# Impact of Non-Gaussian Beam Profiles in the Performance of Hadron Colliders

S. Papadopoulou  European Organization for Nuclear Research (CERN), CH-1211
Geneva 23, Switzerland  Department of Physics, University of Crete, P.O. Box
2208, GR-71003 Heraklion, Greece  Institute of Theoretical and Computational
Physics (ITCP), GR-71003 Heraklion, Greece;  F. Antoniou  European
Organization for Nuclear Research (CERN), CH-1211 Geneva 23, Switzerland  T.
Argyropoulos  European Organization for Nuclear Research (CERN), CH-1211
Geneva 23, Switzerland  M. Hostettler  European Organization for Nuclear
Research (CERN), CH-1211 Geneva 23, Switzerland  Y. Papaphilippou  European
Organization for Nuclear Research (CERN), CH-1211 Geneva 23, Switzerland  G.
Trad  European Organization for Nuclear Research (CERN), CH-1211 Geneva 23,
Switzerland

###### Abstract

At the Large Hadron Collider (LHC), the interplay between a series of effects,
including intrabeam scattering (IBS), synchrotron radiation, longitudinal beam
manipulations, two beam effects (beam-beam, e-cloud) and machine non-
linearities, can change the population of the core and tails and lead to non-
Gaussian beam distributions, at different periods during the beam cycle. By
employing generalised distribution functions, it can be demonstrated that the
modified non-Gaussian beam profiles have an impact in the beam emittance
evolution by itself and thereby to the collider luminosity. This paper focuses
on the estimation of beam distribution modification and the resulting rms beam
size due to the combined effect of IBS and synchrotron radiation by employing
a Monte-Carlo simulation code which is able to track 3D generalised particle
distributions (SIRE). The code is first benchmarked over classical semi-
analytical IBS theories and then compared with measurements from the LHC at
injection and collision energies, including projections for the High-
Luminosity LHC (HL-LHC) high brightness regime. The impact of the distribution
shape on the evolution of the bunch characteristics and machine performance is
finally addressed.

##  I Introduction

The performance of a high-energy hadron collider such as the LHC is heavily
based on the preservation of the injected emittances, under the influence of
several degrading mechanisms. In this respect, an emittance evolution model
was constructed including the effects of intrabeam scattering (IBS),
synchrotron radiation, elastic scattering and luminosity burn-off (while at
collision) Antoniou  _et al._ (2017). This simple model is based on semi-
analytical approaches which assume Gaussian beam distributions, in particular
for IBS. The bunch characteristics evolution predicted by this model revealed
discrepancies, as compared to the measurements, translated to differences in
the luminosity predicted by the model as compared to the experimental
estimations Antoniou  _et al._ (2017); Papadopoulou  _et al._ (2019);
Papadopoulou . One of the possible reasons for these differences could be
attributed to the fact that the bunch profiles appear to be non-Gaussian both
at injection and collision energies, i.e. 450 GeV and 6.5 TeV respectively.
The aim of this study is to quantify the impact of the beam distribution shape
on the emittance and luminosity evolution of hadron colliders. In order to
illustrate this, and employing generalised distribution functions, the
luminosity of non-Gaussian beams is determined in a closed form. The
generalisation of the luminosity estimate for arbitrary distributions does not
only permit its comparison to the usual Gaussian beam estimate but also the
extension of classical results for the impact of non-Gaussian beam
distributions to the luminosity (see Section II). This motivates the
investigation of the emittance evolution beyond the classical analytical
formulas for modelling IBS, which are based on 3D Gaussian beam assumptions
Bjorken and Mtingwa (1982). In this respect, a Monte Carlo multi-particle
simulation code for IBS and Radiation Effects (SIRE) Vivoli and Martini
(2010a); Martini  _et al._ (2016), is employed and compared to LHC data from
Run2. A brief description of the code, and its benchmarking with the existing
IBS analytical approaches and simulations for lepton rings is presented in
Section III. Extending previous benchmarking studies for the LHC with respect
to IBS theories Vivoli and Martini (2010b), a detailed comparison of the
Bjorken-Mtingwa (B-M) IBS theoretical model with the SIRE code for both
injection and collision energies is presented for the nominal LHC using the
Batch Compression Merging and Splitting (BCMS) Garoby (1985); Bejar Alonso
_et al._ (2015) beam and the high luminosity LHC (HL-LHC) Apollinari  _et al._
(2017) beam parameters (Section IV). Finally, measured data from the LHC
corresponding to non-Gaussian longitudinal beam profiles are compared with the
expectations of the rms emittance evolution given by the theoretical B-M
analytical formalism Bjorken and Mtingwa (1982) and the SIRE code (Section V).

##  II Motivation- Impact of non-Gaussian distributions on Luminosity

The performance of a collider is determined by the luminosity which, for two
beams colliding head-on, is given by Herr and Muratori (2003):

${\cal{L}}=2N_{1}N_{2}N_{b}f_{rev}{\int\limits}{\int\limits}{\int\limits}{\int_{-\infty}}^{\infty}\rho_{1x}(x)\rho_{1y}(y)\rho_{1s}(s-s_{0})\rho_{2x}(x)\rho_{2y}(y)\rho_{2s}(s+s_{0})~{}dxdydsds_{0}\>,$
(1)

with $N_{1,2}$ representing the number of particles per bunch for each beam,
$N_{b}$ the total number of colliding bunches, $f_{rev}$ the revolution
frequency and $\rho$ the beam density distribution functions for each plane
and beam.

Based on Eq. (1), assuming Gaussian beams that collide head-on, the luminosity
is expressed as Herr and Muratori (2003):

${\cal{L}}^{G}=\frac{N_{1}N_{2}N_{b}f_{rev}}{4\pi\sigma_{x}^{G}\sigma_{y}^{G}}\>.$
(2)

In order to achieve high luminosity, high intensity bunches and small beam
sizes are required. The horizontal and vertical beam sizes of two colliding
Gaussian bunches are given by:

$\sigma_{x}^{G}=\sqrt{\sigma_{1x}^{2}+\sigma_{2x}^{2}}~{}~{}~{}\mathrm{and}~{}~{}~{}\sigma_{y}^{G}=\sqrt{\sigma_{1y}^{2}+\sigma_{2y}^{2}}\>,$
(3)

where ($\sigma_{1x}$, $\sigma_{1y}$) and ($\sigma_{2x}$, $\sigma_{2y}$) are
the transverse rms beam sizes of beam 1 and beam 2, respectively.

Based on the transverse and longitudinal bunch profile measurements, it has
been observed that the particle distributions in the LHC, both at collision
and injection energies, appear to have shapes that differ from the ones of a
normal distribution Fitterer  _et al._ (2017); Papadopoulou  _et al._ (2017);
Timko  _et al._ (2016). At the LHC injection energy, the emittance evolution
is dominated by the IBS effect, both in the horizontal and in the longitudinal
plane, while no effect is expected in the vertical plane Kuhn (2013-03-12)
where dispersion is minor. From Run 2 data, it is observed that in many cases
the transverse bunch profiles appear to be non-Gaussian during the whole
injection plateau Fitterer  _et al._ (2017). At the LHC collision energy, the
IBS effect is weaker, while synchrotron radiation damping becomes more
pronounced. The bunch profiles at collisions appear to have non-Gaussian
tails, as well. In fact, during the energy ramp, the bunches that are blown up
longitudinally in order to avoid instabilities due to the loss of Landau
damping Baudrenghien and Mastoridis (2013), arrive at the start of collisions
with a clearly non-Gaussian shape Timko  _et al._ (2016).

By assuming that a particle distribution is Gaussian when this is not the
case, not only the rms beam size may be underestimated or overestimated, but
also its impact on performance parameters, such as the luminosity. Therefore,
it is important to use appropriate fitting functions (or some type of
interpolation algorithm) on the beam profile in order to properly address this
discrepancy. A generalized Gaussian function, called the $q$-Gaussian Curado
and Tsallis (1992), can be employed for fitting more accurately bunch profiles
with shapes that differ from the ones of a normal distribution (see Appendix A
for the properties of this distribution function). The parameter $q$ describes
the weight of the tails as compared to the core, ranging from light tailed
ones for $q<1$ (including the square distribution for $q\rightarrow-\infty$)
and extending to a heavy tailed ones for $q>1$, passing through the Gaussian
distributions in the limit of $q\rightarrow 1$. This distribution is actually
a stationary solution of a generalised Fokker-Plank equation which can cover a
full spectrum of statistical behaviors of dynamical systems, from sub to
super-diffusion Levy-type processes Tsallis  _et al._ (2003).

Figure 1: Parameterization of the luminosity variation, normalised to the
corresponding Gaussian luminosity value $\cal{L}$G, with the weight of the
transverse distribution tails given by the parameters $q_{x,y}$, for a
constant q-Gaussian rms beam size.

In view of quantifying the impact of non-Gaussian distributions, the
luminosity is estimated through Eq. (1) by using the specified probability
density functions. Assuming that the two beams are identical and that they
collide head-on, the luminosity for $q$-Gaussian distribution functions in the
transverse plane is given by:

${\cal{L}}^{qG}=\frac{N_{1}N_{2}N_{b}f_{rev}}{4\pi\sigma_{x}^{qG}\sigma_{y}^{qG}}{\cal{I}}_{x}^{qG}{\cal{I}}_{y}^{qG}\>,$
(4)

for $\sigma_{x,y}^{qG}$ being the rms beam sizes (see Appendix A) in the
transverse plane, for both beams. The factors ${\cal{I}}_{x,y}^{qG}$ which
depend on the parameter $q$ in the respective planes and the details of the
calculation are presented in Appendix B, together with the validation of the
luminosity estimation for $q$-Gaussian distributions (shown in Fig. 22). By
comparing this equation to the standard luminosity formula for Gaussian beams
with identical rms sizes, the significance of the tail population contribution
on luminosity can be established and parameterised through $q$.

