Krishnakumari, Cats, and van Lint

# Day-to-day and seasonal regularity of network passenger delay for metro
networks

Panchamy Krishnakumari
Department of Transportation and Planning
Delft University of Technology
P.O. Box 5048  2600 GA Delft  The Netherlands
Phone number: +31 15 27 85279
Email<EMAIL_ADDRESS>
Oded Cats
Department of Transportation and Planning
Delft University of Technology
P.O. Box 5048  2600 GA Delft  The Netherlands
Phone number: +31 15 27 85279
Email<EMAIL_ADDRESS>
Hans van Lint
Department of Transportation and Planning
Delft University of Technology
P.O. Box 5048  2600 GA Delft  The Netherlands
Phone number: +31 15 27 85279
Email<EMAIL_ADDRESS>
##  1 Abstract

In an effort to improve user satisfaction and transit image, transit service
providers worldwide offer delay compensations. Smart card data enables the
estimation of passenger delays throughout the network and aid in monitoring
service performance. Notwithstanding, in order to prioritize measures for
improving service reliability and hence reducing passenger delays, it is
paramount to identify the system components - stations and track segments -
where most passenger delay occurs. To this end, we propose a novel method for
estimating network passenger delay from individual trajectories. We decompose
the delay along a passenger trajectory into its corresponding track segment
delay, initial waiting time and transfer delay. We distinguish between two
different types of passenger delay in relation to the public transit network:
average passenger delay and total passenger delay. We employ temporal
clustering on these two quantities to reveal daily and seasonal regularity in
delay patterns of the transit network. The estimation and clustering methods
are demonstrated on one year of data from Washington metro network. The data
consists of schedule information and smart card data which includes passenger-
train assignment of the metro network for the months of August 2017 to August
2018. Our findings show that the average passenger delay is relatively stable
throughout the day. The temporal clustering reveals pronounced and recurrent
and thus predictable daily and weekly patterns with distinct characteristics
for certain months.

Keywords: passenger delay, estimation, seasonal patterns, daily patterns,
passenger-train assignment, smart card data

##  2 Introduction

Service reliability is known to be one of the most important determinants of
performance, ridership and user satisfaction. In an effort to improve user
satisfaction and transit image, several transit authorities have introduced
policies for offering delay compensations. Such measures include refunds by
Dutch Railways (NS) in The Netherlands [1], by SL in Stockholm, Sweden [2] and
the so-called rush hour promise by WMATA in Washington DC, USA [3]. In order
to provide such services, collecting, monitoring and quantifying the state of
public transportation systems in terms of for example delays, occupancy,
regularity and punctuality is critically important. The data with which many
of these quantities can be directly or indirectly estimated are already
collected by many transport authorities through different sensors and
information sources. Some of the well-known and increasingly used data sources
for public transport network are infrastructure (track segments) and service
network (line) information, time table data (schedule information), automatic
vehicle location (AVL) data which contains real time locations of trains [4]
and by implication the realisation of the schedule, and automatic fare
collection (AFC) or smart card data with origin-destination specific
information of passengers [5].

AFC systems typically come in two forms: tap-in only or tap-in and tap-out.
Tap-in only fare systems registers the boarding station of the passenger only
whereas tap-in-tap-out systems collect both origin and destination locations
of each passenger. Notwithstanding, there are methods to infer the alighting
station of each individual using tap-in only AFC system [6, 7]. An increasing
body of literature is available on the use of smart card data in public
transit; a review of which can be found in [5, 8]. Most of these studies focus
on origin-destination matrix estimation [9, 10]; extracting passenger patterns
[11, 12, 13]; network performance analysis [14, 15] and passenger-train
assignment [16, 17].

Using the readily available timetable and AVL data, it is easy to estimate the
vehicle delay of the individual trains. However, for many applications it is
more interesting and fruitful to investigate the passenger delay for a given
line, network segment or transfer station. Clearly, with the increasing
availability of AFC data and progress in passenger-train assignment research,
it is possible to estimate and quantify this delay for each passenger. A
passenger-train assignment model is necessary here, since AFC systems do not
collect information about the actual trajectory of each passenger. Passenger
delay derived from AFC data is origin-destination specific for each passenger
and hence is valid for a given realised trajectory only. To attribute all
these individual person delays to delay statistics of individual public
transport network elements (routes, lines, nodes) over different periods of
time, we need to aggregate and average the individual passenger delays of all
passengers traversing through the network on each network element.

In this work, we study passenger delay as extract from fusing AVL and AFC data
as the service performance indicator and infer it per network element. This is
the key contribution of this paper: a new data-driven method to derive PT
network delays from individual trajectories. The methodology has similarities
with the data driven OD estimation method we propose in [18], in that we
construct a solvable system of equations utilising all the information at hand
without making more assumptions than strictly needed. Usage of passenger-train
assignment data make this study unique because this type of information is
relatively new. To the authors’ knowledge, this is the first study to explore
the potential application of this unique data set. In this work, we
distinguish two different types of passenger delay in relation to the public
transit network: average passenger delay and total passenger delay. We define
the average passenger delay as the delay incurred by a passenger while
traversing a track segment (link), station (node) or trajectory. The total
passenger delay is the total delay experienced by all the passengers that
traverse that link, node or route. Thus, the total passenger delay is a
function of the number of passengers that traverse that network element during
a given time period.

In the remainder of this paper we show that realised passenger-trajectories
(resulting from AFC data and passenger-train assignment) and schedule
information are sufficient to estimate average and total passenger delays for
all the different network elements. The estimation improves by adding an
additional constraint derived from AVL data, because these are—in our
case—readily available. With these constraints for each passenger and each
vehicle, a solvable system of equations can be formulated. There are various
applications for the resulting network indicators, such as modeling delay
propagation though the network; disruption detection, etc. In this work, we
use these estimated delay patterns to reveal day-to-day or seasonal regularity
by grouping the delay of different days. We demonstrate the framework for the
metro network of Washington DC with one year of data.

The paper is organized as follows: section 3 describes the overall estimation
framework and the clustering to reveal the day-to-day and seasonal regularity
of the delay patterns; in section 4 we outline the network and data used.
These results are presented in section 5. We offer conclusions and a
discussion on further research avenues in section 6.

##  3 Methodology

In this section we first define the notations used in this work. Thereafter,
we describe the network passenger delay estimation problem as a solvable set
of equations and finally, we elaborate the clustering technique used to reveal
the regularities in the delay patterns.

Let $G(S,E,L,V^{trans})$ denote a directed graph representing a public
transport network, in which $S$ is the set of stations; $E$ the set of track
segments between the stations; $L$ the set of ordered pairs of stations
representing a public transport service line; and $V^{trans}$ (subset of $S$)
the set of transfer stations where transfers between lines occur. Passenger
trajectory or route $r_{s_{o},s_{d},n}$ between origin station $s_{o}$ and
destination station $s_{d}$ of passenger $n\in<1,...N>$, where $N$ is the
total number of passengers, is defined using two sets, one of stops and one of
lines, each of which are a subset of $S$ and $L$ respectively. The combination
of $S$ and $L$ allows one to define a third set: the set of track segments
(subset of $E$) that is traversed by the passenger. As a result, we obtain the
initial stop (first element in the stop set); the intermediate transfer stops
(second to one before last element in the stop set) and the link set. The stop
set is defined as $<s_{1},..,s_{i},s_{i+1},..,s_{m}>$, where
$s_{2},..,s_{i},..s_{m-1}$ are the $m$ transfer stations for that route, in
which $s_{1}$ is the origin station $s_{o}$ and $s_{m}$ is the destination
station $s_{d}$. The line set for the same route is defined as
$<l_{1},..,l_{i},..l_{m-1}>$.

###  3.1 Network delay estimation

Assuming that the AFC system is a tap-in-tap-out system with the passenger-
train assignment provided; the origin and destination station, and the
inferred trajectory of each passenger are assumed known. Thus, the passenger
delay $d_{s_{o},s_{d},k,n}$ is defined as:

$d_{s_{o},s_{d},k,n}=\begin{cases}t_{s_{o},s_{d},k,n}-\widetilde{t}_{s_{o},s_{d},k}&if~{}t_{s_{o},s_{d},k,n}>\widetilde{t}_{s_{o},s_{d},k}\\\
0&otherwise\end{cases}$ (1)

where $t_{s_{o},s_{d},k,n}$ is the observed travel time for passenger $n$
departing at time (period) $k$ that can be obtained by finding the difference
between the tap-in and tap-out time of that passenger. Any negative delay
$d_{s_{o},s_{d},k,n}$ is assigned to $0$ as shown in eq.1 since this implies
the passenger reached the destination earlier than expected. The maximum
scheduled route time $\widetilde{t}_{s_{o},s_{d},k}$ can be defined in several
ways. In this work, the following definition has been adopted:

$\widetilde{t}_{s_{o},s_{d},k}=\sum_{l\in r_{s_{o},s_{d}}}\sum_{s_{i}\in
l}\widetilde{t}_{s_{i},s_{i+1},k}^{train}+\sum_{l\in
r_{s_{o},s_{d}}}{h_{l,k}}+\sum_{s_{i}\in
r_{s_{o},s_{d}}}\sum_{s_{i}=s_{1}}^{s_{m-1}}{t_{s_{i}}^{walk}}$ (2)

where $\widetilde{t}_{s_{o},s_{d},k}$ is composed of scheduled running times
and dwell times $\widetilde{t}_{s_{i},s_{i+1},k}^{train}$ assigned to track
segments connecting subsequent stations $s_{i}$ and $s_{i+1}$, headway
$h_{l,k}$ between successive train services for the lines $l$ in that route
and walking time $t_{s_{i}}^{walk}$ at the origin station and transfer
stations denoted by $s_{i}$. The headway between trains are included in
calculating the schedule travel time due to the definition of on-time journey
considered in this study. A passenger is considered to be on-time with regards
to the maximum scheduled route time even he/she just missed a train but was
able to catch the next train of that line and the headway between the trains
account for that fallback time.

