# A Model-Based Approach to
Shot Charts Estimation in Basketball

Luca Scrucca
Department of Economics, Università degli Studi di Perugia
Dimitris Karlis
Department of Statistics, Athens University of Economics  0000-0003-3826-0484
0000-0003-3711-1575

###### Abstract

Shot charts in basketball analytics provide an indispensable tool for
evaluating players’ shooting performance by visually representing the
distribution of field goal attempts across different court locations. However,
conventional methods often overlook the bounded nature of the basketball
court, leading to inaccurate representations, particularly along the
boundaries and corners. In this paper, we propose a novel model-based approach
to shot chart estimation and visualization that explicitly considers the
physical boundaries of the basketball court. By employing Gaussian mixtures
for bounded data, our methodology allows to obtain more accurate estimation of
shot density distributions for both made and missed shots. Bayes’ rule is then
applied to derive estimates for the probability of successful shooting from
any given locations, and to identify the regions with the highest expected
scores. To illustrate the efficacy of our proposal, we apply it to data from
the 2022-23 NBA regular season, showing its usefulness through detailed
analyses of shot patterns for two prominent players.

Keywords: Shot charts; visualization of shooting patterns; density estimation;
transformation-based Gaussian mixtures for bounded data; probability of
successful shooting; expected points scored.

###### Contents

  1. 1 Introduction
  2. 2 Methods
    1. 2.1 Model specification
    2. 2.2 Estimation and model selection
  3. 3 Applications
    1. 3.1 Stephen Curry
    2. 3.2 Joel Embiid
  4. 4 Conclusions

##  1 Introduction

Basketball is among the most popular sports game worldwide. It not only enjoys
widespread popularity as a sport but has also generated substantial economic
benefits through its associated industries. The National Basketball
Association (NBA) is widely recognized as the world’s leading league,
attracting international interest with tremendous amounts spent in related
marketing. In Europe, the Euroleague represents the pinnacle of professional
men’s club basketball competition and is regarded as the top-tier men’s league
on the continent. The increasing interest on basketball has led quite early to
the development of advanced statistical methodologies for measuring
performance (Kubatko et al.,, 2007), while several other proposal have been
made after this. For a broad picture of academic and non-academic research on
basketball analytics we recommend the book of Zuccolotto and Manisera, (2020)
and the broad review paper by Terner and Franks, (2021).

One of the basic characteristics of basketball is that it is a fast-paced
contact game in which the players are constantly moving in heated
confrontations, thus leading to quick transitions from defense to offense or
vice versa. In practice basketball is a game of space (see e.g. Goldsberry,,
2012, p1). The teams that makes better use of spatial aspects can have an
advantage and hence several tactics related to better enhancement of spatio-
temporal game aspects (Sandholtz et al.,, 2020).

Advancements in sports information systems and technology has allowed the
collection of a number of detailed spatio-temporal data that capture various
aspects of basketball Papalexakis and Pelechrinis, (2018); Shortridge et al.,
(2014). Such data can help considerably to understand the game and the effects
of space on that while they also provide interesting information for all
stakeholders of the game, including trainers, team managers, players, scouters
of new players, spectators and journalists. Visualizations of basketball games
can provide important information about the game (Perin et al.,, 2018). An
increasing number of visualization research has been conducted that includes
as visual analysis of player trajectories, visualization of field goals of a
player, and visualization of basic statistics of different players in
different games (Chen et al.,, 2016).

Shots are a key-ingredient of the sport. The final score of a team is defined
by the number of successful shots and their quality, (2 or 3 points plus the 1
point for free throws). As such considerable interest has been made on
understanding and predicting shot tactics and success. For example, Zuccolotto
et al., (2018) utilized several techniques to model scoring probability under
high-pressure conditions in basketball based play-by-play data from the
Italian “Serie A2” Championship 2015/2016. Shortridge et al., (2014) discussed
and proposed different measures about shot efficiency that take into account
the spatial effect and they also proposed visualizations related to shot
efficiency. Oughali et al., (2019) tried to predict shot success based on
several machine learning approaches. Fichman and O’Brien, (2019) discussed the
optimal shot selection strategy for a basketball team. Jiao et al., (2021)
proposed a marked spatial point process for modeling basketball shots based on
the observation that the success rate of a basketball shots may be higher at
locations where a player makes more shots. Related to the spatial aspect are
also the so-called corner 3’s, which are those shots that while producing 3
points are taken closer to the basket, thus allowing for larger probability of
success and distinguished tactic for that shots (Pelechrinis and Goldsberry,,
2021).

Visualizing shots can be a powerful tool for better understanding the
different tactics. Quite early it has been noted that spatial visualizations
like shot charts can be very valuable to reveal the tactical performance of
the teams and hence be a valuable tool in the hands of trainers (Reich et
al.,, 2006). For example, shot charts, that is, maps capturing locations of
(made or missed) shots, and spatio-temporal trajectories for the players on
the court can capture information about the offensive and defensive
tendencies, as well as, schemes used by a team. Characterization of these
processes is important for player and team comparisons, scouting, game
preparation etc. Since then there has been extensive literature related to
shots in basketball including effective visualizations that can produce
insights. Since shots are the most important aspect as it leads to gaining
points, it is quite common to produce statistics related to shot success but
also to shot patterns, including spatio-temporal aspects of shots. The radical
choice of most teams towards different shooting styles that include more
3-points is perhaps partially due to the improved visualizations available.

_Shot charts_ in basketball analytics are a fundamental tool for visually
examining the distribution of players’ field goal attempts and their
efficiency in different court locations. Typically such charts visualize the
locations of all shots made, either by cutting the courts in cells (or
hexagons or other areas) of the same size and representing their frequency by
some color (Chu,, 2010). Ehrlich and Sanders, (2024) proposed alternative ways
to improve the information provided using some model based estimate of the
shot efficiency. See also the work on Fu and Stasko, (2024) about the
importance of visualizing the shooting performance.

However, despite their utility, current shot charts representations face
certain limitations. Predominantly, when constructed from observed data using
hexagons or derived from standard density estimation procedures, they often
fail to take into account the bounded nature of the basketball court. This
limitation can result in misleading representations, especially at the
boundaries and corners of the court. Consequently, the analysis may not fully
account for the contextual constraints imposed by the court’s physical
boundaries, potentially skewing the assessment of shooting patterns and
efficiency, particularly in areas where players are more inclined to attempt
shots due to strategic advantages or positional play. These discrepancies
underscore the importance of refining shot charts methodologies to accurately
depict the nuanced spatial dynamics inherent in basketball shot data.

Figure 1a illustrates the approach commonly used to visualize the spatial
distribution of a player’s shot attempts. Typically, a two-dimensional kernel
density estimate is used (Scott,, 2009). However, if the boundaries of a
basketball court are not taken into account, some artifacts are noticeable,
particularly in the corner 3-point areas, behind the backboard, and in front
of the center 3-point line. In contrast, by adopting the methodology proposed
in this paper we obtain a density estimate that remains confined within the
physical boundaries of the basketball court and, by providing more accurate
spatial estimates, effectively remove the above mentioned artifacts.

Figure 1: Distribution of Stephen Curry’s shot attempts during the 2022-23 NBA
regular season. Panel (a) shows the density estimate obtained using two-
dimensional kernel density estimation, while panel (b) the estimate obtained
by fitting Gaussian mixtures for bounded data, which allows the physical
boundaries of the basketball court to be taken into account.

To summarize, in our proposal we embrace a model-based approach to shot charts
estimation and visualization that: 1) employs Gaussian mixtures to estimate
the density distribution of made and missed shots; 2) takes into account the
physical boundaries of the basketball court; 3) applies Bayes’ rule to derive
estimates for the probability of successful shooting from any location; and 4)
identifies regions with the highest expected scores.

The paper is organized as follows: Section 2 describes the model and the
estimation procedure; Section 3 illustrates the proposed methodology using the
data from the 2022-23 NBA regular season for two players, namely Stephen
Curry, perhaps the GOAT (_Greatest Of All Time_) 3-point shooter, and Joel
Embiid, the MVP (_Most Valuable Player_) for that season; the final section
contains some concluding remarks and potential future extensions to this
paper.

##  2 Methods

Shot charts in basketball analytics provide a visual representation of a
player or team’s shooting performance by analyzing data on shots attempted
from various spots on the court. However, basketball courts come in many
different sizes. In the NBA, the court is 94 by 50 feet (28.7 by 15.2 m),
while under the International Basketball Federation (FIBA) rules, the court is
slightly smaller, measuring 28 by 15 meters (91.9 by 49.2 ft). The 3-point
line is also different, being located at 23 feet 9 inches (7.24 m) from the
center of the basket in the NBA (22 ft or 6.70 m at the corner), and 6.75 m
(22 ft 1.75 in) for FIBA (6.60 m or 21 ft 8 in at the corner). As discussed in
reference to the results shown in Figure 1, these physical constraints on the
basketball court must be given due consideration in density estimation from
shots spatial information.

Figure 2 shows the shots attempted by Stephen Curry (left panel) and Joel
Embiid (right panel) during the 2022-23 NBA regular season with each data
point marked by shot outcome. The significant presence of shots from beyond
the arc of the 3-point line is evident for Curry, while a greater number of
attempts in the mid-range can be traced for Embiid. However, partly because of
the presence of overlapping points, it is difficult to identify the spots from
which the two players preferentially and most effectively shoot at the basket.
Thereby, density estimation becomes crucial for gaining insights into shooting
patterns and optimizing players performance, or to set up an efficient defense
that limits shooting opportunities at preferred positions.

Figure 2: Shots attempted by Stephen Curry and Joel Embiid during the 2022-23
NBA regular season.

Gaussian mixtures (McLachlan and Peel,, 2000; Fraley and Raftery,, 2002) offer
a semiparametric approach to density estimation. In this approach, the density
of the data is expressed as a convex linear combination of one or more
probability density functions. Gaussian mixtures are a popular choice obtained
by using Gaussian densities as components of the mixture.

Gaussian Mixtures Models (GMMs) carry several advantages due to their
intrinsic probabilistic generative nature. In particular, maximum likelihood
estimation of parameters is available via the EM algorithm (see Section 2.2),
with estimates that remain efficient even for multidimensional data. Moreover,
GMMs require no hyperparameters tuning, with the problem of selecting the
complexity of the mixture that can be recast as model selection problem (see
Section 2.2).

Despite the fact that GMMs can approximate any continuous density with
arbitrary accuracy, provided the mixture has an adequate number of components
(see Ferguson,, 1983; Escobar and West,, 1995, among others), it is crucial to
consider the inherent physical constraints of the basketball half-court when
estimating densities in shot charts. This can be achieved by adopting the
transformation-based approach to Gaussian mixture density estimation for
bounded data proposed by Scrucca, (2019). This approach is particularly
suitable for this scenario because it explicitly considers the natural bounds
of the basketball half-court. Next section briefly reviews the methodology of
our proposal.

