scheduling or maintenance. In such cases, either the hyperparameter values of
the models need to be decided in an automated fashion, or the neural design
needs to be generic enough to accommodate new Fog settings, with possibly
different number of nodes, workload characteristics and user demands.

Interpretability. Many state-of-the-art AI methods are being utilized today as
black-box models that give rise to high QoS in Fog systems, but do not have
any transparency on the process that led to the various resource management
level decisions made by an AI agent. This mainly entails explaining the main
reasons for choosing or not choosing certain management decisions and
exploring the unknown state spaces to ensure exhaustive coverage of the
decision space. For sensitive industrial segments, such as healthcare and
autonomous vehicles, it is crucial to expose the underlying patterns and
features in the decision-making process to gain credibility for the end-user.
Building such white-box or explainable models is an emerging field of research
for the use of trustworthy AI models. Many AI models, such as decision trees,
regression algorithms and rule-based systems allow interpretability, but are
not as accurate or scalable as deep-learning based counterparts.

###  6.4 Application Areas

Healthcare. With the rapid increase in connected devices in hospitals, such as
sensors, mobile phones and wearables, the amount of data generated and their
rate of generation is snowballing. This results in a massive increase in the
volume and variety of available health data, paving the way for the
development of more reliable and robust AI models in the areas of proactive
monitoring, disease prevention, and more in smart healthcare [305]. However,
this results in challenges related to ensuring data quality and security,
especially in the context of distributed EdgeAI, particularly in the case of
handling sensitive healthcare information like patient records. To overcome
these challenges, future research is focusing on the convergence of Blockchain
and AI, where Blockchain is used in solving data quality and integration
issues that enable AI to improve the accuracy of data analytics [306].
Moreover, low latency communication technologies like 5G/6G enable novel
technologies like Augmented Reality, Virtual Reality and Tactile Internet,
thus improving and expanding services such as robot-assisted surgery.

Next Generation Networking. Due to the high data rate, reliability, ultra-low
latency, and ultra-low energy consumption provided by 5G/6G, these wireless
communication technologies are identified as the key enablers of future IoT
applications. However, to support the ever-evolving service requirements of
the IoT services, 5G/6G technologies have to be able to observe environment
variations and dynamically self organize the network accordingly [307]. This
can be achieved through AI-empowered management and orchestration of cellular
resources within 5G/6G networks. With the advancements in Software Defined
Networking (SDN) and network function virtualization (NFV), 5G/6G technologies
introduce Network Slicing (NS) to support this. NS is a mechanism for
provisioning virtualized network resources intelligently based on the service
performance metrics [308]. Learning algorithms such as deep learning and
reinforcement learning can improve the dynamism of NS through prediction-based
proactive resource allocation in Edge or Cloud architectures with dynamic
slice creations for different applications [309].

Production and Supply Chain. In the coming age of IIoT, most industrial
pipelines are managed by smart devices. Such devices may utilize AI methods to
self-monitor and predict potential problems in the supply chain to optimize
the overall service efficiency. The COVID-19 pandemic is an example
demonstrating the importance of automation in logistics to avoid service
downtimes [310]. AI based forecasting approaches, such as recurrent neural
models and GANs can be used to predict stock shortage and proactively order
additional stocks to prevent shortage [310]. Similarly, large-scale ML models
can aid the development of smart-manufacturing technologies that utilize
several IoT and Fog devices to collaboratively monitor, control manufacturing
and production related equipment.

Smart Cities. Smart cities aim to utilize IoT to deliver services that can
enhance the living standards within cities. This includes a plethora of
application domains such as smart governance, smart energy, smart
transportation, and smart security [311]. Advancements in wireless
communications such as 5G/6G enable a massive amount of data to be transmitted
towards Edge nodes in real-time. This has resulted in the rise of novel
technologies like crowd-sensing and crowd-sourcing [312]. Distributed
collections and processing of such massive volumes of data demands future
research to focus on distributed and reliable AI, specifically in data-
sensitive applications at the Edge. At the same time, data security becomes
crucial in future smart city services, with the widespread use of crowd-
sensing and crowd-sourcing for data collection. Moreover, ultra-low latency
communication provided future radio access networks to support services
related to hazard avoidance and safety [313], which requires AI models with
higher reliability, accuracy and lower latency.