Given the individual trajectories of the passengers, the aim is to decompose
the delay experienced by the passengers to the corresponding network elements
they traversed along their route. We assume that the observed travel time of a
passenger $t_{s_{o},s_{d},k,n}$ comprises of travel time that can be
attributed to one of the following network elements of the public transport
infrastructure:

  1. 1.

Time spent at the origin stop of the trip

  2. 2.

On-board a train along one of the trip segments

  3. 3.

Time spent at the transfer stations of the trip

Figure 1: Schematic picture of scheduled and observed travel time between
station $s_{2}$ and $s_{5}$.

Similarly, the delay experienced by a passenger is comprised of the delays
occurring at these network elements. The composition of the scheduled and
observed passenger travel time is illustrated in figure 1. Based on this
schematic representation, the passenger delay between stations $(s_{o},s_{d})$
for a given departure time $k$ can be defined as:

$d_{s_{o},s_{d},k,n}\geq d_{s_{o},k}^{wait}+\sum_{l\in
r_{s_{o},s_{d},n}}\sum_{s_{i}\in l}d_{s_{i},s_{i+1},k}^{on-
board}+\sum_{s_{i}\in
r_{s_{o},s_{d},n}}\sum_{s_{i}=s_{1}}^{s_{m-1}}d_{s_{i},k}^{trans}$ (3)

where $d_{s_{o},k}^{wait}$ is the initial waiting delay at the origin station
$s_{o}$, $d_{s_{i},s_{i+1},k}^{on-board}$ is the on-board delay between track
segment $(s_{i},s_{i+1})$ and $d_{s_{i},k}^{trans}$ is the transfer delay at
transfer stop $s_{i}$ in route $r_{s_{o},s_{d},n}$. The inequality in eq.3 is
due to potential non-observable personal travel components which may add
additional delays such as random passenger arrival time or performing an
activity at one of the stations within the gated area. Formulating the
relationship between personal delays and the three network related delay
components by means of an inequality constraint allows us to perform the delay
inference without ignoring unobserved personal delay components. With eq.3,
now each passenger trajectory can be written as a linear combination of three
passenger delay component types. Doing so for all passenger trajectories leads
to a potentially solvable system of equations (inequalities), since each
passenger’s trip between $k$ and $k+t_{s_{o},s_{d},k,n}$ serves as a
constraint for all other trips that traverse one or more common network
elements during this trip. The number of unknowns corresponds to the number of
initial stations, transfer stations and links (connections between successive
stations) in the public transport system.

There is, however, an additional constraint we can formulate, that —as it
turns out—indeed renders the resulting system of equations solvable. This
constraint pertains to the on-board delay component, which must be equal to
the train delay between the corresponding track segments. This train delay can
be directly inferred from the train movement data and the schedule
information. The result is an additional constraint to the system of equations
in eq.3 which reads

$d_{s_{o},s_{d},k}^{train}=\sum_{l\in r_{s_{o},s_{d}}}\sum_{s_{i}\in
l}d_{s_{i},s_{i+1},k}^{on-board},$ (4)

where $d_{s_{o},s_{d},k}^{train}$ is the train delay for a train servicing the
line that starts at $s_{o}$ and ends at $s_{d}$ within time $k$,
$r_{s_{o},s_{d}}$ is the route composed of track segments visited by the train
and the initial waiting time and transfer delay component is 0 as those are
not relevant for a train.

Eq.3 and eq.4 now constitute a solvable system of equations in which
$d_{s_{o},k}^{wait}~{}\forall~{}s_{o}\in S$, $d_{s_{i},s_{i+1},k}^{on-
board}~{}\forall~{}(s_{i},s_{i+1})\in E$ and
$d_{s_{i},k}^{trans}~{}\forall~{}s_{i}\in V^{trans}$ are the unknowns and
$d_{s_{o},s_{d},k,n}$ and $d_{s_{o},s_{d},k}^{train}$ are known. This system
of equations is solved using a constrained linear least square solution method
[19] with a lower bound set to 0 to ensure a non-negative solution.

###  3.2 Temporal clustering

Now that the passenger delay has been decoupled into corresponding delays at
the various network elements (segments, nodes), we can use these values to
compute network performance indicators that are indicative for the state of a
public transit network at any given time. We can define these states for
different time periods / days and use these representations for example to
explore if there is seasonal or daily regularity in the patterns.

To represent the network state (dynamics), we use average passenger delay and
total passenger delay. The average passenger delay equals the estimated delays
($d_{s_{o},k}^{wait}$, $d_{s_{i},s_{i+1},k}^{on-board}$ and
$d_{s_{i},k}^{trans}$) for each corresponding network element respectively,
whereas the total passenger delay equals the average passenger delay weighted
by the number of passengers that experienced that delay. Thus, the average
passenger delay of a given day composed of the three components is represented
using the following feature vector:

$FV^{average}=[d_{s_{o},k}^{wait},~{}d_{s_{i},s_{i+1},k}^{on-
board},~{}d_{s_{i},k}^{trans}]$ (5)

The total passenger delay is represented as follows:

$FV^{total}=[d_{s_{o},k}^{wait}\times
q_{s_{o},k}^{wait},~{}d_{s_{i},s_{i+1},k}^{on-board}\times
q_{s_{i},s_{i+1},k}^{on-board},~{}d_{s_{i},k}^{trans}\times
q_{s_{i},k}^{trans}]$ (6)

where $q_{s_{o},k}^{wait}$, $q_{s_{i},s_{i+1},k}^{on-board}$ and
$q_{s_{i},k}^{trans}$ are the number of passengers that experienced initial
waiting time, on-board and transfer delay respectively.

For studying the dynamics of the passenger delay, we use the vectorized form
of the feature vectors in eq.5 and eq.6. In order to avoid biasing the
clustering from small and large outlier delay values, we normalised the
feature vectors by clustering the delay values into uniform clusters such that
track segments with 0 to 5 mins delays belong to cluster 1 and those with
delays of 5 to 10 mins belong to cluster 2, if the aggregation is 5 mins. Thus
the feature vector is composed of cluster number instead of delay values but
each cluster has a corresponding known range of delay values.

The aim of the clustering is to study if regular patterns emerge for different
days. These insights can be used for developing demand-oriented operational
planning for different days or different months. In this work, we use
hierarchical agglomerative clustering for the temporal clustering [20]. This
is because of the power of such clustering in revealing the distribution of
the feature vectors in the form of a dendrogram [21] which can aid in
determining the optimal number of clusters as shown in [22]. The hierarchy or
distribution of the feature vectors is constructed based on a dissimilarity
metric between the feature vectors. The dissimilarity metric used in this work
is the city block distance.

For each cluster, we build a representative feature vector by taking the
centroid of the feature vectors belonging to that class. Thus, for each class
we have a stacked delay distribution plot of the three network element delay
of the centroid in order to make the analysis informative and comprehensible.
The distribution curve of centroid of each class $c$ for average passenger
delay is defined as:

$dist_{c,k}=\sum_{s_{o}\in S}d_{s_{o},k}^{wait}(c):\sum_{s_{i}\in
S}\sum_{s_{i+1}\in S}d_{s_{i},s_{i+1},k}^{on-board}(c):\sum_{s_{i}\in
S}d_{s_{i},k}^{trans}(c)$ (7)

where $d_{s_{o},k}^{wait}(c)$, $d_{s_{i},s_{i+1},k}^{on-board}(c)$ and
$d_{s_{i},k}^{trans}(c)$ represents the waiting time, on-board delay and
transfer delay of the centroid of class $c$. $dist_{c,k}$ represents the ratio
between all delay components in that cluster $c$ at time $k$. Similarly, a
stacked distribution curve for the total passenger delay is built.

##  4 Data

We demonstrate the estimation and clustering approach on a data set of smart
card data from WMATA for the region of Washington DC, USA. The data is
composed of one year of smart card data from 19 August 2017 to 28 August 2018
from the whole metro network of Washington DC which includes the passenger-
train assignment outputs derived from an application of the so-called ODX
method described in [7]. The data also includes the rail movement data and
schedule information. The metro network is comprised of 6 lines, 91 stations,
186 links and 9 transfer stations as shown in Figure 2.

Figure 2: Washington metro network

The network has an average ridership of $\approx$ 438 000 rides per day and a
total of $\approx$ 157 million rides during the entire study period with an
average travel time of 28 minutes per passenger. 14% of the passengers
experience a mean delay of 6 minutes or longer. 39% of the passenger trips
include transfers with an average of 1.14 transfers per passenger trip.