###  2.1 Model specification

The transformation-based approach for GMMs discussed in Scrucca, (2019)
extends density estimation using mixture modeling to the case of bounded
variables. The basic idea is to carry out density estimation not on the
original data but on appropriately transformed scale. Then, the density for
the original data can be simply obtained by a change of variables.

Let $(x_{i},y_{i})$ denote the coordinates of the position on the court where
a player attempts a shot, for $i=1,\dots,n$, where $n$ is the number of shots
attempted, and $C_{i}=\\{0,1\\}$ the corresponding binary outcome, where $1$
indicates a made shot and $0$ a missed shot. Consider the coordinate-wise
range-logit transformation defined as

$t(x,y)=\begin{bmatrix}t(x)\\\
t(y)\end{bmatrix}=\begin{bmatrix}\log\left(\frac{x-\ell_{x}}{u_{x}-x}\right)\\\\[4.30554pt]
\log\left(\frac{y-\ell_{y}}{u_{y}-y}\right)\\\ \end{bmatrix},$

where $(\ell_{x},u_{x})$ and $(\ell_{y},u_{y})$ are the lower and upper bounds
along, respectively, the $x$-axis and the $y$-axis. Figure 3 shows the
coordinates of the half-court we consider in our study for a 94 by 50 feet NBA
basketball court. Thus, half-court court boundaries are set at
$(\ell_{x}=-25,u_{x}=25)$ and $(\ell_{y}=0,u_{y}=47)$.

Figure 3: NBA half-court dimensions and coordinates (in feet) used in the
present paper.

In the logit-range transformed scale the density of a shot from location
$(x,y)$ can be expressed using the following Gaussian mixture

$h(t(x,y))=\sum_{g=1}^{G}\pi_{g}\;\mathcal{N}(t(x),t(y)\mid\boldsymbol{\mu}_{g},\boldsymbol{\Sigma}_{g}),$
(1)

where $G$ is the number of mixture components, $\pi_{g}$ the mixing
probabilities (with $\pi_{g}>0$ and $\sum_{g=1}^{G}\pi_{g}=1$),
$\boldsymbol{\mu}_{g}$ and $\boldsymbol{\Sigma}_{g}$, respectively, the mean
vector and covariance matrix for Gaussian component $g$. Upon re-expressing it
in the original coordinate scale, the density function can be formulated as
follows:

$f(x,y)=h(t(x,y))\times|\boldsymbol{J}(t(x,y))|,$ (2)

where $|\boldsymbol{J}(t(x,y))|$ is the Jacobian of the transformation.
According to the coordinate-wise transformation approach adopted, the matrix
of first derivatives is diagonal, so the Jacobian reduces to the product of
first derivatives, i.e.

$|\boldsymbol{J}(t(x,y))|=t^{\prime}(x)\times
t^{\prime}(y)=\left(\frac{1}{x-\ell_{x}}+\frac{1}{u_{x}-x}\right)\times\left(\frac{1}{y-\ell_{y}}+\frac{1}{u_{y}-y}\right).$

The density in the transformed coordinates from (1) can be estimated
separately for made ($C=1$) and missed shots ($C=0$), and then back-
transformed in the original scale using (2). Subsequently, the probability of
scoring a basket from a specific location can be calculated using Bayes’
theorem. Specifically, the density at location $(x,y)$ for shot outcome $C=k$,
with $k=\\{0,1\\}$, is given by

$f(x,y\mid
C=k)=\left(\sum_{g=1}^{G_{k}}\pi_{g|k}\;\mathcal{N}(t(x),t(y)\mid\boldsymbol{\mu}_{g|k},\boldsymbol{\Sigma}_{g|k})\right)\times|\boldsymbol{J}(t(x,y))|,$
(3)

where $G_{k}$ represents the number of mixture components for shot outcome
$C=k$. The $\pi_{g|k}$ terms denote the mixing probabilities for outcome $C=k$
($\pi_{g|k}>0$ and $\sum_{g=1}^{G_{k}}\pi_{g|k}=1$), and
$\boldsymbol{\mu}_{g|k}$ along with $\boldsymbol{\Sigma}_{g|k}$ stand for the
mean vectors and covariance matrices for component $g$ of outcome $C=k$.

Once the density is estimated for both made shots, $f(x,y\mid C=1)$, and
missed shots, $f(x,y\mid C=0)$, the probability of a successful shot can be
obtained using Bayes’ rule as:

$\Pr(C=1|x,y)=\frac{\tau_{1}f(x,y\mid C=1)}{\tau_{0}f(x,y\mid
C=0)+\tau_{1}f(x,y\mid C=1)},$ (4)

where $\tau_{1}$ and $\tau_{0}$ are the outcome prior probabilities of,
respectively, made and missed shots.

The estimated probabilities of making shots from various positions on the
court in (4) can be multiplied by the point value of those shots (2 or 3
points) to derive the _expected points scored_ :

$\text{EPS}(x,y)=\begin{cases}2\times\Pr(C=1|x,y)&\text{if $(x,y)$ is within
the 3-point line}\\\ 3\times\Pr(C=1|x,y)&\text{if $(x,y)$ is beyond the
3-point line}\end{cases}$

This represents an important metric which provides valuable insights into
offensive strategies and efficiency from different positions on the court.

###  2.2 Estimation and model selection

Estimation of unknown parameters,
$\pi_{g|k},\boldsymbol{\mu}_{g|k},\boldsymbol{\Sigma}_{g|k}$, for
$g=1,\ldots,G_{k}$ and $k=\\{0,1\\}$, in (3) can be pursued via the EM
algorithm. For details see Scrucca, (2019, Sec. 3.3). Moreover, outcome prior
probabilities, $\tau_{1}$ and $\tau_{0}$, in (4) can be estimated from,
respectively, the proportions of made and missed shots.

Without imposing any constraints on the covariance matrices of Gaussian
components, empirical evidence suggests the inclusion of a Bayesian
regularization prior to increase smoothness of the density estimate over the
basketball court and avoid singularities and degeneracies in maximization of
the likelihood. This can be achieved by adopting the approach of Fraley and
Raftery, (2007), who proposed weekly informative conjugate priors to
regularize the estimation process. The EM algorithm can still be used for
model fitting, but maximum likelihood estimates (MLEs) are replaced by maximum
a posteriori (MAP) estimates. For details see Scrucca et al., (2023, Sec.
7.2).

A crucial aspect in mixture modeling is the choice of the number of mixture
components, $G_{k}$, for each outcome. Typically, the Bayesian Information
Criterion (BIC; Schwarz,, 1978) is used as model selection criterion in finite
mixture models. This choice is justified by Keribin, (2000), who demonstrated
that BIC is consistent for choosing the number of components in a mixture
model, assuming a bounded likelihood (which is guaranteed by the introduction
of the regularized prior mentioned earlier). However, when Bayesian
regularization is introduced a slightly modified version of BIC should be used
for model selection, with the maximized log-likelihood replaced by the log-
likelihood evaluated at the MAP.

##  3 Applications

In this section we analyzed the player-by-player data of some selected NBA
players for the 2022-23 NBA regular season. The data are obtained from the R
package hoopR (Gilani,, 2023), which provides easy access to data available on
ESPN analytics at https://www.espn.com/nba/.

###  3.1 Stephen Curry

Figure 4 shows the estimated densities for made (a) and missed (b) shots,
respectively $f(x,y|C=1)$ and $f(x,y|C=0)$ from (3). Regions are highlighted
by highest density regions (HDRs) corresponding to specific percentages of the
data. Note, however, that these cannot be directly compared, but they can be
used for computing shot probabilities using (4). Required prior probabilities
are estimated using proportions of made and missed shots during the regular
season, giving $\widehat{\tau}_{1}=0.4724$ and $\widehat{\tau}_{0}=0.5276$.

Figure 4: Highest density regions (HDRs) from mixture-based estimated
densities for made (a) and missed (b) shots for Stephen Curry during the
2022-23 NBA regular season.

Figure 5a presents the estimated shot chart highlighting regions of high and
low probability for made shots by Stephen Curry. The chart reveals a
remarkable consistency in Curry’s shooting ability across various regions,
with particularly high probabilities close to the basket and extending well
beyond the three-point line. Notably, two key exceptions emerge: very far
locations and positions approximately 2-3 feet from the three-point line at
the top of the key. Additionally, a closer look suggests a reduced probability
in the right mid-range area.

Building upon the estimated shot chart discussed above, Figure 5b presents the
corresponding graph of expected points scored. This visualization highlights
regions of high scoring efficiency, primarily concentrated around close-range
shots and extending to all areas beyond the three-point arc, with a notable
preference for the left side. Interestingly, these high-efficiency regions
align with areas of higher shot probability observed in Figure 5a, while
regions with lower expected points coincide with areas of lower shot
probability.

Lastly, the table below Figure 5 summarizes key statistics for both two-point
and three-point attempts: number of attempts, observed made shot proportions,
estimated average probabilities, observed average points per attempt, and
estimated expected score. Notably, the empirical and estimated values exhibit
close agreement, highlighting the accuracy of the model. These data showcase
Stephen Curry’s remarkable offensive efficiency beyond the three-point arc,
reflected in an estimated expected score of 1.27 points per attempt compared
to 1.10 points for closer shots.

Figure 5: Shot charts depicting (a) estimated probabilities and (b) expected
points scored per attempt for Stephen Curry during the 2022-23 NBA regular
season. The table below the charts reports a summary of empirical and
estimated key statistics for both two-point and three-point shots.

###  3.2 Joel Embiid

As a second player we analyze Joel Embiid of the Philadelphia 76ers. Compared
to Stephen Curry’s role as shooting guard, Embiid plays as a center, is much
taller and stronger physically, but at the same time has an excellent aptitude
for shooting from mid-range and beyond the arc. During the 2022-23 regular
season Embiid had the highest average points per game (30.6) and won the MVP
award.

Charts in Figure 6 show the highest density regions (HDRs) obtained from
mixture-based estimated densities for made (a) and missed (b) shots. The
majority of shots are concentrated in the paint and near the free-throw line,
while beyond the three-point arc Embiid’s favorite position appears to be the
central one.

Figure 6: Highest density regions (HDRs) from mixture-based estimated
densities for made (a) and missed (b) shots for Joel Embiid during the 2022-23
NBA regular season.

Embiid’s shooting efficiency is very high, as can be seen from the chart in
Figure 7a, with estimated success probabilities well above 50% in almost all
mid-range and close-to-basket positions. For three-point shots, two preferred
positions with very high success rates emerge: in front of the basket and
slightly to the right. In other positions beyond the arc, the estimated
probabilities appear significantly lower.

In terms of expected points scored from different positions, the most
profitable ones are near the basket, thanks to the high shooting percentage,
and those with the highest efficiency beyond the arc, due to the fact that
more points are obtained for each basket made.

Figure 7: Shot charts depicting (a) estimated probabilities and (b) expected
points scored per attempt for Joel Embiid during the 2022-23 NBA regular
season. The table below the charts reports a summary of empirical and
estimated key statistics for both two-point and three-point shots.