###  6.5 Methods

Self-supervised AI. The self-supervised learning technique enables learning
with unlabeled data by solving pretext tasks [314]. In contrast to this,
supervised learning depends on the availability of labeled data. Even though a
massive amount of data gets generated by the sensors, the lack of annotated
data poses an obstacle for using supervised learning. This is specifically
applicable in scenarios where fault-detection or workload scheduling is
required for previously unseen workload or device characteristics in Edge or
Cloud platforms. As generating expert-labeled fault labels or optimal
scheduling decisions is infeasible for large-scale systems, self-supervised
learning offers a possible solution for such scenarios. This approach is
capable of generating a more generalizable model by removing the heavy
dependency on labeled data by automatically generated data annotations,
possibly using a co-simulator [314]. Moreover, self-supervised learning has
the potential to achieve higher reliability due to its robustness to
adversarial examples, label corruption, and input corruptions [315].

Model Driven RL. As Fog environments and service demands are dynamic;
algorithms should have the capability to adapt accordingly. IoT applications
(i.e., healthcare, smart cities, etc.) and their enabling telecommunication
technologies (i.e., 5G, 6G) benefit from RL-based intelligence due to higher
adaptability of RL techniques and their ability to learn without prior
knowledge [186]. However, exploration errors, long learning time and
distributed learning within resource constraint devices in Edge environments
are some of the challenges in utilizing RL techniques within resource-
constrained and distributed Fog environments. A canonical case is the
development of RL based methods for deploying large-scale application
workflows on constrained Edge or Cloud clusters. Addressing these challenges
in future research is vital for the RL-based approaches to reach their full
potential within EdgeAI scenarios. In an attempt to address these challenges,
EdgeAI research is exploring advanced RL approaches such as model-based RL
[316] and co-simulated RL [15].

Analog AI. The current implementation of AI is targeted for digital systems
where the values are stored in a binary format. Herein, the major challenge
posed by digital implementations of DNNs is the linear dependence of memory
footprint with the number of parameters of the neural model. Upcoming analog
memory-based chips present new ways to perform the same operations, but with
orders of magnitude lower amount of memory requirement, computational load and
energy consumed [317]. This is particularly useful in memory-constrained Edge
and MEC devices where sophisticated DNNs need to be executed in AI on Fog
setting or decentralized resource management is required in AI for Fog
settings. However, there are some drawbacks in the loss of precision in
computation across layers within a DNN. The tradeoff offered by such DNN
implementations is similar to model pruning and splitting, but with possibly
more extreme energy/compute benefits. This direction has been explored to a
limited extent and requires further investigation and software development to
efficiently harness the potential of analog hardware accelerators.

Decentralized Modeling. The success of distributed EdgeAI, by utilizing layer
and semantic splitting strategies for AI deployment (discussed in Section 4),
shows some promise in other domains of scheduling and maintenance. For
scheduling applications and maintaining Fog systems at scale, it is possible
to decentralize the training and inference procedures of the resource
management level AI applications across multiple broker nodes. The
decentralized fashion of resource management has a two-fold benefit. Firstly,
there is no single point of failure in the system as the management steps are
run in multiple broker nodes. Secondly, it allows the distribution of resource
management load across multiple computing devices, facilitating the
scalability of the model.

AI Driven Simulations. A major advantage of co-simulators in Fog systems is
the ability to generate new data points for tuning AI models and resolving
issues like exposure bias and data saturation [318]. However, another
important benefit of co-simulators is the ability to run multiple simulations
concurrently and pick the best resource management decisions, allowing
interpretable decision-making. This is due to the ability of co-simulators to
generate a complete execution trace, possible for several future states of the
systems and allowing developers or end-users to visualize the long-term
effects of various decisions. They are able to do this much faster than
executing the decisions on a physical infrastructure and waiting for several
minutes to reflect changes in QoS. This is primarily due to the discrete
event-driven execution style of modern simulators. This is applicable for all
three types of resource management decisions, i.e., deployment, scheduling and
maintenance. Simulations can indicate changes in QoS scores for each model
compression or splitting type, application placement or fault-remediation
steps like preemptive migrations. These signals can facilitate decision
making. However, co-simulators are mere approximations of the entire Fog
system and typically fail to map the entire complexity of real infrastructure.
However, the success of deep surrogate models hints us to build simulators
with possibly millions of parameters and utilize DNNs to estimate optimal
parameter values, such that our simulators resemble the real systems as
closely as possible. An increased number of parameters could, in principle,
give a higher representative capacity to our simulators, now being able to map
complex real-life workloads and device characteristics. Thus, AI-driven
simulators could help improve system performance with the added bonus of
interpretable decision making.

