{p}_{q},p_{2})$ (4)
$\displaystyle\times Q_{q}^{2}\mathcal{A}_{3}^{0,{\rm
id.}\gamma}\left(x,z\right)M_{n}^{0}(\dots,k_{(q\gamma)},\dots;\bar{p}_{q},p_{2})J_{m}^{(n)}(\\{\tilde{k}\\}_{n};z)\,,$

with the $z$-dependent integrated fragmentation antenna function

$\displaystyle\mathcal{A}_{3}^{0,{\rm id.}\gamma}\left(x,z\right)$
$\displaystyle=\frac{1}{C(\epsilon)}\int{\rm
d}\Phi_{2}(q,p_{q};k_{\gamma},k_{q})A_{3}^{0}(k_{q},k^{\,{\rm
id.}}_{\gamma},\check{p}_{q})\delta\left(z-\frac{s_{\check{q}\gamma}}{s_{\check{q}\gamma}+s_{\check{q}q}}\right)$
(5)
$\displaystyle=\frac{Q^{2}}{2}\frac{e^{\gamma_{E}\epsilon}}{\Gamma(1-\epsilon)}\left(Q^{2}\right)^{-\epsilon}\mathcal{J}(x,z)\,A_{3}^{0}(k_{q},k^{\,{\rm
id.}}_{\gamma},\check{p}_{q})$
$\displaystyle=\left(Q^{2}\right)^{-\epsilon}\left(-\frac{1}{2\epsilon}\delta(1-x)P^{(0)}_{\gamma
q}(z)+\frac{1}{2}\delta(1-x)\Big{(}z+P^{(0)}_{\gamma
q}(z)\log\big{(}(1-z)z\big{)}\,\Big{)}\right.$ $\displaystyle\hskip
55.0pt+\left.\frac{1}{2}\mathcal{D}_{0}(x)P^{(0)}_{\gamma
q}(z)-\frac{x+1}{2z}+1\right)+\mathcal{O}(\epsilon)\,$

and the normalisation factor $C(\epsilon)=(4\pi
e^{-\gamma_{E}})^{\epsilon}/(8\pi^{2})$. The Jacobian factor

$\mathcal{J}(x,z)=(1-x)^{-\epsilon}x^{\epsilon}z^{-\epsilon}(1-z)^{-\epsilon}\,$
(6)

originates from expressing the integration over the two-body phase space in
terms of a single integration over $z$. After expressing the invariants in the
antenna function in terms of $x$ and $z$, terms of the form
$(1-x)^{-1-\epsilon}$ are expanded in distributions, where we use the notation

$\mathcal{D}_{n}(x)=\left[\frac{\log^{n}(1-x)}{1-x}\right]_{+}\,\,,\,n\in\mathbb{N}_{0}\,.$
(7)

As can be seen from eq. (5), the quark-photon collinear singularity is
manifest in an $1/\epsilon$-pole at the integrated level. It is cancelled by
the mass factorisation contribution from the quark-to-photon fragmentation
function $D_{q\to\gamma}$, which reads

$\displaystyle{\rm d}\hat{\sigma}^{{\rm MF}}_{q}=$
$\displaystyle\frac{\alpha}{2\pi}{\rm
d}\hat{\sigma}^{B}_{q}\otimes\boldsymbol{\Gamma}_{q\to\gamma}^{(0)}(z,\mu_{A}^{2})$
(8) $\displaystyle=$ $\displaystyle\frac{1}{2}\mathcal{N}^{V}\int_{0}^{1}{\rm
d}z\sum_{{\rm perm.}}{\rm
d}\Phi_{n}(\\{k\\}_{n};p_{q},p_{2})\frac{1}{S_{n}}M_{n}^{0}(\dots,k_{q},\dots)\,Q_{q}^{2}\,\mu_{A}^{-2\epsilon}\,\Gamma_{\gamma
q}^{(0)}(z)\,J_{m}^{(n)}(\\{k\\}_{n};z)\,,$

where $\mu_{A}$ denotes the fragmentation scale and $\Gamma^{(0)}_{\gamma
q}=-(1/\epsilon)P^{(0)}_{\gamma q}$. The Born cross section at hand is