##  5 Results

In this section, we present the results of the delay estimation and the
temporal clustering for the Washington metro network. We chose a temporal
aggregation of 30 minutes since the maximum headway between trains is 20
minutes and choosing an aggregation less than that would imply that there
would be no trains between some OD pairs, hence no passengers and consequently
no system of equations. Having an aggregation of 30 minutes ensures that at
least one train per time slice is included in the system of equations as
represented in eq.3. Figure 3 shows the estimation results of the average
passenger delay for the three network elements for a selected day with an
temporal aggregation of 30 minutes.

Figure 3: Average passenger delay estimation results for 01-09-2017 (a) Track
segment delay (b) Initial waiting time (c) Transfer delay

From figure 3 it can be seen that there is no significant link delay for this
particular day. The waiting time and transfer delay is mapped on a link rather
than a node as we incorporate directionality of the node to distinguish
journeys between different lines. Thus, we can build these three compact 3D
graph for visualising the delay propagation of each day. This can be used to
evaluate the performance of the metro network at the end of the day or
estimate the passenger delay incurred between any given origin-destination
(OD) pair.

A look at the overall distribution of the average and total passenger delays
for the entire analysis period decomposed into different time periods is shown
in figure 4. The average passenger delay, shown in figure 4(a), is stable
compared to the morning and evening peak in the total passenger delay
distribution in figure 4(b). Thus, delays occur at all time periods, more
passengers are affected during the peak periods as can be expected. Figures
4(c) and (d) shows the contribution of each network element delay to the
overall delay. In the case of average passenger delay, 59% of the delay is
associated with the initial waiting time but when the number of passengers
affected is considered, track segment delay contributes the most with 41% of
the delay. Thus, link delay is more correlated with flows than initial waiting
times.

Figure 4: Delay distribution of the whole dataset across the three network
elements for different time periods based on (a) average passenger delay and
(b) total passenger delay. Overall delay distribution of the whole dataset
across the three network elements based on (c) average passenger delay and (d)
total passenger delay

The estimated delays for the three network elements are clustered into uniform
groups which are used to construct the feature vectors and temporally
clustered. The results of the temporal clustering are analysed using a
dendrogram for either different days or different months to extract day-to-day
and seasonal variances respectively. Figure 5 shows the dendrogram of the
temporal clustering for average passenger delay. The number of leaves in the
dendrogram corresponds to the number of days included in the dataset. There
are clearly well-defined clusters with similar days clustered together. We
chose the number of clusters to be 7 as it is a comprehensible number of
clusters as well as we wanted to investigate if setting the number of cluster
to 7 would extract distinct weekday or weekend patterns with clusters
consisting only of Mondays, Tuesdays, etc. We can easily merge or further
divide the clusters based on the insights we gain from these clusters.

Figure 5: Dendrogram of the temporal clustering of average passenger delay

The average passenger delay classes in the dendrogram in figure 5 are numbered
from left to right from 1 to 7 where the distance of 1 represents a delay
range of 0 to 10 minutes. As can be observed the size of the clusters obtained
varies greatly, from 9 to 209 days. The largest class consists of almost 60%
of the data. However, within this large cluster, the distance between the
feature vectors varies from 60 to 120 - implying total passenger delay
differences of up to 600 to 1200 minutes. This means that within a cluster,
some feature vectors have high dissimilarity index whereas others are more
similar.

Similar to the temporal clustering of the average passenger delay, the total
passenger delay can also be clustered. The resulting dendrogram is shown in
Figure 6. We chose the number of cluster as 7 here as well to further analyse
the data within each individual cluster. The total passenger delay classes in
the dendrogram in figure 6 are numbered from left to right from 1 to 7.

Figure 6: Dendrogram of the temporal clustering of total passenger delay Table 1: Description of network states based on average passenger delay. Class | Delay distribution | Distribution of months | Distribution of days
---|---|---|---
1 |  |  |
2 |  |  |
3 |  |  |
4 |  |  |
5 |  |  |
6 |  |  |
7 |  |  |
Table 2: Description of network states based on total passenger delay. Class | Delay distribution | Distribution of months | Distribution of days
---|---|---|---
1 |  |  |
2 |  |  |
3 |  |  |
4 |  |  |
5 |  |  |
6 |  |  |
7 |  |  |

A more in-depth look at each cluster and their corresponding delay
distribution of the average passenger delays is given in Table 1. By looking
at the distribution of the days and months in each cluster of the average
passenger delay in Table 1, we make the following observations:

  * •

There are distinct weekday and weekend patterns.

  * •

For all months, Sundays exhibit distinct patterns which are mainly manifested
in classes 2, 3 and 7 with relatively high average passenger delay.

  * •

In class 3, it is exclusively Sunday but the outlier Monday and Thursday
clustered there corresponds to Christmas, New Year and Thanksgiving day
respectively.

  * •

Class 5 consists exclusively of Saturday except for a Wednesday which
corresponds to a special holiday - 4th of July.

  * •

August and December exhibit distinct patterns from other months.

An in-depth look at the total passenger delay in Table 2 provides insight into
the number of passengers affected on different days and months. We make the
following observations based on Table 2:

  * •

Weekdays and weekends have distinct patterns.

  * •

Weekdays have two distinct peaks in total passenger delay whereas weekends
have more stable passenger delay.

  * •

August and December have distinct patterns compared to other months.

  * •

There is no clear seasonal variance other than these outlier months.

  * •

The total passenger delay is higher in the month of August with an additional
10,000 passenger delay minutes. From Table 1, the delay per passenger for the
month of August is also high. This can be attributed to August being one of
the busiest month in Washington DC.

  * •

Majority of the days belong to classes 2 and 5 which have relatively low
number of passengers affected by delay.

Even though there is a clear division between the delay for different days and
different months, our assumption of setting the number of clusters as 7 for
creating distinct classes containing only Mondays, Tuesdays, etc did not pan
out. This is mainly because of the month of August which biased the
clustering. However, the clustering revealed distinct weekday and weekend
patterns for both average and total passenger delay. Saturdays are clustered
together with weekdays more often than Sundays. Some weekdays are clustered
into the Sunday groups. Our assumption is that this is a special day and we
have confirmed this at least for the main holidays such as Christmas, New year
and Thanksgiving.

##  6 Conclusion

In this study, we propose a new estimation method to map passenger delay into
corresponding network elements. The outputs of our method can aid in measuring
network performance for any given origin-destination pair of the public
transportation network and in prioritizing measures for improving service
robustness. We decompose the delay along a passenger trajectory into its
corresponding track segment delay, initial waiting time and transfer delay. We
demonstrate the method on one year data from Washington metro network. An
initial analysis of the data shows that 14% of the passengers experience a
mean delay of 6 minutes or more in this network when looking across the
different days. The average passenger delay is more or less stable throughout
the day whereas total passenger delay, which accounts for the number of
passengers affected, contains two distinct peaks. The two peaks correspond to
the morning peak and evening peak with the evening peak associated with
greater passenger delay than the morning peak. The initial waiting time
contributes the most to the average passenger delay with 59% whereas the track
segment delay contributes the most to the total passenger delay with 41% of
the total delay being attributed to it. Thus, the track segment delay is more
correlated with flows than the initial waiting times. The results of the
temporal clustering indicate that there are noticeable differences between
weekdays and weekends and recurrent weekly patterns are also manifested.
Interestingly, the seasonal variance is mainly evident for the months of
August and December, with August particularly standing out.

The estimation approach adopted in this study can be transferred to other
locations. It is necessary to ensure that the system of equations is solvable.
Further, a different temporal aggregation might be needed depending on the
headway between the trains. Even though some seasonal variances could be
identified, the month of August biased the clustering approach. Hence, a
further avenue of research is to come up with an iterative hierarchical
clustering to reveal more detailed daily and seasonal variances. Another
direction for future research is to examine if the delay on the road traffic
network has any impact on passenger delay. In [23], 3D speed maps, similar to
our 3D delay maps, are constructed for the urban road network of Amsterdam,
which were clustered to reveal 4 so-called consensual patterns to represent 35
days of data. Having a similar network state representation for both road and
public transport can aid in understanding the interaction between the two
modes.

The insights gained in this study can aid in developing demand-oriented metro
services for different days within different months. Furthermore, it can help
in understanding the delay propagation through a network and thus the
prediction of such delays. This can help in providing proactive services in
case of disruptions, thus improving the service performance by reducing
operational costs and shorten waiting times for passengers.

##  7 Author Contribution

The authors confirm contribution to the paper as follows: study conception and
design: P. Krishnakumari, O. Cats and H. van Lint; data analysis: P.
Krishnakumari; interpretation of results and draft manuscript preparation: P.
Krishnakumari, O. Cats. All authors reviewed the results and approved the
final version of the manuscript.

##  8 Acknowledgement

This research was supported by the SETA project which is financed by the
European Union’s Horizon 2020 Research and Innovation program under the grant
agreement No 688082. The authors would like to thank Washington Metropolitan
Area Transit Authority and in particular Jordan Holt for their valuable
cooperation and providing the data that made this study possible.

