tokeniser performs better on most training task families, but generally worse
for the different types of held out tasks. This mirrors our findings with
smaller models, which showed VQ-GAN to be better for generalisation and
adaptation.

(a) Simulation training tasks.

(b) Simulation held out tasks.

Figure 18: Ablating the VQ-GAN tokeniser vs the patch ResNet used in Gato. The
patch ResNet tokeniser performs better on the training tasks, but worse on the
held out tasks. Figure 19: Tokenisation and obs prediction ablations on the
red_on_green generalisation problem. Unless otherwise specified, we use a
vocabulary size of 4096 and image prediction with a time step shift of $k=5$.
We compare the success rate ($\%$) for train and test tasks to ablate
important design decisions. Left: Predicting future tokens is significantly
better than predicting future pixels. Middle: Increasing VQ-GAN vocabulary
size deteriorates test performance. Right: Predicting the next token in the
current image ($k=0$) is less beneficial than predicting tokens farther in the
future. We find predicting $k=5$ time steps in the future is optimal.

Predicting future image pixels does not give us a similar advantage. We
compared different reconstruction targets in Figure 19. Like the decoder in
MAE [He et al., 2022], we added an extra linear layer whose number of output
channels equals the number of pixel values in a patch. Output tokens
corresponding to observation tokens were processed via this linear layer to
produce a vector of pixel values in a patch. We used mean squared error
between the predicted ($k=5$) and original frame in pixel space. The choice of
prediction target does not seem to affect the performance on training tasks.
However, there is a huge difference in the generalisation capabilities of the
two methods. Qualitatively we found that the policy learnt by predicting
pixels fails (zero success rate) as it seems to ignore the goal and performs a
different task.

Next, we investigated predicting future VQ-GAN tokens for the red_on_green
generalisation task, with different $k$ values, where $k$ represents the
number of time steps ahead at which we are predicting the next observation
token. We sweep from $k=0$, i.e. predicting the next token, to $k=10$ i.e.
predicting the next token of the observation $10$ time steps ahead. As $k$
increases, the performance on the training tasks increases, but we find
diminishing returns as we increase past $k=5$. However, we see a $50\%$
relative improvement in generalisation performance as we increase $k$ from $0$
to $5$. Further increase in prediction horizon results in performance
degradation. These results highlight the importance of designing a challenging
self-supervision task. There is a huge amount of spatial and temporal
redundancy between consecutive image observations especially when the agent is
operating at $20$ Hz. Despite using VQ-GAN tokens, there can be a significant
overlap of tokens among future consecutive frames. Predicting $k$ time steps
in the future alleviates this issue and results in strong generalisation
capabilities in this setting.

Finally, we study the effect of the VQ-GAN vocabulary size in Figure 19.
Increasing the vocabulary size ($V$) has no effect on the performance of the
training tasks. However, the generalisation ability monotonically declines as
we increase $V$.

##  Appendix E RoboCat Training Details

###  E.1 Data processing

The training data for each task contains

  * •

a subset of embodiment proprioception observations (see Table 11);

  * •

the embodiment actions (see Section C.1);

  * •

and two camera images, in particular from a front left and front right view of
the basket (see Section C.2).

Each goal image used in training comes from the front left camera and
correspond to the image obtained at the end of a training trajectory. All
images, including the goal images, are cropped and resized to have dimensions
$128\times 128\times 3$. We apply image augmentation by randomising
brightness, saturation, contrast, and translation to the images, as described
in Lee et al. [2021]. All actions are scaled to be in $[-1,1]$. We trim
trajectories discarding the final steps if no significant difference is
measured with respect to the final images of the scene. This is based on the
intuition that expert actions that do not generate any changes in the images
should not be used at training as they likely do not have an effect on the
task at hand (e.g. the robot arm is not moving or is just hovering around on
the objects). The metric we use to evaluate similarity between images is the
the LPIPS (Learned Perceptual Image Patch Similarity) [Zhang et al., 2018].
The trajectories involving just lifting of objects are also trimmed in order
to have the gripper (and thus the lifted object) visible in the final image
used as goal image.

###  E.2 Data weighting

Based on previous experiments we use the following formula to sample the data
during training. For training the RoboCat-lim generalist, by default every
task variant is sampled equally, regardless of how many episodes are
available. And within each task variant, failed episodes and successful
episodes are also sampled equally. For the successful episodes, the hindsight
goal image is used half the time and a semantically equivalent goal image
(from a different successful episode of the same task variant) is used the
other half of the time. When fine-tuning to a limited number of
demonstrations, we find best performance when only training on successful
episodes. For the final RoboCat generalist we included NIST-i data only from
successful episodes and with a weighting three times bigger to mitigate a
potential imbalance towards stacking and lifting other object sets. Similarly,
after 1 000 000 training steps we decreased by half the weight of all
simulated data to focus the remaining training time towards the real world
tasks. Note that these decisions were preventive and thus keeping all weights
equal may have also been effective.

###  E.3 Training and fine-tuning parameters

For training all RoboCat models we use the AdamW optimiser [Loshchilov and
Hutter, 2017] with a linear warm-up and cosine schedule decay. The linear
warm-up lasts for 15 000 steps, starting from and ending at a different
minimum and maximum learning rates depending on the model (see Table 13). This
learning rate is then cosine decayed by a factor 10 over $2\,000\,000$ steps.
The AdamW optimiser has parameters $\beta_{1}=0.9$, $\beta_{2}=0.95$ and
$\epsilon=1e^{-8}$. We use a batch size of 256 and a sequence length of 1024
tokens for all models. We train with an AdamW weight decay parameter of 0.1.
Additionally, we use stochastic depth [Huang et al., 2016] during pretraining,
where each of the transformer sub-layers (i.e. each Multi-Head Attention and
Dense Feedforward layer) is skipped with a probability of 0.1.

Hyper parameter | RoboCat | RoboCat-lim | 364M
---|---|---|---
Max learning rate | $1e^{-4}$ | $5e^{-6}$ | $1e^{-4}$
Min learning rate | $1e^{-5}$ | $5e^{-7}$ | $1e^{-5}$
Table 13: Learning rates used when training our models. Note that both
RoboCat-lim and RoboCat-lim (patch) used the same learning rates.

For fine-tuning we use the Adam optimiser [Kingma and Ba, 2015] with a
constant learning rate of $1e^{-5}$. The Adam optimiser has parameters
$\beta_{1}=0.9$, $\beta_{2}=0.95$ and $\epsilon=1e^{-8}$. We use a batch size
of 32 and a sequence length of 1024 tokens for all models. We train for up to
$50\,000$ gradient steps. As regularisation, we use dropout [Srivastava et
al., 2014] with a rate of 0.1.

###  E.4 RoboCat-lim training data

In Section 5.2, we evaluate the properties of the RoboCat-lim agent and
systematically measure its ability to generalise and fine-tune across a
diverse set of axes: objects, task variants, embodiments, behaviour sources,
task family, and from sim to real. In this section, we provide more details on
our choice of held-out tasks.

The training tasks for RoboCat-lim are primarily in sim, comprising the
simulated structure building tasks: stacking, pyramid building, and tower
building for the Panda 7DoF; and stacking for the Sawyer 7DoF. The only real-
world data used in training is the Sawyer 5DoF RGB-stacking benchmark.

From these training tasks, we also hold out all blue-on-green stacking tasks
and Sawyer stacking with RGB triplet 5. This provides the first two axes of
generalisation: to the unseen perceptual variation (blue-on-green task, which
has not been observed before) and the held-out objects (triple 5, which has
never been seen with the Sawyer embodiment).

To measure the ability to fine-tune to different behaviour sources we use two
versions of data for blue-on-green stacking in the real-world: from an expert
agent, and from human teleoperation. These have different state and action
distributions, as (for example) human teleoperators tend to favour quick but
constant movements with pauses between ‘stages’ of a task.

We measure sim-to-real capability with two tasks on the real panda 7DoF: one
from the stacking task family and one from tower building.

We also evaluate the ability to fine-tune to an unseen task family: inverted
pyramid building. This task is particularly challenging as it requires careful
and precise motion to balance two objects on top of a third.

Finally, we evaluate embodiment generalisation and fine-tuning using the KUKA
14DoF. While the KUKA 14DoF is a dexterous embodiment with a 3-fingered hand,
it also presents a considerable challenge for RoboCat training. This is due to
the significantly different observation and action spaces that it poses
compared to all other tasks in the RoboCat training. For instance, all
training tasks rely on a 1DoF parallel gripper and have a total of either 5DoF
or 7DoF.

Embodiment | Task Family | Episode Length | Control Frequency | Number of Steps
---|---|---|---|---
Simulation | Panda 7-DoF | Stacking | $20\text{\,}\mathrm{s}\text{/}$ | $20\text{\,}\mathrm{Hz}\text{/}$ | 200
Tower Building
Pyramid Building
Lifting | $120\text{\,}\mathrm{s}\text{/}$ | $10\text{\,}\mathrm{Hz}\text{/}$ | 1200
Inserting | $120\text{\,}\mathrm{s}\text{/}$ | 1200
Sawyer 7-DoF | Stacking | $20\text{\,}\mathrm{s}\text{/}$ | $20\text{\,}\mathrm{Hz}\text{/}$ | 400
Real World | Panda 7-DoF | Stacking | $60\text{\,}\mathrm{s}\text{/}$ | $10\text{\,}\mathrm{Hz}\text{/}$ | 600
Tower Building
Fruit Lifting | $30\text{\,}\mathrm{s}\text{/}$ | 300
Fruit Inserting
Fruit Removing
Gear Lifting |  |
Gear Inserting | $120\text{\,}\mathrm{s}\text{/}$ | 1200
Gear Removing
Shape matching | $30\text{\,}\mathrm{s}\text{/}$ | 300
Sawyer 5-DoF | Stacking | $40\text{\,}\mathrm{s}\text{/}$ | $10\text{\,}\mathrm{Hz}\text{/}$ | 400
Vegetable lifting | $20\text{\,}\mathrm{s}\text{/}$ | $20\text{\,}\mathrm{Hz}\text{/}$ | 400
Tower Building | $60\text{\,}\mathrm{s}\text{/}$ | 1200
Inverted Pyramid | $120\text{\,}\mathrm{s}\text{/}$ | 2400
KUKA 14-DoF | Gear lifting | $60\text{\,}\mathrm{s}\text{/}$ | $10\text{\,}\mathrm{Hz}\text{/}$ | 600
Peg lifting
Table 14: Evaluation setup. Episode length, control frequency and episode
number of steps per task on simulated and real robots.

##  Appendix F Evaluation and Real-World Deployment

In this section, we provide more details about the evaluation procedure used
in the paper.

We evaluated RoboCat and the baselines with 100 (or more, if specified)
episodes for each evaluation task, both in simulation and on the real robot.
For the fine-tuning experiments, we ran shorter evaluations of 25 episodes for
different fine-tuned checkpoints. We then selected the best checkpoint and ran
the 100 episode evaluation with it.

In simulation, the initial state is randomised by moving the arm to a
randomised initial pose and by dropping the objects in the basket. In the real
world, the arm position is randomised similarly. The objects initial positions
are either randomised via a scripted reset or with alternative approaches. We
detailed these processes in Section F.2.

For all our evaluations, we use environments with the control frequencies and
episode lengths that match the ones used for collecting the data for each
task. Table 14 reports this information for our simulated and real robot
tasks.

In the next two sections, we detail the evaluation procedure used the real-
world tasks for the RGB tasks and all other tasks.

###  F.1 Evaluation for real-world RGB tasks

To evaluate our agent on tasks with the RGB objects we used the infrastructure
implemented in Lee et al. [2021]. We rely on the scripted resets mentioned in
Section C.3 and follow the same evaluation procedure, apart from the reward
function. We noticed that the reward function was missing some clear
successes. This is because the reward function defined in Lee et al. [2021]
requires the arm height to end withing some thresholds. While such a criteria
for success can be fair for RL agents trained to optimise such a reward, we
thought it didn’t represent fairly the ability of RoboCat to stack the correct
objects and move the arm away to _any_ position. For this reason, we decided
to visually count the successes during evaluation. For fairness we also
visually recounted the evaluations of Lee et al. [2021] and Reed et al.
[2022].

###  F.2 Evaluation for real-world tasks with no rewards or scripted resets

For all the other real tasks in this work we did not implement a scripted
reward model nor a reset policy. This greatly simplified the implementation of
the tasks, but added significant complexity to evaluation. Our evaluation
system thus consists of two components. In place of scripted rewards, we train
success detectors from human annotations to detect when each task has been
completed. For resets we have a general solution that groups several policies
for different tasks together into a _policy pool_ , with each policy in the
pool serving as a component of the reset policy for its peers. For instance a
policy pool containing the insertion and removal policies for the same object
will allow the removal policy to serve as a reset for the insertion policy and
vice versa.

