# Group DETR: Fast DETR Training with Group-Wise One-to-Many Assignment

Qiang Chen1∗, Xiaokang Chen2 , Jian Wang1
Haocheng Feng1, Junyu Han1, Errui Ding1, Gang Zeng2, Jingdong Wang1
1Baidu VIS
2Key Laboratory of Perception (MoE), School of AI, Peking University
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>Equal contribution.

###### Abstract

Detection Transformer (DETR) relies on One-to-One assignment, i.e., assigning
one ground-truth object to only one positive object query, for end-to-end
object detection and lacks the capability of exploiting multiple positive
object queries. We present a novel DETR training approach, named Group DETR,
to support Group-wise One-to-Many assignment. We make simple modifications
during training: (i) adopt $K$ groups of object queries; (ii) conduct decoder
self-attention on each group of object queries with the same parameters; (iii)
perform One-to-One label assignment for each group, leading to $K$ positive
object queries for each ground-truth object. In inference, we only use one
group of object queries, making no modifications to DETR architecture and
processes. We validate the effectiveness of the proposed approach on DETR
variants, including Conditional DETR, DAB-DETR, DN-DETR, and DINO. Code will
be available.

##  1 Introduction

In object detection, label assignment is a crucial and classic topic. It
defines positive and negative samples for detector training (Zhang et al.,
2020), which affects model learning and impacts detection performance. Many
label assignment methods (Cao et al., 2020; Zhang et al., 2020; Kim & Lee,
2020; Zhu et al., 2020a; Ge et al., 2021) are proposed to achieve significant
improvements over various detectors (Ren et al., 2015a; He et al., 2017; Lin
et al., 2017; Tian et al., 2019). They can be categorized into two main types
based on the number of matched positives per ground-truth (gt) object: (1)
One-to-Many assignment: each gt object can be assigned to multiple positive
samples, and (2) One-to-One assignment: one gt object can only be assigned to
exactly one positive sample.

Among them, One-to-Many assignment is widely adopted in modern detectors (Ren
et al., 2015a; He et al., 2017; Lin et al., 2017; Tian et al., 2019; Chen et
al., 2021), achieving dominant performances and enabling fast training
convergences. One undesired effect of One-to-Many assignment is that it
produces duplicate predictions, preventing the removal of post-processing
steps such as non-maximum suppression (NMS) (Hosang et al., 2017). DETR
(Carion et al., 2020) breaks this convention and achieves end-to-end training
by introducing object queries, applying transformer layers (Vaswani et al.,
2017), and adopting One-to-One assignment (Kuhn, 1955). But it suffers from
the slow convergence issue. Previous attempts (Zhu et al., 2019; Gao et al.,
2021; Meng et al., 2021; Wang et al., 2021b; Liu et al., 2022; Chen et al.,
2022) alleviate this issue from the perspective of object query design and
attention module design in transformer layers. Although a $10\times$ speed up
(from 500 epochs to 50 epochs) is achieved, DETR-based methods still produce
unpleasant results with a standard $1\times$ training schedule (12 epochs). In
this paper, we attribute these results to One-to-One assignment111The reasons
are three folds: (1) detectors (Lin et al., 2017; Tian et al., 2019; Chen et
al., 2021) with One-to-Many assignment usually converge fast; (2) DETR-based
methods (Carion et al., 2020) (with One-to-One assignment) suffer from slow
convergence issue; (3) other end-to-end detectors (Wang et al., 2021a; Sun et
al., 2021a), which are achieved by applying One-to-One assignment, need more
training epochs ($3\times$ schedule) to reach comparable results with their
counterparts with One-to-Many assignment. and provide a further speed up on
training convergence by exploring label assignment methods on DETR-based
detectors.

Figure 1: Experimental results with Conditional-DETR-C5 on COCO. We show the
results with One-to-One, One-to-Many, and ours Group-wise One-to-Many
assignment. Increasing the number of object queries does not significantly
affect the capability of end-to-end training with the One-to-One assignment.
When we adopt the One-to-Many assignment, the detection performance improves
with the help of NMS but quickly falls to a very low level ($<10$ mAP) without
NMS. Group-wise One-to-Many assignment enjoys the merits of both One-to-One
assignment and One-to-Many assignment. In the figure, the $x$-axis shows the
number of object queries and the $y$-axis presents the detection performance
(mAP).

To investigate how label assignment methods affect the training convergence,
we conduct experiments on Conditional DETR (Meng et al., 2021) with a
$1\times$ schedule. We gradually increase the number of total object queries
and matched positive object queries per gt object. According to the
experimental results in Figure 1, detection performance improves significantly
with the increment of total object queries and matched positive object
queries, while we can only obtain minor gains when only increasing the number
of total object queries. The above evidence suggests that multiple positive
object queries are key factors to fast training convergence. Despite multiple
positive object queries (One-to-Many assignment) producing higher performance,
they bring duplicate predictions, making NMS necessary for post-processing.
Given that One-to-One assignment is a crucial factor in achieving end-to-end
training on object detection (Wang et al., 2021a; Sun et al., 2021a; b), we
show a simple way to combine the merits of these two assignment methods.

In this paper, we propose Group DETR to support multiple positive object
queries in an end-to-end manner. To avoid duplicate predictions, Group DETR
introduces Group-wise One-to-Many assignment. It matches multiple positive
object queries per gt object (One-to-Many) in general but decouples them into
multiple independent groups and keeps only one positive per object (One-to-
One) in each group. To achieve it, we make simple modifications during
training: (i) adopt $K$ groups of object queries; (ii) conduct decoder self-
attention on each group of object queries with the same parameters; (iii)
perform One-to-One assignment for each group, leading to $K$ positive object
queries for each gt object. The design achieves fast training convergence,
removes duplicate predictions, and enables end-to-end training. Moreover, we
only use one group of object queries in inference and do not modify either
architecture or processes, bringing no extra cost compared with the original
model.

Our Group DETR is a general training method that can be applied to various
DETR-based models. Extensive experiments on MS COCO (Lin et al., 2014) prove
that our method is effective in achieving fast training convergence. With a
$1\times$ training schedule, Group DETR gives a 5.0 mAP/4.4 mAP/1.5 mAP/2.6
mAP/1.0 mAP gains on Conditional DETR (Meng et al., 2021)/DAB-DETR (Liu et
al., 2022)/DAB-Deformable-DETR (Zhu et al., 2019; Liu et al., 2022)/DN-DETR
(Li et al., 2022)/DINO (Zhang et al., 2022). The non-trivial improvements hold
when we adopt longer training schedules (e.g., $50$ epochs). Label assignment
methods are widely discussed in previous dense detectors (Lin et al., 2017;
Tian et al., 2019; Chen et al., 2021) but few in DETR-based methods (Carion et
al., 2020; Zhu et al., 2019; Meng et al., 2021; Liu et al., 2022). We hope our
Group-wise One-to-Many assignment will motivate researchers to explore more in
this direction.