## References

  * [1] “Refunds for delays by NS, The Netherlands”, https://www.ns.nl/en/customer-service/refunds/refunds-for-delays-on-a-domestic-route.html
  * [2] “Refunds for delays and disruptions by SL, Sweden”, https://sl.se/en/eng-info/contact/delay-compensation/
  * [3] “Rush hour promise by WMATA, Washington DC, USA”, https://www.wmata.com/fares/smartrip/rush-hour-promise.cfm
  * [4] Luis Moreira-Matias, João Mendes-Moreira, Jorge Freire Sousa and João Gama  “Improving mass transit operations by using AVL-based systems: A survey”  In _IEEE Transactions on Intelligent Transportation Systems_ 16.4 IEEE, 2015, pp. 1636–1653
  * [5] Marie-Pier Pelletier, Martin Trépanier and Catherine Morency  “Smart card data use in public transit: A literature review”  In _Transportation Research Part C: Emerging Technologies_ 19.4 Elsevier, 2011, pp. 557–568
  * [6] Martin Trépanier, Nicolas Tranchant and Robert Chapleau  “Individual trip destination estimation in a transit smart card automated fare collection system”  In _Journal of Intelligent Transportation Systems_ 11.1 Taylor & Francis, 2007, pp. 1–14
  * [7] Gabriel E Sánchez-Martínez  “Inference of public transportation trip destinations by using fare transaction and vehicle location data: Dynamic programming approach”  In _Transportation Research Record_ 2652.1 SAGE Publications Sage CA: Los Angeles, CA, 2017, pp. 1–7
  * [8] Haris N Koutsopoulos, Peyman Noursalehi, Yiwen Zhu and Nigel HM Wilson  “Automated data in transit: Recent developments and applications”  In _2017 5th IEEE international conference on models and technologies for intelligent transportation systems (MT-ITS)_ , 2017, pp. 604–609  IEEE
  * [9] Neema Nassir et al.  “Transit stop-level origin–destination estimation through use of transit schedule and automated data collection system”  In _Transportation research record_ 2263.1 SAGE Publications Sage CA: Los Angeles, CA, 2011, pp. 140–150
  * [10] Jason B Gordon, Harilaos N Koutsopoulos, Nigel HM Wilson and John P Attanucci  “Automated inference of linked transit journeys in London using fare-transaction and vehicle location data”  In _Transportation Research Record_ 2343.1 SAGE Publications Sage CA: Los Angeles, CA, 2013, pp. 17–24
  * [11] Xiaolei Ma et al.  “Mining smart card data for transit riders’ travel patterns”  In _Transportation Research Part C: Emerging Technologies_ 36 Elsevier, 2013, pp. 1–12
  * [12] Ashish Bhaskar and Edward Chung  “Passenger segmentation using smart card data”  In _IEEE Transactions on intelligent transportation systems_ 16.3 IEEE, 2014, pp. 1537–1548
  * [13] Xiaolei Ma et al.  “Understanding commuting patterns using transit smart card data”  In _Journal of Transport Geography_ 58 Elsevier, 2017, pp. 135–145
  * [14] Zhen-Liang Ma, Luis Ferreira, Mahmoud Mesbah and Ahmad Tavassoli Hojati  “Modeling bus travel time reliability with supply and demand data from automatic vehicle location and smart card systems”  In _Transportation Research Record_ 2533.1 SAGE Publications Sage CA: Los Angeles, CA, 2015, pp. 17–27
  * [15] Zhenliang Ma, Sicong Zhu, Haris N Koutsopoulos and Luis Ferreira  “Quantile regression analysis of transit travel time reliability with automatic vehicle location and farecard data”  In _Transportation Research Record_ 2652.1 SAGE Publications Sage CA: Los Angeles, CA, 2017, pp. 19–29
  * [16] Juanjuan Zhao et al.  “Estimation of passenger route choice pattern using smart card data for complex metro systems”  In _IEEE Transactions on Intelligent Transportation Systems_ 18.4 IEEE, 2016, pp. 790–801
  * [17] Yiwen Zhu, Haris N Koutsopoulos and Nigel HM Wilson  “A probabilistic Passenger-to-Train Assignment Model based on automated data”  In _Transportation Research Part B: Methodological_ 104 Elsevier, 2017, pp. 522–542
  * [18] P. Krishnakumari, H. Lint, T. Djukic and O. Cats  “A data driven method for OD matrix estimation”  In _Transportation Research Part C: Emerging Technologies_ , 2019  DOI: 10.1016/j.trc.2019.05.014
  * [19] Anna Altman and Jacek Gondzio  “Regularized symmetric indefinite systems in interior point methods for linear and quadratic optimization”  In _Optimization Methods and Software_ 11.1-4 Taylor & Francis, 1999, pp. 275–302
  * [20] Lior Rokach and Oded Maimon  “Clustering methods”  In _Data mining and knowledge discovery handbook_ Springer, 2005, pp. 321–352
  * [21] B Everitt and A Skrondal  “The Cambridge dictionary of statistics 2002”  In _Cambridge, Cambridge_ , 2002
  * [22] Panchamy Krishnakumari et al.  “Understanding Network Traffic States using Transfer Learning”  In _2018 21st International Conference on Intelligent Transportation Systems (ITSC)_ , 2018, pp. 1396–1401  IEEE
  * [23] Clélia Lopez et al.  “Revealing the day-to-day regularity of urban congestion patterns with 3D speed maps”  In _Scientific Reports_ 7.1 Nature Publishing Group, 2017, pp. 14029


# About a peculiar extra $U(1):Z^{\prime}$ discovery limit
Muon anomalous magnetic moment
Electron electric dipole moment

Jae Ho Heo<EMAIL_ADDRESS>Physics Department, University of Illinois at
Chicago, Chicago, Illinois 60607, USA

###### Abstract

The model (Lagrangian) with a peculiar extra $U(1)$ Bar05  is clearly
presented. The assigned extra $U(1)$ gauge charges give a strong constraint to
build Lagrangians. The $Z^{\prime}$ discovery limits are estimated and
predicted at the Tevatron and the LHC. The new contributions of the muon
anomalous magnetic moment are investigated at one and two loops, and we
predict that the deviation from the standard model may be explained. The
electron electric dipole moment could also be generated because of the
explicit CP violation effect in the Higgs sector, and a sizable contribution
is expected for a moderately sized CP phase (argument of the CP-odd Higgs),
$0.1\leq\sin\delta\leq 1$ $(6^{\circ}\leq\arg(A)\leq 90^{\circ})$.

###### pacs:

12.15.-y, 12.60.Cn, 12.60.Fr

††preprint:

##  I Introduction

An extra $U(1)$ (or a few extra $U(1)^{\prime}$s) may arise in the context of
grand unified theories Agu90 , superstring theories Ros88  or generically
emerge as simple extensions of the standard model (SM). Therefore, the models
with an extra $U(1)$ (or a few extra $U(1)^{\prime}$s) have been extensively
considered. Recently, Barr and Dorsner Bar05  suggested another possibility
for an extra $U(1)$ gauge, which satisfies all anomaly constraints in a
maximally economical way, whatever its origin111The origin is close to the
Pati-Salam model with an extra $U(1)$Bar05 . is. In the standard model, all
the possible anomalies from triangle diagrams of three gauge bosons must be
canceled if the Ward identities of the gauge theory are to be satisfied. The
existence of an extra $U(1)$ brings six additional anomaly cancellation
conditions, $U(1)_{Y}^{2}\times U(1)_{X}$, $U(1)_{Y}\times U(1)_{X}^{2}$,
$U(1)_{X}^{3}$, $SU(2)^{2}\times U(1)_{X}$, $SU(3)^{2}\times U(1)_{X}$ ,
gravity $\times U(1)_{X}$. These anomaly cancellations are nontrivial222The
general analyses about these cancellations can be found in Ref.Ema02 ., but
Barr and Dorsner showed a remarkably trivial solution Bar05  with a single
extra lepton triplet per family. These gauge anomalies are exactly canceled
for the fermion gauge charges listed at Table I.

In this letter the model (Lagrangian) is clearly presented. With an extra
lepton triplet, an additional Higgs singlet is necessary to provide masses of
the exotic leptons and the extra gauge boson $Z^{\prime}$. The Higgs singlet
would involve the extra $U(1)$ gauge symmetry breaking, and we assume that the
symmetry is broken near the weak scale. A Higgs triplet333The Higgs triplet is
introduced because of the need to induce the CP violating interaction in this
work. We can add additional scalars, such as nongauged Higgs singlets or
doublets, in other ways, but adding the gauged scalars is more generic. If we
only consider neutrino mass generation without electric dipole and dark matter
(we need to impose the discrete $Z_{2}$ symmetry ; see the next section)
phenomenology, the additional Higgs triplet is unnecessary. The interactions
induced by the Higgs triplet could also give a significant contribution to the
anomalous magnetic moment of the muon (see Sec. IV). with the required gauge
charges is added to explain our interesting phenomenology. The new gauge boson
$Z^{\prime}$ that generically emerges as gauging an extra $U(1)$ is the
intrinsic particle that explains the existence of an extra $U(1)$, so its
discovery limits at the Tevatron and LHC are estimated and predicted. The muon
anomalous magnetic moment, $a_{\mu}\equiv(g_{\mu}-2)/2$, has been a powerful
tool to account for new physics because of its importance. We investigate the
contributions involving the new particles at the one- and two-loop levels. One
can also see the explicit CP violation that generates the electric dipole
moment (EDM) of the electron $d_{e}$, and a sizable contribution is expected
via Barr-Zee two-loop mechanism for the moderate size of the CP-phase
(argument of the CP-odd Higgs), $0.1\leq\sin\delta\leq 1$
$(6^{\circ}\leq\arg(A)\leq 90^{\circ})$.

