# Using Learnable Physics for Real-Time Exercise Form Recommendations

Abhishek Jaiswal<EMAIL_ADDRESS>0000-0003-4795-6205 Indian
Institute of Technology, KanpurKalyanpurKanpurUttar PradeshIndia208016 ,
Gautam Chauhan<EMAIL_ADDRESS>0000-0003-4957-9140 Indian
Institute of Technology, KanpurKalyanpurKanpurUttar PradeshIndia208016 and
Nisheeth Srivastava<EMAIL_ADDRESS>0000-0001-9272-8418 Indian Institute
of Technology, KanpurKalyanpurKanpurUttar PradeshIndia208016

(2023)

###### Abstract.

Good posture and form are essential for safe and productive exercising. Even
in gym settings, trainers may not be readily available for feedback.
Rehabilitation therapies and fitness workouts can thus benefit from
recommender systems that provide real-time evaluation. In this paper, we
present an algorithmic pipeline that can diagnose problems in exercises
technique and offer corrective recommendations, with high sensitivity and
specificity, in real-time. We use MediaPipe for pose recognition, count
repetitions using peak-prominence detection, and use a learnable physics
simulator to track motion evolution for each exercise. A test video is
diagnosed based on deviations from the prototypical learned motion using
statistical learning. The system is evaluated on six full and upper body
exercises.These real-time recommendations, counseled via low-cost equipment
like smartphones, will allow exercisers to rectify potential mistakes making
self-practice feasible while reducing the risk of workout injuries.

real-time exercise pose recommendations, physics-inspired neural networks

††journalyear: 2023††copyright: acmlicensed††conference: Seventeenth ACM
Conference on Recommender Systems; September 18–22, 2023; Singapore,
Singapore††booktitle: Seventeenth ACM Conference on Recommender Systems
(RecSys ’23), September 18–22, 2023, Singapore, Singapore††price: 15.00††doi:
10.1145/3604915.3608816††isbn: 979-8-4007-0241-9/23/09††ccs: Computing
methodologies Interactive simulation††ccs: Computing methodologies Real-time
simulation††ccs: Human-centered computing Human computer interaction (HCI)

##  1\. Introduction

Sedentary lifestyles and physical inactivity are prominent risk factors for
cardiovascular diseases worldwide. Evidence also suggests that physical
activity has dipped considerably over time (Ozemek et al., 2019). Exercise
improves life expectancy and has an effective therapeutic impact on physical
and mental health (Jiménez-Pavón et al., 2020). Assistance in performing
physical exercises (Fletcher et al., 2018) , therefore, has a vital role to
play in improving health on a global scale.

Expert supervision in performing exercises is not readily available and even
if present, self practice needs further assistance. Consequently, digital
technologies have become instrumental in improving accessibility to expert
supervision. In gym settings, Sensor-based methods (Velloso et al., 2013;
Spina et al., 2013) have been prolifically used for pose assessment focusing
on exercise detection (Chang et al., 2007; Šeketa et al., 2015; Seeger et al.,
2011), rep counting (Soro et al., 2019; Spina et al., 2013), incorrect pose
diagnosis (Yurtman and Barshan, 2014; Giggins et al., 2014; Lee et al., 2020;
Kowsar et al., 2016) and suggestions (Zhao et al., 2014; Velloso et al., 2013;
Spina et al., 2013). However, sensors can be obtrusive, expensive, and
difficult to calibrate correctly, and so may be best suited for high-
performance settings (Khurana et al., 2018). More appropriate for less
intensive settings, vision-based methods (Wang et al., 2019; Liu and Chu,
2020; Gharasuie et al., 2021; Wang et al., 2021) have recently gained
prominence due to improvements in deep learning techniques and mobile camera
technology. This direction of research is very promising because it allows for
the possibility of entirely sensor-free tracking of exercise performance.

At present, however, such proposals face considerable difficulties. Most
vision-based approaches to exercise tracking work with predetermined heuristic
parameters which vary across exercises and participants, requiring
considerable hand-crafting (Liu and Chu, 2020; Chen and Yang, 2020). While
vision-based approaches to exercise type recognition and rep-counting are
plentiful, approaches that seek to track exercise form are limited to simple
upper body exercises with relatively little body movement (Liu and Chu, 2020;
Kowsar et al., 2016; Chen and Yang, 2020). Further, most such approaches offer
exercise diagnoses retrospectively after processing entire recorded exercise
sessions (Khurana et al., 2018; Soro et al., 2019).

Taking the conventional idea of recommendation from collaborative filtering a
step further, we frame the task of exercise supervision as that of
recommending correct forms. To this end, we identify the over-general nature
of the deep learning architectures used in vision-based exercise tracking
pipelines as a key problem blocking progress in this area. Rather than use
generic neural network architectures, we propose using a specific variety of
neural networks, specifically designed to learn relationships between physical
objects, as the base inference engine in such recommender systems. Using one
such architecture - Interaction Networks (Battaglia et al., 2016) \- we
describe a novel recommender system for real-time exercise form correction in
this paper. We show that our solution works with very high sensitivity and
specificity for a wide variety of full-body and upper-body exercises.

Section 2 places our proposal in line with existing approaches of exercise-
specific recommendations. We explain the working of our physics inference
model in Section 3 and the components of our recommender system in Section 4.
The value of our system is demonstrated through experiments on different
exercises in Section 5. Finally, we conclude by highlighting the salient
points of our recommender system and identifying directions for future work,
in Section 6.

[It shows the working of Interaction Network ]The image shows an example of
how an Interaction Network works. Given three objects, o1,o2, and o3, as nodes
and two relations, R1 and R2, as edges going from o1 to o3 and o2 to o3,
respectively. Using this data, function f_r evaluates effects e1 and e2
corresponding to R1 and R2. Function f_o uses these effects and object’s state
to predict the next state.

Figure 1. Example showing working of Interaction Network. Vector b includes
sender and receiver object details, Vector e is the resulting effect of an
interaction. Vector c includes interaction effects and the exercised object.

##  2\. Related Work

Several recently proposed systems (Ng, 2020; Vyas, 2019; Wang et al., 2021)
have used state-of-the-art pose estimation techniques to craft heuristic joint
angle thresholds for pose feedback. Recently, real-time pose diagnosis was
done by Alatiah and Chen (2020) using pre-calculated range of motion.
Similarly, Ying et al. (2021) used pre-stored correct exercises to detect
incorrect moves. Such systems offer only binary feedback without any
corrective recommendations.

More granular diagnoses are possible in a system recently proposed by Liu and
Chu (2020), who learned three joint angle indicators for each rep using a RNN
and provided visual diagnosis for two simple upper body dumbbell exercises
with high accuracy. However, their approach requires per frame annotation for
training. Similarly, Gharasuie et al. (2021) developed a low-cost system using
AlphaPose(Fang et al., 2017) based arm angles for upper-body exercises to
count reps and also quantify exercise phase parameters to estimate user
fatigue levels. While such heuristic based methods provide helpful textual
feedback in some instances, they tend to work well only for isolation arm
exercises involving only few joints and do not achieve significant diagnostic
accuracy without extensive frame-level annotation. Our system, in contrast,
with a more sophisticated inference engine, works well for compound exercises
using only video-level annotation.

Closer technically to our approach, Pose Trainer (Chen and Yang, 2020) uses
OpenPose (Cao et al., 2021) on dumbbell exercises along with Dynamic Time
Warping against template moves for rep diagnosis. They use angular heuristics
for exercise feedback. Similarly, AI Coach (Wang et al., 2019) compares sports
trajectories against pre-annotated bad poses, recommending an exemplar-based
video for all identified bad pose frames. This is in contrast with our system,
wherein holistic, real-time, body-focused textual feedback is offered to
users, immediately enabling them to correct any unsafe exercise posture.

Table 1. Body Landmarks for four full body exercises.

Squats | Sit-ups | Push-ups | Lunges
---|---|---|---
Nose | Nose | Shoulder | Shoulder
Left Hip | Shoulder | Hand | Hip
Right Hip | Hip | Hip | Front Knee
Left Knee | Knee | Toe | Back Knee
Right Knee | - | - | Back Toe
- | - | - | Front Heel

[Landmarks for each exercise.]The table shows the landmarks used for each
exercise. For squats, it’s Nose, Left hip, right hip, left knee, and right
knee. For sit-ups, it’s the nose, shoulder, hip, and knee. For push-ups, it’s
shoulder, hand, hip, and toe. For lunges, it’s shoulder, hip, front knee, back
knee, back toe, and front heel.

[Flowchart of our pipeline.]This image shows the general workflow of our
pipeline. We begin with extracting body landmarks through Mediapipe API on
test video. Then representative landmarks for each exercise are chosen and
normalized. We feed this data to the Interaction Network to get MSE time
series for the landmarks. These time series are transformed to frequency
domain using DTFT and a Random Forest classifier is used to classify the test
video into correct class or one of the incorrect classes with proper
recommendation displayed to the user.

Figure 2. Flowchart illustrating our Interactive System’s workflow.

##  3\. Physics prediction using Interaction Network

Interaction Network (IN) (Battaglia et al., 2016) is a Graph Networks based
general-purpose physics engine simulator. The graph nodes represent the
landmark joints (Table 1), and the edges represent the natural joint
relations. The IN (Figure 1) entails a relation-centric function $f_{R}$ for
predicting interaction effects and an object-centric function $f_{O}$ for
predicting the next step dynamics using the calculated interaction effects. It
feeds the sender landmarks’ properties($O_{S}$), the receiver landmark
properties($O_{R}$), and their relational properties($R_{a}$) as matrices at
the current time to the relation-centric function $f_{R}$, which outputs the
effect matrix $E$, therefore we have

$E=f_{R}(O_{S};O_{R};R_{a})\>\>\>\>\textit{where ; means concatenation}.$

Product of the effect matrix with the binary receiver matrix $R_{r}$ yields
$\bar{E}$ = $ER_{r}^{T}$ assimilating the net effects on each receiver object
in its columns. This, along with the object matrix O, is fed to the object-
centric function $f_{O}$ to predict the next state $P$ of each landmark.

$P=f_{O}(O;\bar{E})$

For more information, readers are requested to refer to the IN paper
(Battaglia et al., 2016).

##  4\. Proposing Pose Correcting Recommendations

We first outline the overall methodology of our pipeline, followed by a
detailed description of its sub-components. To begin with, we feed a recorded
or live video to our pipeline, which predicts per-frame keypoints for 25
joints through Mediapipe API (Bazarevsky et al., 2020). Depending on motion
evolution, we select exercise-specific landmarks (Table 1) followed by
normalization and smoothing for physics modeling. The ML model predicts the
motion rollouts for all the landmarks with visibility of only the initial rep
state. Using these predictions, we calculate the Mean Squared Error (MSE) for
individual landmarks and then transform them to the frequency domain for
further processing, as described in subsequent sections. Essentially, we use
frequency domain information from the MSE signals to classify exercise reps as
either correct or incorrect (in one of the multiple predefined modes of
failure) using a Random Forest multi-class classifier (Figure 2). Thus, our
pipeline receives visual input from the user side and emits textual
recommendations from the model side.

###  4.1. Rep Counting using Peak Prominence

We exploit cyclic movements within each exercise for rep segregation. To that
end, we find peaks in the periodic landmark displacement plot. These peaks may
contain extraneous motion data, such as fragments between successive reps and
other discontinuities from tired and distracted performers. To detect genuine
peaks, we find peaks’ importance using peak prominence and use its standard
deviation as a cutoff for our high pass filter. Displacement values above the
cutoff delineate the start and stop of a valid rep.Figure 3 shows the result
of rep counting for a single lunges video.

[A plot of displacement time-series with peaks.]This image shows a plot of
vertical displacement time-series of a chosen landmark’s y values, peaks, and
peak prominence calculated to separate two reps and the start and end of each
rep.

Figure 3. Peak prominence over vertical periodicity for rep counting.

[Figure showing an exerciser doing push-ups along with his referenced
landmarks’ positions]This image shows an exerciser doing push-ups exercise.
The referenced landmarks for push-ups : Hand, Shoulder, Hip, Foot are shown
with a stick figure plot.

Figure 4. Push-ups - stick figure and corresponding video frame.

###  4.2. Preprocessing

MediaPipe API provides 3D positional time series data for 25 body landmarks
for each exercise. We transform these coordinates for unidirectional facing
and use the resulting view along with the landmark’s displacement amplitude to
fix the representative landmarks for an exercise.

We apply Locally Weighted Scatterplot Smoothing (LOWESS) to each time series
(Cleveland, 1981) to reduce noise, discard reps with significant pose
estimation errors and Min-Max normalize the coordinates to induce
translational invariance, finally outputting a stick figure representations
for each rep (Figure 4).

