# The Impact of Differential Privacy on Group Disparity Mitigation

Victor Petrén Bach Hansen 1 2, Atula Tejaswi Neerkaje3 4††footnotemark: ,
Ramit Sawhney 3,
Lucie Flek 3, Anders Søgaard1
1Department of Computer Science, University of Copenhagen, Denmark
2Topdanmark A/S, Denmark
3Conversational AI and Social Analytics (CAISA) Lab, University of Marburg,
Germany
4Manipal Institute of Technology, India
Authors contributed equally

###### Abstract

The performance cost of differential privacy has, for some applications, been
shown to be higher for minority groups; fairness, conversely, has been shown
to disproportionally compromise the privacy of members of such groups. Most
work in this area has been restricted to computer vision and risk assessment.
In this paper, we evaluate the impact of differential privacy on fairness
across four tasks, focusing on how attempts to mitigate privacy violations and
between-group performance differences interact: Does privacy inhibit attempts
to ensure fairness? To this end, we train
$(\varepsilon,\delta)$-differentially private models with empirical risk
minimization and group distributionally robust training objectives. Consistent
with previous findings, we find that differential privacy increases between-
group performance differences in the baseline setting; but more interestingly,
differential privacy reduces between-group performance differences in the
robust setting. We explain this by reinterpreting differential privacy as
regularization.

##  1 Introduction

Classification tasks in computer vision and natural language processing face
the challenge of balancing performance with the need to prevent discrimination
against protected demographic subgroups, satisfying fairness principles. In
some tasks, we train our classifiers on private data and therefore also need
our models to satisfy privacy guarantees.

Privacy-preserving algorithms, however, may tend to disproportionally affect
members of minority classes (Farrand et al., 2020). E.g., Bagdasaryan et al.
(2019), show the performance cost of differential privacy Dwork et al. (2006)
in face recognition is higher for minority groups, suggesting that privacy and
fairness are fundamentally at odds (Chang and Shokri, 2021; Agarwal, 2021). In
this paper, we evaluate two hypotheses at scale: (a) that the performance cost
of differential privacy is unevenly distributed across demographic groups
(Ekstrand et al., 2018; Cummings et al., 2019; Bagdasaryan et al., 2019;
Farrand et al., 2020), and (b) that such effects can be mitigated by more
robust learning objectives (Sagawa et al., 2020a; Pezeshki et al., 2020).

#### Contributions

We build upon previous work suggesting that differential privacy and fairness
are at odds: Differential privacy may often hurt minority groups the most, and
reducing the fairness gap by focusing on minority groups during training
typically puts their privacy at risk. We evaluate this hypothesis at scale by
measuring the impact of differential privacy in terms of fairness across (1) a
baseline empirical risk minimization and (2) under a group distributionally
robust optimization. We conduct our experiments across four tasks of different
modalities, assuming the group membership information is available at training
time, but not at test time: face recognition (CelebA), topic classification,
volatility forecasting based on earning calls, and sentiment analysis of
product reviews. Our results confirm that differential privacy compromises
fairness in the baseline setting; however, we demonstrate that differential
privacy not only mitigates the decrease but also improves  fairness compared
to non-private experiments for 4/5 datasets in the distributionally robust
setting. We explain this by reinterpreting differential privacy as an
approximation of Gaussian noise injection, which is equivalent to strategies
previously shown to determine the efficacy of group-robust learning.

##  2 Fairness and Privacy

Fair machine learning aims to ensure that induced models do not discriminate
against individuals with specific values in their protected attributes (e.g.,
race, gender). We represent each data point as
$z=(x,g,y)\in\mathcal{X}\times\mathcal{G}\times\mathcal{Y}$, with
$g\in\mathcal{G}$ encoding its protected attribute(s).111In practice our
protected attributes in §3 will be age and gender. Both are protected under
the Equality Act 2010. Let $\mathcal{D}^{g}_{y}$ denote the distribution of
data with protected attribute $g$ and label $y$.

Several definitions of group fairness exist in the literature (Williamson and
Menon, 2019), but here we focus on a generalization of approximately constant
conditional (equalized) risk (Donini et al., 2018):222In the fairness
literature, approximate fairness is referred to as $\delta$-fairness, but
below we will use lower case $\delta$ to refer to
$(\varepsilon,\delta)$-differential privacy, and we refer to $\Delta$-fairness
to avoid confusion.

######  Definition 2.1 ($\Delta$-Fairness).

Let $\ell^{g_{i}}(\theta)=\mathbb{E}[\ell(\theta(x),y)|g=g_{i}]$ be the risk
of the samples in the group defined by $g_{i}$, and $\Delta\in[0,1]$. We say
that a model $\theta$ is $\Delta$-fair if for any two values of $g$, say
$g_{i}$ and $g_{j}$, $|\ell^{g_{i}}(\theta)-\ell^{g_{j}}(\theta)|<\Delta$.

Note that if $\ell$ coincides with the performance metric of a task, and
$\delta=0$, this is identical to performance or classification parity (Yuan et
al., 2021).333Performance or classification parity has been argued to suffer
from statistical limitations in Corbett-Davies and Goel (2018), which remind
us that when risk distributions differ, standard error metrics are poor
proxies of individual equity. This is known as the problem of infra-
marginality. Note, however, that this argument does not apply to binary
classification problems. Such a notion of fairness can be derived from John
Rawls’ theory on distributive justice and stability, treating model
performance as a resource to be allocated. Rawls’ difference principle,
maximizing the welfare of the worst-off group, is argued to lead to stability
and mobility in society at large (Rawls, 1971). $\Delta$ directly measures
what is sometimes called Rawlsian min-max fairness (Bertsimas et al., 2011).
In our experiments, we measure $\Delta$-fairness as the absolute difference
between performance of the worst-off and best-off subgroups.

Recall the standard definition of $(\varepsilon,\delta)$-privacy:

######  Definition 2.2.

$\theta$ is $(\varepsilon,\delta)$-private iff
$\mbox{Pr}[\theta(\mathcal{X})]\leq\exp(\varepsilon)\times\mbox{Pr}[\theta(\mathcal{X}^{\prime})]+\delta$
for any two distributions, $\mathcal{X}$ and $\mathcal{X^{\prime}}$, different
at most in one row.

Differential privacy thereby ensures that an algorithm will generate similar
outputs on similar data sets. Note the multiplicative bound
$\exp(\varepsilon)$ and the additive bound $\delta$ serve different roles: The
$\delta$ term represents the possibility that a few data points are not
governed by the multiplicative bound, which controls the level of privacy
(rather than its scope). Note that it also follows directly that if
$\varepsilon=0$ and $\delta=0$, absolute privacy is required, leading $\theta$
to be independent of the data.

Several authors have shown that differential privacy comes at different costs
for minority subgroups (Ekstrand et al., 2018; Cummings et al., 2019;
Bagdasaryan et al., 2019; Farrand et al., 2020). The more private the model is
required to be, the larger group disparities it may exhibit444Note this is a
different trade-off than the fairness-privacy trade-off which results from the
need for collecting sensitive data to learn fair models; the latter is
discussed at length in Veale and Binns (2017).. This happens because
differential privacy distributes noise where it is needed to reduce the
influence of individual examples. Since outlier examples are likely to have
disproportional influence on output distributions (Campbell, 1978; Chernick
and Murthy, 1983), they are also disproportionally affected by noise injection
in differential privacy.

Agarwal (2021) show that, in fact, a $(\varepsilon,0)$-private and fully fair
model – using equalized odds as the definition of fairness – will be unable to
learn anything. To see this, remember that a fully private model is
independent of the data and unable to learn from correlations between input
and output. If $\theta$ is, in addition, required to be fair, it is thereby
required to be fair for all distributions, which prevents $\theta$ from
encoding any prior beliefs about the output distribution. Note this finding
generalizes straight-forwardly to equalized risk, and even to approximate
fairness (since even for finite distributions, we can define a $\Delta>0$,
such that preserving absolute privacy would lead to a constant $\theta$).

######  Theorem 1.

For sufficiently small values of $\Delta$, a fully $(\varepsilon,0)$-private
model $\theta$ that is also $\Delta$-fair, will have trivial performance.

###### Proof.

This follows directly from the above. ∎

While we do not strictly require an absolute privacy in our experiments
(setting $\delta=10^{-5}$), intuitively, privacy potentially compromises
fairness by adding more noise to data points of minority group members than to
those of majority groups. Fairness, on the other hand, leads to over-sampling
or over-attending to data points of minority group members, more likely
compromising their privacy.

Pannekoek and Spigler (2021) show, however, that it is possible to learn
somewhat private and somewhat fair classifiers. They combine differential
privacy with reject option classification. Their results nevertheless suggest
that privacy and fairness objectives are fundamentally at odds, as fairness
decreases with the introduction of differential privacy.

##  3 Experiments

Figure 1: Examples of the different subgroups that appear in a subset of the
datasets we train on. CelebA (left) contains images of celebrities, using
hair-color as our target variable and gender as our protected attribute. Blog
Authorship Corpus (right) contains text-based blogposts on two topics
{Technology, Arts} our targets, using
$\mathcal{G}:\\{\mbox{Man},\mbox{Woman}\\}\times\\{\mbox{Young},\mbox{Old}\\}$
as our protected subgroups.

This section describes the algorithms and datasets involved in our
experiments, and presents the results of these.

###  3.1 Algorithms

#### Empirical Risk Minimization

For a model parameterized by $\theta$, in our baseline Empirical Risk
Minimization (ERM) setting, we minimize the expected loss
$\mathbb{E}[\ell(\theta(x),y)]$ with data
$(x,g,y)\in\mathcal{X}\times\mathcal{G}\times\mathcal{Y}$ drawn from a dataset
$\mathcal{D}$:

$\hat{\theta}_{ERM}=\operatorname*{argmin}_{\theta}\mathbb{E}_{\hat{\mathcal{D}}}[\ell(\theta(x),y)]$
(1)

Here $\hat{\mathcal{D}}$ denotes the empirical training distribution. Note
that we disregard any group information in our data. In an overparameterized
setting, ERM is prone to overfitting spurious correlations, which are more
likely to hurt performance on minority groups (Sagawa et al., 2020b).

