# Multiplicity dependence freeze-out scenarios in pp collisions at $\sqrt{s}$
= 7 TeV

Susil Kumar Panda<EMAIL_ADDRESS>Department of Physics,Ravenshaw
University, Cuttack - 753003, India  Sandeep Chatterjee
<EMAIL_ADDRESS>Department of Physical Sciences, Indian
Institute of Science Education and Research, Berhampur, Transit Campus,
Government ITI, Berhampur 760010, Odisha, India  Ajay Kumar Dash
<EMAIL_ADDRESS>School of Earth and Planetary Sciences, National Institute
of Science Education and Research, HBNI, Jatni - 752050, India  Bedangadas
Mohanty<EMAIL_ADDRESS>School of Physical Sciences, National Institute
of Science Education and Research, HBNI, Jatni, 752050, India  Rita Paikaray
<EMAIL_ADDRESS>Department of Physics,Ravenshaw University, Cuttack
- 753003, India  Subhasis Samanta<EMAIL_ADDRESS>Institute of
Physics, Jan Kochanowski University, 25-406 Kielce, Poland  Ranbir Singh
<EMAIL_ADDRESS>School of Physical Sciences, National Institute of
Science Education and Research, HBNI, Jatni, 752050, India

###### Abstract

The data on transverse momentum integrated hadron yields in different
multiplicity classes of p+p collisions at $\sqrt{s}=7$ TeV have been analyzed
to extract the chemical freeze-out parameters using a thermal model. The
chemical freeze-out parameters have been extracted for three different freeze-
out schemes: i. unified freeze-out for all hadrons in complete thermal
equilibrium (1CFO), ii. unified freeze-out for all hadrons with an additional
parameter $\gamma_{S}$ which accounts for possible out-of-equilibrium
production of strange hadrons (1CFO$+\gamma_{S}$), and iii. separate freeze-
out for hadrons with and without strangeness content (2CFO). It has been
observed that 1CFO$+\gamma_{S}$ scheme gives the best description of the
hadronic yields at midrapidity when multiplicity ($\langle
dN_{ch}/d\eta\rangle$) of the collision is less than 10. This indicates that
the strangeness is out of equilibrium in most of the multiplicity classes of
p+p collisions. All the three parameters of this CFO scheme, temperature
($T$), radius of the fireball ($R$) and strangeness suppression factor
($\gamma_{S}$) increase with the increase of $\langle dN_{ch}/d\eta\rangle$.
Further, we have compared applicability of different CFO schemes considering
two more colliding system p+Pb at $\sqrt{s_{\rm NN}}$ = 5.02 and Pb+Pb at
$\sqrt{s_{\rm NN}}$ = 2.76 TeV along with p+p collisions at $\sqrt{s}=7$ TeV.
We observe a freeze-out volume (or multiplicity) dependence of CFO schemes
regardless of colliding ions. The 1CFO+$\gamma_{S}$, 1CFO and 2CFO schemes
provide best description of the data when the dimension less quantity $VT^{3}$
approximately satisfies the conditions $VT^{3}<50$, $50<VT^{3}<100$ and
$VT^{3}>100$ respectively or the corresponding multiplicity satisfies the
conditions $\langle dN_{ch}/d\eta\rangle<30$, $30<dN_{ch}/d\eta<60$ and
$\langle dN_{ch}/d\eta\rangle>100$ respectively.