###  4.3. Learning exercise dynamics

We represent body landmarks as graph nodes and the natural body connections as
the edges, one each for the forward and backward direction. The choice of
landmarks depended on our understanding of the biomechanics of each exercise.
We consider only 2D position and velocity as nodes’ attributes. Relational
attributes include joint-to-joint distances and angles, with positive and
negative unity indicating edge direction. A top and a bottom stationary
reference point is added to graph nodes with x coordinates 0.5 and y values as
0 and 1. Velocity is approximated as the difference between current and
previous coordinates. The two feed-forward neural networks - the relation-
centric model $f_{R}$ and the object-centric model $f_{O}$ consist of three
and four hidden layers respectively with each layer having 256 units with ReLU
activation and dropout value 0.5. The output layers have linear activation. We
train the model for 2500 epochs with early stopping using AdamW optimizer
(Loshchilov and Hutter, 2017) with 1cycle learning rate policy (Smith, 2018).
and a learning rate of 3e-4.

Table 2. Recommendations offered for full body exercises.

Lunges
---
Keep your knees behind the toes
Keep your legs closer, they are too wide apart
Squats
Keep your knees behind the toes
Don’t bend your knees inward
Keep your Feet shoulder-width apart
Situps
Your back should rise up completely
Pushups
Keep your Knees-hips-Shoulders in a straight line
Lower your chest to align it with hip
Lower your hips
Your chest should not touch the ground

[Sample of recommendations our pipeline provides]The table shows the
recommendations we provide for different anomalies for Lunges and Squats
label.

Table 3. F1 scores of training with different classifiers.

Classifier | Squats | Push-ups | Lunges | Sit-ups
---|---|---|---|---
SVM | 0.81 | 0.96 | 0.93 | 0.97
KNN | 0.75 | 0.87 | 0.94 | 0.92
Naive Bayes | 0.74 | 0.95 | 0.92 | 0.98
Logistic Reg. | 0.85 | 0.97 | 0.95 | 0.98
LDA | 0.87 | 0.98 | 0.97 | 0.97
Random Forest | 0.94 | 0.98 | 0.97 | 0.98

[Performance with different classifiers]This table shows the performance of
our pipeline with different classifiers using weighted F1 scores and standard
deviations over five training runs. We tested six classifiers - Support Vector
Machine, K Nearest Neighbours, Naive Bayes, Logistic Regression, Linear
Discriminant Analysis, and Random Forest. Random Forest gave the best results
in all cases, occasionally matched by one of the other classifiers.

Figure 5. DTFT representations of error signature of different squats labels.
Note how the phase plots for the incorrect squats differ from each other as
well as from a correct squat systematically.

[DTFT error representation for the different incorrect labels of squats.]The
figure shows three images, and each image shows two parts - the DTFT error
signature of squats on the left and the exerciser on the right. Three images
have squats with incorrect labels of ”feet should be shoulder width apart” and
”knee should not bend inward” and there is one correct exercise with the label
”Correct”.

###  4.4. Error Analysis and Rep Classification

The MSE time series emitted by the IN informs the subsequent stages. We
hypothesize that our physics engine predicts the correct method of exercising,
such that considerable deviation from it would hint at an incorrect rep.
Further, the specific combination of MSE from different body components would
hint at the precise mistake made by the exerciser.

To extract this information, we transform the MSE time series from all the
representative landmarks to the frequency domain using the discrete-time
Fourier transform (DTFT). DTFT provides magnitude and phase values for each
time series. Conversion to the frequency domain helps in two ways. It gives a
fixed-sized representation of the variable-length time series. It also helps
to extract the features of the time series. This output of the DTFT, called
the error signature (Figure 5), is a vector representation of an exercise rep
of variable duration. For our case, we take the principal 11 amplitudes and
the corresponding phase values to build the error signature of each exercise
rep. At the final stage of our pipeline, we use a Random Forest classifier for
classification, operated in a multi-class classification setting, with
recommendation labels used as shown in Table 2. We found Random Forest was the
most consistent classifier across all the exercises. (Table 3) and tuned its
hyperparameters using randomized search cross-validation

Table 4. Baselines comparisons for four full body exercises (left) and two
upper body exercises(right). Classification results reported using weighted F1
score with standard deviations over five train-test runs (*Shoulder Press
results reported for two incorrect classes).

Model | Squats | Push-ups | Lunges | Sit-ups
---|---|---|---|---
MLP | 0.91 $\pm$ 0.02 | 0.98 $\pm$ 0.03 | 0.95 $\pm$ 0.03 | 0.99 $\pm$ 0.01
RNN | 0.85 $\pm$ 0.04 | 0.98 $\pm$ 0.01 | 0.94 $\pm$ 0.01 | 0.98 $\pm$ 0.02
GRU | 0.87 $\pm$ 0.03 | 0.98 $\pm$ 0.01 | 0.93 $\pm$ 0.02 | 0.94 $\pm$ 0.04
IN | 0.94 $\pm$ 0.02 | 0.98 $\pm$ 0.01 | 0.97 $\pm$ 0.01 | 0.98 $\pm$ 0.01

Model | ShoulderPress* | FrontRaise
---|---|---
Ng (Ng, 2020) | 0.90 | 0.77
PoseTrainer(Chen and Yang, 2020) | 0.49 | 0.76
MLP | 0.99 $\pm$ 0.01 | 0.82 $\pm$ 0.04
RNN | 0.99 $\pm$ .01 | 0.79 $\pm$ .05
GRU | 0.95 $\pm$ .06 | 0.80 $\pm$ .04
IN | 0.98 $\pm$ 0.01 | 0.88 $\pm$ 0.03

[Weighted F1 scores for baselines and IN]The Table shows F1 scores for GRU,
NN, RNN and IN for 6 exercises - Squats, Push-ups, Lunges, Sit-ups, Shoulder
Press, Front Raise. IN performs best for Squats,Push-ups, Front Raise and
Lunges and comparatively for Sit-ups and Shoulder Press.

##  5\. Empirical Evaluation

Figure 6. Average rollout prediction errors over exercise reps(MSE) for
Baseline Models and Interaction Network. Even though MLP have good F1 scores
(Table 4) in some cases, high prediction error makes their performance
unreliable.

[MSE for baselines and IN]The image shows MSE scores for GRU, NN, RNN and IN
for four exercises - Push-ups, Squats, Lunges, and Sit-ups. IN has least MSE
for all cases, showing that it can accurately learn the physics of the
exercise.

###  5.1. Data

For the evaluation of full-body exercises, we used a proprietary dataset from
E-Trainer Analytics Wizard Pvt. Ltd. This dataset contains the front and side
view of seven exercisers performing more than 150 reps for four exercises -
squats, push-ups, lunges, and sit-ups. Each exercise has one correct class,
whereas incorrectly done exercises could belong to multiple classes. Incorrect
videos were annotated with corrective recommendations by expert physical
trainers.

We used a train test split of 60%-40% to train the classifier on incorrect
classes. Similarly, 60% of the correct class data was used to train our
physics engine. To compare against existing approaches of form prediction, we
conducted evaluation using annotated data for shoulder press and front raise
from a publicly available dataset (Ng, 2020)

###  5.2. Baseline Comparisons

Among available learnable dynamics predictors, we exploit IN for their
interpretability and simplicity. Our pipeline can also function with other
motion predictors to the degree that they can accurately mimic the dynamics of
the exercise. To test this hypothesis, we evaluate our model against several
baselines.

The Multi layer perceptron (MLP) Baseline, with three 256-length hidden layers
and ReLU activation, has all information to learn the interaction dynamics,
requiring it to assimilate relation indices implicitly without explicit scene
factorization. The Recurrent Neural Network (RNN) and Gated Recurrent Unit
(GRU), capable of modeling posture evolution, have three recurrent units with
three features in the hidden state. The final hidden state output is fed to a
fully connected layer to predict future dynamics. All the above baselines use
flattened node attributes as input.

Additionally, we compare our pipeline against popular heuristic techniques
((Ng, 2020), (Chen and Yang, 2020)), which examine reps using geometric
thresholds over joint features. Next, we describe ablation modifications on
IN’s architecture and input.

The Attribute Hidden IN uses the same IN architecture but with an empty
relation attribute matrix which, in principle, could be deduced from position
data demanding estimation of complex distance and inclination functions. The
Independent object IN simulates removing the relation-centric component by
zeroing out the interaction effects, which incapacitates it from modeling
object-object interactions. The Fully Connected (FC) IN connects each joint
with every other joint, procuring the same capacity as the IN but involving
additional irrelevant inputs. The Global Connection (GC) IN, additionally,
connects all the landmark points to the two stationary reference points,
modeling both local and global interactions for superior information
propagation.

###  5.3. Results

We evaluate our diagnostic system on six exercises using three criteria: Rep
Counting , Posture diagnosis, and Real-time prediction. Our peak-prominence
based algorithm perfectly counted all the reps in the full-body exercises. For
each rep detected, we measured recommendation accuracy using weighted F1
scores in a multi-class classification setting.

####  5.3.1. Posture diagnosis

Our results (Table 4) indicate that a pipeline endowed with physics learning
capability outperforms all baselines effectively differentiating correct and
incorrect exercise reps. Performance can be comparable for fewer prediction
classes (e.g., Sit-ups, Shoulder Press) or simple exercises with a small range
of joint motion(e.g., Push-ups). However, the performance gain is evident as
the number of incorrect classes increases (Front Raise - Table 5).

Since classification results are an indirect measure of physics modeling, we
also explore the MSE for the next state prediction (Figure 6). In all cases, a
physics learning engine best describes motion dynamics. Even though the MLP
showed good classification (Table 4), it significantly deviates from the
actual exercise, causing deviant performance, especially with increasing
exercise complexity (Table 5). Even an ill-suited model can classify well if
the error signatures are separable. However, such arbitrary performance gains
do not scale well as the number of classes increases.

Our ablation study on IN’s performance gain indicates that relational
attributes are critical for learning the physics of interactions (Figure 7).
Reasonable variations in the joint-to-joint links (as with FC-IN and GC-IN)
show similar effects, with GC-IN edging slightly over the vanilla IN, probably
due to faster information propagation. Stochastic interactions with
involuntary factors (like fatigue and distractions) possibly contain the
expected performance gain from global propagation. The FC-IN with redundant
information in its irrelevant relations competes well, presumably because the
IN learns to weigh the importance of exercise-specific relations. This
architecture may, thus, bypass the requirement of an explicit relational
matrix, provided that the pose estimation accurately detects all body
landmarks. As expected, the independent object IN without relation function
$f_{r}$ finds it challenging to model the motion dynamics. Similarly, the
Attribute Hidden IN is unable to exploit the joints’ information without
relational attributes.

Table 5. Baselines Comparison for Front Raise with increasing number of
classes . Methods with poor dynamics modeling show significant performance
drop.

FrontRaise | 2 Classes | 4 Classes | 6 Classes
---|---|---|---
MLP | 0.96 | 0.91 | 0.82
RNN | 0.93 | 0.90 | 0.79
GRU | 0.93 | 0.89 | 0.80
IN | 0.96 | 0.91 | 0.88

[Baseline comparison with Increasing Exercise Complexity in Front Raise]This
table shows that as the complexity of exercise increases, the performance
decreases. The performance drop is least for IN when we increase number of
classes from 2 to 6 and most for MLP which had the highest MSE among all
baselines.

####  5.3.2. Diagnosis latency

The MediaPipe Android API, fed with the exercise’s camera feed, outputs the
joints’ coordinates time series. After rep segregation, each rep data is
passed to the server, where our pipeline classifies it as correct or diagnoses
it as a mistake of a particular type. A corrective recommendation specific to
the estimated diagnosis is displayed to the user through our mobile
application. For exercisers operating at normal tempo, this feedback arrives
before their next rep is halfway complete prompting the user to instantly
correct any mistakes in technique (see Table 7 for a quantitative summary and
Table 7 for examples of live recommendations videos).

Table 6. Lag time(seconds) for new rep recognition.

Exercise | Mean(sec) | Standard deviation(sec)
---|---|---
Squats | 0.55 | 0.13
Sit-ups | 0.39 | 0.07
Push-ups | 0.36 | 0.11
Lunges | 0.54 | 0.09

[Lag time in new rep recognition]This table shows the time lag between the
user finishing a rep and its detection. As the rep is detected it is sent to
our server for labelling and appropriate suggestions is conveyed to the user.

Table 7. Link to videos of interaction network predictions rolled out over
time and of exercise sessions diagnosed using our system in real-time.

Data type | Link
---|---
IN predictions simulations | Link
Exercise Demo | Link

[Links to different video folders.]The table shows the links to interaction
networks prediction simulations and the exercise demo.

Figure 7. Average rollout prediction error over exercise reps(MSE) for
Ablation Models and Interaction Network. Models without relation information
experience significant drop in performance for dynamics prediction.

[MSE values for ablation models and IN]The image shows MSE for 4 ablation
model - Attribute Hidden IN, Independent Object IN, Global Connection IN and
Fully Connected IN. Attribute hidden IN and Independent Object IN perform
poorly as they do not have Interaction attributes and relation information
respectively. Other ablation models have similar performance.

##  6\. Discussion and Limitations

Self-training is gaining popularity as more and more people feel disinclined
to commit to a dedicated gym routine. By providing accurate real-time
recommendations, we can benefit such users without compromising their day-to-
day schedule. Our recommender system focuses on rep counting and diagnosis,
assuming that the exercise performed is known (or is easily knowable). For
instance, Moran et al. (2022) used MediaPipe API (Bazarevsky et al., 2020) for
pose recognition to detect the type of exercise performed in real-time, a
capability that could easily inform exercise type in our pipeline.