$\displaystyle{\rm d}\hat{\sigma}^{B}_{q}$
$\displaystyle=\mathcal{N}^{LO}_{{\rm jet}}\sum_{\rm perm.}{\rm
d}\Phi_{n}(\\{k\\}_{n};p_{q},p_{2})\frac{1}{S_{n}}M_{n}^{0}(\dots,k_{q},\dots)J_{m}^{(n)}(\\{k\\}_{n};z)\,.$
(9)

In here, the jet function depends on $z$ because the quark momentum $k_{q}$
denotes a quark-photon cluster containing a photon with momentum fraction $z$.
The normalisation factors are related through
$\mathcal{N}^{V}=C(\epsilon)\mathcal{N}^{R}=2\,C(\epsilon)(4\pi\alpha)\mathcal{N}^{LO}_{\mathrm{jet}}$.
In eq. (8) the factor 1/2 originates from different normalisation conventions
of photonic and jet matrix elements. The full initial-final virtual
subtraction term is then given by

$\displaystyle{\rm d}\hat{\sigma}^{T}_{q(\gamma)}$
$\displaystyle=-\mathcal{N}^{V}\int_{0}^{1}{\rm d}z\int_{0}^{1}\frac{{\rm
d}x}{x}\frac{1}{S_{n}}\sum_{\rm perm.}{\rm
d}\Phi_{n}(\dots,k_{(q\gamma)},\dots;\bar{p}_{q},p_{2})$ (10)
$\displaystyle\times Q_{q}^{2}\boldsymbol{J}_{2}^{(1),{\rm
id.}\gamma}(k_{(q\gamma)},\bar{p}_{q};x,z)M_{n}^{0}(\dots,k_{(q\gamma)},\dots;\bar{p}_{q},p_{2})J_{m}^{(n)}(\\{\tilde{k}\\}_{n};z)\,.$

Combination with the mass factorisation term $\Gamma^{(0)}_{\gamma
q}\left(z\right)$ of the quark-to-photon fragmentation function yields an
$\epsilon$-finite integrated fragmentation dipole:

$\displaystyle\boldsymbol{J}_{2}^{(1),{\rm
id.}\gamma}(k_{(q\gamma)},\bar{p}_{q};x,z)$
$\displaystyle=\mathcal{A}_{3}^{0,{\rm
id.}\gamma}\left(x,z\right)-\frac{1}{2}\mu_{A}^{-2\epsilon}\,\Gamma^{(0)}_{\gamma
q}\left(z\right)\delta(1-x)\,.$ (11)

##  3 Ingredients at NNLO

At NNLO, double unresolved photonic limits of double-real matrix elements as
well as single unresolved photonic limits of real-virtual matrix elements have
to be subtracted. The former limits correspond to triple collinear $q\parallel
g\parallel\gamma$ and $q\parallel\gamma\parallel\bar{q}$ configurations. They
can be subtracted using $\tilde{A}^{0}_{4}(\bar{q},g,\gamma,q)$ and
$\tilde{E}^{0}_{4}(q^{\prime},q,\gamma,\bar{q})$ antenna functions
respectively. Their unintegrated versions have the same form as the already
known antenna functions [15]. We use these antenna functions in the initial-
final configuration, which allows to use the initial-state momentum as
reference direction in the definition of the collinear momentum fraction. The
double-real subtraction term for the $q\parallel g\parallel\gamma$ limit reads

$\displaystyle\text{d}\hat{\sigma}^{S,b_{1}}_{q(\gamma)}=\mathcal{N}^{RR}$
$\displaystyle\sum_{{\rm
perms}}\text{d}\Phi_{n+2}(...,k_{q},k_{g},k_{\gamma},...;p_{q},p_{2})\frac{1}{S_{n+2}}$
(12)
$\displaystyle\times\tilde{A}^{0}_{4}(\check{p}_{q},k_{g},k_{\gamma}^{{\rm
id.}},k_{q})\,Q_{q}^{2}\,M^{0}_{n+2}(...\,,k_{(q\gamma
g)},\,...)\,\emph{J}^{(n)}_{m}(\\{\tilde{k}\\}_{n};z)\,,$

where the momentum fraction is given by
$z=z_{4}(\check{p}_{\check{q}},k_{\gamma}^{{\rm id.}},k_{g},k_{j})$ with the
definition of the NNLO momentum fraction

$z_{4}\left(\check{k}_{a},k_{b}^{{\rm
id.}},k_{c},k_{d}\right)=\frac{s_{ab}}{s_{ab}+s_{ac}+s_{ad}}\,.$ (13)