Finally, it is interesting to compare the different shooting choices of
Stephen Curry and Joel Embiid, and their relative effectiveness and efficiency
(see tables at the bottom of Figures 7 and 5). Curry favors long-distance
shots, attempting approximately 70% more three-point shots (640 attempts
compared to 376), while Embiid notably focuses on within the arc shots (1082
attempts compared to 209). Curry exhibits high estimated probabilities of
scoring for both 2-point (54%) and, especially, 3-point (42%) shots, whereas
Embiid demonstrates a higher percentage in the mid- and close-range shots
(58%), but only a moderate 3-point percentage (33%), which is nonetheless
excellent for his role. These translate into excellent expected points for
both 2-point and 3-point attempts, with Curry astonishingly averaging about
1.27 points per 3-point attempt.

##  4 Conclusions

The availability of good quality spatial data in sports has increased a lot
their usage, including spatial visualizations. For example, we are all
familiar with heatmaps that represent the location density of players as an
attempt to describe their playing behavior but also to identify tactics. Shot
charts are pivotal tools in basketball analytics, offering valuable insights
into players’ shooting tendencies and efficiencies across different areas of
the court. Existing shot chart representations often fall short in accurately
capturing shooting spatial distribution, primarily due to their inability to
account for the bounded nature of the basketball court. In the present paper
we proposed a new approach that employs Gaussian mixtures to estimate the
density distribution using a transformation-based approach that takes into
account the physical boundaries of the court. We demonstrate the effectiveness
of our methodology through case studies involving real-world data from the
2022-23 NBA regular season.

The easiness of applying and fitting Gaussian mixtures to estimate the spatial
distribution creates additional opportunities. An explicit extension of the
proposed work relates to all other sports where spatial location data are
used. Recall also that this may extend to other non-sport related applications
where boundaries need to be taken into account. As a proposal for further
investigation, we also mention the use of mixture models as the basis for
_conditional heatmaps_. So far, most of the sports visualization based on
tracking data is based on the position of a player in the court. Sometimes it
is interesting to visualize the conditional heatmap, i.e. the position of a
player conditional on the position of some other player. For example, in
basketball (but also in football and other team sports) this can reveal
important tactical aspects and space creation strategies for the teams, which
is an important ingredient of the game. Gaussian mixtures allow easily to work
on that since one can easily obtain/estimate the joint distribution of the
location of two players as the joint distribution in 4 dimensions, allowing
also for dependence. From the joint distribution one can estimate the
conditional density in a straightforward manner and thus produce a conditional
heatmap.

## References

  * Chen et al., (2016) Chen, W., Lao, T., Xia, J., Huang, X., Zhu, B., Hu, W., and Guan, H. (2016).  Gameflow: narrative visualization of NBA basketball games.  IEEE Transactions on Multimedia, 18(11):2247–2256.
  * Chu, (2010) Chu, S. (2010).  Information visualization in the NBA: The shot chart.  Technical report, University of California, Berkeley.
  * Ehrlich and Sanders, (2024) Ehrlich, J. and Sanders, S. (2024).  Estimating NBA team shot selection efficiency from aggregations of true, continuous shot charts: A generalized additive model approach.  Available at SSRN: https://ssrn.com/abstract=4697111.
  * Escobar and West, (1995) Escobar, M. D. and West, M. (1995).  Bayesian density estimation and inference using mixtures.  Journal of the American Statistical Association, 90(430):577–588.
  * Ferguson, (1983) Ferguson, T. (1983).  Bayesian density estimation by mixtures of normal distributions.  In Rizvi, M. H., Rustagi, J. S., and Siegmund, D., editors, Recent Advances in Statistics, pages 287–302. Academic Press.
  * Fichman and O’Brien, (2019) Fichman, M. and O’Brien, J. R. (2019).  Optimal shot selection strategies for the NBA.  Journal of Quantitative Analysis in Sports, 15(3):203–211.
  * Fraley and Raftery, (2002) Fraley, C. and Raftery, A. E. (2002).  Model-based clustering, discriminant analysis, and density estimation.  Journal of the American Statistical Association, 97(458):611–631.
  * Fraley and Raftery, (2007) Fraley, C. and Raftery, A. E. (2007).  Bayesian regularization for normal mixture estimation and model-based clustering.  Journal of Classification, 24(2):155–181.
  * Fu and Stasko, (2024) Fu, Y. and Stasko, J. (2024).  HoopInSight: Analyzing and comparing basketball shooting performance thr visualization.  IEEE Transactions on Visualization & Computer Graphics, 30(1):858–868.
  * Gilani, (2023) Gilani, S. (2023).  hoopR: Access Men’s Basketball Play by Play Data.  R package version 2.1.0.
  * Goldsberry, (2012) Goldsberry, K. (2012).  CourtVision: New visual and spatial analytics for the NBA.  In 2012 MIT Sloan Sports Analytics Conference, volume 9, pages 12–15.
  * Jiao et al., (2021) Jiao, J., Hu, G., and Yan, J. (2021).  A Bayesian marked spatial point processes model for basketball shot chart.  Journal of Quantitative Analysis in Sports, 17(2):77–90.
  * Keribin, (2000) Keribin, C. (2000).  Consistent estimation of the order of mixture models.  Sankhya Ser. A, 62(1):49–66.
  * Kubatko et al., (2007) Kubatko, J., Oliver, D., Pelton, K., and Rosenbaum, D. T. (2007).  A starting point for analyzing basketball statistics.  Journal of Quantitative Analysis in Sports, 3(3).
  * McLachlan and Peel, (2000) McLachlan, G. J. and Peel, D. (2000).  Finite Mixture Models.  Wiley, New York.
  * Oughali et al., (2019) Oughali, M. S., Bahloul, M., and El Rahman, S. A. (2019).  Analysis of NBA players and shot prediction using random forest and XGBoost models.  In 2019 International Conference on Computer and Information Sciences (ICCIS), pages 1–5. IEEE.
  * Papalexakis and Pelechrinis, (2018) Papalexakis, E. and Pelechrinis, K. (2018).  thoops: A multi-aspect analytical framework for spatio-temporal basketball data.  In Proceedings of the 27th ACM International Conference on Information and Knowledge Management, pages 2223–2232.
  * Pelechrinis and Goldsberry, (2021) Pelechrinis, K. and Goldsberry, K. (2021).  The anatomy of corner 3s in the NBA: What makes them efficient, how are they generated and how can defenses respond?  arXiv preprint arXiv:2105.12785.
  * Perin et al., (2018) Perin, C., Vuillemot, R., Stolper, C. D., Stasko, J. T., Wood, J., and Carpendale, S. (2018).  State of the art of sports data visualization.  Computer Graphics Forum, 37(3):663–686.
  * Reich et al., (2006) Reich, B. J., Hodges, J. S., Carlin, B. P., and Reich, A. M. (2006).  A spatial analysis of basketball shot chart data.  The American Statistician, 60(1):3–12.
  * Sandholtz et al., (2020) Sandholtz, N., Mortensen, J., and Bornn, L. (2020).  Measuring spatial allocative efficiency in basketball.  Journal of Quantitative Analysis in Sports, 16(4):271–289.
  * Schwarz, (1978) Schwarz, G. (1978).  Estimating the dimension of a model.  The Annals of Statistics, 6(2):461–464.
  * Scott, (2009) Scott, D. W. (2009).  Multivariate Density Estimation: Theory, Practice, and Visualization.  John Wiley & Sons, 2nd edition.
  * Scrucca, (2019) Scrucca, L. (2019).  A transformation-based approach to Gaussian mixture density estimation for bounded data.  Biometrical Journal, 61(4):873–888.
  * Scrucca et al., (2023) Scrucca, L., Fraley, C., Murphy, T. B., and Raftery, A. E. (2023).  Model-Based Clustering, Classification, and Density Estimation Using mclust in R.  Chapman & Hall/CRC.
  * Shortridge et al., (2014) Shortridge, A., Goldsberry, K., and Adams, M. (2014).  Creating space to shoot: quantifying spatial relative field goal efficiency in basketball.  Journal of Quantitative Analysis in Sports, 10(3):303–313.
  * Terner and Franks, (2021) Terner, Z. and Franks, A. (2021).  Modeling player and team performance in basketball.  Annual Review of Statistics and Its Application, 8:1–23.
  * Zuccolotto and Manisera, (2020) Zuccolotto, P. and Manisera, M. (2020).  Basketball data science: With applications in R.  CRC Press.
  * Zuccolotto et al., (2018) Zuccolotto, P., Manisera, M., and Sandri, M. (2018).  Big data analytics for modeling scoring probability in basketball: The effect of shooting under high-pressure conditions.  International Journal of Sports Science & Coaching, 13(4):569–589.


# Propagation of angular momentum in charged pion decay and related processes

Bowen Wang<EMAIL_ADDRESS>School of Physics, Hangzhou Normal University,
Hangzhou, Zhejiang 311121, China

###### Abstract

There are confusions about angular momentum propagation in scattering or decay
processes involving the transition between particle systems that appear to
transform differently under Lorentz transformations. This paper provides an
analysis of the transformation properties of the states and interactions for a
few typical processes within the standard model of particle physics, and
performs explicit calculations showing how angular momentum transfers in these
processes. We shall show that angular momentum conservation is completely
consistent with the calculation in quantum field theory.

##  I Introduction

Within the standard model (SM), the decay of a charged pion111Or more
generally, a charged pseudoscalar meson. (take $\pi^{-}$ to be definite) into
a lepton $l^{-}$ and an anti-neutrino $\bar{\nu}_{l}$ occurs through weak
interactions mediated by a W boson. This process is described in Fig. 1.

Figure 1: The decay of a charged pion into a lepton and an anti-neutrino:
$\pi^{-}\rightarrow l^{-}\bar{\nu}_{l}$. The light blob represents purely
perturbative interactions in the graph, while the dark blob also includes non-
perturbative contributions.