Table 1:  Fermion gauge charges. $T_{3}$ is the weak isospin, $Y$ is the hypercharge, $X$ is the extra $U(1)_{X}$ charge, and $Q=T_{3}+Y$ is the electric charge. The charges for the right handed fermions can also be assigned in the identical way. ($f_{L}^{c}\equiv(f_{L})^{c}$ in this letter, so $(f^{c})_{L}$ implies the antiparticle of $f_{R}$). | $u_{L}$ | $d_{L}$ | $(u^{c})_{L}$ | $(d^{c})_{L}$ | $\nu_{L}$ | ${\ell}_{L}$ | $({\ell}^{c})_{L}$ | $E^{+}_{L}$ | $E^{0}_{L}$ | $E^{-}_{L}$
---|---|---|---|---|---|---|---|---|---|---
T3 | ${1\over 2}$ | $-{1\over 2}$ | 0 | 0 | ${1\over 2}$ | $-{1\over 2}$ | 0 | 1 | 0 | $-1$
Y | ${1\over 6}$ | ${1\over 6}$ | $-{2\over 3}$ | ${1\over 3}$ | $-{1\over 2}$ | $-{1\over 2}$ | 1 | 0 | 0 | 0
X | 1 | 1 | $-1$ | $-1$ | 1 | 1 | 1 | $-1$ | $-1$ | $-1$

##  II Model description (Lagrangian)

With the new particle content, the Yukawa potential for the lepton sector can
have the following enlarged form without the ad hoc imposition of lepton
number conservation.

$y_{1}\mathrm{{Tr}}(\overline{E_{L}^{c}}E_{L})\eta+y_{2}\overline{L_{L}}\phi\ell_{R}+y_{3}\overline{L_{L}^{c}}i\sigma_{2}\chi
L_{L}+y_{4}\overline{L_{L}^{c}}i\sigma_{2}E_{L}\phi+y_{5}\mathrm{{Tr}}(\overline{E_{L}}\chi)\ell_{R}+h.c.,$
(1)

where $\eta$ and $\chi$ denote a Higgs singlet and a Higgs triplet,
$\phi=(\phi^{+},\phi^{0})^{T}$ is a Higgs doublet, and
$L=\left(\nu_{\ell},\ell\right)^{T}$ is the lepton doublet. The bi doublet
representation is taken for the additional lepton triplet and the Higgs
triplet is also taken in the form of a $2\times 2$ matrix transforming under
$SU(2)$ as $\chi\rightarrow U\chi U^{{\dagger}}$.

$E_{L}=\begin{pmatrix}\frac{1}{\sqrt{2}}E^{0}&E^{+}\\\
E^{-}&-\frac{1}{\sqrt{2}}E^{0}\end{pmatrix}_{L},\text{ \
}\chi=\begin{pmatrix}\frac{1}{\sqrt{2}}\chi^{+}&\chi^{++}\\\
\chi^{0}&-\frac{1}{\sqrt{2}}\chi^{+}\end{pmatrix}.$ (2)

The lepton triplet must be a Majorana combination. It should be noted that the
antisymmetric tensor $i\sigma_{2}$ follows from the antisymmetric property of
the charge conjugation.

Since the gauge charges of the leptons are already assigned by anomaly
constraints, the gauge charges of the Higgses are assigned by the combinations
with leptons in the Yukawa potential under the $SU(2)_{L}\times U(1)_{Y}\times
U(1)_{X}$ gauge invariance. If we introduce the assignment of $U(1)_{X}$
charges for the Higgses, the singlet $\eta$ must have a $U(1)_{X}$ charge of
$2$ from the $y_{1}$-term since $E$ has $-1$; the doublet $\phi$ may have a
charge of $2$ from the $y_{2}$-term and $0$ from the $y_{4}$-term ; and the
triplet $\chi$ may have a charge of $-2$ from the $y_{3}$-term and $0$ from
the $y_{5}$-term. The other gauge charges may be assigned in an analogous way,
and the assigned charges of the Higgses are listed at Table II. Notice that
the Higgs doublet and triplet may have two distinctive $U(1)_{X}$ charges.

Table 2:  Higgs gauge charges. $T_{3}$ is the weak isospin, Y is the hypercharge, $X$ is the $U(1)_{X}$ charge, and $Q=T_{3}+Y$ is the electric charge. | $\phi^{+}$ | $\phi^{0}$ | $\eta$ | $\chi^{++}$ | $\chi^{+}$ | $\chi^{0}$
---|---|---|---|---|---|---
$T_{3}$ | ${1\over 2}$ | $-{1\over 2}$ | 0 | 1 | 0 | $-1$
$Y$ | ${1\over 2}$ | ${1\over 2}$ | 0 | 1 | 1 | 1
$X$ | $(0,2)$ | $(0,2)$ | 2 | $(0,-2)$ | $(0,-2)$ | $(0,-2)$

The Yukawa potential with distinct charges takes the form.

$y_{1}\mathrm{{Tr}}\left(\overline{E_{L}^{c}}E_{L}\right)\eta_{(2)}+y_{2}\overline{L_{L}}\phi_{(2)}\ell_{R}+y_{3}\overline{L_{L}^{c}}i\sigma_{2}\chi_{(-2)}L_{L}+y_{4}\overline{L_{L}^{c}}i\sigma_{2}E_{L}\phi_{(0)}+y_{5}\mathrm{{Tr}}\left(\overline{E_{L}}\chi_{(0)}\right)\ell_{R}+h.c.,$
(3)

where the indices in the lower brackets of the Higgses denote $U(1)_{X}$
charges of the Higgses.

A discrete $Z_{2}$ symmetry could be imposed to explain a certain
phenomenology, dark matter. If $E$ is odd and all other particles are even
under $Z_{2}$ symmetry, this would prevent the exotic leptons from coupling
with the ordinary leptons and the neutral lepton $E^{0}$ becomes stable, and
thus could be a dark matter candidate. The Yukawa potential with $Z_{2}$
symmetry is given by

$y_{1}\mathrm{{Tr}}\left(\overline{E_{L}^{c}}E_{L}\right)\eta_{(2)}+y_{2}\overline{L_{L}}\phi_{(2)}\ell_{R}+y_{3}\overline{L_{L}^{c}}i\sigma_{2}\chi_{(-2)}L_{L}+h.c..$
(4)

The Yukawa potential for the quark sector may be built in an analogous way.

$y_{6}\overline{Q_{L}}\widetilde{\phi_{(0)}}u_{R}+y_{7}\overline{Q_{L}}\phi_{(0)}d_{R}+h.c.,$
(5)

where $Q=\left(u,d\right)^{T}$ is the quark doublet and
$\widetilde{\phi_{(0)}}=i\sigma_{2}\phi_{(0)}^{\ast}$. The hypercharge
combinations in the potential prohibits the couplings between quarks and Higgs
triplets. Since quarks receive masses only from $\phi_{(0)}$, there is no tree
level flavor-changing neutral currents. Note that leptons and quarks interact
with two distinct Higgs doublets, which is different from the standard two
Higgs doublet model (2HDM) where one Higgs couples to the up-type quarks and
the other couples to the charged leptons and down-type quarks.

The size of the couplings and vacuum expectation values (VEVs) may be
approximately constrained with the known experimental measurements. The VEV of
$\phi_{(0)}$, $\left\langle\phi_{(0)}\right\rangle$, must be of the order of
$100$ GeV to meet the top quark mass, and $\
\left\langle\phi_{(2)}\right\rangle$ must be $1\sim 100$ GeV to satisfy the
$\tau$-lepton mass and the known SM
VEV444$\left\langle\phi\right\rangle=\sqrt{\left\langle\phi_{(0)}\right\rangle^{2}+\left\langle\phi_{(2)}\right\rangle^{2}}\simeq
174$ GeV.. The Higgs triplet VEV $\left\langle\chi\right\rangle$ must be very
small compared to the Higgs doublet VEV, since the $\rho$ parameter predicted
by the SM is consistent with the experimental measurement in high precision
Cer07 . The neutrino mass may be generated at the tree level in this model.
The mass matrix of neutral leptons is

$\mathcal{M}_{\nu
E}=\begin{pmatrix}m_{\nu}&y_{4}\left\langle\phi_{(0)}\right\rangle\\\
y_{4}\left\langle\phi_{(0)}\right\rangle&M_{E}\end{pmatrix},$

where $m_{\nu}\equiv y_{3}\left\langle\chi_{(-2)}\right\rangle$ and
$M_{E}\equiv y_{1}\left\langle\eta\right\rangle$. The nature of the neutrino
is not known; however, we have approximately predicted the size of the
neutrino mass. We take the exotic lepton $E$ at the weak scale, so the
coupling $y_{4}$ must be very small555According to the famous canonical seesaw
mechanism, the order unity coupling is assumed, with the scale of new physics
of 1013GeV. However, we relax the constraint, as the coupling $y_{4}$ could
approximately be of the order of the electron Yukawa coupling ($\sim
10^{-6}$). since $\left\langle\phi_{(0)}\right\rangle$ is of the order of
$100$ GeV. The seesawlike mechanism is applicable to generate the neutrino
mass. If $Z_{2}$ symmetry is imposed, $y_{4}=0$. The neutrino mass may be
taken as $m_{\nu}$, where $y_{3}$ and/or
$\left\langle\chi_{(-2)}\right\rangle$ be sized for the neutrino mass. For
either case, we predict Majorana-type neutrinos in this model. Since
$\left\langle\chi\right\rangle$ and the coupling $y_{4}$ are small, we can
consider that the massive leptons are in the mass eigenstates for the Yukawa
potential of (3).