##  2 Related Works

##### Label Assignment in Object Detection.

Assignment between the gt objects and training samples is a crucial and
classic topic in object detection (Redmon et al., 2016; Ren et al., 2015b; He
et al., 2017; Lin et al., 2017). Anchor-based detectors (Liu et al., 2016; Lin
et al., 2017) rely on the Intersection-over-Union (IoU) between anchors and gt
boxes to select positive and negative samples. When the maximum IoU between an
anchor and all gt boxes exceeds the hand-craft IoU threshold, the anchor will
be matched to the maximum IoU gt box. Anchor-free detectors (Tian et al.,
2019; Zhang et al., 2020) consider spatial and scale constraints when setting
the points as positives or negatives. In ATSS (Zhang et al., 2020), the
authors provide evidence that the applied assignment methods cause the
differences in performance between anchor-based and anchor-free detectors
during training. Its followers (Kim & Lee, 2020; Zhu et al., 2020a; Ge et al.,
2021) further explore this direction and bring considerable improvements to
object detection. The above methods can be divided into One-to-Many
assignment, which needs NMS for post-processing. DETR (Carion et al., 2020)
explores an alternative way and achieves end-to-end training, removing the
need for NMS. Recent studies (Wang et al., 2021a; Sun et al., 2021a) show that
One-to-One assignment is a key factor in achieving end-to-end training. They
provide evidence to make previous detectors to be end-to-end with One-to-One
assignment. In this paper, we also focus on the assignment methods but on
DETR-based methods, which are less studied in previous works.

##### Accelerating DETR Training.

Since DETR (Carion et al., 2020) was proposed, its slow convergence issue has
been a critical problem that many researchers try to address. Several works
(Meng et al., 2021; Zhu et al., 2020b; Gao et al., 2021; Wang et al., 2021b;
Sun et al., 2021c) achieve $10\times$ speed up for DETR. They mainly focus on
proposing better cross-attention modules, designing new types of object
queries, and updating methods for predicting boxes. Recently, DN-DETR (Li et
al., 2022) and DINO (Zhang et al., 2022) attribute the slow convergence issue
to the instability of hungarian matching (Kuhn, 1955). They propose query
denoising (Li et al., 2022) and contrastive query denoising (Zhang et al.,
2022) to speed up the model training further and achieve plausible results
with only $1\times$ training schedule. Different from previous methods, this
paper aims to support one-to-many assignment and only makes simple
modifications on the DETR decoder during training.

##  3 Methodology

Figure 2: Architecture of Group DETR. Here we show one decoder layer for an
example. We feed $K$ groups of queries to the decoder and conduct self-
attention on each group of queries with the shared parameters. Then we make
One-to-One assignment for each group. We omit the details of the cross-
attention design in this figure.

###  3.1 Overview

We follow detection transformer (DETR) (Carion et al., 2020) and adopt the
transformer encoder and decoder structure. The image features are extracted
with a backbone, then fed to the encoder to model the long-range dependencies.
The decoder takes object queries as input and search for objects from the
encoder feature with the cross-attention module. We illustrate the approach
details in the following parts.

###  3.2 Group DETR

DETR relies on One-to-One assignment for end-to-end object detection but lacks
of the capability of exploiting multiple positive queries. The decoder takes
only one group of $N$ queries (shown in Equation 1) as the input in the
original DETR (Carion et al., 2020) and various DETR variants (Zhu et al.,
2020b; Meng et al., 2021; Liu et al., 2022).

$\displaystyle\mathbb{G}^{1}=\\{\mathbf{q}_{1}^{1},\dots,\mathbf{q}_{N}^{1}\\},$
(1)

Each query in the decoder is responsible for predicting either a gt object
(with class and bounding box) or a “no object” class. The query may come in
different forms, including the high-dimensional feature vector (Carion et al.,
2020; Meng et al., 2021), the anchor point coordinates (Wang et al., 2021b),
and the box coordinates (Liu et al., 2022). During training, the Hungarian
algorithm (Kuhn, 1955) is employed to find an optimal One-to-One assignment
$\hat{\sigma}$ between predicted and ground-truth objects:

$\displaystyle\hat{\sigma}$
$\displaystyle=\operatorname*{arg\,min}_{\sigma\in\xi_{N}}\sum_{i=1}^{N}\mathcal{C}_{\operatorname{match}}(y_{i},\hat{y}_{\sigma(i)}),$
(2) $\displaystyle\mathcal{C}_{\operatorname{match}}$
$\displaystyle=\sum_{i=1}^{M}[\mu_{\operatorname{cls}}\ell_{\operatorname{cls}}(\mathbf{p}_{\hat{\sigma}(i)},\bar{c}_{i})+\ell_{\operatorname{box}}(\mathbf{b}_{\hat{\sigma}(i)},\bar{\mathbf{b}}_{i})].$
(3)

where $\xi_{N}$ is the set of permutations of $N$ elements and
$\mathcal{C}_{\operatorname{match}}(y_{i},\hat{y}_{\sigma(i)})$ is the
matching cost between the ground truth $y_{i}$ and the prediction with index
$\sigma(i)$. Each element of the ground truth can be seen as
$y_{i}={(\bar{c}_{i},\bar{\mathbf{b}}_{i})}$ where $c_{i}$ is the target class
label and $\mathbf{b}_{i}$ is the vector that defines the gt box. Similarly,
the prediction is defined as
$\hat{y}_{\hat{\sigma}(i)}=(\mathbf{p}_{\hat{\sigma}(i)},\mathbf{b}_{\hat{\sigma}(i)})$.
$M$ is the number of ground-truth objects. $\ell_{\operatorname{cls}}$ is the
focal loss (Zhu et al., 2020b; Meng et al., 2021; Liu et al., 2022) and
$\mu_{\operatorname{cls}}=2$ is the trade-off coefficient. $c_{i}$ is the
class of the $i$-th ground-truth box and $c_{i}=\operatorname{null}$ if $i>M$.
$\mathbf{b}_{i}$ is the $i$-th gt box and $\ell_{\operatorname{box}}$ is a
combination of $\ell_{1}$ loss and GIoU loss (Rezatofighi et al., 2019) and
the loss weights are $5$ and $2$, respectively.

##### Group-wise One-to-Many Assignment.

To enlarge the number of positive queries while avoiding duplicate
predictions, we propose the Group-wise One-to-Many assignment strategy. Given
the number of positive queries $K$ that we want to match with each gt object,
we generate $K$ groups of queries (including the original group):

$\displaystyle\mathbb{G}^{1}$
$\displaystyle=\\{\mathbf{q}_{1}^{1},\dots,\mathbf{q}_{N}^{1}\\},$ (4)
$\displaystyle\qquad\cdots\cdots$ $\displaystyle\mathbb{G}^{K}$
$\displaystyle=\\{\mathbf{q}_{1}^{K},\dots,\mathbf{q}_{N}^{K}\\},$ (5)

The total $K$ groups are fed to the decoder in parallel, but each group does
not interact with other groups when performing decoder self-attention with the
same parameters. Then One-to-One assignment is applied to the output of each
group independently and we could get $K$ matching results:
$\hat{\sigma}_{1},\cdots,\hat{\sigma}_{K}$. The modifications during training
are present in Figure 2 compared with the original DETR methods. During
inference, only the first group is kept222According to our experiments, every
group can achieve similar results. Here we simply use the first one. thus
there is no extra cost compared with the original model.