The corresponding matrix element is

$\displaystyle(2\pi)^{4}\delta^{(4)}(k-k_{1}-k_{2})i\mathcal{M}(\pi^{-}(k)\rightarrow
l^{-}(k_{1})\bar{\nu}_{l}(k_{2}))$
$\displaystyle=-\frac{i\sqrt{2\pi\alpha}V_{ud}}{\sin\theta_{W}}\int
d^{4}x\langle l^{-}(k_{1})\bar{\nu}_{l}(k_{2})|W^{+}_{\mu}(x)\
|0\rangle\times\langle 0|J_{q}^{\mu+}(x)|\pi^{-}(k)\rangle,$ (1)

where $\alpha$ is the fine structure constant, $V_{ud}$ the “$ud$” element of
the quark mixing matrix, and $\theta_{W}$ the weak mixing angle. The
definition of the amplitude $\mathcal{M}$ follows the convention of Ref.
Peskin:1995ev . The pion and leptons are in “in” and “out” states
respectively, and the fields between the states are all renormalized. The
polarizations of the fermions are not labeled explicitly. The $V-A$ quark
current is defined by

$\displaystyle
J_{q}^{\mu+}(x)\equiv\overline{\Psi^{u}}(x)\gamma^{\mu}\frac{1-\gamma_{5}}{2}\Psi^{d}(x).$
(2)

The integrand in Eq. 1 is in a factorized form. The first factor gives the
vertex function of two external fermions and an off-shell propagator
Weinberg:1995mt ; Sterman:1993hfp  with all order corrections including the
self-energy graphs.222Only half of the self-energy contribution should be
included for the external lines in accord with the reduction of the S-matrix
Weinberg:1995mt ; Sterman:1993hfp . These are denoted by the blob in light
grey in Fig. 1. The second factor has a simple tensor structure determined by
the properties of the quark current and of the states under translations:

$\displaystyle\langle
0|J_{q}^{\mu+}(x)|\pi^{-}(k)\rangle=\frac{i}{2}f_{\pi}k^{\mu}e^{-ik\cdot x},$
(3)

where $f_{\pi}$ is the pion decay constant (in the convention of the particle
data group ParticleDataGroup:2022pth ). It receives contributions from
perturbative and non-perturbative interactions within the $\pi^{-}$ induced by
the quark current $J_{q}^{\mu+}(x)$. These interactions result in the decay of
the pion and are represented by the blob in dark grey in Fig. 1.

Perturbative calculations can be done for the branching ratio
$R_{e/\mu}^{\pi}\equiv\Gamma(\pi^{-}\rightarrow
e^{-}\bar{\nu}_{e})/\Gamma(\pi^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu})$, which
is independent of $f_{\pi}$. The agreement between the SM prediction of
$R_{e/\mu}^{\pi}$ (with radiative corrections) and measurements has reached a
level within 0.5%, and is limited by the accuracy of the experiment
Pocanic:2014jka . This provides a good precision test of the SM and
perturbative calculations in quantum field theory. Furthermore, $f_{\pi}$ can
be calculated in lattice QCD (see Ref. ParticleDataGroup:2022pth , Table
72.1). It can also be extracted from experiments once the W decay subprocess
(i.e., the first factor of the integrand on the r.h.s. of Eq. 1) is computed
(with parameters such as $V_{ud}$ taken from other measurements). Comparison
of results from both approaches shows an agreement at the percent level
ParticleDataGroup:2022pth . This is also the level of uncertainty of the
Lattice QCD result.

On the other hand, measurements of the angular distributions of pion decay
products in the pion rest frame were done in many early experiments. While
most of them obtained isotropic results (e.g., Ref. Abashian:1957zz ;
PhysRevLett.14.745 ; Frota-Pessoa:1969jeg ), small asymmetry were found in
some of the studies (e.g., Ref. Hulubei:1963zza  ). There are no reports of
significant deviation from a uniform angular distribution in more recent
experiments of pion decays MINOS:2008fnv ; MINOS:2012ozn . The limits for the
sizes of possible Lorentz violating terms are set to the level of $10^{-5}\sim
10^{-4}$ relative to the SM terms Altschul:2013yja ; Altschul:2013ykb ;
Noordmans:2014bua .

It is a consensus that no anisotropic angular distributions can be deduced
from the charged pion’s SM decays, since the pion is a pseudoscalar and the
interactions involved are all Lorentz invariant. All studies seek to explain
possible Lorentz violating effects with extensions of the SM Altschul:2013yja
; Altschul:2013ykb ; Diaz:2013saa ; Noordmans:2013dha ; Noordmans:2014bua . It
may be conceptually confusing, however, that a spin-0 initial state propagates
via a spin-1 vector boson, and remains isotropic before being measured. This
is usually explained by noting that there is a longitudinal polarization mode
of the massive vector boson that behaves like a scalar with 0 angular momentum
Peskin:1995ev ; Fayyazuddin:1994wh ; Barger:1987nn . In Ref. Nakanishi:2002sv
the decay is considered as occuring via a spin-0 goldstone boson to avoid non-
conservation of angular momentum. Nevertheless, it is still helpful to analyze
the decay amplitude using symmetry principles to show explicitly how angular
momentum conservation manifests in the process.

In the following we shall first explore the rotational and Lorentz symmetries
in production processes of a vector boson, and then illustrate the results
based on symmetry considerations by simple tree level calculations. A summary
and discussion of the results will be given at the end.

##  II Angular momentum transfer in the production of a vector boson

It is helpful to first work out the angular momentum of a vector boson created
from a collision of two spin-$1/2$ fermions. This is somewhat more complicated
than the case of charged pion decay, but it provides a richer picture as to
how angular momentum is transferred between different parts of an amplitude.
To make the discussion general, we start by considering the matrix element

$\displaystyle\langle
0|J^{\mu}(x)|p_{a},\lambda_{a};p_{b},\lambda_{b}\rangle=e^{-i(p_{a}+p_{b})\cdot
x}\langle 0|J^{\mu}(0)|p_{a},\lambda_{a};p_{b},\lambda_{b}\rangle,$ (4)

where $J^{\mu}$ denotes the SM current that tranforms like a vector, axial-
vector, or a combination of them, e.g.,
$J^{\mu}=\overline{\Psi^{e}}\gamma^{\mu}\Psi^{e}$,
$\overline{\Psi^{e}}_{L}\gamma^{\mu}\Psi^{\nu}_{L}$, etc. The momenta and
helicities of the fermions are indicated in the state ket. This matrix element
represents the all order contribution to the fermion-vector vertex shown in
Fig. 2 (a) (with the vector boson part excluded, as indicated by the vertical
bar).

(a)

(b)

Figure 2: Vertices that produce a vector boson in the processes (a): fermion-
fermion scattering, and (b): decay (of e.g., $\pi^{\pm}$, $\Psi$, etc.). The
light blob represents purely perturbative interactions in the graph, while the
dark blob also includes non-perturbative contributions.

The initial state is made up of two fermions with definite momenta and
helicities, as indicated in the state ket in Eq. 4. This is not an eigenstate
of the angular momentum. To analyse the angular structure of the process, we
expand the initial state as a combination of states with definite $J$, $M$,
$\lambda_{a}$, $\lambda_{b}$,
$\mathbb{P}^{+}\equiv\mathbb{p}_{a}+\mathbb{p}_{b}$, and
$E^{+}\equiv\sqrt{\mathbb{p}_{a}^{2}+m_{a}^{2}}+\sqrt{\mathbb{p}_{b}^{2}+m_{b}^{2}}$
Jacob:1959at ; Gibson:1976wp . $J$ and $M$ are respectively the quantum
numbers for the total angular momentum of the fermions, and for its projection
on the 3-axis in the center-of-mass (CM) frame of the fermions, in which the
3-direction is defined by the direction of the 3-momentum of a:
$\mathbb{p}\equiv\mathbb{p}_{a}=-\mathbb{p}_{b}$. It is convenient to work in
this CM frame and express the expansion as Jacob:1959at ; Gibson:1976wp

$\displaystyle|p_{a},\lambda_{a};p_{b},\lambda_{b}\rangle=\sum_{J}C_{J}|J,M=\lambda_{a}-\lambda_{b},\lambda_{a},\lambda_{b},|\mathbb{p}|\rangle,$
(5)

where $\mathbb{P}^{+}=0$ is suppressed, and $E^{+}$ is replaced by
$|\mathbb{p}|$ in the states on the r.h.s. of Eq. 5. $J$ runs over the
integers $0,1,\ldots$. There is no sum over $M$, because the 3-component of
the angular momentum coincides with the difference of the helicities. For an
anti-fermion, the physical spin in the direction of its momentum is opposite
to its helicity, so an extra minus sign333This extra minus sign is not to be
confused with the minus sign in $M=\lambda_{a}-\lambda_{b}$. is to be included
in its helicity $\lambda_{a}$ or $\lambda_{b}$ when obtaining
$M=\lambda_{a}-\lambda_{b}$. It is shown Jacob:1959at ; Gibson:1976wp  that
$C_{J}=\sqrt{(2J+1)/(4\pi)}$ in order to be consistent with the normalization
of the momentum and angular momentum eigenstates:

$\displaystyle\langle
p_{a}^{\prime},\lambda_{a}^{\prime};p_{b}^{\prime},\lambda_{b}^{\prime}|p_{a},\lambda_{a};p_{b},\lambda_{b}\rangle=(2\pi)^{6}2E_{a}2E_{b}\delta^{3}(\mathbb{p}_{a}^{\prime}-\mathbb{p}_{a})\delta^{3}(\mathbb{p}_{b}^{\prime}-\mathbb{p}_{b})\delta_{\lambda_{a}^{\prime}\lambda_{a}}\delta_{\lambda_{b}^{\prime}\lambda_{b}},$
(6) $\displaystyle\langle
J^{\prime},M^{\prime},\lambda_{a}^{\prime},\lambda_{b}^{\prime},|\mathbb{p}^{\prime}||J,M,\lambda_{a},\lambda_{b},|\mathbb{p}|\rangle=(2\pi)^{6}\frac{E^{+}}{|\mathbb{p}|}\delta^{3}(\mathbb{P}^{+\prime})\delta(E^{+\prime}-E^{+})\delta_{J^{\prime}J}\delta_{M^{\prime}M}\delta_{\lambda_{a}^{\prime}\lambda_{a}}\delta_{\lambda_{b}^{\prime}\lambda_{b}}.$
(7)

Note that Eq. 7 is expressed in the CM frame of $\mathbb{P}^{+}$, in which
$\mathbb{P}^{+\prime}$ may be non-zero. For a non-zero $\mathbb{P}^{+\prime}$,
Eq. 7 gives 0 irrespective of the 3-direction chosen in the CM frame of
$\mathbb{P}^{+\prime}$ to define $M^{\prime}$. Whereas if
$\mathbb{P}^{+\prime}=\mathbb{P}^{+}=0$, both $M$ and $M^{\prime}$ are defined
with respect to the direction of $\mathbb{p}$. However, in Eq. 7, $M$ and
$M^{\prime}$ are not restricted to $M=\lambda_{a}-\lambda_{b}$ and
$M^{\prime}=\lambda_{a}^{\prime}-\lambda_{b}^{\prime}$, since the states are
not restricted to those in the expansion in Eq. 5. The matrix element in Eq. 4
can also be expanded as

$\displaystyle\langle
0|J^{\mu}(0)|p_{a},\lambda_{a};p_{b},\lambda_{b}\rangle=\sum_{J}C_{J}\langle
0|J^{\mu}(0)|J,M=\lambda_{a}-\lambda_{b},\lambda_{a},\lambda_{b},|\mathbb{p}|\rangle,$
(8)

or, for a full process with final state $|f\rangle$,

$\displaystyle\langle
f|p_{a},\lambda_{a};p_{b},\lambda_{b}\rangle=\sum_{J}C_{J}\langle
f|J,M=\lambda_{a}-\lambda_{b},\lambda_{a},\lambda_{b},|\mathbb{p}|\rangle.$
(9)

