# A Cautionary Tail

A Framework and Case Study for Testing Predictive Model Validity

Peter C. Casey  The Lab @ DC
Government of the
District of Columbia1350 Pennsylvania Avenue NWWashingtonD.C.20004
<EMAIL_ADDRESS>,  Kevin H. Wilson  The Lab @ DC
Government of the
District of Columbia1350 Pennsylvania Avenue NWWashingtonD.C.20004
<EMAIL_ADDRESS>and  David Yokum  The Lab @ DC
Government of the
District of Columbia1350 Pennsylvania Avenue NWWashingtonD.C.
<EMAIL_ADDRESS>
(2018)

###### Abstract.

Data scientists frequently train predictive models on administrative data.
However, the process that generates this data can bias predictive models,
making it important to test models against their intended use. We provide a
field assessment framework that we use to validate a model predicting rat
infestations in Washington, D.C. The model was developed with data from the
city’s 311 service request system. Although the model performs well against
new 311 data, we find that it does not perform well when predicting the
outcomes of inspections in our field assessment. We recommend that data
scientists expand the use of field assessments to test their models.

government, public health, machine learning, applied, data science, urban,
rats

††copyright: rightsretained††doi: 10.475/123_4††isbn:
123-4567-24-567/08/06††conference: KDD Conference; August 2018; London,
England††journalyear: 2018

##  1\. Introduction

Recently, there has been mounting criticism of the use of predictive analytics
in government actions. Much of this is concerned with the notion of fairness
(Chouldechova, 2016; Hardt et al., 2016; Kleinberg et al., 2016) with a
special focus on inequities in data collection (Buolamwini and Gebru, 2018;
Garvie et al., 2016; Jackson and Marx, 2016; Larson et al., 2017). In
particular, this research has shown that biased misclassifications can have
negative consequences for historically disadvantaged groups when the outcomes
being predicted vary with protected classes, like race or gender. This
research has typically dealt with criminal justice, education, and employment.
In this paper, we deal with similar issues, but in a whole new area of
government action: finding rats.

It may not be possible to extrapolate from even a well-performing model
trained on administrative data. The process that generates administrative data
is usually designed to fill a specific purpose, different from the purpose for
which the data scientist is using it. For this reason, it is important to
validate predictive models against their intended use. To do this, we
developed a framework for testing predictive models in the field and used it
to assess the performance of a model predicting rat infestations in the
District of Columbia.

After describing our framework, we present the results of two validations of a
model predicting rat infestations. We trained the model with data from D.C.’s
311 service request system, which residents can use to request city services,
including rodent abatement. We validated the model against both newly-received
311 requests and the results of a field assessment of 100 inspections. We show
that, while our model performs well when predicting the outcomes of new 311
requests, it does not predict the outcomes of inspections well in our field
assessment. We encourage applied data scientists to expand the use of field
assessments to validate models, especially for government actions.

##  2\. Background

Rats are a persistent health risk. Rats carry a wide range of diseases that
infect humans (Firth et al., 2014). Sudden increases in cases of Leptospirosis
following recent hurricanes in Puerto Rico and Dominica highlight the impact
that rats can have on public health (Nedelman, 2017; Dominica News Online,
2015). Rats are especially problematic in urban areas, where large, dense
human populations produce waste and contribute to conditions that support rat
colonies (Himsworth et al., 2014b).

In D.C., requests for rodent abatement via the city’s 311 system more than
doubled from 2,123 in fiscal year 2015 to 5,015 in fiscal year 2017 (Figure
1). When the city receives a 311 request for rodent abatement, the city’s
rodent control team is deployed to inspect the location.111Details of rodent
control’s workflow are based on conversations with the program staff and a
ride-along with city pest controllers. Inspectors generally inspect the entire
block where a request was made, searching for burrows that rats use for
shelter. If inspectors find rat burrows, they treat the burrows with
rodenticide and close the burrows with dirt. Not all requests for rodent
abatement lead inspectors to rat burrows. Less than half (46%) of inspections
from August 2015 to August 2017 resulted in rodent control finding rat
burrows.

Figure 1. Number of 311 requests for rodent abatement by month from Oct. 2014
to Sep. 2017. Requests more than doubled from FY2015 to FY2017.

While the 311 system provides rich crowd-sourced data about rodent activity,
it is not necessarily an accurate or complete picture of where rodents are in
the city. Even when residents witness rodent activity, they may not notify
311. Residents may not know about the city’s rodent abatement services, the
opportunity cost of contacting 311 may be too high, or they may not believe
311 is an effective way to address the issue. Research has shown that a
person’s propensity to use 311 at all varies with demographic characteristics
that could lead already vulnerable populations to remain underserved (Lerman
and Weaver, 2014; Levine and Gershenson, 2014). In Washington, D. C., requests
for rodent abatement vary geographically (Figure 2). Requests are most common
in the city’s densely-populated inner wards, like Ward 1 and Ward 2, and least
common in its less dense outer wards, including Wards 3, 7, and 8.

Figure 2. Requests for rodent abatement in the District of Columbia from FY15
through FY17.

In order to identify locations where rat infestations may go unreported, we
developed a predictive model to extrapolate from incoming 311 data to the
environmental factors that contribute to rodent infestations. Our goal was to
identify locations in the city where rodent infestations are likely, so that
they can be abated regardless of whether or not a 311 request is received. By
addressing rodent infestations that go unreported, we hoped to contain and
reduce the citywide rodent population and improve the health and safety of
District residents.

##  3\. Field Assessment Framework

Because we rely on administrative data from D.C.’s 311 system, we were
concerned that differences in people’s propensity to utilize 311 may bias our
model. To ensure that our model performed well in the field, we developed a
framework for assessing it.

First, we selected a number N of locations to inspect based on the model’s
predictions. We selected N = 100, because this was the number of additional
inspections the rodent control team said they could add to their
workload.222The rodent control team committed four code enforcement inspectors
to complete the field assessment. Code enforcement inspectors are trained to
inspect for rodent activity. However, unlike the inspectors who respond to 311
requests, they are not certified pest controllers. Over the course of the
field assessment, rodent control received 311 requests for 25 of the 100
locations selected for the field assessment. Both code enforcement inspectors
and certified pest controllers inspected these locations separately. Pest
controllers found rat burrows at 16 of the locations, code enforcement found
burrows at 14, and the two teams founds burrows at 11 locations in common. We
then selected a range of the distribution of predicted probabilities to test.
It was important to have a wide enough range to allow for variation in
outcomes across the model’s predictions. However, we also wanted to minimize
the number of inspections where rodent control did not find rat burrows. For
these reasons, we chose to draw our sample from a range of predicted
probabilities from 0.5 to 0.9.333We used a Bayesian test to see if
$\mathrm{Pr}(p_{1}<p_{2})>0.95$ where the priors on $p_{1}$ and $p_{2}$ are
both $\mathrm{Unif}[0,1]=\mathrm{Beta}(1,1)$. Given these priors, if we see
$s$ successes and $N-s$ failures in our data $X_{1}$, then the posterior
$p_{1}|X_{1}\sim\mathrm{Beta}(s+1,N-s+1)$. To test if we had enough power, we
modeled $p^{\mathrm{true}}_{1i}\sim\mathrm{Unif}[0.5,0.6]$ and
$p^{\mathrm{true}}_{2j}\sim\mathrm{Unif}[0.8,0.9]$ for $i,j\in[1..N]$ and
wrote $X_{k}=\sum_{i}\mathrm{Bern}(p_{ki})$. We then computed
$\mathbb{E}_{X_{1},X_{2}}\left[\mathrm{Pr}(p_{1}|X_{1}<p_{2}|X_{2})>0.95\right]$.
This value was approximately $0.78$ for $N=25$.

We compared the outcomes of the field asessment to predicted probabilities by
mapping them on decile plots, where the $x$-axis represents the decile of the
predicted probabilities and the $y$-axis represents the percentage of
locations in that decile where rat infestations were found. We compared this
to a similar mapping of the outcomes of new 311 requests for rodent abatement
made over the same period.

##  4\. Model Development

Past models have predicted 311 requests for rodent abatement, relying
primarily on features derived from the history of 311 requests in an area
(Thornton, 2013). Such an approach risks doubling down on people’s uneven
propensity to request city services through 311. Instead of predicting the
locations of 311 requests, we predict whether or not rat burrows will be found
when responding to 311. Rather than use features based on 311 history to
predict whether rat burrows will be found, we draw from research in urban
rodentology and conversations with the city’s rodent control team to develop
features based on the environmental factors that contribute to rodent
infestations. It was our hope that this approach to modeling rodent
infestations would avoid biasing our results towards areas of the city where
more city services are requested.

###  4.1. Outcome

Our model predicts the likelihood that inspectors will find rat burrows
responding to a 311 request for rodent abatement on a Census block over three
months.444We tested whether the model performed better predicting outcomes
over 1, 3, or 6 months during cross-validation. We found that the model always
performs better predicting outcomes over 3 months. We identified the outcomes
of inspections by using keywords and phrases to recode inspectors’ service
notes. If an inspector said they found rat burrows on an inspection, then the
inspection was assigned 1, otherwise it was 0. After coding the outcomes of
inspections, we aggregated the results to the Census block. If a rat burrow
was found responding to at least one 311 request over three months, the block
was assigned 1, otherwise 0.

We chose Census blocks as our unit of analysis because rats have very limited
home ranges, and tend not to cross natural or human-made barriers, such as
rivers or roads (Traweger and Slotta-Bachmayr, 2005). Research has found that
even blocks with large infestations may have neighboring blocks with no
evidence of rat activity (Himsworth et al., 2014a). Since Census blocks tend
to correspond to these same natural and human-made barriers, they are a
convenient unit of analysis. Census blocks also correspond to the area that
rodent control inspects when they respond to 311 requests.

Restricting our model to predicting the outcomes of 311 requests yields a more
balanced sample than predicting the locations of 311 requests, more generally.
Requests for rodent abatement only come from about 3% of the city’s Census
blocks each month, and less than 30% of Census blocks from August 2015 to
August 2017, the data we use to train the model. In contrast, rodent control
finds rat burrows at about 46% of Census blocks where requests are made.

###  4.2. Feature Selection

We intentionally exclude features derived from 311 requests to avoid biasing
our model towards locations where residents may have a higher propensity to
request city services. Instead, we limit our features to aspects of the urban
environment that contribute to rodent infestation but are unlikely to be
correlated with residents’ propensity to use 311.555All features were
aggregated to the Census block. For a full list of features, see the model
code. Our decisions about what data to include in the model were guided by
research on urban rodentology and conversations with experts, including the
city’s rodent control team.666We also consulted extensively with Robert
Corrigan, a noted expert on urban rodentology.

Like all mammals, rats prefer environments based on the availability of food,
water, and shelter (Feng and Himsworth, 2014). Poorly-secured food waste is
among the most common sources of food for urban rats (Promkerd et al., 2008;
Traweger et al., 2006). Denser human populations, the number of residential
buildings, and the number of units per building all contribute to trash
production and thereby food sources for rodents (Himsworth et al., 2014b). A
mix of residential and commercial zoning can also contribute to rat
infestations by increasing trash production, especially when there are nearby
restaurants and other food vendors (Himsworth et al., 2012).

Norwegian rats, which are the most common in North American cities, like D.C.,
burrow underground for shelter. Areas with soft earth or deteriorating
infrastructure create opportunities for shelter (de Masi et al., 2009;
Himsworth et al., 2014b), so we included features measuring impervious surface
area and alleyway conditions. Older buildings in poor condition also
contribute to rat harborage (Himsworth et al., 2014b), so we included data on
building age and condition. Rats will take shelter in sewers, so we included
data on the locations of sewer grates. We also included data about the
locations of parks and community gardens where rats may find both food and
shelter.

Table 1. Best Performing Model by Model Type Model | Mean | Min | Max | Mean
---|---|---|---|---
Type | P@N | P@N | P@N | ROC-AUC
Logistic | 0.56 | 0.50 | 0.64 | 0.60
Regression |  |  |  |
Gradient | 0.68 | 0.61 | 0.72 | 0.67
Boosting |  |  |  |
Random | 0.75 | 0.68 | 0.82 | 0.70
Forests |  |  |  |

Figure 3. Decile plots from the last four months of the cross-validation show
that the model is well-calibrated to the outcomes of 311 requests. In each
subplot, the $x$-axis represents decile bins of the predicted probabilities
for each location. The $y$-axis represents the percentage of locations where
rat burrows were found. The red line represents the percentage of inspections
in each decile bin that rat burrows were found. A perfect relationship between
the two axes is represented by the dotted black line. If the model was well-
calibrated to the predicted probabilities, we would expect the red line and
dotted line to coincide.

A wide range of other factors are believed to contribute to rodent
infestations. One factor that is commonly cited is construction. We included
the locations of recent construction permits in the model. We also included
dummies for month and year in order to capture potential seasonal effects.
While there is clear seasonal variation in the number of 311 requests for
rodent abatement (see Figure 1), research in other cities suggests that rodent
populations do not vary significantly seasonally (de Masi et al., 2009;
Himsworth et al., 2014a).

###  4.3. Model Selection

We selected our model type and tuned hyperparameters using temporal cross-
validation. Our models were trained with data from August 2015 to August 2017.
For each month from August 2016 to August 2017, we trained a model on data
from all prior months and tested against the outcomes of inspections over the
following three months. We used precision at N (P@N) to evaluate the models,
where N = 100. We chose P@N to fit the goal of our model: to maximize the
probability that rodent control would find rat infestations at the model’s top
N targets each month. We chose N = 100 to fit the rodent control team’s
capacity for additional inspections each month.

We chose temporal cross-validation for three reasons. First, it replicates our
goal of predicting the location of rat infestations in the coming months.
Second, we wanted our model to perform well year-round, not only in seasons
when requests for rodent abatement are high. Third, since we are always trying
to predict the outcomes of inspections on the same set of Census blocks, and
the environmental factors in our model features do not change rapidly,
autocorrelation is not a particular concern.777Most of our features are
elements of the urban environment that change fairly slowly or not at all. We
include proxies for seasons, but the percentage of inspections where rat
burrows are found does not vary much seasonally.

We tested random forest, gradient boosting, and logistic regression models. We
found that a random forest classifier performed best (see Table 1). On
average, inspectors found rat burrows at about 74 of the 100 locations
identified as most likely to have rat infestations by our chosen model, a more
than 60% improvement over the baseline. The model is also well-calibrated.
Figure 3 presents decile plots from the last four months of the temporal
cross-validation. In each case, the percentage of locations where rat burrows
were found fits closely to the predicted probabilities generated from the
model.