AI Driven Co-Design. Currently, almost all resource management solutions for
Fog systems solve only a specific problem from the three domains of
deployment, scheduling or maintenance. However, for holistic performance
enhancement, it is crucial to develop AI models that can concurrently take
decisions across multiple facets of the management of Fog resources to
efficiently exploit the synergy across these decision domains. Research in AI-
based augmentation of Fog systems may benefit from other efforts in system co-
design [319] to improve upon the existing management solutions. This is
crucial in Fog systems particularly due to the constraints certain decision
types impose on other resource management control knobs. For instance,
provisioning decision constrains the devices on which incoming tasks can be
scheduled on or the active tasks may be migrated to. There is a need to build
end-to-end AI models that rely on multi-modal data and can take multiple
decision types simultaneously for data privacy and improved system
performance.

##  7 Conclusions

This work conducts an extensive literature review of the methods concerned
with AI-based augmentation of Fog systems. We discuss diverse state-of-the-art
techniques for Fog resource management, specifically for optimal AI
deployment, workload scheduling and system maintenance. There is significant
overlap across different decision domains in terms of the used AI models. This
overlap suggests the importance of certain design decisions over others and
hints at the possible gaps of current research. We have highlighted the
importance of a more comprehensive research style that not only considers
specific aspects of resource management but distills historical knowledge
gathered from the myriad of AI-based decision-making methods to develop well
informed AI models and eclectic management solutions. The various advances in
the field of computing need to be considered in tandem to bolster AI research
and build holistic AI-based methods for emerging application areas, future
technologies and next-generation users.

## Acknowledgments

This work is supported by the the President’s Ph.D. Scholarship at Imperial
College London and Australian Research Council Discovery Project. We thank
Shikhar Tuli, Zifeng Niu, Runan Wang, Matthew Sheldon and William Plumb for
helpful discussions.

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Shreshth Tuli is a President’s Ph.D. Scholar at the Department of Computing,
Imperial College London, UK. Prior to this he was an undergraduate student at
the Department of Computer Science and Engineering at Indian Institute of
Technology - Delhi, India. He has worked as a visiting research fellow at the
CLOUDS Laboratory, School of Computing and Information Systems, the University
of Melbourne, Australia. His research interests include Fog Computing and Deep
Learning. For further information, visit https://shreshthtuli.github.io/.

Fatemeh Mirhakimi is a current Master of Research student at Western Sydney
University, Australia. She received B.S and M.S. in Computer Engineering from
the Azad University of Arak, respectively. She has conducted a research
project about power-consuming management in hybrid vehicles based on fuzzy-
genetic methods. Her research interests include cloud and edge computing, the
internet of things, and reliability and fault tolerance.

Samodha Pallewatta is a Ph.D. student at the Cloud Computing and Distributed
Systems (CLOUDS) Laboratory, Department of Computing and Information Systems,
The University of Melbourne, Australia. Her research interests encompass
Fog/Edge Computing, Internet of Things (IoT), Microservice architecture and
Distributed Systems. She is one of the contributors of the iFogSim simulator,
used extensively for resource management research in Fog/Edge computing.

Syed Zawad received the B.Sc. degree in computer science and engineering from
BRAC University, Dhaka, Bangladesh. He is currently pursuing the Ph.D. degree
with the Department of Computer Science and Engineering, University of Nevada,
Reno, NV, USA. He has interned as a Researcher with Baidu, Sunnyvale, CA, USA.
He also has three years of work experience as a Software Engineer for Web
applications. His research area of interest is in high performance computing,
deep learning, federated learning, and neural architecture search.