We are particularly interested in analysing the contribution to the matrix
element in Eq. 8 from each value of $J$. A vanishing contribution from a
specific $J$ signifies that the propagation of this angular momentum
eigenvalue is forbidden. To determine the allowed $J$ values we should
consider the transformation properties of both the operator $J^{\mu}$ and the
state
$|J,M=\lambda_{a}-\lambda_{b},\lambda_{a},\lambda_{b},|\mathbb{p}|\rangle$.
The angular momentum eigenstates with fixed $J$ furnish an irreducible
representation under the spatial rotation $R(\alpha,\beta,\gamma)$, which is a
3 by 3 matrix that rotates the spatial coordinates and is described by the
Eular angles $\alpha,\beta$ and $\gamma$:

$\displaystyle
U(R(\alpha,\beta,\gamma))|J,\lambda_{a}-\lambda_{b},\lambda_{a},\lambda_{b},|\mathbb{p}|\rangle=\sum_{M^{\prime}}\mathscr{D}^{J}_{M^{\prime},\lambda_{a}-\lambda_{b}}(R(\alpha,\beta,\gamma))|J,M^{\prime},\lambda_{a},\lambda_{b},|\mathbb{p}|\rangle.$
(10)

Note that $\lambda_{a}$, $\lambda_{b}$, and $|\mathbb{p}|$ are invariant under
rotations. $M$ (on the l.h.s.) is fixed to $\lambda_{a}-\lambda_{b}$, but is
changed by a rotation to $M^{\prime}$ (on the r.h.s.). The subscripts of the
Wigner $\mathscr{D}$-matrix $\mathscr{D}^{J}_{M^{\prime},M}$ (The explicit
form can be found in e.g., Jacob:1959at ; Leader:2011vwq ) take the values
$-J,-J+1,\ldots,J$. (All helicity values $\lambda_{a}$ and $\lambda_{b}$ are
allowed as long as $-J\leq M=\lambda_{a}-\lambda_{b}\leq J$). The current
operator $J^{\mu}(0)$ is a 4-vector that transforms according to

$\displaystyle
U(\Lambda)J^{\mu}(0)U^{-1}(\Lambda)=(\Lambda^{-1})^{\mu}_{\nu}J^{\nu}(0)$ (11)

under an arbitrary Lorentz transformation $\Lambda$ (The $x$ dependence of
$J^{\mu}$ is removed by translation in Eq. 4 so that the transformation of the
current is simpler). On the r.h.s. of Eq. 11 $\mu$ and $\nu$ label
respectively the row and column of the matrix $\Lambda^{-1}$. This is
equivalent to the self-representation of the Lorentz group, which can be seen
from the property for an arbitrary Lorentz transformation matrix $\Lambda$

$\displaystyle g\Lambda g^{-1}=(\Lambda^{-1})^{T},$ (12) $\displaystyle
g=g^{-1}=\text{diag}\\{1,-1,-1,-1\\}.$ (13)

Therefore $J^{\mu}(0)$ furnish a 4-dimensional irreducible representation of
the Lorentz group. However, upon restriction to its spatial rotation subgroup
$SO(3)$, the current operators are no longer irreducible. Clearly, $J^{0}$ is
invariant under rotations, while $J^{1}$, $J^{2}$, and $J^{3}$ are mixed. This
reduces the 4 components to an $SO(3)$ singlet labeled by $J=0$ and a triplet
in the SO(3) self-representation labeled by $J=1$:

$\displaystyle U(R)J^{0}(0)U^{-1}(R)=J^{0}(0)D^{J=0}_{0,0}(R),\quad
D^{J=0}(R)=1,$ (14) $\displaystyle
U(R)J^{i}(0)U^{-1}(R)=J^{j}(0)D^{J=1}_{j,i}(R),\quad
D^{J=1}_{j,i}(R)=(R^{-1})_{i,j}=R_{j,i},\quad i,j=1,2,3.$ (15)

As representations of the $SO(3)$ group with the same dimension are
equivalent, the $D^{J=0}$ and $D^{J=1}$ matrices above are equivalent to the
representations $\mathscr{D}^{0}$ and $\mathscr{D}^{1}$ in Eq. 10,
respectively. Hence the only non-vanishing contribution in Eq. 8 is from $J=0$
and $J=1$ according to Wigner-Eckart theorem.444The rule of adding two angular
momenta dictates that $J^{\mu}(0)$ and
$|J,M=\lambda-\lambda^{\prime},\lambda,\lambda^{\prime},|\mathbb{p}|\rangle$
must be in equivalent representations in order to be combined by Clebsch-
Gordon coefficients to form a singlet under rotations. This is not exactly
right because they could still vanish due to constraint from further
symmetries (See section IV for an example), but at least all final states with
$J\geq 2$ cannot be produced via a vector boson coupling to the vertex in Fig.
2 (a).

One may want to start by considering the transformation of $J^{\mu}(0)$ and
$|J,M=\lambda-\lambda^{\prime},\lambda,\lambda^{\prime},|\mathbb{p}|\rangle$
under the full Lorentz group, rather than its subgroup. However, it is not
straightforward to see the transformation law of the angular momentum
eigenstates in this case, except that
$|J=0,M=\lambda-\lambda^{\prime}=0,\lambda,\lambda^{\prime},|\mathbb{p}|\rangle$
will no longer be invariant (a boost changes the eigenvalues $\mathbb{P}^{+}$
and $E^{+}$). Nonetheless, the analysis above shows that a non-vanishing
result is obtained when projecting this state onto the self representation of
the Lorentz group.

The analysis above can be carried over immediately to the case of the charged
pion decay (or more generally, charged pseudoscalar meson decay) via
production of a W boson, as in Fig. 2 (b). The initial state pion is a
pseudoscalar with $J=M=0$, and the amplitude $\langle
0|J_{q}^{\mu+}(x)|\pi^{-}(k)\rangle$ in Eq. 3 is non-zero from the same
reasoning as given above of the $J=0$ contribution being non-zero in two
fermion scatterings.

##  III Amplitudes with angular momentum eigenstates

Now we derive formulae for practical calculations of the amplitudes on the
r.h.s. of Eq. 9. The spatial part of $p_{a}$ in Eq. 5 can be rotated by
$R(\phi,\theta,0)$ to a direction determined by the azimuthal angle $\phi$ and
polar angle $\theta$, with respect to the original axes. Denoting the initial
state under this rotation by

$\displaystyle|\phi,\theta,\lambda_{a},\lambda_{b},|\mathbb{p}|\rangle\equiv
U(R(\phi,\theta,0))|p_{a},\lambda_{a};p_{b},\lambda_{b}\rangle=|R(\phi,\theta,0)p_{a},\lambda_{a};R(\phi,\theta,0)p_{b},\lambda_{b}\rangle,$
(16)
$\displaystyle|p_{a},\lambda_{a};p_{b},\lambda_{b}\rangle=|\phi=0,\theta=0,\lambda_{a},\lambda_{b},|\mathbb{p}|\rangle,$
(17)

we obtain the expansion of the momentum eigenstates in an arbitrary direction
by applying the rotation on Eq. 5:

$\displaystyle|\phi,\theta,\lambda_{a},\lambda_{b},|\mathbb{p}|\rangle=\sum_{J,M}C_{J}\mathscr{D}^{J}_{M,\lambda_{a}-\lambda_{b}}(R(\phi,\theta,0))|J,M,\lambda_{a},\lambda_{b},|\mathbb{p}|\rangle.$
(18)

The inverse expansion can be obtained using the properties of the
$\mathscr{D}$-matrices:

$\displaystyle|J,M,\lambda_{a},\lambda_{b},|\mathbb{p}|\rangle=C_{J}\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta\sin\theta\mathscr{D}^{J*}_{M,\lambda_{a}-\lambda_{b}}(R(\phi,\theta,0))|\phi,\theta,\lambda_{a},\lambda_{b},|\mathbb{p}|\rangle.$
(19)

This can be used to compute perturbatively the amplitude for the production of
a vector boson from a state with a particular $J$:

$\displaystyle\langle
0|J^{\mu}(0)|J,M,\lambda_{a},\lambda_{b},|\mathbb{p}|\rangle=C_{J}\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta\sin\theta\mathscr{D}^{J*}_{M,\lambda_{a}-\lambda_{b}}(R(\phi,\theta,0))\langle
0|J^{\mu}(0)|\phi,\theta,\lambda_{a},\lambda_{b},|\mathbb{p}|\rangle,$ (20)

or, for a full process with final state $|f\rangle$,

$\displaystyle\langle
f|J,M,\lambda_{a},\lambda_{b},|\mathbb{p}|\rangle=C_{J}\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta\sin\theta\mathscr{D}^{J*}_{M,\lambda_{a}-\lambda_{b}}(R(\phi,\theta,0))\langle
f|\phi,\theta,\lambda_{a},\lambda_{b},|\mathbb{p}|\rangle.$ (21)

The Feynman rules for the momentum eigenstates on the r.h.s. of Eqs. 20, 21
are well known. In the following we shall compute a few simple processes to
show the propagation of $J$ values.

##  IV Example processes

(a) $e^{-}_{L}\bar{\nu}_{eL}\rightarrow W^{-*}\rightarrow
e^{-}_{L}\bar{\nu}_{eL}$.

(b)
$e^{-}_{L}e^{+}_{L}\rightarrow\gamma^{*}\rightarrow\mu^{-}_{L}\mu^{+}_{L}$.

Figure 3: Example 2 to 2 processes at tree level with the production and decay
of a vector boson.

We consider 2 to 2 tree level fermion scattering processes in $s$ channel
mediated by a vector boson. The two processes in Fig. 3 will be treated in
sequence. In these calculations, we would like to see explicitly the
contribution from the $J=0$ and $J=1$ eigenstates, in accord with the generic
result obtained in section II. Therefore the initial and final state fermions
are taken to have definite helicities such that the angular structure of the
processes are easily revealed. Specifically, we require that the two fermions
in the initial state have the opposite spin components along the direction of
the incoming beams. This is necessary to ensure that the $J=0$ partial wave is
non-zero in the expansion of the momentum eigenstates in Eq. 5. In contrast,
if the fermions have the same spin along the beam, then the total spin of the
system is $S=1$, which cannot be canceled by the orbital angular momentum to
give $J=0$ because the orbital angular momentum is perpendicular to the beam
direction. In this case, the partial wave expansion only contains terms with
$J\geq 1$.

###  IV.1 $e^{-}_{L}(p_{a})\bar{\nu}_{eL}(p_{b})\rightarrow W^{-*}\rightarrow
e^{-}_{L}(p_{c})\bar{\nu}_{eL}(p_{d})$.

Here the subscript “L” for all the four fermions refers to the helicity
configuration $\lambda_{a}=-\lambda_{b}=\lambda_{c}=-\lambda_{d}=-1$, rather
than definite chiralities. Recall that when obtaining
$M=\lambda_{a}-\lambda_{b}$ we give an extra minus sign to $\lambda_{b}$ for
the anti-neutrino, so that in this case $M=0$ instead of $-2$. In the massless
limit, the helicity and chirality eigenstates coincide, but this amplitude
vanishes by helicity conservation. However, we shall keep the fermion masses
so that the $J=0$ mode is allowed, as will be seen below. Recall that we are
examining the propagation of each $J$-mode in principle rather than evaluating
the practical importance of the contribution from each $J$.