$z_{4}$ faithfully reproduces the momentum fraction of the photon in all
double unresolved limits. The $\tilde{A}^{0}_{4}$ antenna still carries a
single unresolved photon limit which has to be subtracted to guarantee a
cancellation of the singularities of the double-real matrix element (for
details see [16]). To remove this singularity an additional subtraction term
is introduced which takes the form

$\displaystyle\text{d}\hat{\sigma}^{S,b_{2}}_{q(\gamma)}$
$\displaystyle=-\mathcal{N}^{RR}\sum_{{\rm
perms}}\text{d}\Phi_{n+2}(...,k_{q},k_{g},k_{\gamma},...;p_{q},p_{2})\frac{1}{S_{n+2}}$
(14) $\displaystyle\times A^{0}_{3}(\check{p}_{q},k_{\gamma}^{{\rm
id.}},k_{q})\,A^{0}_{3}(\check{\bar{p}}_{q},k_{g},k^{{\rm id.}}_{(\gamma
q)})\,Q_{q}^{2}\,M^{0}_{n+2}(...\,,k_{((q\gamma)g)},\,...)\,\emph{J}^{(n)}_{m}(\\{\tilde{\tilde{k}}\\}_{n};z=uv)\,.$

In the first mapping corresponding to the leftmost antenna function the photon
is identified and a momentum fraction $u$ of the photon in the quark-photon
cluster is reconstructed. In the second mapping the quark-photon cluster is
identified and a momentum fraction $v$ of the cluster in the limit
$g\parallel(\gamma q)$ is calculated. The momentum fraction which is used in
the jet function $\emph{J}^{(n)}_{m}$ to reconstruct the photon momentum is
then given by

$z=uv=z_{3}\left(\check{p}_{\check{q}},k_{\gamma}^{{\rm
id.}},k_{q}\right)\,z_{3}\left(\check{\bar{p}}_{\check{q}},k_{(\gamma
q)}^{{\rm id.}},k_{g}\right)=z_{4}\left(\check{p}_{\check{q}},k_{\gamma}^{{\rm
id.}},k_{g},k_{q}\right)\,,$ (15)

where we used the definition of the NLO momentum fraction in eq. (2) and the
form of the initial-final mapping [17]. As the two momentum fractions in eq.
(12) and in eq. (14) coincide, the cancellation of the single unresolved
photon limit between the two terms is guaranteed. At NNLO, additional terms
are needed for an overall subtraction of all unresolved limits of the matrix
element. In these terms more $X^{0}_{3}$ fragmentation antenna functions
appear where a photon-parton cluster is identified (as the second antenna
function in eq. (14)). All these fragmentation antenna functions have been
integrated in the initial-final and final-final configuration [18].
For the subtraction of the single collinear $q\parallel\gamma$ limit of one-
loop matrix elements only a single one-loop antenna function, the
$\tilde{A}^{1}_{3}(\bar{q},\gamma,q)$ antenna function [15], is needed. We use
it exclusively in the initial-final configuration. The corresponding
subtraction term takes the same form as eq. (1) but with the replacement
$\mathcal{N}^{R}\to\mathcal{N}^{RV}$ and $A^{0}_{3}\to\tilde{A}^{1}_{3}$.
Since we use the fragmentation antenna functions $\tilde{A}^{0}_{4}$,
$\tilde{E}^{0}_{4}$ and $\tilde{A}^{1}_{3}$ in their unintegrated form in the
subtraction terms, they have to be added back in their integrated form at the
double-virtual level of the calculation.
The initial-final antenna functions are kinematically described by the
scattering process

$q(q^{2})+p_{i}\rightarrow k_{j}+k_{k}(+k_{l})\,,$ (16)

where $k_{j}^{2}=k_{k}^{2}=k_{l}^{2}=p_{i}^{2}=0$ and $q^{2}=-Q^{2}<0$.
We give a short outline of the integration of these classes of fragmentation
antenna functions.

###  3.1 Integration of the $X^{0}_{4}$ Fragmentation Antenna Functions

The inclusive integrated initial-final antenna functions $\mathcal{X}^{0}_{4}$
are obtained by integration over the corresponding three-body phase space
[19], i.e.