#### Distributionally Robust Optimization

Several authors have suggested to mitigate the effects of such overfitting by
explicitly optimizing for out-of-distribution mixtures of sub-populations (Hu
et al., 2018; Oren et al., 2019; Sagawa et al., 2020a). In this work we focus
on Group-aware Distributionally Robust Optimization (Group DRO) (Sagawa et
al., 2020a).

Under the assumption that the training distribution $\mathcal{D}$ is a mixture
of a discrete number of groups, $\mathcal{D}_{g}$ for $g\in\mathcal{G}$, we
define the worst-case loss as the maximum of the group-specific expected
losses:

$\ell(\theta)_{worst}=\operatorname*{max}_{g\in\mathcal{G}}\mathbb{E}_{\hat{\mathcal{D}_{g}}}[\ell(\theta(x),y)]$
(2)

In Group DRO – in contrast with ERM – we exploit our knowledge of the group
membership of data points ($x,g,y$). The overall objective is for minimizing
the empirical worst-case loss is therefore:

$\hat{\theta}_{DRO}=\operatorname*{argmin}_{\theta}\Big{[}\ell(\hat{\theta})_{worst}:=\operatorname*{max}_{g\in
G}\mathbb{E}_{\hat{\mathcal{D}}_{g}}[\ell(\theta(x),y)]\Big{]}$ (3)

Note, again, that the knowledge of group membership $g$ is only available at
training time, not at test time. Unlike Sagawa et al. (2020a), we do not
employ heavy $\ell_{2}$ regularization during our experiments, but rather use
it with the same parameters as proposed in Koh et al. (2021).

#### Differentially Private Stochastic Gradient Descent (DP-SGD)

We implement differential privacy (Dwork et al., 2006) using DP-SGD, as
presented in Abadi et al. (2016). DP-SGD limits the influence of training
samples by (i) clipping the per-batch gradient where its norm exceeds a pre-
determined clipping bound $C$, and by (ii) adding Gaussian noise $\mathcal{N}$
characterized by a noise scale $\sigma$ to the aggregated per-sample
gradients. We control this influence with a privacy budget $\varepsilon$,
where lower values for $\varepsilon$ indicates a more strict level of privacy.
DP-SGD has remained popular, among other things because it generalizes to
iterative training procedures (McMahan et al., 2018), and supports tighter
bounds using the Rényi method (Mironov, 2017).

Differential privacy generally comes at a performance cost, leading to
privacy-preserving models performing worse compared to their non-private
counterparts (Alvim et al., 2011). However, we follow Kerrigan et al. (2020)
and finetune the private models, which are first pretrained (without
differential privacy) on a large public dataset. This protocol generally seems
to provide a better trade-off between accuracy and privacy (Kerrigan et al.,
2020), leading to better-performing, yet private models. The only exception to
this setup is the volatility forecasting task, where our models were trained
from scratch, as those rely on PRAAT audio features.

|  | Performance at $\varepsilon$-Privacy
---|---|---
|  | No DP | $\varepsilon_{1}$ | $\varepsilon_{2}$ | $\varepsilon_{3}$
|  | Score | $\varepsilon$ | Score | $\varepsilon$ | Score | $\varepsilon$ | Score | $\varepsilon$
Celeb | ERM | $0.954\pm 0.000$ | - | $0.943\pm 0.001$ | $9.50$ | $0.940\pm 0.002$ | $5.17$ | $0.932\pm 0.001$ | $0.99$
DRO | $0.953\pm 0.001$ | - | $0.899\pm 0.006$ | $9.50$ | $0.891\pm 0.014$ | $5.17$ | $0.873\pm 0.007$ | $0.99$
Blog | ERM | $0.699\pm 0.002$ | - | $0.661\pm 0.003$ | $9.25$ | $0.661\pm 0.003$ | $5.03$ | $0.648\pm 0.005$ | $1.02$
DRO | $0.692\pm 0.001$ | - | $0.651\pm 0.001$ | $9.25$ | $0.650\pm 0.005$ | $5.03$ | $0.630\pm 0.003$ | $1.02$
Vol. | ERM | $0.756\pm 0.036$ | - | $0.778\pm 0.073$ | $9.32$ | $0.794\pm 0.046$ | $6.42$ | $0.778\pm 0.039$ | $0.96$
DRO | $0.814\pm 0.061$ | - | $0.798\pm 0.042$ | $9.32$ | $0.815\pm 0.056$ | $6.42$ | $0.833\pm 0.093$ | $0.96$
T-UK | ERM | $0.933\pm 0.008$ | - | $0.919\pm 0.002$ | $9.39$ | $0.916\pm 0.001$ | $4.94$ | $0.889\pm 0.009$ | $1.02$
DRO | $0.931\pm 0.004$ | - | $0.893\pm 0.006$ | $9.39$ | $0.873\pm 0.015$ | $4.94$ | $0.820\pm 0.015$ | $1.02$
T-US | ERM | $0.894\pm 0.007$ | - | $0.817\pm 0.014$ | $10.71$ | $0.812\pm 0.009$ | $5.10$ | $0.666\pm 0.019$ | $1.01$
DRO | $0.899\pm 0.009$ | - | $0.569\pm 0.132$ | $10.71$ | $0.437\pm 0.112$ | $5.10$ | $0.342\pm 0.012$ | $1.01$

|  | Group-disparity at $\varepsilon$-Privacy
---|---|---
|  | No DP | $\varepsilon_{1}$ | $\varepsilon_{2}$ | $\varepsilon_{3}$
|  | GD | $\varepsilon$ | GD | $\varepsilon$ | GD | $\varepsilon$ | GD | $\varepsilon$
Celeb | ERM | $0.556\pm 0.021$ | - | $0.746\pm 0.032$ | $9.50$ | $0.734\pm 0.025$ | $5.17$ | $0.770\pm 0.013$ | $0.99$
DRO | $0.514\pm 0.042$ | - | $0.039\pm 0.018$ | $9.50$ | $0.080\pm 0.031$ | $5.17$ | $\mathbf{0.056}\pm 0.027$ | $0.99$
Blog | ERM | $0.108\pm 0.013$ | - | $0.149\pm 0.006$ | $9.25$ | $0.140\pm 0.004$ | $5.17$ | $0.136\pm 0.011$ | $0.99$
DRO | $0.078\pm 0.009$ | - | $0.056\pm 0.020$ | $9.25$ | $0.070\pm 0.013$ | $5.17$ | $\mathbf{0.077}\pm 0.027$ | $0.99$
Vol. | ERM | $0.302\pm 0.042$ | - | $0.328\pm 0.067$ | $9.32$ | $0.557\pm 0.050$ | $6.42$ | $0.573\pm 0.050$ | $0.96$
DRO | $0.221\pm 0.062$ | - | $0.320\pm 0.085$ | $9.32$ | $0.371\pm 0.058$ | $6.42$ | $0.421\pm 0.083$ | $0.96$
T-UK. | ERM | $0.018\pm 0.005$ | - | $0.022\pm 0.006$ | $9.39$ | $0.020\pm 0.014$ | $4.94$ | $0.037\pm 0.006$ | $1.02$
DRO | $0.030\pm 0.008$ | - | $0.030\pm 0.004$ | $9.39$ | $0.039\pm 0.023$ | $4.94$ | $\mathbf{0.025}\pm 0.010$ | $1.02$
T-US | ERM | $0.055\pm 0.006$ | - | $0.048\pm 0.019$ | $10.71$ | $0.054\pm 0.015$ | $5.10$ | $0.109\pm 0.017$ | $1.01$
DRO | $0.036\pm 0.007$ | - | $0.118\pm 0.040$ | $10.71$ | $0.078\pm 0.030$ | $5.10$ | $\mathbf{0.021}\pm 0.030$ | $1.01$

Table 1: Performance (top) and $\Delta$-Fairness (bottom) of ERM and Group DRO
across different degrees of differential privacy ($\varepsilon$).
$\varepsilon_{1}$, $\varepsilon_{2}$ and $\varepsilon_{3}$ corresponds to
$\varepsilon$-values of roughly 10, 5 and 1 respectively (see table for exact
values). We report F1 scores for sentiment and topic classification, accuracy
for face recognition and MSE for volatility forecasting. Group disparity (GD)
is measured by the absolute difference between the best and worst performing
sub-group ($\Delta$-Fairness; see Definition 2.1). The performance and
corresponding uncertainties are based on several individual runs of each
configuration, see §6.2 in the Appendix for further details. Differential
privacy consistently hurts fairness for ERM. For Group DRO, we bold-face
numbers where strict differential privacy ($\varepsilon_{3}$) increases
fairness; this happens in 4/5 datasets. We see large increases for face
recognition and small increases for topic classification and sentiment
analysis.

###  3.2 Tasks and architectures

To study the impact of differential privacy on fairness, in ERM and Group DRO,
we evaluate increasing levels of differential privacy across five datasets
that span four tasks and three different modalities: speech, text and vision.

#### Facial Attribute Detection

We study facial attribute recognition with the CelebFaces Attributes Dataset
(CelebA) (Liu et al., 2015). It contains faces of celebrities annotated with
attributes, such as hair color, gender and other facial features. Following
Sagawa et al. (2020a), we use the hair color as our target variable, with
gender being the demographic attribute (see Figure 1 (left)). The dataset
contains $\sim 163K$ datapoints, where the smallest group (blond males) only
counts $1387$. We finetune a publicly pretrained ResNet50, a standard model
for image classification tasks, on the CelebA dataset and evaluate model
performances as accuracies over 3 individual seeds.

#### Topic Classification

For topic classification, we use the Blog Authorship Corpus (Schler et al.,
2006). The Blog Authorship Corpus contains weblogs written on 19 different
topics, collected from the Internet before August 2004. The dataset contains
self-reported demographic information about the gender and age of the authors.
The dataset is limited to gender information in the binary form555Note that
all binary gender assumptions in this work are unwillingly inherited from the
datasets.. We binarize age, distinguishing between young ($=<35$) and older
($>35$) authors,666Older authors tend to be underrepresented in web data.
resulting in four different group combinations (see Figure 1 (right)). We
chose two topics of roughly equal size (Technology and Arts), reducing the
topic classification task to a binary classification task. For our
experiments, we finetune a pretrained English DistilBERT model (Sanh et al.,
2019). To reduce the overall added computational cost of DP-SGD, we freeze our
model, except for the outer-most Transformer encoder layer as well as the
classification layer. We report model performances as F1 scores over 3
individual seeds.