The initial momenta are

$\displaystyle p_{a}=(E_{a},0,0,p),\quad E_{a}=\sqrt{m_{a}^{2}+p^{2}},$ (22)
$\displaystyle p_{b}=(E_{b},0,0,-p),\quad E_{b}=\sqrt{m_{b}^{2}+p^{2}},$ (23)

where $m_{a}\equiv m_{e}$, and $m_{b}\equiv m_{\nu_{e}}$. The tree level
vertex for the subprocess $e^{-}_{L}(p_{a})\bar{\nu}_{Le}(p_{b})\rightarrow
W^{-*}$ is (excluding the W propagator and an overall factor
$-i\sqrt{2\pi\alpha}/\sin\theta_{W}$ at the vertex)

$\displaystyle V^{\mu}(e^{-}_{L}(p_{a})\bar{\nu}_{eL}(p_{b})\rightarrow
W^{-*})=\langle
0|J^{\mu}(0)|p_{a},\lambda_{a};p_{b},\lambda_{b}\rangle_{\text{tree}}=\bar{v}^{\lambda_{b}}(p_{b})\gamma^{\mu}\frac{1-\gamma^{5}}{2}u^{\lambda_{a}}(p_{a}),$
(24)

where
$J^{\mu}(0)=\overline{\Psi^{\nu_{e}}}\gamma^{\mu}\frac{1-\gamma_{5}}{2}\Psi^{e}$,
and the particle species are omitted in the states and spinors. Here we follow
the method in Ref. Peskin:1995ev  to compute the amplitude for polarized
states and employ the Weyl representation for the Dirac $\gamma$-matrices:

$\displaystyle\gamma^{0}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\quad\gamma^{i}=\begin{pmatrix}0&\sigma_{i}\\\
-\sigma_{i}&0\end{pmatrix},\quad\gamma^{5}=\begin{pmatrix}-1&0\\\
0&1\end{pmatrix},$ (25)

in which each element is a 2 by 2 matrix and $\sigma_{i}$ are the Pauli
matrices. The corresponding spinors for the spin-up electron and spin-down
anti-neutrino (both with respect to the 3-direction) are

$\displaystyle[u^{\lambda_{a}}(p_{a})]^{T}=(0,\sqrt{E_{a}+p},0,\sqrt{E_{a}-p}),$
(26)
$\displaystyle[v^{\lambda_{b}}(p_{b})]^{T}=(0,\sqrt{E_{b}-p},0,\sqrt{E_{b}+p}).$
(27)

Inserting these expressions into Eq. 24 gives

$\displaystyle V^{\mu}(e^{-}_{L}(p_{a})\bar{\nu}_{eL}(p_{b})\rightarrow
W^{-*})=\sqrt{(E_{b}-p)(E_{a}+p)}(1,0,0,1).$ (28)

The vertex for $W^{-*}\rightarrow e^{-}_{L}(p_{c})\bar{\nu}_{eL}(p_{d})$ is

$\displaystyle V^{\mu}(W^{-*}\rightarrow
e^{-}_{L}(p_{c})\bar{\nu}_{eL}(p_{d}))=\bar{u}^{\lambda_{c}}(p_{c})\gamma^{\mu}\frac{1-\gamma^{5}}{2}v^{\lambda_{d}}(p_{d})=\left[\bar{v}^{\lambda_{d}}(p_{d})\gamma^{\mu}\frac{1-\gamma^{5}}{2}u^{\lambda_{c}}(p_{c})\right]^{*}.$
(29)

The term in the square brackets is of the same form as in Eq. 24, except the
momenta $\mathbb{p}_{a}=-\mathbb{p}_{b}$ are rotated from the 3-direction to
$\mathbb{p}_{c}=-\mathbb{p}_{d}$. Lorentz covariance dictates that it must
transform like a 4-vector constructed from $p_{c}$ and $p_{d}$, the only
possible form being

$\displaystyle\bar{v}^{\lambda_{d}}(p_{d})\gamma^{\mu}\frac{1-\gamma^{5}}{2}u^{\lambda_{c}}(p_{c})=Ap_{c}^{\mu}+Bp_{d}^{\mu},$
(30)

where $A$ and $B$ are Lorentz invariants. In the center-of-mass frame, the
spatial part becomes $(A-B)\mathbb{p}_{c}$, which is consistent with the
spatial part on the r.h.s. of Eq. 28 when $\mathbb{p}_{c}$ coincides with
$\mathbb{p}_{a}$. This also gives

$\displaystyle A-B=\sqrt{(E_{b}-p)(E_{a}+p)}/p.$ (31)

Noting that the temporal part of Eq. 30 is invariant under rotations and can
be obtained from Eq. 28, we conclude

$\displaystyle V^{\mu}(W^{-*}\rightarrow
e^{-}_{L}(p_{c})\bar{\nu}_{eL}(p_{d}))=\sqrt{(E_{b}-p)(E_{a}+p)}(1,\mathbb{p}_{c}/p),$
(32)

where $E_{a}=E_{c}$ and $E_{b}=E_{d}$.

Now restore the couplings (and other overall factors) at the vertices and
contract with the W propagator in unitary gauge555We would like to show that
the propagation of zero angular momentum is possible without resorting to the
unphysical Goldstone boson.

$\displaystyle\frac{-i}{q^{2}-M_{W}^{2}}(g_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{M_{W}^{2}})$
(33) $\displaystyle(q\equiv p_{a}+p_{b})$

to get the amplitude for the full process

$\displaystyle\mathcal{M}(e^{-}_{L}(p_{a})\bar{\nu}_{eL}(p_{b})\rightarrow
W^{-*}\rightarrow
e^{-}_{L}(p_{c})\bar{\nu}_{eL}(p_{d}))=\frac{2\pi\alpha}{\sin^{2}\theta_{W}}\frac{(E_{b}-p)(E_{a}+p)}{(E_{a}+E_{b})^{2}-M_{W}^{2}}\left[1-\frac{(E_{a}+E_{b})^{2}}{M_{W}^{2}}-\cos\theta\right],$
(34)

where $M_{W}$ is the mass of the W boson. The dependence of the result on the
external momenta is only through the angle $\theta$ between the initial and
final fermion beams.

It is straightforward to compute the amplitude with an angular momentum
eigenstate using Eq. 21, in the form

$\displaystyle\langle\phi,\theta,\lambda_{c}=\lambda_{a},\lambda_{d}=\lambda_{b},p|J,M,\lambda_{a},\lambda_{b},p\rangle$
$\displaystyle=C_{J}\int_{0}^{2\pi}d\phi^{\prime}\int_{0}^{\pi}d\theta^{\prime}\sin\theta^{\prime}\mathscr{D}^{J*}_{M,\lambda_{a}-\lambda_{b}}(R(\phi^{\prime},\theta^{\prime},0))\langle\phi,\theta,\lambda_{c}=\lambda_{a},\lambda_{d}=\lambda_{b},p|\phi^{\prime},\theta^{\prime},\lambda_{a},\lambda_{b},p\rangle.$
(35)

The matrix element on the r.h.s. of Eq 35 is computed with the initial and
final states

$\displaystyle|i\rangle\equiv|p_{a}^{\prime},\lambda_{a};p_{b}^{\prime},\lambda_{b}\rangle=|\phi^{\prime},\theta^{\prime},\lambda_{a},\lambda_{b},|\mathbb{p}_{a}^{\prime}|=p\rangle,$
(36)
$\displaystyle|f\rangle\equiv|p_{c},\lambda_{c};p_{d},\lambda_{d}\rangle=|\phi,\theta,\lambda_{c}=\lambda_{a},\lambda_{d}=\lambda_{b},|\mathbb{p}_{c}|=p\rangle,$
(37) $\displaystyle\langle f|i\rangle_{\text{tree}}=\langle
p_{c},\lambda_{c};p_{d},\lambda_{d}|p_{a}^{\prime},\lambda_{a};p_{b}^{\prime},\lambda_{b}\rangle_{\text{tree}}=N_{ab}\left[1-\frac{(E_{a}+E_{b})^{2}}{M_{W}^{2}}-\cos\theta_{ac}\right],$
(38)

where on the r.h.s. of Eq. 38 (and below), we suppress the factor
$i(2\pi)^{4}$ and the delta function that implements the overall momentum
conservation. $N_{ab}$ represents the coefficient in Eq. 34 multiplying the
angular factor $1-\cos\theta$, and $\theta_{ac}$ is the angle between
$\mathbb{p}_{a}^{\prime}$ and $\mathbb{p}_{c}$:

$\displaystyle\cos\theta_{ac}=\sin\theta\cos\phi\sin\theta^{\prime}\cos\phi^{\prime}+\sin\theta\sin\phi\sin\theta^{\prime}\sin\phi^{\prime}+\cos\theta\cos\theta^{\prime}.$
(39)

There is no contradiction between Eq. 38 and Eq. 6, because Eq. 38 is a
product of “in” and “out” states that makes a transition amplitude in the
presence of interactions, while Eq. 6 is a normalization condition when both
states are “in”, “out”, or free states. We do not label these states with
“in”, “out”, or “free” since they can be distinguished by context.

Inserting the explicit expressions for the $\mathscr{D}$-matrices Jacob:1959at
; Leader:2011vwq , the angular integration of Eq. 35 can be trivially
performed to give

$\displaystyle\langle\phi,\theta,\lambda_{c}=\lambda_{a},\lambda_{d}=\lambda_{b},p|J,M,\lambda_{a},\lambda_{b},p\rangle_{\text{tree}}=N_{ab}\times\left\\{\begin{aligned}
&-\sqrt{\frac{4\pi}{3}}\cos\theta,\,&\text{for}\,J=1,M=0,\\\
&\sqrt{4\pi}\left[1-\frac{(E_{a}+E_{b})^{2}}{M_{W}^{2}}\right],\,&\text{for}\,J=0,M=0.\end{aligned}\right.$
(40)

Hence, both $J=0$ and $J=1$ partial waves contribute to the amplitude Eq. 34.
666The smallness of the neutrino mass $m_{b}$ makes the factor $E_{b}-p$ close
to zero in the expression for the amplitude (which does not affect our
arguments of course). If one chooses to compute alternatively the amplitude
for the same process but with all external fermions right-handed, a factor
$E_{a}-p$ will appear instead of $E_{b}-p$, and the non-zero contribution is
still from $J=0$ and $J=1$. Indeed, plugging the result of Eq. 40 into the
expansion Eq. 9 recovers Eq. 34 immediately. Note the $J\geq 2$ contribution
vanishes. This can be seen from the form of the terms on the r.h.s. of Eq. 38.
Those terms proportional to $1$ and $\cos\theta^{\prime}$ can be expressed by
the matrix elements $\mathscr{D}^{J=0}_{00}$ and $\mathscr{D}^{J=1}_{00}$,
which are orthogonal to the $\mathscr{D}^{J*}_{00}$ with $J\geq 2$ in the
integration in Eq. 35. The other terms are proportional to eigher
$\cos\phi^{\prime}$ or $\sin\phi^{\prime}$, which vanish after the integral
over $\phi^{\prime}$. This is consistent with our general argument that states
with $J\geq 2$ transform differently from the transformation of the current
under rotations. Note that the argument was made for the vertex in Fig. 2 (a),
with all order contributions. However, the vanishing of the vertex must be
order by order since the perturbation series is zero for arbitrary values of
the coupling constants involved in the interactions.