Giuliano Casale joined the Department of Computing at Imperial College London
in 2010, where he is currently a Reader. Previously, he worked as a research
scientist and consultant in the capacity planning industry. He teaches and
does research in performance engineering and cloud computing, topics on which
he has published more than 100 refereed papers. His research work has received
multiple awards, recently the best paper award at ACM SIGMETRICS. He serves on
the editorial boards of IEEE TNSM and ACM TOMPECS and as current chair of ACM
SIGMETRICS.

Bahman Javadi is an Associate Professor in Networking and Cloud Computing at
Western Sydney University, Australia. Prior to this appointment, he was a
Research Fellow at the University of Melbourne and a Postdoctoral Fellow at
the INRIA Rhone-Alpes, France. His research interests include Cloud computing,
Edge computing, performance evaluation of large-scale distributed computing
systems, and reliability and fault tolerance. He is a Senior Member of ACM,
Senior Member of IEEE and Senior Fellow of Advance HE of UK. His website is:
https://staff.cdms.westernsydney.edu.au/~bjavadi/.

Feng Yan is an Assistant Professor of Computer Science and Engineering at
University of Nevada, Reno and the director of the Intelligent Data and
Systems Lab. He received M.S. and Ph.D. degrees in Computer Science from the
College of William and Mary and worked at Microsoft Research and HP Labs. His
research bridges the fields of big data, machine learning, and systems. He is
a recipient of the Best Student Paper Award of IEEE CLOUD 2018, the Best Paper
Award of CLOUD 2019, and the Best Student Paper Award of ITNG 2021, as well as
the NSF CAREER Award, the NSF CRII Award, and the Regents’ Rising Researcher
Award.

Rajkumar Buyya is a Redmond Barry Distinguished Professor and Director of the
Cloud Computing and Distributed Systems (CLOUDS) Laboratory at the University
of Melbourne, Australia. He has authored over 625 publications and seven text
books including ”Mastering Cloud Computing” published by McGraw Hill, China
Machine Press, and Morgan Kaufmann for Indian, Chinese and international
markets respectively. He is one of the highly cited authors in computer
science and software engineering worldwide (h-index=154, g-index=322, 124,800+
citations).

Nicholas R. Jennings is the Vice-Chancellor and President of Loughborough
University. He is an internationally-recognised authority in the areas of AI,
autonomous systems, cyber-security and agent-based computing. He is a member
of the UK government’s AI Council, the governing body of the Engineering and
Physical Sciences Research Council, and chair of the Royal Academy of
Engineering’s Policy Committee. Before Loughborough, he was the Vice-Provost
for Research and Enterprise and Professor of Artificial Intelligence at
Imperial College London, the UK’s first Regius Professor of Computer Science
(a post bestowed by the monarch to recognise exceptionally high quality
research) and the UK Government’s first Chief Scientific Advisor for National
Security.


# Why mean pT is interesting

Michal Praszalowicz

###### Abstract

We discuss energy dependence of mean $p_{\rm T}$ correlation with $N_{\rm ch}$
basing on general features of high energy collisions such as saturation and
geometrical scaling. We use Color Glass Condensate calculation of an effective
interaction radius that scales as a third root of multiplicity, and then
saturates. With this model input we construct scaling variable for $\langle
p_{\rm T}\rangle(N_{\rm ch})$ at different energies both for pp and pPb
collisions, and show that recent ALICE data indeed does exhibit this scaling
property. We discuss energy dependence of the interaction radius and argue
that since the radius cannot grow too large, a universal behavior of $\langle
p_{\rm T}\rangle$ for large multiplicities is expected.