####  F.2.1 Success detection

We treat success detection for each task as a per-time step binary
classification problem. For each task we annotate several episodes with per-
time step success labels using a annotation-efficient annotation procedure.
Our annotation procedure takes advantage of the following simple observation:
If a task is solved (or not) at time $t$ it is overwhelmingly likely that it
is also solved (or not) at time $t+1$. Put more simply, there are relatively
few time steps where a task transitions from unsolved to solved (or vice
versa). Rather than collecting labels for every frame directly, we ask
annotators to mark transition points between solved and unsolved states and
extend these to per-time step labels by painting the transition label forward
in time until the next annotation. This strategy dramatically reduces the
number of annotations compared to annotating every frame. Using this strategy
labelling a 1200 time step episode typically requires fewer than five
annotations (and frequently requires only one).

Using data annotated in this way we train success detectors for each task.
Each success detector is a ResNet-18 backbone with a binary classification
head. To handle occlusions we use the three basket cameras in the reward model
in order to observe the scene from multiple perspectives. Each camera image is
scaled to $132\times 132$ and further cropped to $128\times 128$ before being
processed separately by the ResNet backbone. The resulting embeddings from
each camera are averaged before being fed to the binary classification head.
At train time we use random cropping and image augmentation, while at
deployment time we disable the augmentations and take a central crop of each
image.

Once trained the success detectors are used together with policy pools for
resetting our environments as we describe in Section F.2.2. We did not use
success detectors for reporting RoboCat performance, as they have an accuracy
around $90\%$. To ensure we counted all success from RoboCat we visually
assessed when an experiment has succeeded or failed. In future work, we aim at
improving accuracy of the reward models in order to automatise also this step
of the evaluation.

####  F.2.2 Policy pools as generalised reset policies

Policy pools are our general solution to resets in the real world. Their role
in our set up is three-fold: i) policy pool provides resets for non-self-
resetting tasks, ii) generates diverse initial conditions for evaluation, and
iii) makes efficient use of robot time by interleaving the evaluation of
multiple policies at a time.

A policy pool groups together a set of diverse tasks and allows us to schedule
episodes of different tasks to be run in sequence. In this simplest case, we
can replicate the standard evaluation setting by building a pool comprised of
a forward task and its reset, allowing the schedule to alternate between them.
However, the policy pool machinery allows us to extend beyond this simple
setting in several ways.

Each prop set typically affords several tasks, for example the NIST-i gears
set affords nine distinct tasks: lift, insert, and remove for each of the
three gears. A policy pool allows us to group all nine of these tasks together
and interleave their execution in arbitrary orders. This is desirable for two
reasons which we discuss next.

From a global perspective this procedure makes efficient use of robot time,
because every episode is an evaluation of some policy on some task. There is
no time wasted by running a specialised behaviour that resets the environment;
instead the evaluation episodes for each task simultaneously serves as the
reset behaviour for other tasks in the pool.

From a local perspective this procedure generates a wider variety of initial
conditions than a single fixed reset policy. From the perspective of each task
its reset is provided by a composition of all the other tasks in the pool.
This has the nice effect that larger pools (with more diverse tasks) tend to
generate more diverse initial conditions within each task.

Another extension of the basic setting is that we can control the ordering of
the schedule of tasks within a policy pool. In practice, this is how we handle
non-self-resetting tasks. In the general case not all tasks are eligible to be
run from all reachable states of the environment. Returning to the gears
example, an insert task is eligible to be run only if the corresponding gear
is not already inserted on the target shaft (similarly a remove task is only
eligible when the corresponding gear is inserted). At the beginning of each
episode our scheduling system for the policy pool uses the success detectors
(Section F.2.1) to determine which tasks are eligible to be run given the
state of the environment, and then randomly chooses one of these tasks to
actually run.

The mapping between tasks and policies within a policy pool is arbitrary, and
need not be 1-1. The same machinery can be used to implement single task
evaluation (by providing a different policy for each task in the pool) or
multi-task evaluation (by defining a pool where every task is executed by the
same policy). Given RoboCat is generalist, when using policy pool we only use
the policy from RoboCat to evaluate any task.

##  Appendix G Additional Experiments

###  G.1 Generalist capabilities and expert performance

In this section, we report more details on the performance of RoboCat on
training and fine-tuning tasks, and we compare it to the success rate of the
training data, defined as percentage of the successful episodes among all the
training episodes. Table 15 shows the overall RoboCat performance per each
task variant.

Training | Embodiment | Task Family |  Object Set | Variations
---|---|---|---|---
Simulation |  Sawyer 7-DoF |  |  |  RB |  RG |  GR |  GB |  BR |  BG
Stacking |  RGB objects, triplet 1 |  62% |  86% |  85% |  – |  73% |  92%
RGB objects, triplet 2 |  79% |  87% |  85% |  82% |  85% |  83%
RGB objects, triplet 3 |  90% |  86% |  80% |  90% |  89% |  91%
RGB objects, triplet 4 |  92% |  95% |  91% |  82% |  81% |  85%
RGB objects, triplet 5 |  69% |  80% |  – |  87% |  63% |  89%
|  NIST-i gears |  80% |  – |  80% |  62% |  79% |  82%
Panda 7-DoF |  |  |  RB |  RG |  GR |  GB |  BR |  BG
Stacking |  RGB objects, triplet 1 |  69% |  82% |  77% |  80% |  74% |  85%
RGB objects, triplet 2 |  77% |  84% |  82% |  76% |  77% |  77%
RGB objects, triplet 3 |  77% |  72% |  89% |  91% |  63% |  85%
RGB objects, triplet 4 |  88% |  90% |  93% |  85% |  83% |  90%
RGB objects, triplet 5 |  92% |  77% |  83% |  80% |  80% |  80%
|  NIST-i gears |  67% |  69% |  66% |  72% |  82% |  69%
|  |  RBG |  RGB |  GRB |  GBR |  BRG |  BGR
Tower building |  RGB objects, triplet 2 |  – |  – |  – |  – |  – |  57%
RGB objects, triplet 4 |  – |  58% |  71% |  66% |  68% |  72%
RGB objects, triplet 5 |  34% |  58% |  – |  – |  – |  –
|  RGB objects, triplet 6 |  59% |  – |  – |  – |  44% |  73%
|  |  RBG |  RGB |  GRB |  GBR |  BRB |  BGR
Pyramid building |  RGB objects, triplet 1 |  71% |  69% |  52% |  64% |  79% |  75%
RGB objects, triplet 2 |  63% |  61% |  51% |  44% |  73% |  55%
RGB objects, triplet 3 |  66% |  59% |  62% |  68% |  48% |  58%
RGB objects, triplet 4 |  61% |  55% |  67% |  61% |  80% |  78%
|  RGB objects, triplet 5 |  71% |  48% |  77% |  68% |  75% |  77%
|  | Small | Medium | Large
Lifting |  NIST-i gears | 86% | 81% | 72%
Insertion-peg |  NIST-i gears | 56% | 79% | 79%
Real World |  Sawyer 5-DoF |  |  |  RB |  RG |  GR |  GB |  BR |  BG
Stacking |  RGB objects, triplet 1 |  87% |  – |  – |  – |  – |  45%
RGB objects, triplet 2 |  70% |  – |  – |  – |  – |  –
RGB objects, triplet 3 |  82% |  – |  – |  – |  – |  –
RGB objects, triplet 4 |  93% |  – |  – |  – |  – |  –
|  RGB objects, triplet 5 |  68% |  – |  – |  – |  – |  –
|  |  Carrot |  Cucumber |  Pepper |  Potato |  – |  –
Lifting |  YCB-i vegetables |  50% |  50% |  49% |  67% |  – |  –
Tower building |  RGB objects, triplet 5 | 23%
Inverted pyramid building |  RGB objects, custom triplet | 17%
Panda 7-DoF |  |  |  Apple |  Banana |  Lemon |  Peach |  – |  –
Lifting |  YCB fruits |  40% |  59% |  60% |  57% |  – |  –
|  | Small | Medium | Large
Lifting |  NIST-i gears | 92% | 94% | 95%
Insertion-peg |  NIST-i gears | 65% | 78% | 88%
| Removal-peg |  NIST-i gears | 96% | 97% | 98%
Fine-tuning |  Real World |  Panda 7-DoF |  |  | Lemon | Peach | Strawberry
Insertion-bowl |  YCB fruits and YCB-i bowl | 84% / 82% | 76% / 86% | 92% / 84%
Removal-bowl |  YCB fruits and YCB-i bowl | 60% / 62% | 69% / 76% | 64% / 77%
|  | Circle | Pentagon | Square
Insertion-base |  Shape-matching objects |  07% / 19% |  06% / 10% |  04% / 11%
Removal-base |  Shape-matching objects | 47% / 72% | 87% / 94% | 75% / 79%
KUKA 14-DoF | Lifting |  NIST-i gears | 56% / 86%
Table 15: Per-task RoboCat performance on training and fine-tuning tasks. This
table expands Table 1 by providing the success rate for each task variant
instead of the average per task families. Fine-tuning results are reported for
500 and 1000 demonstrations respectively, separated by a slash.

####  G.1.1 Training tasks

We use RL-trained experts for all of the structure-building tasks in sim
(Panda 7-DoF stacking, pyramid, and tower building; Sawyer 7-DoF stacking,
Figure 20) and in real (Sawyer 5-DoF stacking, Figure 21). These experts are
state-based for the sim tasks, and vision-based for the real-world tasks. In
the sim case, the vision-based RoboCat generalist has a performance reduction
of 10% or less compared to the state-based single-task experts (Figure 20),
for all families except tower building. It is important to note that these
experts were trained via RL with privileged state information which included
the poses of the target objects. In contrast, our vision-based generalist
agent discards any privileged state information in order to perform
effectively in the real-world. As such, we would not expect RoboCat to improve
upon or even match the success rate of the data for all tasks. On the other
hand, when RoboCat has access to same information as the experts that
generated the data, as for the real tasks Figure 21, it is able to overcome
the performance of the agents generating data.

Human-teleoperated demonstrations were used for the NIST-i tasks in both sim
and real. Figure 22 and Figure 23 compare the success rate of the human
teleoperators and RoboCat separately for each task variant. In both cases, we
can see that RoboCat on average matches the human teleoperator performance.

Self-generated data were used together with a limited number of human-
teleoperated demonstrations to train RoboCat on additional real robot tasks.
Given a limited number of demonstrations for a new task, we first fine-tuned
previous versions of RoboCat to solve this new task. We then deployed the
fine-tuned RoboCat in the real world to collect more self-generated data for
the specific task. The newly generated data together with the successful human
teleoperations were used in the final generalist. Figure 24 and Figure 25 aim
to contextualise the performance of RoboCat on these tasks. The ‘generated +
demos’ numbers refer to the percentage of successful episodes among the total
number of episodes used to train RoboCat on these tasks. Successful episodes
include successful self-generated episodes and successful teleoperations. We
use failed self-generated episodes, but not failed demonstrations. On average
RoboCat performance matches the success rate of the training data, overcoming
it in some cases (see lifting banana, peach, strawberry Figure 24 and potato
Figure 25).

####  G.1.2 Fine-tuning performance

Figure 28 compares the success rate of the human teleoperators to
contextualise the fine-tuning results. In general, we see how fine-tuning on
more demonstrations in general improves performance, sometimes reaching
similar performance as the human teleoperators. This does not apply to the
insert tasks, which result very challenging for our agents due to the required
reorientation and shape understanding.

Figure 20: RoboCat vs experts performance: Panda 7-DoF structure-building
training tasks (sim). RoboCat performance compared to the success rate of the
training data for each task family. Figure 21: RoboCat vs experts performance:
Sawyer 5-DoF RGB stacking tasks (real). This plot compares the performance of
RoboCat on the real stacking tasks with respect to the overall success rate of
the training data available for each task variant. Figure 22: RoboCat vs
teleoperation performance: Panda 7-DoF NIST-i tasks (sim). RoboCat compared
with the success rate of the data collected by human teleoperators, for each
task variant.  Figure 23: RoboCat vs teleoperation performance: Panda 7-DoF
NIST-i tasks (real). RoboCat compared with the success rate of the data
collected by human teleoperators, for each task variant. Figure 24: RoboCat vs
training data: Panda 7-DoF YCB lifting tasks (self-improvement, real). This
plot compares the performance of RoboCat on the self-improvement tasks with
respect to the overall success rate of the training data available for these
tasks (self-generated data and successful human teleoperations). Figure 25:
RoboCat vs training data: Sawyer 5-DoF RGB and YCB tasks (self-improvement,
real). This plot compares the performance of RoboCat on the self-improvement
tasks with respect to the overall success rate of the training data available
for these tasks (self-generated data and successful human teleoperations).