##  5\. Results

After model selection, we trained the best-performing model on all Census
blocks with a 311 request between August 2015 and September 2017 and generated
predictions for every Census block for the following three months from October
to December. We then selected 100 locations for inspection following the field
assessment framework outlined above. From October 10 to November 20, 2017,
rodent control completed all 100 inspections and collected data on whether or
not they found rat burrows via an online form. We compared the outcomes of
these 100 inspections to the model’s predictions to assess the its
performance. We also compared our model’s predictions to the outcomes of 311
requests for rodent abatement over the same period.

###  5.1. Field Assessment

Figure 4. Decile plot comparing predicted probabilities to field assessment
outcomes. See Figure 3. The red line does not coincide with the dotted line,
showing that the model is poorly-calibrated to the outcomes of the field
assessment.

Figure 4 is a decile plot comparing the outcomes of the 100 inspections in our
field assessment to the predicted probabilities generated by the
model.888Running the model 100 times with different random seeds, showed that
predicted probabilities were nearly perfectly correlated. The model’s
predictions do not fit the outcomes of our field assessment.999We also not
that the AUC was 0.5, which means it was no better than random on the subset
of locations inspected. However, because we restricted our field assessment to
locations with a high probability of finding rat burrows, we do not believe
that AUC is a useful metric. The percentage of locations where rat burrows
were found varies little across the decile bins. Inspectors found rat burrows
at 48% of the locations inspected in the field assessment, ranging from 48% in
decile (0.5, 0.6] to 46% in decile (0.8, 0.9].

###  5.2. Validation against New 311 Requests

Figure 5. Decile plot comparing predicted probabilities to 311 request
outcomes from Oct. 10 to Nov. 20, 2017. See Figure 3. The red line fits
closely to the dotted line, showing that the model is well-calibrated to new
311 requests.

Figure 6. Distribution of errors by ward in D.C. Errors on the x-axis are the
difference between the outcome and the predicted probability for each block in
the ward. The y-axis is the number of blocks with an error in that range.
Errors are positive where rat burrows were found and negative where rat
burrows were not found. Errors below -0.5 indicate False Positives, where as
errors above 0.5 indicate False negatives (red). Errors between -0.5 and 0.0
indicated True Negatives, while errors between 0.0 and 0.5 indicated True
Positives (blue).

Figure 5 is a decile plot comparing the model’s predictions to the outcomes of
311 requests during the field assessment from October 10 to November 20, 2017.
The model was fairly accurate, with AUC at 0.71, and well-calibrated. Pest
controllers responding to 311 requests were more likely to find rat burrows in
locations with higher predicted probabilities. Rat burrows were found in 56%
of locations in decile (0.5, 0.6] and 74% of locations in decile (0.8, 0.9].

Figure 6 shows the distribution of model errors by ward in D.C. when compared
to the outcomes of 311 requests from October 10 to November 20. The skew of
the errors varies by ward, indicating that errors vary geographically. Wards 1
and 2 are densely-populated and located in the downtown center of the city. In
both these wards, errors tend to be left-skewed, indicating that the model
under-estimated the probability of finding rat burrows in those areas, even
though the predicted probabilities in those areas were relatively high. In
contrast, errors are right-skewed in Wards 7 and 8, southeast of the Anacostia
River, where it is less common to receive reports of rats or find rat burrows.
However, when the model is incorrect (i.e., returns a false positive or false
negative), errors tend to be slightly left-skewed in those wards. In other
words, while the model correctly predicts that rat infestations are uncommon
in Wards 7 and 8, where the model is wrong it tends to under-predict the
likelihood of finding rat burrows.

##  6\. Discussion

Our results highlight the importance of field testing predictive models. Our
model is accurate and well-calibrated when compared to new 311 requests, but
the model does not perform well predicting the outcomes of our field
assessment. This illustrates the pitfalls of relying on administrative data to
both train and validate predictive models. If the data is biased by the
process that generates it, then models trained on that data may not be useful
in the field. When applied to government actions, biased models can lead to
inequitable service delivery.

We presented a field assessment framework to help address the problems
introduced by training models on biased administrative data. To date, there
has been little work done to develop similar methods for validating models in
the field. We encourage more data scientists to test their models using field
assessments, and hope that our methods can serve as a model for such
validations.

###### Acknowledgements.

The authors would like to thank Mayor Muriel Bowser and her administration,
especially City Administrator Rashad Young, Director Jennifer Reed, Director
LaQuandra Nesbitt, and interim Chief Technology Officer Barney Krucoff. Also
special thanks to Gerard Brown, Sharon Lewis, and the Rodent and Vector
Control Division at the District of Columbia’s Department of Health for their
hard work and expertise, without which this project would not have been
possible. Thanks also to Michael Bentivegna, Sam Quinney, Jennifer Doleac,
Ryan Moore, Katherine Gan, the Data Team at the Office of the Chief Technology
Officer, and The Lab @ DC for their feedback and contributions to the
development of this study. Finally, thanks to Robert Corrigan and Donald Green
for their expert advice on this project.

## References

  * (1)
  * Buolamwini and Gebru (2018) Joy Buolamwini and Timnit Gebru. 2018.  Gender shades: Intersectional accuracy disparities in commercial gender classification.  _Proceedings of Machine Learning Research_ 81 (2018), 1–15.
  * Chouldechova (2016) Alexandra Chouldechova. 2016\.  Fair prediction with disparate impact: A study of bias in recidivism prediction instruments.  (2016).  arXiv:1610.07524.
  * de Masi et al. (2009) Eduardo de Masi, Pedro Vilaça, and Maria Tereza Pepe Razzolini. 2009\.  Environmental conditions and rodent infestation in Campo Limpo district, Sao Paulo municipality, Brazil.  _International Journal of Environmental Health Research_ 19, 1 (2009), 1–16.
  * Dominica News Online (2015) Dominica News Online. 2015\.  Warning in light of suspected cases of Leptospirosis.  (2015).
  * Feng and Himsworth (2014) Alice Y.T. Feng and Chelsea G Himsworth. 2014.  The secret life of the city rat: a review of the ecology of urban Norway and black rats (Rattus norvegicus and Rattus rattus).  _Urban Ecosystems_ 17, 1 (2014), 149–162.
  * Firth et al. (2014) Cadhla Firth, Meera Bhat, Matthew A. Firth, Simon H. Williams, Matthew J. Frye, Peter Simmonds, Juliette M. Conte, James Ng, Joel Garcia, Nishit P. Bhuva, et al. 2014\.  Detection of zoonotic pathogens and characterization of novel viruses carried by commensal _Rattus norvegicus_ in New York City.  _MBio_ 5, 5 (2014).
  * Garvie et al. (2016) Clare Garvie, Alvaro Bedoya, and Jonathan Frankle. 2016\.  The perpetual line-up.  _Georgetown Law Center on Privacy & Technology_ (2016).
  * Hardt et al. (2016) Moritz Hardt, Eric Price, and Nathan Srebro. 2016.  Equality of Opportunity in Supervised Learning.  (2016).  arXiv:1610.02413.
  * Himsworth et al. (2012) Chelsea Himsworth, Alice Feng, Kirbee Parsons, Thomas Kerr, and David Patrick. 2012.  Using experiential knowledge to understand urban rat ecology: A survey of Canadian pest control professionals.  _Urban Ecosystems_ 16 (2012).
  * Himsworth et al. (2014a) Chelsea Himsworth, Claire Jardine, Kirbee Parsons, Alice Feng, and David Patrick. 2014a.  The characteristics of wild rat (Rattus spp.) populations from an inner-city neighborhood with a focus on factors critical to the understanding of rat-associated zoonoses.  _PLOS One_ 9 (2014).
  * Himsworth et al. (2014b) Chelsea Himsworth, Kirbee Parsons, Alice Feng, Thomas Kerr, Claire Jardine, and David Patrick. 2014b.  A mixed methods approach to exploring the relationship between Norway rat (Rattus norvegicus) abundance and features of the urban environment in an inner-city neighborhood of Vancouver, Canada.  _PlOS One_ 9 (2014).
  * Jackson and Marx (2016) David Jackson and Gary Marx. 2016.  Data mining program designed to predict child abuse proves unreliable, DCFS says.  _Chicago Tribune_ (2016).
  * Kleinberg et al. (2016) Jon M. Kleinberg, Sendhil Mullainathan, and Manish Raghavan. 2016\.  Inherent Trade-Offs in the Fair Determination of Risk Scores.  (2016).  arXiv:1609.05807.
  * Larson et al. (2017) Jeff Larson, Mattu Surya, Lauren Kirchner, and Julia Angwin. 2017\.  How we analyzed the COMPAS recidivism algorithm.  _ProPublica_ (2017).
  * Lerman and Weaver (2014) Amy E. Lerman and Vesla Weaver. 2014.  Staying out of sight? Concentrated policing and local political action.  _The Annals of the American Academy of Political and Social Science_ 651, 1 (2014), 202–219.
  * Levine and Gershenson (2014) Jeremy R. Levine and Carl Gershenson. 2014.  From political to material inequality: Race, immigration, and requests for public goods.  _Sociological Forum_ 29, 3 (2014), 607–627.
  * Nedelman (2017) Michael Nedelman. 2017\.  Suspected leptospirosis cases increasing in Puerto Rico after Hurricane Maria.  _CNN_ (2017).
  * Promkerd et al. (2008) Prasartthong Promkerd, Yuvaluk Khoprasert, Phongthep Virathavone, Manivone Thoummabouth, Ouane Sirisak, and Thomas Jaekel. 2008\.  Factors explaining the abundance of rodents in the city of Luang Prabang, Lao PDR, as revealed by field and household surveys.  _Integrative Zoology_ 3, 1 (2008), 11–20.
  * Thornton (2013) Sean Thornton. 2013\.  Using predictive analytics to combat rodents in Chicago.  _Data-Smart City Solutions_ (2013).
  * Traweger and Slotta-Bachmayr (2005) Doris Traweger and Leopold Slotta-Bachmayr. 2005.  Introducing GIS-modelling into the management of a brown rat (Rattus norvegicus Berk.) (Mamm. Rodentia Muridae) population in an urban habitat.  _Journal of Pest Science_ 78, 1 (2005), 17–24.
  * Traweger et al. (2006) Doris Traweger, Rita Travnitzky, Cornelia Moser, Christian Walzer, and Guenther Bernatzky. 2006\.  Habitat preferences and distribution of the brown rat (Rattus norvegicus Berk.) in the city of Salzburg, Austria: Implications for an urban rat management.  _Journal of Pest Science_ 79, 3 (2006), 113–125.


# Local Optical Probe of Motion and Stress in a multilayer graphene NEMS

Antoine Reserbat-Plantey  Laëtitia Marty  Olivier Arcizet  Nedjma Bendiab
Vincent Bouchiat  Institut Néel, CNRS et Université Joseph Fourier, BP 166,
F-38042 Grenoble Cedex 9, France

Nanoelectromechanical systems (NEMSs)reviewNEMS are emerging nanoscale
elements at the crossroads between mechanics, optics and electronics, with
significant potential for actuation and sensing applications. The reduction of
dimensions compared to their micronic counterparts brings new effects
including sensitivity to very low massbachtoldbalance ; zettlbalance ,
resonant frequencies in the radiofrequency range zettlNTRF , mechanical non-
linearitiesBachtoldNonlinear and observation of quantum mechanical
effectsquantumNEMS . An important issue of NEMS is the understanding of
fundamental physical properties conditioning dissipation mechanisms, known to
limit mechanical quality factors and to induce aging due to material
degradation. There is a need for detection methods tailored for these systems
which allow probing motion and stress at the nanometer scale. Here, we show a
non-invasive local optical probe for the quantitative measurement of motion
and stress within a multilayer graphene NEMS provided by a combination of
Fizeau interferences, Raman spectroscopy and electrostatically actuated
mirror. Interferometry provides a calibrated measurement of the motion,
resulting from an actuation ranging from a quasi-static load up to the
mechanical resonance while Raman spectroscopy allows a purely spectral
detection of mechanical resonance at the nanoscale. Such spectroscopic
detection reveals the coupling between a strained nano-resonator and the
energy of an inelastically scattered photon, and thus offers a new approach
for optomechanics.

Graphene’sNovoselov2004p2970 outstanding mechanical HoneAFM2008 , electrical
castroneto2009 and optical Nair2008 properties, make it an ideal material for
flexible, conductive and semi-transparent films. Multilayer graphene (MLG),
which has a thickness of several tens of atomic layers, is sufficiently stiff
Booth2008 to produce free-standing cantilevers, with an unprecedented aspect
ratio. Such structures can be used to make suspended mirrors, with a mass
ranging from tens to hundreds of femtograms. When suspended over silica, such
cantilevers form optical cavities which can be electrostatically actuated and,
are thus ideal for the implementation of NEMSBunch . Previous attempts to
probe local motion of graphene resonatorsBachtoldAFM have reached nanometer
scale but cannot measure directly the stress and remained confined to a
limited range of pressure and temperature. Nevertheless, recent studiesBolotin
based on hybrid graphene-metallic cantilevers has brought promising results on
static stress graphene using optical profilometry. In the present work, we use
Raman spectroscopy to probe the local stress within a MLG cantilever. We
explore mechanical regimes from DC up to MHz frequencies by taking advantage
of the large dynamical range of optical detection. The MLG displacement is
considerably greater than previously reported, and optical interferences allow
self-calibration of displacements, while Raman spectroscopy gives quantitative
analysis of the local stress within the structure.

Figure 1: Fizeau fringes in a MLG cantilever overhanging silicon oxide. a:
typical SEM micrograph of a sample, consisting in two reflectors: an oxidized
silicon back-mirror and a MLG cantilever, having a semi-transparent behavior.
b: Schematic view of the device : electrical excitation and optical detection,
either with a photodiode (intensity) or a Raman spectrometer (intensity and
spectral data). c: White light optical image of a device showing iridescence.
d: Reflectance profile is measured along the dashed-line of the inset. The
reduction in signal strength observed at large distances from the hinge is due
to reduced spatial mode matching. The fringe contrast is however preserved.
Inset: Reflectance confocal (X,Y) scan at 633 nm. The scale length is 5
$\mu$m.