Figure 2: The $q$-Gaussian density distribution function for a light tailed
(blue, $q<1$), a Gaussian (black, $q=1$) and a heavy tailed (red, $q>1$) bunch
profile, having identical $q$-Gaussian rms beam sizes.

This is illustrated in Figure 1 where the luminosity variation normalised to
the corresponding one for Gaussian beams ($\cal{L}$G) is parameterized with
the parameter $q$ of the $q$-Gaussian fitting function, characterising the
weight of the tails in the transverse plane, for fixed $q$-Gaussian rms beam
sizes in all planes, assuming head on collisions (i.e. no dependence of the
luminosity on the longitudinal beam size, see Appendix B). The bunch profiles
corresponding to a light tailed ($q<1$), a Gaussian ($q=1$) and a heavy tailed
($q>1$) distribution, having identical beam sizes, are plotted in Fig. 2. As
$q$ (and $\beta^{qG}$) vary for fixed $q$-Gaussian rms beam sizes (based on
Eq. (15) in Appendix A), the luminosity varies as well with respect to the one
estimated for purely Gaussian beams. Practically, if the tails of a
distribution differ by $10\%$ compared to the ones of a Gaussian distribution
(i.e. $q=0.9$ or $q=1.1$), the luminosity value can be overestimated or
underestimated by $5\%$. It is then clear that, for two beams colliding head-
on, the shape of the transverse distributions has a significant impact to the
estimated luminosity, in particular for the LHC experiments which target a
$\sim 2~{}\%$ accuracy in their estimates Eftymiopoulos .

Figure 3: In order to estimate the divergence of the luminosity for non-
Gaussian distributions from the one for Gaussian densities, the
${\cal{L}}_{w}<w^{2}>^{1/2}$ (see Eq. 5) is plotted versus the weight of the
tails $q$, for light tailed (left) and the heavy tailed (right) distributions,
i.e. $q<1$ and $q>1$, respectively. The points corresponding to a parabola and
a Gaussian distribution, firstly presented in Hereward (1969); Potter (1992),
are plotted.

The impact of non-Gaussian distributions on the luminosity was firstly
discussed by Hereward Hereward (1969). In particular, the luminosity integrals
were calculated for several distributions as the rectangular and the parabolic
which correspond to a $q$-Gaussian with $q\rightarrow-\infty$ and $q=0$,
respectively. Assuming that the two beams are identical and that they collide
head-on, these distributions were used as examples to estimate the discrepancy
from the luminosity for Gaussian densities. This discrepancy is calculated
after solving the ${\cal{L}}_{w}$ that is the integral over the density
functions in one plane, being identical for both beams, as:

${\cal{L}}_{w}<w^{2}>^{1/2}=\sigma_{w}\int^{\infty}_{-\infty}\rho(w)^{2}dw\>,$
(5)

with $w=x,y$ and for $\sigma_{w}$ being the transverse rms beam size, since
these solutions correspond to the transverse plane. It was found that Hereward
(1969); Potter (1992):

${\cal{L}}_{w}<w^{2}>^{1/2}=\begin{array}[]{cc}\Bigg{\\{}&\begin{array}[]{l}0.2887\>,~{}\mathrm{for~{}a~{}rectangular~{}distribution}\\\
0.2683\>,~{}\mathrm{for~{}a~{}parabolic~{}distribution}{}\\\
\frac{1}{2\sqrt{\pi}}=0.2821\>,~{}\mathrm{for~{}a~{}Gaussian~{}distribution}{}\end{array}\end{array}\>$
(6)

In fact, this approach already identified the existence of a minimum for a
light tailed parabolic distribution, which becomes obvious by employing the
$q$-Gaussian, as observed in Fig. 3, where ${\cal{L}}_{w}<w^{2}>^{1/2}$ is
plotted versus $q$ for $q<1$, i.e. light tails (left), and $q>1$, i.e. heavy
tails (right). The results for $q$-Gaussian distributions (grey curves) are in
perfect agreement with the case studies discussed in Hereward (1969). This is
also true for a rectangular distribution which corresponds to a $q$-Gaussian
with $q\rightarrow-\infty$ and is beyond the range of the left plot of Fig. 3.
For heavy tailed distributions, there is no upper limit for the constant of
Eq. (6), as already inferred by Hereward Hereward (1969). In Fig. 3 (right)
the case of a heavy tailed distribution with $q=1.65$ is denoted by a red
square. The extreme case of $q\rightarrow 5/3$ corresponds to a $q$-Gaussian
whose rms size goes to infinity (i.e. Levy distributions, see Appendix A).

The sensitivity of the luminosity on the distribution as generalised by
employing the $q$-Gaussian function justifies the need of carefully studying
the evolution of distributions in hadron colliders. For the LHC, a luminosity
model was constructed Antoniou  _et al._ (2017). The evolution of the
emittance includes the effects of IBS, synchrotron radiation, elastic
scattering, betatron coupling, noise and burn-off. Although this model has a
relative agreement with respect to the measured luminosities by the
experiments (i.e. ATLAS Aad  _et al._ (2012) and CMS Chatrchyan  _et al._
(2012)), there is still some room for improvement Papadopoulou . Indeed, the
emittance evolution for IBS was based on the module of MAD-X Antoniou and
Zimmermann (2012) following the B-M theory which assumes Gaussian beam
distributions. The extension of this to non-Gaussian distributions as observed
in the LHC may shed light to the origin of the remaining discrepancy between
the model and the measurements.

##  III Intrabeam scattering theories, observations and simulations

One of the statistical processes causing a spreading of particles in phase
space or a continuous increase of beam emittance is the small angle multiple
Coulomb scattering, called Intrabeam scattering (IBS), which plays an
important role in $e^{+}/e^{-}$ damping rings, high intensity/low energy light
sources Bane (2012) and high intensity hadron C. Bhat and Colestock (1999) and
ion Fischer  _et al._ (2001) circular machines. The IBS theory for
accelerators was firstly introduced by Piwinski Piwinski (1974a) and extended
by Martini Martini (1984), establishing a formulation called the standard
Piwinski method. Later, Bjorken and Mtingwa (BM) Bjorken and Mtingwa (1982)
used a different approach to describe the effect, taking into account the
strong focusing effect. The Modified Piwinski method Bane  _et al._ (2002)
that includes the strong focusing effect, was developed by Bane. Some
approximations of these theories are the high energy one by Bane Bane  _et
al._ (2002) and the completely integrated modified Piwinski Kubo  _et al._
(2005). A different approach developed by Lebedev for hadron beams is based on
a Boltzmann type integro-differential equation and includes betatron coupling
Lebedev (2005).

The analytical models that describe the IBS effect Bjorken and Mtingwa (1982);
Piwinski (1974b) assume Gaussian beam distributions. The stationary solution
of the Fokker-Planck equation is a particle distribution that is Gaussian in
the phase space. However, taking into account the effects of IBS, radiation
damping and quantum excitation but also other diffusive mechanisms Jowett
(1986), there is no evidence that the distribution remains Gaussian.
Therefore, it is important to develop analytical formulas and simulation tools
that calculate the interplay between these effects for any distribution.

For the Relativistic Heavy Ion Collider (RHIC), the IBS growth rates were
calculated and benchmarked with experimental data using the distribution
function evolution (based on the Fokker-Planck equations), extending the usual
approach of employing a conventional Gaussian-like distribution Wei  _et al._
(2005). In this respect, IBS growth rates were calculated for a bi-gaussian
distribution, which was interesting for studying the possibility of using
electron cooling in RHIC Parzen (2004). Later, a model which is suitable for
IBS calculations for arbitrary distribution functions and its comparison to
experimental data was presented in Fedotov  _et al._ (2010). The IBS effect
was also studied for high-brightness electron linac beams which appear to be
non-Gaussian, especially in the longitudinal plane Xiao and Borland (2010).
For low-emittance high-intensity electron storage rings, the interplay between
intrabeam scattering and wake-field forces is discussed in Venturini (2001).
In particular, the calculation of the IBS growth rates and the estimation of
the emittance growth is discussed in detail in Lebedev and Shiltsev (2014),
where the importance of knowing the formation of the distribution tails is
underlined, referring also to the use of Monte-Carlo methods.

In order to simulate the impact of a distribution shape on the emittance
evolution when considering IBS and radiation effects the SIRE (Software for
IBS and Radiation Effects) Vivoli and Martini (2010a); Martini  _et al._
(2016) code was developed by Vivoli and Martini at CERN. A similar Monte Carlo
approach (IBStrack Biagini  _et al._ (2011)) was implemented also in the
collective effects simulation tool CMAD Pivi (2007); Sonnad  _et al._ (2012).
Both algorithms have as their basis MOCAC (MOnte CArlo Code), a Monte-Carlo
code developed by Zenkevich and collaborators Zenkevich  _et al._ (2005,
2006), which calculates the IBS effect for arbitrary distributions, by
representing the beam through a large number of macro-particles occupying
points in the 6-dimensional phase space. Being an extension of MOCAC, SIRE was
developed to simulate the evolution of the beam particle distributions, taking
into account the effects of IBS, synchrotron radiation and quantum excitation.
For the evolution of the LHC bunch parameters and the shape of the bunch
profiles presented in this paper, the simulations are performed using SIRE. As
the physics and implementation details of the code are extensively explained
in Martini  _et al._ (2016), we only briefly summaries some of its key
features.