Thus, to summarize, this paper offers a novel system for recommending form
corrections to exercisers performing rep-based training in real-time with high
precision. We introduce the use of learnable physics engines to model body
physics, a task for which they are very well-suited. The success of our
physics model permits downstream classifiers to accurately diagnose modes of
failure of exercises using differential prediction error residuals between the
model prediction and actual observations. Empirical evaluations show that our
system diagnoses defective techniques in complex full-body exercises with high
sensitivity and specificity. We expect the adoption of such interactive
systems to help healthcare providers scale up access to supervised physical
exercise.

We conclude with a brief exploration of the limitations of our system, and
possible directions for future work. The most critical technical limitation of
the present system is its reliance on pre-defined relational attributes for
each exercise’s Interaction Network. These attributes depend on the nature of
human biomechanics and must be decided beforehand. Learning relational
attributes from data could improve this performance even further, a clear
direction for future work. Our system is currently tested only for exercises
with significant vertical periodicity, an artifact of our peak-prominence
based rep-counting scheme, though vertical periodicity also exists in many
other exercises. Replacing this with a more sophisticated rep-counting method
could extend our system’s capabilities to a more general set of exercises. In
particular, given the known diagnostic value of gait analysis in predicting
health outcomes for the elderly (Cesari et al., 2005; Verghese et al., 2009),
extending this system’s digital diagnostic capabilities to monitoring and
diagnosing gait-related problems presents a very promising direction for
future work.

## References

  * (1)
  * Alatiah and Chen (2020) Talal Alatiah and Chen Chen. 2020.  Recognizing exercises and counting repetitions in real time.  _arXiv preprint arXiv:2005.03194_ (2020).
  * Battaglia et al. (2016) Peter Battaglia, Razvan Pascanu, Matthew Lai, Danilo Jimenez Rezende, et al. 2016\.  Interaction networks for learning about objects, relations and physics.  _Advances in neural information processing systems_ 29 (2016).
  * Bazarevsky et al. (2020) Valentin Bazarevsky, Ivan Grishchenko, Karthik Raveendran, Tyler Zhu, Fan Zhang, and Matthias Grundmann. 2020.  Blazepose: On-device real-time body pose tracking.  _arXiv preprint arXiv:2006.10204_ (2020).
  * Cao et al. (2021) Zhe Cao, Gines Hidalgo, Tomas Simon, Shih-En Wei, and Yaser Sheikh. 2021.  OpenPose: Realtime Multi-Person 2D Pose Estimation Using Part Affinity Fields.  _IEEE Transactions on Pattern Analysis & Machine Intelligence_ 43, 01 (2021), 172–186.
  * Cesari et al. (2005) Matteo Cesari, Stephen B Kritchevsky, Brenda WHJ Penninx, Barbara J Nicklas, Eleanor M Simonsick, Anne B Newman, Frances A Tylavsky, Jennifer S Brach, Suzanne Satterfield, Douglas C Bauer, et al. 2005\.  Prognostic value of usual gait speed in well-functioning older people—results from the Health, Aging and Body Composition Study.  _Journal of the American Geriatrics Society_ 53, 10 (2005), 1675–1680.
  * Chang et al. (2007) Keng-hao Chang, Mike Y Chen, and John Canny. 2007.  Tracking free-weight exercises. In _International Conference on Ubiquitous Computing_. Springer, 19–37.
  * Chen and Yang (2020) Steven Chen and Richard R Yang. 2020.  Pose Trainer: correcting exercise posture using pose estimation.  _arXiv preprint arXiv:2006.11718_ (2020).
  * Cleveland (1981) William S Cleveland. 1981\.  LOWESS: A program for smoothing scatterplots by robust locally weighted regression.  _American Statistician_ 35, 1 (1981), 54.
  * Fang et al. (2017) Hao-Shu Fang, Shuqin Xie, Yu-Wing Tai, and Cewu Lu. 2017\.  Rmpe: Regional multi-person pose estimation. In _Proceedings of the IEEE international conference on computer vision_. 2334–2343.
  * Fletcher et al. (2018) Gerald F Fletcher, Carolyn Landolfo, Josef Niebauer, Cemal Ozemek, Ross Arena, and Carl J Lavie. 2018\.  Promoting physical activity and exercise: JACC health promotion series.  _Journal of the American College of Cardiology_ 72, 14 (2018), 1622–1639.
  * Gharasuie et al. (2021) Morteza Mohammady Gharasuie, Noah Jennings, and Shubham Jain. 2021\.  Performance Monitoring for Exercise Movements using Mobile Cameras. In _Proceedings of the Workshop on Body-Centric Computing Systems_. 1–6.
  * Giggins et al. (2014) Oonagh M Giggins, Kevin T Sweeney, and Brian Caulfield. 2014\.  Rehabilitation exercise assessment using inertial sensors: a cross-sectional analytical study.  _Journal of neuroengineering and rehabilitation_ 11, 1 (2014), 1–10.
  * Jiménez-Pavón et al. (2020) David Jiménez-Pavón, Ana Carbonell-Baeza, and Carl J Lavie. 2020\.  Physical exercise as therapy to fight against the mental and physical consequences of COVID-19 quarantine: Special focus in older people.  _Progress in cardiovascular diseases_ 63, 3 (2020), 386\.
  * Khurana et al. (2018) Rushil Khurana, Karan Ahuja, Zac Yu, Jennifer Mankoff, Chris Harrison, and Mayank Goel. 2018\.  GymCam: Detecting, recognizing and tracking simultaneous exercises in unconstrained scenes.  _Proceedings of the ACM on Interactive, Mobile, Wearable and Ubiquitous Technologies_ 2, 4 (2018), 1–17.
  * Kowsar et al. (2016) Yousef Kowsar, Masud Moshtaghi, Eduardo Velloso, Lars Kulik, and Christopher Leckie. 2016.  Detecting unseen anomalies in weight training exercises. In _Proceedings of the 28th Australian Conference on Computer-Human Interaction_. 517–526.
  * Lee et al. (2020) Jaehyun Lee, Hyosung Joo, Junglyeon Lee, and Youngjoon Chee. 2020\.  Automatic classification of squat posture using inertial sensors: Deep learning approach.  _Sensors_ 20, 2 (2020), 361.
  * Liu and Chu (2020) An-Lun Liu and Wei-Ta Chu. 2020.  A posture evaluation system for fitness videos based on recurrent neural network. In _2020 International Symposium on Computer, Consumer and Control (IS3C)_. IEEE, 185–188.
  * Loshchilov and Hutter (2017) Ilya Loshchilov and Frank Hutter. 2017.  Decoupled weight decay regularization.  _arXiv preprint arXiv:1711.05101_ (2017).
  * Moran et al. (2022) Alex Moran, Bart Gebka, Joshua Goldshteyn, Autumn Beyer, Nathan Johnson, and Alexander Neuwirth. 2022.  Muscle Vision: Real Time Keypoint Based Pose Classification of Physical Exercises.  _arXiv preprint arXiv:2203.12111_ (2022).
  * Ng (2020) Jiunn Ng. 2020.  _Posture evaluation for variants of weight-lifting workouts recognition_.  Ph. D. Dissertation. UTAR.
  * Ozemek et al. (2019) Cemal Ozemek, Carl J Lavie, and Øivind Rognmo. 2019\.  Global physical activity levels-Need for intervention.  _Progress in cardiovascular diseases_ 62, 2 (2019), 102–107.
  * Seeger et al. (2011) Christian Seeger, Alejandro P Buchmann, and Kristof Van Laerhoven. 2011.  myHealthAssistant: a phone-based body sensor network that captures the wearer’s exercises throughout the day.. In _BodyNets_. 1–7.
  * Šeketa et al. (2015) Goran Šeketa, Dominik Džaja, Sara Žulj, Luka Celić, Igor Lacković, and Ratko Magjarević. 2015.  Real-time evaluation of repetitive physical exercise using orientation estimation from inertial and magnetic sensors. In _First European Biomedical Engineering Conference for Young Investigators_. Springer, 11–15.
  * Smith (2018) Leslie N Smith. 2018\.  A disciplined approach to neural network hyper-parameters: Part 1–learning rate, batch size, momentum, and weight decay.  _arXiv preprint arXiv:1803.09820_ (2018).
  * Soro et al. (2019) Andrea Soro, Gino Brunner, Simon Tanner, and Roger Wattenhofer. 2019.  Recognition and repetition counting for complex physical exercises with deep learning.  _Sensors_ 19, 3 (2019), 714.
  * Spina et al. (2013) Gabriele Spina, Guannan Huang, Anouk Vaes, Martijn Spruit, and Oliver Amft. 2013.  COPDTrainer: a smartphone-based motion rehabilitation training system with real-time acoustic feedback. In _Proceedings of the 2013 ACM international joint conference on Pervasive and ubiquitous computing_. 597–606.
  * Velloso et al. (2013) Eduardo Velloso, Andreas Bulling, Hans Gellersen, Wallace Ugulino, and Hugo Fuks. 2013\.  Qualitative activity recognition of weight lifting exercises. In _Proceedings of the 4th Augmented Human International Conference_. 116–123.
  * Verghese et al. (2009) Joe Verghese, Roee Holtzer, Richard B Lipton, and Cuiling Wang. 2009.  Quantitative gait markers and incident fall risk in older adults.  _The Journals of Gerontology: Series A_ 64, 8 (2009), 896–901.
  * Vyas (2019) Parth Vyas. 2019\.  Pose estimation and action recognition in sports and fitness.  (2019).
  * Wang et al. (2019) Jianbo Wang, Kai Qiu, Houwen Peng, Jianlong Fu, and Jianke Zhu. 2019.  Ai coach: Deep human pose estimation and analysis for personalized athletic training assistance. In _Proceedings of the 27th ACM International Conference on Multimedia_. 374–382.
  * Wang et al. (2021) Lehao Wang, Yucheng Chen, and Weijun Duan. 2021.  Monocular Keypoint based Pull-ups Measurement on Strict Pull-ups Benchmark. In _2021 4th International Conference on Computer Science and Software Engineering (CSSE 2021)_. 307–311.
  * Ying et al. (2021) Haocong Ying, Tie Liu, Mingxin Ai, Jiali Ding, and Yuanyuan Shang. 2021.  AICoacher: A System Framework for Online Realtime Workout Coach. In _Proceedings of the 29th ACM International Conference on Multimedia_. 3787–3790.
  * Yurtman and Barshan (2014) Aras Yurtman and Billur Barshan. 2014.  Automated evaluation of physical therapy exercises using multi-template dynamic time warping on wearable sensor signals.  _Computer methods and programs in biomedicine_ 117, 2 (2014), 189–207.
  * Zhao et al. (2014) Wenbing Zhao, Roanna Lun, Deborah D Espy, and M Ann Reinthal. 2014\.  Realtime motion assessment for rehabilitation exercises: Integration of kinematic modeling with fuzzy inference.  _Journal of Artificial Intelligence and Soft Computing Research_ 4, 4 (2014), 267–285.


# Spectroscopic implications from the combined analysis of processes with
pseudoscalar mesons111Supported by the Votruba-Blokhintsev Program for
Cooperation of the Czech Republic with JINR (Dubna), the Grant Agency of the
Czech Republic (Grant No.202/08/0984), the Slovak Scientific Grant Agency
(Grant VEGA No.2/0034/09), and the Bogoliubov-Infeld Program for Cooperation
of Poland with JINR (Dubna). 222Talk given at XIII International Conference
Selected Problems of Modern Theoretical Physics, Bogoliubov Laboratory of
Theoretical Physics, JINR, Dubna, Russia, June 23-27, 2008.

Yu.S. Surovtsev333E-mail address<EMAIL_ADDRESS>Bogoliubov
Laboratory of Theoretical Physics, JINR, Dubna, Russia  P. Bydžovský444E-mail
address<EMAIL_ADDRESS>Nuclear Physics Institute, ASCR, Řež near Prague,
Czech Republic  M. Nagy555E-mail address<EMAIL_ADDRESS>Institute of
Physics, SAS, Bratislava, Slovakia

(19. 11. 2008)

###### Abstract

In the analysis a status and parameters of the scalar, vector, and tensor
mesonic resonances are obtained and compared with other results. Possible
classification of the resonance states in terms of the SU(3) multiplets is
discussed.

###### pacs:

11.55.Bq, 13.75.Lb, 14.40.Cs

Outline:

  * •

Motivation

  * •

Method of analysis

  * •

Analysis of the isoscalar-scalar sector

  * •

Analysis of the isovector $P$-wave of $\pi\pi$ scattering

  * •

Analysis of the isoscalar-tensor sector

  * •

Spectroscopic implications from the analysis

##  I Motivation

The spectroscopy of light mesons plays an important role in understanding the
strong interactions at low energies. Among possibilities to study the spectrum
of light mesons, analysis of the $\pi\pi$ interaction is particularly useful
and, therefore, it has always been an object of continuous theoretical and
experimental investigation PDG08 . Here, we present results of the coupled-
channel analysis of data on processes
$\pi\pi\to\pi\pi,K\overline{K},\eta\eta,\eta\eta^{\prime}$ in the channels
with $I^{G}J^{PC}=0^{+}0^{++}$ and $0^{+}2^{++}$ and on the $\pi\pi$
scattering in the channel with $1^{+}1^{--}$.