Samples were prepared from micron-sized MLG planar flakes clamped on one side
by a gold film and overhanging silicon oxide (see Methods). Typical samples
had a thickness of approximately 100 monolayers (ca. 30 nm), as verified by
atomic force microscopy (cf Supp. Info.). Their thicknesses were adjusted so
as to prevent collapse whilst maintaining semi-transparency with an optical
reflectance (transmission) coefficient $R$ = 0.22 ($T$ = 0.61) for a 30 nm
thick MLGSkulason . Some flakes tend to stick up after the fabrication process
(see Fig. 1a), at a wedge angle $\alpha\rm{\in[5^{\circ};35^{\circ}]}$, and
these leave a wedge gap of length $h(x,y)$ in the range between 0.3 and 3
microns. The resulting structures form an optical cavity, characterized by a
low optical interference order ($2h/\lambda$¡10, where $\lambda$ is the probe
wavelength), with MLG top mirror of extremely low mass (10-100 fg), and of
high mechanical resonance frequencies (1-100 MHz). With approximately 100
measured samples, we have observed a variety of geometries (Fig. 1), allowing
us to explore various mechanical regimes, with a wide range of wedge angles
$\alpha$, sizes, and shapes. Iridescence is observed under white light
illumination (Fig. 1c), and the interference pattern observed under
monochromatic light (Fig. 1d) has contrasted equal-thickness fringes (so-
called Fizeau fringes, see Supp. Info.). Unlike conventional graphene-based
optical cavities with fixed geometriesLing2010 , the optical length of the
cavity increases linearly along the cantilever, which allows the observation
of multiple interference fringes (cf. Fig. 1cd). Interference patterns are
observed both from the reflection of the pump laser and from Raman scattered
light (see Supp. Info.), the latter having the considerable advantage of
carrying local informations related to the material (stress, doping, defects,
temperature).

Furthermore, the optical length of the cavity can be adjusted through
electrostatic actuation of the cantilever, thus producing rigid shift of the
interference fringes pattern (see video in Supp. Info.). This is achieved by
applying a DC or AC voltage $V$ (typically up to $30$ Volts) to the clamp
electrode (Fig. 1b) while the SiO2 capped silicon substrate is grounded Bunch
. This results in an attractive electrostatic force $F$, which produces
reduction of the cavity length with respect to the equilibrium position
$h_{0}$ in absence of driving. We measure the response of a harmonic drive,
which create of force quadratic in voltage $F(2\omega)\propto V(\omega)^{2}$
through the local light intensity variation, $\Delta I(x,y,2\omega,h_{0})$:

$\Delta
I(x,y,2\omega,h_{0})\propto\chi_{mec}(x,y,2\omega)\chi_{opt}(x,y,h_{0})\
V(\omega)^{2},$ (1)

where $\chi_{mec}$ is the mechanical susceptibility (see Supp. Info.) and
$\chi_{opt}$ is the optical susceptibility defined as
$\chi_{opt}(x,y,h_{0})=\partial g_{opt}/\partial h$, where $g_{opt}$ is a
periodic interferometric function of $h(x,y)$ defined as the normalized
reflected light ($I_{r}$) or Raman scattered light ($I_{G}$)
$I_{r,G}/I_{0}=g_{opt}(h)$ (see Fig. 1d).

Figure 2: Quasi static actuation and stress mapping of wedged MLG NEMS. a:
Variations in G peak intensity (black) and position (red) under MLG actuation,
revealing peak softening. The lower dashed line represents the drive voltage.
b: Map of G peak produced by confocal (X,Z) scan mapping of the cantilever
cross section. Inset: G peak position along the cantilever (purple, and along
the same sample following collapse of the cantilever onto the silica substrate
(green). Black marks indicate the hinge position. The scale bars represent a
length of 5 $\rm{\mu}$m.

The quadratic dependence of $\Delta I$ upon voltage is systematically
observed, both for reflected light (cf. Fig. S5) and for the MLG Raman lines
(cf. Fig. 2a). Since $g_{opt}$ is $\lambda/2$ periodic, a precise calibration
of the low frequency motion response under electrostatic actuation can be
obtained and is found to be of the order of 20 nm.V-2. (cf. Supp. Info.).
Interestingly, energy of the stress-sensitive optical phonon (so called Raman
G peak) also follows quadratic behavior. The G peak Raman shift is indeed
synchronized with the interferential response $I_{G}(t)$ (Fig. 2a), and
exhibits softening of approximately 1.9 cm-1 at the maximum cantilever
deflection. This Raman peak softening cannot be interpreted as a doping effect
since the doping level necessary to induce the observed Raman shifts would
correspond to surface charge incompatible with the one induced by the gate
driveKim . Moreover, the doping induced during AC gating would directly follow
gate variation and therefore be $\omega$ periodic which is in disagreement
with the observed $2\omega$ periodic Raman shift (cf. Fig. 2a). This Raman
peak softening is interpreted as a stress/strain effect and, by analogy with
strained graphene measurementsOtakar2010 ; Otakar2011 ; Hone2009 , it is thus
possible to extract a corresponding strain value of $0.06\%$ at maximum
deviation resulting from a quasi static stress of 600 MPa. For such low
strain, the G band splitting is not resolved. Besides, the stress exerted at
the hinge scales like $LF/[2t^{2}\rm{sin}(2\alpha)]$ where $F$, $L$ and $t$
are the electrostatic force, the cantilever length and thickness, respectively
(see Supp. Info.). For our large aspect ratio structures ($L/t\gg 1$), the
local stress can be very intense and reaches hundreds of MPa for electrostatic
forces estimated here, which are about 25 nN.$\mu$m-1. This value is in
agreement with the quasi-static stress of 600 MPa deduced above.

Nevertheless, the MLG Raman signature depends on the position along the flake.
A micro-Raman confocal (X,Z) scan (Fig. 2b) reveals a linear increase in the
position of the G peak along the cantilever axis, from the free-end of the
cantilever to the hinge, which is not observed when the MLG is collapsed
(inset of Fig. 2b). This linear shift could be interpreted as a continuously
increasing electrostatic field effectKim ; Ferrari , due to charge within the
substrate which also influences the position of the Raman G peakBerciaud2009 .
However, in this experiment, the G peak shows local hardening around the hinge
position, in the suspended case, whereas local softening is observed at the
same location, after collapse. Indeed, uniaxial strain in MLG also induces
symmetry breaking of the Raman G peak, leading to a mode splitting and, each
component (G+, G-) softens or hardens under tensile or compressive strain,
respectivelyOtakar2010 ; HoneAFM2008 . That stress induced Raman shift is
characterized by an average mode shift rate aboutOtakar2011 -3.2 cm-1.GPa-1.
This outcome is in agreement with a maximum compressive strain at the hinge in
the suspended case and, a transition toward a tensile strain when it
collapses. By converting these Raman shifts into stress at the hinge, this
gives an equivalent built-in stress of 300 MPa.

Figure 3: Detection of mechanical resonance by Fizeau interferometry. a:
Amplitude at $2\omega$ versus drive frequency for different rf drive voltages
showing the non-linear behavior of the fundamental and the two first harmonic
mechanical modes. Laser probe is focused close to the hinge. Schematics of the
deformed shape are associated to each mode and dark dots represent the laser
probe position. b: Amplitude at $2\omega$ versus drive frequency, for an
increasing range of drive voltages (bottom to top), revealing signal folding
due to optical interferences. The laser spot is located close to the free end
of the cantilever. c: Evolution of MLG cantilever resonance frequency
$\omega_{0}$ (red) and its associated dissipation $Q^{-1}$(blue) as a function
of the temperature of the optical cryostat. Measurements a-c are performed
under vacuum and the laser spot is positioned at the edge of a fringe
($\chi_{opt}$ is optimal).

Interestingly, changing both the laser spot position and the drive amplitude
allows probing in a separated fashion the non-linearities arising from
mechanical (Fig. 3a) and from optical (Fig. 3b) origins. It is worth noting
that optical non-linearities are observed when the probe is far from the hinge
(see Fig. 3b) where, due to the lever-arm effect, oscillation amplitude of
$h(x,y)$ becomes comparable to the probe wavelength. Like in quasi-static
regime, it is possible to calibrate the displacement amplitude with respect to
the driving excitation $\delta V_{AC}$ by using the periodic nature of
$\chi_{opt}$. Peak folding in indeed observed for increasing drive beyond 3V
(cf. Fig. 3b). Assuming mechanical linear response, the drive increase to
produce two successive foldings (corresponding to $\lambda/4$ in amplitude)
provides calibration of the drive efficiency, which equals 150 nm.V-2 in the
case presented in Fig. 3b. Interestingly, the entire signature of the optical
non-linearities is visible for a restricted range of drive voltage which
ensures to neglect mechanical non-linearities.

Close to the hinge, optical non-linearities are extinguished due to smaller
variations of $h$ and thus reveal non-linearities of mechanical origin, which
are observed on each mode for drive voltages higher than 4V (cf. Fig. 3a).
This measurement highlights the wide range of mechanical non-linearities
observed in MLG structuresLifshitz ; BachtoldNonlinear ; Landau , and it is
worth noting that the detection efficiency strongly depends on the mode
profile since it is based on Fizeau fringes pattern modulation.
$\chi_{mac}(x,y,2\omega)$ can exhibit important variations due to the spatial
nature of the probed vibration. In particular, $\chi_{mec}(x,y,2\omega)$ can
be strongly reduced when the laser probe is focused at a node of the
mechanical resonance. As an example, the first harmonic ($\omega_{1}$), found
to be a torsional mode via finite elements analysis, implies cantilever
deformation with singular position where the cavity length does not vary
(typically, a node region ($x_{n},y_{n}$)). Thus, according to Eq. 1,
$\chi_{mec}(x_{n},y_{n},2\omega_{1})\ll\chi_{mec}(x_{n},y_{n},2\omega_{0})$,
whereas focusing the laser at a different position allows to enhance the local
optical response. This particular extinction feature of the detection has a
great potential for further mapping of MLG deformation associated with a
single mechanical mode.

In order to investigate the influence of the laser probe in our all-optical
method, cryogenic measurements has been carried out as shown in Fig. 3c. The
fundamental resonant frequency exhibits a linear upshift upon cooling from
300K to 70K, below which it saturates due to extrinsic heating (see Supp.
Info.). In contrast to doubly clamped graphene-based NEMSBunch ;
BachtoldNonlinear ; Mandar , it is not possible to discuss the frequency
hardening observed in Fig. 3c in terms of cantilever tensioning induced by
differential thermal expansion since we study a simply clamped geometry. An
important feature of any resonator is the measurement of the quality factor,
defined as $Q=\omega/\Delta\omega$, which characterizes the high sensitivity
(high $Q$) of the resonator to its environment. A linear decrease of the
dissipation, $Q^{-1}$, is observed upon cooling to 70K. Both effects,
frequency hardening and decrease of the dissipation, are possibly a
consequence of the stiffening of the clamp electrode. Further measurements
will allow to investigate both extrinsic effects (clamp stiffening losses) and
mechanical intrinsic properties of MLG which should bring new insights to
understand damping mechanisms in NEMS. Effective substrate temperature is
obtained by measuring Stokes and Anti-Stokes Raman intensities ratio (Supp.
Info.) and indicates a temperature threshold of 70K. Interestingly, all the
physical quantities (resonant frequencies, quality factor $Q$, Raman shift)
are sensitive to the environmental temperature until 70K. This demonstrates
experimentally that room temperature experiments discussed in this letter are
not altered by laser heating. Concerning absorption of mechanical energy at
resonance, we have seen no change in the Raman Stokes/Anti-Stokes measurements
when sweeping the excitation frequency through the mechanical resonance,
indicating no increase of phonon bath temperature.

Figure 4: Detection of mechanical resonance and dynamic stress using Raman
spectroscopy. a: Raman spectra of the G peak under MLG actuation at mechanical
resonance frequency (red) and off resonance (detuned by 360 kHz) (blue). For
the resonating case, signal to noise ratio is smaller than off-resonance case
due to larger oscillation amplitude at resonance which takes the resonator out
of focus. b-c: Lock-in amplitude at $2\omega$ (dark line) as a function of
drive frequency, for a 5V rf drive voltage under vacuum (b) and in air (c).
The position of the Raman G peak (green) shows a softening which coincides
with the mechanical resonance. This softening is even more marked under
vacuum. Blue and red circles, shown in caption b, correspond to the Raman
spectra plotted in Fig. 3a. This sample is the same as presented in Fig. 3a.

To demonstrate the spectral detection of mechanical resonance, Raman response
of MLG cantilever under vacuum is plotted (Fig. 4a) at fundamental mechanical
resonance $\omega_{0}$ = 1.2 MHz (red curve) and off resonance (blue curve).
At $\omega_{0}$, G peak softening is -5 cm-1 in position and about +10 cm-1 in
width (peak’s FWHM) (Fig. 4a) which takes into account the averaging induced
broadening (see Supp. Info.). This Raman softening estimated at -1 cm-1.V-2,
is attributed to corresponding variation in stress within the cantilever,
induced at mechanical resonance according to universal stress behavior in sp2
carbon materialsOtakar2011 ; Hone2009 ; Ferrari2009 (shift rate: 0.003
cm-1.MPa-1). This dynamical stress is thus about 1.6 GPa, and therefore
provides a quantitative means of detecting NEMS resonances stress effects. It
is worth noting that the measured stress in MLG cantilever at mechanical
resonance is more than one order of magnitude higher than previously
reportedPomeroy in silicon-based MEMS devices.

In Figs. 4b-c, we have detected the fundamental mechanical resonance of this
MLG cantilever using both reflected and Raman scattered light under different
experimental conditions. As the lifetime of optical phonons is much shorter (1
ps) than $\omega_{0}^{-1}$ ($\rm{\sim\ 100\ ns}$), the Raman scattering
process provides instantaneous information related to stress in the vibrating
cantilever. For each excitation frequency, we record a Raman spectrum (1s
averaging), which reflects stress at the cantilever position. For several
samples, we were able to check that this softening behavior (green curve, Fig.
4b-c), observed under mechanical excitation, coincides with the mechanical
resonance width irrespective of the chamber pressure (cf. Fig. 4b-c).

In contrast to the vacuum case (Fig. 4b), where the quality factor is about
$Q_{vac}$ = 26.1, the same sample in air (Fig. 4c) has a reduced quality
factor ($Q_{air}$ = 2.3) as well as a shifted Raman G peak, which indicates
that the dynamical stress depends on the oscillation amplitude (as also
suggested by the Fig. S9 in Supp. Info.). The value of $Q_{air}$ agrees with
typical viscous damping modelHosaka for that particular geometry, thus
confirming that viscous damping is the predominant mechanism for limiting the
quality factor in air (see Supp. Info.). Nevertheless, this damping mechanism
is no longer the main one under vacuum where the dissipation may be governed
by clamping losses. One can compare the ratio of the drive efficiency at low
frequency (20 nm.V-2) and at resonance (150 nm.V-2) with the ratio of the G
peak shift sensitivity at low frequency (1 cm-1.V-2) and at resonance (5
cm-1.V-2). Both ratios equal to the measured quality factor $Q$ ($\rm{\sim
6}$), as expected for a mechanical resonatorLandau . To demonstrate the
versatility of the Raman-based spectral detection of the mechanical resonance,
we have investigated a similar effect on two other types of NEMS (Si nano
cantilevers and SiC nanowires, see Supp. Info.).