##  I Introduction

In a high-energy ion collision, a fireball is produced. When the energy
density at the core of the fireball is sufficiently high the quark-gluon
plasma (QGP), a deconfined phase of quarks and gluons is formed. The fireball
expands resulting in a decrease in the temperature. A QGP to hadronic phase
transition occurs when the temperature drops below the transition temperature.
At the initial stage, hadrons interact both elastically and inelastically in
the hadronic medium. Inelastic interaction among the hadrons ceases at
chemical freeze-out (CFO). At this stage, hadronic yields get fixed and do not
change afterwards. The elastic interaction continues until the kinetic freeze-
out is reached. Hadrons then stream outwards freely and eventually reach the
detector. Experimental data on hadronic yields in ion collisions are
traditionally described by the hadron resonance gas (HRG) model. In the most
simplified formulation of the HRG model, it is assumed that the CFO of all the
hadrons and resonances occurs at the same temperature, baryon chemical
potential and volume of the freeze-out surface. We call this unified CFO
scheme 1CFO. This 1CFO HRG model successfully describes the hadron yields in
nucleus-nucleus collision across a wide range of centre-of-mass energy Braun-
Munzinger  _et al._ (1996); Yen and Gorenstein (1999); Braun-Munzinger  _et
al._ (1999); Andronic  _et al._ (2006). To describe strange hadrons in
elementary collisions like $e^{+}e^{-},pp$ and $p\bar{p}$ Becattini (1996);
Becattini  _et al._ (1996); Becattini (1996); Becattini and Heinz (1997), an
additional parameter $\gamma_{s}$ was introduced. The parameter $\gamma_{s}$
accounts for the deviation from chemical equilibrium in the strange sector.
This CFO scheme is referred to as 1CFO+$\gamma_{s}$. The Pb-Pb collision data
at 2.76 TeV indicated a separation of CFO between light and strange quark
hadrons Abelev  _et al._ (2012a). The lattice QCD study also suggested a
separate CFO for the strange hadrons Bellwied  _et al._ (2013). Later, a
flavor dependent sequential freeze-out scheme with different freeze-out
surfaces for hadrons with zero and non-zero strangeness content was proposed
Chatterjee  _et al._ (2013). This CFO scheme is referred to as 2CFO. Thus,
despite the phenomenological success of HRG models, it is not clear which CFO
scheme is suitable for which data. As a result, the understanding of thermal
and chemical equilibration is still an open issue. In Ref. Chatterjee  _et
al._ (2017) it was shown that hadron yield for Pb+Pb collision at
$\sqrt{s_{\rm NN}}$ = 2.76 TeV Abelev  _et al._ (2013a, b, 2014a, 2015) can be
described well with the 2CFO scheme whereas minimum bias events of p+p
collision at $\sqrt{s}$ = 7 TeV Abelev  _et al._ (2012b, c); Adam  _et al._
(2015) prefer 1CFO+$\gamma_{S}$ scheme. In this present work, we have done a
similar analysis using the newly available data of multiplicity dependence of
hadron yield created in p+p collision at $\sqrt{s}$ = 7 TeV Acharya  _et al._
(2019). We have also compared multiplicity dependence of different CFO schemes
at LHC considering p+p collisions at $\sqrt{s}=7$ TeV, p+Pb collisions at
$\sqrt{s_{\rm NN}}$ = 5.02 and Pb+Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76
TeV. Some recent work on multiplicity dependence of freeze-out parameters at
the LHC energy can be found in Refs. Sharma  _et al._ (2019); Vovchenko  _et
al._ (2019).

The paper is arranged in the following way. In Sec. II we discussed different
CFO schemes used for this study. The results from the model and data are
compared in Sec. III. Finally, we summarise our findings in Sec. IV.

##  II Different chemical freeze-out schemes

###  II.1 1CFO scheme

In the 1CFO scheme, it is assumed that the fireball at the chemical freeze-out
is in thermal and chemical equilibrium. The logarithm of the partition
function of multi-component gas of hadrons and resonances in the grand
canonical ensemble can be written as

$\ln Z=\sum_{i}\ln Z_{i},$ (1)

where the sum is over all the species. For $i$-th hadrons

$\ln
Z_{i}=\pm\frac{Vg_{i}}{2\pi^{2}}\int_{0}^{\infty}p^{2}\,dp\ln[1\pm\exp(-(E_{i}-\mu_{i})/T)],$
(2)

where the upper and lower sign of $\pm$ corresponds to fermions and bosons,
respectively. In the last expression, $V$ is the fireball volume, $T$ is the
chemical freeze-out temperature. $g_{i}$, $m_{i}$, $\mu_{i}$ are the
degeneracy factor, mass, chemical potential of the $i$-th hadron respectively.
The chemical potential of the $i$-th hadron can be written as
$\mu_{i}=B_{i}\mu_{B}+S_{i}\mu_{S}+Q_{i}\mu_{Q}$ where $B_{i},S_{i},Q_{i}$ are
respectively the baryon number, strangeness and electric charge of the hadron.
All the chemical potentials are not independent. The $\mu_{S}$ and $\mu_{Q}$
can be extracted applying the following constraints

$\displaystyle\text{Net}S$ $\displaystyle=$ $\displaystyle 0$ (3)
$\displaystyle\text{Net}B/\text{Net}Q$ $\displaystyle=$ $\displaystyle r$ (4)

For A+A collisions, the value of $r$ is $\sim 2.5$ while it is 1 for p+p
collisions. These constituents come from the conservation of baryon,
strangeness and charge quantum numbers of the colliding ions.