###  3.3 Discussion

####  3.3.1 Each group learns the distribution of object positions in the
dataset

We visualize the distribution of normalized object query positions (which are
reference points in Conditional DETR (Meng et al., 2021)) in Figure 4. We find
that, the distributions of different groups are similar and object queries
from different groups are usually clustered together. We guess this is because
they both learn the distribution of object positions in the dataset.

Figure 3: Distribution of multiple group object queries. We show the
normalized reference points of all groups based on the Conditional-DETR-C5.
Different color represents different groups and the number of groups is $11$.
Best view in color.

Figure 4: Two distance metrics ($\mathbf{PD}$ and $\mathbf{MD}$) as a function
of training steps. The $x$-axis presents the training steps and the $y$-axis
presents the relative length to the size of the original image. We use $3$
groups in this experiment.

####  3.3.2 Each ground-truth box is assigned to queries from different
regions

We define two distance metrics: perturbation distance $\mathbf{PD}$ and
matching distance $\mathbf{MD}$. Perturbation distance is the average distance
between object queries from the one group to the closest object queries from
the other group. Here, we take the case of only 2 groups as an example:

$\displaystyle\mathbf{PD}=\frac{1}{2N}\sum_{i=1}^{N}(\lVert\mathcal{P}(\mathbf{q}^{1}_{i}),\mathcal{P}(\mathcal{N}_{2\rightarrow
1}(\mathbf{q}^{1}_{i})\rVert+\lVert\mathcal{P}(\mathbf{q}^{2}_{i}),\mathcal{P}(\mathcal{N}_{1\rightarrow
2}(\mathbf{q}^{2}_{i})\rVert),$ (6)

where $\mathcal{P}()$ is the position of a object query,
$\mathcal{N}_{2\rightarrow 1}(\mathbf{q}^{1}_{i})$ represents the closest
object query of $\mathbf{q}^{1}_{i}$ in group $\mathbb{G}^{2}$. $\lVert
a,b\rVert$ is the euclidean distance between two positions $a$ and $b$.

Matching distance is the average distance between object queries in one group
and object queries from other groups that match with the same gt object:

$\displaystyle\mathbf{MD}=\frac{1}{M}\sum_{i=1}^{M}\lVert(\mathcal{P}(\mathbf{q}^{1}_{\hat{\sigma_{1}}(i)}),\mathcal{P}(\mathbf{q}^{2}_{\hat{\sigma_{2}}(i)})\rVert,$
(7)

where $M$ is the number of gt objects. ${\hat{\sigma_{1}}(i)}$ is the object
query index in group $\mathbb{G}^{1}$ that is matched with the $i$-th gt
object. We plot these two distances with respect to training iterations in
Figure 4. We find that $\mathbf{MD}$ is significantly larger than
$\mathbf{PD}$. This evidence suggests that our Group-wise One-to-Many
assignment can generate multiple positive object queries for each gt object.

##  4 Experiments

###  4.1 Setting

Dataset. We perform the experiments on the COCO $2017$ (Lin et al., 2014)
detection dataset, which contains about $118$K training (train) images and
$5$K validation (val) images. Following the common practice, we report the
standard mean average precision (mAP) result on the COCO validation dataset
under different IoU thresholds and object scales.

Training. We follow the DETR training protocol (Carion et al., 2020). The
backbone is the ImageNet-pretrained model from TORCHVISION with batchnorm
layers fixed, and the transformer parameters are initialized using the Xavier
initialization scheme (Glorot & Bengio, 2010). We train the model on the COCO
training set for $12/50$ epochs, with the AdamW (Loshchilov & Hutter, 2017)
optimizer. The learning rate is dropped by a factor of $10$ after $11/40$
epochs. The weight decay is set to be $1\times e^{-4}$. The learning rates for
the backbone and the transformer are initialized as $1\times e^{-5}$ and
$1\times e^{-4}$, respectively. The number of object queries is set according
to different baseline models.

We use the augmentation scheme same as DETR (Carion et al., 2020): resize the
input image such that the short side is at least $480$ and at most $800$
pixels and the long side is at most $1333$ pixels; randomly crop the image
such that a training image is cropped with a probability $0.5$ to a random
rectangular patch.

###  4.2 Improvements on DETR-based methods

We apply our Group-wise One-to-Many assignment to different DETR-based methods
to verify the effectiveness, $e.g.$, Conditional DETR (Meng et al., 2021),
DAB-DETR (Liu et al., 2022), DN-DETR (Li et al., 2022), DINO (Zhang et al.,
2022) and Mask2Former (Cheng et al., 2021).