###  G.2 Impact of model size

To evaluate the effect of model size, we compare the 1.18B-parameter RoboCat-
lim model from Section 5.2 with a smaller 364M model on the simulated
structure-building tasks. The results in Figure 26 demonstrate that the larger
model yields an improvement of 6-21% in the success rate for the different
task families. In particular, while the smaller model only slightly degrades
performance for the easier stacking tasks, it is much poorer for the more
challenging pyramid and tower-building tasks.

Figure 26: Impact of model size. This plot compares the duccess rate of a
full-sized 1.18B-parameter model with that of a smaller 364M model, on the
RoboCat-lim simulated structure-building tasks.

(a) Simulated NIST-i environment.

(b) Fixed base NIST-i environment.

(c) Inverted fixed base NIST-i environment.

(d) Moving base NIST-i environment.

Figure 27: The different types of NIST-i based environments we ablate
performance against. Note, in the main paper we report performance against
environments from (a) and (d) only.

(a) Panda 7-DoF insert and remove tasks with YCB and YCB-i objects (real).

(b) Panda 7-DoF insert and remove tasks with shape matching objects (real).

(c) KUKA 14-DoF gear lifting task (real).

Figure 28: Final performance on each fine-tuning task.

###  G.3 Visual foundation baselines

We consider four different network architectures as baseline methods for
manipulation tasks: ResNet [He et al., 2016], normaliser-free networks
(NFNets) [Brock et al., 2021], vision transformers (ViTs) [Dosovitskiy et al.,
2020], and the ViT variant Swin transformers [Liu et al., 2021]. Within each
category, we used the following models: ResNet-50, ResNet-101, ResNet-200,
NFNet-f0, NFNet-f3, NFNet-f6, ViT-b, ViT-l, Swin-b, Swin-l, and Swin-s. Each
model was pretrained on a selection of datasets: ImageNet 1k [Deng et al.,
2009], ImageNet 21k [Ridnik et al., 2021], Microsoft COCO [Lin et al., 2014],
and the JFT Dataset [Riquelme et al., 2021]. We also evaluated the use of CLIP
[Radford et al., 2021], masked auto-encoder (MAE) [He et al., 2022], SimCLR
[Chen et al., 2020b], BYOL [Grill et al., 2020], DetCon [Hénaff et al., 2021],
Odin [Hénaff et al., 2022], and DINO [Caron et al., 2021].

One common way to use pretrained visual models for control tasks is to add a
learned policy head [Nair et al., 2022, Parisi et al., 2022]. The image input
is first embedded by the foundation model. The output embedding is then
concatenated with any proprioception information and fed into the policy head.
The policy head consists of two multi-layer perceptrons (MLPs), each with 256
parameters. The output of the top MLP is then used by a linear policy head to
generate the robot action. While this means that the VFM baselines only use
the current timestep (versus approximately 3 timesteps for the transformer-
based RoboCat agent), prior experiments with stacking tasks found that
incorporating context, via either LSTMs or observation stacking of 4 previous
timesteps, did not improve performance. Finally, we use behaviour cloning with
mean squared loss as the optimisation objective.

model | obj | dataset | pretrain
---|---|---|---
nfnet-f0 | Supervised | ImageNet-1k | imagenet1k-supervised
nfnet-f0 | Supervised | JFT-4B | jft4b-8epochs-supervised
nfnet-f0 | CLIP | ImageNet-1k | f0-clip-4dataset-lowres
nfnet-f0 | CLIP | ImageNet-1k | f0-clip-align
nfnet-f0 | CLIP | ImageNet-1k | f0-clip-4dataset
nfnet-f0 | CLIP | ImageNet-1k | f0-clip-align-stock
nfnet-f1 | Supervised | ImageNet-1k | imagenet1k-supervised
nfnet-f1 | Supervised | JFT-4B | jft4b-8epochs-supervised
nfnet-f2 | Supervised | ImageNet-1k | imagenet1k-supervised
nfnet-f3 | Supervised | JFT-4B | jft4b-8epochs-supervised
nfnet-f3 | Supervised | ImageNet-1k | imagenet1k-supervised
nfnet-f3plus | Supervised | JFT-4B | jft4b-8epochs-supervised
nfnet-f4 | Supervised | ImageNet-1k | imagenet1k-supervised
nfnet-f5 | Supervised | ImageNet-1k | imagenet1k-supervised
nfnet-f5 | CLIP | ImageNet-1k | f5-clip-align
nfnet-f6 | CLIP | ImageNet-1k | f6-clip-align-stock-sum_grads
nfnet-f6 | CLIP | ImageNet-1k | f6-clip-4dataset
nfnet-f6 | CLIP | ImageNet-1k | f6-clip-align
nfnet-f6 | Supervised | ImageNet-1k | imagenet1k-supervised
nfnet-f7 | Supervised | JFT-4B | jft4b-4epochs-supervised
nfnet-f7plus | Supervised | JFT-4B | jft4b-4epochs-supervised
ResNet-101 | Supervised | ImageNet-1k | imagenet1k-supervised
ResNet-101 | DetCon | ImageNet-1k | imagenet1k-detcon
ResNet-101 | BYOL | ImageNet-1k | imagenet1k-byol
ResNet-200 | DetCon | ImageNet-1k | imagenet1k-detcon
ResNet-200 | BYOL | ImageNet-1k | imagenet1k-byol
ResNet-50 | ODIN | ImageNet-1k | imagenet1k-odin
ResNet-50 | Supervised | ImageNet-1k | imagenet1k-supervised
ResNet-50 | DetCon | ImageNet-1k | imagenet1k-detcon-coco-finetune
ResNet-50 | BYOL | ImageNet-1k | imagenet1k-byol-lowres
ResNet-50 | DetCon | ImageNet-1k | imagenet1k-detcon
ResNet-50 | BYOL | ImageNet-1k | imagenet1k-byol
swin-b | Supervised | ImageNet-1k | imagenet1k-supervised
swin-e | CLIP | JFT-4B | swin-e-clip-align-stock-jft
swin-h | CLIP | ImageNet-1k | swin-h-clip-align-stock-highres
swin-h | CLIP | ImageNet-1k | swin-h-clip-align-stock-lowres
swin-l | CLIP | ImageNet-1k | swin-l-clip-align-stock
swin-s | Supervised | ImageNet-1k | imagenet1k-supervised
swin-s | ODIN | ImageNet-1k | imagenet1k-odin
swin-t | ODIN | ImageNet-1k | imagenet1k-odin
swin-t | Supervised | ImageNet-1k | imagenet1k-supervised
vit-b | Supervised | JFT-4B | b16-224-jft-pretrain
vit-b | Supervised | ImageNet-21k | b16-224-i21k-pretrain-augreg
vit-b | Supervised | ImageNet-21k | b16-224-i21k-pretrain
vit-b | Supervised | ImageNet-1k | b16-224-i21k-pretrain-i1k-finetune
vit-b | Supervised | ImageNet-1k | b16-384-i21k-pretrain-i1k-finetune
vit-b | MAE | ImageNet-1k | b16-224-i1k-mae-pretrain
vit-b | Supervised | ImageNet-1k | b16-224-i1k-pretrain
vit-deit-b | MAE | ImageNet-1k | b16-224-i1k-mae-pretrain
vit-deit-l | MAE | ImageNet-1k | l16-224-i1k-mae-pretrain
vit-dino-b | MAE | ImageNet-1k | b16-224-i1k-mae-pretrain
vit-l | Supervised | ImageNet-1k | l16-224-i1k-pretrain
vit-l | Supervised | ImageNet-1k | l16-384-i21k-pretrain-i1k-finetune
vit-l | Supervised | ImageNet-21k | l16-224-i21k-pretrain
vit-l | Supervised | ImageNet-1k | l16-384-jft-pretrain-i1k-finetune
vit-l | MAE | ImageNet-1k | l16-224-i1k-mae-pretrain
vit-l | Supervised | JFT-4B | l16-224-jft-pretrain
vit-l | Supervised | ImageNet-1k | l16-224-i21k-pretrain-i1k-finetune
vit-mae-b | MAE | ImageNet-1k | b16-224-i1k-mae-pretrain
Table 16: Details of the baseline methods: including the model name, training
objectives and pretaining dataset.

We fine-tuned a total of 59 combinations of the above methods and evaluated on
a subset of RoboCat tasks. Then, we selected the top representatives of each
of the network architecture categories and evaluated them on more tasks. For
the sake of clarity, we only report the top two methods in the main paper. We
use the average final policy success rate as the performance metric in our
experiments. Each experiment ran for 100 episodes, with each episode lasting
400 time steps.

We fine-tuned the models with different data limitations: 500 demonstrations,
1000 demonstrations, and more than $10\,000$ demonstrations. Training took
$200\,000$ training steps for all models. We validated the models in
simulation during training and selected the best model snapshot based on the
validation success rate for further comprehensive evaluation. We tried both
freezing and non-freezing scenarios for the parameters of the pretrained
models. Since the non-frozen pretrained model performed much better, we only
report the results for non-frozen models.

|  | final_success
---|---|---
| Model | nfnet-f0 | nfnet-f3 | nfnet-f6 | swin-b | swin-l | swin-s | vit-dino-b | vit-l
| Training Objective | CLIP | Supervised | CLIP | Supervised | CLIP | Supervised | MAE | MAE
| Dataset | ImageNet-1k | JFT-4B | ImageNet-1k | ImageNet-1k | ImageNet-1k | ImageNet-1k | ImageNet-1k | ImageNet-1k
# Demo | Task |  |  |  |  |  |  |  |
more than $10\,000$ | panda_7-DoF_stacking_BG_set_1 | 0.92 | 0.86 | 0.87 | 0.91 | 0.93 | 0.91 | 0.81 | 0.87
| panda_7-DoF_stacking_RG_set_1 | 0.83 | 0.51 | 0.78 | 0.81 | 0.92 | 0.81 | 0.75 | 0.62
| sawyer_5-DoF_stacking_set_5_RB | 0.79 | 0.79 | 0.78 | 0.79 | 0.87 | 0.76 | 0.71 | 0.70
| sawyer_7-DoF_stacking_BG_set_1 | 0.83 | 0.80 | 0.88 | 0.91 | 0.90 | 0.84 | 0.77 | 0.82
1000 | panda_7-DoF_inv_pyra_GRB_set_4 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
| panda_7-DoF_stacking_BG_set_1 | 0.25 | 0.40 | 0.60 | 0.24 | 0.49 | 0.20 | 0.02 | 0.07
| panda_7-DoF_stacking_RG_set_1 | 0.20 | 0.31 | 0.33 | 0.25 | 0.28 | 0.25 | 0.03 | 0.12
| panda_7-DoF_tower_GRB_set_4 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
| panda_7-DoF_tower_RBG_set_5 | 0.04 | 0.08 | 0.14 | 0.08 | 0.16 | 0.06 | 0.01 | 0.03
| sawyer_5-DoF_stacking_set_5_RB | 0.66 | 0.76 | 0.74 | 0.62 | 0.70 | 0.43 | 0.24 | 0.30
| sawyer_7-DoF_stacking_BG_set_1 | 0.39 | 0.50 | 0.56 | 0.36 | 0.58 | 0.36 | 0.06 | 0.11
500 | panda_7-DoF_stacking_BG_set_1 | 0.09 | 0.19 | 0.36 | 0.08 | 0.06 | 0.09 | 0.01 | 0.03
| sawyer_5-DoF_stacking_set_5_RB | 0.33 | 0.55 | 0.64 | 0.18 | 0.16 | 0.08 | 0.03 | 0.05
| sawyer_7-DoF_stacking_BG_set_1 | 0.10 | 0.24 | 0.37 | 0.13 | 0.20 | 0.14 | 0.01 | 0.02
training set | panda_7-DoF_pyramid_GRB_set_3 | 0.28 | 0.28 | 0.35 | 0.27 | 0.33 | 0.21 | 0.15 | 0.19
| panda_7-DoF_stacking_BG_set_1 | 0.63 | 0.69 | 0.70 | 0.62 | 0.73 | 0.58 | 0.41 | 0.46
| panda_7-DoF_stacking_GB_set_2 | 0.48 | 0.61 | 0.64 | 0.52 | 0.56 | 0.50 | 0.38 | 0.41
| panda_7-DoF_tower_GRB_set_5 | 0.37 | 0.40 | 0.55 | 0.37 | 0.42 | 0.40 | 0.22 | 0.31

Table 17: Evaluation of baselines on training and held-out tasks with limited
number of demonstrations.