The scalar sector is problematic up to now especially as to an assignment of
the discovered mesonic states to quark-model configurations in spite of a big
amount of work devoted to these problems (see, e.g., Ref. Ani06  and
references therein). An exceptional interest to this sector is supported by
the fact that there, possibly indeed, we deal with a glueball $f_{0}(1500)$
(see, e.g., Ref. PDG08 ; Ams96 ).

Investigation of vector mesons is up-to-date subject due to their role in
forming the electromagnetic structure of particles and because our knowledge
about these mesons is still too incomplete (e.g., in the Particle Data Group
tables PDG08  (PDG) the mass of $\rho(1450)$ is ranging from 1250 to 1582
MeV).

In the tensor sector, among the thirteen discussed resonances, the nine states
($f_{2}(1430)$, $f_{2}(1565)$, $f_{2}(1640)$, $f_{2}(1810)$, $f_{2}(1910)$,
$f_{2}(2000)$, $f_{2}(2020)$, $f_{2}(2150)$, $f_{2}(2220)$) must be confirmed
in various experiments and analyses. For example, in the analysis of
$p\overline{p}\to\pi\pi,\eta\eta,\eta\eta^{\prime}$, five resonances –
$f_{2}(1920)$, $f_{2}(2000)$, $f_{2}(2020)$, $f_{2}(2240)$ and $f_{2}(2300)$ –
have been obtained, one of which, $f_{2}(2000)$, is a candidate for the
glueball Ani05 .

In our analysis, we have used both a model-independent method KMS96 , based on
the first principles (analyticity and unitarity) directly applied to analysis
of experimental data, and the multichannel Breit–Wigner forms. The former
approach permits us to introduce no theoretical prejudice to extracted
parameters of resonances, however, it is limited with the possibility to use
only three coupled channels. Therefore, in more general cases, one has to use,
e.g., the Breit–Wigner approach. Considering the obtained disposition of
resonance poles on the Riemann surface, obtained coupling constants with
channels, and resonance masses we draw particular conclusions about nature of
the investigated states.

##  II Method of analysis

In both methods of analysis, we parametrized the $S$-matrix elements
$S_{\alpha\beta}$ where $\alpha,\beta=1,2,\cdots,n$ denote channels, using the
Le Couteur-Newton relations LeCou . This relations express the $S$-matrix
elements of all coupled processes in terms of the Jost matrix determinant
$d(k_{1},\cdots,k_{n})$ that is a real analytic function with the only square-
root branch-points at the channel momenta $k_{\alpha}=0$.

In the model-independent approach, the $S$-matrix is determined on the 4- and
8-sheeted Riemann surfaces for the 2- and 3-channel cases, respectively. The
matrix elements $S_{\alpha\beta}$ have the right-hand cuts along the real axis
of the $s$ complex plane ($s$ is the invariant total energy squared), starting
at the coupled-channels thresholds $s_{i}$ ($i=1,2,3$), and the left-hand cuts
related to the crossed channels. The Riemann-surface sheets are numbered
according to the signs of analytic continuations of the channel momenta
$k_{i}=\sqrt{s-s_{i}}/{2}~{}~{}~{}~{}(i=1,2,3)$, as shown in Table 1.

Table 1: Signs of channel momenta on the eight sheets of the Rieman surface in the 3-channel case. sheet: | I | II | III | IV | V | VI | VII | VIII
---|---|---|---|---|---|---|---|---
Im $k_{1}$ | $+$ | $-$ | $-$ | $+$ | $+$ | $-$ | $-$ | $+$
Im $k_{2}$ | $+$ | $+$ | $-$ | $-$ | $-$ | $-$ | $+$ | $+$
Im $k_{3}$ | $+$ | $+$ | $+$ | $+$ | $-$ | $-$ | $-$ | $-$

The model-independent method which essentially utilizes an uniformizing
variable can be used only for the 2-channel case and under some conditions for
the 3-channel one. Only in these cases we obtain a simple symmetric (easily
interpreted) picture of the resonance poles and zeros of the $S$-matrix on an
uniformization plane. The important branch points, corresponding to the
thresholds of the coupled channels and to the crossing ones, are taken into
account in the uniformizing variable.

The resonance representations on the Riemann surfaces are obtained with the
help of formulas from Ref. KMS96 , expressing analytic continuations of the
$S$-matrix elements to unphysical sheets in terms of those on sheet I that
have only the zeros of resonances (beyond the real axis), at least, around the
physical region. Then, starting from the resonance zeros on sheet I, one can
obtain an arrangement of poles and zeros of resonance on the whole Riemann
surface.

In the 2-channel case, we obtain three types of resonances described by a pair
of conjugate zeros on sheet I: (a) in $S_{11}$, (b) in $S_{22}$, (c) in each
of $S_{11}$ and $S_{22}$.

In the 3-channel case, we obtain seven types of resonances corresponding to
seven possible situations when there are resonance zeros on sheet I only in
$S_{11}$ – (a); $S_{22}$ – (b); $S_{33}$ – (c); $S_{11}$ and $S_{22}$ – (d);
$S_{22}$ and $S_{33}$ – (e); $S_{11}$ and $S_{33}$ – (f); and $S_{11}$,
$S_{22}$, and $S_{33}$ – (g).
A resonance of every type is represented by a pair of complex-conjugate
clusters (of poles and zeros on the Riemann surface). Note that whereas the
cases (a), (b) and (c) can be simply related to the representation of
resonances by the Breit-Wigner forms, the cases (d), (e), (f) and (g) are
practically lost at that description. The cluster type is related to the
nature of state. For example, if we consider the $\pi\pi$, $K\overline{K}$,
and $\eta\eta$ channels, then a resonance which is coupled relatively more
strongly to the $\pi\pi$ channel than to the $K\overline{K}$ and $\eta\eta$
ones is described by the cluster of type (a). If the resonance is coupled more
strongly to the $K\overline{K}$ and $\eta\eta$ channels than to the $\pi\pi$
one, then it is represented by the cluster of type (e) (say, the state with
the dominant $s{\bar{s}}$ component). The flavour singlet (e.g., glueball)
must be represented by the cluster of type (g) (of type (c) in the 2-channel
consideration) as a necessary condition for the ideal case, if this state lies
above the thresholds of considered channels.

We can distinguish, in a model-independent way, a bound state of colourless
particles (e.g., $K\overline{K}$ molecule) and a $q{\bar{q}}$ bound state.
Just as in the 1-channel case, the existence of the particle bound-state means
the presence of the pole on the real axis under the threshold on the physical
sheet, so in the 2-channel case, the existence of the particle bound-state in
channel 2 ($K\overline{K}$ molecule) that, however, can decay into channel 1
($\pi\pi$ decay), would imply the presence of a pair of complex conjugate
poles on sheet II under the second-channel threshold without the corresponding
shifted pair of poles on sheet III.

In the 3-channel case, the bound-state in channel 3 ($\eta\eta$) that,
however, can decay into channels 1 ($\pi\pi$ decay) and 2 ($K\overline{K}$
decay), is represented by the pair of complex conjugate poles on sheet II and
by shifted poles on sheet III under the $\eta\eta$ threshold without the
corresponding poles on sheets VI and VII. This test KMS96 ; MPe93  is a
multichannel analogue of the known Castillejo–Dalitz–Dyson poles in the one-
channel case. According to this test, earlier in Ref. KMS96 , the
interpretation of the $f_{0}(980)$ state as the $K\overline{K}$ molecule has
been rejected because this state is represented by the cluster of type (a) in
the 2-channel analysis of processes $\pi\pi\to\pi\pi,K\overline{K}$ and,
therefore, it does not satisfy the necessary condition to be the
$K\overline{K}$ molecule.

##  III Analysis of the isoscalar-scalar sector

Considering the $S$-waves of processes
$\pi\pi\to\pi\pi,K\overline{K},\eta\eta,\eta\eta^{\prime}$ in the model-
independent method, we performed two variants of the 3-channel analysis:
variant I: the combined analysis of
$\pi\pi\to\pi\pi,K\overline{K},\eta\eta\,$;
variant II: analysis of $\pi\pi\to\pi\pi,K\overline{K},\eta\eta^{\prime}$.
Influence of the $\eta\eta^{\prime}$-channel in variant I and the
$\eta\eta$-channel in variant II are taken into account via the background.
Here, the left-hand cuts are neglected in the Riemann-surface structure
assuming that contributions on these cuts are also included in the background.

Under neglecting the $\pi\pi$-threshold branch point (however, unitarity on
the $\pi\pi$-cut is taken into account), the uniformizing variable is

$w=\frac{k_{2}+k_{3}}{\sqrt{m_{\eta}^{2}-m_{K}^{2}}}~{}~{}~{}~{}{\rm
for~{}variant~{}I},$ (1)

and

$w^{\prime}=\frac{k_{2}^{\prime}+k_{3}^{\prime}}{\sqrt{\frac{1}{4}(m_{\eta}+m_{\eta^{\prime}})^{2}-m_{K}^{2}}}~{}~{}~{}~{}{\rm
for~{}variant~{}II}.$ (2)

The quantities related to variant II are primed.

On the $w$-plane, the Le Couteur-Newton relations are 666 Other authors have
also used the parameterizations with the Jost functions in analyzing the
$S$-wave $\pi\pi$ scattering in the one-channel approach Boh80  and in the
two-channel one MPe93 .

$\displaystyle
S_{11}=\frac{d^{*}(-w^{*})}{d(w)},~{}~{}~{}~{}~{}~{}~{}~{}S_{22}=\frac{d(-w^{-1})}{d(w)},~{}~{}~{}~{}~{}~{}~{}~{}S_{33}=\frac{d(w^{-1})}{d(w)},$
(3) $\displaystyle
S_{11}S_{22}-S_{12}^{2}=\frac{d^{*}({w^{*}}^{-1})}{d(w)},~{}~{}~{}~{}~{}~{}~{}~{}S_{11}S_{33}-S_{13}^{2}=\frac{d^{*}(-{w^{*}}^{-1})}{d(w)}\,,$
(4)

where the $d$-function is assumed in the form

$d=d_{B}d_{res},$ (5)

and the resonance part is

$d_{res}(w)=w^{-\frac{M}{2}}\prod_{r=1}^{M}(w+w_{r}^{*})$ (6)

with $M$ the number of resonance zeros. The background part is taken as

$d_{B}=\mbox{exp}[-i\sum_{n=1}^{3}\frac{k_{n}}{m_{n}}(\alpha_{n}+i\beta_{n})],$
(7)

where

$\displaystyle\alpha_{n}=a_{n1}+a_{n\sigma}\frac{s-s_{\sigma}}{s_{\sigma}}\theta(s-s_{\sigma})+a_{nv}\frac{s-s_{v}}{s_{v}}\theta(s-s_{v}),$
(8)
$\displaystyle\beta_{n}=b_{n1}+b_{n\sigma}\frac{s-s_{\sigma}}{s_{\sigma}}\theta(s-s_{\sigma})+b_{nv}\frac{s-s_{v}}{s_{v}}\theta(s-s_{v})$
(9)

with $s_{\sigma}$ the $\sigma\sigma$ threshold and $s_{v}$ a combined
threshold of many opened channels in the vicinity of 1.5 GeV (e.g.,
$\eta\eta^{\prime},~{}\rho\rho,~{}\omega\omega$).

In variant II, the terms

$a_{n\eta}^{\prime}\frac{s-4m_{\eta}^{2}}{4m_{\eta}^{2}}\theta(s-4m_{\eta}^{2})~{}~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}~{}b_{n\eta}^{\prime}\frac{s-4m_{\eta}^{2}}{4m_{\eta}^{2}}\theta(s-4m_{\eta}^{2})$
(10)

should be added to $\alpha^{\prime}_{n}$ and $\beta^{\prime}_{n}$ to account
for an influence of the $\eta\eta$-channel.

As the data, we use the results of phase analyses given for phase shifts of
the amplitudes $\delta_{ab}$ and for moduli of the $S$-matrix elements
$\eta_{ab}=|S_{ab}|$ ($a,b=$1-$\pi\pi$, 2-$K\overline{K}$, 3-$\eta\eta$ or
$\eta\eta^{\prime}$):

$S_{aa}=\eta_{aa}e^{2i\delta_{aa}},~{}~{}~{}~{}~{}S_{ab}=\eta_{ab}e^{i\phi_{ab}}.$
(11)

If below the $\eta\eta$-threshold there is the 2-channel unitarity, then the
relations

$\eta_{11}=\eta_{22},~{}~{}\eta_{12}=(1-{\eta_{11}}^{2})^{1/2},~{}~{}\phi_{12}=\delta_{11}+\delta_{22}$
(12)

are fulfilled in this energy region.