The yield of the $i$-th hadron in the grand canonical ensemble can be written
as

$N_{i}=T\frac{\partial\ln
Z_{i}}{\partial\mu_{i}}=\frac{g_{i}V}{2\pi^{2}}\sum_{k=1}^{\infty}(\pm)^{k+1}\frac{m_{i}^{2}T}{k}K_{2}\left(\frac{km_{i}}{T}\right)e^{\frac{k\mu_{i}}{T}},$
(5)

where $K_{2}$ is the modified Bessel function of the second kind. In the last
expression, the plus sign is for bosons and the minus sign is for fermions.

Hadronic yields measured by the detectors in experiments include feed-down
from heavier hadrons and resonances. Therefore, the total hadronic yield is
obtained by including the resonance decay contribution to the above primordial
yield

$N_{i}^{total}=N_{i}+\sum_{j}B.R._{ij}N_{j},$ (6)

where $B.R._{ij}$ is the branching ratio for the $j$-th hadron species to
$i$-th hadron species.

###  II.2 1CFO + $\gamma_{S}$ scheme

In this CFO scheme, incomplete strangeness equilibration is assumed. The
hadronic yield in this CFO scheme can be written as Das  _et al._ (2017)

$N_{i}=\frac{g_{i}V}{2\pi^{2}}\sum_{k=1}^{\infty}(\pm)^{k+1}\frac{m_{i}^{2}T}{k}K_{2}\left(\frac{km_{i}}{T}\right)e^{\frac{k\mu_{i}}{T}}\gamma_{S}^{k|S_{i}|},$
(7)

where $|S_{i}|$ is the number of valence strange quarks and anti-quarks in the
hadron species $i$. The value $\gamma_{S}=1$ corresponds to complete
strangeness equilibration. Compared to the 1CFO scheme (Eq. 5), only
$\gamma_{S}$ factor is extra here.

###  II.3 2CFO scheme

The expression of hadronic yield in this CFO scheme is as Eq. 6. However,
there are two different sets of parameter for hadrons with and without
strangeness content.

Within the GCE ensemble, conservation laws for quantum or particle numbers are
enforced on average through the temperature and chemical potentials. It allows
fluctuations of conserved averages.

###  II.4 Canonical and strangeness canonical ensemble

Apart from the grand canonical ensemble, people have also used other ensemble
like canonical and strangeness canonical ensemble. In canonical ensemble (CE)
all the charges are exactly conserved (CE) Wheaton and Cleymans (2009) whereas
in strangeness canonical ensemble (SCE) Wheaton and Cleymans (2009); Kraus
_et al._ (2009) only strangeness is exactly conserved. Both CE and SCE are
implemented in the THERMUS program Wheaton and Cleymans (2009).

The available experimental data are measured in unit rapidity, so in this work
the exact conservation is enforced across only a single unit of rapidity.