The Higgs potential is also amenable to the gauge invariance with the extra
$U(1)$.

$\displaystyle V$ $\displaystyle\supset
V_{2HDM}+\left\\{\mu_{1}\phi_{(0)}^{T}\chi_{(0)}^{{\dagger}}\phi_{(0)}+\mu_{2}\phi_{(2)}^{{\dagger}}\chi_{(-2)}^{{\dagger}}\phi_{(0)}+h.c.\right\\}$
$\displaystyle+\left\\{\lambda_{1}\phi_{(2)}^{{\dagger}}\phi_{(0)}\mathrm{{Tr}}\left(\chi_{(-2)}^{{\dagger}}\chi_{(0)}\right)+\lambda_{2}\phi_{(2)}^{{\dagger}}\sigma^{a}\phi_{(0)}\mathrm{{Tr}}\left(\chi_{(-2)}^{{\dagger}}\sigma^{a}\chi_{(0)}\right)+h.c.\right\\},$
(6)

where $V_{2HDM}$ stands for the Higgs potential involving only Higgs doublets,
and the functional form is the same as the 2HDM with $Z_{2}$ symmetry. In
addition to the two complex trilinear couplings, the two complex quartic
couplings are possible, those involving the CP violation phenomenology. The
phenomenology with two complex trilinear coupings can be found in Ref.Ema98
666They assigned the Higgs triplets of the order of $10^{13}$GeV to explain
neutrino masses and Baryogenesis via Leptogenesis. However the Higgs triplets
are assumed to have masses of the order of weak scale to explain the
interesting phenomenology in our scenario., and the complex quartic couplings
are related to the electric dipole moment of fermions which will be discussed
as a part of this letter. The other interaction terms are trivial and almost
irrelevant to the phenomenology.

##  III $Z^{\prime}$ discovery limit

The interactions of the $Z^{\prime}$ boson with the fermions are described by

${\displaystyle\sum\limits_{f}}z_{f}^{\prime}g_{Z^{\prime}}Z_{\mu}^{\prime}\overline{f}\gamma^{\mu}f,$
(7)

where $f=E_{L},Q_{L},L_{L},u_{R},d_{R},e_{R}$ are the lepton and quark fields
and $z_{f}^{\prime}$ is the gauge charge corresponding to the fermion.

The leptonic decays $Z^{\prime}\rightarrow\ell^{+}\ell^{-}(e^{+}e^{-}$ and
$\mu^{+}\mu^{-})$ provide the most distinctive signature for observing the
$Z^{\prime}$ signal at the hadron colliders. The cross section of the
$p{\bar{p}}$ collision in the $\ell^{+}\ell^{-}$ channel can be calculated at
the narrow width $Z^{\prime}$ pole in the center-of-momentum (CM) frame. The
hadronic cross section is given by

$\sigma(Z^{\prime})=K\sum_{q,{\bar{q}}}\int_{0}^{1}dx_{1}dx_{2}(f_{q}^{p}(x_{1})f_{\overline{q}}^{\overline{p}}{(x_{2})}+f{{}_{\overline{q}}^{p}(x_{1})}f_{q}^{\overline{p}}(x_{2})){\hat{\sigma}(Z^{\prime}),}$
(8)

where ${\hat{s}=x}_{1}x_{2}s$ is the partonic fraction of $s$, $f(x)$’s are
the partonic distribution functions (PDFs) and the sum is performed over all
the light quarks. $K$ is the QCD correction factor ($\sim$ 1.3) Mcar04 , which
accounts for higher order QCD corrections. The partonic cross secion
${\hat{\sigma}(Z^{\prime})}$ is calculated in a sum over the spins of the
final states and an average over the spins and colors of the initial states.

${\hat{\sigma}}(Z^{\prime})={\frac{\pi z_{f}^{\prime
2}g_{Z^{\prime}}^{2}}{48}}\delta({\hat{s}}-M_{Z^{\prime}}^{2}).$ (9)

Eq.(8) and (9) lead to the hadronic cross section in the $\ell^{+}\ell^{-}$
channel.

$\sigma(Z^{\prime})\cdot Br_{\ell^{+}\ell^{-}}=K{\frac{\pi z_{f}^{\prime
2}g_{Z^{\prime}}^{2}}{48s}}\sum_{q,{\bar{q}}}\int_{\frac{m_{Z^{\prime}}^{2}}{s}}^{1}{\frac{dx}{x}}\left(f_{q}^{p}(x)f_{\overline{q}}^{\overline{p}}\left({\frac{M_{Z^{\prime}}^{2}}{xs}}\right)+f{{}_{\overline{q}}^{p}(x)}f_{q}^{\overline{p}}\left({\frac{M_{Z^{\prime}}^{2}}{xs}}\right)\right)\cdot
Br_{\ell^{+}\ell^{-}},$ (10)

where $Br_{\ell^{+}\ell^{-}}$ is the branching ratio of $Z^{\prime}$ to
$\ell^{+}\ell^{-}$. We may take $z_{f}^{\prime 2}\simeq 1,$ since precision
measurements of $Z$-pole observables predict the small $Z-Z^{\prime}$ mixing
angle$(\leq 10^{-3})$ Cer07 .

Figure 1: The $Z^{\prime}$ discovery limit at the Tevatron ($\sqrt{s}=1.96$
TeV and $L=1.3$ $fb^{-1}$) and the LHC($\sqrt{s}=14$ TeV and $L=100$
$fb^{-1}$). The horizontal lines indicate the experimental sensitivities, and
the bold lines are predictions of the cross section. The predictions are for
the coupling, $g_{Z^{\prime}}=0.1$ and $0.7$ (SM coupling). MRST LO PDFs Mar02
are used. The intersections of the curves determine the lower mass limits.

For $pp$ collision at the LHC, the proton PDF takes the place of the
antiproton PDF. Fig.1 shows the predicted cross sections with the present
experimental sensitivity at the Tevatron Run II777The CDF Collaboration CDF07
has set the better luminosity for $\sigma(Z^{\prime})\cdot
Br_{\ell^{+}\ell^{-}}$ than the DØ D007  in some reason, so it is considered
for the CDF collider. and the projected experimental sensitvity at the LHC
LHC05 . The actual experimental analysis shows an experimental line with a
more complicated structure than the horizontal line in the figure. For a
nonzero background888The non-zero background is roughly taken from Ref.CDF07 ,
which all the expected backgrounds are considered. The most significant source
of background in this channel is the SM Drell-Yan process via
$Z/\gamma^{\ast}$ as reported in Ref. CDF07 ., $N_{Z^{\prime}}=3$ events are
excluded at the Tevatron. The $Z^{\prime}$ discovery limits are $300$ GeV,
$870$ GeV for $g_{Z^{\prime}}=0.1,0.7$ at the Tevatron, and the LHC may probe
$Z^{\prime}$ upto $3.1$ TeV, $5.7$ TeV for $g_{Z^{\prime}}=0.1,0.7$. Since the
$U(1)_{X}$ gauge charge of the Higgs singlet $\eta$ is $2$,
$M_{Z^{\prime}}\simeq 2g_{Z^{\prime}}\left\langle\eta\right\rangle$. We
predict the lower limit of the extra $U(1)$ symmetry breaking to be around
$200\sim 800$ GeV at the Tevatron (CDF detector).

##  IV muon anomalous magnetic moment

The deviation of the current experimental value from the SM prediction is
approximately $3.0\sigma$ and the numerical deviation is $\Delta
a_{\mu}=27.5(8.4)\times 10^{-10}$ Mda07  or $27.7(9.3)\times 10^{-10}$ Fdo08 .
The experimental value is the measurement of the BNL experiment Muo06 . We
investigate one- and two-loop contributions.

Figure 2: The one-loop contributions to $a_{\mu}$ involving the extra
particles, $E,\chi,$ and $Z^{\prime}$.