#### Volatility Forecasting

For the stock volatility forecasting task, we use the Earnings Conference
Calls dataset by Qin and Yang (2019). This consists of 559 public earnings
calls audio recordings for 277 companies in the S&P 500 index, spanning over a
year of earnings calls. The self-reported genders of the CEOs was scraped by
Sawhney et al. (2021) from
Reuters,777https://www.thomsonreuters.com/en/profiles.html
Crunchbase,888https://www.crunchbase.com/discover/people and the WikiData
API.999https://query.wikidata.org/ The extracted genders were found to be in
the binary form Sawhney et al. (2021), with 12.3% of speakers being female and
87.7% of speakers being male, a highly skewed distribution. Since our primary
focus with this task is to explore the impact of differential privacy on
speech, we use only audio features without the call transcripts. For each
audio recording $A$ of a given earning call $E$, the goal is to predict the
company’s stock volatility as a regression task. Following Qin and Yang
(2019), we calculate the average log volatility $\tau$ days (temporal window)
following the day of the earnings call. For each audio clip belonging to a
given call, we extract 26-dimensional features with PRAAT (Boersma and Van
Heuven, 2001). Each audio embedding of the call is fed sequentially to a
BiLSTM, followed by an attention layer and two fully-connected layers. The
model is trained by optimizing the Mean Square Error (MSE) between the
predicted and true stock volatility. For all results, we report MSE on the
test set for a 70:10:20 temporal split of the data.The results are averaged
over 5 seeds.

#### Sentiment Analysis

For our sentiment analysis task, we use the Trustpilot Corpus (Hovy et al.,
2015)101010https://bitbucket.org/lowlands/release/src/master/WWW2015/data/. It
consists of text-based user reviews from the Trustpilot website, rating
companies and services on a 1 to 5 star scale. The reviews spans 5 different
countries; Germany, Denmark, France, United Kingdom and USA, however, we only
consider the English reviews, i.e. UK and US. The Trustpilot contains
demographic information about the gender, age and geographic location of the
users, but as with the topic classification task, we only concern ourselves
with the gender and age of the users. As with the topic classification task,
we finetune DistilBERT on the UK and US English parts of the Trustpilot
Corpus, freezing all parameters but the final encoder layer, as well as the
classification layer. Classification performance is measured as F1 scores and
the results are averaged over 3 seeds.

Our implementation is a PyTorch extension of the WILDS
repository111111https://github.com/p-lambda/wilds/ (Koh et al., 2021) using
the DP-SGD implementation provided by the Opacus Differential Privacy
framework121212https://opacus.ai/. For further details about data and
training, see §6.2 in the Appendix. We release the code for our experiments
at: https://github.com/vpetren/fair_dp.

Figure 2: Face Attribute Detection: Performance of individual groups of
increasing levels of $\varepsilon$. Comparing baseline ERM to Group DRO, we
find that Group DRO performance on the minority group (blond males) perform
much better under privacy constraints; we return to this in §3.4.

Figure 3: Topic Classification: Performance of individual groups of increasing
levels of $\varepsilon$. Group DRO, compared to baseline ERM, results in a
more balanced performance across all groups, even on a low privacy budget.

###  3.3 Results

Our results are presented in Table 1. The top half of the table presents
standard (average) performance numbers across multiple runs of ERM and Group
DRO at different privacy levels. Recall that performance for sentiment
analysis as well as topic classification is measured in F1, volatility
forecasting is measured in MSE and face recognition is measured in accuracy.
The accuracy of our ERM face attribute detection classifier is 0.954 in the
non-private setting, for example.

Our first observation is that, as hypothesized earlier, differential privacy
hurts model performance. For our smallest text-based dataset (T-US),
performance becomes very poor at the strictest privacy level. This is however
associated with a high amount of variance between seeds, see Figure 5 in the
Appendix. The above face attribute detection classifier, which had an accuracy
of 0.954 in the non-private setting, has a performance of 0.932 at this level.

#### Differential privacy hurts fairness in ERM

The effect on differential privacy on fairness (bottom half of Table 1) is
also quite consistent. The gap between the majority group and the minority
group (or, more precisely, the best-performing and the worst-performing
demographic subgroup) widens with increased privacy. In face recognition, for
example, the accuracy gap between the two groups is 0.556 without differential
privacy, but 0.770 at the strictest privacy level.

Figure 4: Volatility Forecasting: A comparison of group-disparity between
subgroups for increasing temporal volatility windows ($\tau$) and privacy
budgets ($\varepsilon$), over 5 independent runs.

#### Differential privacy increases fairness in Group DRO

For Group DRO, we see the opposite effect. For 4/5 datasets, we see that
differential privacy leads to an increase in fairness. For face recognition,
for example, the gap goes from 0.514 in the non-private setting to 0.056 in
the strictest, basically disappearing. This is also illustrated in the bar
plots in Figure 2. See Figure 3 for similar bar plots of the topic
classification results; we include similar plots for other tasks in the
Appendix. We do also observe that this increase in privacy can be expensive in
terms of overall performance (e.g. Trustpilot-US). Note that the increase in
fairness at higher privacy levels is seemingly at odds with previous results
suggesting that privacy and fairness conflict, e.g., Agarwal (2021). We return
to this question in §3.4.

Note also that the only exception to the latter trend is for volatility
forecasting, where differential privacy hurts fairness both in ERM and Group
DRO (though Group DRO mitigates the disparity). This speech-based prediction
is the only regression task, and the only task for which we do not rely on
pretrained models trained on public data.

For this task, we further analyze group disparity for varying temporal windows
($\tau$) used to calculate target volatility values, along with increasingly
strict privacy budgets ($\varepsilon$) in Figure 4. The disparity between
subgroups widens with stricter privacy guarantees (Bagdasaryan et al., 2019).
This gap is significant for lower values of $\tau$, strengthening the
hypothesis that short-term volatility forecasting is much harder than long-
term (Qin and Yang, 2019), especially for minority classes due to the
disproportionate impact of noise. Comparing ERM and Group DRO, we find Group
DRO mitigates this disparity gap. We observe disparity reduces with increasing
temporal window, since stock prices over a larger time frame are comparatively
more stable (Qin and Yang, 2019). As a consequence, the influence of Group DRO
for higher $\tau$ ($6,7$) is reduced, despite facilitating faster convergence.
Most importantly, we observe the power of Group DRO in mitigating the
disparity caused by strict privacy safeguards ($\varepsilon=0.96$) for crucial
short term prediction ($\tau=3)$ tasks.

###  3.4 Discussion

It is well-known that differential privacy comes with a performance cost
(Shokri and Shmatikov, 2015).131313A multitude of algorithmic improvements
have been proposed to mitigate the overall accuracy drop caused by the
increased privacy protection -– including private sampling from hyperbolic
word representation spaces (Feyisetan et al., 2019), Gaussian $f$-differential
privacy (Bu et al. 2020), and gradient denoising (Nasr et al., 2020). It is
yet to be examined, if the empirical application of such utility preservation
techniques affects the disparate impact issue. However, recent work has
additionally shown that differential privacy is at odds with most, if not all,
definitions of fairness, including equalized risk (Ekstrand et al., 2018;
Cummings et al., 2019; Bagdasaryan et al., 2019; Farrand et al., 2020). Our
work makes two important contributions: (a) We evaluate and confirm this
hypothesis at a larger scale than previous studies for standard empirical risk
minimization; and (b) we point out that the opposite holds true in the context
of Group Distributionally Robust Optimization: Here, adding differential
privacy improves fairness (equalized risk).

While (b) at first seems to contradict the very hypothesis that (a) confirms –
namely that privacy is at odds with fairness – we believe the explanation is
quite simple, namely that we are observing two opposite trends (at the same
time): On one hand, differential privacy adds disproportionate noise to
minority group examples; but on the other hand, it adds Gaussian noise which
acts as a regularizer to improve robust optimization.

In their evaluation of Group Distributionally Robust Optimization, Sagawa et
al. (2020a) observe that robustness is only achieved in the context of heavy
regularization; specifically, they show fairness improvements when they add
$\ell_{2}$ regularization or early stopping. The $\ell_{2}$ regularization and
early stopping did not increase fairness under ERM, but seemed to ’activate’
Group DRO. This makes intuitive sense: Since regularized models cannot
perfectly fit the training data, heavily regularized Group DRO sacrifices
average performance for worst-case performance and obtain better
generalization. In the absence of regularization, however, Group DRO is less
effective.

In our experiments (§3), we add minimal regularization to Group DRO, following
the implementation in Koh et al. (2021), but differential privacy, we argue,
provides that additional regularization. To see this, remember that DP-SGD
works by Gaussian noise injection. Gaussian noise injection is known to be
near-equivalent to $\ell_{2}$-regularization and early stopping (Bishop,
1995). DP-SGD simply makes the trade-off more urgent.

##  4 Related Work

#### Fair machine learning

Early work on mitigating group-level disparities included oversampling (Shen
et al., 2016; Guo and Viktor, 2004) and undersampling (Drumnond, 2003;
Barandela et al., 2003), as well as instance weighting (Shimodaira, 2000).
Other proposals modify existing training algorithms or cost functions to
obtain fairness (Khan et al., 2017; Chung et al., 2015). In the context of
large-scale deep neural networks, Group DRO is a particularly interesting
approach to mitigating group-level disparities (Creager et al., 2021). See
Williamson and Menon (2019) and Corbett-Davies and Goel (2018) for interesting
discussions of how fairness has been measured. More recent alternatives to
Group DRO include Invariant Risk Minimization (Arjovsky et al., 2020),
Spectral Decoupling (Pezeshki et al., 2020) and Adaptive Risk Minimization
(Zhang et al., 2021). We ran experiments with both Invariant Risk Minimization
and Spectral Decoupling, but they performed much worse than Group DRO.