There exists a fundamental coupling between the device position (directly
given by the cavity length $h(x,y)$) and a spectroscopic property (measured by
the Raman peak shift). In our case, the flake displacement generates a
mechanical stress that causes a shift of stress-sensitive Raman peaks. We
therefore extract a coupling constant that is the ratio between the Raman peak
shift (in Hz) and the estimated displacement (in meters). The novel
optomechanical coupling linking the Raman G peak position shift to the
cantilever displacement reaches $\rm{\sim 10^{17}}$ Hz.m-1 in magnitude, which
compares favorably to similar quantities involving other optomechanical
systemsPRL arcizet 2006 . This large optomechanical coupling, in which all the
isotropically scattered Raman photons carry informations on the nano-resonator
dynamics, enables mechanical stress information to be spectrally encoded.
Interestingly, this provides an efficient rejection of background signal even
in a backscattering configuration for on-chip devices. For the detection of
submicron NEMS, this generates in many cases better signal to noise ratio,
compared to diffraction-limited elastic optical detection techniques. Finally,
the resonant nature of Raman scattering in graphene preserves a large
interaction cross-section, allowing the optomechanical coupling to be
maintained even when working with nanosized oscillators, which is not the case
in standard optomechanical approachesPRL arcizet 2006 in which only a small
fraction of the detected photons carries the optomechanical information.

In conclusion, we demonstrate a non invasive and high bandwidth optical probe,
enabling imaging of dynamical stress and motion in a NEMS. This probe,
combining Raman spectroscopy with Fizeau interferometry, is applied to
multilayer graphene NEMS and is found to be compatible with two other types of
NEMSs. Calibrated motion and stress can be measured and mechanical resonances
can be detected through optical mode shifting and mapped as a local stress
along a vibrating cantilever. The reliability of Raman spectroscopy in this
context finds its origins in the large optomechanical coupling between strain
modulation and mechanical displacements. We demonstrate the coupling between
flexural vibrational modes and optical phonons. This localized probe of
material stress is furthermore expected to preserve its large coupling
strength when working with even smaller oscillators. Due to its high
stiffness, semi-transparency behavior and, extremely low mass, MLG emerges as
an ultra-sensitive platform for the simultaneous exploration of the spatial,
temporal and spectral properties of NEMS, this system is thus promising for
the detection of ultralow forces and could be used as carbon-based molecular
sensors. Moreover, this probe allows low temperature measurements, thus paving
the way for stress mapping of other high quality factor resonators and
understanding of dissipation factors in such systems.

###### Acknowledgements.

This work is partially supported by ANR grants (MolNanoSpin, Supergraph,
Allucinan), ERC Advanced Grant No. 226558 and, the Nanosciences Foundation of
Grenoble. Samples were fabricated in the NANOFAB facility of the Néel
Institute. We thank A. Allain, D. Basko, C. Blanc, E. Bonet, O. Bourgeois, E.
Collin, T. Crozes, L. Del-Rey, M. Deshmukh, E. Eyraud, C. Girit, R. Haettel,
C. Hoarau, D. Jeguso, D. Lepoittevin, R. Maurand, J-F. Motte, R. Piquerel, Ph.
Poncharal, V. Reita, A. Siria, C. Thirion, P. Vincent, R. Vincent and W.
Wernsdorfer for help and discussions.
Methods Multilayered Graphene flakes are deposited on 280 nm thick oxidized
silicon wafer by micro-mechanical exfoliationNovoselov2004p2970 of Kish
graphite. Electrical contacts are made using deep UV lithography and e-beam
deposition of 50 nm Au film. Samples are suspended by etching (buffered
hydrogen fluoride at concentration 1:3 HF/NH4F) and drying using CO2 critical
point drying. Experiments have been performed on approximately 100 samples
(see Supp. Info). Micro-Raman spectroscopy was performed with a commercial
Witec Alpha 500 spectrometer setup with a dual axis XY piezo stage in a back-
scattering/reflection configuration. Grating used has 1800 lines/mm which
confer a spectral resolution of 0.01 cm-1 for 10 s integration time. Two laser
excitation wavelengths are used, 633 nm (He-Ne) and 532 nm (Solid state argon
diode). Raman spectra are recorded in air with a Nikon x100 objective
($NA=0.9$) focusing the light on a 320 nm diameter spot (532 nm light) and,
with a Mitutoyo x50 objective (M plan APO NIR) in vacuum. All measurements
made under vacuum (Fig. 3, Fig. 4a-b) are under active pumping at residual
pressure equals to 10-6 bar. For Raman (reflectance) measurements laser power
is kept below 1mW.$\mu$m-2 (1 $\mu$W.$\mu$m-2). For rf measurements in air or
vacuum, optical response is recorded with a silicon fast photodiode and a
lock-in detector synchronized at twice the rf drive frequency. The signal is
maximum when 2$\ \omega_{AC}$ coincides with the fundamental mechanical
resonance frequency $\omega_{0}$ of the cantilever. Cryogenic measurements
involve an optical continuous He flow Janis cryostat with electrical contacts.

## References

  * (1) Ekinci, K. L. Roukes, M. L. Nanoelectromechanical systems. Rev. Sci. Instrum. 76, 061101 (2005).
  * (2) Lassagne, B. Garcia-Sanchez, D. Aguasca, A. Bachtold A. Ultrasensitive Mass Sensing with a Nanotube Electromechanical Resonator. Nano Letters 8, 11 (2008).
  * (3) Jensen, K. Kim, K. Zettl, A. An atomic-resolution nanomechanical mass sensor. Nature Nanotechnology 3, 533 - 537 (2008) .
  * (4) Peng, H. B. Chang, C.W. Aloni, S. Yuzvinsky, T. D. Zettl, A. Ultrahigh Frequency Nanotube Resonators. Phys. Rev. Lett. 97, 087203, (2006).
  * (5) Eichler, A. et al. Nonlinear damping in mechanical resonators based on graphene and carbon nanotubes. To appear in Nature Nanotechnology DOI 10.1038/NNANO.2011.71. Arxiv: 1103.1778 (2011).
  * (6) O’Connell, A.D. et al. Quantum Ground State and Single-Phonon Control of a Mechanical Oscillator. Nature 464, 697 (2010).
  * (7) Novoselov, K. S. et al. Electric field effect in atomically thin carbon films. Science 306, 2970 (2004).
  * (8) Lee, C. Wei, X. Kysar, J. Hone, J. Measurement of the Elastic Properties and Intrinsic Strength of Monolayer Graphene. Science 321, 385 (2008).
  * (9) Castro Neto, A. Guinea, F. Peres, N.M.R. Novoselov, K.S. Geim, A.K. The electronic properties of graphene. Reviews of Modern Physics 1, 81 (2009).
  * (10) Nair, R.R. et al. Fine Structure Constant Defines Visual Transparency of Graphene. Science 320, 5881 (2008).
  * (11) Booth, T. J. et al. Macroscopic graphene membranes and their extraordinary stiffness. Nano letters 8, 8 (2008).
  * (12) Bunch, S.J. et al. Electromechanical Resonators from Graphene Sheets. Science 315, 490 (2007).
  * (13) Garcia-Sanchez, D. et al. Imaging Mechanical Vibrations in Suspended Graphene Sheets. Nano Letters 8,5 (2008).
  * (14) Conley, H., Lavrik, N.V., Prasai, D. andBolotin, K.I. Graphene bimetallic-like cantilevers: probing graphene/substrate interactions. Nano letters doi:10.1021/nl202562u (2011).
  * (15) Blake, P. et al. Making graphene visible. Applied Physics Letters 91, 063124 (2007).
  * (16) Skulason, H.S. Gaskell, P.E. Szkopek, T. Optical reflection and transmission properties of exfoliated graphite from a graphene monolayer to several hundred graphene layers. Nanotechnology 21, 29 (2010).
  * (17) Ling, X. Zhang, J. Interference Phenomenon in Graphene-Enhanced Raman Scattering. J. Phys. Chem C. 115, 6 (2010).
  * (18) Yan, J. Zhang, Y. Kim, P Pinczuk, A. Electric Field Effect Tuning of Electron-Phonon Coupling in Graphene. Phys. Rev. Lett. 98, 16 (2007).
  * (19) Otakar, F. et al. Compression Behavior of Single-Layer Graphenes. ACS Nano 4, 6 (2010).
  * (20) Otakar, F. et al. Development of a universal stress sensor for graphene and carbon fibre. Nature Communications 2, 255 (2011).
  * (21) Huang, M. et al. Phonon softening and crystallographic orientation of strained graphene studied by Raman spectroscopy. Proc. Natl. Acad. Sci. 106, 18 (2009).
  * (22) Mohiuddin, T.M.G. et al. Uniaxial strain in graphene by Raman spectroscopy: G peak splitting, Grüneisen parameters, and sample orientation. Phys. Rev. B. 79, 20 (2009).
  * (23) Das, A. et al. Monitoring dopants by Raman scattering in an electrochemically top-gated graphene transistor. Nature Nanotechnology 3, 4 (2008).
  * (24) Berciaud, S. Ryu, S. Brus, L.E. and Heinz, T.F. Probing the intrinsic properties of exfoliated graphene: Raman spectroscopy of free-standing monolayers. Nano letters 9, 346-52 (2009).
  * (25) Lifshitz, R. and Cross, M.C. Nonlinear Dynamics of Nanomechanical and Micromechanical Resonators. Review of Nonlinear Dynamics and Complexity Wiley (2008).
  * (26) Landau, L. D. Lifshitz, E.M. Theory Of Elasticity. Pergamon Press. (1960).
  * (27) Singh, V. et al. Probing thermal expansion of graphene and modal dispersion at low-temperature using graphene nanoelectromechanical systems resonators. Nanotechnology 21, 165204 (2010).
  * (28) Hosaka, H. Itao, K. and Kuroda, S. Damping characteristics of beam-shaped micro-oscillators. Sens. Actuators A 49, 87 (1995).
  * (29) Pomeroy, J.W. et al. Dynamic Operational Stress Measurement of MEMS Using Time-Resolved Raman Spectroscopy. Jour. of Micro. syst. 17, 6 (2008).
  * (30) Arcizet, O. et al. High-Sensitivity Optical Monitoring of a Micromechanical Resonator with a Quantum-Limited Optomechanical Sensor. Phys. Rev. Lett. 97, 13 (2006).


# Semileptonic decays of doubly charmed baryons with bag model

Chao-Qiang Geng, Chia-Wei Liu, Aowen Zhou and Xiao Yu
School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute
for Advanced Study, UCAS, Hangzhou 310024, China
University of Chinese Academy of Sciences, 100190 Beijing, China

###### Abstract

We study the semileptonic decays of $B_{cc}$ ${\rightarrow}$
$B_{c}\ell^{+}\nu_{\ell}$ with the bag model, where $\ell$ = $(e,\mu)$,
$B_{cc}$ = $(\Xi_{cc}^{++}$, $\Xi_{cc}^{+}$, $\Omega_{cc}^{+}$), and $B_{c}$
are the singly charmed baryons with $J^{P}=1/2^{+}$. We obtain the decay
widths of
$\Gamma(\Xi_{cc}^{++}{\rightarrow}\Xi_{c}^{+}e^{+}\nu_{e},\Xi_{c}^{\prime+}e^{+}\nu_{e},\Lambda_{c}^{+}e^{+}\nu_{e},\Omega_{c}^{+}e^{+}\nu_{e})=(5.1\pm
0.1,11\pm 1,0.34\pm 0.06,0.76\pm 0.06)\times 10^{-14}$ GeV,
$\Gamma(\Xi_{cc}^{+}\rightarrow\Xi_{c}^{0}e^{+}\nu_{e},\Xi_{c}^{\prime
0}e^{+}\nu_{e},\Sigma_{c}^{0}e^{+}\nu_{e})=(5.1\pm 0.6,11\pm 1,1.5\pm
0.1)\times 10^{-14}$ GeV, and
$\Gamma(\Omega_{cc}^{+}\rightarrow\Omega_{c}^{0}e^{+}\nu_{e},\Xi_{c}^{0}e^{+}\nu_{e},\Xi_{c}^{\prime
0}e^{+}\nu_{e})=(22\pm 2,0.32\pm 0.04,0.77\pm 0.06)\times 10^{-14}$ GeV. We
also get that $\Gamma$($B_{cc}$ ${\rightarrow}$
$B_{c}\mu^{+}\nu_{\mu}$)/$\Gamma$($B_{cc}$ ${\rightarrow}$
$B_{c}e^{+}\nu_{e}$) = $0.97\sim 1.00$. In addition, we discuss the $SU(3)$
flavor breaking effects, classified into three aspects: phase space
differences, spectator quarks, and overlappings of the transited quarks. In
particular, we show that the breaking effects are dominated by the phase space
differences, which can be as large as 25%. Explicitly, we find that
$\Gamma(\Xi_{cc}^{++}\to\Lambda_{c}^{+}e^{+}\nu_{e})V_{cs}^{2}/\Gamma(\Xi_{cc}^{++}\to\Xi_{c}^{+}e^{+}\nu_{e})V_{cd}^{2}=1.24$,
which is expected as $1$ under the exact $SU(3)$ flavor symmetry.

##  I Introduction

In 2002, the SELEX collaboration reported a resonant structure in
$\Lambda_{c}^{+}K^{-}\pi^{+}$ and $pD^{+}K^{-}$ SELEX:2002wqn ; SELEX:2004lln
, which can be potentially caused by $\Xi_{cc}^{+}(3620)$. However, the same
structure was not confirmed by the FOCUS, BABAR and BELLE collaborations
Ratti:2003ez ; BaBar:2006bab ; Belle:2013htj . Eventually, the long awaited
evidence finally arrived in 2017 via
$\Xi_{cc}^{++}\to\Lambda_{c}^{+}K^{-}\pi^{+}\pi^{+}$ at LHCb LHCb:2017iph ,
where the mass is determined to be

$M_{\Xi_{cc}^{++}}=(3621.40\pm 0.72\pm 0.27\pm 0.14)~{}\text{MeV}.$ (1)

This encouraging finding was soon accompanied by the lifetime measurement of
$\Xi_{cc}^{++}$ LHCb:2018zpl  as well as the observation of $\Xi_{cc}^{++}$
$\rightarrow$ $\Xi_{c}^{+}\pi^{+}$ LHCb:2018pcs . One can reasonably expect
much more experimental results in the future, providing opportunities to
deepen our knowledge of hadron physics.