###  III.1 Software for IBS and Radiations Effects (SIRE)

For the IBS calculations, SIRE Vivoli and Martini (2010a); Martini  _et al._
(2016) uses the classical Rutherford cross section, which is closer to the
Piwinski formalism Piwinski (1974b). It uses as input the optics functions at
different locations of the lattice in order to determine the trajectories of
the particles in phase space. Instead of using the 6 coordinates for position
and momentum, the two Courant-Snyder and longitudinal invariants and the 3
phases (betatron and synchrotron) are used. For a linear ring, the 3
invariants are conserved between points around the lattice and can only be
changed by the effects of IBS, synchrotron radiation and quantum excitation,
while the phases are chosen randomly at each given point of the lattice. The
time steps for which the IBS and radiation effects are called should be
specified such that they are larger than the revolution time and smaller than
the damping/growth times. Dividing the total time by the time steps shows how
frequently the IBS, synchrotron radiation and quantum excitation routines are
called. The quantum excitation is implemented by adding to the 6 coordinates
of each macro-particle a Gaussian random noise component.

Depending on the elapsed time, the synchrotron radiation damping acts on the
invariants of the macro-particles as an exponential decrement. The routine
introduced for this reason is called after the calculation of the IBS effect
at each iteration. Using small iteration time steps $dt$ (which are much
smaller than the damping times and for which the emittances change
adiabatically), the evolution of the transverse emittance and energy spread
due to the effects of IBS and synchrotron radiation can be obtained by solving
the coupled differential equations:

$\displaystyle\dfrac{d\epsilon_{x,y}}{dt}$
$\displaystyle=\dfrac{-2(\epsilon_{x,y}-\epsilon_{{x,y}0})}{\tau_{x,y}}+\dfrac{2\epsilon_{x,y}}{T_{x,y}}\>,$
(7) $\displaystyle\dfrac{d\sigma_{p}}{dt}$
$\displaystyle=\dfrac{-(\sigma_{p}-\sigma_{{p}0})}{\tau_{p}}+\dfrac{\sigma_{p}}{T_{p}}\>,$

with $\epsilon_{{x,y}0}$ and $\sigma_{{p}0}$ being the zero current (without
the effect of IBS) equilibrium transverse emittances and energy spread,
respectively. The $\tau_{x,y}$, $\tau_{p}$ are the synchrotron radiation
damping times and $T_{x,y}$, $T_{p}$ the IBS growth times.

The algorithm SIRE uses to calculate IBS is similar to that implemented in
MOCAC, where the beam is represented by a large number of macro-particles
occupying points in the 6-dimensional phase space. The default distribution
defined in SIRE by using a random number generator, is the Gaussian and is
given in action angle variables. In order to introduce a different
distribution, the proper random deviates should be generated or the action
angle variables of all macroparticles for the desired distribution should be
provided. After specifying the total beam population and the number of macro-
particles, the initial distribution can be tracked. The particle distribution
in all planes can be saved as often as requested during the simulation time.
Currently, the output file gives the evolution of the emittance in all planes
for the specified time steps.

After providing the beam distribution and the optics along a lattice, the beam
is geometrically divided according to the specified number of cells for each
plane. The macro-particles are assigned to each cell according to their
geometrical position. For each lattice point defined in the optics file, the 3
phases of each macro-particle are randomly chosen and the position and
momentum of the macro-particles are calculated. Based on the classical
Rutherford cross section, intra-beam collisions between pairs of macro-
particles are calculated in each cell. The momentum of particles is changed
due to scattering. The number of macro-particles and cells, i.e. the number of
collisions each macro-particle experiences, is chosen so as to give accurate
results for a reasonable computational time. The scattering angles for each
collision are determined. In order to get the mean value of the emittance and
momentum deviation changes for all particles, we have to integrate over the
phase space volume of betatron coordinates, momentum deviations and azimuthal
positions of the interacting particles. The beam distribution is then updated
based on the new invariants of the macro-particles. For a specified number of
time steps which practically shows how frequently the IBS and synchrotron
radiation routines are called, the beam distribution is updated and the rms
beam emittances are recomputed, giving finally as output the emittance
evolution in time. The simulation proceeds to the next lattice point and
continues until the end time is reached.

A lattice compression technique named “lattice recurrences” has been
implemented to speed up the calculations Vivoli and Martini (2010a). Since the
increase of the invariants due to IBS is linear to the first order in the
traveling time along an element, the IBS kicks with optics functions differing
less than a specified precision value are considered equivalent. For the
corresponding group of elements, the IBS effect is evaluated only once,
resulting to a reduction of the computational time.

###  III.2 The logarithmic Coulomb factor

The IBS growth times have a complicated dependence on the beam properties, due
to the coupling of the three planes through dispersion. Some of these
properties are the bunch charge and energy, the beam optics and the
equilibrium rms horizontal, vertical emittances and the energy spread. The IBS
growth times depend also on a logarithmic Coulomb factor which is used to
include the contribution of events having a very large and very small impact
parameter. The typical way of computing a log factor overemphasises the
importance of the very small impact parameter events, for which the tails of
the steady-state bunch distributions are non-Gaussian. In the high energy
approximation by Bane Bane (2012), in order to describe the size of the core
of the bunch, the Coulomb log factor is calculated as was first proposed by
Raubenheimer Raubenheimer (1991), i.e. based on a boundary between the
contribution to the core and the tails. In B-M, Bane and CIMP methods, the
Coulomb factor is defined as the ratio of the maximum $r_{\mathrm{max}}$ to
the minimum $r_{\mathrm{min}}$ impact parameter in the collision of two
particles in the bunch, that is
$(log)\equiv\ln(r_{\mathrm{max}}/r_{\mathrm{min}})$. For typical flat beams,
the $r_{\mathrm{max}}$ is taken to be equal to the vertical beam size
$\sigma_{y}$, while $r_{\mathrm{min}}$ is taken to be
$r_{\mathrm{min}}=r_{0}\beta_{x}/(\gamma^{2}\epsilon_{x})$, with $r_{0}$ being
the classical particle radius. Then, the Coulomb factor can be written as:

$(log)=\mathrm{ln}\left(\frac{\gamma^{2}\epsilon_{x}\sqrt{\beta_{y}\epsilon_{y}}}{r_{0}\beta_{x}}\right)\>.$
(8)

The formalism by Piwinski always seems to underestimate the IBS effect with
respect to the other theoretical models. What diversifies Piwinski’s method,
is the different definition of the Coulomb factor. In that approach, the
maximum impact parameter which is typically taken as the vertical beam size
appears. In the high energy limit, with $d$ being the maximum impact
parameter, the Coulomb $(log)$ for Piwinski can be written as Bane (2002):

$(log)=\mathrm{ln}\left(\frac{d\sigma_{x}^{2}}{4r_{0}\alpha^{2}}\right)\>,$
(9)

where $a=\frac{\sigma_{x}}{\gamma}\sqrt{\frac{\beta_{x}}{\epsilon_{x}}}$.
Comparing the $(log)$ factors of Eq. (8) and (9), we find that condition
$d=4\sigma_{y}$ in order for the Piwinski approach agreeing with the other
models. SIRE uses the “binary collision map” algorithm, conceived by
Zenkevich, which allows to reduce the effects of the continuous time dynamical
IBS system to a discrete time map in momentum space. For the binary collision
events, the maximum impact parameter is taken as the beam height.

###  III.3 Benchmarking of IBS theoretical model with SIRE

The performance of hadron machines is limited by the IBS effect causing
emittance growth. For lepton machines such as future linear collider Damping
Rings, new generation light sources and B-factories, the IBS effect can also
be predominant. It is thus important to study the IBS theories in the presence
of synchrotron radiation and quantum excitation and benchmark the existing
theoretical models and tracking codes with experimental data. In this way, the
codes limitations can be identified so that to apply the necessary
improvements in order to get better predictions for a machine’s operation.

A benchmarking of the IBS theoretical models with Monte-Carlo codes is
presented in Antoniou (2012) for lepton rings. The comparison between
different theoretical models and SIRE is discussed for the Compact Linear
Collider (CLIC) damping ring (DR) Papaphilippou  _et al._ (2012), having
ultra-low emittances which are strongly dominated by IBS, in the presence of
synchrotron radiation and quantum excitation. Results of this comparison are
presented in Fig. 4, for one turn of the DR lattice. Due to the fact that in
SIRE the generation of the distribution is based on a random number generator,
the tracking simulations were performed several times, resulting in the one
standard deviation error-bars (plotted in green). The classical formalism of
Piwinski (red colored) and SIRE are in perfect agreement, as was expected,
since SIRE uses the classical Rutherford cross section which is closer to the
Piwinski formalism. The Bjorken-Mtingwa (black colored) and Bane (magenta
colored) formalisms overestimate the effect compared to Piwinski method mainly
to a mismatch of the Coulomb factor used in the different approaches (see Eq.
(9)).

Figure 4: One turn comparison for the horizontal (left) and vertical (middle)
emittance and energy spread (right) between the tracking code SIRE and the
theoretical models Bjorken-Mtingwa (B-M), Piwinski and Bane for the CLIC DR
lattice.

The IBS theoretical models have been studied in detail and benchmarked with
experimental data also for hadron beams over the years C. Bhat and Colestock
(1999); Fischer  _et al._ (2001). A comparison of the LHC data with
simulations performed with SIRE is discussed in Vivoli and Martini (2010b);
Papadopoulou  _et al._ (2017). In this paper, the SIRE simulations, as well as
the benchmarking with the B-M formalism and experimental data are discussed in
detail for the LHC and the High Luminosity LHC (HL-LHC) Apollinari  _et al._
(2017).