#### Fairness and privacy

Recent studies suggest that privacy-preserving methods such as differential
privacy tend to disproportionately affect minority class samples (Ekstrand et
al., 2018; Cummings et al., 2019; Bagdasaryan et al., 2019; Farrand et al.,
2020). Pannekoek and Spigler (2021) show that it is possible to learn somewhat
private and somewhat fair classifiers, in their case by combining differential
privacy and reject option classification. Jagielski et al. (2019) introduced
the so-called DP-oracle-learner, derived from an oracle-efficient algorithm
(Agarwal et al., 2018), which satisfies equalized odds, an alternative notion
of fairness (Williamson and Menon, 2019). Lyu et al. (2020) introduced
Differentially Private GANs (DPGANs), while Tran et al. (2020) utilize
Lagrangian duality to integrate fairness constraints to protected attributes.
Group DRO has, to the best of our knowledge, not been studied under
differential privacy before.

##  5 Conclusions

In §2, we summarized previous work suggesting that differential privacy and
fairness are at odds. In §3, we then confirmed this hypothesis at scale,
across five datasets, spanning four tasks and three modalities, showing that
for Empirical Risk Minimization, stricter levels of privacy consistently hurt
fairness. This holds true even after pretraining on large-scale public
datasets (Kerrigan et al., 2020). In the context of Group-aware
Distributionally Robust Optimization (Group DRO) (Sagawa et al., 2020a),
however, which is designed to mitigate group-level performance disparities
(optimizing for equalized risk), we saw the opposite effect: Strict levels of
differential privacy were associated with an increase in fairness. In §3.4, we
discuss how this aligns well with the observation that Group DRO works best in
the context of heavy $\ell_{2}$ regularization, keeping in mind that Gaussian
noise injection is near-equivalent to $\ell_{2}$ regularization (Bishop,
1995).

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##  6 Appendix

###  6.1 Additional Figures

This section contains group-specific bar-plots for the performance on
individual groups in the Trustpilot Corpus. For barplots on CelebA and Blog
Authorship, see Figure 2 and 3.

Figure 5: Performance of individual groups of increasing levels of
$\varepsilon$ for the Trustpilot-US corpus. Error bars show standard deviation
over 3 individual seeds.

Figure 6: Performance of individual groups of increasing levels of
$\varepsilon$ for the Trustpilot-UK corpus. Error bars show standard deviation
over 3 individual seeds.

###  6.2 Experimental Details

This section contains additional details surrounding the experiments described
in §3.

#### CelebA

We use the same processed version of the CelebA dataset as Sagawa et al.
(2020a) and Koh et al. (2021), that is, we use the same train/val/test splits
as Liu et al. (2015) with the Blond Hair attribute as the target with the Male
attribute being the spuriously correlated variable. See group distribution in
the training data in Table 2.

Non-Blond, Man | Blond, Man | Non-Blond, Woman | Blond, Woman
---|---|---|---
$66874$ | $1387$ | $71629$ | $22880$
Table 2: Group distribution in the training set of CelebA

#### Blog Authorship Corpus

In addition to the preprocessing described in §3, we split the data into a
60/20/20 train/val/test split (you can find the exact seed that generates the
splits in our code). See group distribution in the training data in Table 3.

Group | Young, Man | Old, Man | Young, Woman | Old, Woman
---|---|---|---|---
Count | $27222$ | $2295$ | $12750$ | $2435$
Table 3: Group distribution in the training set of Blog Authorship corpus

The Blog Authorship Corpus can be downloaded at:
https://www.kaggle.com/rtatman/blog-authorship-corpus

#### Earnings Conference Calls

Out of the 559 calls, we only include 535 datapoints that contain self-
reported demographic attributes about gender. See Table 4 for group
distributions for the training data. The target stock volatility variable is
calculated following Kogan et al. (2009); Qin and Yang (2019), defined by:

$v_{[t-\tau,t]}=\text{ln}\biggl{(}\sqrt{\frac{\sum_{i=0}^{\tau}(r_{t-i}-\bar{r})^{2}}{\tau}}\biggr{)}$
(4)

Here $r_{t}$ is the return price at day $t$ and $\bar{r}$ the mean of return
prices over the period of $t-\tau$ to $t$. We refer to $\tau$ as the temporal
volatility window in our experiments. The return price $r_{t}$ is defined as
$r_{t}=\frac{P_{t}}{P_{t-1}}-1$ where $P_{t}$ is the closing price on day $t$.

Group | Man | Woman
---|---|---
Count | $333$ | $42$
Table 4: Group distribution in the training set of Earnings Conference Calls

#### Trustpilot

We only include the datapoints that contains complete demographic attributes,
i.e. the gender, age and location, but as with our topic classification
experiments, we only study the group that we can define based on age and
gender. All attributes are self-reported. For training we divide the reviews
into the four resulting groups (Old-Man, Young-Woman, etc.) and downsample the
largest groups to match the size of the smallest group. For validation as well
as testing, we withhold 200 samples from each demographic with an even
distribution among the ratings (1 to 5). The review scores are then binarized
by grouping positive (4 and 5 stars) and negative (1 and 2 stars) and
discarding neutral ones (3 stars). For a similar use of this binarization
scheme, see Gupta et al. (2020) and Desai et al. (2019). See the group
distributions for the training data in Table 5 and 6 for the US and UK tasks
respectively.

Group | Young, Man | Old, Man | Young, Woman | Old, Woman
---|---|---|---|---
Count | $7242$ | $7210$ | $7222$ | $7255$
Table 5: Group distribution in the training set of Trustpilot-US Group | Young, Man | Old, Man | Young, Woman | Old, Woman
---|---|---|---|---
Count | $18464$ | $18693$ | $18554$ | $18693$
Table 6: Group distribution in the training set of Trustpilot-UK

#### BiLSTM

The BiLSTM model was trained using a Nvidia Tesla K80 GPU. We use a learning
rate of $1e^{-2}$ and train using DP-SGD for 30 epochs using a virtual batch
size of 32. The average sequence length of the audio embeddings is 159. We set
the maximum sequence length to 150 as we did not observe a performance
increase for higher values. We run 5 individual seeds for each configuration.

In our differentially private experiments with the BiLSTM (i.e Earnings
Conference Calls), we fix the gradient clipping $C$ to $0.8$. By specifying
various approximate target levels of $\varepsilon\in\\{1,5,10\\}$ a
corresponding noise multiplier $\sigma$ is computed with the Opacus framework,
based on the batch size and number of training epochs.

#### DistilBERT

DistilBERT is a small Transformer model trained by distilling BERT (Devlin et
al., 2019) (bert-base-uncased). It has 3/5th of the parameters of bert-base-
uncased, runs 60% faster, while preserving over 95% of the performance of
bert-base-uncased, as measured on the GLUE language understanding benchmark
(Wang et al., 2018).

We finetune DistilBERT on the Trustpilot corpus and Blog Authorship corpus for
20 epochs each, using a batch size of 8, accumulating gradient for a total
virtual batch size of 16 using the built in Opcaus functionality. We limit the
number of tokens in a sequence to 256 and use a learning rate of $5e^{-4}$
with the AdamW optimizer in addition to a weight decay of $0.01$. Otherwise we
use the default parameters defined in the Huggingface Transformers python
package (version 4.4.2). The models are trained using a single Nvidia TitanRTX
GPU and each configuration takes between 5 and 14 hours to run, depending on
the size of that dataset and if DP is used or not. We run 3 individual seeds
for each configuration.

In our differentially private experiments with DistilBERT (i.e. Blog
Authorship and Trustpilot), we fix the gradient clipping $C$ to $1.2$ and by
specifying various target levels of $\varepsilon\in\\{1,5,10\\}$ a
corresponding noise multiplier $\sigma$ is computed with the Opacus framework,
based on the batch size and number of training epochs.

#### Resnet50

ResNet50 is a variant of the ResNet model (He et al., 2015), which has 48
convolution layers along with 1 max pooling and 1 average pooling layer. It
has 3.8 x $10^{9}$ floating points operations.

We finetune our Resnet50 model on the CelebA dataset for 20 epochs using a
batch size of 64. We optimize the model using standard stochastic gradient
descent (SGD) with a learning rate of $1e^{-3}$, momentum of $0.9$ and no
weight decay. We train our models using a single Nvidia TitanRTX GPU and each
configuration takes between 6 and 8 hours to run, depending on if DP is used
or not. We run 3 individual seeds for each configuration.

As with the differentially private DistilBERT experiments, we also here fix
the gradient clipping $C$ to $1.2$ and by specifying various target levels of
$\varepsilon\in\\{1,5,10\\}$ a corresponding noise multiplier $\sigma$ is
computed with the Opacus framework, based on the batch size and number of
training epochs.


# W boson mass in the NP models with extra $U(1)$ gauge group

Jin-Lei Yang1,2111jlyang@hbu.edu.cn,Zhao-Feng Ge3222gezhaofeng@itp.ac.cn,Xiu-
Yi Yang4333yxyruxi@163.com,Sheng-Kai Cui1,2,Tai-Fu<EMAIL_ADDRESS>Department of Physics, Hebei University, Baoding, 071002, China1
Key Laboratory of High-precision Computation and Application of Quantum Field
Theory of Hebei Province, Baoding, 071002, China2
CAS Key Laboratory of Theoretical Physics, School of Physical Sciences,
University of Chinese Academy of Sciences, Beijing 100049, China3
School of science, University of Science and Technology Liaoning, Anshan,
114051, China4

###### Abstract

The precise measurement of the W boson mass is closely related to the
contributions of new physics (NP), which can significantly constrain the
parameter space of NP models, particularly those with an additional $U(1)$
local gauge group. The inclusion of a new $Z^{\prime}$ gauge boson and gauge
couplings in these models can contribute to the oblique parameters $S$, $T$,
$U$ and W boson mass at tree level. Taking into account the effects of kinetic
mixing, we calculate and analyze the oblique parameters $S$, $T$, $U$ and W
boson mass in such NP models in this study. It is found that the kinetic
mixing effects can make significant contributions to the W boson mass, which
can satisfy the recently measured W boson mass at CDF II or ATLAS by choosing
appropriate values of gauge coupling constants and extra $U(1)$ group charges
of leptons or scalar doublets. In addition, if the leptonic Yukawa couplings
are invariant under the extra $U(1)$ local gauge group, these contributions
can be eliminated by redefining the gauge boson fields through eliminating the
neutral currents involving charged leptons, even with nonzero kinetic mixing
effects.