$\mathcal{X}^{0}_{i,jkl}(x)=\frac{1}{C(\epsilon)^{2}}\int\text{d}\Phi_{3}(k_{j},k_{k},k_{l};p_{i},q)\frac{Q^{2}}{2\pi}X^{0}_{i,jkl}\,,$
(17)

with $x=\frac{Q^{2}}{2p\cdot q}$ and the normalisation factor
$C(\epsilon)=(4\pi e^{-\gamma_{E}})^{\epsilon}/(8\pi^{2})$. For initial-final
fragmentation antenna functions the same normalisation as in eq. (17) is used
but the integration remains differential in the final-state momentum fraction
$z$, i.e.

$\mathcal{X}^{0,\,{\rm
id.}j}_{i,jkl}(x,z)=\frac{1}{C(\epsilon)^{2}}\int\text{d}\Phi_{3}(k_{j},k_{k},k_{l};p_{i},q)\,\delta\left(z-x\frac{(p_{i}+k_{j})^{2}}{Q^{2}}\right)\frac{Q^{2}}{2\pi}X^{0}_{i,jkl}\,.$
(18)

The final-state momentum fraction is fixed by the additional
$\delta$-distribution and it describes the fraction of energy carried by
particle $j$ in the unresolved limit. In the definition of the momentum
fraction the initial-state momentum $p_{i}$ is used as a reference momentum.
The momentum fraction can be rewritten as $z=z_{4}(\check{p}_{i},k_{j}^{\rm
id.},k_{k},k_{l})$. As discussed above, all double-unresolved identified
photon limits are contained in two fragmentation antenna functions:
$\tilde{A}^{0}_{4}(\hat{q},g,\gamma^{{\rm id.}},q)$ and
$\tilde{E}^{0}_{4}(\hat{q}^{\prime},q,\gamma^{{\rm id.}},\bar{q})$. To
integrate these fragmentation antenna functions, we use the reduction to
master integrals technique. Using unitarity relations, we can express the
$\delta$-distributions in eq. (18) by cut propagators [20]. In this way we
rewrite the phase-space integrals as $2\to 2$ three-loop integrals in forward
scattering kinematics.
The reduction to master integrals is performed using the program Reduze2 [21],
which uses the Laporta algorithm [22] to solve the system of equations between
the different integrals obtained by integration-by-parts techniques [23, 24]
and Lorentz invariance [25].
The initial-final scattering kinematics gives rise to 12 propagators from
which four are cut propagators. In the reduction to master integrals we
require all integrals to contain the four cut propagators in the denominator.
For the integration of the two photonic $X^{0}_{4}$ fragmentation antenna
functions we find nine master integrals. The master integrals are calculated
using their differential equations in the two kinematic variables $x$ and $z$.
The boundary conditions are fixed by integrating the solution of the
differential equations over $z$ and comparing the result with the inclusive
master integrals in [19]. The expressions for the integrated fragmentation
antenna functions are very lengthy [18] so that we do not quote them here.

###  3.2 Integration of the $X^{1}_{3}$ Fragmentation Antenna Functions

The inclusive integrated one-loop antenna functions in the initial-final
configuration are defined as [19]

$\mathcal{X}^{1}_{i,jk}(x)=\frac{1}{C(\epsilon)}\int\text{d}\Phi_{2}(k_{j},k_{k};p_{i},q)\frac{Q^{2}}{2\pi}X^{1}_{i,jk}\,,$
(19)

where $X^{1}_{i,jk}$ is the unintegrated one-loop antenna function and
$\text{d}\Phi_{2}$ the two-particle phase space. We define the integrated
initial-final one-loop fragmentation antenna functions in line with eq. (19)
as

$\displaystyle\mathcal{X}^{1,{\rm id.}j}_{i,jk}(x,z)$
$\displaystyle=\frac{1}{C(\epsilon)}\int\text{d}\Phi_{2}(k_{j},k_{k};p_{i},q)\,\delta\left(z-\frac{s_{ij}}{s_{ij}+s_{ik}}\right)\frac{Q^{2}}{2\pi}X^{1}_{i,jk}$
(20)
$\displaystyle=\frac{Q^{2}}{2}\frac{e^{\gamma_{E}\epsilon}}{\Gamma(1-\epsilon)}\left(Q^{2}\right)^{-\epsilon}\mathcal{J}(x,z)\,X^{1}_{i,jk}\,.$