Table 17 shows the success rate of all baseline methods on a held-out task.
Then we took the top representative methods of each network architecture and
further evaluated them on other tasks with limited data: 500, 1000, and more
than $10\,000$. We also evaluated these methods on a subset of the training
tasks.

###  G.4 NIST-i extended study

We evaluate the complexity and some design decisions made for the NIST-i tasks
using RoboCat. We find that fixing the base in the physical world can have
interesting implications, more cameras can lead to improved performance and
using data from largely unrelated tasks can lead to increased performance on
NIST-i, suggesting indications of positive skill transfer.

####  G.4.1 Task complexity

So far, we considered a fixed base in simulation and a freely moving base in
the physical world (see Figure 12 for details). We argue that a moving base
leads to both a harder task in the physical world and provides a more diverse
set of tasks for training. We challenge this argument in this section. We do
this by fixing the base to the basket in the same orientation as what we do
for simulation and measure if this new setting is easier to solve. Figure 27
illustrates the different environments. Note that at train time we only use
demonstrations from 27(a) and 27(d).

Base Location | NIST-i task
---|---
Insert | Remove
Fixed base | 64% | 100%
Inverted fixed base | 40% | 99%
Moving base | 38% | 97%
Table 18: Real-world NIST-i task ablation. Average success over all three
NIST-i gear sizes. A 364M agent performs much better on the fixed base
environment.

Table 18 reports the performance of a smaller, 364M agent, trained on the same
data as RoboCat-lim. We observe that despite not having fixed base data from
the physical world, the agent performs a lot better on the fixed-based task as
opposed to the other two tasks. This suggests that the fixed base environment
is easier possibly due to its similarity with the simulated environment,
suggesting positive skill transfer from simulation data.

####  G.4.2 Data bias

Section G.4.1 showed that there may be some positive transfer across similar
tasks on a smaller 364M parameter agent. However, the performance reported in
that section could also be due to data bias. That is, it could be that the
collected demonstrations from raters might be primarily centred around the
middle of the basket with the peg base orientated similarly to the one in the
simulated data. This itself would interfere with the hypothesis for positive
skill transfer in favour of having data bias. To test this, we utilise
specialist behaviour cloning agents (Specialist BC) that we separately train
on each NIST-i task using only the subset of NIST-i training data belongs to
that specific task. Behaviour cloning agents are known to overfit to the
training data, leading to poor generalisation [Osa et al., 2018]. If the
reported performance in Table 18 is due to data bias, then we expect to see
the same type of improved performance of a specialist BC agent when evaluated
on the fixed based tasks as opposed to the moving base tasks.

Base Location | Specialist BC
---|---
NIST-i Insert Task
Fixed base | 8%
Moving base | 13%
Table 19: Real-world NIST-i data bias. Average success over all three NIST-i
gear sizes. The agents have similar perform on both fixed and moving base
environments. No evidence for data bias.

Table 19 shows that having a fixed or moving base results in a negligible
difference in the performance for the specialist BC agents. In fact, the
relationship between the performance on fixed and moving base is inverted (all
training data is collected while having a moving base). This indicates that
for both the fixed and moving base tasks there is no particular data bias
towards the peg base being situated in the centre and oriented in the same way
as in simulation.

####  G.4.3 Skill transfer

So far, we showed that there are some indications of skill transfer for the
smaller 364M generalist. In this section, we study to what extent this holds
for our final RoboCat agent. Notably, we use the same NIST-i insertion data
for training both the 364M and RoboCat models. However, our final version of
RoboCat is trained on a larger number of tasks and is of bigger capacity
(1.2B). In this sense, a successful skill transfer would close the gap between
the achieved performance on the fixed and moving base environments for the
final generalist. This is what we observe in Table 20. Specifically, moving
from the 364M model to the full RoboCat agent eliminates just under two thirds
of the gear insertion failures in both the fixed base and the moving base
setting. This indicates that indeed training a bigger model on more data that
involves different types of bring to pose tasks can be beneficial for solving
the moving base insertion tasks. Moreover, the additive performance gap for
RoboCat depending on the base state is much smaller than the additive
performance gap for the 364M agent suggesting again indications of positive
skill transfer.

NIST-i Insert Task | 364M error | RoboCat error | Reduction factor
---|---|---|---
Fixed base | 0.36 | 0.13 | 2.77
Moving base | 0.62 | 0.23 | 2.70

Table 20: Skill transfer analysis. Average accumulated error over all three
NIST-i gear sizes. Moving from the 364M model to the full RoboCat agent
eliminates just under two thirds of the gear insertion failures in both the
fixed and the moving base settings, indicating a smaller performance gap for
the RoboCat.

####  G.4.4 Additional observations

Finally, in the process of finding the best performing specialist BC agent we
noticed that including the wrist cameras to the observation of the agent
significantly improved the performance of our specialists. That is, instead of
using just two camera observations, as we do for RoboCat, using a total of
four camera observations - two from the basket and two from the wrist of the
robot, was very beneficial for a specialist BC agent (see Table 21).

Performance of | NIST-i task
---|---
Insert | Remove
Specialist BC (2 cameras) | 13% | 24%
Specialist BC (4 cameras) | 36% | 54%
Table 21: Dependency on the number of cameras. Average performance over all
three NIST-i gear sizes. More cameras have a significant effect on the
specialist’s performance.

An exciting direction for future research is to understand the effect of
different camera observations for contact-rich manipulation tasks in the
context of RoboCat.


# QCD parameters and SM-high precision from $e^{+}e^{-}\to$ Hadrons : Updated
111This is an update of the results in Ref. [1], where some preliminary ones
have been presented at the alphas 2024 workshop (5-9 february 2024, ECT*
Trento-IT).

Stephan Narison  Laboratoire Univers et Particules de Montpellier (LUPM),
CNRS-IN2P3,
Case 070, Place Eugène Bataillon, 34095 - Montpellier, France
and
Institute of High-Energy Physics of Madagascar (iHEPMAD)
University of Ankatso, Antananarivo 101, Madagascar<EMAIL_ADDRESS>
###### Abstract

1. I report an update of my previous comparison of the theoretical value of the muon anomaly $a_{\mu}\equiv\frac{1}{2}(g-2)_{\mu}$ with the new measurement. One finds : $\Delta a_{\mu}\equiv a_{\mu}^{exp}-a_{\mu}^{th}=(143\pm 42_{th}\pm 22_{exp})\times 10^{-11}$ indicating about 3$\sigma$ discrepancy between the SM predictions and experiment.

2. I improve the estimate of QCD power corrections up to dimension $D=12$ and provide a new estimate of the ones up to $D=20$ within the Shifman-Vainshtein-Zahkarov (SVZ) expansion by combining the ratio of the SVZ Borel/Laplace sum rules (LSR) with the Braaten-Pich and the author (BNP) $\tau$-like decay moments for the $I=1$ vector current. The results summarized in Table 1 confirm a violation of the factorization of the four-quark condensates and the value of the gluon one $\langle\alpha_{s}G^{2}\rangle$ from some other sources. Up to $D=20$, I do not observe any factorial nor exponential growth of the size of these power corrections.

3. I use these new values of power corrections to extract $\alpha_{s}$ from the BNP lowest moment. To order $\alpha_{s}^{4}$, I find within the SVZ expansion : $\alpha_{s}(M_{\tau})=0.3081(50)_{fit}(71)_{\alpha_{s}^{5}}$ [resp. $0.3260(47)_{fit}(62)_{\alpha_{s}^{5}}]$ implying $\alpha_{s}(M_{Z})=0.1170(6)(3)_{evol}$ [resp. $0.1192(6)(3)_{evol}$] for Fixed Order (FO) [resp. Contour Improved (CI)] PT series. They lead to the mean: $\alpha_{s}(M_{\tau})|_{SVZ}=0.3180(58)_{fit}(99)_{syst}$ and $\alpha_{s}(M_{Z})|_{SVZ}=0.1182(14)(3)_{evol}$ where the systematic error(syst) takes into account the discrepancy between the FO and CI results.

4. Some (eventual) contributions beyond the SVZ expansion ($1/Q^{2}$, instantons and duality violation) are discussed in Sections 10 and 11.

###### keywords:

Muon anomaly, QCD spectral sum rules, QCD parameters,
$e^{+}e^{-},\,\tau$-decay.

##  1 Introduction

Precise determinations of the hadronic contributions to the muon anomaly
$a_{\mu}\equiv\frac{1}{2}(g-2)_{\mu}$ and the QCD parameters (power
corrections and $\alpha_{s}$) are important inputs for testing the Standard
Model (SM) and for QCD and hadrons phenomenolgy.

In this paper, I shall revisit the results obtained in [1] (refered hereafter
as SN23) due to the new experimental measurement of the positive charged muon
anomaly [2] received after the publication of the papers in SN23.

I shall also re-estimate the QCD condensates and some higher dimension ones by
combining the analysis from the ratio of the SVZ [3] 222For reviews, see e.g.
[4, 5, 6, 7] and references quoted in [1]. LSR [8, 9]333For a recent review,
see e.g. [10]. with the one from some higher BNP $\tau$-like moments [11, 12,
13]. These new values of QCD condensates will be used as inputs for
deterimining $\alpha_{s}$ from the BNP lowest moment.

In so doing, I shall use, like in SN23 [1], the PDG 22 compilation of the
$e^{+}e^{-}\to$ Hadrons [14] $\oplus$ the recent CMD3 data [15] for the pion
form factor and the value of gluon condensate $\langle\alpha_{s}G^{2}\rangle$
from heavy quarkonia and some other sources.

##  2 The $I=1$ isovector two-point function

We shall be concerned with the two-point correlator :

$\Pi^{\mu\nu}_{H}(q^{2})=i\int d^{4}x~{}e^{-iqx}\langle 0|{\cal
T}{J^{\mu}_{H}}(x)\left({J^{\nu}_{H}}_{H}(0)\right)^{\dagger}|0\rangle~{}~{}=-(g^{\mu\nu}q^{2}-q^{\mu}q^{\nu})\Pi_{H}(q^{2})$
(1)

built frrom the T-product of the bilinear $I=1$ vector current :

$J^{\mu}_{H}(x)=\frac{1}{2}{[}:\bar{\psi}_{u}\gamma^{\mu}\psi_{u}-\bar{\psi}_{d}\gamma^{\mu}\psi_{d}:{]},$
(2)

It obeys the dispersion relation:

$\Pi_{H}(q^{2})=\int_{t>}^{\infty}\frac{dt}{t-q^{2}-i\epsilon}\frac{1}{\pi}\,{\rm
Im}\Pi_{H}(t)+\cdots,$ (3)

where $\cdots$ are subtraction constants polynomial in $q^{2}$ and $t>$ is the
hadronic threshold.. From the optical theorem, the spectral function ${\rm
Im}\Pi_{H}(t)$ can be related to the $e^{+}e^{-}\to\,{\rm Hadrons}$ total
cross-section as:

$R^{I=1}_{ee}\equiv\frac{\sigma(e^{+}e^{-}\to\,{\rm
Hadrons}}{\sigma(e^{+}e^{-}\to\mu^{+}\mu^{-})}=\left(\frac{3}{2}\right)8\pi\,{\rm
Im}\Pi_{H}(t).$ (4)

The data handlings have been discussed in details in Ref. [1] and will not be
repeated here. We use the compilation of PDG22 [14] which take into account
the correlations among different data to which while we use the new CMD3 [15]
data for the pion form factor below 1 GeV. In this paper, we shall be
concerned with the data below 1.875 GeV.

##  3 Comparison of the experimental and theoretical values of the muon
anomaly $a_{\mu}$

We start the paper by updating our previous comparison (1st reference in Ref.
[1]) of the theoretical and new experimental values of the muon anomaly
$a_{\mu}$ [2] :

$a_{\mu}^{exp}=116592059(22)\times 10^{-11},$ (5)

which improves the accuracy of previous results in Refs. [16] and FNAL [17] by
a factor 1.86.