###### Keywords:

Saturation, geometrical scaling, interaction radius

###### :

13.85.Hd, 12.38.Aw

Correlations are always interesting since they are sensitive to the fine
details of interactions. For example standard PYTHIA Monte Carlo fails in the
case of $\left\langle p_{\rm{T}}\right\rangle$ correlation with $N_{\rm ch}$
(see e.g. Refs. Abelev:2013ala ; Abelev:2013bla ), while it does describe well
one particle spectra and total multiplicities. New effect called color
recombination (see e.g. Gustafson:2008zz ) has to be added to PYTHIA to take
care of rather strong rise of $\left\langle p_{\rm{T}}\right\rangle$ with
$N_{\rm ch}$. On the other hand newer MC generator EPOS Pierog:2013ria  that
has saturation effects built in does not require special tuning to describe
$\left\langle p_{\rm{T}}\right\rangle$ in function of $N_{\rm ch}$
Abelev:2013ala ; Abelev:2013bla . In this talk that is based on Refs.
McLerran:2013oju ; McLerran:2014apa  (and where a complete list of references
can be found) we shall study consequences of saturation and geometrical
scaling (GS) for mean transverse momenta at the LHC. Here the energy must be
really large since the character of $\left\langle p_{\rm{T}}\right\rangle$
dependence on $N_{\rm ch}$ changes dramatically (see Fig. 1 in Ref.
Armesto:2008zz ) from the ISR energies (where it decreases or stays constant)
to the LHC energies (where it rises). Theoretical interest in $\left\langle
p_{\rm{T}}\right\rangle$ correlation with $N_{\rm ch}$ goes back to the paper
by L. van Hove VanHove:1982vk  where he pointed out that qualitative change is
expected in the presence of the QCD phase transition. Today this motivation is
perhaps less important since the QCD deconfining transition is believed to be
a soft crossover.

From the saturation point of view high energy central rapidity production of
moderate $p_{\rm T}$ particles can be viewed as a result of a collision of two
gluonic clouds characterized by one saturation scale that depends on gluon
longitudinal momenta $x_{1}\sim x_{2}$ denoted in the following as $x$:

$Q_{\rm{s}}^{2}(x)=Q_{0}^{2}\left(\frac{x_{0}}{x}\right)^{\lambda}$ (1)

where $Q_{0}$ is an arbitrary scale parameter for which we take 1 GeV$/c$, and
for $x_{0}$ we take $10^{-3}$.

An immediate consequence of (1) is GS of particle (or strictly speaking gluon)
spectra McLerran:2010ex :

$\frac{dN_{\rm g}}{dyd^{2}p_{\bf{T}}}=S_{\bot}\mathcal{F}(\tau)$ (2)

where

$\tau=\frac{p_{\bf{T}}^{2}}{Q_{\bf{s}}^{2}(x)}$ (3)

is the scaling variable. Here $S_{\bot}$ is a transverse area which will be
specified later. Logarithmic corrections due the running of the strong
coupling constant are neglected in Eq. (2).

In order to integrate (2) over $d^{2}p_{\rm T}$ we have to change integration
variable to $d\tau$ which gives

$\frac{dN_{\rm g}}{dy}=A\,S_{\bot}\bar{Q}_{\mathrm{s}}^{2}(W)$ (4)

where $A$ is an energy independent integral of the universal function
$\mathcal{F}(\tau)$ over $d\tau$. Here

$\bar{Q}_{\mathrm{s}}(W)=Q_{0}\left(\frac{W}{Q_{0}}\right)^{\lambda/(2+\lambda)}$
(5)

is an average saturation scale which gives (provided that $S_{\bot}$ is energy
independent) power like growth of multiplicity with the scattering energy $W$
– a fact well confirmed by recent LHC data on multiparticle production.

An immediate consequence of Eq. (4) is that McLerran:2014apa

$\left\langle p_{\rm{T}}\right\rangle\sim\bar{Q}_{\mathrm{s}}(W)\,,$ (6)

which means that $\left\langle p_{\rm{T}}\right\rangle$ rises with energy as
$W^{\lambda/(2+\lambda)}$. In order to find numerical value of $\lambda$ we
have performed simple analysis McLerran:2014apa  based on GS hypothesis,
requiring that ratios

$\mathcal{R}_{ik}(\tau)=\frac{dN(W_{i},\tau)}{dyd^{2}p_{\rm{T}}}/\frac{dN(W_{k},\tau)}{dyd^{2}p_{\rm{T}}}$
(7)

are as close to unity as possible over the widest range of $\tau$, with the
result $\lambda\approx 0.22$. As a consequence mean $p_{\rm T}$ rises with
energy as a power of energy $W$, which, as shown in Fig. 1, is in perfect
agreement with high energy data Khachatryan:2010us .