The integration takes the same form as for the $X^{0}_{3}$ fragmentation
antenna functions, see eq. (5) above. The Jacobian factor $\mathcal{J}$ is
given in eq. (6). As can be seen from eq. (20), no actual integration has to
be performed to obtain the integrated fragmentation antenna functions
$\mathcal{X}^{1,{\rm id.}j}_{i,jk}$. However, to express the integrated
fragmentation antenna functions in terms of distributions in $(1-x)$ and in
$z$ we first have to cast the unintegrated antenna functions in a form
suitable for this expansion. Therefore, deriving the integrated initial-final
one-loop fragmentation antenna functions follows the steps of the derivation
of the integrated initial-initial one-loop antenna functions presented in
[26]. In contrast to the NLO $X^{0}_{3}$ antenna functions which only contain
rational terms in the invariants, the one-loop antenna functions $X^{1}_{3}$
also contain logarithms and polylogarithms in the invariants. These functions
have branch cuts in the limits $x\to 1$ and $z\to 0$. Therefore, the expansion
in distributions in $z=0$ and $x=1$ cannot be performed directly. We follow
the strategy of [26] and express the one-loop antenna functions in terms of
one-loop master integrals.
The one-loop master integrals appearing in the expressions for the one-loop
antenna functions are the one-loop bubble ${\rm Bub}(s_{ij})$ and the one-loop
${\rm Box}(s_{ij},s_{ik})$ in all kinematic crossings. Both master integrals
are well-defined in the Euclidean region, in which all invariants are smaller
than 0. The master integrals have to be analytically continued from this
kinematic region, to the kinematic region under consideration given by

$s_{ij}<0\quad,\quad s_{ik}<0\quad,\quad s_{jk}>0\quad,\quad
s_{ijk}=-Q^{2}<0\,.$ (21)

While the analytic continuation of the bubble master integrals is trivial,
care has to be taken in case of the box integrals where hypergeometric
functions appear. In the analytic continuation of these box integrals the
hypergeometric functions must not have a branch cut in the kinematic endpoints
$x=1$ and $z=0$ so that an expansion in distributions can be performed. The
$z$-integration of the resulting expressions recovers the known real-virtual
initial-final master integrals [19] and enabled us to identify an error in
their numerical implementation for jet production in deep-inelastic scattering
[27].
The relevant one-loop integrated fragmentation antenna function for photon
production is $\mathcal{\tilde{A}}^{1,\,{\rm id.}\gamma}_{3}(x,z)$. Its
expression is very lengthy [18] such that we do not quote it here.

###  3.3 Coefficient Functions for Semi-Inclusive Deep Inelastic Scattering

The antenna functions $A^{0}_{3}$, $A^{1}_{3}$ and $\tilde{A}^{1}_{3}$ are
derived from the squared matrix element of the scattering $\gamma^{\star}\to
qg\bar{q}$ at tree level and at one-loop respectively and the antenna
functions $A^{0}_{4}$ and $\tilde{A}^{0}_{4}$ are derived from the squared
matrix element of the scattering process $\gamma^{\star}\to qgg\bar{q}$.
Therefore, the results for the integrated version of these fragmentation
antenna function can be compared to the coefficient functions of semi-
inclusive deep inelastic scattering. Using the notation of [28], the expansion
of the semi-inclusive coefficient functions reads

$\omega^{i}_{f^{\prime}f}=\omega_{f^{\prime}f}^{i,(0)}(x,z)+\frac{\alpha_{s}}{\pi}\omega^{i,(1)}_{f^{\prime}f}(x,z)+\left(\frac{\alpha_{s}}{\pi}\right)^{2}\omega^{i,(2)}_{f^{\prime}f}(x,z)+\mathcal{O}(\alpha_{s}^{3})\,,$
(22)

with $i=T,L$. $f^{\prime}$ corresponds to the parton which fragments into the
final-state hadron and $f$ corresponds to the parton in the initial state. We
find that the combination of coefficient functions corresponding to the
integrated fragmentation antenna functions is
$\omega^{T}-\omega^{L}/(2-2\epsilon)$. The results for the one-loop
coefficient functions are well known [29]. Comparing the results of the
integrated $X^{0}_{3}$ fragmentation antenna functions to the results stated
in [30], we find