##  I Introduction

The maintaining of gauge invariant in the standard model (SM) requires the
introducing of gauge bosons. Due to the fact that the Lagrangian involving
Higgs doublet in the SM is invariant under the $SU(2)_{L}\otimes U(1)_{Y}$
gauge transformation, the electroweak gauge bosons acquire nonzero masses
through the spontaneous symmetry breaking (SSB). This SSB mechanism also
provides mass to all massive particles in the SM. Therefore, high-precision
measurements of gauge boson masses are crucial for testing SM and searching
for NP phenomena. Recently, the CDF II measurement CDF:2022hxs  of the W boson
mass in 2022 is approximately $7\sigma$ away from the SM prediction
Haller:2022eyb

$\displaystyle M_{W}^{\rm SM}=80354\pm 7\;{\rm MeV},$ (1) $\displaystyle
M_{W}^{\rm CDF\;II}=80433.5\pm 9.4\;{\rm MeV}.$ (2)

And $M_{W}^{\rm CDF\;II}$ is not compatible with the world average

$\displaystyle M_{W}^{\rm Average}=80377\pm 12\;{\rm MeV},$ (3)

which is based on the measurements at LEP-2 ALEPH:2013dgf , Tevatron
CDF:2012gpf ; D0:2013jba  and the LHC ATLAS:2017rzl ; LHCb:2021bjt . The
measured W boson mass anomaly at CDF II has prompted numerous studies
attempting to explain it within various NP models Athron:2022uzz ; Lu:2022bgw
; DAlise:2022ypp ; Strumia:2022qkt ; Zhu:2022tpr ; Athron:2022qpo ;
deBlas:2022hdk ; FileviezPerez:2022lxp ; Athron:2022isz ; Babu:2022pdn ;
Bahl:2022xzi ; Sakurai:2022hwh ; Nagao:2022dgl ; Du:2022brr ; Zeng:2022lkk ;
Barger:2022wih ; Primulando:2022vip ; Kawamura:2022fhm ; VanLoi:2022eir ;
Cao:2022mif ; Heckman:2022the ; Alguero:2022est ; Allanach:2022bik ;
Senjanovic:2022zwy ; Mandal:2023oyh . In a recent update, the ATLAS
collaboration reported a measurement of the W boson mass based on data from
proton-proton (pp) collisions at a center-of-mass energy $\sqrt{s}=7\;{\rm
TeV}$, the result reads ATLAS:2023fsi

$\displaystyle M_{W}^{\rm ATLAS}=80360\pm 16\;{\rm MeV}.$ (4)

Interestingly, the updated result from the ATLAS collaboration shows agreement
with the SM prediction, with no observed deviation from the SM expectation.
This underscores the importance of obtaining a proper W boson mass that can
satisfy experimental measurements in NP models, which is crucial for NP
phenomenology studies.

Among the various NP models, those with an extra $U(1)$ gauge group, such as
the $U(1)_{B-L}$ extended SM, the minimal supersymmetric model (MSSM) with
local $B-L$ gauge symmetry Khalil:2008ps ; Basso:2012gz ; Elsayed:2012ec ;
Khalil:2015naa ; Yang:2019aao ; Yang:2020ebs ; Yang:2020bmh ; Yang:2021duj ,
the MSSM with $U(1)_{X}$ local gauge group Bandyopadhyay:2011qm ;
Belanger:2011rs ; Belanger:2015cra ; Belanger:2017vpq ; Zhao:2019ggv  and etc,
have attracted the attention of physicists. In these NP models, the
introduction of a new $Z^{\prime}$ gauge boson corresponding the extra $U(1)$
gauge group and new introduced gauge couplings can make contributions to the
oblique parameters $S$, $T$, $U$ at the tree level, and consequently impact
the theoretical predictions on the W boson mass. Therefore, determining the
appropriate gauge coupling strength and the new $U(1)$ gauge group charges of
the particle fields that can satisfy the high-precision measurements of the W
boson mass is crucial in these NP models. In this work, we present a general
analysis of the W boson mass in the NP models with a new extra $U(1)$ gauge
group, and the obtained results can be applied to all such NP models.

The paper is organized as follows. The $S$, $T$, $U$ parameters and W boson
mass in the NP models with extra $U(1)$ local gauge group are calculated in
Sec. II. The numerical results are presented and analyzed in Sec. III.
Finally, a summary is made in Sec. IV.

##  II W boson mass in the NP models with extra $U(1)$ local gauge group

For convenience, the newly introduced $U(1)$ local gauge group can be defined
as $U(1)_{X}$ ($X=B,\;L,\;B-L$ and etc, where $B$ refers to the baryon number,
$L$ refers to the lepton number), i.e. the local gauge group of this kind of
NP model is extended to $SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}\otimes
U(1)_{X}$. Due to the presence of two Abelian groups, the gauge kinetic mixing
can occur and can be induced through RGEs even if it is set to zero at $M_{\rm
GUT}$ Holdom:1985ag ; Matsuoka:1986ig ; delAguila:1987st ; delAguila:1988jz ;
Foot:1991kb ; Babu:1997st . This leads to the covariant derivatives in this
type of NP model being expressed as

$\displaystyle
D_{\mu}=\partial_{\mu}-i\left(\begin{array}[]{c}Y,\;\;X\end{array}\right)\left(\begin{array}[]{c}g_{Y},\;\;\;g_{YX}\\\
g_{XY},\;\;\;\;g_{X}\end{array}\right)\left(\begin{array}[]{c}B_{\mu}\\\
Z^{\prime}_{\mu}\end{array}\right),$ (10)

where $Y,\;X$ are the hypercharge and $U(1)_{X}$ charge respectively, $g_{Y}$
is the measured hypercharge coupling constant, $g_{X}$ is the coupling
constant corresponding to $U(1)_{X}$ gauge group, $g_{YX}$ is the coupling
constant arises from the gauge kinetic mixing effect, while $g_{XY}$ can
always be rotated to $0$ as long as the two Abelian gauge groups are unbroken.

The kinetic terms of vector bosons $\Pi_{33}(p^{2})$, $\Pi_{00}(p^{2})$,
$\Pi_{30}(p^{2})$, $\Pi_{WW}(p^{2})$ can be defined by the effective
Lagrangian Marandella:2005wd ; Cacciapaglia:2006pk

$\displaystyle\mathcal{L}_{\rm
oblique}=-\frac{1}{2}W_{\mu}^{3}\Pi_{33}(p^{2})W^{3\mu}-\frac{1}{2}B_{\mu}\Pi_{00}(p^{2})B^{\mu}-W_{\mu}^{3}\Pi_{30}(p^{2})B^{\mu}$
$\displaystyle\qquad\qquad\;-W_{\mu}^{+}\Pi_{WW}(p^{2})W^{-\mu}.$ (11)

$\Pi_{ij}(p^{2})$ ($ij=33,\;00,\;30,\;WW$) in Eq. (11) can be expanded at
$p^{2}=0$ because the NP is generally considered to be very heavy, and
$\Pi_{ij}(p^{2})$ can be written as

$\displaystyle\Pi_{ij}(p^{2})=\Pi_{ij}(0)+p^{2}\Pi^{\prime}_{ij}(0)+\cdot\cdot\cdot,$
(12)

where the higher order terms can be neglected safely. The oblique parameters
$S,\;T,\;U$ are defined as Cacciapaglia:2006pk

$\displaystyle
S=4s_{W}c_{W}\Pi^{\prime}_{30}(0),\;T=\frac{\Pi_{33}(0)-\Pi_{WW}(0)}{M_{W}^{2}},\;U=4s_{W}^{2}[\Pi^{\prime}_{33}(0)-\Pi^{\prime}_{WW}(0)],$
(13)

where $s_{W}\equiv\sin\theta_{W},\;c_{W}\equiv\cos\theta_{W}$, and
$\theta_{W}$ is the Weinberg angle.

Generally, the $\Pi$ matrix of neutral vector bosons can be written in the
basis $(B_{\mu},\;W^{3}_{\mu},\;Z^{\prime}_{\mu})$ as

$\displaystyle\left(\begin{array}[]{ccc}p^{2}-M_{W}^{2}t^{2}&M_{W}^{2}t&-M_{W}^{2}tt^{\prime}\\\
M_{W}^{2}t&p^{2}-M_{W}^{2}&M_{W}^{2}t^{\prime}\\\
-M_{W}^{2}tt^{\prime}&M_{W}^{2}t^{\prime}&p^{2}-M_{Z^{\prime}}^{2}\end{array}\right),$
(17)

where $t=g_{Y}/g_{2}$, $t^{\prime}=(g_{YX}+2X_{H}g_{X})/g_{2}$. The gauge
couplings involving charged leptons can be written as

$\displaystyle\mathcal{L}=-i[(-\frac{1}{2}g_{2}W^{3}_{\mu}+Y_{L}g_{Y}B_{\mu}+Y_{L}g_{YX}Z^{\prime}_{\mu}+X_{L}g_{B}Z^{\prime}_{\mu})\bar{e}_{L}e_{L}$
$\displaystyle\qquad+(g_{Y}Y_{E}B_{\mu}+g_{B}X_{E}Z^{\prime}_{\mu}+g_{YX}Y_{E}Z^{\prime}_{\mu})\bar{e}_{R}e_{R}]$
(18)

$Y_{L}$, $Y_{E}$ are the hypercharges of the left-handed and right-handed
components of leptons respectively, they can be normalized as
$Y_{L}=-\frac{1}{2}$, $Y_{E}=-1$. And $X_{L},\;X_{E},\;X_{H}$ are the
$U(1)_{X}$ charges of left-handed components of leptons, right-handed
components of leptons and scalar doublets respectively. Since the most strict
constraints come from the precision measurements performed at $e^{+}e^{-}$
colliders (such as LEP1, LEP2 and etc), we choose to eliminate the neutral
currents involving charged leptons in Eq. (18), which can be done by
redefining the vector fields