On the theoretical aspect, the low-lying charmed baryons are categorized by
the representations of the flavor $SU(3)$ ($SU(3)_{F}$) symmetry, given in
FIG. 1. Under $SU(3)_{F}$, the doubly charmed baryons ($B_{cc}$) form a
triplet, while the singly charmed baryons ($B_{c}$) consist of a antitriplet
and a sextet. To deal with the weak decays of $B_{cc}$, lots of approaches
have been performed Wang:2017mqp ; Cerri:2018ypt ; Meng:2017udf ; Wang:2017azm
; Gutsche:2017hux ; Li:2017pxa ; Guo:2017vcf ; Lu:2017meb ; Xiao:2017udy ;
Sharma:2017txj ; Ma:2017nik ; Yu:2017zst ; Meng:2017dni ; Hu:2017dzi ;
Shi:2017dto ; Xiao:2017dly ; Yao:2018ifh ; Ozdem:2018uue ; Ali:2018ifm ;
Dias:2018qhp ; Li:2018epz ; Zhao:2018mrg ; Xing:2018bqt ; Ali:2018xfq ;
Liu:2018euh ; Xing:2018lre ; Dhir:2018twm ; Berezhnoy:2018bde ; Jiang:2018oak
; Zhang:2018llc ; Meng:2018zbl ; Gutsche:2018msz ; Shi:2019hbf ; Perez-
Marcial:1989sch ; Albertus:2011xz ; Hu:2020mxk ; Shi:2019fph ; Hu:2019bqj . In
the diquark approach, two of the three quarks are grouped as a diquark
cluster, simplifying the problem to a two-body one Zhao:2018mrg ; Xing:2018bqt
; Zhu:2018epc ; Ali:2018xfq ; Liu:2018euh ; Xing:2018lre . Nonetheless, it is
unclear which quarks shall form a diquark cluster. On the other hand, the
problem does not exist in the MIT bag model (MBM), as a diquark cluster is
unnecessary.

Figure 1: Quark states of the charmed baryons, where (a) represents an
$SU(3)_{F}$ triplet with the doubly charmed baryons, while (b) and (c)
correspond to the $SU(3)_{F}$ antitriplet and sextet with the singly charmed
baryons, respectively.

The MBM describes hadrons at rest as localized objects. Along with the bag and
zero point energies, the model is suitable to explain the mass spectra.
However, it becomes problematic in the decays due to the unwanted center-of-
mass motion. This problem can be understood by the Heisenberg uncertainty
principle, which states that a localized object can not possess a definite
momentum. If we treat a bag state as a baryon at rest, the calculations will
not respect the energy-momentum conservation. The problem was tackled a few
years ago by taking the linear superposition of infinite bags in Ref
Geng:2020ofy . This approach has been applied to various decay systems Bag1 ;
Bag2 ; Bag3 ; Bag4 ; Bag5 .

This paper is organized as follows. We present the formalism of the decay
branching fractions in terms of the helicity amplitudes in Sec. II. In Sec.
III, we give our numerical results and compare them with those in the
literature. We summarize this work in Sec. IV.

##  II Formalism

The effective Hamiltonian for the transitions of $c\to f\ell^{+}\nu$ ($f=d,s$)
at the quark level is given as

$\mathcal{H}_{eff}=\frac{G_{F}}{\sqrt{2}}V_{cf}{\bar{\ell}}{\gamma^{\mu}}(1-\gamma_{5}){\nu_{\ell}}\bar{f}{\gamma}_{\mu}(1-\gamma_{5})c,$
(2)

where $G_{F}$ is the Fermi constant, and $V_{cf}$ correspond to the Cabibbo-
Kobayashi-Maskawa matrix elements. The weak transition amplitudes of the
doubly charmed baryons are then given as

$\mathcal{A}(B_{cc}{\rightarrow}B_{c}{\ell^{+}}{\nu}_{\ell})=\frac{G_{F}}{\sqrt{2}}V_{cq}{\bar{\ell}}{\gamma^{\mu}}(1-\gamma_{5}){\nu_{\ell}}{\bra{B_{c},p_{f}}}\bar{f}{\gamma}_{\mu}(1-\gamma_{5})c{\ket{B_{cc},p_{i}}}\,,$
(3)

with the baryon matrix elements parameterized by

$\displaystyle\bra{B_{c},p_{f},\lambda_{f}}\bar{f}{\gamma}_{\mu}(1-\gamma_{5})c{\ket{B_{cc},p_{i},\lambda_{i}}}$
(4) $\displaystyle=$
$\displaystyle\bar{u}_{f}(p_{f},\lambda_{f})\left[\gamma_{\mu}f_{1}(q^{2})-i\sigma_{\mu\nu}\frac{q^{\nu}}{M_{i}}f_{2}(q^{2})+f_{3}(q^{2})\frac{q_{\mu}}{M_{i}}\right]u_{i}(p_{i},\lambda_{i})$
$\displaystyle-\bar{u}_{f}(p_{f},\lambda_{f})\left[\gamma_{\mu}g_{1}(q^{2})-i\sigma_{\mu\nu}\frac{q^{\nu}}{M_{i}}g_{2}(q^{2})+g_{3}(q^{2})\frac{q_{\mu}}{M_{i}}\right]{\gamma_{5}}u_{i}(p_{i},\lambda_{i}),$

where $f_{123}(q^{2})$ and $g_{123}(q^{2})$ are the form factors,
${\sigma^{\mu\nu}}=i\left[{\gamma}^{\mu},{\gamma}^{\nu}\right]/2$,
$q_{\mu}=p_{i}^{\mu}-p_{f}^{\mu}$, and $\lambda_{f(i)}$, $p^{\mu}_{f(i)}$,
$M_{f(i)}$ and $u_{f(i)}$ are the helicity, four-momentum, mass and Dirac
spinor of $B_{c(c)}$, respectively.

In order to calculate the decay widths, we introduce a set of helicity
amplitudes $H^{V(A)}_{\lambda_{f}\lambda_{W}}$, where $\lambda_{f}$ and
$\lambda_{W}$ represent the helicity quantum numbers of $B_{c}$ and the off-
shell $W^{+}$ boson, respectively. Relations between the helicity amplitudes
and form factors are given by Kadeer:2005aq

$\displaystyle
H_{\frac{1}{2}1}^{V}=\sqrt{2Q_{-}}\left(-f_{1}(q^{2})-\frac{M_{i}+M_{f}}{M_{i}}f_{2}(q^{2})\right),$
(5) $\displaystyle
H_{\frac{1}{2}0}^{V}=\frac{\sqrt{Q_{-}}}{q^{2}}\left((M_{i}+M_{f})f_{1}(q^{2})+\frac{q^{2}}{M_{i}}f_{2}(q^{2})\right),$
$\displaystyle
H_{\frac{1}{2}t}^{V}=\frac{\sqrt{Q_{+}}}{q^{2}}\left((M_{i}-M_{f})f_{1}(q^{2})+\frac{q^{2}}{M_{i}}f_{3}(q^{2})\right),$
$\displaystyle
H_{\frac{1}{2}1}^{A}=\sqrt{2Q_{+}}\left(g_{1}(q^{2})-\frac{M_{i}-M_{f}}{M_{i}}g_{2}(q^{2})\right),$
$\displaystyle
H_{\frac{1}{2}0}^{A}=\frac{\sqrt{Q_{+}}}{q^{2}}\left(-(M_{i}-M_{f})g_{1}(q^{2})+\frac{q^{2}}{M_{i}}g_{2}(q^{2})\right),$
$\displaystyle
H_{\frac{1}{2}t}^{A}=\frac{\sqrt{Q_{-}}}{q^{2}}\left(-(M_{i}+M_{f})g_{1}(q^{2})+\frac{q^{2}}{M_{i}}g_{3}(q^{2})\right),$

where $Q_{\pm}=(M_{i}\pm M_{f})^{2}-q^{2}$ and
$H_{{\lambda_{f}}{\lambda_{W}}}^{V(A)}=(-)H_{-{\lambda_{f}}-{\lambda_{W}}}^{V(A)}$.
The differential decay widths are given in terms of the helicity amplitudes as
Geng:2019bfz ; Kadeer:2005aq ; Korner:1994nh

$\displaystyle\partial_{q}\Gamma=$
$\displaystyle\frac{\partial\Gamma}{\partial
q^{2}}=\frac{1}{3}\frac{G_{F}^{2}}{(2\pi)^{3}}|V_{fc}|^{2}\frac{(q^{2}-m_{\ell}^{2})^{2}p}{8M_{i}^{2}q^{2}}\left[\left(1+\frac{m_{\ell}^{2}}{2q^{2}}\right)\right.$
(6)
$\displaystyle\left.\left(\left|H_{\frac{1}{2}1}\right|^{2}+\left|H_{-\frac{1}{2}-1}\right|^{2}+\left|H_{\frac{1}{2}0}\right|^{2}+\left|H_{-\frac{1}{2}0}\right|^{2}\right)+\frac{3m_{\ell}^{2}}{2q^{2}}\left(\left|H_{\frac{1}{2}t}\right|^{2}\left|H_{-\frac{1}{2}t}\right|^{2}\right)\right],$

where $p=\sqrt{Q^{+}Q^{-}}/2M_{B_{i}}$,
$H_{\lambda_{f}\lambda_{W}}=H_{\lambda_{f}\lambda_{W}}^{V}-H_{\lambda_{f}\lambda_{W}}^{A}$
and $m_{\ell}$ is the lepton mass.

In this work, we evaluate the form factors with the homogeneous bag model
(HBM) Bag5 . The baryon wave functions of $B_{cc}$ are given as

$|B_{cc},\updownarrow\rangle=\int\frac{1}{2\sqrt{3}}\epsilon^{\alpha\beta\gamma}q_{a\alpha}^{\prime\dagger}(\vec{x}_{1})c_{b\beta}^{\dagger}(\vec{x}_{2})c_{c\gamma}^{\dagger}(\vec{x}_{3})\Psi_{A_{\updownarrow}(ucc)}^{abc}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})[d^{3}\vec{x}]|0\rangle\,,$
(7)

where $q^{\prime}=(u,d,s)$ for
$B_{cc}=(\Xi_{cc}^{++},\Xi_{cc}^{+},\Omega_{cc}^{+})$, $q^{\prime\dagger}$ and
$c^{\dagger}$ represent the creation operators of quarks, the Latin and Greek
letters stand for the Dirac spinor and color indices, and $\Psi_{A}$ are the
spatial wave functions defined in Refs. Bag3 ; Bag4 ; Bag5 , respectively. On
the other hand, the wave functions of $B_{c}$ can be found in Ref. Bag3 .

We choose the Breit frame to calculate the baryon matrix elements, where
$B_{c}$ and $B_{cc}$ have the opposite velocities $\vec{v}=v\hat{z}$ and
$-\vec{v}$. The baryon matrix elements of the current operators are then
governed by

$\displaystyle\langle{B_{c}(\vec{v})},\lambda_{f}|f^{\dagger}\Upsilon
c(0)|B_{cc}(-\vec{v}),\lambda_{i}\rangle=\mathcal{N}_{B_{c}}{\mathcal{N}_{B_{cc}}}\int
d^{3}\vec{x}_{\triangle}\Upsilon_{fc}^{{\lambda_{f}}{\lambda_{i}}}(\vec{x}_{\Delta})\prod_{q=c,q^{\prime}}D_{q}^{v}(\vec{x}_{\triangle}),$
(8) $\displaystyle D_{q}^{v}(\vec{x}_{\triangle})=\frac{1}{\gamma}\int
d^{3}\vec{x}\phi_{q}^{\dagger}\left(\vec{x}+\frac{1}{2}\vec{x}_{\triangle}\right)\phi_{q}\left(\vec{x}-\frac{1}{2}\vec{x}_{\triangle}\right)e^{-2iE_{q}vz},$
$\displaystyle\Upsilon_{fc}^{{\lambda_{f}}{\lambda_{i}}}(\vec{x}_{\triangle})=\sum_{{\lambda_{q}}{\lambda_{c}}}N^{{\lambda_{f}}{\lambda_{i}}}_{{\lambda_{q}}{\lambda_{c}}}\int
d^{3}\vec{x}\phi_{f{\lambda_{q}}}^{\dagger}\left(\vec{x}^{+}\right)S_{\vec{v}}{\Upsilon}S_{-\vec{v}}\phi_{c{\lambda_{c}}}\left(\vec{x}^{-}\right)e^{2i(E_{q^{\prime}}+E_{c})\vec{v}\cdot\vec{x}}\,,$

where $\mathcal{N}_{B_{c},B_{cc}}$ are the normalization constants, $\Upsilon$
is an arbitrary Dirac matrix, $\vec{x}^{\pm}$ = $\vec{x}$ ${\pm}$
$\vec{x}_{\triangle}/2$, $\phi_{q}$ are the bag wave functions in the MBM,
$S_{\pm v}=a_{+}\pm a_{-}\gamma^{0}\gamma^{3}$ with
$a_{\pm}=\sqrt{1\pm\gamma^{2}}$ and $\gamma=1/\sqrt{1-v^{2}}$, and
$\lambda_{q,c}\in\\{\uparrow,\downarrow\\}$. The derivations of Eq. (8) and
the explicit forms of $\phi_{q}$ are given in Ref. Bag5 . The first line of
Eq. (8) is the total overlapping between $B_{cc}$ and $B_{c}$ induced by
$f^{\dagger}\Upsilon c(0)$ at the quark level, while the second and third
terms are interpreted as :

  * •

The spectator quark effects are governed by $D_{q}^{v}(\vec{x}_{\Delta})$,
which describe the overlapping of $q$ in two bag states separated by
$\vec{x}_{\Delta}$.

  * •

The quark transitions are described by
$\Upsilon_{fc}^{\lambda_{f}\lambda_{i}}$, where
$N_{\lambda_{q},\lambda_{c}}^{\lambda_{f},\lambda_{i}}$ are the spin-flavor
overlapping coefficients.