The diagrams of Fig.2 display one-loop contributions involving the new
particles, $E$, $\chi,$ and $Z^{\prime}$. The relevant interaction Lagrangian
for diagrams (a) and (b) of Fig. 2 comes from the $y_{5}$-term of the Yukawa
potential of (3). The states $\chi_{(0)},\chi_{(-2)}$ may be rotated into the
mass eigenstates $\chi_{\ell},\chi_{h}$, where $\chi_{\ell}$ and $\chi_{h}$
are the light and heavy mass eigenstates. The rotational angle is determined
in the Higgs potential. However, the couplings with the Higgs triplet are free
parameters, so we redefine the new couplings in the mass eigenstates. The
relevant Lagrangian for the light scalar state $\chi_{\ell}$ is given by

$-y\left(\overline{E_{L}^{0}}\chi_{\ell}^{+}\mu_{R}+\overline{E_{L}^{-}}\chi_{\ell}^{0}\mu_{R}-\overline{E_{L}^{+}}\chi_{\ell}^{++}\mu_{R}+h.c.\right),$
(11)

where $y$ is the Yukawa coupling in the mass eigenstates of $\chi$. The
Lagrangian for the heavy mass eigenstates can be given in the same fashion.
The $y_{4}$-term with which the Higgs doublet is involved is neglected due to
the small coupling constrained by the neutrino mass.

The contribution of Fig.2(a) is negligible, since the particles
($E^{+}$,$E^{-}$) on the line which are hooked up by the photon have opposite
electric charges. We calculated the contribution of Fig.2(b), and it is given
by

$\Delta
a_{\mu}^{(\text{one})}=\frac{3y^{2}}{8\pi^{2}}\left(\frac{m_{\mu}}{M_{E}}\right)f\left(\frac{M_{\chi}^{2}}{M_{E}^{2}}\right)\simeq
4.03\times 10^{-6}\cdot
y^{2}\left(\frac{1\text{TeV}}{M_{E}}\right)f\left(\frac{M_{\chi}^{2}}{M_{E}^{2}}\right),$
(12)

where the prefactor of 3 comes from the electric charges of
$\chi^{\pm},\chi^{\pm\pm}$.

The corresponding one-loop function is

$f(z)=\int_{0}^{1}dx\frac{(1-x)x}{zx+1-x}=\frac{1-z^{2}+2z\ln z}{2(1-z)^{3}}$
(13)

which has asymptotic behaviors,

$f(z)\longrightarrow\left\\{\begin{array}[c]{cc}\frac{1}{6}&\text{as }z=1\
,\\\ {\frac{1}{2z}-}\frac{\ln z}{z^{2}}&\text{ for }z\gg 1\ ,\\\
\frac{1}{2}+z\ln z&\text{for }z\ll 1\ .\end{array}\right..$ (14)

We neglect the contribution from the other scalars (called the heavy scalars),
since those scalars are split into light and heavy mass eigenstates, in
general, and the one-loop function behaves $f(z)\rightarrow 0$ as
$z\rightarrow\infty$. Furthermore, the large splitting is necessary to
generate the sizable electric dipole moment, that will be discussed in the
next section. The $M_{\chi}$ or $M_{\chi_{\ell}}$ implies the mass of the
light scalar in this letter.

Figure 3: $\Delta a_{\mu}$ as a function of the exotic lepton mass $M_{E}$ for
various values of $M_{\chi}$ at the one-loop level.

Fig.3 shows the predictions of the anomalous magnetic moment for
$0.1$TeV$<M_{E},M_{\chi}<1$TeV. The range of deviations from the SM is
presented in the dark ”allowed” band Mda07 . The predictions are in the
allowed band around the Yukawa coupling $y=0.05$. Since $\Delta
a_{\mu}^{(\text{one})}\sim y^{2}/M_{E},$ the Yukawa coupling $y$ is very
sensitive to the deviation $\Delta a_{\mu}$. Besides the above region, a
possible scenario is $M_{E}\approx M_{\chi}>1$TeV for the Yukawa coupling
$y>0.06$. The contribution by the $Z^{\prime}$ gauge boson of Fig.2(c) is
negligible, since $\Delta a_{\mu}\sim m_{\mu}^{2}/M_{Z^{\prime}}^{2}$.

Figure 4: Two-loop contributions to $a_{\mu}$ $(d_{e})$ (mirror graphs are not
displayed.).

If $Z_{2}$ symmetry is imposed, there is no one-loop contribution to explain
the deviation. We consider the two-loop contribution via the Barr-Zee type of
mechanism, which is depicted in Fig.4. The relevant Lagrangian to induce the
Barr-Zee two-loop contribution is given by

$-\frac{\sqrt{2}m_{\mu}r_{\mathcal{H}}}{v}\overline{\mu}\mathcal{H}\mu-\frac{\lambda_{+}v}{\sqrt{2}}\mathcal{H}\left(\chi_{\ell}\chi_{\ell}+\chi_{h}\chi_{h}\right),$

where $\mathcal{H}=h$ or $H$, $v=\sqrt{2}\left\langle\phi\right\rangle,$ and
$\lambda_{+}$ is the coupling in the mass eigenstates of $\chi$ for
$\mathcal{H}$. The rotational angles999Coventionally, the rotational angle
between the neutral Higgses in 2HDM is denoted by the symbol $\alpha$. But in
this letter, the symbol $\alpha$ is used for the electric fine structure
constant, so we use the symbol $\beta_{h}$ for the rotational angle.
$r_{h}=-\sin\beta_{h}/\cos\beta$ and $r_{H}=\cos\beta_{h}/\cos\beta$ to the
muon are the same as in the standard 2HDM, since the scalar $\phi_{(2)}$,
which is consistent with the scalar to couple to the charged leptons in the
2HDM, couples to the muon. There is no contribution from the CP-odd Higgs
(pseudoscalar) $A$ because the interaction with the CP-odd Higgs violates CP
symmetry, so the effect of the CP-odd Higgs involves the electric dipole
moment.

The contribution of two loops is given by

$\displaystyle\Delta a_{\mu}^{(\text{two})}$
$\displaystyle\simeq-\sum_{\mathcal{H},\chi}\frac{\alpha
m_{\mu}^{2}}{16\pi^{3}}\frac{Q_{\chi}^{2}r_{\mathcal{H}}\lambda_{+}}{m_{\mathcal{H}}^{2}}\left[F\left(\frac{M_{\chi_{\ell}}^{2}}{m_{\mathcal{H}}^{2}}\right)+F\left(\frac{M_{\chi_{h}}^{2}}{m_{\mathcal{H}}^{2}}\right)\right]$
$\displaystyle=-2.07\times 10^{-11}\cdot\sum_{\mathcal{H}=h,H}\lambda
r_{\mathcal{H}}\left(\frac{200\text{GeV}}{m_{\mathcal{H}}}\right)^{2}\left[F\left(\frac{M_{\chi_{\ell}}^{2}}{m_{\mathcal{H}}^{2}}\right)+F\left(\frac{M_{\chi_{h}}^{2}}{m_{\mathcal{H}}^{2}}\right)\right].$
(15)

Note that $\sum Q_{\chi}^{2}=5$ due to singly and doubly charged scalars in
the inner loop. The two-loop function is

$F\left(z\right)=\int_{0}^{1}dx\frac{x(1-x)}{z-x(1-x)}\ln\left[\frac{x(1-x)}{z}\right]$
(16)

which has asymptotic behaviors,

$F(z)\longrightarrow\left\\{\begin{array}[c]{cc}-{0.344}&\text{as }z=1\ ,\\\
-{\frac{1}{6z}}\ln z-{\frac{5}{18z}}&\text{ for }z\gg 1\ ,\\\ (2+\ln
z)&\text{for }z\ll 1\ .\end{array}\right.\ .$ (17)

The Barr-Zee two-loop contributions, according to Eq.(15), are suppressed by
the muon mass and the loop factor, and thus the large $r_{\mathcal{H}}$ and
the small $m_{\mathcal{H}}$ are necessary. The lower limit of the light Higgs
boson mass is around $44$ GeV for $r_{h}\simeq\tan\beta$ from the LEP Opal01 ,
but the light Higgs boson keeps the same lower limit of the SM Higgs boson,
$113.5$ GeV, for $r_{H}\simeq\tan\beta.$ The case for $r_{h}\simeq\tan\beta$
is taken. We can approach these analyses in the 2HDM since the VEVs of the
Higgs triplets have the small size. Besides, the doubly charged scalar
$\chi^{++}$ in the inner loop gives the main contribution to the deviaton due
to its double electric charge. The lower limit of the doubly charged scalar,
around $120$ GeV from the Tevatron CDF04  and the LEP L303 , is considered.

Figure 5: $\Delta a_{\mu}$ as a function of the light Higgs boson mass $m_{h}$
at the two-loop level for various values of $\tan\beta$.

Fig.5 shows the predictions for the Barr-Zee two-loop contribution, $\Delta
a_{\mu}^{(\text{two})}$, as a function of the light Higgs boson mass $m_{h}$.
To predict the two-loop contribution $\Delta a_{\mu}^{(\text{two})}$, we
assume a coupling $\lambda_{+}$ of the same size as the SM Higgs quartic
coupling for the SM Higgs of 120 GeV. The predictions barely reside in the
allowed region.