$\displaystyle
B_{\mu}=\tilde{B}_{\mu}-c_{Y}Z^{\prime}_{\mu},\;\;W_{\mu}^{3}=\tilde{W}_{\mu}^{3}-c_{W}Z^{\prime}_{\mu},$
$\displaystyle
Z^{\prime}_{\mu}=\tilde{Z}^{\prime}_{\mu}+\frac{1}{p^{2}-M_{Z^{\prime}}^{2}}\\{[(p^{2}-M_{W}^{2}t^{2})A_{Y}+M_{W}^{2}tA_{W}+M_{W}^{2}tt^{\prime}]\tilde{B}_{\mu}+[M_{W}^{2}tA_{Y}$
$\displaystyle\qquad\;\;+(p^{2}-M_{W}^{2})A_{W}-M_{W}^{2}t^{\prime}]\tilde{W}^{3}_{\mu}\\},$
$\displaystyle
A_{Y}=\frac{g_{X}X_{E}+g_{YX}Y_{E}}{g_{1}Y_{E}},\;\;A_{W}=\frac{2}{g_{2}}(Y_{L}g_{Y}c_{Y}-X_{L}g_{B}-Y_{L}g_{YX})$
(19)

Then the oblique parameters $S,\;T,\;U$ can be obtained from Eq. (13) as

$\displaystyle
S=\frac{8s_{W}^{2}M_{W}^{2}g_{X}}{M_{Z^{\prime}}^{2}}[\frac{1}{g_{Y}^{2}}(g_{X}X_{E}-g_{YX})+\frac{g_{X}}{g_{2}^{2}}(X_{E}-2X_{L})](X_{E}-X_{L}+X_{H}),$
(20) $\displaystyle
T=\frac{4g_{X}^{2}M_{W}^{2}}{g_{2}^{2}M_{Z^{\prime}}^{2}}(X_{E}-X_{L}+X_{H})^{2},$
(21) $\displaystyle
U=\frac{-8g_{X}^{2}M_{W}^{2}}{g_{2}^{2}M_{Z^{\prime}}^{2}}(X_{E}-2X_{L})(X_{E}-X_{L}+X_{H}).$
(22)

As can be seen in Eqs. (20-22) that $S=T=U=0$ when $X_{E}-X_{L}+X_{H}=0$, it
indicates the leptonic Yukawa couplings are invariant under the $U(1)_{X}$
symmetry in this case which avoids the need of a model to generate the lepton
masses. And for nonzero $X_{E}-X_{L}+X_{H}=0$, other mechanisms are needed to
generate the lepton masses, such as the $U(1)_{H}$ model Mira:1999fx  and etc.
In addition, we neglect the $Z$ couplings to neutrinos or quarks in our
calculations approximately, because they are measured much less accurately
than $Z$ couplings to charged leptons, and all effects involving charged
leptons are included in the approximation. The W boson mass with local
$U(1)_{X}$ gauge group can be written as Peskin:1990zt ; Peskin:1991sw

$\displaystyle M_{W}^{2}=(M_{W}^{\rm
SM})^{2}+\frac{c_{W}^{2}M_{Z}^{2}}{c_{W}^{2}-s_{W}^{2}}\Big{(}-\frac{1}{2}S+c_{W}^{2}T+\frac{c_{W}^{2}-s_{W}^{2}}{4s_{W}^{2}}U\Big{)}.$
(23)

Eqs. (20-22) indicate that the $U(1)_{X}$ charges of leptons, denoted as
$X_{E}$ and $X_{L}$, can affect the theoretical predictions for the $W$ boson
mass. Consequently, we will consider $X_{E}$ and $X_{L}$ as free parameters to
explore their effects in the following analysis. Typically, $X_{E}$ and
$X_{L}$ are subject to constraints enforced by chiral anomaly cancellations.
For NP models with an extra $U(1)_{X}$ local gauge group, there are four
distinctive chiral anomaly cancellation patterns for per generation
Nagao:2022dgl

$\displaystyle\sum_{f}X_{f}=\sum_{f}(X_{f})^{3}=\sum_{f}(X_{f})^{2}Y_{f}=\sum_{f}X_{f}(Y_{f})^{2}=0.$
(24)

Here, $f$ represents various fermion species, encompassing the left-handed
lepton doublet $\hat{L}$, the left-handed quark doublet $\hat{Q}$, the right-
handed lepton singlets $\hat{E}$ and $\hat{N}$ (heavy neutrinos), and the
right-handed quark singlets $\hat{u}$ and $\hat{d}$. It is evident from Eq.
(24) that the phenomena of chiral anomaly cancellations can be realized by
assigning approximate values to $X_{N}$, $X_{Q}$, $X_{u}$, and $X_{d}$ while
considering $X_{E}$ and $X_{L}$ as unconstrained. We focus on illustrating the
effects of extra $U(1)_{X}$ local gauge group on the $W$ boson mass, hence we
do not explore the chiral anomaly cancellations detailedly for different
$X_{E}$, $X_{L}$ in specific NP models.

##  III Numerical analyses

Based on the calculations in Sec. II, the numerical results are computed and
presented in this section. As input parameters PDG , $M_{W}^{\rm SM}$ is taken
as $80.354{\rm GeV}$, the $Z$ boson mass is $M_{Z}=91.1876\;{\rm GeV}$, the
fine-structure constant $\alpha_{em}(m_{Z})=1/128.9$, and the fermion coupling
constant $G_{F}=1.1664\times 10^{-5}{\rm GeV}^{-2}$. The experimental
constraint on $Z^{\prime}$ boson mass reads $M_{Z^{\prime}}>4.05\;{\rm TeV}$
at $95\%$ confidence level (CL) ATLAS:2016cyf .

Figure 1: Taking $X_{E}=-1/2$, $X_{L}=-1$, the results of $M_{W}$ versus
$g_{YX}$ are plotted for $X_{E}-X_{L}+X_{H}=1/2$ (a), $1$ (b), $3/2$ (c),
where the gray areas denote the ATLAS $2\sigma$ interval, the green areas
denote the CDF II $2\sigma$ interval, the black, red, blue lines denote the
results for $M_{Z^{\prime}}=4.2,\;7,\;10\;{\rm TeV}$ respectively, and the
solid, dashed, dotted lines denote the results for $g_{X}=0.1.\;0.4,\;0.7$
respectively.

The kinetic mixing effect appears in any NP models with two Abelian groups, we
focus on the effects of $g_{YX}$ on $M_{W}$ with eliminating the neutral
currents involving charged leptons firstly. Taking $X_{E}=-1/2$, $X_{L}=-1$,
the results of $M_{W}$ versus $g_{YX}$ are plotted for
$X_{E}-X_{L}+X_{H}=1/2,\;1,\;3/2$ in Fig. 1 (a), (b), (c) respectively, where
the gray areas denote the ATLAS $2\sigma$ interval, the green areas denote the
CDF II $2\sigma$ interval, the black, red, blue lines denote the results for
$M_{Z^{\prime}}=4.2,\;7,\;10\;{\rm TeV}$ respectively, and the solid, dashed,
dotted lines denote the results for $g_{X}=0.1.\;0.4,\;0.7$ respectively. The
picture shows that large $g_{X}$, $g_{YX}$ and $X_{E}-X_{L}+X_{H}$ combined
with small $M_{Z^{\prime}}$ can well explain the measured $M_{W}$ at CDF II,
while the one measured at ATLAS prefers small $g_{X}$, $g_{YX}$,
$X_{E}-X_{L}+X_{H}$ and large $M_{Z^{\prime}}$. In addition, a positive
$g_{YX}$ can increase the theoretical prediction of $M_{W}$, while a negative
$g_{YX}$ suppresses the contributions from $U(1)_{X}$ gauge group to $M_{W}$.
The effects of $g_{YX}$ are affected by the values of $M_{Z^{\prime}}$,
$g_{X}$, $X_{E}-X_{L}+X_{H}$ obviously, where $g_{YX}$ affects $M_{W}$
drastically when $M_{Z^{\prime}}$ is small and $g_{X}$, $X_{E}-X_{L}+X_{H}$
are large. The fact is well to be understood because the effects of
introducing $U(1)_{X}$ local gauge group are highly suppressed by large
$M_{Z^{\prime}}$ or very small $g_{X}$, and the leptonic Yukawa couplings
approach to invariant under the $U(1)_{X}$ symmetry as $X_{E}-X_{L}+X_{H}$
approaches to $0$. Since the kinetic mixing constant $g_{YX}$ may affects the
theoretical predictions on $M_{W}$ significantly, the constraints on
$M_{Z^{\prime}}/g_{X}$ from high-precision W boson mass in the NP models with
extra local $U(1)$ gauge group would be relaxed partly by considering the
contributions from $g_{YX}$.

Figure 2: Taking $g_{B}=0.6$, $M_{Z^{\prime}}=5\;{\rm TeV}$, the allowed
results of $X_{E},\;X_{L}$ for $X_{E}-X_{L}+X_{H}=1/2$ (a1), $1$ (b1), $3/2$
(c1) are plotted, where the blue (red) points are obtained by considering the
W boson mass measured at CDF II (ATLAS) in $2\sigma$ interval. Similarly, the
results obtained for $g_{YX}$ scanning in the range $g_{YX}=(-0.4,0.4)$ are
plotted in Fig. 2 (a2), (b2), (c2).

In order to see the effects of $X_{E},\;X_{L},\;X_{H}$ on the W boson masses,
we take $g_{B}=0.6$, $M_{Z^{\prime}}=5\;{\rm TeV}$ and scan the following
parameter space

$\displaystyle
g_{YX}=(-0.8,0.8),\;X_{E}=(-\frac{3}{2},\frac{3}{2}),\;X_{L}=(-\frac{3}{2},\frac{3}{2}),\;X_{H}=(-\frac{3}{2},\frac{3}{2}).$
(25)

The allowed results of $X_{E},\;X_{L}$ for $X_{E}-X_{L}+X_{H}=1/2,\;1,\;3/2$
are plotted in Fig. 2 (a1), (b1), (c1) respectively, where the blue points are
obtained by considering the W boson mass measured at CDF II in $2\sigma$
interval, the red points are obtained by considering the W boson mass measured
at ATLAS in $2\sigma$ interval. To illustrate the effects of kinetic mixing on
the allowed results of $X_{E},\;X_{L}$, the results for scanning $g_{YX}$ in
the range $g_{YX}=(-0.4,0.4)$ are plotted in Fig. 2 (a2), (b2), (c2)
similarly. As can be seen by comparing Fig. 2 (a1), (b1), (c1) with Fig. 2
(a2), (b2), (c2), the allowed ranges of $X_{E},\;X_{L}$ for
$g_{YX}=(-0.4,0.4)$ are narrower than the ones for $g_{YX}=(-0.8,0.8)$,
because in the chosen parameter space $g_{YX}$ affects the numerical results
obviously as concluded above. For NP models with extra $U(1)$ local gauge
group, the ranges of $X_{E}$, $X_{L}$ would be limited by the W boson mass
measured at CDF II more strictly for smaller $X_{E}-X_{L}+X_{H}$, while the
ranges of $X_{E}$, $X_{L}$ would be limited by the one measured at ATLAS more
strictly for larger $X_{E}-X_{L}+X_{H}$. And the W boson mass measured at CDF
II or ATLAS can be satisfied for appropriate values of $X_{E}$, $X_{L}$,
$X_{H}$, $g_{YX}$, $g_{X}$, $M_{Z^{\prime}}$.