###  IV.2
$e^{-}_{L}(p_{a})e^{+}_{L}(p_{b})\rightarrow\gamma^{*}\rightarrow\mu^{-}_{L}(p_{c})\mu^{+}_{L}(p_{d})$.

A calculation similar to the one carried out above gives

$\displaystyle\mathcal{M}(e^{-}_{L}(p_{a})e^{+}_{L}(p_{b})\rightarrow\gamma^{*}\rightarrow\mu^{-}_{L}(p_{c})\mu^{+}_{L}(p_{d}))=-\frac{4\pi\alpha
m_{a}m_{c}}{E_{a}^{2}}\cos\theta,$ (41)

where $m_{a}=m_{e}$ and $m_{c}=m_{\mu}$. In place of Eq. 40 we have

$\displaystyle\langle\phi,\theta,\lambda_{c}=\lambda_{a},\lambda_{d}=\lambda_{b},p|J,M,\lambda_{a},\lambda_{b},p\rangle_{\text{tree}}=N_{ab}\times\left\\{\begin{aligned}
&-\sqrt{\frac{4\pi}{3}}\cos\theta,\,&\text{for}\,J=1,M=0,\\\ &\qquad
0,\,&\text{for}\,J=0,M=0,\end{aligned}\right.$ (42)

where now $N_{ab}$ stands for the factor multiplying $-\cos\theta$ in Eq. 41.
This result shows that only the $J=1$ mode contributes to the angular
distribution of the final state, which by itself recovers Eq. 41. The
amplitudes with $J\geq 2$ are zero by the similar arguments given above. The
new feature compared to the $e$-$\bar{\nu}_{eL}$ scattering is the vanishing
of $J=0$, which is not a coincidence and has a deeper origin.

To see this we note that a positronium state with definite spin $S$ and
orbital angular momentum $L$ is an eigenstate of the charge conjugation
operator $C$, with the eigenvalue $\eta_{C}=(-1)^{L+S}$ Perkins:2000xb . This
property can be carried over to analyzing our process with an unbounded
electron-positron pair. The state $|J,M,\lambda_{a},\lambda_{b},p\rangle$ with
$J=M=0$ has a definite $J$ rather than $L$ or $S$. However, the rule of adding
two angular momenta requires that $J=0$ can only be obtained by combining
states with definite $L$ and $S$ that satisfy $L=S$. For all these states,
$(-1)^{L+S}=1$, and $|J=0,M=0\rangle$ is therefore a state that is even under
charge conjugation:

$\displaystyle C|J=0,M=0\rangle=|J=0,M=0\rangle,$ (43)

where the quantum numbers other than $J$ and $M$ are suppressed in the state.
Since the electromagnetic current is C-odd, we have

$\displaystyle\langle 0|J^{\mu}(0)|J=0,M=0\rangle$ $\displaystyle=\langle
0|C^{-1}CJ^{\mu}(0)C^{-1}C|J=0,M=0\rangle$ $\displaystyle=-\langle
0|J^{\mu}(0)|J=0,M=0\rangle$ $\displaystyle=0,$ (44)

which merely reflects the fact that $J^{\mu}$ and $|J=0,M=0\rangle$ belong to
different representations of the charge conjugation operation (one C-even and
the other C-odd). So the $J=0$ mode does vanish.777The result here and in
section II are obtained for the vertex functions. However, as we are focusing
on s-channel graphs with a vector boson, these vertices are parts of the
amplitudes. Furthermore, even though the contribution from $J^{\mu}$, with
$\mu=1,2,$ or $3$ is also zero from rotational invariance of the states , this
does not ensure they are zero outside the CM frame of the electron-positron
pair (where the rotational invariance of the initial state is lost. See Eq. 3
for a similar instance), unless the contribution from $J^{0}$ is also zero.
Apparently, this argument cannot be applied to the case of
$e^{-}\bar{\nu}_{e}$ scattering or charged pion decay, because the initial
state of either process has a net electric charge that is reversed by C, and
is therefore not an eigenstate of C. The initial state must then have a non-
zero projection onto the C-odd space of the states, such that the amplitude
with the current is not restricted to zero by charge conjugation.

##  V Conclusion

In this study, we analyze the propagation of angular momentum via a vector
boson in several processes by expanding the initial states into a
superposition of angular momentum eigenstates of the system. Through a general
discussion based on symmetry principles and two practical calculations, we are
able to show that both the singlet ($J=0$) and triplet ($J=1$) representations
of the rotation group can be transferred to the final state via an
intermediate vector boson, unless constraints from further symmetries are
present to forbid it.

The crucial point to understand the $J=0$ case is that an initial state that
is invariant under spatial rotations is not a singlet under a generic Lorentz
transformation. The initial state can be changed by boosts in any directions
and has a non-zero projection onto the self representation of the Lorentz
group. There is no Lorentz violation in the production of a vector field
(i.e., a gauge boson) from such a state via interactions that are Lorentz
invariant. The spin of a vector boson as a virtual particle has to be measured
via its decay product and is not necessarily equal to its spin as a free
field.

##  VI Acknowledgement

This work was supported by National Science Foundation of China (12105068) and
Hangzhou Normal University Start-up Funds.

## References

  * (1) M. E. Peskin and D. V. Schroeder, An Introduction to quantum field theory.  Addison-Wesley, Reading, USA
  * (2) S. Weinberg, The Quantum theory of fields. Vol. 1: Foundations.  Cambridge University Press
  * (3) G. F. Sterman, An Introduction to quantum field theory.  Cambridge University Press
  * (4) Particle Data Group, R. L. Workman et al., “Review of Particle Physics,” PTEP 2022 (2022) 083C01
  * (5) D. Pocanic, E. Frlez, and A. van der Schaaf, “Experimental study of rare charged pion decays,” J. Phys. G 41 (2014) 114002, arXiv:1407.2865 [hep-ex]
  * (6) A. Abashian et al., “Angular Distributions of Positrons from pi+-mu+-e+ Decays Observed in a Liquid Hydrogen Bubble Chamber,” Phys. Rev. 105 (1957) 1927–1928
  * (7) S. Taylor, E. L. Koller, T. Huetter, P. Stamer, and J. Grauman, “Search for anomalous ${\pi}^{+}$ decay among ${\tau}^{+}$ decay secondaries,” Phys. Rev. Lett. 14 (May, 1965) 745–746. https://link.aps.org/doi/10.1103/PhysRevLett.14.745
  * (8) E. Frota-Pessoa, “Isotropy in pi-minus mu decays,” Phys. Rev. 177 (1969) 2368–2370
  * (9) H. Hulubei, J. S. Auslander, E. M. Friedlander, and S. Titeica, “Angular Distribution of Muons in pi-mu Decay at Rest,” Phys. Rev. 129 (1963) 2789–2801. [Erratum: Phys.Rev. 131, 2841 (1963)]
  * (10) MINOS, P. Adamson et al., “Testing Lorentz Invariance and CPT Conservation with NuMI Neutrinos in the MINOS Near Detector,” Phys. Rev. Lett. 101 (2008) 151601, arXiv:0806.4945 [hep-ex]
  * (11) MINOS, P. Adamson et al., “Search for Lorentz invariance and CPT violation with muon antineutrinos in the MINOS Near Detector,” Phys. Rev. D 85 (2012) 031101, arXiv:1201.2631 [hep-ex]
  * (12) B. Altschul, “Contributions to Pion Decay from Lorentz Violation in the Weak Sector,” Phys. Rev. D 88 (2013) 076015, arXiv:1308.2602 [hep-ph]
  * (13) B. Altschul, “Neutrino Beam Constraints on Flavor-Diagonal Lorentz Violation,”Phys. Rev. D 87 9, (2013) 096004, arXiv:1302.2598 [hep-ph]
  * (14) J. P. Noordmans and K. K. Vos, “Limits on Lorentz violation from charged-pion decay,”Phys. Rev. D 89 10, (2014) 101702, arXiv:1404.7629 [hep-ph]
  * (15) J. S. Díaz, A. Kostelecký, and R. Lehnert, “Relativity violations and beta decay,”Phys. Rev. D 88 7, (2013) 071902, arXiv:1305.4636 [hep-ph]
  * (16) J. P. Noordmans, H. W. Wilschut, and R. G. E. Timmermans, “Limits on Lorentz violation from forbidden $\beta$ decays,”Phys. Rev. Lett. 111 17, (2013) 171601, arXiv:1308.5570 [hep-ph]
  * (17) . Fayyazuddin and . Riazuddin, A Modern Introduction To Particle Physics (3rd Edition).  Oxford University Press
  * (18) V. D. Barger and R. J. N. Phillips,  COLLIDER PHYSICS
  * (19) N. Nakanishi, “T*-product and false nonconservation of angular momentum in the pion decay,” Mod. Phys. Lett. A 17 (2002) 89–93
  * (20) M. Jacob and G. C. Wick, “On the General Theory of Collisions for Particles with Spin,” Annals Phys. 7 (1959) 404–428
  * (21) W. M. Gibson and B. R. Pollard, Symmetry Principles in Elementary Particle Physics.  Cambridge University Press
  * (22) E. Leader, Spin in particle physics,  vol. 15
  * (23) D. H. Perkins, Introduction to high energy physics.  Cambridge University Press, 4th ed.


# Generating Uniform Magnetic Fields Using Pulsed Field Magnetisation

Dian Weerakonda
Department of Engineering
University of Cambridge
Cambridge, CB2 1PZ
U.K.
<EMAIL_ADDRESS>
Benjamin Bryant
Oxford Instruments NanoScience
Abingdon
Oxfordshire, OX13 5QX
U.K.
<EMAIL_ADDRESS>
Anthony Dennis
Department of Engineering
University of Cambridge
Cambridge, CB2 1PZ
U.K.
<EMAIL_ADDRESS>
Tobia Nava
Department of Engineering
University of Cambridge
Cambridge, CB2 1PZ
U.K.
<EMAIL_ADDRESS>
John Durrell
Department of Engineering
University of Cambridge
Cambridge, CB2 1PZ
U.K.
<EMAIL_ADDRESS>

(February 9, 2024)

###### Abstract

Bulk high-temperature superconductors (HTS) are capable of generating very
strong magnetic fields while maintaining a relatively compact form factor.
Solenoids constructed using stacks of ring-shaped bulk HTS have been
demonstrated to be capable of nuclear magnetic resonance (NMR) spectroscopy
and magnetic resonance imaging (MRI). However, these stacks were magnetised
via field cooling (FC), which typically requires a secondary superconducting
charging magnet capable of sustaining a high magnetic field for a long period.
A more economical alternative to FC is pulsed field magnetisation, which can
be carried out with a magnet wound from a normal conductor, such as copper. In
this work, we present a technique we have developed for iteratively
homogenising the magnetic field within a stack of ring-shaped bulk HTS by
manipulating the spatial profile of the applied pulsed field.