This experimental value is to be compared with the theoeretical results
compiled in Table 1 of Ref. [18] and in Table 8 of Ref. [19] to which we add
the new estimate of the lowest order hadronic contributions to the vacuum
polarization:

$a_{\mu}|^{hvp}_{l.o}=(7036.5\pm 38.9)\times 10^{-11}.$ (6)

We notice that our predicition using the CMD3 data is larger by an amount of
about $100\times 10^{-11}$ compared to the previous estimates using KLOE data
as compared in Fig. 17 of SN23 [1] and the discussion in Section 10 of this
paper. Therefore, we obtain [1] :

$a_{\mu}^{th}=116591916(42)\times 10^{-11}.$ (7)

This leads to (2nd reference in Ref. [1]) :

$\Delta a_{\mu}\equiv a_{\mu}^{exp}-a_{\mu}^{th}=(143\pm 42_{th}\pm
22_{exp})\times 10^{-11},$ (8)

which indicates about $3\,\sigma$ discrepancy between experiment and the SM
predictions.

##  4 The standard SVZ expansion

Within the SVZ [3] Operator Product Expansion (OPE), QCD condensates with
higher and higher dimensions are assumed to approximate the not yet known QCD
non-perturbative contributions. The SVZ-expansion reads ($q^{2}\equiv-Q^{2}$)
:

$8\pi^{2}\Pi_{H}(-Q^{2},m_{q}^{2},\mu)=\sum_{D=0,2,..}\frac{C_{D}(q^{2},m_{q}^{2},\mu)\langle
O_{D}(\mu)\rangle}{(Q^{2})^{D/2}}~{},$ (9)

where $m_{q}$ is the quark mass, $\mu$ is the subtraction scale which
separates the long and short distance dynamics. $C_{D}$ are perturbatively
calculable Wilson coefficients while $\langle O_{D}(\mu)\rangle$ are non-
perturbative QCD condensates of dimension $D$. In the phenomenological
analysis, the OPE is often truncated at $D=6,8$ where the approach gives a
satisfactory explanation of different data (see e.g. [4, 5, 6, 7, 8, 9]).

In addition, one should note that the contributions of higher dimension
condensates are not under a good control due to the large number of Feynman
diagram ones, to the inaccurate estimate of their size and to the difficulty
to built a renormalizaion group invariant (RGI) condensate due to their mixing
under renormalization [20].

In SN23, one uses the PT expression up to order $\alpha_{s}^{4}$ while the OPE
is truncated at $D=6,8$.

One should mention that, besides the well-known $\langle\bar{\psi}\psi\rangle$
quark condensate, the gluon condensates have been determined from the heavy
quark mass-splittings and some other sum rules [21, 22]:

$\langle\alpha_{s}G^{2}\rangle=(6.39\pm 0.35)\times 10^{-2}\,{\rm
GeV^{4}},~{}~{}~{}~{}~{}~{}~{}~{}~{}{\langle
g^{3}G^{3}\rangle}/{\langle\alpha_{s}G^{2}\rangle}=8.2(1.0)\,[{\rm GeV^{4}}],$
(10)

while different analysis of the light meson systems lead to the value of the
four-quark condensate (see e.g. the papers quoted in [7] and [1]) :

$\rho\alpha_{s}\langle\bar{\psi}\psi\rangle^{2}=5.8(9)\times 10^{4}\,[\rm
GeV^{6}].$ (11)

##  5 The Laplace sum rules (LSR) and their ratios

###  $\bullet~{}$Form of the LSR and their ratio

In Ref. [23] and SN23 [1], the dimension $D=4,6$ and 8 condensates appearing
in the OPE of the two-point vector correlator have been estimated using the
ratio of Laplace sum rule moments [3, 8, 9]444For a recent review, see e.g.
[10].:

${\cal R}^{c}_{10}(\tau)\equiv\frac{{\cal L}^{c}_{1}}{{\cal
L}^{c}_{0}}=\frac{\int_{t>}^{t_{c}}dt~{}e^{-t\tau}t\,R_{ee}^{I=1}(t,\mu)}{\int_{t>}^{t_{c}}dt~{}e^{-t\tau}R_{ee}^{I=1}(t,\mu)},$
(12)

where $\tau$ is the LSR variable, $t>$ is the hadronic threshold. Here $t_{c}$
is the threshold of the “QCD continuum” which parametrizes, from the
discontinuity of the Feynman diagrams, the spectral function ${\rm
Im}\,\Pi_{H}(t,m_{q}^{2},\mu^{2})$. $m_{q}$ is the quark mass and $\mu$ is an
arbitrary subtraction point. The exponential weight enhances the contribution
of the low-energy region of the spectral function accessible eperimentally. We
shall see in Eq.14 that it introduces a factorial factor in front of each
condensates which accelerates the convergence of the OPE.

###  $\bullet~{}$The PT QCD expression of the LSR

To ordrer $\alpha_{s}^{4}$, the perturbative (PT) expression of the lowest
moment reads [24]:

${\cal
L}^{PT}_{0}(\tau)=\frac{3}{2}\tau^{-1}\Big{[}1+a_{s}+2.93856\,a_{s}^{2}+6.2985\,a_{s}^{3}+22.2233\,a_{s}^{4}\Big{]}.$
(13)

Then, taking its derivative in $\tau$, one gets ${\cal L}_{1}(\tau)$ and then
their ratio ${\cal R}_{10}(\tau)$.

###  $\bullet~{}$The NPT QCD expression of the LSR

From Eq. 9, one can deduce the lowest moment LSR:

${\cal
L}^{NPT}_{0}(\tau)=\frac{3}{2}\tau^{-1}\sum_{D}\frac{d_{D}}{(D/2-1)!}\tau^{D/2}~{},$
(14)

from which one can deduce ${\cal L}^{NPT}_{1}$ and ${\cal R}_{10}$ where
$d_{D}\equiv C_{D}\langle O_{D}\rangle$.

##  6 The $\tau$-decay-like moment sum rules

###  $\bullet~{}$The BNP lowest moment

Figure 1: Cauchy contour integral in the complex $Q^{2}\equiv|q^{2}|$-plane

Initially introduced by BNP [12, 11], the lowest original $\tau$-decay moment
reads in the massless quark limit :

${\cal
R}^{ee}_{0}=12\pi\int_{t>}^{x_{0}}dt\,(1-x_{0})^{2}(1+2x_{0})\,2\,R_{ee}^{I=1}(x_{0}),$
(15)

where : $x_{0}\equiv t/M_{0}^{2}$ with $M_{0}$ is the $\tau$-like lepton mass.
The moment can be conveniently reformulated in terms of the Cauchy integral
along the contour in Fig. 1 à la Shankar [25] 555Some other applications in
QCD can e.g. by found in [26].:

${\cal
R}^{ee}_{0}=6i\,\pi\int_{|x_{0}|=1}dx_{0}\,(1-x_{0})^{2}(1+2x_{0})\,2\,R_{ee}^{I=1}(x_{0}),$
(16)

with a radius $|x_{0}|=1$. To order $\alpha_{s}^{4}$, the PT corrections to
the lowest BNP moment reads [12, 13, 27, 28]:

$\displaystyle\delta^{(0)}_{0}$ $\displaystyle=$ $\displaystyle
a_{s}+5.2026\,a_{s}^{2}+26.368\,a_{s}^{3}+127.085\,a_{s}^{4}+{\cal
O}(a_{s}^{5})~{}~{}~{}~{}~{}{\rm~{}~{}~{}~{}(FO)}$ (17) $\displaystyle=$
$\displaystyle
1.364\,a_{s}+2.54\,a_{s}^{2}+9.71\,a_{s}^{3}+64.29\,a_{s}^{4}+{\cal
O}(a_{s}^{5})~{}~{}~{}~{}~{}{\rm~{}~{}~{}~{}(CI)}$

for Fixed Order (FO) and Contour Improved (CI) PT series.

Unlike different Finite Energy Sum Rule (FESR) used in the current literature
[29] and its pinched versions [30], the main advantage of the $\tau$-decay-
like moment is the presence of the weight factor $(1-x_{0})^{2}$ which
suppresses the contribution near the real axis where the data are inaccurate.
This approach has lead to an accurate determination of $\alpha_{s}$ at the
$\tau$ mass.

###  $\bullet~{}$The $\tau$-like high moments

Inspired by the original work of [13], different forms of high-moments have
been ptoposed in the literature for extracting simultaneously $\alpha_{s}$ and
the QCD condensates [31, 32, 33, 34]. In this paper, we shall be concerned
with the moments of the form:

${\cal
R}_{n}^{ee}=6i\,\pi\int_{|x_{0}|=1}dx_{0}\,(1-x_{0})^{2}(1+2x_{0})(x_{0}^{n})\,2\,R_{ee}^{I=1}(x_{0}),$
(18)

where $n=1$ to 6.

###  $\bullet~{}$Perturbative corrections to the $\tau$-like high moments at
order $\alpha_{s}^{4}$

The perturbative contributions read to order $\alpha_{s}^{4}$ for fixed order
(FO) perturbation series:

$\displaystyle\delta^{(0)}_{1}$ $\displaystyle=$ $\displaystyle
a_{s}+3.7776\,a_{s}^{2}+6.8810\,a_{s}^{3}-66.7235\,a_{s}^{4}+{\cal
O}(a_{s}^{5})$ $\displaystyle\delta^{(0)}_{2}$ $\displaystyle=$ $\displaystyle
a_{s}+3.2151\,a_{s}^{2}+1.0687\,a_{s}^{3}-98.2293\,a_{s}^{4}+{\cal
O}(a_{s}^{5})$ $\displaystyle\delta^{(0)}_{3}$ $\displaystyle=$ $\displaystyle
a_{s}+2.8990\,a_{s}^{2}-1.7710\,a_{s}^{3}-108.034\,a_{s}^{4}+{\cal
O}(a_{s}^{5})$ $\displaystyle\delta^{(0)}_{4}$ $\displaystyle=$ $\displaystyle
a_{s}+2.6928\,a_{s}^{2}-3.4664\,a_{s}^{3}-111.868\,a_{s}^{4}+{\cal
O}(a_{s}^{5})$ $\displaystyle\delta^{(0)}_{5}$ $\displaystyle=$ $\displaystyle
a_{s}+2.4365\,a_{s}^{2}-5.4048\,a_{s}^{3}-119.087\,a_{s}^{4}+{\cal
O}(a_{s}^{5})$ $\displaystyle\delta^{(0)}_{6}$ $\displaystyle=$ $\displaystyle
a_{s}+2.5464\,a_{s}^{2}-4.5966\,a_{s}^{3}-108.552\,a_{s}^{4}+{\cal
O}(a_{s}^{5})$ (19)

###  $\bullet~{}$$\tau$-like high moments with non-Perturbative corrections at
lowest order for $m_{u,d}=0$

To lowest order in $\alpha_{s}$ and in the massless quark limit, the non-
perturbative corrections to the different moments give:

$\displaystyle{\cal R}^{ee}_{0}$ $\displaystyle=$
$\displaystyle\frac{3}{2}\Bigg{[}1+\delta^{(0)}_{0}-6\frac{d_{6}}{M_{0}^{6}}-{4}\frac{d_{8}}{M_{0}^{8}}\Bigg{]}:~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\delta^{(D)}_{n}=0~{}~{}~{}(for~{}~{}D\geq
5),$ $\displaystyle{\cal R}^{ee}_{1}$ $\displaystyle=$
$\displaystyle\frac{9}{20}\Bigg{[}1+\delta^{(0)}_{1}-\frac{20}{3}\frac{d_{4}}{M_{0}^{4}}+20\frac{d_{8}}{M_{0}^{8}}+\frac{40}{3}\frac{d_{10}}{M_{0}^{10}}\Bigg{]}:~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\delta^{(D)}_{n}=0~{}~{}~{}(for~{}~{}D\geq
6),$ $\displaystyle{\cal R}^{ee}_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{5}\Bigg{[}1+\delta^{(0)}_{2}+15\frac{d_{6}}{M_{0}^{6}}-45\frac{d_{10}}{M_{0}^{10}}-30\frac{d_{12}}{M_{0}^{12}}\Bigg{]}:~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\delta^{(D)}_{n}=0~{}~{}~{}(for~{}~{}D\geq
7),$ $\displaystyle{\cal R}^{ee}_{3}$ $\displaystyle=$
$\displaystyle\frac{3}{28}\Bigg{[}1+\delta^{(0)}_{3}-28\frac{d_{8}}{M_{0}^{8}}+84\frac{d_{12}}{M_{0}^{12}}+56\frac{d_{14}}{M_{0}^{14}}\Bigg{]}:~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\delta^{(D)}_{n}=0~{}~{}~{}(for~{}~{}D\geq
8),$ $\displaystyle{\cal R}^{ee}_{4}$ $\displaystyle=$
$\displaystyle\frac{9}{140}\Bigg{[}1+\delta^{(0)}_{4}+\frac{140}{3}\frac{d_{10}}{M_{0}^{10}}-140\frac{d_{14}}{M_{0}^{14}}-\frac{280}{3}\frac{d_{16}}{M_{0}^{16}}\Bigg{]}:~{}~{}~{}~{}\delta^{(D)}_{n}=0~{}~{}~{}(for~{}~{}D\geq
9),$ $\displaystyle{\cal R}^{ee}_{5}$ $\displaystyle=$
$\displaystyle\frac{1}{24}\Bigg{[}1+\delta^{(0)}_{5}-72\frac{d_{12}}{M_{0}^{12}}-216\frac{d_{16}}{M_{0}^{16}}-144\frac{d_{18}}{M_{0}^{18}}\Bigg{]}:~{}~{}~{}~{}~{}~{}~{}~{}\delta^{(D)}_{n}=0~{}~{}~{}(for~{}~{}D\geq
10),$ $\displaystyle{\cal R}^{ee}_{6}$ $\displaystyle=$
$\displaystyle\frac{1}{35}\Bigg{[}1+\delta^{(0)}_{6}+75\frac{d_{14}}{M_{0}^{14}}-225\frac{d_{18}}{M_{0}^{18}}-150\frac{d_{20}}{M_{0}^{20}}\Bigg{]}:~{}~{}~{}~{}~{}~{}~{}~{}\delta^{(D)}_{n}=0~{}~{}~{}(for~{}~{}D\geq
11),$ (20)

##  7 Extraction of the QCD condensates

###  $\bullet~{}$Condensates from the $\tau$-decay-like high moments

In so doing we fix the value of $\alpha_{s}$ from the world average [14].
Then, we are looking for stability in the change of $M_{0}$ for extracting the
optimal values of the condensates.