Model | w/ Group DETR | #Groups | Schedule | mAP | APs | APm | APl
---|---|---|---|---|---|---|---
Conditional DETR-C$5$ |  | $1$ |  $12$e | $32.6$ | $14.7$ | $35.0$ | $48.3$
Conditional DETR-C$5$ | ✓ | $2$ |  $12$e | $34.4$ | $15.1$ | $37.1$ | $51.8$
Conditional DETR-C$5$ | ✓ | $3$ |  $12$e | $35.3$ | $15.7$ | $38.5$ | $53.3$
Conditional DETR-C$5$ | ✓ | $4$ |  $12$e | $35.9$ | $15.9$ | $38.9$ | $53.8$
Conditional DETR-C$5$ | ✓ | $5$ |  $12$e | $36.5$ | $15.8$ | $39.7$ | $54.6$
Conditional DETR-C$5$ | ✓ | $6$ |  $12$e | $36.4$ | $16.6$ | $39.7$ | $54.5$
Conditional DETR-C$5$ | ✓ | $7$ |  $12$e | $36.8$ | $16.8$ | $40.2$ | $54.8$
Conditional DETR-C$5$ | ✓ | $9$ |  $12$e | $37.2$ | $16.5$ | $40.5$ | $54.8$
Conditional DETR-C$5$ | ✓ | $11$ |  $12$e | 37.6(+5.0) | $18.2$ | $40.7$ | $55.9$
Conditional DETR-C$5$ |  | $1$ |  $50$e | $40.9$ | $20.5$ | $44.2$ | $59.6$
Conditional DETR-C$5$ | ✓ | $11$ |  $50$e | 43.4(+2.5) | $23.0$ | $47.3$ | $62.3$
Table 1: Experimental results based on Conditional DETR (Meng et al., 2021). We show the improvements with different training schedules, e.g., 12 epochs and 50 epochs.  Model | w/ Group DETR | Schedule | mAP | APs | APm | APl
---|---|---|---|---|---|---
DN-DETR-C$5$ |  |  $12$e | $38.6$ | $17.9$ | $41.6$ | $57.7$
DN-DETR-C$5$ | ✓ |  $12$e | 40.6(+2.0) | $19.8$ | $43.9$ | $59.4$
DN-DETR-DC$5$ |  |  $12$e | $41.9$ | $22.2$ | $45.1$ | $59.8$
DN-DETR-DC$5$ | ✓ |  $12$e | 44.5(+2.6) | $25.9$ | $48.2$ | $62.2$
DAB-DETR-C$5$ |  |  $12$e | $35.2$ | $16.7$ | $38.6$ | $51.6$
DAB-DETR-C$5$ | ✓ |  $12$e | 39.1(+3.9) | $19.7$ | $42.5$ | $56.8$
DAB-DETR-DC$5$ |  |  $12$e | $37.5$ | $19.4$ | $40.6$ | $53.2$
DAB-DETR-DC$5$ | ✓ |  $12$e | 41.9(+4.4) | $23.3$ | $45.6$ | $58.4$
DAB-Deformable-DETR |  |  $12$e | $44.2$ | $27.5$ | $47.1$ | $58.6$
DAB-Deformable-DETR | ✓ |  $12$e | 45.7(+1.5) | $28.1$ | $49.0$ | $60.6$
DINO-Deformable-DETR |  |  $12$e | $48.8$ | $31.0$ | $52.0$ | $62.4$
DINO-Deformable-DETR | ✓ |  $12$e | 49.8(+1.0) | $32.4$ | $53.0$ | $64.2$
DAB-DETR-C$5$ |  |  $50$e | $42.2$ | $21.5$ | $45.7$ | $60.3$
DAB-DETR-C$5$ | ✓ |  $50$e | 44.5(+2.3) | $24.2$ | $48.5$ | $63.2$
DAB-DETR-DC$5$ |  |  $50$e | $44.5$ | $25.3$ | $48.2$ | $62.3$
DAB-DETR-DC$5$ | ✓ |  $50$e | 46.7(+2.2) | 27.6 | 50.9 | 64.0
DAB-Deformable-DETR |  |  $50$e | $48.1$ | $31.4$ | $51.4$ | $63.4$
DAB-Deformable-DETR | ✓ |  $50$e | 49.7(+1.6) | $31.4$ | $52.5$ | $65.6$
Table 2: Experimental results based on DAB-DETR (Liu et al., 2022), DN-DETR (Li et al., 2022) and DINO (Zhang et al., 2022) on COCO (Lin et al., 2014). Note that we do not use multiple patterns (Wang et al., 2021b) in our experiments. For DN-DETR in the table, we use the improved version of DN (dynamic DN groups (Zhang et al., 2022)) and set the DN number to 100. Thus, we can achieve a slightly better baseline result than the one (with $3$ patterns) reported in the original DN-DETR. For DINO-Deformable-DETR (Zhang et al., 2022) in the table, we only adopt $3$ groups when training with Group DETR.  Model | w/ Group DETR | Schedule | mAPm | AP${}^{m}_{s}$ | AP${}^{m}_{m}$ | AP${}^{m}_{l}$
---|---|---|---|---|---|---
Mask2Former |  |  $12$e | $38.5$ | $17.6$ | $41.4$ | $60.4$
Mask2Former | ✓ |  $12$e | 39.5(+1.0) | $18.8$ | $42.4$ | $60.8$
Table 3: Experimental results based on Mask2Former (Cheng et al., 2021) on
COCO (Lin et al., 2014). The metric mAPm in the table represents the mask AP
for instance segmentation.

Conditional DETR. Conditional DETR uses a high-dimensional vector as the
object query. We enlarge the number of groups from $1$ to $11$ for our Group
DETR in Table 1. Please note that only the first group is kept during
inference. We observe consistent improvements over baseline models. The model
with $11$ groups achieves $37.6$ mAP, which is $5.0$ mAP higher than the
baseline. Moreover, the gains ($+2.5$ mAP) hold when we train the model with a
longer training schedule (50 epochs). In the following experiments, unless
specified, we adopt $11$ groups for Group DETR by default.

DAB-DETR, DN-DETR, and DINO. DAB-DETR uses a $4$-D vector that represents an
anchor box as the object query. Results are listed in Table 2. For the C$5$
model, The model trained with Group DETR obtains $3.9$ mAP and $2.3$ mAP gains
over the baseline under the $12$ epochs and $50$ epochs training schedules.
For the DC$5$ model, the model trained with Group DETR outperform the baseline
model by $4.4$ mAP and $2.2$ mAP under the $12$ epochs and $50$ epochs
settings, respectively. We apply the proposed method to a stronger baseline
DAB-Deformable-DETR (Liu et al., 2022) which use multi-scale features and
deformable attention (Zhu et al., 2020b). Group DETR improves the strong
baseline by $1.6$ AP under the $50$ epochs setting. We also conduct
experiments on DN-DETR and DINO. Results show that the proposed Group DETR
could obtain non-trivial improvements ($+2.6$ mAP and $+1.0$ mAP) over these
methods, which verifies the generality.

##### Mask2Former.

We apply the proposed Group DETR to the Mask2Former (Cheng et al., 2021) to
see whether it could generalize to image segmentation tasks. We adopt the
standard setting (deformable encoder (Zhu et al., 2019) \+ multi-scale
decoder) of Mask2Former as our baseline on MS COCO (Lin et al., 2014). The
results in Table 3 show the effectiveness ($+1.0$ mAPm).

##  5 Conclusion

In this paper, we present Group DETR to support multiple positive object
queries. Group DETR introduces Group-wise One-to-Many label assignment to
achieve end-to-end object detection. In detail, we decouple the multiple
positive object queries into multiple independent groups and conduct One-to-
One assignment in each group. During testing, only one group is kept and there
is no extra computational cost brought compared to the original model. We hope
our Group-wise One-to-Many assignment will motivate researchers to explore
more in this direction.

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# Fermionic Electroweak NNLO Corrections to $e^{+}e^{-}\to ZH$ with Polarized
Beams and Different Renormalization Schemes

Ayres Freitas<EMAIL_ADDRESS>Qian Song<EMAIL_ADDRESS>Keping Xie
<EMAIL_ADDRESS>Pittsburgh Particle-physics Astro-physics & Cosmology
Center(PITT-PACC)
Department of Physics & Astronomy, University of Pittsburgh, Pittsburgh, PA
15260, USA

###### Abstract

Recently, the next-to-next-to-leading order (NNLO) electroweak corrections
with fermion loops to the Higgsstrahling process were computed. Here we
present numerical results for polarized electron/positron beams, as well as
for two input parameter schemes known as the $\alpha(0)$ and $G_{\mu}$
schemes. The size of the NNLO corrections strongly depends on the beam
polarization, leading to an increase of the $ZH$ cross-section by 0.76% for
$e^{+}_{\rm L}e^{-}_{\rm R}$ beams, and a decrease of 0.04% for $e^{+}_{\rm
R}e^{-}_{\rm L}$ beams. Furthermore, inclusion of the NNLO corrections is
found to significantly reduce the discrepancy between the results in the
$\alpha(0)$ and $G_{\mu}$ schemes. Using the remaining difference, together
with other methods, the theory uncertainty from missing bosonic electroweak
corrections is estimated to be less than 0.3%.