_Keywords_ bulk high-temperature superconductor $\cdot$ pulsed field
magnetisation $\cdot$ trapped field magnet

##  1 Introduction

Bulk high-temperature superconductors (HTS) promise to replace permanent
magnets in multiple applications where performance is directly correlated with
the strength of the magnetic field [1, 2, 3]. Among these is nuclear magnetic
resonance (NMR) spectroscopy. While NMR spectrometers typically make use of
magnets made from low-temperature superconductors (LTS) operating above 9 T,
permanent magnet-based benchtop NMR spectrometers are not uncommon. They
require far less lab space, and the purchase and maintenance costs are only a
fraction of that of their superconducting counterparts [4]. However, the
sensitivity and resolution of these systems are limited by their modest
magnetic fields, where the current state-of-the-art is unable to reach fields
above 3 T [5].

On the other hand, single-grain (RE)Ba2Cu3O7-δ (REBCO), where RE is a rare-
earth element, ring-shaped bulks have been used by Nakamura et al. to
construct a bulk HTS magnet for NMR spectroscopy that was able to operate at
4.7 T [6]. While the homogeneity of this magnet was much lower than that which
typical commercial systems achieve, it demonstrated that bulk HTS has the
potential to replace permanent magnets in benchtop NMR systems. However,
perhaps the biggest downside to their implementation, from a practical
standpoint, was the method of magnetisation—field cooling (FC). FC requires a
secondary charging magnet that can fully envelop the magnet being charged,
which, in their case, was a large-bore superconducting magnet. Should such a
magnetising technique be implemented in a commercial system, the large
charging magnet will need to be shipped with the spectrometer magnet for
charging onsite. Alternatively, the spectrometer magnet will need to be
magnetised in the factory and somehow kept at its operating temperature (< 95
K) during shipping and installation. Both of these methods pose significant
difficulties for implementation that are not present if a pulsed field is used
for magnetisation.

Pulsed field magnetisation (PFM) of bulk HTS can be performed by discharging a
bank of capacitors into a magnet wound from a normal conductor, such as
copper, to generate a strong magnetic field over a small period of time (< 200
ms) [7, 8]. The use of copper significantly reduces the manufacturing and
operating costs of the charging magnet. Furthermore, the small footprint of
the charging magnet, resulting from the absence of stringent cooling demands
from the copper, allows the charging magnet to be integrated into the main
spectrometer magnet assembly. In this work, we demonstrate that the spatial
profile of the pulsed field applied to a solenoid constructed out of a stack
of ring-shaped EuBCO bulks can be varied to iteratively modify (and
homogenise) the trapped field.

##  2 Experimental Method

###  2.1 Field Cooling

Four EuBCO ring-shaped bulks with a 60 mm outer diameter, 36 mm inner
diameter, and 17 mm high were stacked together to form the bulk HTS solenoid.
Each bulk was fully encapsulated in a ring of copper with a 5 mm wall
thickness to minimise any thermal gradients between different sections of the
solenoid. This stack was then field-cooled at 77 K with an applied field of
3.5 T. Upon the removal of the applied field, a Hall probe mounted to a linear
stage was used to map the $z$-component (the direction along the stack) of the
magnetic field along the central cylindrical axis of the solenoid in 1 mm
steps. The magnet used for FC has a sufficiently large uniform region that we
may assume each of the four bulks was fully magnetised following the
procedure.

###  2.2 Pulsed Field Magnetisation

The setup used for PFM is shown in Fig. 1. The bulks were assembled on the
coldhead of a Gifford-McMahon cryocooler inside a 316L stainless steel vacuum
chamber with a room temperature bore going through the centre of the stack.
Heaters were used at the top and bottom of the stack to control the
temperature. To generate the pulsed field, a copper solenoid connected to a
bank of capacitors was used, with an IGBT acting as a switch that connects the
capacitors to the solenoid. Details of this system can be found in [9]. The
solenoid was placed on a translating table such that its vertical position
could be adjusted as required to modify the spatial profile of the pulse
applied to the stack. During operation, the copper solenoid was cooled with
liquid nitrogen to minimise its electrical resistance and hence maximise
efficiency. Five Hall sensors were placed inside the room temperature bore to
monitor the magnetic field during and after the pulse application.

Figure 1: Pulsed field magnetisation setup. Four EuBCO rings are assembled
inside a 316L stainless steel vacuum chamber with a room temperature bore, on
the coldhead of a GM cryocooler. A liquid nitrogen-cooled solenoid, whose
vertical position is adjustable, is placed around the vacuum chamber to
generate the pulsed field.

Starting with an initial pulse with the centre of the coil aligned with bottom
bulk, pulses with peak field values in the range of 1.3-1.7 T were applied
with the pulse coil aligned at various positions of the stack to minimise the
trapped magnetic field gradient along the $z$-axis of the stack. This allowed
the trapped field within the stack to be iteratively homogenised, with only a
small delay (5 minutes) between subsequent pulses to allow for thermal
gradients induced by the preceding pulse to dissipate. Once the process of
iterative homogenisation was completed, the Hall sensors inside the bore were
removed and a single Hall sensor mounted to a linear stage was used to measure
the $z$-component of the trapped magnetic field in the stack along the central
cylindrical axis in 1 mm steps.

##  3 Results & Discussion

The normalised axial profiles obtained with field cooling and PFM are shown in
Fig. 2. The central field achieved with FC is 2.0 T, while with PFM, it is
0.30 T. However, the total variation in field across the length of the
solenoid in the PFM case is 4$\times$ lower. This variant of PFM, where the
spatial profile of the trapped field is modified iteratively with a very small
delay between iterations, could be highly instrumental when magnetising bulk
HTS solenoids for applications where the uniformity of the field is crucial.

Further development of this technique should be towards developing an
automated routine incorporating high-resolution magnetic field maps to
determine the exact spatial profile of the pulse required to shim the trapped
field. Currently, there is only one degree of freedom for the spatial profile
of the pulse, but, for example, a segmented pulse coil where each segment can
be activated independently will allow the span of the pulse to also be
controlled, which should allow for more precise tuning of the trapped field.

Figure 2: Normalised axial magnetic field profiles obtained with field cooling
and PFM using a Hall sensor attached to a linear stage. The central field with
FC was 2.0 T and the central field with PFM was 0.3 T. The field variation
across the length of the stack is 4$\times$ lower in the iteratively
homogenised PFM case.

##  4 Conclusion

The ability of REBCO bulks to trap strong magnetic fields at relatively high
temperatures has been utilised in multiple applications. Our results
demonstrate that the PFM technique has benefits beyond economy and footprint.
PFM could be used for not just magnetisation, but also the shimming of bulk
HTS magnets for NMR spectroscopy or MRI. Further development of the PFM
technique to reach trapped fields comparable to that of FC could provide a
compelling case for the adoption of PFM as the standard technique of
magnetisation of bulk HTS for applications that require precise control of the
generated field.

## Acknowledgments

This work was supported by funding from Oxford Instruments NanoScience and the
Engineering and Physical Sciences Research Council. This work was also
supported by the Engineering and Physical Sciences Research Council’s
Cambridge Royce facilities grant EP/P024947/1 and Sir Henry Royce Institute –
recurrent grant EP/R00661X/1.

## References

  * [1] J. H. Durrell, M. D. Ainslie, D. Zhou, P. Vanderbemden, T. Bradshaw, S. Speller, M. Filipenko, and D. A. Cardwell, “Bulk superconductors: a roadmap to applications,” _Superconductor Science and Technology_ , vol. 31, no. 10, p. 103501, 10 2018. [Online]. Available: https://iopscience.iop.org/article/10.1088/1361-6668/aad7ce
  * [2] M. D. Ainslie, A. George, R. Shaw, L. Dawson, A. Winfield, M. Steketee, and S. Stockley, “Design and market considerations for axial flux superconducting electric machine design,” _Journal of Physics: Conference Series_ , vol. 507, no. 3, p. 032002, 5 2014.
  * [3] K. Nakagawa, F. Mishima, Y. Akiyama, and S. Nishijima, “Study on magnetic drug delivery system using HTS bulk magnet,” _IEEE Transactions on Applied Superconductivity_ , vol. 22, no. 3, 2012.
  * [4] T. A. van Beek, “Low-field benchtop NMR spectroscopy: status and prospects in natural product analysis†,” _Phytochemical Analysis_ , vol. 32, no. 1, pp. 24–37, 1 2021. [Online]. Available: https://onlinelibrary.wiley.com/doi/full/10.1002/pca.2921https://onlinelibrary.wiley.com/doi/abs/10.1002/pca.2921https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/10.1002/pca.2921
  * [5] “Q Magnetics | 125 MHz Benchtop NMR Spectrometer.” [Online]. Available: https://www.qmagnetics.com/
  * [6] T. Nakamura, D. Tamada, Y. Yanagi, Y. Itoh, T. Nemoto, H. Utumi, and K. Kose, “Development of a superconducting bulk magnet for NMR and MRI,” _Journal of Magnetic Resonance_ , vol. 259, pp. 68–75, 10 2015.
  * [7] U. Mizutani, T. Oka, Y. Itoh, Y. Yanagi, M. Yoshikawa, and H. Ikuta, “Pulsed-field magnetization applied to high-Tc superconductors,” _Applied Superconductivity_ , vol. 6, no. 2-5, pp. 235–246, 1998.
  * [8] Y. Tsui, D. A. Moseley, A. R. Dennis, Y. H. Shi, M. R. Beck, V. Cientanni, D. A. Cardwell, J. H. Durrell, and M. D. Ainslie, “Trapped Fields >1 T in a Bulk Superconducting Ring by Pulsed Field Magnetization,” _IEEE Transactions on Applied Superconductivity_ , vol. 32, no. 6, 9 2022.
  * [9] Y. K. Tsui, D. Moseley, A. R. Dennis, Y. Shi, M. Beck, V. Cientanni, D. Cardwell, J. Durrell, and M. Ainslie, “Waveform Control Pulsed Field Magnetization of RE-Ba-Cu-O Bulk Superconducting Rings,” _IEEE Transactions on Applied Superconductivity_ , vol. 32, no. 4, pp. 1–5, 6 2022.