####  $\diamond~{}$${\cal R}^{ee}_{0}$ moment

– Using a two-parameter ($d_{6},d_{8}$) fit in the lowest BNP moment, we do
not find a conclusive result. We interpret this fact due to the opposite sign
of the two contributions which tend to cancel out. However, this feature is
welcome for determining $\alpha_{s}$ as the sum of the non-perturbative
contriutions are small but may indicate that it is not a good place for
determining accurately the condensates .

– We impose more constraints by using a one-parameter fit with as an initial
value the one :

$d_{6}=-(20.5\pm 2)\times 10^{-2}~{}{\rm GeV}^{6},$ (21)

obtained recently in SN23 [1] from the ratio of LSR into ${\cal R}^{ee}_{0}$
in an attempt to fix $d_{8}$. The resulting value of $d_{8}$ does not show any
$M_{0}$ stability as also observed in SN23. Then, we conclude that the
analysis from ${\cal R}_{0}^{ee}$ is not also conclusive.

####  $\diamond~{}$${\cal R}^{ee}_{1}$ moment

In the following we shall only work with a one-parameter fit $d_{0}$ and use
as initial values of $\langle\alpha_{s}G^{2}\rangle$ in Eq. 10 obtained from
heavy quarks mass-splittings and some other sources and the value of $d_{8}$
fromSN23:

$d_{8}=(4.7\pm 3.5)\times 10^{-2}~{}{\rm GeV}^{8}.$ (22)

We extract $d_{10}$. Then, we repeat the analysis by fixing $d_{10}$ and
$d_{6}$ and deduce $d_{8}$.

####  $\diamond~{}$${\cal R}^{ee}_{2}$ moment

We use the value of $d_{10}$ into ${\cal R}^{ee}_{2}$ and extract $d_{12}$.
Using this value of $d_{12}$, we re-iterate the analysis to extract $d_{10}$
and continue the iterations.

####  $\diamond~{}$Higher moments $n=2$ to 6

We repeat the procedure for higher moments and extract $d_{14}$ to $d_{20}$.
The quoted values are obtained without any iterations as they are already
quite precise.

####  $\diamond~{}$Conclusions from $\tau$-like moments

The values of $d_{8}$ to $d_{20}$ from the $\tau$-like moments are shown in
Figs. 2 and 3 where we notice nice stabilities of the results versus $M_{0}$.
The optimal values are quoted in Table 1

a) b)

Figure 2: a) Value of the condensate $d_{8}$ from ${\cal R}_{1}^{ee}$ versus
$M_{0}$; b) Values of the condensates $d_{10}$ and $d_{12}$ from ${\cal
R}_{1}^{ee}$ and ${\cal R}_{2}^{ee}$ versus $M_{0}$

a) b)

Figure 3: a) Value of the condensate $d_{1}4$ and $d_{16}$ from ${\cal
R}_{3}^{ee}$ and ${\cal R}_{4}^{ee}$ versus $M_{0}$; b) Value of the
condensates $d_{18}$ and $d_{20}$ from ${\cal R}_{5}^{ee}$ and ${\cal
R}_{6}^{ee}$ versus $M_{0}$

###  $\bullet~{}$$d_{4}$ and $d_{6}$ from the ratio of LSR

Once we obtain the previous values of the $d=8$ to $d=20$ condensates, we
inject them to the ratio ${\cal R}_{10}$ of LSR for re-extracting $d_{6}$ and
$d_{4}$. To strongly constrain the parameters, we continue to use a one-
parameter fit. Fixing again $\langle\alpha_{s}G^{2}\rangle$ as in Eq. 10, we
re-extract $d_{6}$ by including into the QCD expression of ${\cal R}_{10}$ the
contributions of condensates up to $d_{20}$. The analysis is shown in Fig. 4.

a) b)

Figure 4: a) Value of the condensate $d_{6}$ from ${\cal R}_{10}^{ee}$ versus
$M_{0}$; b) Value of the condensate $d_{4}$ from ${\cal R}_{10}^{ee}$ versus
$M_{0}$

The results are summarized in Table 1 and compared with the other estimates
quoted in Table 2. One can notice a good agreement with some published results
but the values obtained in this work are more accurate.

$\color[rgb]{.1,.5,0.3}\langle\alpha_{s}G^{2}\rangle$ | $\color[rgb]{.1,.5,0.3}-d_{6}$ | $\color[rgb]{.1,.5,0.3}d_{8}$ | -$\color[rgb]{.1,.5,0.3}d_{10}$ | -$\color[rgb]{.1,.5,0.3}d_{12}$ | $\color[rgb]{.1,.5,0.3}d_{14}$ | $\color[rgb]{.1,.5,0.3}-d_{16}$ | $\color[rgb]{.1,.5,0.3}-d_{18}$ | $\color[rgb]{.1,.5,0.3}d_{20}$
---|---|---|---|---|---|---|---|---
$7.8\pm 3.5$ | $23.6\pm 3.7$ | $18.2\pm 0.6$ | $6.1\pm 0.2$ | $18.4\pm 3.8$ | $59.2\pm 7.1$ | $118.3\pm 13.2$ | $202.0\pm 20.7$ | $421.3\pm 44.3$
Table 1:  Values of the QCD condensates of dimension $D$ in units of $10^{-2}$ GeVD from the present analysis. $\langle\alpha_{s}G^{2}\rangle$ | $-d_{6}$ | $d_{8}$ | $-d_{10}$ | $-d_{12}$ | Refs.
---|---|---|---|---|---
$0.67\pm 0.89$ | $15.2\pm 2.2$ | $22.3\pm 2.5$ |  |  |  ALEPH [31]
$5.34\pm 3.64$ | $14.2\pm 3.5$ | $21.3\pm 2.5$ |  |  |  OPAL [32]
$3.5^{+2.2}_{-3.8}$ | $19.7^{+11.8}_{-7.9}$ | $23.7^{+11.8}_{-15.8}$ | $11.8\pm 19.7$ | $7.9\pm 19.7$ |  (${d_{14,16}=0}$) Pich-Rodriguez [33]
Table 2:  Values of the QCD condensates from some other $\tau$-moments at
Fixed Order (FO) of the PT series.

###  $\bullet~{}$Comments on the results

$\diamond~{}$The value of $\langle\alpha_{s}G^{2}\rangle$ obtained here is
compatible with the one in Eq. 10 but less accurate. However, we consider this
result as an improvement of the ones from some other $\tau$-decay moments
given in Table 2 using FO perturbative series. It disfavours some negative
values obtained from CI perturbative series. This value is also consistent
with the one $(8\pm 4)\times 10^{-2}$ GeV4 obtained from a matching of the
short and long distances contributions to the Adler function at large $N_{c}$
[35].

$\diamond~{}$The value of $d_{6}$ is in fair agreement with the one in Eq. 21
from ratio of LSR obtained in SN23 [1] where the OPE has been truncated at
$D=8$. The inclusion of higher dimension condensates has increased slightly
$|d_{6}|$ by about 13.6%. The ratio
$\langle\alpha_{s}G^{2}\rangle/\rho\alpha_{s}\langle\bar{\psi}\psi\rangle^{2}=95.4$
is in the range of the ones obtained using different methods quoted in [36].

$\diamond~{}$The value of $d_{8}$ is larger than the one in Eq. 22 from ratio
of LSR. As the extraction of $d_{8}$ from LSR in SN23 has been done by
neglecting all higher dimension condensates included in the present analysis,
we interpret the value of $d_{8}$ obtained in SN23 as an effective condensate
including all high dimension ones. To check this argument, we evaluate the NP
contributions to ${\cal R}_{10}^{ee}$ at the typical optimization sum rule
scale $\tau\approx 1$ GeV-2. We neglect all higher dimension condensates and
find :

$\sum_{4}^{8}\delta^{NP}=5.1\,\times
10^{-2},~{}~{}~{}~{}\sum_{4}^{6}\delta^{NP}=-1.0\,\times 10^{-2}\rightarrow
d_{8}\approx 6.1\,\times 10^{-2}\,{\rm GeV}^{8}$ (23)

in perfect agreement with the fitted value in Eq. 22.

$\diamond~{}$Using the previous values of the condensates, we check the
convergence of the OPE. We obtain at $\tau\simeq 1$ GeV-2:

$\sum_{10}^{20}\delta^{NP}=-0.4\,\times 10^{-2},$ (24)

which is negligible. This feature justifies (a posteriori) the
phenomenological success of the SVZ sum rules by truncating the OPE up to
dimension $D=6,8$ condensates.

$\diamond~{}$From Table 1, we observe that there is no signal of a factorial
nor/and exponential growth of the size of the condensates up to $D=20$ where
the OPE is truncated. Similar observation can be made in the $V-A$ channel for
the OPE up to $D=18$ [37, 38].

###  $\bullet~{}$Comments on the ${\cal L}_{0}$ LSR moment

We have also tried to use the ${\cal L}_{0}$ LSR moment to fix the $D\leq 8$
dimension condensates 666Some other LSR moments with weights are discussed in
[37, 39].. Our analysis is unconclusive and confirms the observation done in
SN23. The same feature is observed for an attempt to extract $\alpha_{s}$.

##  8 $\alpha_{s}$ from the lowest BNP moment using $e^{+}e^{-}\to$ Hadrons
data

Armed with these improved values of the QCD condensates, we extract
$\alpha_{s}$ from the lowest BNP moment ${\cal R}^{ee}_{0}$. For convenience,
we rescale at $M_{\tau}$ the value of $\alpha_{s}$ obtained for different
values of $M_{0}$

###  $\bullet~{}$Standard SVZ expansion

###  $\diamond~{}$Fit of the data points

The analysis is shown in Fig. 5 (lower set of points) for (FO) and (CI)
perturbative series to order $\alpha_{s}^{4}$. We observe a stability for
$M_{0}$ between 1.5 to 1.7 GeV. A least square fit of the data points gives:

$\displaystyle\alpha_{s}(M_{\tau})$ $\displaystyle=$ $\displaystyle
3081(49)_{fit}~{}~{}~{}~{}~{}~{}{\rm(FO)}$ (25) $\displaystyle=$
$\displaystyle 3260(47)_{fit}~{}~{}~{}~{}~{}~{}{\rm(CI)}$

Figure 5: a) Value of $\alpha_{s}(M_{\tau})$ from the BNP lowest moment for
different values of $M_{0}$. The red (green) points correspond to (FO) [resp.
(CI)] PT series. The lower curves correspond to the SVZ expansion. The upper
ones to the case of zero value of the QCD condensates.