##  I Introduction

A high-luminosity $e^{+}e^{-}$ collider operating at center-of-mass energies
of 240–250 GeV can perform model-independent precision measurements of the
properties of the recently discovered Higgs boson. Several collider concepts
have been proposed for this purpose: the International Linear Collider (ILC)
ilc1 ; ilc2 , the Future Circular Collider (FCC-ee) fccee , and the Circular
Electron-Positron Collider (CEPC) cepc . The cross-section for the main
production process, $e^{+}e^{-}\to ZH$, is projected to be measured with per-
cent level precision at these facilities (1.2$\%$ at ILC, 0.4$\%$ at FCC-ee,
and 0.5$\%$ at CEPC). Any deviations from the Standard Model (SM) expectation
can be interpreted as sign of new physics beyond the SM.

Such an interpretation requires sufficiently accurate theoretical predictions
for $ZH$ production within the SM, including higher-order radiative
corrections. While the next-to-leading order (NLO) corrections have been known
since many years nlo1 ; nlo2 ; nlo3 , the mixed electroweak-QCD NNLO
corrections have been computed more recently ewqcd1 ; ewqcd2 . Very recently,
the fermionic electroweak NNLO corrections have been completed Freitas:2022hyp
, where “fermionic” denotes contributions from diagrams with closed fermion
loops. There is also ongoing work to compute the full electroweak NNLO
corrections, including “bosonic” contributions Chen:2022mre .

This article expands on the work of Ref. Freitas:2022hyp  by including the
effect of beam polarization and different renormalization schemes. Beam
polarization can help to disentangle the nature of potential new physics
effects in $ZH$ production. The SM NLO corrections to $e^{+}e^{-}\to ZH$ with
polarized beams have been presented in Ref. Bondarenko:2018sgg . Here, this is
extended by also computing the fermionic electroweak NNLO corrections.

In addition, we present NNLO results for two different renormalization
schemes. One scheme, called the $\alpha(0)$ scheme, uses the electromagnetic
coupling, together with particle masses, as inputs to specify the SM
parameters. This scheme has been used in Ref. Freitas:2022hyp . Alternatively,
the $G_{\mu}$ scheme instead uses the Fermi constant as an input to define the
electromagnetic coupling strength. The numerical difference between the
predictions in the two schemes can be taken as a proxy for the impact of
missing higher-order corrections.

##  II Polarized cross-sections

The perturbative expansion of the squared matrix element ${\cal M}$ takes the
following form:

$\displaystyle|{\cal M}|^{2}=\,$ $\displaystyle|{\cal M}_{(0)}|^{2}$ (LO)
$\displaystyle+2\,\text{Re}\\{{\cal M}_{(0)}^{*}{\cal M}_{(1)}\\}$ (NLO)
$\displaystyle+|{\cal M}_{(1)}|^{2}+2\,\text{Re}\\{{\cal M}_{(0)}^{*}{\cal
M}_{(2)}\\}$ $\displaystyle\text{(NNLO)},$ (1)

where the subscript indicates the loop order. For the process $e^{+}e^{-}\to
ZH$, the matrix element has the form

$\displaystyle{\cal
M}_{(n)}=\bar{v}(p_{e^{+}})\Gamma^{\mu}_{(n)}u(p_{e^{-}})\,\varepsilon_{\mu}\,,$
(2)

where $p_{e^{\pm}}$ are the momenta of the incoming electron and positron,
respectively, and $\varepsilon_{\mu}$ is the polarization vector of the
outgoing Z boson. For the unpolarized cross-section, one needs to average of
initial spins, which is accomplished by the Carsimir trace technique:

$\displaystyle\frac{1}{4}\sum_{e^{\pm}\text{ spins}}{\cal M}^{*}_{(m)}{\cal
M}_{(n)}=\frac{1}{4}\,\text{Tr}\bigl{\\{}\overline{\Gamma}^{\mu}_{(m)}\,\cancel{p}_{e^{+}}\,\Gamma_{\mu,(n)}\,\cancel{p}_{e^{-}}\bigr{\\}},$
(3)

where the electron mass has been neglected.

On the other hand, the results for left-/right-handed polarized electron and
positron beams are obtained by inserting polarization projectors $P_{\rm
R,L}=(1\pm\gamma^{5})/2$:

$\displaystyle\begin{aligned} {\cal M}^{*}_{(m)}{\cal
M}_{(n)}\big{|}_{e^{+}_{j}e^{-}_{k}}=\frac{1}{4}\,\text{Tr}\bigl{\\{}\overline{\Gamma}^{\mu}_{(m)}\,\cancel{p}_{e^{+}}\,P_{j}\,\Gamma_{\mu,(n)}\,P_{k}\,\cancel{p}_{e^{-}}\bigr{\\}}&\\\
(j,k=\text{L,R}),&\end{aligned}$ (4)

After this step, the remaining calculation proceeds just in the same way as
for the unpolarized cross-section. In particular, the non-trivial two-loop
integrals can be computed with the techniques described in Refs. Song:2021vru
; Freitas:2022hyp : One of the of the two subloops simplified using Feynman
parameters and then expressed in terms of an dispersion integral. This allows
one to solve the second subloop integral analytically in terms of well-known
one-loop Passarino-Veltman functions. Ultraviolet (UV) divergencies are
removed with suitable subtraction terms, which can be integrated analytically
and then added back. Additional technical aspects of the computation of the
subtraction terms are given in the appendix. Finally, the finite remainder
integral is integrated numerically over the Feynman parameters and the
dispersion variable. See Refs. Song:2021vru ; Freitas:2022hyp  for more
details.

Instead of separately going through all the steps of the computation for left-
and right-handed polarized beams, one can alternatively derive the polarized
matrix elements from the unpolarized one, which seems more efficient for
calculating the two-loop diagrams. This is achieved by grouping certain types
of diagrams. Considering a two-loop diagram where the incoming fermion line
connects with $N$ gauge bosons($\gamma$ or $Z$), the polarized matrix elements
satisfy the relationships

$\displaystyle M_{(2)}^{V_{1}\cdots
V_{N}*}M_{(0)}\big{|}_{e^{+}_{R}e^{-}_{L}}$
$\displaystyle=\frac{4\,g_{L}^{eeZ}\,g_{L}^{eeV_{1}}\cdots
g_{L}^{eeV_{N}}}{g_{L}^{eeZ}\,g_{L}^{eeV_{1}}\cdots
g_{L}^{eeV_{N}}+g_{R}^{eeZ}\,g_{R}^{eeV_{1}}\cdots
g_{R}^{eeV_{N}}}\bigl{[}M_{(2)}^{V_{1}\cdots V_{N}*}M_{(0)}\bigr{]}_{\rm
unpol}\,,~{}$ $\displaystyle M_{(2)}^{V_{1}\cdots
V_{N}*}M_{(0)}\big{|}_{e^{+}_{L}e^{-}_{R}}$
$\displaystyle=\frac{4\,g_{R}^{eeZ}\,g_{R}^{eeV_{1}}\cdots
g_{R}^{eeV_{N}}}{g_{L}^{eeZ}\,g_{L}^{eeV_{1}}\cdots
g_{L}^{eeV_{N}}+g_{R}^{eeZ}\,g_{R}^{eeV_{1}}\cdots
g_{R}^{eeV_{N}}}\bigl{[}M_{(2)}^{V_{1}\cdots V_{N}*}M_{(0)}\bigr{]}_{\rm
unpol}\,,~{}$ $\displaystyle M_{(2)}^{V_{1}\cdots
V_{N}*}M_{(0)}\big{|}_{e^{+}_{L}e^{-}_{L}}$
$\displaystyle=M_{(2)}^{V_{1}\cdots
V_{N}*}M_{(0)}\big{|}_{e^{+}_{R}e^{-}_{R}}=0.$ (5)

where $g_{L(R)}^{eeV}$ is the left(right)-handed coupling of $eeV$ vertex.