The $\pi\pi$ scattering data, which range from the threshold up to 1.89 GeV,
are taken from Ref. Hya73 ; expd1 777Note that there are alternative data,
e.g., one of the solutions of the phase analysis in Ref. Grayer  and the
recent phase analysis in Ref. Kaminski_data  which are in accordance with each
other, but which differ from those used here, especially in the $f_{0}(980)$
region of energy. Analysis with these data should be performed separately.
This work is in progress.. For $\pi\pi\to K\overline{K}$, practically all the
accessible data are used expd2 . For $\pi\pi\to\eta\eta$, we used data for
$|S_{13}|^{2}$ from the threshold to 1.72 GeV expd3 . For
$\pi\pi\to\eta\eta^{\prime}$, the data for $|S_{13}|^{2}$ from the threshold
to 1.813 GeV are taken from Ref. expd4 . We included all the five resonances
discussed below 1.9 GeV.

In variant I, we got satisfactory description: for the $\pi\pi$ scattering,
$\chi^{2}/\mbox{NDF}\approx 1.35$; for $\pi\pi\to K\overline{K}$,
$\chi^{2}/\mbox{NDF}\approx 1.77$; for $\pi\pi\to\eta\eta$,
$\chi^{2}/\mbox{N.exp.points}\approx 0.86$. The total $\chi^{2}/\mbox{NDF}$ is
$345.603/(301-40)\approx 1.32$. From possible resonance representations by
pole-clusters, the analysis selects the following one: the $f_{0}(600)$ is
described by the cluster of type (a); $f_{0}(1370)$, type (c); $f_{0}(1500)$,
type (g); $f_{0}(1710)$, type (b); and the $f_{0}(980)$ is represented only by
the pole on sheet II and shifted pole on sheet III in both variants. The
background parameters are: $a_{11}=0.2006$, $a_{1\sigma}=0.0146$, $a_{1v}=0$,
$b_{11}=0$, $b_{1\sigma}=-0.01025$, $b_{1v}=0.0542$, $a_{21}=-0.6986$,
$a_{2\sigma}=-1.4207$, $a_{2v}=-5.958$, $b_{21}=0.047$, $b_{2\sigma}=0$,
$b_{2v}=6.888$, $b_{31}=0.6511$, $b_{3\sigma}=0.3404$, $b_{3v}=0$;
$s_{\sigma}=1.638~{}{\rm GeV}^{2}$, $s_{v}=2.084~{}{\rm GeV}^{2}$.

In variant II, we got the following description: for the $\pi\pi$ scattering
$\chi^{2}/\mbox{NDF}\approx 1.0$! for $\pi\pi\to K\overline{K}$
$\chi^{2}/\mbox{NDF}\approx 1.62$; for $\pi\pi\to\eta\eta^{\prime}$
$\chi^{2}/\mbox{N.exp.points}\approx 0.36$. The total $\chi^{2}/\mbox{NDF}$ is
$282.682/(293-38)\approx 1.11$! In this case, the $f_{0}(600)$ is described by
the cluster of type (a′); $f_{0}(1370)$, type (b′); $f_{0}(1500)$, type (d′);
and $f_{0}(1710)$, type (c′). The background parameters are:
$a_{11}^{\prime}=0.0111$, $a_{1\eta}^{\prime}=-0.058$,
$a_{1\sigma}^{\prime}=0$, $a_{1v}^{\prime}=0.0954$,
$b_{11}^{\prime}=b_{1\eta}^{\prime}=b_{1\sigma}^{\prime}=0$,
$b_{1v}^{\prime}=0.047$, $a_{21}^{\prime}=-3.439$,
$a_{2\eta}^{\prime}=-0.4851$, $a_{2\sigma}^{\prime}=1.7622$,
$a_{2v}^{\prime}=-5.158$, $b_{21}^{\prime}=0$, $b_{2\eta}^{\prime}=-0.7524$,
$b_{2\sigma}^{\prime}=2.6658$, $b_{2v}^{\prime}=1.836$,
$b_{31}^{\prime}=0.5545$, $s_{\sigma}=1.638~{}{\rm GeV}^{2}$,
$s_{v}=2.126~{}{\rm GeV}^{2}$.

Figure 1: The phase shift and module of the $S$-matrix element in the $S$-wave
$\pi\pi$-scattering. The solid curve corresponds to variant I and the dashed
curve to variant II.

Figure 2: The phase shift and module of the $S$-matrix element in $S$-wave of
$\pi\pi\to K\overline{K}$. The solid curve corresponds to variant I and the
dashed curve to variant II.

Figure 3: The squared modules of the $\pi\pi\to\eta\eta$ (upper figure) and
$\pi\pi\to\eta\eta^{\prime}$ (lower figure) $S$-wave matrix elements.

In Figures 1-3, we show results of fitting to the experimental data and in
Table 2 we indicate the obtained pole clusters for resonances on the eight
sheets of the complex energy plane $\sqrt{s}$, on which the 3-channel
$S$-matrix is determined ($\\!\sqrt{s_{r}}={\rm E}_{r}-i\Gamma_{r}\\!$).

Table 2: Pole clusters for the $f_{0}$-resonances in variants I and II. Sheet | II | III | IV | V | VI | VII | VIII
---|---|---|---|---|---|---|---
variant I
$\\!\\!f_{0}(600)\\!\\!$ | $\\!\\!{\rm E}_{r}\\!\\!$ |  598.2$\pm$13 |  585.8$\\!\pm\\!$14 |  |  |  505.8$\\!\pm\\!$16 |  518.2$\\!\pm\\!$15 |
| $\\!\Gamma_{r}\\!$ |  583$\\!\pm\\!$18 |  583$\\!\pm\\!$18 |  |  |  583$\\!\pm\\!$18 |  583$\\!\pm\\!$18 |
$\\!f_{0}(980)\\!$ | $\\!{\rm E}_{r}\\!$ |  1013.1$\\!\pm\\!$4 |  983.6$\\!\pm\\!$9 |  |  |  |  |
| $\\!\Gamma_{r}\\!$ |  34.1$\\!\pm\\!$6 |  57.4$\\!\pm\\!$10 |  |  |  |  |
$\\!f_{0}(1370)\\!$ | $\\!{\rm E}_{r}\\!$ |  |  |  |  1398.2$\\!\pm\\!$16 |  1398.2$\\!\pm\\!$18 |  1398.2$\\!\pm\\!$18 |  1398.2$\\!\pm\\!$13
| $\\!\Gamma_{r}\\!$ |  |  |  |  287.4$\\!\pm\\!$17 |  270.6$\\!\pm\\!$15 |  155$\\!\pm\\!$9 |  171.8$\\!\pm\\!$7
$\\!f_{0}(1500)\\!$ | $\\!{\rm E}_{r}\\!$ |  1502.6$\\!\pm\\!$11 |  1479.5$\\!\pm\\!$13 |  1502.6$\\!\pm\\!$12 |  1496.7$\\!\pm\\!$12 |  1498$\\!\pm\\!$16 |  1496.8$\\!\pm\\!$12 |  1502.6$\\!\pm\\!$10
| $\\!\Gamma_{r}\\!$ |  357.1$\\!\pm\\!$15 |  139.4$\\!\pm\\!$12 |  238.7$\\!\pm\\!$13 |  139.9$\\!\pm\\!$14 |  191.2$\\!\pm\\!$17 |  87.36$\\!\pm\\!$11 |  356.5$\\!\pm\\!$14
$\\!f_{0}(1710)\\!$ | $\\!{\rm E}_{r}\\!$ |  |  1708.2$\\!\pm\\!$12 |  1708.2$\\!\pm\\!$10 |  1708.2$\\!\pm\\!$13 |  1708.2$\\!\pm\\!$15 |  |
| $\\!\Gamma_{r}\\!$ |  |  142.3$\\!\pm\\!$9 |  160.3$\\!\pm\\!$8 |  323.3$\\!\pm\\!$14 |  305.3$\\!\pm\\!$13 |  |
variant II
$\\!f_{0}(600)\\!$ | $\\!{\rm E}_{r}\\!$ |  616.5$\\!\pm\\!$8 |  621.8$\\!\pm\\!$10 |  |  |  598.3$\\!\pm\\!$11 |  593$\\!\pm\\!$12 |
| $\\!\Gamma_{r}\\!$ |  563$\\!\pm\\!$11 |  563$\\!\pm\\!$12 |  |  |  563$\\!\pm\\!$14 |  563$\\!\pm\\!$13 |
$\\!f_{0}(980)\\!$ | $\\!{\rm E}_{r}\\!$ |  1009.3$\\!\pm\\!$3 |  986$\\!\pm\\!$6 |  |  |  |  |
| $\\!\Gamma_{r}\\!$ |  32$\\!\pm\\!$4 |  58$\\!\pm\\!$5.5 |  |  |  |  |
$\\!f_{0}(1370)\\!$ | $\\!{\rm E}_{r}\\!$ |  |  1394.3$\\!\pm\\!$9 |  1394.3$\\!\pm\\!$11 |  1412.7$\\!\pm\\!$13 |  1412.7$\\!\pm\\!$14 |  |
| $\\!\Gamma_{r}\\!$ |  |  236.3$\\!\pm\\!$10 |  255.7$\\!\pm\\!$12 |  255.7$\\!\pm\\!$12 |  236.3$\\!\pm\\!$19 |  |
$\\!f_{0}(1500)\\!$ | $\\!{\rm E}_{r}\\!$ |  1498.3$\\!\pm\\!$11 |  1502.4$\\!\pm\\!$9 |  1498.3$\\!\pm\\!$12 |  1498.3$\\!\pm\\!$13 |  1494.6$\\!\pm\\!$11 |  1498.3$\\!\pm\\!$14 |
| $\\!\Gamma_{r}\\!$ |  198.8$\\!\pm\\!$14 |  236.8$\\!\pm\\!$11 |  193$\\!\pm\\!$9 |  198.8$\\!\pm\\!$11 |  194$\\!\pm\\!$8 |  193$\\!\pm\\!$10 |
$\\!f_{0}(1710)\\!$ | $\\!{\rm E}_{r}\\!$ |  |  |  |  1726.1$\\!\pm\\!$12 |  1726.1$\\!\pm\\!$13 |  1726.1$\\!\pm\\!$12 |  1726.1$\\!\pm\\!$10
| $\\!\Gamma_{r}\\!$ |  |  |  |  140.2$\\!\pm\\!$9 |  111.6$\\!\pm\\!$8 |  84.2$\\!\pm\\!$8 |  112.8$\\!\pm\\!$7

The $f_{0}(1370)$ and $f_{0}(1710)$ are represented by the pole clusters
corresponding to states with the dominant $s\bar{s}$ component; $f_{0}(1500)$,
with the dominant glueball component.

Note a surprising result obtained for the $f_{0}(980)$. This state lies
slightly above the $K\overline{K}$ threshold and is described by the pole on
sheet II and by the shifted pole on sheet III under the $\eta\eta$ threshold
without the corresponding poles on sheets VI and VII, as it was expected for
standard clusters. This corresponds to the description of the $\eta\eta$ bound
state.

Masses and total widths of states should be calculated from the pole
positions. If, when calculating these quantities, the resonance part of
amplitude is taken in the form

$T^{res}=\frac{\sqrt{s_{r}}\Gamma_{el}}{m_{res}^{2}-s_{r}-i\sqrt{s_{r}}\Gamma_{tot}},$
(13)

we obtain values of masses and total widths of the $f_{0}$-resonances,
presented in Table 3.

Table 3: Masses and total widths of the $f_{0}$-resonances (all in MeV). | Variant I | Variant II
---|---|---
State | $m_{res}$ | $\Gamma_{tot}$ | $m_{res}$ | $\Gamma_{tot}$
$f_{0}(600)$ | 835.3 | 1166 | 834.9 | 1126
$f_{0}(980)$ | 1013.7 | 68.2 | 1009.8 | 64
$f_{0}(1370)$ | 1408.7 | 343.6 | 1417.5 | 511
$f_{0}(1500)$ | 1544 | 714 | 1511.4 | 398
$f_{0}(1710)$ | 1715.7 | 321 | 1729.8 | 225.6

##  IV Analysis of the isovector $P$-wave of $\pi\pi$ scattering

In this sector we applied both the model-independent method and multichannel
Breit–Wigner forms. We analyzed data in Ref. expd5 ; Hya73 , for the
inelasticity parameter ($\eta$) and phase shift of the $\pi\pi$-scattering
amplitude ($\delta$) ($S(\pi\pi\to\pi\pi)=\eta\exp(2i\delta)$), introducing
three ($\rho(770)$, $\rho(1250)$ and $\rho(1550-1780)$), four (the indicated
ones plus $\rho(1860-1910)$) and five (the indicated four plus $\rho(1450)$)
resonances SB_NPA08 .

###  IV.1 The Model-Independent Analysis

Since in the data for the $P$-wave $\pi\pi$ scattering a deviation from
elasticity is observed in the near-threshold region of the $\omega\pi$
channel, we considered explicitly the thresholds of the $\pi\pi$ and
$\omega\pi$ channels and the left-hand one at $s=0$ in the uniformizing
variable:

$v=\frac{(m_{\omega}+m_{\pi^{0}})/2~{}\sqrt{s-4m_{\pi^{+}}^{2}}+m_{\pi^{+}}~{}\sqrt{s-(m_{\omega}+m_{\pi^{0}})^{2}}}{\sqrt{s\left[\left((m_{\omega}+m_{\pi^{0}})/2\right)^{2}-m_{\pi^{+}}^{2}\right]}}.$
(14)

Influence of other channels which couple to the $\pi\pi$ one is supposed to be
taken into account via the background.