##  IV Simulations performed with SIRE for the LHC

In order to understand the evolution of the bunch characteristics, based on
the bunch profile observations, it is important to study the interplay between
IBS and radiation effects (synchrotron radiation and quantum excitation)
during the full LHC cycle. This is done using SIRE for two cases which are
important for the current and future machine performance; the nominal BCMS
Garoby (1985); Bejar Alonso  _et al._ (2015) and the HL-LHC Apollinari  _et
al._ (2017) parameters. Despite some blow-up in the LHC during the ramp, it is
observed that the BCMS beam gives an increase in peak luminosity of around
20%. The HL-LHC is the major LHC upgrade aiming to increase integrated
luminosity by at least a factor of 10 compared to the nominal LHC design value
(from 300 to 3000 fb-1). In order to achieve that, the bunch population needs
to be increased and the transverse beam size at the collision points has to be
lowered.

Apart from the IBS and synchrotron radiation which are the dominant effects
for the emittance evolution in the LHC, a combination of other diffusion
mechanisms, like the beam-beam effect, electron-cloud, noise (due to the power
converters, the transverse damper, the crab cavities, etc.), and other non-
linearities cause emittance growth and/or particle losses Lamont and Johnson
(2014). Despite the fact that these mechanisms are not included in SIRE, it is
possible to add empirically (i.e. based on observations) their contribution.
Practically, there is the option of adding or complementing the variation of
the bunch parameters in time. In fact, the simulation studies presented in
this paper for the LHC are focused on the 3$\sigma$ range of the particle
distributions and therefore, mechanisms which concern the far tail regime are
not taken into account as they are more important for the lifetime of the beam
then on distribution evolution .

###  IV.1 Reduced lattice

As mentioned earlier, one of the inputs required by SIRE are the optical
functions along the ring. As the LHC is a very long accelerator of about 27
km, with a very large number of elements in the sequence (more than 11000),
SIRE requires an extremely long computational time to track the distribution
for all the elements along the ring. Aiming to reduce the computational time,
a study was first performed in order to identify the optimal minimum number of
critical IBS kicks around the lattice, without affecting the overall effect.
The IBS growth rates were firstly calculated for the full optics of the LHC,
using the IBS module of the Methodical Accelerator Design code (MAD-X)
Antoniou and Zimmermann (2012) which is based on the Bjorken-Mtingwa
formalism.

Figure 5: The IBS growth rates along the LHC in all three planes: longitudinal
(green), the horizontal (blue) and the vertical (magenta). The IBS kicks that
are noted with red dashed lines, represent the positions of the 92 elements
that compose the reduced lattice.

Figure 5 shows the IBS growth rates in the longitudinal (green), the
horizontal (blue) and the vertical (magenta) plane. Taking into account the
strong IBS kicks along the ring, various lattices with a reduced number of
elements, including the case of the smooth lattice approximation, were tested.
Then, using the IBS module of MAD-X, the emittance evolution was calculated
for several sets of beam parameters to assure that the choice of the elements
is valid for a wide range of regimes, for which the IBS impact may be weaker
or stronger. Finally, the optimal lattice chosen consists of only 92 elements
whose positions are denoted by red dashed lines in Fig. 5.

Figure 6: The growth of the horizontal emittance (left) and bunch length
(right) due to IBS, as computed by MAD-X with the Bjorken-Mtingwa formalism,
in a time period of 30 min at injection, when considering the full lattice
(black solid line), the reduced lattice (red dashed line) and the smooth
lattice approximation- mean optics (magenta dotted line).

Figure 6 shows the emittance (left) and the bunch length (right) growth during
30 min at injection energy, for the nominal BCMS beams, with initial emittance
and $4\sigma$ bunch length that are respectively $\epsilon_{x0}=1.5$ $\mu
m$$rad$ and $\sigma_{s0}=1$ ns, having a bunch population of $1.2\times
10^{11}$ protons. The black solid line refers to the case of the full lattice,
while the red dashed one to the reduced lattice with the 92 elements. The
magenta dotted line corresponds to the case of the smooth lattice
approximation for which a lattice with a unique element, having the optics
that represent in the best possible way the mean optics of the full lattice,
is considered. The agreement of the full and the reduced lattice is very good
in all planes. On the other hand, by using the smooth lattice approximation
the IBS effect is slightly underestimated, in particular, in the longitudinal
plane. In the next, since the results for the reduced and the full lattice
agree also in SIRE, the reduced lattice is used as an input for the simulation
code. After choosing the optimal number of cells and macro-particles, the
computational time in the case of the reduced lattice is almost 20 times
shorter than the one of the full LHC lattice.

###  IV.2 Convergence studies

For a specified set of input beam parameters, various scans should be
performed for different combinations of number of macro-particles and cells in
order to find the optimal values which provide a fast tracking and at the same
time, guarantee that the scattering process leads to accurate results. In
these terms, in order to avoid having a very small number of macro-particles
per cell, the total number of cells is calculated based on the optimal minimum
number of macro-particles per cell. For $n_{x}$, $n_{y}$, $n_{z}$ being the
number of cells in the horizontal, vertical and longitudinal plane,
respectively, it is assumed that in the transverse plane there is a
correlation between the number of cells ratio and the beam sizes ratio,
meaning that $n_{x}/n_{y}$=$\sigma_{x}$/$\sigma_{y}$. Therefore, for $n_{mp}$
being the total number of macro-particles, the number of macro-particles per
cell is:

$n_{mp}/cell=\dfrac{n_{mp}}{n_{x}n_{y}n_{z}}=\dfrac{n_{mp}}{n_{x}^{2}(\dfrac{\sigma_{y}}{\sigma_{x}})n_{z}}\>.$
(10)

A scanning of the total number of cells is performed for an example set of
beam parameters to be used as an input for tracking. Based on Eq. (10), by
keeping the total number of macro-particles constant, the different
combinations of cell numbers determines the number of macro-particles per
cell. Figure 7 (left) shows the dependence of the emittance variation (ratio
of final versus initial value) in the horizontal (blue) and longitudinal
(green) plane on the number of macro-particles per cell, for a specified time
duration. The value of the number of macro-particles per cell after which the
variation of the emittances in both planes remains constant is chosen as the
optimal minimum value. After specifying this value, a scanning is performed
for a fixed number of macro-particles, in order to choose the number of cells
to be used, firstly in the longitudinal plane.

Figure 7: The dependence of the horizontal (blue) and longitudinal (green)
emittance variation on the number of macro-particles per cell (left), on the
number of cells in the longitudinal plane (middle) and in the horizontal plane
(right), for a specific time period.

Then, the number of cells in the horizontal plane can be calculated using Eq.
(10), when knowing the ratio of the beam sizes in the transverse plane 111Here
it is assumed that the ratio of the transverse beam sizes is initially
$\sigma_{x}$/$\sigma_{y}=1$.. In Fig. 7, the dependence of the emittances
variation is plotted versus the number of cells in the longitudinal (middle)
and horizontal (right) plane, for a specific time duration. It can be noticed
that the variation of the emittances remains constant after a certain number
of cells in the longitudinal and horizontal plane that is, for the example set
of beam parameters, 350 and 13 cells, respectively. Finally, the number of
cells in the vertical plane can be calculated by
$n_{y}=n_{x}/(\sigma_{x}/\sigma_{y})$.

###  IV.3 Benchmarking of SIRE with the B-M IBS theoretical model

SIRE has the advantage to accept any type of distribution as an input. If
requested, it also gives as output the distribution at any stage of the
tracking. In order to benchmark the code with the analytical formulation of
B-M for the LHC, a Gaussian distribution was tracked for two sets of bunch
parameters which are summarized in Table 1 for both the injection ($450$ GeV)
and collision energy ($6.5$ TeV). The first case corresponds to the nominal
BCMS Garoby (1985); Bejar Alonso  _et al._ (2015) LHC beams, having a
significantly lower transverse beam size with respect to the nominal
production scheme. The second case corresponds to the HL-LHC Apollinari  _et
al._ (2017); Métral  _et al._ (2018) parameters, for which the bunch
population is very high. The input optics functions used for tracking
correspond to the ones of the aforementioned reduced lattice.

Table 1: Nominal (BCMS) and HL-LHC parameters, at injection energy ($450$ GeV) and at collision energy ($6.5$ TeV). IBS growths | Injection energy | Collision energy
---|---|---
LHC (BCMS) | HL-LHC | LHC (BCMS) | HL-LHC
$\epsilon_{x,y}$ [$\mu$m.rad] | 1.5 | 2.0 | 2.5 | 2.5
4$\sigma$ bunch length [ns] | 1.0 | 1.2 | 1.0 | 1.2
Bunch population [$10^{11}$] | 1.2 | 2.3 | 1.1 | 2.2

####  IV.3.1 At the LHC injection energy (450 GeV)

The evolution of the horizontal emittance (left), the vertical emittance
(middle) and energy spread (right) after 1 h at injection energy ($450$ GeV),

Figure 8: The growth of the horizontal (left) and vertical (middle) emittance
and energy spread (right) due to IBS, in a time period of 1 h at the injection
energy of the LHC ($450$ GeV) for the nominal parameters, as computed by the
SIRE code (blue line) and the Bjorken-Mtingwa analytical formalism in MAD-X
(red line).