##  IV Summary

The contributions to the oblique parameters $S$, $T$, $U$ and W boson mass are
calculated in the new physics (NP) models with extra $U(1)$ local gauge group.
In such NP models, the new introduced $Z^{\prime}$ gauge boson and leptonic
gauge couplings can make significant contributions to $S$, $T$, $U$ and W
boson mass at the tree level if we choose to eliminate the neutral currents
involving charged leptons. Hence, the precise measurements of W boson mass are
related closely to such NP models. Considering the kinetic mixing effects
(which arises in any NP models with two Abelian groups) in the calculations,
the analytical results are presented in this work, and the presented results
can be applied to all such NP models. Based on the numerical results analyzed
above, the kinetic mixing effects can affect the contributions to W boson mass
significantly, and the W boson mass measured at CDF II or ATLAS can be
satisfied for proper values of $X_{E}$, $X_{L}$, $X_{H}$, $g_{YX}$, $g_{X}$,
$M_{Z^{\prime}}$. Importantly, if the leptonic Yukawa couplings are invariant
under the extra $U(1)$ local gauge group, the contributions to $S$, $T$, $U$
and W boson mass in such NP models can be eliminated by redefining the gauge
boson fields through eliminating the neutral currents involving charged
leptons, even with nonzero kinetic mixing effects.

###### Acknowledgements.

The work has been supported by the National Natural Science Foundation of
China (NNSFC) with Grants No. 12075074, No. 12235008, No. 11535002, No.
11705045, Hebei Natural Science Foundation for Distinguished Young Scholars
with Grant No. A2022201017, Natural Science Foundation of Guangxi Autonomous
Region with Grant No. 2022GXNSFDA035068, and the youth top-notch talent
support program of the Hebei Province.

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Application of Lorentzian CFT Principal Series Representation to Near Forward
Scattering

Pulkit Agarwal1, Richard C. Brower2, Timothy G. Raben3 and Chung-I Tan4

1 National University of Singapore, Singapore

2 Boston University, Boston, MA 02215

3 Michigan State University

4 Brown University, Providence, RI 02912

50th International Symposium on Multiparticle Dynamics
(ISMD2021)
12-16 July 2021 10.21468/SciPostPhysProc.?

## Abstract

We present a discussion on recent progress in high energy diffraction from the
perspective of AdS/CFT, through which a unified treatment for both
perturbative and non-perturbative Pomeron emerges. By working with Unitary
Irreducible Representation of Conformal group, a frame is provided in
extending AdS/CFT to both forward and near-forward scattering. We present an
analysis involving an exact solution to conformal blocks in Minkowski CFT and
discuss possible applications. Phenomenological applications can range from
forward scattering to DIS/DVCS/TMD at LHC energies and beyond.

##  1 Introduction

Study of Conformal Field Theories (CFTs) has recently gained traction within
various domains of physics. Quantum Chromodynamics (QCD) has numerous limits
where the dynamics appear near conformal [1]. The use of conformal methods for
QCD are numerous111We highlight here a very brief selection of a vast
literature. and prove especially useful in the high energy limit away from any
relevant mass scale (e.g. [2]),and in the strong coupling limit using
holography (e.g. [3, 4]). The latter example is of particular interest where a
unified description of the perturbative and non-perturbative Pomeron emerges
[5]. The canonical AdS/CFT approach is formulated using Euclidean CFTs. In
order to make claims about systems like particle scattering, the results would
have to be analytically continued to Lorentzian space—and many nontrivial
connections between Euclidean and Minkowski CFTs have been found [6, 7]. A
direct treatment of Minkowski CFTs was done in [8], which emphasized the
importance of Regge behavior for the associated conformal blocks. It deals
with the study of scattering within the CFT framework where it was shown that
an invariant function over $SO(d,2)$ can be represented via the unitary
principal series as

$F(g)=\sum_{m,m^{\prime}}\int_{-i\infty}^{i\infty}\frac{d\widetilde{\ell}}{2\pi
i}\int_{-i\infty}^{i\infty}\frac{d\widetilde{\Delta}}{2\pi
i}\,\,a_{m,m^{\prime}}(\widetilde{\ell},\widetilde{\Delta})\,{\mathcal{G}}_{(\widetilde{\ell},\widetilde{\Delta};m,m^{\prime})}(g).$
(1)

The representation functions (or group harmonics)
$\mathcal{G}_{\Delta,\ell}(g)$ can be associated with the defined Minkowski
conformal blocks.

This shift from a Euclidean metric to a Minkowski one can be thought of as
moving to a different real form of the same complex extension:
$SO(d+1,1)\rightarrow SO(d,2)$. The aim is to provide an unitary, irreducible
representation for $SO(d,2)$ using a convenient induced representation. Under
Iwasawa decomposition, $G=KAN$, [9], there is a rank 2 Maximum Abelian
Subgroup $A=SO(1,1)\times SO(1,1)$ of $SO(d,2)$ which carries the physics of
the two independent non-compact generators in the group. These generators are
chosen to be a dilatation and a longitudinal Lorentz boost. For CFT, the group
harmonics $\mathcal{G}_{\Delta,\ell}(g)$, which are purely kinematical, are
found through standard procedure via the unitarized Casimir equation using
appropriate Regge boundary conditions and are a function of the two non-
compact parameters (which will be related to the two familiar conformal cross
ratios). These group harmonics are generalized spherical functions that are
obtained after integrating over the maximal compact subgroup $K=SO(d)\times
SO(2)$, thus reducing Eq. (1) to the limit of $m=m^{\prime}=0$ only [10].

With applications to diffractive physics as our goal, we will start by setting
up the relevant kinematics. We will discuss how the two conformal cross ratios
can be combined to make connection with our group theoretic interpretation.
After "diagonalizing" the Casimir operator, we will state our solutions of the
group harmonics for the special cases of $d=2,4$, which will be most relevant
for phenomenological applications to diffractive physics. We will conclude
with a brief discussion of these applications.

##  2 Kinematics

In order to discuss scattering in CFT, the process we consider is that of off-
shell scattering of four-currents222For illustrative purpose, we shall treat
external states as conformal scalars.

$\gamma^{\star}(1)+\gamma^{\star}(3)\rightarrow\gamma^{\star}(2)+\gamma^{\star}(4).$
(2)

Scattering amplitudes for such a process can be written in terms of a partial
wave decomposition where the dynamics are encoded in the partial wave
amplitudes $a(\ell,\Delta)$. These amplitudes can in principle be found
exactly in a theory with a Lagrangian. In a theory without a Lagrangian, like
our Regge treatment, a fundamental physical assumption is that of meromorphy.
The success of a CFT treatment of QCD lies in an agreement between these two
approaches.

In this set-up, we can choose a causal structure that makes studying the
$t$-channel OPE most natural as follows. For the $s$-channel scattering
process Eq. 2, points $(1,4)$ and $(2,3)$ are timelike separated333In
contrast, all distances in a Euclidean setting are necessarily spacelike
separated. $(x^{2}_{14},x^{2}_{23}<0)$ whereas all other distances are
spacelike (eg. $x_{13}^{2},x_{24}^{2}>0$). Therefore, we designate our
$2\rightarrow 2$ scattering $s$-channel process as $1+3\rightarrow 2+4$. The
corresponding process $1+\bar{2}\rightarrow\bar{3}+4$ is the $t$-channel and
$1+\bar{4}\rightarrow 2+\bar{3}$ is $u$-channel. In terms of cross ratios:
$u=({x_{12}^{2}}{x_{34}^{2}})/({x_{13}^{2}}{x_{24}^{2}})$ and
$v=({x_{23}^{2}}{x_{14}^{2}})/({x_{13}^{2}}{x_{24}^{2}})$ we are restricted to
the first quadrant $(u,v>0)$. Since this is also the case for Euclidean CFT,
it is useful to consider instead the $(\sqrt{u},\sqrt{v})$444The $(u,v)$ plane
has a double sheet structure that the $(\sqrt{u},\sqrt{v})$ plane resolves.
plane such that
$\sqrt{u}=(\sqrt{x_{12}^{2}}\sqrt{x_{34}^{2}})/(\sqrt{x_{13}^{2}}\sqrt{x_{24}^{2}})$
and
$\sqrt{v}=(\sqrt{x_{23}^{2}}\sqrt{x_{14}^{2}})/(\sqrt{x_{13}^{2}}\sqrt{x_{24}^{2}})$.