On the $v$-plane, the resonance part of the 2-channel $S$-matrix element of
$\pi\pi$-scattering $S_{res}$ has no cuts and has the form

$S_{res}=\frac{d(-v^{-1})}{d(v)},$ (15)

where $d(v)$ represents the contribution of resonances SB_NPA08 .

The background part is

$\displaystyle
S_{bg}=\exp\left[2i\left(\sqrt{\frac{s-4m_{\pi^{+}}^{2}}{s}}\right)^{3}\left(\alpha_{0}+\alpha_{1}~{}\frac{s-s_{1}}{s}~{}\theta(s-s_{1})+\right.\right.$
$\displaystyle\left.\left.\alpha_{2}~{}\frac{s-s_{2}}{s}~{}\theta(s-s_{2})\right)\right]\,,$
(16)

where $\alpha_{i}=a_{i}+ib_{i}$, $s_{1}$ is the threshold of 4$\pi$ channel
noticeable in the $\rho$-like meson decays and $s_{2}$ is the threshold of
$\rho 2\pi$ channel. Due to allowing for the left-hand branch-point at $s=0$
in the $v$-variable, $a_{0}=b_{0}=0$. Furthermore, $b_{1}=0$ which is related
to the experimental fact that the $P$-wave $\pi\pi$ scattering is elastic also
above the 4$\pi$-channel threshold up to about the $\omega\pi^{0}$ threshold.

In Figure 4 we present results of fitting to the data with three, four and
five resonances.

Figure 4: The phase shift of amplitude and module of the $S$-matrix element
for the $P$-wave $\pi\pi$-scattering in the model-independent approach.

We obtained satisfactory description with the total $\chi^{2}/\mbox{NDF}$
equal to $~{}291.76/(183-15)=1.74$, $~{}278.50/(183-19)=1.70$, and
$~{}266.14/(183-23)=1.66$ for the case of three, four and five resonances,
respectively.

The $\rho(770)$ is described by the cluster of type (a) and the others by type
(b). The background parameters are: $a_{1}=0.0093\pm 0.0199$, $a_{2}=0.0618\pm
0.0305$, and $b_{2}=-0.0135\pm 0.0371$ for the three-resonance,
$a_{1}=0.0017\pm 0.2118$, $a_{2}=0.0433\pm 0.3552$, and $b_{2}=-0.0044\pm
0.4782$ for the four-resonance, and $a_{1}=0.0256\pm 0.0186$, $a_{2}=0.0922\pm
0.0335$, and $b_{2}=0.0011\pm 0.0478$ for the five-resonance descriptions. The
positive sign of $b_{2}$ in the last case is more natural from the physical
point of view.

Though the description can be considered, practically, as the same in all
three cases, careful comparison of the obtained parameters and energy
dependence of the fitted quantities suggests that the resonance $\rho(1900)$
is desired and that the $\rho(1450)$ might be also included improving slightly
the description (at all events, its existence does not contradict to the
data).

In Table 4, we show the pole clusters of the $\rho$-like states on the lower
$\sqrt{s}$-half-plane (in MeV) (the conjugate poles on the upper half-plane
are not shown).

Table 4: Pole clusters distributed on the sheets II, III, and IV for the case with five $\rho$-like resonances. $\sqrt{s_{r}}$ in MeV is given. | II | III | IV
---|---|---|---
$\rho(770)$ | $\\!765.8\pm 0.6-i(73.3\pm 0.4)\\!$ | $\\!778.2\pm 9.1-i(68.9\pm 3.9)\\!$ |
$\rho(1250)$ |  | $\\!1251.4\pm 11.3-i(130.9\pm 9.1)\\!$ | $\\!1251\pm 11.1-i(130.5\pm 9.2)\\!$
$\rho(1470)$ |  | $\\!1469.4\pm 10.6-i(91\pm 12.9)\\!$ | $\\!1465.4\pm 12.1-i(99.8\pm 15.6)\\!$
$\rho(1600)$ |  | $\\!1634\pm 20.1-i(144.7\pm 23.8)\\!$ | $\\!1592.9\pm 7.9-i(73.7\pm 11.7)\\!$
$\rho(1900)$ |  | $\\!1882.8\pm 24.8-i(112.4\pm 25.2)\\!$ | $\\!1893\pm 21.9-i(93.4\pm 19.9)\\!$

Masses and total widths of the obtained $\rho$-states can be calculated from
the pole positions on sheets II and IV for resonances of type (a) and (b),
respectively. The obtained values are shown in Table 5.

Table 5: Calculated masses and total widths of the $\rho$-states (all in MeV). | $m_{res}$ | $\Gamma_{tot}$
---|---|---
$\rho(770)$ |  769.3$\pm$0.6 |  146.6$\pm$0.9
$\rho(1250)$ |  1257.8$\pm$11.1 |  261$\pm$18.3
$\rho(1470)$ |  1468.8$\pm 12.1$ |  199.6$\pm$31.2
$\rho(1600)$ |  1594.6$\pm 8$ |  147.4$\pm$23.4
$\rho(1900)$ |  1895.3$\pm$21.9 |  186.8$\pm$39.8

###  IV.2 The Breit–Wigner Analysis

We used the 5-channel Breit–Wigner forms in constructing the Jost matrix
determinant $d(k_{1},\cdots,k_{5})$. The resonance poles and zeros in the
$S$-matrix are generated utilizing the Le Couteur–Newton relation

$S_{11}=\frac{d(-k_{1},\cdots,k_{5})}{d(k_{1},\cdots,k_{5})}\;,$ (17)

where $k_{1}$, $k_{2}$, $k_{3}$, $k_{4}$, and $k_{5}$ are the momenta of
$\pi\pi$, $\pi^{+}\pi^{-}2\pi^{0}$, $2\pi^{+}2\pi^{-}$, $\eta 2\pi$, and
$\omega\pi^{0}$ channels, respectively. The Jost function is taken as

$d=d_{res}d_{bg}\,,$ (18)

where the resonance part is

$d_{res}(s)=\prod_{r}\left[M_{r}^{2}-s-i\sum_{j=1}^{5}\rho_{rj}^{3}~{}R_{rj}~{}f_{rj}^{2}\right]$
(19)

with $\rho_{rj}=k_{j}(s)/k_{j}(M_{r}^{2})$ and $f_{rj}^{2}/M_{r}$ the partial
width of a resonance of mass $M_{r}$. $R_{rj}$ is a Blatt–Weisskopf barrier
factor:

$R_{rj}=\frac{1+\frac{1}{4}(\sqrt{M_{r}^{2}-4m_{j}^{2}}~{}r_{rj})^{2}}{1+\frac{1}{4}(\sqrt{s-4m_{j}^{2}}~{}r_{rj})^{2}}$
(20)

with radius $r_{rj}=0.7035$ fm for all resonances in all channels as a result
of our analysis. Furthermore, we have assumed that the widths of resonance
decays to $\pi^{+}\pi^{-}2\pi^{0}$ and $2(\pi^{+}\pi^{-})$ channels are
related each other by relation: $f_{r2}=f_{r3}/\sqrt{2}$. This relation is
well justified with a 5-10% accuracy, for example, by calculations of the
$\rho^{0}$-meson decays in some variant of the chiral model Ach05 .

The background part of the Jost function is

$d_{bg}=\exp\left[-i\left(\sqrt{\frac{s-4m_{\pi^{+}}^{2}}{s}}\right)^{3}\left(\alpha_{0}+\alpha_{1}~{}\frac{s-s_{1}}{s}~{}\theta(s-s_{1})\right)\right]\;,$
(21)

where $\alpha_{i}=a_{i}+ib_{i}$ and $s_{1}$ is the threshold of the $\rho
2\pi$ channel.

In Figure 5, results of fitting to the data are shown and in Table 6, the
$\rho$-like resonance parameters are presented. We obtained equally reasonable
description in all three cases: the total
$\chi^{2}/\mbox{NDF}=316.21/(183-17)=1.87$, $314.69/(183-22)=1.92$, and
$303.10/(183-27)=1.91$ for the case of three, four, and five resonances,
respectively.

Figure 5: The phase shift of amplitude and module of the $S$-matrix element for the $P$-wave $\pi\pi$-scattering for the case of five resonances in the Breit-Wigner approach. Table 6: The $\rho$-like resonance parameters in the Breit-Wigner analysis (all in MeV). State | $\rho(770)$ | $\rho(1250)$ | $\rho(1450)$ | $\rho(1600)$ | $\rho(1900)$
---|---|---|---|---|---
$M$ | 777.69$\\!\pm\\!$0.32 | 1249.8$\\!\pm\\!$15.6 | 1449.9$\\!\pm\\!$12.2 | 1587.3$\\!\pm\\!$4.5 | 1897.8$\\!\pm\\!$38
$f_{r1}$ | 343.8$\\!\pm\\!$0.73 | 87.7$\\!\pm\\!$7.4 | 56.9$\\!\pm\\!$5.4 | 248.2$\\!\pm\\!$5.2 | 47.3$\\!\pm\\!$12
$f_{r2}$ | 24.6$\\!\pm\\!$5.8 | 186.3$\\!\pm\\!$39.9 | 100.1$\\!\pm\\!$18.7 | 240.2$\\!\pm\\!$8.6 | 73.7
$f_{r3}$ | 34.8$\\!\pm\\!$8.2 | 263.5$\\!\pm\\!$56.5 | 141.6$\\!\pm\\!$26.5 | 339.7$\\!\pm\\!$12.5 | 104.3
$f_{r4}$ |  | 231.8$\\!\pm\\!$111 | 141.2$\\!\pm\\!$98 | 141.8$\\!\pm\\!$33 | 9
$f_{r5}$ |  | 231$\\!\pm\\!$115 | 150$\\!\pm\\!$95 | 108.6$\\!\pm\\!$40.4 | 10
$\Gamma_{tot}$ |  $\approx$154.3 |  $>$175 |  $>$52 |  $>$168 |  $>$10

The background parameters for the five-resonance description are:
$a_{0}=-0.00121\pm 0.0018$, $a_{1}=-0.1005\pm 0.011$, and $b_{1}=0.0012\pm
0.006$. The background parameters for the other two cases can be found in Ref.
SB_NPA08 .

In order to look at consistency of the description, we checked if the obtained
formula for the $\pi\pi$-scattering amplitude gives a value of the scattering
length consistent with the results of other approaches (Table 7). It seems
that the satisfactory agreement we obtained is not accidental, because in the
energy region from the $\pi\pi$ threshold to about 500 MeV (where the
experimental data appear) there are no opened channels. Therefore, at the
adequate representation of the amplitude, its continuation to the threshold is
unique.

Table 7: Comparison of the $\pi\pi$ scattering length from various approaches. $a_{1}^{1}[10^{-3}m_{\pi^{+}}^{-3}]$ |  References |  Remarks
---|---|---
$33.9\pm 2.02$ | This paper | Breit–Wigner analysis
$34$ | NJL92  | Local NJL model
$37$ | Osi06  | Non-local NJL model
$37.9\pm 0.5$ | Cap08  | Roy equations using ChPT
$39.6\pm 2.4$ | Kam03  | Roy equations
$38.4\pm 0.8$ | Pel05  | Forward dispersion relations

##  V Analysis of isoscalar-tensor sector

In analysis of the processes $\pi\pi\to\pi\pi,K\overline{K},\eta\eta$, we
considered explicitly also the channel $(2\pi)(2\pi)$. Here it is impossible
to use the uniformizing-variable method. Therefore, using the Le Couteur-
Newton relations, we generate the resonance poles by some 4-channel Breit-
Wigner forms. The $d(k_{1},k_{2},k_{3},k_{4})$-function is taken as
$d=d_{B}d_{res}$, where the resonance part is

$d_{res}(s)=\prod_{r}\left[M_{r}^{2}-s-i\sum_{j=1}^{4}\rho_{rj}^{5}R_{rj}f_{rj}^{2}\right]$
(22)

with $\rho_{rj}=2k_{j}/\sqrt{M_{r}^{2}-4m_{j}^{2}}$ and $f_{rj}^{2}/M_{r}$ the
partial width. The Blatt–Weisskopf barrier factor for a tensor particle is

$R_{rj}=\frac{9+\frac{3}{4}(\sqrt{M_{r}^{2}-4m_{j}^{2}}~{}r_{rj})^{2}+\frac{1}{16}(\sqrt{M_{r}^{2}-4m_{j}^{2}}~{}r_{rj})^{4}}{9+\frac{3}{4}(\sqrt{s-4m_{j}^{2}}~{}r_{rj})^{2}+\frac{1}{16}(\sqrt{s-4m_{j}^{2}}~{}r_{rj})^{4}},$
(23)

with radii of 0.943 fm for all resonances in all channels, except for
$f_{2}(1270)$ and $f_{2}(1960)$ for which they are: for $f_{2}(1270)$, 1.498,
0.708, and 0.606 fm in the channels $\pi\pi$, $K\overline{K}$, and $\eta\eta$,
respectively; for $f_{2}(1960)$, 0.296 fm in the channel $K\overline{K}$.