Figure 9: The growth of the horizontal (left) and vertical (middle) emittance
and energy spread (right) due to IBS, in a time period of 1 h at the injection
energy of the LHC ($450$ GeV) for the HL-LHC parameters, as computed by the
SIRE (blue line) and the Bjorken-Mtingwa analytical formalism in MAD-X (red
line).

where the IBS effect is dominant, are presented in Fig. 8 for the nominal BCMS
case and in Fig. 9 for the HL-LHC parameters. The red and the blue lines
correspond to the analytical calculations of the MAD-X MAD-X homepage  IBS
routine (based on the B-M formalism) and to the SIRE results, respectively.

Table 2: IBS growths of the transverse emittances and energy spread during 1 h at injection energy ($450$ GeV). IBS growths | Nominal (BCMS) | HL-LHC
---|---|---
$MAD-X$ | $SIRE$ | $MAD-X$ | $SIRE$
$d\epsilon_{x}/\epsilon_{x0}$ [%] | 24.6 | 24.1 | 20.1 | 19.6
$d\epsilon_{y}/\epsilon_{y0}$ [%] | 0.2 | 0.4 | 0.2 | 0.3
$d\sigma_{l}/\sigma_{l0}$ [%] | 21.4 | 20.8 | 16.8 | 16.2

Due to the fact that in SIRE the generation of the distribution is based on a
random number generator, the tracking simulations were performed several
times, resulting in the two standard deviation spread that are plotted in
light blue. Table 2 summarizes the IBS growth of the transverse emittances and
energy spread, for the nominal BCMS and HL-LHC parameters, as computed by the
SIRE code and the B-M analytical formalism in MAD-X.

Figure 10: The initial and final (after 1 h) distributions in the horizontal
(left), vertical (middle) and longitudinal (right) plane, for the nominal BCMS
bunch parameters at injection energy ($450$ GeV), are denoted by blue and red
stars, respectively. They are fitted with the Gaussian (dashed line) and the
q-Gaussian (solid line) functions.

Figure 11: The initial and final (after 1 h) distributions in the horizontal
(left), vertical (middle) and longitudinal (right) plane, for the HL-LHC bunch
parameters at injection energy ($450$ GeV), are denoted by blue and red stars,
respectively. They are fitted with the Gaussian (dashed line) and the
q-Gaussian (solid line) functions.

In the horizontal and longitudinal plane the IBS effect is dominant, while in
the vertical plane, it is minor. Even though the SIRE simulation algorithm and
the B-M analytical formalism make use of different approaches to calculate the
IBS effect (SIRE uses the classical Rutherford cross section which is closer
to the Piwinski formalism), they seem to agree very well during the 1 h time
at injection energy. In the longitudinal plane, there is a small difference
observed for longer time-spans. Such differences can be explained by the fact
that SIRE reshapes the beam distributions, in a self-consistent way, after
each collisional process, while the B-M IBS formalism assumes Gaussian beam
distributions throughout the calculation.

The variation of the initially Gaussian particle distributions within 1 h at
injection energy is shown in logarithmic scale in Fig. 10 and Fig. 11 for the
nominal BCMS and the HL-LHC case, respectively. The initial and final (after 1
h) distributions in the horizontal (left), vertical (middle) and longitudinal
(right) plane, are denoted by blue and red stars, respectively. They are
fitted with the Gaussian (dashed line) and the $q$-Gaussian (solid line)
functions. The fitting results of the initial and final distributions are
presented in Table 3 for the nominal BCMS case and in Table 4 for the HL-LHC
case.

As was expected from the results shown in Fig. 8-9 concerning the IBS growth,
the horizontal and longitudinal rms beam sizes get larger as time evolves,
while the vertical one does not change. The vertical distributions remain
Gaussian since $q\approx 1$. For both the nominal and the HL-LHC case, the $q$
parameter of the horizontal and longitudinal distributions is decreased. This
can be explained by the fact that, due to IBS, the core of the distributions
is blown up in such a way that it covers up the initially Gaussian tails of
the input distributions, which remain less affected. In the longitudinal plane
the decrease in $q$ is more significant for the HL-LHC case. This indicates
that the stronger IBS is, the more the core is blown up. Since for a light
tailed distribution ($q<1$) the Gaussian fit overestimates the rms value, the
resulted beam sizes are slightly larger than in the case of the $q$-Gaussian
fit. Comparing the root mean square error (RMSE) values of the two fitting
functions for the final non-Gaussian bunch profiles shows that the
$q$-Gaussian fit is better, in particular for the horizontal plane.

Table 3: Initial and final (after 1 h) fit results for the horizontal, vertical and longitudinal bunch profiles shown in Fig. 10, for the nominal BCMS parameters case at injection energy ($450$ GeV). Fit Parameters |  | Horizontal distribution | Vertical distribution | Longitudinal distribution
---|---|---|---|---
|  | $Initial$ | $Final$ | $Initial$ | $Final$ | $Initial$ | $Final$
Gaussian | $\sigma_{rms}\pm 10^{-3}$ |  | $0.19~{}[mm]$ | $0.22~{}[mm]$ | $0.19~{}[mm]$ | $0.19~{}[mm]$ | $0.25~{}[ns]$ | $0.33~{}[ns]$
$RMSE~{}[10^{-3}]$ |  | 1 | 14 | 1 | 1 | 1 | 10
q-Gaussian | $\sigma_{rms}\pm 10^{-3}$ |  | $0.19~{}[mm]$ | $0.21~{}[mm]$ | $0.19~{}[mm]$ | $0.19~{}[mm]$ | $0.25~{}[ns]$ | $0.32~{}[ns]$
$q\pm dq$ |  | $1.024\pm 0.003$ | $0.893\pm 0.002$ | $0.970\pm 0.007$ | $0.967\pm 0.006$ | $0.992\pm 0.002$ | $0.941\pm 0.001$
$RMSE~{}[10^{-3}]$ |  | 1 | 1 | 1 | 1 | 1 | 6
Table 4: Initial and final (after 1 h) fitting results for the horizontal, vertical and longitudinal bunch profiles shown in Fig. 11, for the HL-LHC parameters case at injection energy ($450$ GeV). Fit Parameters |  | Horizontal distribution | Vertical distribution | Longitudinal distribution
---|---|---|---|---
|  | $Initial$ | $Final$ | $Initial$ | $Final$ | $Initial$ | $Final$
Gaussian | $\sigma_{rms}\pm 10^{-3}$ |  | $0.22~{}[mm]$ | $0.25~{}[mm]$ | $0.22~{}[mm]$ | $0.22~{}[mm]$ | $0.30~{}[ns]$ | $0.37~{}[ns]$
$RMSE~{}[10^{-3}]$ |  | 1 | 14 | 1 | 1 | 3 | 13
q-Gaussian | $\sigma_{rms}\pm 10^{-3}$ |  | $0.22~{}[mm]$ | $0.24~{}[mm]$ | $0.22~{}[mm]$ | $0.22~{}[mm]$ | $0.30~{}[ns]$ | $0.36~{}[ns]$
$q\pm dq$ |  | $0.992\pm 0.003$ | $0.891\pm 0.004$ | $0.995\pm 0.003$ | $0.987\pm 0.003$ | $1.019\pm 0.005$ | $0.885\pm 0.001$
$RMSE~{}[10^{-3}]$ |  | 1 | 1 | 1 | 1 | 3 | 4

####  IV.3.2 At the LHC collision energy (6.5 TeV)

Since at collision energy IBS becomes weaker and synchrotron radiation starts
playing an important role, it is the interplay between these effects that
determines the evolution of the bunch characteristics. In this respect, for
the benchmarking of the B-M IBS theoretical model with SIRE at collision
energy, apart from the IBS, the radiation effects (synchrotron radiation and
quantum excitation) are also taken into account. It should be mentioned that
for the results presented in the following plots the intensity is assumed to
be constant (which is actually true for the few non-colliding bunches, during
physics fills).

Figure 12 shows the horizontal emittance (left), the vertical emittance
(middle) and energy spread (right) evolution after 10 h at collision energy
for the nominal BCMS case, while Fig. 13 shows the evolutions for the HL-LHC
parameters. The red and the blue lines correspond to the analytical
calculations of the MAD-X MAD-X homepage  IBS routine (based on the B-M
formalism) and to the SIRE results, respectively. The two standard deviation
error-bars for the simulation results are plotted in light blue. Table 5
summarizes the variation of the transverse emittances and energy spread during
10 h at the collision energy of the LHC, for the nominal BCMS and HL-LHC
parameters, as computed by the SIRE code and the B-M analytical formalism in
MAD-X.

Figure 12: The evolution of the horizontal (left) and vertical (middle)
emittance and energy spread (right) due to IBS and radiation effects, in a
time period of 10 h at the collision energy of the LHC ($6.5$ TeV) for the
nominal BCMS parameters, as computed by the SIRE code (blue line) and the
Bjorken-Mtingwa analytical formalism in MAD-X (red line).