If the fermion line connects only with W bosons, which only interact with
left-handed fermions, thus we end up with very simple equations

$\displaystyle M_{(2)}^{W_{1}\cdots
W_{N}*}M_{(0)}\big{|}_{e^{+}_{R}e^{-}_{L}}$
$\displaystyle=4\bigl{[}M_{(2)}^{W_{1}\cdots W_{N}*}M_{(0)}\bigr{]}_{\rm
unpol}\,,~{}$ $\displaystyle M_{(2)}^{W_{1}\cdots
W_{N}*}M_{(0)}\big{|}_{e^{+}_{L}e^{-}_{R}}$ $\displaystyle=0.$ (6)

Numerical results for the polarized cross-section are presented in Tab. 1,
using the following input parameters:

$\displaystyle m_{W}^{\rm exp}=80.379~{}\text{GeV}$
$\displaystyle\Rightarrow\quad m_{W}=80.352~{}\text{GeV},$ $\displaystyle
m_{Z}^{\rm exp}=91.1876~{}\text{GeV}$ $\displaystyle\Rightarrow\quad
m_{Z}=91.1535~{}\text{GeV},$ $\displaystyle m_{H}=125.1~{}\text{GeV},$
$\displaystyle m_{t}=172.76~{}\text{GeV},$ $\displaystyle\alpha^{-1}=137.036,$
$\displaystyle\Delta\alpha=0.059,$ $\displaystyle\sqrt{s}=240~{}\text{GeV}.$
(7)

where $\sqrt{s}$ represents the center-of-mass energy, and the masses of all
the other fermions are set to be 0. Furthermore,
$\Delta\alpha=1-\alpha(m_{Z})/\alpha(0)$ accounts for the running of the
electromagnetic coupling between the Thomson limit and the weak scale, and it
includes non-perturbative hadronic effects.

Since the Z and W bosons have sizeable decay widths, their masses ($m_{Z}$ and
$m_{W}$) are defined via the complex pole of the gauge-boson propagators. This
definition is theoretically well-defined but it differs from the one that is
commonly used in experimental studies. The relation between the two is given
by Bardin:1988xt

$\displaystyle m_{Z}=m_{Z}^{\rm exp}\,[1+(\Gamma_{Z}^{\rm exp}/m_{Z}^{\rm
exp})^{2}]^{-1/2},$ (8) $\displaystyle\qquad\Gamma_{Z}=\Gamma_{Z}^{\rm
exp}\,[1+(\Gamma_{Z}^{\rm exp}/m_{Z}^{\rm exp})^{2}]^{-1/2}.$ (9)

The numerical results presented in this paper do not include QED initial-state
radiation (ISR). At the order that we are working, QED ISR factorizes and can
be taken into account through convolution with a universal structure function.

| $e^{+}_{\rm R}e^{-}_{\rm L}$ | $e^{+}_{\rm L}e^{-}_{\rm R}$
---|---|---
$\sigma^{\text{LO\phantom{NN}}}$ [fb] | 541.28 | 350.55
$\sigma^{\text{NLO\phantom{N}}}$ [fb] | 507.92 | 411.66
$\sigma^{\text{NNLO}}$ [fb] | 507.51 | 418.68
$\qquad\mathcal{O}(\alpha^{2}_{N_{f}=2})$ | 1.75 | 5.77
$\qquad\mathcal{O}(\alpha^{2}_{N_{f}=1})$ | $-2.15$ | 1.25
Table 1: Numerical results for the integrated $ZH$ production cross section,
in fb, at LO, NLO and fermionic electroweak NNLO, for different beam
polarizations. The results are for the $\alpha(0)$ renormalization scheme and
$\sqrt{s}=240$ GeV. The electroweak NNLO corrections are also listed
individually according to the number of fermion loops symbolized as $N_{f}$.

As can be seen from Tab. 1, the electroweak NNLO corrections depend strongly
on the beam polarization. The contributions with two closed fermions loops
($N_{f}=2$) are significantly larger for right-handed electron polarization
and left-handed positron polarization than for the opposite case. The
contribution with one closed fermion loop ($N_{f}=1$) has opposite signs for
the two polarization, which leads to an accidental cancellation for the
unpolarized cross-section.

The higher-order corrections also modify the shape of the differential cross-
section, and the specific form of this modification depends on the beam
polarization. This is illustrated in Fig 1, which shows the cross-section as a
function of the scattering angle $\theta$ at different orders of perturbation
theory for two beam polarizations. As can be seen from the figure, the shape
distortions from the NNLO corrections are not very large, but non-negligible.
They are more significant for the $e^{-}_{\rm L}e^{+}_{\rm R}$ polarization,
which is plausible due to the impact of box diagrams with W bosons in this
case.

Figure 1: Differential cross-sections, as a function of the scattering angle
$\theta$, for two different beam polarizations. As in Tab. 1, the plots are
based on the $\alpha(0)$ renormalization scheme and $\sqrt{s}=240$ GeV.

##  III Results for different renormalization schemes

The on-shell (OS) renormalization scheme defines the masses of all elementary
particles via the (complex) propagator pole. In our calculation, this
prescription is applied to the gauge boson masses, $m_{Z,W}$, the Higgs boson
mass, $m_{H}$, and the top-quark mass, $m_{t}$. All other fermion masses are
neglected. As already mentioned in the previous section, the finite width of
the Z and W bosons needs to be taken into account, so that the poles of their
propagators become complex. In deriving the $Z/W$ mass counterterms, the power
counting $\Gamma_{Z,W}/m_{Z,W}\sim\mathcal{O}(\alpha)$ is used, _i.e._ one
performs a simultaneous expansion in the coupling constant and the gauge-boson
widths. Since the Higgs-boson width is very small and the top quark only
appear inside of loop contributions, their widths can be neglected. More
details on the mass renormalization can be found, _e.g._ , in Ref.
Freitas:2002ja .

In addition, one needs a renormalization prescription for the electroweak
coupling strength. In the following, two different schemes for this purpose
are compared. One scheme, called the $\alpha(0)$ scheme, defines
$\alpha=e^{2}/(4\pi)$ as the electromagnetic coupling at zero momentum, and
the weak coupling is defined via

$\displaystyle
g=\frac{e}{\sin\theta_{W}}=\frac{e}{\sqrt{1-m_{W}^{2}/m_{Z}^{2}}}$ (10)

to all orders in perturbation theory. This scheme is sensitive to the running
of $\alpha(Q)$ from $Q=0$ to the weak scale, and it needs $\Delta\alpha$ as a
numerical input.