The background part has the form

$d_{B}=\mbox{exp}\left[-i\sum_{n=1}^{3}\left(\frac{2k_{n}}{\sqrt{s}}\right)^{5}(a_{n}+ib_{n})\right]$
(24)

with

$\displaystyle
a_{1}=\alpha_{11}+\frac{s-4m_{K}^{2}}{s}~{}\alpha_{12}~{}\theta(s-4m_{K}^{2})+\frac{s-s_{v}}{s}~{}\alpha_{10}~{}\theta(s-s_{v})),$
(25) $\displaystyle
b_{n}=\beta_{n}+\frac{s-s_{v}}{s}~{}\gamma_{n}~{}\theta(s-s_{v}).$ (26)

$s_{v}\approx 2.274$ GeV2 is a combined threshold of the channels
$\eta\eta^{\prime}$, $\rho\rho$, and $\omega\omega$.

The data for the $\pi\pi$ scattering are taken from an energy-independent
analysis by Hyams et al. Hya73 . The data for $\pi\pi\to
K\overline{K},\eta\eta$ are taken from works Lin92 .

We obtained a satisfactory description with ten resonances $f_{2}(1270)$,
$f_{2}(1430)$, $f_{2}^{\prime}(1525)$, $f_{2}(1580)$, $f_{2}(1730)$,
$f_{2}(1810)$, $f_{2}(1960)$, $f_{2}(2000)$, $f_{2}(2240)$, and $f_{2}(2410)$
(the total $\chi^{2}/\mbox{NDF}=161.147/(168-65)\approx 1.56$) and with eleven
states adding one more resonance $f_{2}(2020)$ which is needed in the combined
analysis of processes $p\overline{p}\to\pi\pi,\eta\eta,\eta\eta^{\prime}$
Ani05 . In our analysis, the description with eleven resonances is practically
the same as that with ten resonances: the total
$\chi^{2}/\mbox{NDF}=156.617/(168-69)\approx 1.58$.

The obtained resonance parameters are shown in Table 8 for the cases of ten
and eleven states.

Table 8: The resonance parameters in the tensor sector for ten and eleven states (in MeV). State |  $M$ | $f_{r1}$ | $f_{r2}$ | $f_{r3}$ | $f_{r4}$ | $\Gamma_{tot}$
---|---|---|---|---|---|---
ten states
$f_{2}(1270)$ |  1275.3$\\!\pm\\!$1.8 |  470.8$\\!\pm\\!$5.4 |  201.5$\\!\pm\\!$11.4 |  90.4$\\!\pm\\!$4.76 |  22.4$\\!\pm\\!$4.6 |  $\approx$212
$f_{2}(1430)$ |  1450.8$\\!\pm\\!$18.7 |  128.3$\\!\pm\\!$45.9 |  562.3$\\!\pm\\!$142 |  32.7$\\!\pm\\!$18.4 |  8.2$\\!\pm\\!$65 |  $>$230
$f_{2}^{\prime}(1525)$ |  1535$\\!\pm\\!$8.6 |  28.6$\\!\pm\\!$8.3 |  253.8$\\!\pm\\!$78 |  92.6$\\!\pm\\!$11.5 |  41.6$\\!\pm\\!$160 |  $>$49
$f_{2}(1565)$ |  1601.4$\\!\pm\\!$27.5 |  75.5$\\!\pm\\!$19.4 |  315$\\!\pm\\!$48.6 |  388.9$\\!\pm\\!$27.7 |  127$\\!\pm\\!$199 |  $>$170
$f_{2}(1730)$ |  1723.4$\\!\pm\\!$5.7 |  78.8$\\!\pm\\!$43 |  289.5$\\!\pm\\!$62.4 |  460.3$\\!\pm\\!$54.6 |  107.6$\\!\pm\\!$76.7 |  $>$182
$f_{2}(1810)$ |  1761.8$\\!\pm\\!$15.3 |  129.5$\\!\pm\\!$14.4 |  259$\\!\pm\\!$30.7 |  469.7$\\!\pm\\!$22.5 |  90.3$\\!\pm\\!$90 |  $>$177
$f_{2}(1960)$ |  1962.8$\\!\pm\\!$29.3 |  132.6$\\!\pm\\!$22.4 |  333$\\!\pm\\!$61.3 |  319$\\!\pm\\!$42.6 |  65.4$\\!\pm\\!$94 |  $>$119
$f_{2}(2000)$ |  2017$\\!\pm\\!$21.6 |  143.5$\\!\pm\\!$23.3 |  614$\\!\pm\\!$92.6 |  58.8$\\!\pm\\!$24 |  450.4$\\!\pm\\!$221 |  $>$299
$f_{2}(2240)$ |  2207$\\!\pm\\!$44.8 |  136.4$\\!\pm\\!$32.2 |  551$\\!\pm\\!$149 |  375$\\!\pm\\!$114 |  166.8$\\!\pm\\!$104 |  $>$222
$f_{2}(2410)$ |  2429$\\!\pm\\!$31.6 |  177$\\!\pm\\!$47.2 |  411$\\!\pm\\!$196.9 |  4.5$\\!\pm\\!$70.8 |  460.8$\\!\pm\\!$209 |  $>$170
eleven states
$f_{2}(1270)$ |  1276.3$\\!\pm\\!$1.8 |  468.9$\\!\pm\\!$5.5 |  201.6$\\!\pm\\!$11.6 |  89.9$\\!\pm\\!$4.79 |  7.2$\\!\pm\\!$4.6 |  $\approx$210.5
$f_{2}(1430)$ |  1450.5$\\!\pm\\!$18.8 |  128.3$\\!\pm\\!$45.9 |  562.3$\\!\pm\\!$144 |  32.7$\\!\pm\\!$18.6 |  8.2$\\!\pm\\!$63 |  $>$230
$f_{2}^{\prime}(1525)$ |  1534.7$\\!\pm\\!$8.6 |  28.5$\\!\pm\\!$8.5 |  253.9$\\!\pm\\!$79 |  89.5$\\!\pm\\!$12.5 |  51.6$\\!\pm\\!$155 |  $>$49.5
$f_{2}(1565)$ |  1601.5$\\!\pm\\!$27.9 |  75.5$\\!\pm\\!$19.6 |  315$\\!\pm\\!$50.6 |  388.9$\\!\pm\\!$28.6 |  127$\\!\pm\\!$190 |  $>$170
$f_{2}(1730)$ |  1719.8$\\!\pm\\!$6.2 |  78.8$\\!\pm\\!$43 |  289.5$\\!\pm\\!$62.6 |  460.3$\\!\pm\\!$545. |  108.6$\\!\pm\\!$76. |  $>$182.4
$f_{2}(1810)$ |  1760$\\!\pm\\!$17.6 |  129.5$\\!\pm\\!$14.8 |  259$\\!\pm\\!$32. |  469.7$\\!\pm\\!$25.2 |  90.3$\\!\pm\\!$89.5 |  $>$177.6
$f_{2}(1960)$ |  1962.2$\\!\pm\\!$29.8 |  132.6$\\!\pm\\!$23.3 |  331$\\!\pm\\!$61.5 |  319$\\!\pm\\!$42.8 |  62.4$\\!\pm\\!$91.3 |  $>$118.6
$f_{2}(2000)$ |  2006$\\!\pm\\!$22.7 |  155.7$\\!\pm\\!$24.4 |  169.5$\\!\pm\\!$95.3 |  60.4$\\!\pm\\!$26.7 |  574.8$\\!\pm\\!$211 |  $>$193
$f_{2}(2020)$ |  2027$\\!\pm\\!$25.6 |  50.4$\\!\pm\\!$24.8 |  441$\\!\pm\\!$196.7 |  58$\\!\pm\\!$50.8 |  128$\\!\pm\\!$190 |  $>$107
$f_{2}(2240)$ |  2202$\\!\pm\\!$45.4 |  133.4$\\!\pm\\!$32.6 |  545$\\!\pm\\!$150.4 |  381$\\!\pm\\!$116 |  168.8$\\!\pm\\!$103 |  $>$222
$f_{2}(2410)$ |  2387$\\!\pm\\!$33.3 |  175$\\!\pm\\!$48.3 |  395$\\!\pm\\!$197.7 |  24.5$\\!\pm\\!$68.5 |  462.8$\\!\pm\\!$211 |  $>$168

The background parameters for ten resonances are: $\alpha_{11}=-0.07805$,
$\alpha_{12}=0.03445$, $\alpha_{10}=-0.2295$, $\beta_{1}=-0.0715$,
$\gamma_{1}=-0.04165$, $\beta_{2}=-0.981$, $\gamma_{2}=0.736$,
$\beta_{3}=-0.5309$, $\gamma_{3}=0.8223$; and for eleven resonances are:
$\alpha_{11}=-0.0755$, $\alpha_{12}=0.0225$, $\alpha_{10}=-0.2344$,
$\beta_{1}=-0.0782$, $\gamma_{1}=-0.05215$, $\beta_{2}=-0.985$,
$\gamma_{2}=0.7494$, $\beta_{3}=-0.5162$, $\gamma_{3}=0.786$.

In Figures 6 and 7 we show results of fitting to the data.

Figure 6: The phase shift and module of the $\pi\pi$-scattering $D$-wave
$S$-matrix element.

Figure 7: The squared modules of the $\pi\pi\to K\overline{K}$ (upper figure)
and $\pi\pi\to\eta\eta$ (lower figure) $D$-wave $S$-matrix elements.

##  VI Spectroscopic implications from the analysis

In the combined model-independent analysis of data on the
$\pi\pi\to\pi\pi,K\overline{K},\eta\eta,\eta\eta^{\prime}$ processes in the
channel with $I^{G}J^{PC}=0^{+}0^{++}$, an additional confirmation of the
$\sigma$-meson with mass 835 MeV is obtained (the pole position on sheet II is
$598-i583$ MeV). This value of mass corresponds most near to the one ($\sim
860$ MeV) of Ref. Tornqvist  and rather accords with prediction
($m_{\sigma}\approx m_{\rho}$) on the basis of mended symmetry by S. Weinberg
Wei90 . Note that our values of $E_{r}$ and $\Gamma_{r}$ for the
$f_{0}(600)$-pole position are larger than those obtained in the dispersive
analysis of data on only the $\pi\pi$ scattering, see Ref. G-MKP08  and
reference therein.

Indication for $f_{0}(980)$ to be the $\eta\eta$ bound state is obtained. From
the point of view of the quark structure, this is the 4-quark state. Maybe,
this is consistent somehow with arguments in favour of the 4-quark nature of
$f_{0}(980)$ 20 .

The ${f_{0}}(1370)$ and $f_{0}(1710)$ have the dominant $s{\bar{s}}$
component. Conclusion about the ${f_{0}}(1370)$ agrees quite well with the one
drawn by the Crystal Barrel Collaboration 21  where the ${f_{0}}(1370)$ is
identified as $\eta\eta$ resonance in the $\pi^{0}\eta\eta$ final state of the
${\bar{p}}p$ annihilation at rest. Conclusion about the $f_{0}(1710)$ is quite
consistent with the experimental facts that this state is observed in
$\gamma\gamma\to K_{S}{\bar{K}}_{S}$ 22  and not observed in
$\gamma\gamma\to\pi^{+}\pi^{-}$ 23 .

As to the $f_{0}(1500)$, we suppose that it is practically the eighth
component of octet mixed with the glueball being dominant in this state. Its
biggest width among the enclosing states tells also in behalf of its glueball
nature 24 .

We propose the following assignment of scalar mesons below 1.9 GeV to lower
nonets, excluding the $f_{0}(980)$ as the $\eta\eta$ bound state. The lowest
nonet: the isovector $a_{0}(980)$, the isodoublet $K_{0}^{*}(900)$, and
$f_{0}(600)$ and $f_{0}(1370)$ as mixtures of the eighth component of octet
and the SU(3) singlet. Then the Gell-Mann–Okubo (GM-O) formula

$3m_{f_{8}}^{2}=4m_{K_{0}^{*}}^{2}-m_{a_{0}}^{2}\,,$ (27)

gives $m_{f_{8}}=872$ MeV ($m_{\sigma}=835\pm 14$ MeV). In the relation for
masses of nonet

$m_{\sigma}+m_{f_{0}(1370)}=2m_{K_{0}^{*}}\,,$ (28)

the left-hand side is about 25 % bigger than the right-hand one.

The next nonet: $a_{0}(1450)$, $K_{0}^{*}(1450)$, and $f_{0}(1500)$ and
$f_{0}(1710)$. From the GM-O formula, we get $m_{f_{8}}\approx 1450$ MeV. In
the relation

$m_{f_{0}(1500)}+m_{f_{0}(1710)}=2m_{K_{0}^{*}(1450)}\,,$ (29)

the left-hand side is about 12 % bigger than the right-hand one.

Now an adequate mixing scheme should be found.