Figure 13: The evolution of the horizontal (left) and vertical (middle) emittance and energy spread (right) due to IBS and radiation effects, in a time period of 10 h at the collision energy of the LHC ($7$ TeV) for the HL-LHC parameters, as computed by the SIRE (blue line) and the Bjorken-Mtingwa analytical formalism in MAD-X (red line). Table 5: Variation of the transverse emittances and energy spread during 10 h at FT energy. IBS growths | Nominal (BCMS) | HL-LHC
---|---|---
$MAD-X$ | $SIRE$ | $MAD-X$ | $SIRE$
$d\epsilon_{x}/\epsilon_{x0}$ [%] | -0.1 | -1.4 | 7.4 | 5.0
$d\epsilon_{y}/\epsilon_{y0}$ [%] | -26.2 | -26.1 | -31.4 | -31.2
$d\sigma_{l}/\sigma_{l0}$ [%] | -0.1 | -2.6 | -12.6 | -14.7

After a few hours at collisions, the B-M analytical formalism and the
simulations start differentiating. In order to understand whether these
differences are explained by the fact that SIRE reshapes the beam
distributions after each collisional process and the B-M IBS formalism assumes
always Gaussian beam distributions, the bunch parameters given by SIRE at 5 h
are used as input for the IBS and synchrotron radiation calculations in MAD-X
(Gaussian bunches). The red dotted lines in Fig. 12 and Fig. 13 represent the
results of these tests. Even if giving as input to MAD-X exactly the same
bunch parameters as in SIRE, there is clear divergence of the MAD-X results
(red dotted lines) with SIRE right after the 5 h at collisions. This
divergence is much larger than the one observed during the first hours at
collisions. After 5 h at collision energy, the beam in SIRE has been reshaped
enough so that IBS and radiation processes act differently as compared to
Gaussian MAD-X distributions. Consequently, the differences observed between
the B-M analytical formalism and the simulations are expected because MAD-X
assumes always Gaussian distribution, in contrast to SIRE that takes into
account the variation of the bunch shape throughout the calculation.

Due to the fact that the IBS effect is minor in the vertical plane, the strong
synchrotron radiation damping mechanism leads to a clear reduction of the
vertical emittance. However, the variation of the horizontal emittance and
energy spread is determined by the interplay of IBS growth with synchrotron
radiation damping. For the nominal BCMS parameters, these variations are very
small after 10 h at collision energy (Table 5). For the HL-LHC case, having
the same initial horizontal emittance but the double bunch population compared
to the nominal BCMS parameters (Table 1), the IBS effect prevails over
synchrotron radiation in the horizontal plane after almost 3 h (Fig. 13
(left)). As can be seen in Fig. 13 (right) this in not the case for the
longitudinal plane, for which the initial bunch length of 1.2 ns compared to
the 1 ns in the nominal case, renders IBS weaker than synchrotron radiation
and, results in the decrease of the energy spread.

Figure 14: The initial and final (after 10 h) distributions in the horizontal
(left), vertical (middle) and longitudinal (right) plane, for the nominal BCMS
bunch parameters at collision energy ($6.5$ TeV), are denoted by blue and red
stars, respectively. They are fitted with the Gaussian (dashed line) and the
q-Gaussian (solid line) functions.

Figure 15: The initial and final (after 10 h) distributions in the horizontal (left), vertical (middle) and longitudinal (right) plane, for the HL-LHC bunch parameters at collision energy ($7$ TeV), are denoted by blue and red stars, respectively. They are fitted with the Gaussian (dashed line) and the q-Gaussian (solid line) functions. Table 6: Initial and final (after 10 h) fit results for the horizontal, vertical and longitudinal bunch profiles shown in Fig. 14, for the nominal BCMS parameters case at collision energy ($6.5$ TeV). Fit Parameters |  | Horizontal distribution | Vertical distribution | Longitudinal distribution
---|---|---|---|---
|  | $Initial$ | $Final$ | $Initial$ | $Final$ | $Initial$ | $Final$
Gaussian | $\sigma_{rms}\pm 10^{-4}$ |  | $0.064~{}[mm]$ | $0.067~{}[mm]$ | $0.064~{}[mm]$ | $0.056~{}[mm]$ | $0.25~{}[ns]$ | $0.26~{}[ns]$
$RMSE~{}[10^{-3}]$ |  | 1 | 25 | 1 | 1 | 1 | 30
q-Gaussian | $\sigma_{rms}\pm 10^{-4}$ |  | $0.064~{}[mm]$ | $0.064~{}[mm]$ | $0.064~{}[mm]$ | $0.055~{}[mm]$ | $0.25~{}[ns]$ | $0.24~{}[ns]$
$q\pm dq$ |  | $1.004\pm 0.003$ | $0.856\pm 0.005$ | $0.982\pm 0.004$ | $0.971\pm 0.004$ | $1.007\pm 0.004$ | $0.830\pm 0.006$
$RMSE~{}[10^{-3}]$ |  | 1 | 1 | 1 | 1 | 1 | 1
Table 7: Initial and final (after 10 h) fitting results for the horizontal, vertical and longitudinal bunch profiles shown in Fig. 15, for the HL-LHC parameters case at collision energy ($7$ TeV). Fit Parameters |  | Horizontal distribution | Vertical distribution | Longitudinal distribution
---|---|---|---|---
|  | $Initial$ | $Final$ | $Initial$ | $Final$ | $Initial$ | $Final$
Gaussian | $\sigma_{rms}\pm 10^{-4}$ |  | $0.062~{}[mm]$ | $0.067~{}[mm]$ | $0.062~{}[mm]$ | $0.052~{}[mm]$ | $0.30~{}[ns]$ | $0.28~{}[ns]$
$RMSE~{}[10^{-3}]$ |  | 1 | 27 | 2 | 2 | 2 | 17
q-Gaussian | $\sigma_{rms}\pm 10^{-4}$ |  | $0.062~{}[mm]$ | $0.063~{}[mm]$ | $0.062~{}[mm]$ | $0.052~{}[mm]$ | $0.30~{}[ns]$ | $0.27~{}[ns]$
$q\pm dq$ |  | $1.005\pm 0.004$ | $0.852\pm 0.004$ | $0.991\pm 0.005$ | $0.977\pm 0.005$ | $0.990\pm 0.003$ | $0.825\pm 0.001$
$RMSE~{}[10^{-3}]$ |  | 1 | 1 | 1 | 1 | 1 | 1

The evolution of the initially Gaussian (in all planes) particle distributions
within 10 h at collision energy is shown in logarithmic scale in Fig. 14 and
Fig. 15 for the nominal BCMS and the HL-LHC case, respectively. The initial
and final (after 10 h) distributions in the horizontal (left), vertical
(middle) and longitudinal (right) plane, are denoted by blue and red stars,
respectively. They are fitted with the Gaussian (dashed line) and the
$q$-Gaussian (solid line) functions. The fitting results of the initial and
final distributions are presented in Table 6 for the nominal BCMS case and in
Table 7 for the HL-LHC case. The RMSE values of the two fitting functions show
that when the final bunch profiles are strongly non-Gaussian, the q-Gaussian
fitting results should be considered. In this respect, the evolution of the
particle distributions in all planes for the nominal BCMS and HL-LHC cases is
discussed based on the $q$-Gaussian results.

The horizontal beam sizes do not change after 10 h at collision energy because
the blow up caused by IBS is balanced out by the synchrotron radiation
damping. However, there is a transformation of the horizontal distributions’
shape for which the tails become less populated ($q<1$). In the longitudinal
plane both the beam size and the $q$ parameter are reduced, meaning that
synchrotron radiation prevails over IBS and the core is blown up due to IBS
giving underpopulated tails. In the vertical plane, the dominant synchrotron
radiation damping results in a smaller beam size without changing much the
formation of the tails, so the distribution remains Gaussian.

##  V Bunch profile measurements in the LHC

The transverse diagnostic instruments for measuring the bunch profiles in the
LHC are the betatron matching monitor Burger  _et al._ (2013), the Beam Gas
Ionization (BGI) monitor Sapinski  _et al._ (2012), the Beam Gas Vertex (BGV)
monitor, the Beam Wire Scanners (WS) Bosser  _et al._ (1985) and the Beam
Synchrotron Light Monitor (BSRT) Méot  _et al._ (2002). Compatibly with high
intensity and high energy operation, the BSRT is the only instrument offering
non-invasive, continuous and bunch-by-bunch measurements of the LHC beams. The
BSRT is calibrated with respect to the WS during dedicated low beam intensity
runs. 222The WS can measure the emittance throughout the full LHC machine
cycle including the energy ramp, provided that the total intensity in the
machine is limited to 240 nominal bunches at 450 GeV and 12 nominal bunches at
6.5 TeV. The LHC is equipped with two synchrotron radiation monitors (one per
beam) used to characterize the transverse and longitudinal beam distributions.
Due to the significant diffraction patterns at the tails of the BSRT
transverse distributions there is no clear picture about the shape of the
tails and so, they are often assumed to be Gaussian.

A parameter that is generally used to measure the longitudinal emittance in
circular accelerators is the bunch length. The bunch length is given by the
projection of the distribution function on the phase axis, which is known as
the bunch profile or line density. It is operationally measured by the LHC
Beam Quality Monitor (BQM) Papotti  _et al._ (2011) which uses a wall current
monitor pick-up (WCM) Bohl and Malo (2009) to acquire the longitudinal
profiles. Additionally, the longitudinal synchrotron radiation monitor (BSRL)
Jeff  _et al._ (2012), which uses the same synchrotron light source as the
BSRT, continuously measures the longitudinal distribution of charges in the
beams. The scopes connected to the WCM pick-ups can acquire longitudinal bunch
profiles of both beams during a full LHC cycle.

###  V.1 Comparison between experimental data, the SIRE and the B-M analytical
formalism, at the LHC collision energy (6.5 TeV)

The longitudinal bunch manipulations performed during the ramp to avoid
instabilities due to the loss of Landau damping Baudrenghien and Mastoridis
(2013), produce bunches that arrive at collision energy with a clearly non-
Gaussian longitudinal shape. In addition, the transfer functions of the
pickups and cables were measured and are used for deconvolution Esteban Müller
(2016), resulting in some cases in tails which are asymmetric or have ripples.