Figure 1: Mean transverse momentum as a function of energy with the power-like
fit $0.227\times W^{0.099}$ (solid line). Compilation of data taken from Ref.
Khachatryan:2010us . The dashed line corresponds to the CMS logarithmic fit
aimed at describing also low energy data: $0.413-0.0171\,\ln
s+0.00143\,\ln^{2}s$, with $W^{2}=s$. (Figure from Ref. McLerran:2014apa .)

Replacing $\bar{Q}_{\mathrm{s}}(W)$ in Eq. (1) by (4) at fixed $W=W_{0}$ and
adding a constant piece, which takes into account nonperturbative effects and
contributions from the particle masses, we arrive at McLerran:2013oju :

$\left.\left\langle
p_{\rm{T}}\right\rangle\right|_{W_{0}}=\alpha+\beta\,\sqrt{\frac{\pi\,N_{\rm{ch}}}{\left.S_{\bot}(\sqrt[3]{\gamma
N_{\rm{ch}}})\right|_{W_{0}}}}\,.$ (8)

Here $\alpha$ and $\beta$ are constants that do not depend on energy and
$\gamma$ is a parameter that, using parton-hadron duality assumption, relates
number of gluons to the number of charged particles.

Three important comments concerning Eq. (8) are in order. First, transverse
size entering Eqs. (2) and (4) in a situation where we fix number of particles
corresponds to the overlap between two hadrons colliding at given impact
parameter $b$. As such it does depend on the multiplicity itself. Second, in
Eq. (8) we have assumed explicit relation between $S_{\bot}$ and $N_{g}=\gamma
N_{\rm ch}$ that has been calculated in the CGC effective theory Bzdak:2013zma
, which essentially says that particle multiplicity is proportional to the
interaction volume. Explicitly: it rises with volume linearly up to some
maximal $V_{\rm max}$ and then at fixed volume it rises further on only due to
fluctuations. This allows to define interaction radius $R\sim S_{\bot}^{1/2}$
whose explicit dependence on $N_{g}$ can be found in Refs. McLerran:2013oju ;
McLerran:2014apa . And finally, the CGC calculations have been performed for
fixed scattering energy $W_{0}$, so in principle in order to find mean $p_{\rm
T}$ at a different energy one should recalculate $S_{\bot}$ in Eq. (8). In the
following we shall argue that this is essentially not necessary.

Indeed, although after Eq. (5) we have argued that $S_{\bot}$ does not depend
on energy, in a situation when we change energy at fixed multiplicity, we have
to vary $S_{\bot}$ accordingly, so that Eq. (4) remains satisfied. As a result
at higher energies the same number of particles comes from the smaller
overlap, _i.e._ from larger $b$. Taking into account this induced energy
dependence of the interaction radius

$\left.R\right|_{W}=\left(\frac{W_{0}}{W}\right)^{\lambda/(2+\lambda)}\left.R\right|_{W_{0}}$
(9)

we arrive at

$\left.\left\langle
p_{\rm{T}}\right\rangle\right|_{W}={\alpha}+{\beta}\,\frac{\sqrt{N_{\rm{ch}}}}{\left.R(\sqrt[3]{{\gamma}N_{\rm{ch}}})\right|_{W}}={\alpha}+{\beta}\,\left(\frac{W}{W_{0}}\right)^{\lambda/(2+\lambda)}\frac{\sqrt{N_{\rm{ch}}}}{\left.R(\sqrt[3]{{\gamma}N_{\rm{ch}}})\right|_{W_{0}}}$
(10)

where $W_{0}$ corresponds to the energy for which the interaction radius has
been computed in some explicit model. In the CGC theory that we use here
Bzdak:2013zma  $W_{0}=7$ TeV for pp and $5.02$ TeV for pPb.

It follows from Eq. (10) that there exists a universal scaling variable

$\left(W/W_{0}\right)^{\lambda/(2+\lambda)}\sqrt{N_{\rm{ch}}}/{\left.R(\sqrt[3]{{\gamma}N_{\rm{ch}}})\right|_{W_{0}}}$

for which all mean $p_{\rm T}$ data should fall on one curve. This scaling
property is clearly visible in Fig. 2.