###  $\diamond~{}$Estimate of the $\alpha_{s}^{5}$ corrections

Noting that the coefficient of $\alpha_{s}$ for the Adler $D$-function grows
geometrically [47]:

$D(Q^{2})=\sum_{n}{a_{s}^{n}c_{n}}~{}:~{}c_{0}=c_{1}=1,~{}~{}c_{2}=1.656,~{}c_{3}=6.37,~{}c_{4}=49.09$
(26)

we deduce:

$c_{4}\approx
c_{3}^{2}~{}\longrightarrow~{}c_{5}\simeq\left({c_{3}}/{c_{2}}\right)c_{4}^{2}\simeq
228.$ (27)

This value of $c_{5}$ is comparable with some other estimates from FAC :
$\simeq 275$ [28], conformal mappings : $=255$ [48], Padé approximants :
$=(277\pm 51)$[49] and a linear behavaviour of FO coefficients : $\simeq 283$
[50]. Writing the QCD expression of the moment as:

${\cal R}^{ee}_{0}=\sum_{n}a_{s}^{n}(g_{n}+c_{n})~{},$ (28)

where $g_{n}$ comes from a RG-resummation [13, 51]. One obtains :

$\displaystyle g_{5}$ $\displaystyle=$
$\displaystyle-780~{}~{}~{}\longrightarrow~{}~{}~{}\Delta\alpha^{5}_{s}(M_{\tau})\simeq\pm
71\times 10^{-4}~{}~{}~{}{\rm(FO)},$ (29) $\displaystyle=$ $\displaystyle
0~{}~{}~{}~{}~{}~{}~{}~{}\longrightarrow~{}~{}~{}\Delta\alpha^{5}_{s}(M_{\tau})\simeq\pm
62\times 10^{-4}~{}~{}~{}~{}{\rm(CI)}\vspace*{-0.5cm}$

###  $\diamond~{}$Final value of $\alpha_{s}$ using the SVZ expansion

Adding the different corrections, we deduce:

$\displaystyle\alpha_{s}(M_{\tau})$ $\displaystyle=$ $\displaystyle
0.3081(49)_{fit}(71)_{\alpha_{s}^{5}}~{}~{}~{}\longrightarrow~{}~{}~{}\alpha_{s}(M_{Z})=0.1170(6)(3)_{evol}~{}~{}~{}~{}~{}~{}{\rm(FO)}$
(30) $\displaystyle=$ $\displaystyle
0.3260(47)_{fit}(62)_{\alpha_{s}^{5}}~{}~{}~{}\longrightarrow~{}~{}~{}\alpha_{s}(M_{Z})=0.1192(6)(3)_{evol}~{}~{}~{}~{}~{}~{}{\rm(CI)},$

where the sum of non-perturbative corrections at the $\tau$-mass is given in
Table 3. We notice that the increase of $|d_{6}|$ has slightly decreased the
value of $\alpha_{s}$ compared to the one in [1]. From the previous values, we
attempt to esimate the mean:

$\alpha_{s}(M_{\tau})|_{SVZ}=0.3194(52)(113)_{syst}~{}~{}~{}\longrightarrow~{}~{}~{}\alpha_{s}(M_{Z})|_{SVZ}=0.1184(15)(3)_{evol},$
(31)

where the systematics is the distance between the mean central value to the
FO/CI ones. This result can be compared with the PDG23 weighted average:

$\alpha_{s}(M_{Z})|_{PDG}=0.1178(5)$ (32)

###  $\bullet~{}$QCD models without power corrections

Here, we analyze a QCD model without power corrections. We show the result in
Fig 5 (upper set of curves). We obtain at the minimum $M_{0}\simeq 1.75$ of
the curves:

$\displaystyle\alpha_{s}(M_{\tau})$ $\displaystyle=$ $\displaystyle
0.3579(115)_{fit}(34)_{\alpha_{s}^{5}}~{}~{}~{}\longrightarrow~{}~{}~{}\alpha_{s}(M_{Z})=0.1228(12)(3)_{evol}~{}~{}~{}~{}~{}~{}{\rm(FO)}$
(33) $\displaystyle=$ $\displaystyle
0.3855(117)_{fit}(83)_{\alpha_{s}^{5}}~{}~{}~{}\longrightarrow~{}~{}~{}\alpha_{s}(M_{Z})=0.1255(13)(3)_{evol}~{}~{}~{}~{}~{}~{}{\rm(CI)},$

which corresponds to the mean:

$\alpha_{s}(M_{\tau})|_{PT}=0.3692(92)(163)_{syst}~{}~{}~{}\longrightarrow~{}~{}~{}\alpha_{s}(M_{Z})|_{PT}=0.1239(18)(3)_{evol},$
(34)

This result is about 4.6 $\sigma$ higher than the PD23 weighted average and
disfavours some QCD models without power corrections.

##  9 Comparison with $\alpha_{s}$ from $\tau$-decay data within the SVZ
expansion

Estimates of $\alpha_{s}$ using the vector component of $\tau$-decay data
within different $\tau$-decay moments are available in the literature [31, 32,
33, 34] as shown in Table 3. One cna notice that the central value of
$\alpha_{s}(M_{\tau})$ from $e^{+}e^{-}\to$ Hadrons data using the lowest BNP
moment is slightly lower than the one from $\tau$-decay data but agrees within
the errors. We also note that our value of $\delta^{(NP)}$ at the $\tau$-mass
agrees with the OPAL one but higher than the ones using ALEPH data.

| This work | ALEPH [31] | OPAL [32] | PR [33] | ALEPH [34] |
---|---|---|---|---|---|---
(F0) | 0.3081(86) | 0.3200(220) | 0.3230(160) | 0.3200(150) | – |
(CI) | 0.3260(78) | 0.3400(230) | 0.3470(220) | 0.3370(200) | 0.3460(110) |
$\delta^{(NP)}\,\times 10^{2}$ | 3.8(8) | 2.0(3) | 3.6(4) | 1.7(3) | 2.0(3) |
Table 3:  $\alpha_{s}(M_{\tau})$ within the SVZ-expansion from the vector
component of $\tau$-decay data using some high-$\tau$-moments. For a better
comparison with our result where the condensates have been estimated wihin
(FO), we quote the value of $\delta^{(NP)}$ corresponding to (FO) at
$M_{\tau}$.

##  10 The quest of $1/Q^{2}$ and tachyonic gluon mass beyond the SVZ
expansion

$\bullet~{}$The quest of $1/Q^{2}$

A priori, this question is iirrelevant as one does not expect to have a
dimension-two condensate $\langle A^{2}\rangle$ ($A_{\mu}$is the gluon field)
due to gauge invariance (see however [40]). However, this term is present in
some holographic models [41] and in a lattice calculation of the QCD potential
[42] and gluon condensates [43, 44]. It has been also proposed [46, 45] for a
phenomenological parametrization of UV renormalon as an alternative to the
large $\beta$-approximation. The large $\eta$-approximation is not yet fully
justified due to the non-observation of the factorial growth and alternate
signs of the calculated coefficients of the Adler $D$-function and some other
QCD observables known to order $\alpha_{s}^{4}$. Instead, these calculated
coefficents grow as a geometric sum [47].

$\bullet~{}$Tachyonic gluon mass $\lambda^{2}$

$\diamond~{}$In [52, 53], a systematic way to account for a such term is the
introduction of a tachyonic gluon mass squared $\lambda^{2}$. An estimate of
this mass using $e^{+}e^{-}\to$ Hadrons data and some other data lead to,[54,
36, 52, 55] :

$a_{s}\lambda^{2}=-(7\pm 3)\times 10^{-2}~{}{\rm GeV}^{2},$ (35)

which we re-estimate here from the ratio of LSR ${\cal R}_{10}^{ee}$. The
result is shown in Fig. 6 from which we deduce to order $\alpha_{s}^{4}$ at
the stability regions :

$a_{s}\lambda^{2}=-(6.7\pm 1.5)\times 10^{-2}~{}{\rm GeV}^{2}.$ (36)

Figure 6: The tachyonic gluon mass squared $a_{s}\lambda^{2}$ versus $M_{0}$.

in perfect agreement with the previous result.

$\diamond~{}$A such value of $\lambda^{2}$ resolves the hierarchy of the
$\pi,.\rho$ mesons and glueball sum rule scales [52]. However, a direct
inclusion of this effect to the $\tau$-decay moment is quite delicate. It has
been argued in Ref.[52] that this term is dual to the large order terms of the
PT series and decreases when more terms in the PT series are added. To check
this argument, we study the variation of $\lambda^{2}$ versus the number of
$\alpha_{s}$ term added in the PT series of ${\cal R}_{10}$. We show the
analysis in Fig. 7a where we, indeed, see that $|\lambda^{2}|$ decreases when
more $\alpha_{s}^{n}$ terms are added in the PT series even at low values of
$n$. A similar feature is observed at large orders in the calculation of the
gluon condensate $\langle\alpha_{s}G^{2}\rangle$ on the lattice [43, 44].

a) b)

Figure 7: a) $a_{s}\lambda^{2}$ for different truncations of the PT series; b)
$a_{s}\lambda^{2}$ for different values of the gluon condensate
$\langle\alpha_{s}G^{2}\rangle$: the blue point is the SVZ value of
$\langle\alpha_{s}G^{2}\rangle$, the red point is the value in Eq. 10, the
green point is the value quoted in Table 1.

We have also checked that the change of $d_{6}$ does not almost affect the
size of $\lambda^{2}$ while we show in Fig. 7b the effect of
$\langle\alpha_{s}G^{2}\rangle$. From the above results, we consider that the
$\lambda^{2}$ effects to the BNP moment is:

$\delta^{(2)}_{\lambda^{2}}\leq 4\%,$ (37)

of the PT lowest order one which is much less than the size of sum of the
$\alpha_{s}$ corrections.

$\diamond~{}$Assuming that our estimate of $\alpha_{s}^{5}$ correction to the
$\tau$-decay moment is a good approximation of the contribution of higher
order terms and using the duality property with $\lambda^{2}$ advocated by
Ref. [47], we do not add this $\lambda^{2}$ correction in ${\cal R}_{0}^{ee}$
to avoid a double counting.

##  11 Some other contributions beyond the SVZ expansion

###  $\bullet~{}$Small size nstantons

$\diamond~{}$Ref. [56] found that these corrections can be parametrized as:

$\delta^{inst}\simeq\left(\frac{3.64\,\Lambda}{M_{\tau}}\right)^{9}\left(\frac{\hat{m}_{u}\hat{m}_{d}\hat{m}_{s}}{M_{\tau}^{3}}\right)\simeq
1.6\,\times 10^{-8}$ (38)

where [57] $\hat{m}_{u}=3.1(2)$ MeV, $\hat{m}_{d}=6.1(4)$ MeV and
$\hat{m}_{s}=(114.3\pm 6.4)$ MeV are the renormalization group invariant light
quark masses and we use $\Lambda=0.34$ GeV. This contribution is completely
negligible.

$\diamond~{}$In Ref. [58], the instanton effect is introduced via the
constituent quark mass:

$m_{const}=m_{current}-\frac{2}{3}\pi^{2}\langle\bar{\psi}\psi\rangle\rho^{2},$
(39)

where $\rho$ is the instanton size. The authors conclude that at the
$\tau$-mass, the effect behaves as $1/M_{\tau}^{18}$ and is :

$\delta^{inst}\simeq 0.03\sim 0.05,$ (40)

which is comparable with the $D=6$ dimension condensates contribution.
However, one should notice that this effect depends crucially on the value of
the instanton size $\rho$ which is not under a good control.

$\diamond~{}$Ref. [59] has tested the validity of Ref. [58] prediction by
noting that Ref. [58] result contributes to the $V-A$ $\tau$-decay rate as :

$\delta_{V-A}^{inst}\simeq 0.09\sim 0.15,$ (41)

while the ALEPH data [31] only allows the range of values :

$\delta_{V-A}^{inst}\simeq(1\pm 3)\,\times 10^{-3}$ (42)

Using the (accepted) relation:

$\delta_{V+A}^{inst}\simeq\frac{\delta_{V-A}^{inst}}{20},$ (43)

one concludes that :

$\delta_{V}^{inst}\simeq(0.5\pm 1.5)\,\times 10^{-3},$ (44)

which is less than 5% of the $d_{6}$ contribution to the ${\cal
R}_{\tau}^{ee}$ decay rate. Moreover, the result of [58] is much larger than
the direct fit of the dimension $D=18$ term in the OPE quoted in Table 1.

###  $\bullet~{}$Duality violation

$\diamond~{}$This effect is maintained by Ref. [30]. It corresponds to an
additional QCD contribution to the spectral function introduced by hand as:

$\Delta{\rm Im}\,\Pi(t)|_{DV}\sim
e^{-(\delta+\gamma\,t)}\,\sin(\alpha+\beta\,t).$ (45)

above a certain threshold : $t_{c}\equiv\hat{s}_{0}\simeq 1.5$ GeV2 where
$\delta,~{}\gamma,~{}\alpha,~{}\beta$ are free unknown fitted parameters.