$\displaystyle\frac{1}{4Q_{q}^{2}C_{F}}\left(\omega^{T,(1)}_{gq}-\frac{1}{2(1-\epsilon)}\omega^{L,(1)}_{gq}\right)$
$\displaystyle=\mathcal{A}^{0,{\rm
id.}\gamma}_{3}\left(x,z\right)-\mu_{A}^{-2\epsilon}\frac{1}{2}\Gamma^{(0)}_{\gamma
q}(z)\,,$ (23)
$\displaystyle\frac{1}{4Q_{q}^{2}C_{F}}\left(\omega^{T,(1)}_{qq}-\frac{1}{2(1-\epsilon)}\omega^{L,(1)}_{qq}\right)$
$\displaystyle=\mathcal{A}^{0,{\rm
id.q}}_{3}\left(x,z\right)-\mu_{F}^{-2\epsilon}\Gamma^{(1)}_{qq}(x)-\mu_{A}^{-2\epsilon}\Gamma^{(1)}_{qq}(z)+\mathcal{A}^{1}_{2}\,,$

where $\mathcal{A}^{1}_{2}$ is the one-loop vertex factor [15] in space-like
kinematics. The NNLO coefficient functions $\omega_{f^{\prime}f}^{(2)}$ are
unknown, however, recently an approximation of the NNLO coefficient functions
$\omega^{i,(2)}_{qq}$ was derived using the threshold resummation formalism
[28].
With the results of our integrated fragmentation antenna functions we are able
to predict the subleading color contribution to the above combinations of the
semi-inclusive coefficient functions $\omega^{i,(2)}_{gq}$. Expressed in terms
of the integrated antenna functions and mass factorisation kernels it reads

$\displaystyle\frac{1}{4Q_{q}^{2}C_{F}}\left(\omega^{T,(2)}_{gq}-\frac{1}{2(1-\epsilon)}\omega^{L,(2)}_{gq}\right)\bigg{\rvert}_{1/N}$
(24) $\displaystyle=\tilde{\mathcal{A}}^{0,{\rm
id.\gamma}}_{4}(x,z)+\tilde{\mathcal{A}}^{1\,U,{\rm
id.}\gamma}_{3}(x,z)-\mu_{A}^{-2\epsilon}\frac{1}{2}\Gamma^{(0)}_{\gamma
q}(z)\otimes\left(\mathcal{A}^{0,{\rm
id.}q}_{3}(x,z)+\mathcal{A}^{1}_{2}\right)$
$\displaystyle-\mu_{F}^{-2\epsilon}\Gamma^{(1)}_{qq}(x)\otimes\mathcal{A}^{0,{\rm
id.}\gamma}_{3}(x,z)+\frac{1}{2}\mu_{A}^{-2\epsilon}\Gamma^{(0)}_{\gamma
q}(z)\otimes\left(\mu_{F}^{-2\epsilon}\Gamma^{(1)}_{qq}(x)+\mu_{A}^{-2\epsilon}\Gamma^{(1)}_{qq}(z)\right)$
$\displaystyle-\frac{1}{2}\left(\mu_{A}^{2}\right)^{-2\epsilon}\Gamma^{(1)}_{\gamma
q}(z)\,,$

where the notation $...|_{1/N}$ means that only terms proportional to $1/N$
are considered. The expressions for the initial-state mass factorisation
kernel $\Gamma^{(1)}_{qq}$ can be found in [16] and the one-loop photon
fragmentation mass factorisation kernel $\Gamma^{(1)}_{\gamma q}$ is given by
[14]:

$\Gamma^{(1)}_{\gamma q}(z)=\frac{1}{4\epsilon^{2}}\left(P_{qq}^{(0)}\otimes
P_{\gamma q}^{(0)}\right)(z)-\frac{1}{4\epsilon}P^{(1)}_{\gamma q}(z)\,.$ (25)

##  4 Conclusion

In this talk, we have described the extension of the antenna subtraction
method to account for photon fragmentation processes up to NNLO, deriving all
newly required phase space factorisations and integrated antenna functions
[18]. New results on NNLO semi-inclusive DIS coefficient functions were
obtained as a by-product. Our extension of the antenna subtraction method will
allow to compute NNLO corrections to processes with final-state photons at
hadron colliders for the realistic fixed-cone isolation prescriptions used in
the experimental measurements, thereby improving upon previous results for the
idealised dynamical-cone isolation.

## Acknowledgements

We would like to thank Marius Höfer and Alexander Huss for interesting
discussions on the work presented here. This work was supported by the Swiss
National Science Foundation (SNF) under contract 200020-175595.

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