##  V Electron Electric Dipole Moment

The EDM of fermions predicted by the standard model is extremely small
compared to the present experimental bounds. Another mechanism beyond the SM
has been required to induce the sizable EDM. There are also explicit CP
violation interactions related to the Barr-Zee two-loop mechanism Smba90 ;
Dar99 ; Jheo08  for the EDM in this model. Since the interaction must involve
the CP violation, it is comprised of only the CP-odd Higgs (pseudoscalar) $A$.
The irreducible CP phase appears in the diagonalization101010The detailed
process for diagonalization of mass matrix by the unitary transformation can
be found in Ref.Jheo08 . of the mass matrix for the Higgs triplets in the
Higgs potential of (6). If we introduce the new phenomenological couplings,
the relevant interaction Lagrangian is given by

$\frac{\sqrt{2}m_{\mu}r_{A}}{v}\overline{e}i\gamma^{5}Ae-\frac{\lambda_{-}v}{\sqrt{2}}A\left(\chi_{\ell}\chi_{\ell}-\chi_{h}\chi_{h}\right),$

where $r_{A}=\tan\beta$ is the rotational angle, and
$\lambda_{-}=\lambda\sin\delta$ where $\sin\delta$ is the CP-violation effect
which comes from combinations of the complex quartic couplings in the
potential of (6). The Barr-Zee diagrams were well calculated in many papers to
induce the sizable electric dipole moment, and the result is identical to the
Barr-Zee two-loop contribution of the anomalous magnetic moment, except for
CP-violation effect. The electron electric dipole moment results in

$\displaystyle\left(\frac{d_{e}}{e}\right)^{\gamma}$
$\displaystyle=-\sum_{\chi}\frac{\alpha
m_{e}}{32\pi^{3}}\frac{Q_{\chi}^{2}r_{A}\lambda_{-}}{m_{A}^{2}}\left[F\left(\frac{M_{\chi_{\ell}}^{2}}{m_{A}^{2}}\right)-F\left(\frac{M_{\chi_{h}}^{2}}{m_{A}^{2}}\right)\right]$
$\displaystyle=-9.25\times
10^{-27}\cdot\left(\frac{200\text{GeV}}{m_{A}}\right)^{2}\lambda\sin\delta\tan\beta\left[F\left(\frac{M_{\chi_{\ell}}^{2}}{m_{A}^{2}}\right)-F\left(\frac{M_{\chi_{h}}^{2}}{m_{A}^{2}}\right)\right]\text{
\ }(cm),$ (18)

where the two-loop function is given in Eq.(16) ; also note that $\sum
Q_{\chi}^{2}=5$ due to singly and doubly charged scalars from the Higgs
triplets. The electron EDM results in the difference between two contributions
from the light and heavy scalars, $\chi_{\ell}$ and $\chi_{h}$. The
contribution from the heavy scalar is neglected, since the two-loop function
behaves like $F\left(z\right)\rightarrow 0$ as $z\rightarrow\infty$.

Figure 6: Numerical estimates of the EDMs as a function of the CP-odd (or
pseudoscalar) Higgs boson mass for various values of $\tan\beta$ and
$M_{\chi}$. Also shown the predictions for CP-phase, $0.1\leq\sin\delta\leq 1$
$(6^{\circ}\leq\arg(A)\leq 90^{\circ})$. The horizonal line indicates the
current 90% C.L. experimental bound Bcr02 .

In order to predict the electron electric dipole moment numerically, we also
assume the coupling $\lambda$ of the same size as the SM Higgs quartic
coupling for the SM Higgs of 120 GeV. Fig.6 shows the predictions of the
electron electric dipole moment as a function of the CP-odd Higgs (or
pseudoscalar) mass with the current 90% C.L. experimental bound Bcr02 . The
sizable contributions are expected for the moderate size of the CP phase,
$0.1\leq\sin\delta\leq 1$ $(6^{\circ}\leq\arg(A)\leq 90^{\circ})$.

##  VI Conclusions

The model (Lagrangian) with a peculiar extra $U(1)$, that Barr and Dosner
suggested, has clearly been presented. The gauge charges of the extra $U(1)$
give a strong constraint to build the Lagrangians. $Z^{\prime}$ discovery
limits are estimated and predicted at the Tevatron and the LHC. The discovery
limit at the Tevatron (CDF detector) gives the lower limit of the extra $U(1)$
symmetry breaking scale, approximately $200\sim 800$ GeV. The muon anomalous
magnetic moment could be explained at the one-loop level for a Yukawa coupling
around $0.05$. If we allow masses of the new particles to be more than 1 TeV,
the larger Yukawa coupling is possible. However, smaller Yukawa couplings are
prohibited by the discovery limits of new particles at the Tevatron and the
LEP. The muon anomalous magnetic moment could also be explained at the two-
loop level, but the region of parameters is very narrow. There are explicit
CP-violation interactions in this model. A sizable electron electric dipole
moment is expected for a moderately sized CP phase, $0.1\leq\sin\delta\leq 1$,
$(6^{\circ}\leq\arg(A)\leq 90^{\circ})$ via the Barr-Zee mechanism.

## References

  * (1) S.M. Barr and I. Dorsner, Phys. Rev. D 72, 015011 (2005) [arXiv:hep-ph/0503186] ; S.M. Barr and A. Khan, Phys. Rev. D 74, 085023 (2006) [arXiv:hep-ph/0607054].
  * (2) F. del Aguila, J.M. Moreno and M. Quiros, Nucl. Phys. Proc. Suppl. 16, 621 (1990) ; Ugo Amaldi, et al., Phys. Rev. D 36, 1385 (1987) ; D. London and J.L. Rosner, Phys. Rev. D 34, 1530 (1986).
  * (3) Vernon D. Barger, N.G. Deshpande and K. Whisnant, Phys. Rev. Lett. 56, 30 (1986) ; G. Cleaver, M. Cvetic, J.R. Espinosa, L.L. Everett and P. Langacker, Nucl. Phys. B 525, 3 (1998) [arXiv:hep-th/9711178] ; J.L. Hewett and T.G. Rizzo, Phys. Rept. 183, 193 (1984) ; M. Cvetic and P. Langacker, Mod. Phys. Lett. A 11, 1247 (1996) [arXiv:hep-ph/9602424].
  * (4) E. Ma, Mod. Phys. Lett. A 17, 535 (2002) [arXiv:hep-ph/0112232] ; E. Ma and D.P. Roy, Nucl. Phys. B 644, 290 (2002). [arXiv:hep-ph/0206150] ; S.M. Barr, B. Bednarz and C. Benesh, Phys. Rev. D 34, 235 (1986).
  * (5) ALEPH, DELPHI, L3, OPAL Collaborations and LEP Electroweak Working Group, arXiv:0712.0929 [hep-ex]; G.B. Gelmini and M. Roncodelli, Phys. Lett. B 99, 411 (1981).
  * (6) E. Ma and U. Sarkar, Phys. Rev. Lett. 80, 5716 (1998) [arXiv:hep-ph/9802445].
  * (7) M. Carena, A. Daleo, B.A. Dobrescu and T.M.P. Tait, Phys. Rev. D 70, 093009 (2004) [arXiv:hep-ph/0408098].
  * (8) A.D. Martin, R.G. Roberts, W.J. Stirling and R.S. Thorne, Phys. Lett. B 531, 216 (2002) [arXiv:hep-ph/0201127].
  * (9) T. Aaltonen et al. [CDF Collaboration], Phys. Rev. Lett. 99, 171802 (2007) [arXiv:hep-ex/0707.2524].
  * (10) V.M. Abazov et al. [DØ Collaboration], Phys. Rev. D 76, 012003 (2007) [arXiv:hep-ex/0702025].
  * (11) F. Gianotti, M.L. Mangano and T. Virdee et al., Eur. Phys. J. C 39, 293 (2005) [arXiv:hep-ph/0204087].
  * (12) M. Davier, Nucl. Phys. Proc. Suppl. 169, 288 (2007) [arXiv:hep-ph/0701163].
  * (13) F. Domingo and U. Ellwanger, J. High Energy Phys. 0807, 079 (2008) [arXiv:hep-ph/0806.0733].
  * (14) H.N. Brown et al. [muon (g-2) Collaboration], Phys. Rev. D 73, 072003 (2006) [arXiv:hep-ex/0602035].
  * (15) G. Abbiendi et al. [OPAL Collaboration], Eur. Phys. J. C 18, 425 (2001) [arXiv:hep-ex/0007040].
  * (16) D.E. Acosta et al. [CDF Collaboration], Phys. Rev. Lett. 93, 221802 (2004) [arXiv:hep-ex/0406073]; Phys. Rev. Lett. 95, 071801 (2005) [arXiv:hep-ex/0503004].
  * (17) P. Achard et al. [L3 collaboration], Phys. Lett. B 576, 18 (2003) [arXiv:hep-ex/0309076].
  * (18) S.M. Barr and A. Zee, Phys. Rev. Lett. 65, 21 (1990) [Erratum-ibid. 65, 2920 (1990)].
  * (19) Darwin Chang, Wai-Yee Keung, and Apostolos Pilaftsis, Phys. Rev. Lett. 82, 900 (1999) [Erratum-ibid. 83, 3972 (1999)] [arXiv:hep-ph/9811202].
  * (20) Jae Ho Heo and Wai-Yee Keung, Phys. Lett. B 661, 259 (2008) [arXiv:hep-ph/0801.0231].
  * (21) B. C. Regan, E.D. Commins, C.J. Schmidt and D. DeMille, Phys. Rev. Lett. 88, 071805 (2002).