Figure 16: The evolution of the $q$ parameter during 11.5 h at collisions (6.5
TeV), for a train of bunches in the longitudinal plane.

By assuming that these profiles are Gaussian may lead in underestimating or
overestimating the actual bunch length. For the studies presented in this
paper, these profiles are fitted using the $q$-Gaussian function. An example
showing the evolution of the $q$ parameter for the longitudinal profile of a
bunch train during 11.5 h at collisions (6.5 TeV) in the LHC is presented in
Fig. 16. It is clear that with such $q$ parameter values, corresponding to
non-Gaussian tails, the rms beam size cannot be accurately estimated by using
the Gaussian function.

Figure 17: The evolution of a longitudinal bunch profile during 11.5 h at
collisions (6.5 TeV).

The increase of the $q$ parameter means that the longitudinal distributions
with the underpopulated tails ($q<1$) at the start of collisions, become more
Gaussian ($q\rightarrow 1$) as time evolves. This is a general statement that
can be made for the longitudinal distribution observed at the collision energy
of the LHC. The evolution of the longitudinal particle distribution of a
single bunch that is picked out of the train of bunches is shown in Fig. 17
for the time period of 11.5 h. The initial bunch profile (plotted in blue) is
fitted with the Gaussian and the q-Gaussian functions that give different rms
beam sizes because of the dependence of the standard deviation on the $q$
parameter (Eq. (15)). The fitting results are used to generate a Gaussian and
a q-Gaussian distribution to be tracked in SIRE in order to compare the
experimental observations with the results of the code.

Figure 18: The initial (at the start of collisions) and the final (after 11.5 h) longitudinal bunch profiles as observed in the LHC (left) and as calculated by the SIRE (right), in logarithmic scale, are denoted by blue and red stars, respectively. They are fitted with the Gaussian (dashed line) and the $q$-Gaussian (solid line) functions. Table 8: Fitting results for the initial (at the start of collisions) and the final (after 11.5 h) longitudinal bunch distribution shown in Fig. 18, as was observed in the LHC and as was calculated by the SIRE code. Fit Parameters |  | Initial (t=0) | Final (t=11.5 h)
---|---|---|---
|  | $DATA$ | $SIRE$ | $DATA$ | $SIRE$
Gaussian | $\sigma_{rms}\pm d\sigma_{rms}~{}[ns]$ |  | $0.299\pm 0.003$ | $0.297\pm 0.002$ | $0.233\pm 0.002$ | $0.237\pm 0.002$
$RMSE~{}[10^{-3}]$ |  | 22 | 19 | 18 | 20
q-Gaussian | $\sigma_{rms}\pm d\sigma_{rms}~{}[ns]$ |  | $0.286\pm 0.004$ | $0.290\pm 0.001$ | $0.227\pm 0.002$ | $0.235\pm 0.001$
$q\pm dq$ |  | $0.88\pm 0.03$ | $0.85\pm 0.01$ | $0.93\pm 0.03$ | $0.86\pm 0.01$
$RMSE~{}[10^{-3}]$ |  | 10 | 3 | 10 | 4

In Fig. 18, the initial (at the start of collisions) and the final (after 11.5
h) longitudinal bunch profiles, as observed in the LHC (left) and as
calculated by SIRE (right) for an initially $q$-Gaussian simulated profile,
are denoted by blue and red stars, respectively. They are plotted in
logarithmic scale and they are fitted with the Gaussian (dashed line) and the
$q$-Gaussian (solid line) functions. The reduction of the bunch population
with time due to burn-off and the extra (on top of IBS) transverse emittance
blow-up observed in the machine, are taken into account for the simulation.
The transverse distributions are assumed to be Gaussian, since at collisions
the shape of their tails is not clear due to diffraction. The fitting results
are presented in Table 8. Even if there seems to be no change at the tails of
the simulated distribution, in reality the profiles become more Gaussian.
Within 11.5 h at collisions, the rms beam size of the measured bunch profile
and of the corresponding tracked distribution is reduced by 21% and by 19%,
respectively. This shows a very good agreement between the experimental data
and the simulations performed with SIRE.

Figure 19 shows in logarithmic scale the initial (blue stars) and the final
(red stars) horizontal (left) and vertical (right) bunch profiles as
calculated by SIRE, fitted with the Gaussian (dashed line) and the
$q$-Gaussian (solid line) functions. As can be seen in Table 9, the
simulations showed no change in the transverse beam sizes and that is because
the extra (on top of IBS) transverse emittance blow-up is included. The effect
of IBS together with the extra blow-up assumed, widens the core of the
horizontal bunch in such a way that the $q$ parameter is decreased by around
10% within these 11.5 h. Since IBS is negligible in the vertical plane, the
fact that the vertical bunch profile remains Gaussian indicates that the
interplay between the synchrotron radiation damping and the extra blow-up do
not change the tails of the distribution.

The $4\sigma$-bunch length evolution when assuming Gaussian (left) and
$q$-Gaussian (right) initial distributions is shown in Figure 20. The blue
line corresponds to the SIRE calculations and the red line to the results
given by the IBS module of MAD-X Antoniou and Zimmermann (2012) which is based
on the analytical formulation of B-M and always assumes Gaussian
distributions. The bunch length evolution, together with the two standard
deviation error-bars, when fitting the data with the Gaussian and the
q-Gaussian functions is represented by a black and a grey line, respectively.
The bunch length values differ for the two distribution functions used due to
the fact that, for a light tailed distribution the rms value is overestimated
by fitting a Gaussian. When assuming a Gaussian distribution, the bunch length
evolution calculated by the code gets closer to the measured data. For the
$q$-Gaussian case the agreement between data and simulations is excellent.
This is a remarkable result, taking into account the fact that no assumptions
are being made in the simulations apart from identical initial conditions with
respect to the experimental ones. In agreement with the results presented in
the previous section, the divergence between the SIRE and the MAD-X for longer
time-spans is something to be expected since the distribution shape in SIRE is
updated, while in MAD-X it remains Gaussian.

Figure 19: The initial (at the start of collisions) and the final (after 11.5 h) horizontal (left) and vertical (right) bunch profiles as calculated by the SIRE, in logarithmic scale, are denoted by blue and red stars, respectively. They are fitted with the Gaussian (dashed line) and the q-Gaussian (solid line) functions. Table 9: Fitting results for the initial (at the start of collisions) and the final (after 11.5 h) transverse bunch distributions shown in Fig. 19, as was calculated using the SIRE code. Fit Parameters |  | Horizontal distribution | Vertical distribution
---|---|---|---
|  | $Initial$ | $Final$ | $Initial$ | $Final$
q-Gaussian | $\sigma_{rms}\pm 10^{-4}~{}[mm]$ |  | $0.076$ | $0.076$ | $0.076$ | $0.076$
$q\pm dq$ |  | $0.990\pm 0.004$ | $0.893\pm 0.005$ | $0.992\pm 0.003$ | $0.983\pm 0.003$
$RMSE~{}[10^{-3}]$ |  | 3 | 3 | 3 | 3

Figure 20: The bunch length ($4\sigma$) evolution during several hours in
collisions, as computed by the SIRE code (blue), the B-M analytical formalism
(red) and as measured by the longitudinal profile monitors when assuming a
Gaussian (left) and a q-Gaussian (right) distribution.

At collisions, the divergence between the luminosity model Antoniou  _et al._
(2017) and the measured luminosity by the experiments becomes more pronounced
as time evolves Papadopoulou . Actually, the predicted luminosity by the model
is always larger compared to the measured one by the end of collisions. As
calculated by SIRE, the weight of the horizontal bunch profile tails is
decreased in time (see Table 9) and as explained in Section II, for lighter
tails and a constant beam size, the luminosity is expected to become lower. It
is then clear that by taking into account the luminosity change due to the
variation of the transverse distribution tails, the model predictions can be
significantly improved.

##  VI Summary and outlook

In the LHC, the interplay between a series of effects can lead to
distributions with non-Gaussian tails. Since the rms value of a distribution
can be underestimated or overestimated by using a simple Gaussian function,
the use of appropriate fitting functions to accurately estimate the beam size
and the behavior of the tails is necessary. The impact of non-Gaussian
distribution shapes on the estimated luminosity is discussed. One of the next
steps is to improve the luminosity model, that is currently based on Gaussian
distributions, by taking into account the actual shape of the bunch profiles.
In this way, it is possible to get more accurate luminosity predictions.
Already, for the operational scenario of the High Luminosity LHC upgrade
Métral  _et al._ (2018), a non-Gaussian bunch length estimation is being
considered.

The way IBS and radiation effects act depends on the shape of the bunch
profiles. Aiming to quantify the impact of the distribution’s shape on the
emittance evolution, a multi-particle tracking code called SIRE, is used. The
benchmarking of the B-M analytical formalism with SIRE showed a very good
agreement for the first couple of hours at the injection (450 GeV) and
collision (6.5 TeV) energies of the LHC, even if they make use of different
approaches to calculate the IBS effect. The differences observed for longer
time-spans are expected, since in SIRE the particle distributions are updated,
while MAD-X always assumes Gaussian distributions. The results obtained from
the simulations encourage the idea of using the code for tracking
distributions coming from experimental data, in order to study the impact of
the distribution’s shape on the evolution of the bunch characteristics. After
the comparison with experimental data, the fact that SIRE takes into account
the change of the particle distribution showed that it is a very useful tool
for estimating the actual bunch parameters evolution in the machine. The
contribution of effects such as betatron coupling, noise and electron-cloud,
to the emittance growth are planned to be included in the simulation code, in
order to complement the existing semi-analytical emittance evolution model.