The second scheme, called the $G_{\mu}$ scheme, relates the weak coupling to
the Fermi coupling $G_{\mu}$,

$\displaystyle\frac{G_{\mu}}{\sqrt{2}}=\frac{g^{2}}{8m_{W}^{2}}(1+\Delta r).$
(11)

The numerical value for $G_{\mu}$ is extracted from the measured muon lifetime
pdg . The quantity $\Delta r$ contains radiative corrections that are
determined by matching the muon decay matrix element in the Fermi theory and
the full SM. We use NLO and fermionic NNLO results for $\Delta r$ from Refs.
Freitas:2000gg ; Freitas:2002ja  (also see Refs. Awramik:2003ee ). The
electromagnetic coupling in the $G_{\mu}$ scheme is derived from $g$ by again
using eq. (10). Note that this scheme does not depend on the shift
$\Delta\alpha$ of the running electromagnetic coupling.

Using the input parameters in eq. (7), together with $G_{\mu}=1.1663787\times
10^{-5}$, the results obtained in the two renormalization schemes are shown in
Tab 2. As can be seen from the table, the numerical agreement between the
results in the two schemes improves with each order in perturbation theory, as
expected.

|  $\alpha(0)$ scheme |  $G_{\mu}$ scheme
---|---|---
$\sigma^{\text{LO\phantom{NN}}}$ [fb] | 222.96 | 239.18
$\sigma^{\text{NLO\phantom{N}}}$ [fb] | 229.89 | 232.08
$\sigma^{\text{NNLO}}$ [fb] | 231.55 | 232.74
$\qquad\mathcal{O}(\alpha^{2}_{N_{f}=2})$ | 1.88 | 0.73
$\qquad\mathcal{O}(\alpha^{2}_{N_{f}=1})$ | $-0.23$ | $-0.07$
Table 2: Numerical results for the unpolarized integrated $ZH$ production
cross section, in fb, for two different renormalization schemes. Results are
given for $\sqrt{s}=240$ GeV at LO, NLO and fermionic electroweak NNLO. For
the latter, the contributions from two ($N_{f}=2$) and one ($N_{f}=1$) closed
fermion loops are also shown individually.

In fact, this convergence is further improved when including the mixed
electroweak-QCD two-loop corrections ewqcd1 ; ewqcd2 . We use numerical
results for this contribution from Ref. ewqcd2 . In order to do so, we have to
compute our electroweak corrections for the same input parameters used there.
The results are shown in Tab. 3.

|  $\alpha(0)$ scheme |  $G_{\mu}$ scheme
---|---|---
$\sigma^{\text{LO}}$ [fb] | 223.14 | 239.64
$\sigma^{\text{NLO}}$ [fb] | 229.78 | 232.46
$\sigma^{\text{NNLO,EW$\times$QCD}}$ [fb] | 232.21 | 233.29
$\sigma^{\text{NNLO,EW}}$ [fb] | 233.86 | 233.98
Table 3: Similar to Tab. 2, but using input values and mixed EW-QCD
corrections from Ref. ewqcd2 .

The prediction for the cross-section including all available results agrees
very well between the two renormalization schemes, with a difference of 0.12
fb. This difference is due to missing higher-order corrections, where the
dominant impact is expected from the bosonic electroweak NNLO corrections,
_i.e._ from two-loop contributions without closed fermion loops.

Therefore, one can use the difference between the two renormalization schemes
as an order-of-magnitude estimate of the perturbative theory uncertainty.
Since this estimate is only a lower bound on the size of missing higher-order
contributions, we conservatively multiply it by a factor 2, to arrive at an
error estimate of 0.24 fb.

An alternative estimate of the bosonic NNLO corrections could be obtained by
considering a subset of the latter, namely those stemming from $|{\cal
M}_{(1,\rm bos)}|^{2}$, where ${\cal M}_{(1,\rm bos)}$ is the matrix element
of the bosonic NLO corrections. This leads to a contribution of 0.65 fb to the
cross-section. The contribution from genuine bosonic two-loop diagrams,
$2\,\text{Re}\\{{\cal M}_{(0)}^{*}{\cal M}_{(2,\rm bos)}\\}$, is expected to
be smaller than this, since the Born matrix element ${\cal M}_{(0)}$ contains
several suppression factors: (a) the e-e-Z couplings in the initial state are
smaller than the e-$\nu$-W couplings, which appear in the 1-loop box diagrams,
by a factor $2^{-3/2}\sim 0.35$; (b) the s-channel Z propagator produces a
factor $m_{Z}^{2}/(s-m_{Z}^{2})\sim 0.17$ for $\sqrt{s}=240$ GeV.

Thus it seems plausible that the missing bosonic electroweak NNLO corrections
have an impact between 0.24 and 0.65 fb on the SM prediction for the $ZH$
production cross-section. These theory error estimates are lower than the
anticipated experimental precision (0.4–1%), but a direct calculation of these
missing contributions is still desirable.

##  IV Conclusions

In this article, we present the calculation of the $e^{+}e^{-}\to ZH$ cross
section with polarized beams, while also addressing the renormalization scheme
dependence. The electroweak NNLO corrections exhibit a strong dependence on
the beam polarizations. The corrections are found to be large for
$e_{L}^{+}e_{R}^{-}$ beam polarization, while small for $e_{R}^{+}e_{L}^{-}$
case due to numerical cancellation. By computing the cross section in the
$\alpha(0)$ and $G_{\mu}$ schemes, we have shown that the renormalization
scheme dependence decreases by including the two-loop electroweak corrections,
and reduces further by adding mixed EW-QCD corrections. Renormalization scheme
dependence can be utilized to estimate missing higher order corrections.
Combining this with partial results for the missing bosonic electroweak NNLO
corrections, we estimate the latter to be about $0.1-0.3\%$, thus lower than
the anticipated experimental precision ($0.4-1\%$).

## Acknowledgments

This work has been supported in part by the National Science Foundation under
grant no. PHY-2112829.