Figure 2: Mean $\left\langle p_{\rm{T}}\right\rangle$ in p-p collisions at 7
TeV (full black circles), 2.76 TeV (full red down-triangles), 0.9 TeV (full
blue up-triangles) and in p-Pb collisions at 5.02 TeV (full brown diamonds)
plotted in terms of scaling variable
$(W/W_{0})^{\lambda/(2+\lambda)}\sqrt{N_{\mathrm{ch}}/S_{\bot}}$. For p-p
$W_{0}=7$ TeV and for p-Pb $W_{0}=5.02$ TeV. (Figure from Ref.
McLerran:2014apa .)

To obtain plots in Fig. 2 we have used parametrization of $R(\sqrt[3]{N_{g}})$
at $W_{0}=7$ TeV from Ref. McLerran:2013oju . It is however clear from Eq. (9)
that for energies smaller than $W_{0}$, but still large, the interaction
radius can become exceedingly large, especially for large multiplicities. We
expect therefore that for large multiplicities interaction radii at different
energies should tend to a common limiting value McLerran:2014apa . The details
of this saturation can be extracted from the data. To this end we have used
first equation in formula (10) to extract shape of $R(N_{\rm ch})$ (assuming
$\beta=1$). In order to fix the only remaining parameter $\alpha$ we have
required scaling law (9) to be valid over the largest possible range of
$N_{\rm ch}$ with the result $\alpha=0.285$. The interaction radii for pp
collisions at different energies obtained in this way are plotted in Fig. 3.
We see that indeed they show a kind of turnover for large multiplicities – a
sign of possible convergence to the scaling value at $N_{\rm
ch}\rightarrow\infty$. This may be verified by future measurements of mean
$p_{\rm T}$. Let us stress, however, that from this point of view,
measurements of $\left\langle p_{\rm{T}}\right\rangle$ for very large $N_{\rm
ch}$ at smaller energies would be even more interesting. On the same plot in
Fig. 3 we also show the CGC calculation of $R(\sqrt[3]{\gamma N_{\rm
ch})}/\beta$ with $\gamma=1.138$ as extracted from the data in Ref.
McLerran:2014apa  (dashed line). We see quite good agreement with the model
independent result at 7 TeV represented by (black) full circles.

Figure 3: Interaction radii for pp collisions at 0.9 TeV (blue) up-triangles,
2.76 TeV (red) down-triangles and 7 TeV (black) full circles extracted from
the mean $p_{\rm T}$ data by means on the first equation of formula (10) with
$\beta=1$ and $\alpha=0.285$ together with experimental errors. Dashed line
corresponds to the CGC parameterization of Ref. McLerran:2014apa .

The analysis presented here shows that an intriguing insight into the
interaction geometry in multiple particle production can be gained by studying
mean $p_{\rm T}$ correlation with $N_{\rm ch}$. A good description of recent
ALICE data is obtained when one assumes that number of particles produced at
given impact parameter is proportional to the overlap volume of the
interacting hadrons rather than to the transverse area alone. This means that
the evolution time in the longitudinal direction is larger for larger
multiplicities. Similar mechanism has been proposed within the framework of
the multipomeron exchange model Armesto:2008zz  where one postulates that the
string tension of the color system stretched after the collision gets larger
for larger multiplicities. In our analysis based on saturation and geometrical
scaling fixed number of particles is produced from smaller volume when the
collision energy increases. This means that when multiplicity is fixed
collisions are more peripheral at high energy and more central at lower
energies. However, since there is a natural cutoff on the size of the
interaction volume corresponding to the size of interacting hadrons, one
expects that for very large multiplicities interaction radii tend to the fixed
limit independently of energy. This trend seems to be seen already in the
existing LHC data. Future data will shed more light on this interesting issue.

Many thanks to the organizers of the International Workshop on Diffraction in
High-Energy Physics 2014, Primos̆ten, Croatia, for the stimulating and
interesting meeting. The research of M.P. has been supported by the Polish NCN
grant 2011/01/B/ST2/00492.

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