In the vector sector, the obtained value of mass for the $\rho(770)$ is
smaller in the model-independent approach, $769.3$ MeV, and a little bit
bigger in the Breit–Wigner one, $777.69\pm 0.32$ MeV, than the averaged value
cited in the PDG tables PDG08 , $775.49\pm 0.34$ MeV. However, it also occurs
in analysis of some reactions (see PDG tables). The obtained value of the
total width in the first case ($146.6$ MeV) is in a good agreement with the
averaged PDG one ($149.4\pm 1.0$ MeV) and it is a little bit bigger in the
second case ($\approx 154.3$ MeV) than the averaged PDG value, however, this
is encountered also in other analyses (see PDG tables). Note that predicted
widths of the $\rho(770)$ decays to the $4\pi$-modes are significantly larger
than, e.g., the ones evaluated in the chiral model of some mesons based on the
hidden local symmetry added with the anomalous terms Ach05 .

The first $\rho$-like meson has the mass 1257.8$\pm$11 MeV in the model-
independent analysis and 1249.8$\pm$15.6 MeV in the Breit–Wigner one. These
values differ significantly from the mass (1459$\pm$11 MeV) of the first
$\rho$-like meson cited in the PDG tables. The $\rho(1250)$ was discussed
actively some time ago 25  and later the evidence for its existence was
obtained in SB_NPA08 ; 26 .

If the $\rho(1250)$ is interpreted as the first radial excitation of the
$1^{+}1^{--}$ $q{\bar{q}}$ state, then it lies down well on the corresponding
linear trajectory with an universal slope on the $(n,M^{2})$ plane (n is the
radial quantum number of the $q{\bar{q}}$ state)27 , whereas the $\rho(1450)$
turns out to be considerably higher than this trajectory. The $\rho(1250)$ and
the isodoublet $K^{*}(1410)$ are well located to the octet of the first radial
excitations. The mass of the latter should be by about 150 MeV larger than the
mass of the former. Then the GM-O formula

$3m_{\omega_{8}^{\prime}}^{2}=4m_{{K^{*}}^{\prime}}^{2}-m_{\rho^{\prime}}^{2}$
(30)

gives $m_{\omega_{8}^{\prime}}=1460$ MeV, that is fairly good compatible with
the mass of the first $\omega$-like meson $\omega(1420)$, for which one
obtains the values in range 1350-1460 MeV (see PDG tables).

Existence of the $\rho(1450)$ (along with $\rho(1250)$) does not contradict to
the data. In the $q{\bar{q}}$ picture, it might be the first ${}^{3}D_{1}$
state with, possibly, the isodoublet $K^{*}(1680)$ in the corresponding octet.
From the GM-O formula, we should obtain the value 1750 MeV for the mass of the
eighth component of this octet. This corresponds to one of the observations of
the second $\omega$-like meson with masses from 1606 to 1840 MeV that is cited
in the PDG tables under the $\omega(1650)$.

The third $\rho$-like meson has the mass about 1600 MeV rather than 1720 MeV
cited in the PDG tables PDG08 .

As to the $\rho(1900)$, in this energy region there are practically no data on
the $P$-wave of $\pi\pi$ scattering. The model-independent analysis testifies
in favour of existence of this state, whereas the Breit–Wigner analysis gives
the same description with and without the $\rho(1900)$.

The suggested picture for the first two $\rho$-like mesons is consistent with
predictions of the quark model 28 . In Ref. 29  the discussed mass spectrum
for radially excited $\rho$ and $K^{*}$ mesons was obtained using rather
simple mass operator. If the existence of the $\rho(1250)$ is confirmed, some
quark potential models, e.g., in Ref. 30 , will require substantial revisions,
because the first $\rho$-like meson is usually predicted about 200 MeV higher
than this state. To the point, the first $K^{*}$-like meson is obtained in the
indicated quark model at 1580 MeV, whereas the corresponding very well
established resonance has the mass of only 1410 MeV.

In the tensor sector, we carried out two analysis – without and with the
$f_{2}(2020)$. We do not obtain $f_{2}(1640)$, $f_{2}(1910)$ and
$f_{2}(2150)$, however, we see $f_{2}(1450)$ and $f_{2}(1730)$ which are
related to the statistically-valued experimental points.

Usually one assigns the states $f_{2}(1270)$ and $f_{2}^{\prime}(1525)$ to the
ground tensor nonet. To the second nonet, one could assign $f_{2}(1600)$ and
$f_{2}(1760)$ though for now the isodoublet member is not discovered. If
$a_{2}(1730)$ is the isovector of this octet and if $f_{2}(1600)$ is almost
its eighth component, then, from the GM-O formula, we expect this isodoublet
mass at about 1633 MeV. Then the relation for masses of nonet would be
fulfilled with a 3% accuracy. Karnaukhov et al. 31  observed the strange
isodoublet with yet indefinite remaining quantum numbers and with mass
$1629\pm 7$ MeV in the mode $K_{s}^{0}\pi^{+}\pi^{-}$. This state might be the
tensor isodoublet of the second nonet.

The states $f_{2}(1963)$ and $f_{2}(2207)$ together with the isodoublet
$K_{2}^{*}(1980)$ could be put into the third nonet. Then in the relation for
masses of nonet

$M_{f_{2}(1963)}+M_{f_{2}(2207)}=2M_{K_{2}^{*}(1980)},$ (31)

the left-hand side is only 5.3 % bigger than the right-hand one. If one
consider $f_{2}(1963)$ as the eighth component of octet, the GM-O formula

$M_{a_{2}}^{2}=4M_{K_{2}^{*}(1980)}^{2}-3M_{f_{2}(1963)}^{2}$ (32)

gives $M_{a_{2}}=2030$ MeV. This value coincides with the one for
$a_{2}$-meson obtained in works 32 . This state is interpreted as a second
radial excitation of the $1^{-}2^{++}$-state on the basis of consideration of
the $a_{2}$ trajectory on the $(n,M^{2})$ plane Ani05 .

As to $f_{2}(2000)$, the presence of the $f_{2}(2020)$ in the analysis with
eleven resonances helps to interpret $f_{2}(2000)$ as the glueball. In the
case of ten resonances, the ratio of the $\pi\pi$ and $\eta\eta$ widths is in
the limits obtained in Ref. Ani05  for the tensor glueball on the basis of the
1/N-expansion rules. However, the $K\overline{K}$ width is too large for the
glueball. At practically the same description of processes with the
consideration of eleven resonances as in the case of ten, their parameters
have varied a little, except for the ones for $f_{2}(2000)$ and $f_{2}(2410)$.
Mass of the latter has decreased by about 40 MeV. As to $f_{2}(2000)$, its
$K\overline{K}$ width has changed significantly. Now all the obtained ratios
of the partial widths are in the limits corresponding to the glueball.

The question of interpretation of the $f_{2}(1450)$, $f_{2}(1730)$,
$f_{2}(2020)$ and $f_{2}(2410)$ is open.

## References

  * (1) C. Amsler et al., Phys. Lett. B667, 1 (2008).
  * (2) V.V. Anisovich, Int. J. Mod. Phys. A 21, 3615 (2006).
  * (3) C. Amsler and F.E. Close, Phys. Rev. D 53, 295 (1996).
  * (4) V.V. Anisovich et al., Int. J. Mod. Phys. A 20, 6327 (2005).
  * (5) D. Krupa, V.A. Meshcheryakov, and Yu.S. Surovtsev, Nuovo Cimento A 109, 281 (1996).
  * (6) K.J. Le Couteur, Proc. R. London, Ser. A 256, 115 (1960); R.G. Newton, J. Math. Phys. 2, 188 (1961); M. Kato, Ann. Phys. 31, 130 (1965).
  * (7) D. Morgan and M.R. Pennington, Phys. Rev. D 48, 1185 (1993).
  * (8) B. Hyams et al., Nucl. Phys. B 64, 134 (1973); ibid. 100, 205 (1975).
  * (9) A. Zylbersztejn et al., Phys. Lett. B 38, 457 (1972); P. Sonderegger, P. Bonamy, in Proc. 5th Int. Conference on Elementary Particles, Lund, 1969, 372; J.R. Bensinger et al., Phys. Lett. B 36, 134 (1971); J.P. Baton et al., Phys. Lett. B 33, 525 (1970); ibid. 33, 528 (1970); P. Baillon et al., Phys. Lett. B 38, 555 (1972); L. Rosselet et al., Phys. Rev. D 15, 574 (1977); A.A. Kartamyshev et al., Pis’ma Zh. Eksp. Theor. Fiz. 25, 68 (1977); A.A. Bel’kov et al., Pis’ma Zh. Eksp. Theor. Fiz. 29, 652 (1979).
  * (10) G. Grayer et al., Nucl. Phys. B 75, 189 (1974).
  * (11) R. Kaminski, L. Lesniak, and K. Rybicki, Z. Phys. C 74, 79 (1997).
  * (12) W. Wetzel et al., Nucl. Phys. B 115, 208 (1976); V.A. Polychronakos et al., Phys. Rev. D 19, 1317 (1979); P. Estabrooks, Phys. Rev. D 19, 2678 (1979); D. Cohen et al., Phys. Rev. D 22, 2595 (1980); G. Costa et al., Nucl. Phys. B 175, 402 (1980); A. Etkin et al., Phys. Rev. D 25, 1786 (1982).
  * (13) F. Binon et al., Nuovo Cimento A 78, 313 (1983).
  * (14) F. Binon et al., Nuovo Cimento A 80, 363 (1984).
  * (15) S.D. Protopopescu et al., Phys. Rev. D 7, 1279 (1973); P. Estabrooks and A.D. Martin, Nucl. Phys. B 79, 301 (1974).
  * (16) Yu.S. Surovtsev and P. Bydžovský, Nucl. Phys. A 807, 145 (2008).
  * (17) J. Bohacik and H. Kühnelt, Phys. Rev. D 21, 1342 (1980).
  * (18) N.N. Achasov and A.A. Kozhevnikov, Phys. Rev. D 71, 034015 (2005).
  * (19) V. Bernard, A.A. Osipov, and U.G. Meissner, Phys. Lett. B 285, 119 (1992).
  * (20) A.A. Osipov, A.E. Radzhabov, and M.K. Volkov, arXiv:hep-ph/0603130.
  * (21) I. Caprini, G. Colangelo, and H. Leutwyler, Int. J. Mod. Phys. A 21, 954 (2006).
  * (22) R. Kamiński, L. Leśniak, and B. Loiseau, Phys. Lett. B551, 241 (2003).
  * (23) J.R. Peláez and F.J. Ynduráin, Phys. Rev. D 71, 074016 (2005).
  * (24) S.J. Lindenbaum and R.S. Longacre, Phys. Lett. B 274, 492 (1992); R.S. Longacre et al., Phys. Lett. B 177, 223 (1986).
  * (25) N.A. Tornqvist and M. Roos, Phys. Rev. Lett. 76, 1575 (1996).
  * (26) S. Weinberg, Phys. Rev. Lett. 65, 1177 (1990).
  * (27) R. García-Martín, R. Kamiński, and J.R. Pelaez, arXiv:0810.1134[hep-ph].
  * (28) N.N. Achasov, Nucl. Phys. A 675, 279c (2000); M.N. Achasov et al., Phys. Lett. B 438, 441 (1998); ibid. 440, 442 (1998).
  * (29) C. Amsler et al., Phys. Lett. B 355, 425 (1995).
  * (30) S. Braccini, Frascati Phys. Series XV, 53 (1999).
  * (31) R. Barate et al., Phys. Lett. B 472, 189 (2000).
  * (32) V.V. Anisovich et al., Nucl. Phys. A (Proc. Suppl.) 56, 270 (1997).
  * (33) N.M. Budnev et al., Phys. Lett. B70, 365 (1977); S.B. Gerasimov and A.B. Govorkov, Z. Phys. C 13, 43 (1982); ibid. 29, 61 (1985).
  * (34) D. Aston et al., Nucl. Phys. B (Proc. Suppl.) 21, 105 (1991); T.S. Belozerova and V.K. Henner, Phys. Elem. Part. Atom. Nucl. 29, 148 (1998); Yu.S. Surovtsev and P. Bydžovský, arXiv:hep-ph/0701274, Frascati Phys. Series, Vol. XLVI, 1535 (2007); I. Yamauchi, T. Komada, Frascati Phys. Series, Vol. XLVI, 445 (2007).
  * (35) A.V. Anisovich, V.V. Anisovich, and A.V. Sarantsev, Phys. Rev. D 62, 051502 (2000).
  * (36) E. van Beveren, G. Rupp, T.A. Rijken, and C. Dullemond, Phys. Rev. D 27, 1527 (1983).
  * (37) S.B. Gerasimov and A.B. Govorkov, Z. Phys. C 13, 43 (1982); ibid. 29, 61 (1985).
  * (38) S. Godfrey and N. Isgur, Phys. Rev. D 32, 189 (1985).
  * (39) V.M. Karnaukhov et al., Yad. Fiz. 63, 652 (2000).
  * (40) A.V. Anisovich et al., Phys. Lett. B 452, 173 (1999); ibid. 452, 187 (1999); ibid. 517, 261 (2001).