Notice that we now cover a larger part of the $(\sqrt{u},\sqrt{v})$ plane
since, for the $s$-channel say, $\sqrt{v}<0$555We have chosen here a ”positive
root” convention meaning that $\sqrt{x^{2}_{ij}}>0$. Similar procedure can be
followed for the $u$-channel to find $\sqrt{u},\sqrt{v}<0$.. We can define a
set of crossing symmetric variables $w,\sigma$ by requiring that
$w\sigma=\frac{1-v}{u}$. For our purposes we choose666A similar choice was
made in the Euclidean lattice treatment of [11].:
$w\equiv\frac{1-\sqrt{v}}{\sqrt{u}}$ and
$\sigma\equiv\frac{1+\sqrt{v}}{\sqrt{u}}$. In order to see why this set is
interesting, we can pick a particular parameterization of our coordinate space
such that the cross ratios can be expressed in the form:

$\sqrt{u}=\frac{2}{\cosh y+\sigma},\quad\sqrt{v}=-\frac{\cosh y-\sigma}{\cosh
y+\sigma}$ (3)

where $\sigma=(r_{1}^{2}+r_{3}^{2}+b^{2})/(2r_{1}r_{3})=\cosh\xi+b^{2}$. Here,
$(y,\xi)$ parametrize our Lorentz boost ($y$ is essentially rapidity) and
dilatation respectively777Notice that this is a parametrization of our MASG
discussed before. See [8] for further discussion.. The cross ratios can be
rewritten in terms of $w\equiv\cosh y$, and
$\sigma\equiv\cosh\zeta=\cosh\xi+b^{2}$. In this frame, causal relation
$x_{23}^{2}=x_{14}^{2}<0$ leads to $\sqrt{v}<0$, which requires that
$1<\cosh\zeta<\cosh y$. For s-channel scattering, the physical region is given
by the constraint $1<\sigma<w<\infty\ $.

Figure 1: Moving from $u,v$ plane to $\sqrt{u},\sqrt{v}$ plane, on to
$w,\sigma$ plane for crossing symmetry. Here, the "Euclidean region" is where
the $t$-channel OPE converges as $|w|<1$. This corresponds to $y\rightarrow
iy$ in Eq. 3, which is essentially a Wick rotation.

##  3 Unitary Conformal Blocks

In order to construct our scattering amplitudes, what we are after is a
representation of our symmetry group $SO(d,2)$. This can be constructed via a
direct group theoretic approach: by starting with a $d+2$ dimensional
embedding space and restricting to a null sub-manifold, we can project down to
our $d$ dimensional spacetime of interest. A representation can then be
induced by the MASG $A$ [9]. In this work, we will focus on an equivalent
approach that doesn’t quite require the full machinery of group theory by
realizing that our partial waves are solutions to the Casimir equation888A
group theoretic treatment involving integrating over the maximal compact
subgroup $K$ to obtain these spherical functions will be presented in [10]..

###  3.1 Unitarized Casimir Equation and Conformal Blocks

In our Minkowski setting, we are interested in the space corresponding to
unitary irreducible representation where the dilatation $D$ and Lorentz boost
$L_{zt}$ are diagonal, with eigenvalues $\widetilde{\Delta}$ and
$\widetilde{\ell}$ respectively. For the invariant conformal scalar function
of interest to us, the Casimir can be expressed as a differential operator in
terms of $(u,v)$999We have multiplied by a factor of 2 from that defined
conventionally, e.g., [12]. We have also adopted the sign convention suitable
for our Minkowski treatment.

$\mathcal{C}=-2[(1-u-v)\partial_{v}(v\partial_{v})+u\partial_{u}(u\partial_{u}-d)-(1+u-v)(u\partial_{u}+v\partial_{v})(u\partial_{u}+v\partial_{v})].$
(4)

It can be shown that the Casimir acting on the representation space of
$SO(d,2)$, labelled by $(\widetilde{\ell},\widetilde{\Delta})$, can be brought
into an explicitly Hermitian form via a Jacobian, taking on the form

$\widetilde{\cal C}=-\partial_{y}^{2}-\partial_{\zeta}^{2}+V(y,\zeta).$ (5)

The Jacobian can be written out explicitly and is chosen to have a simple
large $t=e^{y}$ limit, $J\sim t^{-d/2}\sigma^{-(d-2)/2}$. With this choice,
the potential also takes on a relatively simple form as a function of
$(w,\sigma)$ or $(y,\zeta)$ and contains a constant term $V_{0}$. After
resetting the "ground state energy" to be zero, the effective potential $V$
vanishes in the limit $w\rightarrow\infty$, with $\sigma$ fixed. It follows
that the spectrum is strictly positive and continuous. This brings us to one
of our main (expected) results:

$\widetilde{\ell},\widetilde{\Delta}\rightarrow\text{imaginary}.$ (6)

Solutions to $(\widetilde{\cal C}-V_{0})\Psi(y,\zeta)=\lambda\Psi(y,\zeta)$
are asymptotically plane-wave, with eigenvalues identified with eigenvalues of
boost $L_{zt}$ and dilatation $D$, ($k_{\xi}$ to $k_{\zeta}$),
$\lambda=k_{y}^{2}+k_{\zeta}^{2}\equiv-\widetilde{\ell}^{2}-\widetilde{\Delta}^{2}$.
It is instructive to compare the eigenvalue for the Casimir, $\lambda+V_{0}$,
to that obtained under the Euclidean treatment,
$C(\ell,\Delta)=-[\Delta(\Delta-d)+\ell(\ell+d-2)]$. By continuing both $\ell$
and $\Delta$ complex, one finds they can be identified with the corresponding
Minkowski values

$0<k_{y}^{2}=-\widetilde{\ell}^{2}=-(\ell+(d-2)/2)^{2},\quad{\rm and}\quad
0<k_{\zeta}^{2}=-\widetilde{\Delta}^{2}=-(\Delta-d/2)^{2}\ .$ (7)

That is, the analytically continued $|\widetilde{\ell}|$ goes into eigenvalues
for the Lorentz boost, $L_{zt}$, and $|\widetilde{\Delta}|$ into eigenvalues
for the dilatation, $D$, in the Minkowski setting, with both
$\widetilde{\ell}$ and $\widetilde{\Delta}$ purely imaginary. This is indeed
our expectation101010These unitarized eigenvalues ($\widetilde{\Delta}$,
$\widetilde{\ell}$) are related to the root description of our symmetry group
$SO(d,2)$ [10]. Our Jacobian is due to requiring a Unitary Irreducible
Representation via the Casimir operator. It is related to the group measure
necessary for defining the inner product [13, 9]..

A notable feature of this very simple "potential" scattering form of the
equation is that it factorizes for the special cases of $d=2,4$. There are two
ways to write the solution to this equation: as an infinite series of plane
wave products or as a product of two hypergeometric functions. We can write
the leading behavior as:

$G^{(M)}_{(\widetilde{\ell},\widetilde{\Delta})}(y,\zeta)=J(y,\zeta){\cal
G}_{(\widetilde{\ell},\widetilde{\Delta})}(y,\zeta)\sim
w^{-d/2}\sigma^{-(d-2)/2}(w^{\widetilde{\ell}}\sigma^{-\widetilde{\Delta}})=w^{\ell-1}\sigma^{-\Delta+1}$
(8)

where the right hand side is the Regge limit with the expected form [8].

###  3.2 Conformal Dynamics and Meromorphy in $\ell$ and $\Delta$:

It should be stressed that much of our current study has focussed on the
kinematics of unitary representations for $SO(d,2)$. Dynamics of CFT reside in
the partial-wave amplitudes
$a^{(\alpha)}(\widetilde{\ell},\widetilde{\Delta})$. In particular, CFT
dynamics assume that the singularities consist of poles. Regge asymptotics
further requires that these singularities lie on surfaces in the
$\widetilde{\ell}-\widetilde{\Delta}$ plane, i.e., each surface can be
labelled by
$\widetilde{\Delta}=\widetilde{\Delta}_{\alpha}(\widetilde{\ell})$,
$\alpha=1,2,\cdots$, each defining a spectral curve. As discussed in [5], the
most important singularity is that interpolating the CFT primary associated
with the stress-energy tensor, i.e., for $d=4$, $\ell=2$,
($\widetilde{\ell}=1$), a pole at $\Delta=4$, ($\widetilde{\Delta}=2$). This
singularity, at $d=4$, is historically referred to as the Pomeorn. The
associated spectral curve can be found in strong coupling via AdS/CFT, leading
to a simple parabolic curve.

In order to extract the contribution of the Pomeron at high energies, it is
necessary to extend analytically the representation into the
$\widetilde{\ell}-\widetilde{\Delta}$ plane. This in turn requires one to
continue the conformal harmonics analytically. The continuation is analogous
to the well-known relation between conventional Legendre functions
$\frac{P_{\ell}(z)}{\tan\ell\pi}=Q_{\ell}(z)-Q_{-\ell-1}(z)$. This as well as
other related issues will be examined in [10].

##  4 Applications to Diffractive Physics

The CFT methods described here are most relevant for scattering where the
causal structure is important. However, in order to apply these results to QCD
studies the conformal symmetry must be broken. This is usually done by working
at finite temperature or via a confinement deformation of the dual AdS space:
a cutoff is introduced such that the dual space is only asymptotically AdS but
has a cutoff in the bulk (at a scale $\sim\Lambda_{\text{QCD}}$). Various
thermal, hard wall, and soft wall models for these cutoffs have been developed
to great success.

Our discussion on CFT correlators provides a unified analytic framework that
can work directly with both a Minkowski and a Euclidean metric. The causal
structure set up in Fig. 1 is appropriate for discussing near forward
scattering for $t<0$ and diffractive physics, e.g., examining deep inelastic
scattering and inclusive production (e.g. [14]). In particular, it will be
profitable to apply this framework to the study of deeply virtual Compton
scattering (DVCS) (e.g. [15, 16]).

##  5 Conclusion

A study of scattering in CFT can provide significant conclusions to be drawn
for QCD processes due to its near conformal nature. In a direct group
theoretic approach, we expand the $2\rightarrow 2$ scattering amplitudes in a
partial wave expansion. These partial waves are unitary irreducible
representations of the Lorentzian conformal group $SO(d,2)$ induced by the
maximal abelian subgroup (MASG) $A=SO(1,1)\times SO(1,1)$. As is typical in a
Regge treatment, we are interested in the $t$-channel OPE of the scattering
process $\gamma^{*}+\gamma^{*}\rightarrow\gamma^{*}+\gamma^{*}$ which in turn
lends a natural causal structure to our 4-point functions. With an appropriate
gauge choice, we are able to relate the two conformal cross ratios to the 2
parameters of our MASG. By restricting our partial waves to the space of
generalized spherical functions and defining an appropriate inner product, we
extract the precise leading Regge behavior.

#### Funding information

P.A. is supported by the National University of Singapore Research
Scholarship. This work was supported in part by the U.S. Department of Energy
(DOE) under Award No. DE-SC0015845 for R.C.B. T.G.R is funded through an
internal research fund at Michigan State University.

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