In the heavy constituent quark limit $(m_{u,d,c}\to\infty)$, the formalism is
reduced to

$\displaystyle\langle{B_{c}(\vec{v})},\uparrow|f^{\dagger}c(0)|B_{cc}(-\vec{v}),\uparrow\rangle=\sum_{{\lambda_{q}}{\lambda_{c}}}N^{\uparrow\uparrow}_{{\lambda_{q}}{\lambda_{c}}}\,,$
$\displaystyle\langle{B_{c}(\vec{v})},\uparrow|f^{\dagger}\gamma^{0}\gamma^{1}\gamma_{5}c(0)|B_{cc}(-\vec{v}),\downarrow\rangle=\sum_{{\lambda_{q}}{\lambda_{c}}}N^{\uparrow\downarrow}_{{\lambda_{q}}{\lambda_{c}}}\,.$
(9)

From the angular momentum conservation, we have that

$\displaystyle
N^{\lambda_{f}\lambda_{i}}_{\lambda_{q}\lambda_{c}}=0~{}~{}~{}~{}~{}\text{for}~{}~{}\lambda_{f}-\lambda_{i}\neq\lambda_{c}-\lambda_{q}.$
(10)

It states that if the baryon spin is (un)flipped by the operator, then the
spin of the quark shall also be (un)flipped. In addition, by the Wiger-Eckart
theorem, we find that

$N^{\uparrow\uparrow}_{\lambda_{q}\lambda_{c}}=N^{\downarrow\downarrow}_{-\lambda_{q}-\lambda_{c}}\,,~{}~{}~{}~{}N^{\uparrow\uparrow}_{\uparrow\uparrow}-N^{\uparrow\uparrow}_{\downarrow\downarrow}=N^{\downarrow\uparrow}_{\downarrow\uparrow}=N^{\uparrow\downarrow}_{\uparrow\downarrow}.$
(11)

Consequently, there are only two independent numbers given as

$N_{\text{unflip}}\equiv
N^{\uparrow\uparrow}_{\uparrow\uparrow}+N^{\uparrow\uparrow}_{\downarrow\downarrow}\,,~{}~{}~{}~{}N_{\text{flip}}\equiv
N^{\downarrow\uparrow}_{\downarrow\uparrow}\,,$ (12)

which are collected in TABLE 1.

Table 1:  The spin-flavor overlappings of $B_{cc}\to B_{c}$. $c\to s$ |  $N_{\text{unflip}}$ |  $N_{\text{flip}}$ |  $c\to d$ |  $N_{\text{unflip}}$ |  $N_{\text{flip}}$
---|---|---|---|---|---
$\Xi^{++}_{cc}\rightarrow\Xi^{+}_{c}$ | $\frac{\sqrt{6}}{2}$ | $\frac{\sqrt{6}}{6}$ | $\Xi^{++}_{cc}\rightarrow\Lambda^{+}_{c}$ | $\frac{\sqrt{6}}{2}$ | $\frac{\sqrt{6}}{6}$
$\Xi^{+}_{cc}\rightarrow\Xi^{0}_{c}$ | $\frac{\sqrt{6}}{2}$ | $\frac{\sqrt{6}}{6}$ | $\Omega^{+}_{cc}\rightarrow\Xi^{0}_{c}$ | $\frac{\sqrt{6}}{2}$ | $\frac{\sqrt{6}}{6}$
$\Xi^{++}_{cc}\rightarrow\Xi^{\prime+}_{c}$ | $-\frac{\sqrt{2}}{2}$ | $-\frac{5\sqrt{2}}{6}$ | $\Xi^{++}_{cc}\rightarrow\Sigma^{+}_{c}$ | $-\frac{\sqrt{2}}{2}$ | $-\frac{5\sqrt{2}}{6}$
$\Xi^{+}_{cc}\rightarrow\Xi^{\prime 0}_{c}$ | $-\frac{\sqrt{2}}{2}$ | $-\frac{5\sqrt{2}}{6}$ | $\Omega^{+}_{cc}\rightarrow\Xi^{\prime 0}_{c}$ | $-\frac{\sqrt{2}}{2}$ | $-\frac{5\sqrt{2}}{6}$
$\Omega^{+}_{cc}\rightarrow\Omega^{0}_{c}$ | $-1$ | $-\frac{5}{3}$ | $\Xi^{+}_{cc}\rightarrow\Sigma^{0}_{c}$ | $-1$ | $-\frac{5}{3}$

##  III Numerical results

The values of $V_{cf}$ are given by Workman:2022ynf

$\displaystyle|V_{cs}|=0.987\pm 0.011,\quad|V_{cd}|=0.221\pm 0.004,$ (13)

while the model parameters are taken as Bag5

$R=4.7\pm 0.3\ {\rm
GeV}^{-1}\,,~{}~{}~{}m_{c}=1.655~{}\text{GeV}\,,~{}~{}~{}m_{u,d}=0\,,~{}~{}~{}m_{s}=0.2\pm
0.1\ {\rm GeV}\,.$ (14)

Notice that $B_{cc}$ and $B_{c}$ have different bag radii, found to be around
$4.4$ and $5.0$ GeV-1, respectively Hypothetic3 . However, to simplify the
formalism, we take their bag radii as equal and allow them to vary from $4.4$
to $5.0$ GeV-1.

To illustrate the recoil effects of the form factors, we plot those of
$\Omega_{cc}^{+}\to\Omega_{c}$ in FIG. 2. We define

$\omega\equiv\frac{M_{i}^{2}+M_{f}^{2}-q^{2}}{2M_{i}M_{f}}=\frac{1+v^{2}}{1-v^{2}},$
(15)

so that the zero recoil point $(q^{2}=q_{max}^{2}=(M_{i}-M_{f})^{2})$
corresponds to $\omega=1$ for all the decays. As shown in the figures, $f_{3}$
and $g_{2}$ can be taken as zero practically. The uncertainties of $f_{1}$ and
$g_{1}$ are negligible at the low $q^{2}$ regions, and around $10\%$ at
$\omega=1.06$. As a result, at $\omega=1$, $f_{1}$ and $g_{1}$ are not
polluted by the uncertainties of the quark energies. However, the
uncertainties of $f_{2}$ and $g_{3}$ are large in all regions.

(a)  (b)

Figure 2: The form factors of $\Omega_{cc}^{+}\rightarrow\Omega_{c}$ as
functions of $\omega$, where the center lines and bands correspond to the
central values and uncertainties, respectively.

The form factors of $\Xi_{cc}^{++}\to B_{c}$ at $q^{2}=0$, along with those in
the literature, are given in Table 2. For completeness, we also show all our
calculated values of the form factors for $\Xi_{cc}^{++}\to B_{c}$ with the
HBM in Table 3. Note that the form factors of $\Xi_{cc}^{++}\to\Xi_{c}^{+}$
and $\Xi_{cc}^{+}\to\Xi_{c}^{0}$ are the same due to the isospin symmetry.
Compared to those in the literature, our form factors of
$\Xi_{cc}^{++}\to\Lambda_{c}^{+}$ have an overall minus sign due to the
convention on the baryon wave functions, which does not affect the physical
quantities. Note that our results of the form factors at $q^{2}=0$ are
significantly smaller than those in other approaches.

Table 2:  The form factors of $\Xi_{cc}^{++}$ calculated in the HBM, light-front quark model (LFQM), and QCD sum rule (QCDSR) at $q^{2}=0$. | HBM (This work) | LFQM Wang:2017mqp ; Shi:2019hbf  | QCDSR Shi:2019hbf ; Shi:2019fph ; Hu:2019bqj
---|---|---|---
$f_{1}^{\Xi^{++}_{cc}\rightarrow\Lambda^{+}_{c}}$ | $0.28\pm 0.05$ | $-0.79$ | $-0.59\pm 0.05$
$f_{2}^{\Xi^{++}_{cc}\rightarrow\Lambda^{+}_{c}}$ | $-0.01\pm 0.01$ | $0.008$ | $0.039\pm 0.024$
$f_{3}^{\Xi^{++}_{cc}\rightarrow\Lambda^{+}_{c}}$ | $-0.16\pm 0.02$ | \- - | $0.35\pm 0.11$
$g_{1}^{\Xi^{++}_{cc}\rightarrow\Lambda^{+}_{c}}$ | $0.09\pm 0.02$ | $-0.22$ | $-0.13\pm 0.08$
$g_{2}^{\Xi^{++}_{cc}\rightarrow\Lambda^{+}_{c}}$ | $0.01\pm 0.00$ | $0.05$ | $0.037\pm 0.027$
$g_{3}^{\Xi^{++}_{cc}\rightarrow\Lambda^{+}_{c}}$ | $-0.21\pm 0.02$ | \- - | $0.31\pm 0.09$
$f_{1}^{\Xi^{++}_{cc}\rightarrow\Sigma^{+}_{c}}$ | $-0.24\pm 0.01$ | $-0.47$ | $-0.35\pm 0.04$
$f_{2}^{\Xi^{++}_{cc}\rightarrow\Sigma^{+}_{c}}$ | $-0.53\pm 0.05$ | $1.04$ | $1.15\pm 0.12$
$f^{\Xi^{++}_{cc}\rightarrow\Sigma^{+}_{c}}_{3}$ | $0.03\pm 0.00$ | \- - | $-1.40\pm 0.39$
$g^{\Xi^{++}_{cc}\rightarrow\Sigma^{+}_{c}}_{1}$ | $-0.37\pm 0.05$ | $-0.62$ | $-0.23\pm 0.06$
$g_{2}^{\Xi^{++}_{cc}\rightarrow\Sigma^{+}_{c}}$ | $-0.05\pm 0.00$ | $0.05$ | $-0.26\pm 0.15$
$g_{3}^{\Xi^{++}_{cc}\rightarrow\Sigma^{+}_{c}}$ | $0.89\pm 0.06$ | \- - | $2.68\pm 0.39$
Table 3:  The form factors from the HBM with
$\Xi_{c}^{(\prime)}=(\Xi_{c}^{(\prime)+},\Xi_{c}^{(\prime)0})$ for
$\Xi_{cc}=(\Xi_{cc}^{++},\Xi_{cc}^{+})$.

|  $q^{2}=0$ |  $q^{2}=q_{max}^{2}$ |  |  $q^{2}=0$ |  $q^{2}=q_{max}^{2}$
---|---|---|---|---|---
$f_{1}^{\Xi_{cc}\rightarrow\Xi_{c}}$ |  $0.45\pm 0.05$ |  $1.36\pm 0.00$ | $g_{1}^{\Xi_{cc}\rightarrow\Xi_{c}}$ |  $0.14\pm 0.02$ |  $0.38\pm 0.00$
$f_{2}^{\Xi_{cc}\rightarrow\Xi_{c}}$ |  $-0.03\pm 0.00$ |  $-0.15\pm 0.01$ | $g_{2}^{\Xi_{cc}\rightarrow\Xi_{c}}$ |  $0.01\pm 0.00$ |  $0.03\pm 0.01$
$f_{3}^{\Xi_{cc}\rightarrow\Xi_{c}}$ |  $-0.20\pm 0.04$ |  $-0.56\pm 0.03$ | $g_{3}^{\Xi_{cc}\rightarrow\Xi_{c}}$ |  $-0.28\pm 0.04$ |  $-0.79\pm 0.02$
$f_{1}^{\Xi_{cc}\rightarrow\Xi^{\prime}_{c}}$ |  $-0.31\pm 0.03$ |  $-0.70\pm 0.00$ | $g_{1}^{\Xi_{cc}\rightarrow\Xi^{\prime}_{c}}$ |  $-0.49\pm 0.04$ |  $-1.10\pm 0.01$
$f_{2}^{\Xi_{cc}\rightarrow\Xi^{\prime+}_{c}}$ |  $-0.63\pm 0.07$ |  $-1.48\pm 0.02$ | $g_{2}^{\Xi_{cc}\rightarrow\Xi^{\prime}_{c}}$ |  $-0.06\pm 0.02$ |  $-0.13\pm 0.02$
$f_{3}^{\Xi_{cc}\rightarrow\Xi^{\prime+}_{c}}$ |  $0.03\pm 0.01$ |  $0.06\pm 0.02$ | $g_{3}^{\Xi_{cc}\rightarrow\Xi^{\prime}_{c}}$ |  $1.00\pm 0.12$ |  $2.31\pm 0.05$
$f_{1}^{\Omega^{+}_{cc}\rightarrow\Omega^{0}_{c}}$ |  $-0.48\pm 0.04$ |  $-0.99\pm 0.00$ | $g_{1}^{\Omega^{+}_{cc}\rightarrow\Omega^{0}_{c}}$ |  $-0.69\pm 0.06$ |  $-1.56\pm 0.02$
$f_{2}^{\Omega^{+}_{cc}\rightarrow\Omega^{0}_{c}}$ |  $-0.93\pm 0.10$ |  $-2.17\pm 0.03$ | $g_{2}^{\Omega^{+}_{cc}\rightarrow\Omega^{0}_{c}}$ |  $-0.09\pm 0.02$ |  $-0.20\pm 0.04$
$f_{3}^{\Omega^{+}_{cc}\rightarrow\Omega^{0}_{c}}$ |  $0.05\pm 0.02$ |  $0.11\pm 0.02$ | $g_{3}^{\Omega^{+}_{cc}\rightarrow\Omega^{0}_{c}}$ |  $1.52\pm 0.18$ |  $3.51\pm 0.08$
$f_{1}^{\Xi^{++}_{cc}\rightarrow\Lambda^{+}_{c}}$ |  $0.28\pm 0.05$ |  $1.42\pm 0.02$ | $g_{1}^{\Xi^{++}_{cc}\rightarrow\Lambda^{+}_{c}}$ |  $0.09\pm 0.02$ |  $0.37\pm 0.00$
$f_{2}^{\Xi^{++}_{cc}\rightarrow\Lambda^{+}_{c}}$ |  $-0.01\pm 0.01$ |  $-0.16\pm 0.01$ | $g_{2}^{\Xi^{++}_{cc}\rightarrow\Lambda^{+}_{c}}$ |  $0.01\pm 0.00$ |  $0.02\pm 0.01$
$f_{3}^{\Xi^{++}_{cc}\rightarrow\Lambda^{+}_{c}}$ |  $-0.16\pm 0.02$ |  $-0.73\pm 0.06$ | $g_{3}^{\Xi^{++}_{cc}\rightarrow\Lambda^{+}_{c}}$ |  $-0.21\pm 0.02$ |  $-0.91\pm 0.05$
$f_{1}^{\Omega^{+}_{cc}\rightarrow\Xi^{0}_{c}}$ |  $0.33\pm 0.05$ |  $1.41\pm 0.02$ | $g_{1}^{\Omega^{+}_{cc}\rightarrow\Xi^{0}_{c}}$ |  $0.10\pm 0.02$ |  $0.37\pm 0.00$
$f_{2}^{\Omega^{+}_{cc}\rightarrow\Xi^{0}_{c}}$ |  $-0.01\pm 0.01$ |  $-0.11\pm 0.02$ | $g_{2}^{\Omega^{+}_{cc}\rightarrow\Xi^{0}_{c}}$ |  $0.01\pm 0.00$ |  $0.04\pm 0.01$
$f_{3}^{\Omega^{+}_{cc}\rightarrow\Xi^{0}_{c}}$ |  $-0.20\pm 0.02$ |  $-0.75\pm 0.06$ | $g_{3}^{\Omega^{+}_{cc}\rightarrow\Xi^{0}_{c}}$ |  $-0.26\pm 0.03$ |  $-0.99\pm 0.06$
$f_{1}^{\Xi^{++}_{cc}\rightarrow\Sigma^{+}_{c}}$ |  $-0.24\pm 0.01$ |  $-0.70\pm 0.00$ | $g_{1}^{\Xi^{++}_{cc}\rightarrow\Sigma^{+}_{c}}$ |  $-0.37\pm 0.05$ |  $-1.08\pm 0.01$
$f_{2}^{\Xi^{++}_{cc}\rightarrow\Sigma^{+}_{c}}$ |  $-0.53\pm 0.05$ |  $-1.65\pm 0.7$ | $g_{2}^{\Xi^{++}_{cc}\rightarrow\Sigma^{+}_{c}}$ |  $-0.05\pm 0.00$ |  $-0.16\pm 0.03$
$f_{3}^{\Xi^{++}_{cc}\rightarrow\Sigma^{+}_{c}}$ |  $0.03\pm 0.00$ |  $0.09\pm 0.02$ | $g_{3}^{\Xi^{++}_{cc}\rightarrow\Sigma^{+}_{c}}$ |  $0.89\pm 0.06$ |  $2.71\pm 0.16$
$f_{1}^{\Omega^{+}_{cc}\rightarrow\Xi^{\prime 0}_{c}}$ |  $-0.24\pm 0.03$ |  $-0.70\pm 0.01$ | $g_{1}^{\Omega^{+}_{cc}\rightarrow\Xi^{\prime 0}_{c}}$ |  $-0.37\pm 0.05$ |  $-1.08\pm 0.01$
$f_{2}^{\Omega^{+}_{cc}\rightarrow\Xi^{\prime 0}_{c}}$ |  $-0.56\pm 0.05$ |  $-1.71\pm 0.07$ | $g_{2}^{\Omega^{+}_{cc}\rightarrow\Xi^{\prime 0}_{c}}$ |  $-0.06\pm 0.00$ |  $-0.17\pm 0.03$
$f_{3}^{\Omega^{+}_{cc}\rightarrow\Xi^{\prime 0}_{c}}$ |  $0.04\pm 0.00$ |  $0.11\pm 0.02$ | $g_{3}^{\Omega^{+}_{cc}\rightarrow\Xi^{\prime 0}_{c}}$ |  $0.97\pm 0.07$ |  $2.92\pm 0.18$
$f_{1}^{\Xi^{+}_{cc}\rightarrow\Sigma^{0}_{c}}$ |  $-0.38\pm 0.05$ |  $-0.95\pm 0.03$ | $g_{1}^{\Xi^{+}_{cc}\rightarrow\Sigma^{0}_{c}}$ |  $-0.52\pm 0.07$ |  $-1.47\pm 0.05$
$f_{2}^{\Xi^{+}_{cc}\rightarrow\Sigma^{0}_{c}}$ |  $-0.75\pm 0.07$ |  $-2.23\pm 0.00$ | $g_{2}^{\Xi^{+}_{cc}\rightarrow\Sigma^{0}_{c}}$ |  $-0.07\pm 0.00$ |  $-0.21\pm 0.03$
$f_{3}^{\Xi^{+}_{cc}\rightarrow\Sigma^{0}_{c}}$ |  $0.05\pm 0.01$ |  $0.12\pm 0.03$ | $g_{3}^{\Xi^{+}_{cc}\rightarrow\Sigma^{0}_{c}}$ |  $1.26\pm 0.09$ |  $3.67\pm 0.07$