$\diamond~{}$The eventual presence of a such term in QCD [61] and in some toy-
model [62]) has been discussed at large $N_{c}$ with an infinite number of
narrow resonances in order to modelize the data in the Minkowski space (see
also [60] for QCD in two dimensions). This oscillating term in the Minkowski
space is expected to be dual to an asymptotic exponential behaviour of power
corrections in the Euclidian region [61].

Figure 8: Value of the ratio in Eq. 49.

$\diamond~{}$One can fix $\hat{s}_{0}\equiv t_{c}$ from the lowest FESR moment
[29]:

$F_{0}(t_{c})\equiv\int_{0}^{\hat{t}_{c}}d\,t\,R_{ee}^{I=1}=\frac{3}{2}\hat{s}_{0}\Big{[}1+a_{s}+3.891\,a_{s}^{2}+11.228\,a_{s}^{3}+\cdots\Big{]},$
(46)

which, for a $\rho$-meson dominance, gives [29, 35, 1]:

$\hat{s}_{0}=1.5~{}{\rm GeV}^{2},$ (47)

$\diamond~{}$Using the $e^{+}e^{-}\to$ Hadrons data, until
$\sqrt{t_{c}}\simeq$ 1.875 GeV, the duality condtion requires :

$\hat{s}_{0}\equiv t_{c}=(5.1\sim 5.6)~{}{\rm GeV}^{2},$ (48)

as shown in Fig. 8 from SN23 [1] where we plot the ratio :

$r^{fesr}\equiv\frac{F^{data}_{0}}{F^{qcd}_{0}},$ (49)

where $r^{fesr}=1$ corresponds to the duality region. This value does not
support the one 1.6 GeV2 obtained in [35] using the $e^{+}e^{-}\to$ Hadrons
data (notice that in Ref. [29] a value around 2.1 GeV2 has been obtained).
However, one may (intuitively) expect a value of $t_{c}$ above the data
region.

$\diamond~{}$One can inspect that with a such value of $\hat{s}_{0}$, the
addional contribution due to Duality Violation becomes unobservable when one
uses a complete data thanks to the exponential weight for its parametrization.

$\diamond~{}$A critical analysis of the effects of DV for values of
$\hat{s}_{0}$ around 1.5 GeV2 on the determination of the QCD condensates and
$\alpha_{s}$ using $\tau$-decay ALEPH data is explicitly discussed in Ref.
[63].

$\diamond~{}$One should also notice that power corrections in the Eucldian
space estimated previously in Table 1 up to $D=20$ do not show any exponential
behaviour which disfavours by a duality analytic continuation the presence of
this oscillating DV term in the Minkowski space.

##  12 Summary

In this paper, I have updated the results obtained in SN23 [1].

1. By using the new measurement of Ref. [2] of the positive charge muon anomaly, I found about $3\,\sigma$ deviation of the Standard Model predictions.

2. I have combinedd the uses of ratio of LSR and some high $\tau$-decay like moments to improve the estimates of the QCD condensates of dimension $D\leq 12$ dimensions and give new estimates of the ones $D=14$ to 20. Up to this order, I have not observed any exponential or/and factorial blow up of their contribution to the two-point function. I confirm the violation of four-quark factorization by a factor $\rho\simeq 6.89$ and the value of the gluon condensate $\langle\alpha_{s}G^{2}\rangle$ (though less accurate) found from heavy quark channels and some other sources.

3. Using these new values of the condensates,I extract from the lowest BNP moment the value of $\alpha_{s}(M_{\tau})$ given in Eq. 30 to 31.

4. I discuss, in Sections 10 and 11, some (eventual) contributions beyond the SVZ expansion ($1/Q^{2}$, instantons and Duality Violation). Such effects are expected to be relatively small (see e.g. [64]).

## Acknowledgements

It is a pleasure to thank Toni Pich and Valya Zakharov for some
correspondences.

## References

  * [1] S. Narison, Nucl. Phys. A 1039 (2023) 122744; S. Narison, QCD23 (8-14 july 2023, Montpellier-FR) (to appear in Nucl. Part. Phys. Proc. - Elsevier) : arXiv : 2309.05342 [hep-ph] (2023).
  * [2] D.P. Aguillard et al., Phys. Rev. Lett. 131 (2023) 16, 161802.
  * [3] M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, Nucl. Phys. B147 (1979) 385; ibid, Nucl. Phys. B147 (1979) 448.
  * [4] V.I. Zakharov, Int. J. Mod .Phys. A14, (1999) 4865.
  * [5] S. Narison, QCD spectral sum rules, World Sci. Lect. Notes Phys. 26 (1989) 1.
  * [6] S. Narison, QCD as a theory of hadrons, Cambridge Monogr. Part. Phys. Nucl. Phys. Cosmol. 17. (2004) 1-778 [hep-ph/0205006].
  * [7] S. Narison, Nucl. Part. Phys. Proc.312-317 (2021) 87; ibid, 258-259 (2015) 189.
  * [8] S. Narison and E. de Rafael, Phys. Lett. B 103, (1981) 57.
  * [9] J.S. Bell and R.A. Bertlmann, Nucl. Phys. B187, (1981) 285.
  * [10] For a recent review, see e.g. S. Narison, The Laplace Transform and its Applications, ed. V. Martinez-Lucaes, Nova Science Pub., New-York - 2024 (arXiv: 2309.00258 [hep-ph]).
  * [11] E. Braaten, Phys. Rev. Lett. 60 (1988) 606; E. Braaten, Phys. Rev. D39 (1989) 1458; S. Narison, A. Pich, Phys. Lett. B211 (1988) 183.
  * [12] E. Braaten, S. Narison, A. Pich, Nucl. Phys. B373 (1992) 581.
  * [13] F. Le Diberder, A. Pich, Phys. Lett. B286 (1992) 147; B289 (1992) 165.
  * [14] R.L. Workman et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2022 (2022) 083C01.
  * [15] CMD-3 Collaboration - F.V. Ignatov et al., 2302.08834 [hep-ex] (2023).
  * [16] G. Bennett et al., Phys. Rev. D73 (2006) 072003.
  * [17] B. Abi et al., Phys. Rev. Lett. 126 (2021) 141801; T. Albahri et al., Phys. Rev. D103 (2021) 072002.
  * [18] T. Aoyama et al., Phys. Rept. 887 (2020)1.
  * [19] M. Knecht, S. Narison, A. Rabemananjara, D. Rabetiarivony Phys. Lett. B787 (2018) 111.
  * [20] S. Narison, R. Tarrach, Phys. Lett. B125 (1983) 217; M. Jamin, M. Kremmer, Nucl. Phys. B277 (1986) 349.
  * [21] S. Narison, Int. J. Mod. Phys. A33 (2018) no. 10, 185004; Addendum: Int. J. Mod. Phys. A33 (2018) no.10, 1850045.
  * [22] S. Narison, Phys. Lett. B693 (2010) 559, erratum ibid, B705 (2011) 544; ibid, B706 (2012) 412; ibid, B707 (2012) 259.
  * [23] G. Launer, S. Narison, R. Tarrach, Z. Phys. C26 (1984) 433.
  * [24] M.S.A. Alam Khan (private communication).
  * [25] R. Shankar, Phys. Rev. D15 (1977) 755.
  * [26] E.G. Floratos, S. Narison, E. de Rafael Nucl. Phys. B155 (1979) 115.
  * [27] S.G. Gorishny, A.L. Kataev, S.A. Larin, Phys. Lett. B259 (1991) 144; L.R. Surguladze, M.A. Samuel, Phys. Rev. Lett. 66 (1991) 560.
  * [28] P. A. Baikov, K.G. Chetyrkin.J.H. Kühn, Phys.Rev.Lett. 101(2008) 012002.
  * [29] R.A. Bertlmann, G. Launer and E. de Rafael, Nucl. Phys. B250, (1985) 61; R.A. Bertlmann, C.A. Dominguez, M. Loewe, M. Perrottet and E. de Rafael, Z. Phys. C39 (1988) 231.
  * [30] D. Boito et al., Phys. Rev. D103 (2021) 3, 034028; Phys. Rev. D98 (2018) 074030.
  * [31] The ALEPH Collaboration-R. Barate et al., Eur. Phys. J. C4 (1998) 409.
  * [32] The OPAL Collaboration-K. Ackerstaff et al., Eur. Phys. J. C7 (1999) 571.
  * [33] A. Pich, A. Rodríguez-Sánchez, Phys.Rev. D94 (2016) 3, 034027; Mod.Phys.Lett. A31 (2016) 30, 1630032.
  * [34] M. Davier, A. Höcker, B. Malaescu, C. Z. Yuan, Z. Zhang, Eur. Phys. J. C74 (2014) 2803.
  * [35] S. Peris, M. Perrottet, E. de Rafael, JHEP 05 (1998) 011.
  * [36] S. Narison, Phys. Lett. B361 (1995) 121.
  * [37] S. Narison, Phys. Lett. B624 (2005) 223.
  * [38] A. Rodríguez-Sánchez, M. González-Alonso, A. Pich, Phys. Rev. D94 (2016) 014017.
  * [39] C. Ayala, G. Cvetic and D. Teca, J. Phys. G50 (2023) 045004.
  * [40] L. Stodolsky , P. van Baal, V.I. Zakharov, Phys. Lett. B552(2003) 214; F.V. Gubarev, L. Stodolsky, V.I. Zakharov, Phys. Rev. Lett. 86 (2001) 2220.
  * [41] O. Aharony et al., Phys. Rept. 323 (2000) 183; O. Andreev, V. I. Zakharov, Phys. Rev. D74 (2006) 025023; F. Jugeau, S. Narison, H. Ratsimbarison, Phys. Lett. B722 (2013) 111.
  * [42] S. Bali, Phys. Lett. B460 (1999) 170.
  * [43] P.E.L. Rakow, PoS LAT2005 (2006) 284.
  * [44] R. Horsley, P. E.L. Rakow, G. Schierholz, Nucl. Phys. Proc. Suppl. 106 (2002) 870; A. Di Giacomo, G.C. Rossi, Phys. Lett. B 100 (1981)481; G. Burgio et al., Phys. Lett. B422 (1998) 219.
  * [45] R. Akhoury, V. I. Zakharov, Phys. Lett. B438 (1998) 165; M.N. Chernodub et al., Phys. Lett. B475 (2000) 303.
  * [46] G Altarelli, P. Nason, G. Ridolfi, Z. Phys. C68 (1995) 257; G. Altarelli, Nucl. Phys. (Proc. Suppl.) B40 (1995)59.
  * [47] S. Narison, V.I. Zakharov, Phys. Lett. B679 (2009) 355.
  * [48] I. Caprini, Phys. Rev D100 (2019) 056019;
  * [49] D. Boito, P. Masjuan, F. Oliani, JHEP 08 (2018) 075.
  * [50] M. Beneke, M. Jamin JHEP 09 (2008)044.
  * [51] A. Kataev, V. V. Starshenko, Mod. Phys. Lett. A10 (1995) 235.
  * [52] K.G. Chetyrkin, S. Narison, V. I. Zakharov, Nucl. Phys. B550(1999) 353.
  * [53] For reviews, see e.g.: V.I. Zakharov, Nucl. Phys. Proc. Suppl. 164 (2007) 240; S. Narison, Nucl. Phys. Proc. Suppl. 164 (2007) 225.
  * [54] S. Narison, Phys. Lett. B300 (1993) 293.
  * [55] M. Kozhevnikova, A. Oganesian and O. Teryaev, EPJ Web of Conferences 204 (2019) 02005; M. Kozhevnikova, A. Oganesian, O. Teryaev, O. Solovtsova, Nonlin. Phenom. Complex Syst. 25 (2022) 3.
  * [56] P. Nason, M. Porrati, Nucl. Phys. B421 (1994) 518.
  * [57] S. Narison, Phys. Lett. B738 (2014) 346.
  * [58] I.I. Balitsky, M. Beneke, Vladimir M. Braun, Phys. Lett. B318 (1993) 371.
  * [59] V. Kartvelishvili, M. Margvelashvili, Phys. Lett. B 345 (1995) 161.
  * [60] G. ′t Hooft, Nucl. Phys. B75 (1974) 461.
  * [61] M. Shifman, Nucl. Phys. Proc. Suppl. B207-208 (2010) 298; ibid arXiv: hep-ph/0009131; B. Blok, M. Shifman, Da-Xin Zhang, Phys. Rev. D 57 (1998) 2691; Phys. Rev. D 59 (1999) 019901 (erratum);
  * [62] E. de Rafael, Nucl. Phys. Proc. Suppl. B207-208 (2010) 290.
  * [63] A. Pich, A. Rodríguez-Sánchez, JHEP 07 (2022) 145.
  * [64] S. Narison, Phys. Lett. B673 (2009) 30.