##  Appendix A UV subtraction terms

Figure 2: Two-loop VZH vertex diagram

The UV divergent subtraction terms must be expanded appropriately to get the
correct finite term. As stated in Ref. Freitas:2022hyp , there are three types
of subtraction terms: two for subloop divergences and one more for a global
divergence. The latter corresponds to vacuum diagrams, the analytical formulas
of which can be obtained and expanded to higher orders in $\epsilon$ with TVID
tvid . The general form for the subloop divergences can be expressed as the
multiplication of two one-loop scalar integrals. Taking the two-loop vertex
diagram Fig. 2 as an example, the tensor integral can be written as

$\displaystyle I$
$\displaystyle=\int\frac{d^{D}q_{2}}{i\pi^{2}}\frac{d^{D}q_{1}}{i\pi^{2}}\sum_{n_{0},n_{1},n_{2},i,j}c_{ij}^{n_{0},n_{1},n_{2}}\times\\{p_{i}^{n_{0}},q_{1}^{n_{1}},q_{2}^{n_{2}}\\}_{j}$
$\displaystyle\times\frac{1}{(q_{2}^{2}-m_{V_{2}}^{2})((q_{2}+p)^{2}-m_{V_{1}}^{2})((q_{2}+q_{1})^{2}-m_{f_{1}}^{2})}$
$\displaystyle\times\frac{1}{(q_{1}^{2}-m_{f_{2}}^{2})((q_{1}-p_{h})^{2}-m_{f_{2}}^{2})((q_{1}-p)^{2}-m_{f_{2}}^{2})}\,,$
(12)

where $\\{p_{i}^{n_{0}},q_{1}^{n_{1}},q_{2}^{n_{2}}\\}$ denotes dot products
among external momentum $p_{i}$ and loop momentum $q_{1,2}$, and $n_{i}$
denote the power of each of them. The index $j$ labels all possible dot
product conditions. For example for $n_{0}=0,n_{1}=n_{2}=2$, the possible dot
products read

$\displaystyle\\{p_{i}^{0},q_{1}^{2},q_{2}^{2}\\}_{1}=(q_{1}\cdot
q_{1})(q_{2}\cdot q_{2}),$
$\displaystyle\\{p_{i}^{0},q_{1}^{2},q_{2}^{2}\\}_{2}=(q_{1}\cdot
q_{2})(q_{1}\cdot q_{2}).$ (13)

The SM Feynman rules require that $n_{1}\leq 4,n_{2}\leq
2,n_{0}+n_{1}+n_{2}\leq 6$. $c_{ij}^{n_{0},n_{1},n_{2}}$ is the coefficient of
a dot product, and it is a function of masses and dimension $D$. The integral
(12) contains a subloop divergence from the $q_{1}$ loop, which originate from
the numerators $q_{1}^{n_{1}}$ with $n_{1}\geq 4$. To make the $q_{1}$
integral UV finite, the following subtraction term is constructed:

$\displaystyle I_{\text{subtr}}^{q_{1}}$
$\displaystyle=\int\frac{d^{D}q_{2}}{i\pi^{2}}\,\frac{d^{D}q_{1}}{i\pi^{2}}\sum_{i,j}\Big{[}c_{ij}^{2,4,0}\times\\{p_{i}^{2},q_{1}^{4},q_{2}^{0}\\}_{j}$
$\displaystyle+c_{ij}^{1,4,1}\times\\{p_{i}^{1},q_{1}^{4},q_{2}^{1}\\}_{j}+c_{ij}^{0,4,2}\times\\{p_{i}^{0},q_{1}^{4},q_{2}^{2}\\}_{j}$
$\displaystyle+c_{ij}^{1,4,0}\times\\{p_{i}^{1},q_{1}^{4},q_{2}^{0}\\}_{j}+c_{ij}^{0,4,1}\times\\{p_{i}^{0},q_{1}^{4},q_{2}^{1}\\}_{j}$
$\displaystyle+c_{ij}^{0,4,0}\times\\{p_{i}^{0},q_{1}^{4},q_{2}^{0}\\}_{j}\Big{]}$
$\displaystyle\times\frac{1}{(q_{2}^{2}-m_{V_{2}}^{2})((q_{2}+p)^{2}-m_{V_{1}}^{2})(q_{1}^{2}-m_{f_{1}}^{2})}$
$\displaystyle\times\frac{1}{(q_{1}^{2}-m_{f_{2}}^{2})(q_{1}^{2}-m_{f_{2}}^{2})(q_{1}^{2}-m_{f_{2}}^{2})}\,.$
(14)

From Eq. 14, one can see that the loop integrals of $q_{1}$ and $q_{2}$ are
disentangled. After performing the loop integration, one obtains

$\displaystyle
I_{\text{subtr}}^{q_{1}}=B_{0}(p^{2},m_{V_{2}}^{2},m_{V_{1}}^{2})\times\Big{[}a_{1}A_{0}(m_{f_{1}}^{2})+a_{2}A_{0}(m_{f_{2}}^{2})\Big{]}$
(15)

where $a_{i}$ are functions of masses, external momenta and dimension $D$. A
similar subloop subtraction term needs to be introduced in the vacuum
integrals for the global divergence. Combining the two subloop subtraction
terms, we obtain

$\displaystyle I_{\text{subtr}}$
$\displaystyle=\Big{[}B_{0}(p^{2},m_{V_{2}}^{2},m_{V_{1}}^{2})-B_{0}(0,m_{V_{2}}^{2},m_{V_{1}}^{2})\Big{]}$
$\displaystyle\quad\times\Big{[}a_{1}A_{0}(m_{f_{1}}^{2})+a_{2}A_{0}(m_{f_{2}}^{2})\Big{]}$
(16)

This term can now be expanded in powers of $\epsilon=(4-D)/2$, resulting in
the expressions

$\displaystyle I_{\text{subtr}}^{\text{div}}$
$\displaystyle=\Big{[}B_{0}^{(0)}(p^{2},m_{V_{2}}^{2},m_{V_{1}}^{2})-B_{0}^{(0)}(0,m_{V_{2}}^{2},m_{V_{1}}^{2})\Big{]}$
$\displaystyle\quad\times\Big{[}a_{1}^{(0)}A_{0}^{(-1)}(m_{f_{1}}^{2})+a_{2}^{(0)}A_{0}^{(-1)}(m_{f_{2}}^{2})\Big{]},$
(17) $\displaystyle I_{\text{subtr}}^{\text{fin}}$
$\displaystyle=\Big{[}B_{0}^{(0)}(p^{2},m_{V_{2}}^{2},m_{V_{1}}^{2})-B_{0}^{(0)}(0,m_{V_{2}}^{2},m_{V_{1}}^{2})\Big{]}$
$\displaystyle\quad\times\Big{[}a_{1}^{(0)}A_{0}^{(0)}(m_{f_{1}}^{2})+a_{2}^{(0)}A_{0}^{(0)}(m_{f_{2}}^{2})$
$\displaystyle\qquad+a_{1}^{(1)}A_{0}^{(-1)}(m_{f_{1}}^{2})+a_{2}^{(1)}A_{0}^{(-1)}(m_{f_{2}}^{2})\Big{]}$
$\displaystyle+\Big{[}B_{0}^{(1)}(p^{2},m_{V_{2}}^{2},m_{V_{1}}^{2})-B_{0}^{(1)}(0,m_{V_{2}}^{2},m_{V_{1}}^{2})\Big{]}$
$\displaystyle\quad\times\Big{[}a_{1}^{(0)}A_{0}^{(-1)}(m_{f_{1}}^{2})+a_{2}^{(0)}A_{0}^{(-1)}(m_{f_{2}}^{2})\Big{]}.$
(18)

where $(n)$ denote the expansion order in $\epsilon$. Eq. 18 indicates that
$\mathcal{O}(\epsilon)$ parts of one-loop scalar functions must be taken into
account. Analytical expressions for these can be found in Ref. Nierste:1992wg
.

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