(a)

(b)

(c)

(d)

Figure 3:  The $\omega$ dependencies of the $B_{cc}\rightarrow B_{c}$ form
factors.

In FIGs. 3a and 3b, we compare the form factors of
$\Xi_{cc}^{++}\to\Xi_{c}^{+}$ and $\Xi_{cc}^{++}\to\Lambda_{c}^{+}$,
corresponding to $c\to s$ and $c\to d$ transitions, respectively. These form
factors shall be identical in the limit of the $SU(3)_{F}$ symmetry. The
figures show that for a fixed value of $\omega$, the form factors are
approximately the same. Explicitly, their values deviate by less than $13\%$.
However, the phase space of the $c\to s$ transition is about $30\%$ smaller
than the one of $c\to d$. As shown in the figures, the form factors of
$\Xi_{cc}^{++}\to\Xi_{c}^{+}$ with $\omega>1.07$ are missing as they
correspond to the region of $q^{2}<0$.

On the other hand, to examine the spectator effects, we plot the results of
$\Xi_{cc}^{++}\to\Sigma_{c}^{+}$ and $\Omega_{cc}^{+}\to\Xi_{c}^{0}$ in FIGs.
3c and 3d, corresponding to the $c\to d$ transition with $(c,u)$ and $(c,s)$
as the spectator quarks, respectively. We find that the form factors deviate
less than $11\%$ between the two types of transition. We conclude that the
form factors well respect the $SU(3)_{F}$ symmetry if one uses the variables
of $\omega$ instead of $q^{2}$.

The total decay widths of $\Gamma$ are computed by integrating Eq. (6). To
further examine the results, we decompose the decay widths into four
fragments, given by

$\displaystyle P_{1}$
$\displaystyle=\frac{1}{\Gamma}\int_{m_{e}^{2}}^{\frac{1}{4}q_{max}^{2}}\partial_{q}\Gamma
dq^{2}\,,~{}~{}~{}P_{2}$
$\displaystyle=\frac{1}{\Gamma}\int_{\frac{1}{4}q_{max}^{2}}^{\frac{1}{2}q_{max}^{2}}\partial_{q}\Gamma
dq^{2},$ (16) $\displaystyle P_{3}$
$\displaystyle=\frac{1}{\Gamma}\int_{\frac{1}{2}q_{max}^{2}}^{\frac{3}{4}q_{max}^{2}}\partial_{q}\Gamma
dq^{2}\,,~{}~{}~{}P_{4}$
$\displaystyle=\frac{1}{\Gamma}\int_{\frac{3}{4}q_{max}^{2}}^{q_{max}^{2}}\partial_{q}\Gamma
dq^{2},$

with their values listed in Table 4, respectively. The uncertainties of
$P_{i}$ are tiny compared to the total branching fractions due to the
correlations. In addition, we find that except for
$\Xi_{cc}^{++}\to\Xi_{c}^{+}e^{+}\nu_{e}$, the values of $P_{1}$ are the
smallest among the fragments. In contrast to the others, the decay width of
$\Xi_{cc}^{++}\to\Xi_{c}^{+}e^{+}\nu_{e}$ distributes smoothly among the four
regions.

Table 4:  The total and fragmentary decay widths.

channels |  $\Gamma\times 10^{14}~{}\text{GeV}^{-1}$ |  $P_{1}$ |  $P_{2}$ |  $P_{3}$ |  $P_{4}$
---|---|---|---|---|---
$\Xi^{++}_{cc}\rightarrow\Xi^{+}_{c}e^{+}\nu_{e}$ |  $5.11\pm 0.64$ |  $0.25\pm 0.02$ |  $0.29\pm 0.01$ |  $0.29\pm 0.01$ |  $0.17\pm 0.02$
$\Xi^{++}_{cc}\rightarrow\Xi^{\prime+}_{c}e^{+}\nu_{e}$ |  $10.9\pm 0.8$ |  $0.14\pm 0.01$ |  $0.24\pm 0.01$ |  $0.33\pm 0.00$ |  $0.29\pm 0.02$
$\Omega^{+}_{cc}\rightarrow\Omega^{0}_{c}e^{+}\nu_{e}$ |  $22.1\pm 1.6$ |  $0.14\pm 0.01$ |  $0.24\pm 0.01$ |  $0.33\pm 0.00$ |  $0.29\pm 0.02$
$\Xi^{++}_{cc}\rightarrow\Lambda^{+}_{c}e^{+}\nu_{e}$ |  $0.34\pm 0.06$ |  $0.17\pm 0.03$ |  $0.26\pm 0.01$ |  $0.32\pm 0.00$ |  $0.24\pm 0.04$
$\Xi^{++}_{cc}\rightarrow\Sigma^{+}_{c}e^{+}\nu_{e}$ |  $0.76\pm 0.06$ |  $0.12\pm 0.02$ |  $0.21\pm 0.01$ |  $0.34\pm 0.00$ |  $0.34\pm 0.03$
$\Xi^{+}_{cc}\rightarrow\Sigma^{0}_{c}e^{+}\nu_{e}$ |  $1.52\pm 0.12$ |  $0.12\pm 0.02$ |  $0.21\pm 0.02$ |  $0.34\pm 0.00$ |  $0.34\pm 0.03$

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4:  The partial decay widths for $B_{cc}\rightarrow B_{c}e^{+}\nu_{e}$,
where the dashed lines and band widths are the center values and
uncertainties, while (LABEL:sub@fig:_Omega_decay_widths-1,
LABEL:sub@fig:_Omega_decay_widths-2, LABEL:sub@fig:_Omega_decay_widths-3) and
(LABEL:sub@fig:_Omega_decay_widths-4, LABEL:sub@fig:_Omega_decay_widths-5,
LABEL:sub@fig:_Omega_decay_widths-6) correspond to the $c\to s$ and $c\to d$
transitions, respectively.

(a)

(b)

Figure 5:  The partial decay widths of $P_{C}$ and $P_{C}$ for
$\Xi_{cc}^{++}\to\Xi_{c}^{(\prime)+}e^{+}\nu_{e}$.

In FIG. 4, we plot the differential decay widths. In the high $q^{2}$ areas,
the uncertainties are minor since $f_{1}$ and $g_{1}$ have few errors, as
explained at the beginning of this section. We see that
$\partial_{q}\Gamma(\Xi_{cc}^{++}\to\Xi_{c}^{+}e^{+}\nu_{e})$ is much smoother
than the others. To see the underlying reason, we define the parity conserving
and violating partial decay widths as

$P_{C}\equiv\partial_{q}\Gamma(g_{i}=0)\,,\
P_{V}\equiv\partial_{q}\Gamma(f_{i}=0)\,,\ \partial_{q}\Gamma=P_{C}+P_{V}\,,$
(17)

where in $P_{C(V)}$ we take $g_{1,2,3}(f_{1,2,3})$ as zero, corresponding to
the parity conserving (violating) parts of the decay widths. We show
$P_{C,V}(\Xi_{cc}^{++}\rightarrow\Xi_{c}^{(\prime)+})$ in FIG. 5.

We see that $P_{C,V}$ have very different behaviors. On the one hand, $P_{C}$
contribute to $\Gamma$ mostly in the low $q^{2}$ regions, and their values
decrease quickly as $q^{2}$ goes up. On the other hand, $P_{V}$ behave
oppositely. In $\Xi_{cc}^{+}\to\Xi_{c}^{+}e^{+}\nu_{e}$, the net result is
that the tendencies of $P_{C,V}$ smear out each other in $\partial_{q}\Gamma$.
In contrast, the behavior of $\partial_{q}\Gamma$ is dominated by $P_{V}$ in
$\Xi_{cc}^{++}\to\Xi_{c}^{\prime+}e^{+}v_{e}$. The sharp difference can be
traced back to the spin-flavor overlappings, where we approximately have
$P_{C}\propto N_{\text{unflip}}^{2}$ and $P_{V}\propto N_{\text{flip}}^{2}$.
From TABLE 1, we see that $(N_{\text{flip}}/N_{\text{unflip}})^{2}$ are $1/9$
and $100/36$ for $\Xi_{cc}^{++}\to\Xi_{c}^{+}$ and
$\Xi_{cc}^{++}\to\Xi_{c}^{\prime+}$, respectively, which explains the opposite
behaviors.

To test the lepton universality in the future experiments, we provide the
ratios of

$\mathcal{R}=\Gamma(B_{cc}\rightarrow
B_{c}\mu^{+}v_{\mu})/\Gamma(B_{cc}\rightarrow B_{c}e^{+}v_{e})$

in Table 5. Their values are close to but below 1. Clearly, if $\mathcal{R}>1$
in the future experiment, it will be a signal of new physics.

Table 5:  The ratios $\mathcal{R}$. $~{}c\to s~{}$ | $~{}~{}~{}\mathcal{R}~{}~{}~{}$ | $~{}c\to d~{}$ | $~{}~{}~{}\mathcal{R}~{}~{}~{}$
---|---|---|---
$\Xi^{++}_{cc}\rightarrow\Xi^{+}_{c}$ | 0.99 | $\Xi^{++}_{cc}\rightarrow\Lambda^{+}_{c}$ | 1.00
$\Xi^{+}_{cc}\rightarrow\Xi^{0}_{c}$ | 0.99 | $\Omega^{+}_{cc}\rightarrow\Xi^{0}_{c}$ | 1.00
$\Xi^{++}_{cc}\rightarrow\Xi^{\prime+}_{c}$ | 0.97 | $\Xi^{++}_{cc}\rightarrow\Sigma^{+}_{c}$ | 0.97
$\Xi^{+}_{cc}\rightarrow\Xi^{\prime 0}_{c}$ | 0.97 | $\Omega^{+}_{cc}\rightarrow\Xi^{\prime 0}_{c}$ | 0.97
$\Omega^{+}_{cc}\rightarrow\Omega^{0}_{c}$ | 0.97 | $\Xi^{+}_{cc}\rightarrow\Sigma^{0}_{c}$ | 0.97

To eliminate the uncertainties caused by $V_{cf}$, by defining

$\text{G}_{B_{c}}^{B_{cc}}=\frac{1}{|V_{cf}|^{2}}\Gamma(B_{cc}\rightarrow
B_{c}e^{+}\nu_{e}),$ (18)

we find that

$\displaystyle\text{G}_{\Xi_{c}^{+}}^{\Xi_{cc}^{++}}:\text{G}_{\Xi_{c}^{0}}^{\Xi_{cc}^{+}}:\text{G}_{\Lambda_{c}^{+}}^{\Xi_{cc}^{++}}:\text{G}_{\Xi_{c}^{0}}^{\Omega_{cc}^{+}}$
$\displaystyle=1:0.99:1.24:1.17,$ (19)
$\displaystyle\text{G}_{\Xi_{c}^{\prime+}}^{\Xi_{cc}^{++}}:\text{G}_{\Xi_{c}^{\prime
0}}^{\Xi_{cc}^{+}}:\frac{1}{2}\text{G}_{\Omega_{c}^{0}}^{\Omega_{cc}^{+}}:\text{G}_{\Sigma_{c}^{+}}^{\Xi_{cc}^{++}}:\text{G}_{\Xi_{c}^{\prime
0}}^{\Omega_{cc}^{+}}:\frac{1}{2}\text{G}_{\Sigma_{c}^{0}}^{\Xi_{cc}^{+}}$
$\displaystyle=1:0.99:1.01:1.30:1.32:1.31,$

which are all expected to be 1 in the exact $SU(3)_{F}$ symmetry. As mentioned
in the discussions of the form factors early, the main $SU(3)_{F}$ breaking
effects come from the phase space difference. For instance, from FIG. 4, the
phase space of $\Xi_{cc}^{++}\to\Lambda_{c}^{+}$ is 1.3 times larger than the
one of $\Xi_{cc}^{++}\to\Xi_{c}^{+}$, which partly explains the ratios in Eq.
(19) .