##  Appendix A The q-Gaussian distribution function

The q-Gaussian Curado and Tsallis (1992) which is used to describe more
accurately bunch profiles with tails that differ from the ones of a normal
distribution, has a probability density function given by:

$f(x)={\sqrt{\beta^{qG}}\over C_{q}}e_{q}(-\beta^{qG}x^{2})\;.$ (11)

The q-exponential function is given by:

$e_{q}(x)=\begin{array}[]{cc}\Bigg{\\{}&\begin{array}[]{l}\exp(x)\>~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{},~{}\mathrm{if}~{}~{}~{}q=1\\\
\left(1+\left(1-q\right)x\right)^{1\over 1-q}\;,~{}\mathrm{if}~{}~{}~{}q\neq
1~{}\mathrm{and}~{}\left(1+\left(1-q\right)x\right)>0\\\
0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{},~{}\mathrm{if}~{}~{}~{}q\neq
1~{}\mathrm{and}~{}\left(1+\left(1-q\right)x\right)\leq
0\end{array}\end{array}\>.$ (12)

The parameter $q$ describes the weight of the tails, in the sense that the
larger its value, the heavier the tails become, as presented in Fig. 21. In
the limit of $q\rightarrow 1$, the normal distribution is obtained.

Figure 21: The q-Gaussian distribution function for different $q$ and
$\beta^{qG}$ values.

The distribution is characterized as “light” tailed when $q<1$ and as “heavy”
tailed when $q>1$. The normalization factor $C_{q}$ differs for specific
ranges of the $q$ parameter and it is written as:

$C_{q}=\begin{array}[]{cc}\Bigg{\\{}&\begin{array}[]{l}\frac{2\sqrt{\pi}}{(3-q)\sqrt{1-q}}\frac{\Gamma\left({1\over
1-q}\right)}{\Gamma\left({3-q\over
2(1-q)}\right)}\>,~{}~{}~{}\mathrm{for}~{}~{}~{}-\infty<q<1\\\
\sqrt{\pi}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{},~{}~{}~{}\mathrm{for}~{}~{}~{}q=1\\\
\frac{\sqrt{\pi}}{\sqrt{q-1}}\frac{\Gamma\left({3-q\over
2(q-1)}\right)}{\Gamma\left({1\over
q-1}\right)}~{}~{}~{}~{}~{}~{}\;,~{}~{}~{}\mathrm{for}~{}~{}~{}1<q<3\end{array}\end{array}\>.$
(13)

The parameter $\beta^{qG}$ is a real positive number. As the normal
distribution, the $q$-Gaussian is an even function taking its maximum at
$x=0$, where

$f(0)={\sqrt{\beta^{qG}}\over C_{q}}\;.$ (14)

For a certain $q$ value, the higher is the value of $\beta^{qG}$, the larger
is the maximum of the probability density function, as can be observed in Fig.
21. The standard deviation which also differs for specific ranges of the $q$
parameter, is given by:

$\sigma^{qG}=\begin{array}[]{cc}\Bigg{\\{}&\begin{array}[]{l}\sqrt{\frac{1}{\beta^{qG}(5-3q)}}\>,~{}~{}\mathrm{for}~{}~{}~{}q<5/3\\\
\infty\;\;\;\;\;\;\;\;\;\;\;\,\,,~{}~{}\mathrm{for}~{}~{}~{}5/3\leq q<2\\\
\mathrm{undefined\,,~{}~{}for}~{}~{}~{}2\leq q<3\end{array}\end{array}\>.$
(15)

In the heavy tail regimes, the distribution is equivalent to the Student’s
t-distribution with a direct mapping between $q$ and the degrees of freedom
$\nu$ (Eq. (16)). Statistically the q-Gaussian is a scaled re-parametrization
of the Student’s t-distribution Student (1908) for which the parameter $\nu$
is constrained to be a positive integer related to the sample size. The
advantage of the q-Gaussian function is that, by introducing the parameters
$q$ and $\beta^{qG}$, a generalization of the Student’s t-distribution to
negative and non-integer degrees of freedom is possible, where:

$q=\frac{\nu+3}{\nu+1}~{}~{}~{}\mathrm{with}~{}~{}~{}\beta^{qG}=\frac{1}{3-q}\>.$
(16)

##  Appendix B Luminosity calculation for q-Gaussian density distribution
functions

Using Eq. (11) as the probability density functions, the general luminosity
formula in Eq. (1) is solved for q-Gaussian distributions in all planes. For
the two beams being identical, integrating firstly over $s$ and $s_{0}$:

$I^{qG}_{s}={\int\limits}{\int_{-\infty}}^{\infty}\rho_{1s}(s-s_{0})\rho_{2s}(s+s_{0})~{}dsds_{0}\>,$
(17)

and then, integrating over x and y:

$I^{qG}_{xy}={\int\limits}{\int_{-\infty}}^{\infty}\rho_{1x}(x)\rho_{1y}(y)\rho_{2x}(x)\rho_{2y}(y)~{}dxdy={\int\limits}{\int_{-\infty}}^{\infty}\rho_{x}(x)^{2}\rho_{y}(y)^{2}~{}dxdy\>,$
(18)

keeping in mind that for $w=x,y,s$ it is:

$\displaystyle w$
$\displaystyle\in\left[\pm\dfrac{1}{\sqrt{\beta^{qG}(1-q)}}\right]\>,~{}\mathrm{for}~{}-\infty<q_{w}<1$
(19) $\displaystyle w$
$\displaystyle\in(-\infty,\infty)\>,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathrm{for}~{}~{}1\leq
q_{w}<3\;,$

the solutions of these integrals are found to be:

$I^{qG}_{s}=1\>,$ (20)

and

$I^{qG}_{x,y}=\begin{array}[]{cc}\Bigg{\\{}&\begin{array}[]{l}\dfrac{\beta^{qG}_{x,y}}{C_{q_{x,y}}^{2}}\dfrac{\sqrt{\pi}\Gamma\left({-3+q_{x,y}\over-1+q_{x,y}}\right)}{\sqrt{\beta^{qG}_{x,y}(1-q_{x,y})}\Gamma\left({3q_{x,y}-7\over
2(-1+q_{x,y})}\right)}\>,~{}~{}~{}\mathrm{for}~{}-\infty<q_{x,y}<1\\\
\dfrac{\beta^{qG}_{x,y}}{C_{q_{x,y}}^{2}}\dfrac{\sqrt{\pi}\Gamma\left({-q_{x,y}+5\over
2(-1+q_{x,y})}\right)}{\sqrt{\beta^{qG}_{x,y}(-1+q_{x,y})}\Gamma\left({2\over-1+q_{x,y}}\right)}~{}\;,~{}~{}~{}\mathrm{for}~{}~{}~{}~{}~{}1\leq
q_{x,y}<3\end{array}\end{array}\>,$ (21)

for $I^{qG}_{x}I^{qG}_{y}=I^{qG}_{xy}$ and, for $\beta^{qG}_{xy}$ and
$C_{q_{xy}}$ being the beta parameters and the normalization factors in the
transverse plane. After some simplifications, using also Eq. (13) and Eq.
(15), it is found that the luminosity for q-Gaussian distribution functions
depends on the ${\cal{I}}_{x,y}^{qG}$ (see Eq.(4)) which are defined as:

${\cal{I}}_{x,y}^{qG}=\begin{array}[]{cc}\Bigg{\\{}&\begin{array}[]{l}\dfrac{(2+1/k)^{2}}{2\sqrt{3+2k}}\dfrac{\Gamma\left(1+2k\right)\Gamma\left(1/2+k\right)^{2}}{\Gamma\left(3/2+2k\right)\Gamma\left(k\right)^{2}}\>~{}~{},~{}~{}~{}\mathrm{for}~{}-\infty<q_{x,y}<1\\\
\dfrac{2}{\sqrt{-(3+2k)}}\dfrac{\Gamma\left(-1/2-2k\right)\Gamma\left(-k\right)^{2}}{\Gamma\left(-2k\right)\Gamma\left(-1/2-k\right)^{2}}\>,~{}~{}~{}\mathrm{for}~{}~{}~{}~{}~{}~{}1\leq
q_{x,y}<\dfrac{5}{3}\end{array}\end{array}\>,$ (22)

for $k=\dfrac{1}{1-q_{x,y}}$. As for the Gaussian case (Eq. (2)), the
luminosity for q-Gaussian beams colliding head-on does not depend on the
longitudinal beam size.

Figure 22: Luminosity variation with respect to the transverse beam size
variation, for the q-Gaussian (${\cal{L}}^{qG}$) with $q=1$ (i.e. normal
distribution shape) and the Gaussian case (${\cal{L}}^{G}$), for the same beam
parameters.

In Fig. 22, the variation of the luminosity is plotted with respect to the
transverse beam size for the $q$-Gaussian case (${\cal{L}}^{qG}$) with $q=1$
(i.e. normal distribution shape) and the Gaussian case (${\cal{L}}^{G}$).
Basically, the transverse beam sizes in Eq. (2) and Eq. (4) are being varied
equivalently and the resulted luminosity changes are found using these two
equations. The excellent agreement demonstrates that in the limit of
$q\rightarrow 1$, the luminosity estimation for $q$-Gaussian distributions
(given in Eq. (4)) allows to obtain the exact same result as for Gaussian
distributions. By keeping the $q$ parameter constant (here $q=1$) and varying
the beam size, the $\beta^{qG}$ parameter also varies.

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