Table 6:  The decay widths in units of $10^{-14}$ GeV along with the ones in
the literature.

$B_{cc}\rightarrow B_{c}e^{+}\nu_{e}$ |  This work |  MBM Perez-Marcial:1989sch  |  HQSS Albertus:2011xz  |  LFQM Wang:2017mqp  |  LFQM Hu:2020mxk  |  QCDSR Shi:2019hbf
---|---|---|---|---|---|---
$\Xi^{++}_{cc}\rightarrow\Xi^{+}_{c}e^{+}\nu_{e}$ |  $5.11\pm 0.64$ |  7.36 |  5.78 |  11.50 |  8.74 |  $7.72\pm 3.70$
$\Xi^{+}_{cc}\rightarrow\Xi^{0}_{c}e^{+}\nu_{e}$ |  $5.08\pm 0.64$ |  7.36 |  5.73 |  11.40 |  8.63 |  $7.72\pm 3.70$
$\Xi^{++}_{cc}\rightarrow\Xi^{\prime+}_{c}e^{+}\nu_{e}$ |  $10.92\pm 0.81$ |  17.56 |  9.64 |  12.80 |  14.30 |  $5.31\pm 3.52$
$\Xi^{+}_{cc}\rightarrow\Xi^{\prime 0}_{c}e^{+}\nu_{e}$ |  $10.85\pm 0.81$ |  13.02 |  9.57 |  12.70 |  14.10 |  $5.31\pm 3.52$
$\Omega^{+}_{cc}\rightarrow\Omega^{0}_{c}e^{+}\nu_{e}$ |  $22.09\pm 1.63$ |  26.76 |  18.61 |  25.50 |  28.00 |  $12.50\pm 8.02$
$\Xi^{++}_{cc}\rightarrow\Lambda^{+}_{c}e^{+}\nu_{e}$ |  $0.34\pm 0.06$ |  0.46 |  0.32 |  1.05 |  0.80 |  $0.76\pm 0.37$
$\Omega^{+}_{cc}\rightarrow\Xi^{0}_{c}e^{+}\nu_{e}$ |  $0.32\pm 0.04$ |  0.46 |  0.27 |  0.81 |  0.59 |  $0.61\pm 0.28$
$\Xi^{++}_{cc}\rightarrow\Sigma^{+}_{c}e^{+}\nu_{e}$ |  $0.76\pm 0.06$ |  0.78 |  0.52 |  0.96 |  1.09 |  $0.49\pm 0.29$
$\Omega^{+}_{cc}\rightarrow\Xi^{\prime 0}_{c}e^{+}\nu_{e}$ |  $0.77\pm 0.06$ |  0.91 |  0.49 |  0.93 |  1.03 |  $0.56\pm 0.35$
$\Xi^{+}_{cc}\rightarrow\Sigma^{0}_{c}e^{+}\nu_{e}$ |  $1.52\pm 0.12$ |  1.50 |  1.04 |  1.91 |  2.17 |  $0.99\pm 0.58$

The computed decay widths, along with those in the literature, are shown in
Table 6, where Ref. Perez-Marcial:1989sch  computes the form factors by the
MBM, Ref. Albertus:2011xz  analyzes the decays with the heavy quark spin
symmetry (HQSS), Refs. Wang:2017mqp ; Hu:2020mxk  adopt the LFQM with
different sets of the parameter input, and Ref. Shi:2019hbf  calculates the
decay widths by the QCDSR. The decay width of
$\Xi_{cc}^{++}\to\Xi_{c}^{(\prime)+}e^{+}\nu_{e}$ is slightly larger than the
one of $\Xi_{cc}^{+}\to\Xi_{c}^{(\prime)0}e^{+}\nu_{e}$ as $M_{\Xi_{cc}^{++}}$
is 2 MeV larger than $M_{\Xi_{cc}^{+}}$. For the $c\to s$ transition, our
results of the decay widths are well consistent with the ones of the HQSS but
systematically lower than those in the LFQM Hu:2020mxk  and MBM Perez-
Marcial:1989sch  by a factor of 1.5. We note that in the MBM Perez-
Marcial:1989sch , the $q^{2}$ dependencies of the form factors are put by hand
and independent of the spectator quarks. For instance, the $q^{2}$
dependencies of $\Lambda_{c}\to\Lambda$ and $\Xi_{cc}^{++}\to\Xi_{c}^{+}$ are
taken to be the same in the MBM Perez-Marcial:1989sch . However, we emphasize
that the spectator effects of the charm quark and others are very different,
as shown explicitly in $D_{q}^{v}$ of Eq. (8). The exponential factor of
$\exp(-2iE_{c}vz)$ deviates largely to those of $\exp(-2iE_{u,d,s}vz)$. In
particular, we find that it suppresses the decay widths by more than $40\%$,
which causes the deviations between the results of the MBM and ours.

##  IV Conclusion

We have studied the semileptonic decays of the doubly charmed baryons.
Explicitly, we have found that
$\Gamma(\Xi_{cc}^{++}{\rightarrow}\Xi_{c}^{+}e^{+}\nu_{e},\Xi_{c}^{\prime+}e^{+}\nu_{e},\Lambda_{c}^{+}e^{+}\nu_{e},\Omega_{c}^{+}e^{+}\nu_{e})=(5.1\pm
0.1,11\pm 1,0.34\pm 0.06,0.76\pm 0.06)\times 10^{-14}$ GeV,
$\Gamma(\Xi_{cc}^{+}\rightarrow\Xi_{c}^{0}e^{+}\nu_{e},\Xi_{c}^{\prime
0}e^{+}\nu_{e},\Sigma_{c}^{0}e^{+}\nu_{e})=(5.1\pm 0.6,11\pm 1,1.5\pm
0.1)\times 10^{-14}$ GeV, and
$\Gamma(\Omega_{cc}^{+}\rightarrow\Omega_{c}^{0}e^{+}\nu_{e},\Xi_{c}^{0}e^{+}\nu_{e},\Xi_{c}^{\prime
0}e^{+}\nu_{e})=(22\pm 2,0.32\pm 0.04,0.77\pm 0.06)\times 10^{-14}$ GeV. We
have discussed the $SU(3)_{F}$ breaking effects regarding the aspects of (i)
the phase spaces differences, (ii) the spectator quark effects, and (iii) the
overlappings of the transited quarks. We have shown that the other breaking
effects are negligible compared to the phase space differences. In particular,
the form factors well respect the $SU(3)_{F}$ relations using $\omega$ as the
variables. In addition, we have obtained that
$\Gamma(\Xi_{cc}^{++}\to\Lambda_{c}^{+}e^{+}\nu_{e})V_{cs}^{2}/\Gamma(\Xi_{cc}^{++}\to\Xi_{c}^{+}e^{+}\nu_{e})V_{cd}^{2}=1.24$,
which is expected to be $1$ under the exact $SU(3)_{F}$ symmetry.

The behaviors of the parity violating and conserving partial decay widths have
been examined. Accordingly, we have demonstrated that the partial decay width
of $\Xi_{cc}^{++}\to\Xi_{c}^{+}e^{+}\nu_{e}$ is smoother than others, which
can be testified in future experiments. We have also shown that the spectator
effects of the charm quark suppress the decay widths by $40\%$ and shall not
be taken to be the same as the other quarks.

###### Acknowledgements.

This work is supported in part by the National Key Research and Development
Program of China under Grant No. 2020YFC2201501 and the National Natural
Science Foundation of China (NSFC) under Grant No. 12147103.

## References

  * (1) M. Mattson et al. [SELEX], Phys. Rev. Lett. 89, 112001 (2002).
  * (2) A. Ocherashvili et al. [SELEX], Phys. Lett. B 628, 18 (2005).
  * (3) S. P. Ratti, Nucl. Phys. B Proc. Suppl. 115, 33 (2003).
  * (4) B. Aubert et al. [BaBar], Phys. Rev. D 74, 011103 (2006).
  * (5) Y. Kato et al. [Belle], Phys. Rev. D 89, 052003 (2014).
  * (6) R. Aaij et al. [LHCb], Phys. Rev. Lett. 119, 112001 (2017).
  * (7) R. Aaij et al. [LHCb], Phys. Rev. Lett. 121, 052002 (2018).
  * (8) R. Aaij et al. [LHCb], Phys. Rev. Lett. 121, 162002 (2018).
  * (9) R. Perez-Marcial, R. Huerta, A. Garcia and M. Avila-Aoki, Phys. Rev. D 40, 2955 (1989).
  * (10) C. Albertus, E. Hernández and J. Nieves, Phys. Lett. B 704, 499 (2011).
  * (11) W. Wang, F. S. Yu and Z. X. Zhao, Eur. Phys. J. C 77, 781 (2017).
  * (12) L. Meng, N. Li and S. l. Zhu, Eur. Phys. J. A 54, 143 (2018).
  * (13) W. Wang, Z. P. Xing and J. Xu, Eur. Phys. J. C 77, 800 (2017).
  * (14) T. Gutsche, M. A. Ivanov, J. G. Körner and V. E. Lyubovitskij, Phys. Rev. D 96, 054013 (2017).
  * (15) H. S. Li, L. Meng, Z. W. Liu and S. L. Zhu, Phys. Lett. B 777, 169 (2018).
  * (16) Z. H. Guo, Phys. Rev. D 96, 074004 (2017).
  * (17) Q. F. Lü, K. L. Wang, L. Y. Xiao and X. H. Zhong, Phys. Rev. D 96, 114006 (2017).
  * (18) L. Y. Xiao, K. L. Wang, Q. f. Lu, X. H. Zhong and S. L. Zhu, Phys. Rev. D 96, 094005 (2017).
  * (19) N. Sharma and R. Dhir, Phys. Rev. D 96, 113006 (2017).
  * (20) Y. L. Ma and M. Harada, J. Phys. G 45, 075006 (2018).
  * (21) F. S. Yu, H. Y. Jiang, R. H. Li, C. D. Lü, W. Wang and Z. X. Zhao, Chin. Phys. C 42, 051001 (2018).
  * (22) L. Meng, H. S. Li, Z. W. Liu and S. L. Zhu, Eur. Phys. J. C 77, 869 (2017).
  * (23) X. H. Hu, Y. L. Shen, W. Wang and Z. X. Zhao, Chin. Phys. C 42, 123102 (2018).
  * (24) Y. J. Shi, W. Wang, Y. Xing and J. Xu, Eur. Phys. J. C 78, 56 (2018).
  * (25) L. Y. Xiao, Q. F. Lü and S. L. Zhu, Phys. Rev. D 97, 074005 (2018).
  * (26) D. L. Yao, Phys. Rev. D 97, 034012 (2018).
  * (27) U. Özdem, J. Phys. G 46, 035003 (2019).
  * (28) A. Ali, A. Y. Parkhomenko, Q. Qin and W. Wang, Phys. Lett. B 782, 412 (2018).
  * (29) J. M. Dias, V. R. Debastiani, J. J. Xie and E. Oset, Phys. Rev. D 98, 094017 (2018).
  * (30) R. H. Li and C. D. Lu, arXiv:1805.09064 [hep-ph].
  * (31) R. Dhir and N. Sharma, Eur. Phys. J. C 78, 743 (2018).
  * (32) A. Cerri, V. V. Gligorov, S. Malvezzi, J. Martin Camalich, J. Zupan, S. Akar, J. Alimena, B. C. Allanach, W. Altmannshofer and L. Anderlini, et al. CERN Yellow Rep. Monogr. 7, 867 (2019).
  * (33) A. V. Berezhnoy, A. K. Likhoded and A. V. Luchinsky, Phys. Rev. D 98, 113004 (2018).
  * (34) L. J. Jiang, B. He and R. H. Li, Eur. Phys. J. C 78, 961 (2018).
  * (35) Q. A. Zhang, Eur. Phys. J. C 78, 1024 (2018).
  * (36) L. Meng and S. L. Zhu, Phys. Rev. D 100, 014006 (2019).
  * (37) T. Gutsche, M. A. Ivanov, J. G. Körner, V. E. Lyubovitskij and Z. Tyulemissov, Phys. Rev. D 99, 056013 (2019).
  * (38) X. H. Hu and Y. J. Shi, Eur. Phys. J. C 80, 56 (2020).
  * (39) X. H. Hu, R. H. Li and Z. P. Xing, Eur. Phys. J. C 80, 320 (2020).
  * (40) Y. J. Shi, Y. Xing and Z. X. Zhao, Eur. Phys. J. C 79, 501 (2019).
  * (41) Y. J. Shi, W. Wang and Z. X. Zhao, Eur. Phys. J. C 80, 568 (2020).
  * (42) Z. X. Zhao, Eur. Phys. J. C 78, 756 (2018).
  * (43) Y. Xing and R. Zhu, Phys. Rev. D 98, 053005 (2018).
  * (44) A. Ali, Q. Qin and W. Wang, Phys. Lett. B 785, (2018).
  * (45) M. Z. Liu, Y. Xiao and L. S. Geng, Phys. Rev. D 98, 014040 (2018).
  * (46) Z. P. Xing and Z. X. Zhao, Phys. Rev. D 98, 056002 (2018).
  * (47) R. Zhu, X. L. Han, Y. Ma and Z. J. Xiao, Eur. Phys. J. C 78, 740 (2018).
  * (48) C. Q. Geng, C. W. Liu and T. H. Tsai, Phys. Rev. D 102, 034033 (2020).
  * (49) C. W. Liu and C. Q. Geng, JHEP 01, 128 (2022).
  * (50) C. Q. Geng, X. N. Jin and C. W. Liu, Phys. Rev. D 106, 053006 (2022).
  * (51) C. Q. Geng, X. N. Jin and C. W. Liu, arXiv:2210.15588 [hep-ph].
  * (52) J. Zhang, X. N. Jin, C. W. Liu and C. Q. Geng, arXiv:2210.16825 [hep-ph].
  * (53) C. W. Liu and C. Q. Geng, arXiv:2205.08158 [hep-ph].
  * (54) A. Kadeer, J. G. Korner and U. Moosbrugger, Eur. Phys. J. C 59, 27 (2009).
  * (55) C. Q. Geng, C. W. Liu, T. H. Tsai and S. W. Yeh, Phys. Lett. B 792, 214 (2019).
  * (56) J. G. Korner, M. Kramer and D. Pirjol, Prog. Part. Nucl. Phys. 33, 787 (1994).
  * (57) R. L. Workman et al. [Particle Data Group], PTEP 2022, 083C01 (2022).
  * (58) W. X. Zhang, H. Xu and D. Jia, Phys. Rev. D 104, 114011 (2021).