##  III Result

###  III.1 Multiplicity dependence freeze-out scenarios in p+p collision at
LHC

Figure 1: (Color online) Variation of $\chi^{2}$/ndf for the three different freeze-out schemes (1CFO, 1CFO$+\gamma_{S}$ and 2CFO) with the charge particle multiplicity measured at midrapidity in p+p collisions at $\sqrt{s}$ = 7 TeV Acharya  _et al._ (2019).  Table 1: The chemical freeze-out parameters in the 1CFO, 1CFO+$\gamma_{s}$ and 2CFO schemes in p+p collisions at $\sqrt{s}$ = 7 TeV for for different multiplicity classes. Centrality (%) | 1CFO (1CFO+$\gamma_{s}$) | 2CFO
---|---|---
|  | Strange | Non-Strange
|  $T$ (MeV) |  $R$ (fm) | $\gamma_{s}$ |  $T_{S}$ (MeV) |  $R_{S}$ (fm) |  $T_{NS}$ (MeV) |  $R_{NS}$ (fm)
0-1 | 162$\pm$ 3 (164$\pm$ 3) | 2.29 $\pm$ 0.10 (2.30$\pm$ 0.11) | 0.89 $\pm$ 0.03 | 167$\pm$ 3 | 2.06$\pm$ 0.10 | 145$\pm$ 4 | 3.16$\pm$ 0.20
1-5 | 160$\pm$ 3 (163$\pm$ 3) | 2.15 $\pm$ 0.09 (2.16$\pm$ 0.10) | 0.86 $\pm$ 0.03 | 165$\pm$ 3 | 1.93$\pm$ 0.09 | 145$\pm$ 3 | 2.90$\pm$ 0.15
5-10 | 161$\pm$ 3 (163$\pm$ 3) | 1.99 $\pm$ 0.09 (2.01$\pm$ 0.09) | 0.86 $\pm$ 0.03 | 166$\pm$ 3 | 1.78$\pm$ 0.09 | 145$\pm$ 3 | 2.71$\pm$ 0.13
10-20 | 159$\pm$ 3 (162$\pm$ 3) | 1.89 $\pm$ 0.08 (1.91$\pm$ 0.08) | 0.84 $\pm$ 0.03 | 164$\pm$ 3 | 1.70$\pm$ 0.09 | 146$\pm$ 3 | 2.52$\pm$ 0.14
20-30 | 158$\pm$ 3 (161$\pm$ 3) | 1.77 $\pm$ 0.08 (1.80$\pm$ 0.08) | 0.82 $\pm$ 0.03 | 163$\pm$ 3 | 1.60$\pm$ 0.08 | 146$\pm$ 3 | 2.34$\pm$ 0.13
30-40 | 157$\pm$ 3 (160$\pm$ 3) | 1.69 $\pm$ 0.07 (1.71$\pm$ 0.08) | 0.80 $\pm$ 0.03 | 161$\pm$ 3 | 1.52$\pm$ 0.08 | 146$\pm$ 3 | 2.18$\pm$ 0.12
40-50 | 156$\pm$ 3 (159$\pm$ 3) | 1.58 $\pm$ 0.07 (1.62$\pm$ 0.07) | 0.78 $\pm$ 0.03 | 160$\pm$ 3 | 1.43$\pm$ 0.07 | 145$\pm$ 3 | 2.05$\pm$ 0.12
50-70 | 153$\pm$ 3 (157$\pm$ 3) | 1.47 $\pm$ 0.07 (1.51$\pm$ 0.07) | 0.76 $\pm$ 0.03 | 157$\pm$ 3 | 1.35$\pm$ 0.07 | 144$\pm$ 4 | 1.88$\pm$ 0.12
70-100 | 148$\pm$ 3 (152$\pm$ 3) | 1.33 $\pm$ 0.07 (1.38$\pm$ 0.07) | 0.72 $\pm$ 0.04 | 149$\pm$ 3 | 1.27$\pm$ 0.07 | 140$\pm$ 4 | 1.69$\pm$ 0.13

In this work, first we have studied multiplicity dependence of different CFO
schemes in p+p collisions at $\sqrt{s}$ = 7 TeV. For this, we have used data
on $p_{\rm T}$ integrated hadron yield in p+p collisions measured at mid-
rapidity at $\sqrt{s}$ = 7 TeV Acharya  _et al._ (2019). We have performed our
analysis for grand canonical, canonical and strangeness canonical ensembles
using a thermal model called THERMUS Wheaton and Cleymans (2009). Further, for
the grand canonical ensemble, three different freeze-out schemes- 1CFO,
1CFO$+\gamma_{S}$ and 2CFO are considered. For 1CFO and 1CFO$+\gamma_{S}$
schemes standard version of THERMUS is used and it is extended to include the
2CFO scheme.

To extract the CFO parameters we use available midrapidity yields of the
following hadrons ($\pi^{+}+\pi^{-}$)/2, ($K^{+}+K^{-}$)/2, $K_{s}^{0}$,
($p+\bar{p}$)/2, ($K^{*0}+\bar{K^{*0}}$)/2, $\phi$,
($\Lambda+\bar{\Lambda}$)/2 and ($\Xi+\bar{\Xi}$)/2. Since an almost equal
number of particle and antiparticle are produced at this energy, chemical
potentials are taken as zero for all the CFO schemes. Therefore, the
parameters to be extracted from the thermal fits in 1CFO are only the fireball
volume $V$ and temperature $T$ at the chemical freeze-out. In the
1CFO$+\gamma_{S}$ scheme, we have one extra parameter $\gamma_{S}$. In 2CFO
scheme, different freeze-out volume and temperature ($V_{S}$, $T_{S}$,
$V_{NS}$ and $T_{NS}$) for the non-strange and strange hadrons are there. In
this CFO scheme, the $\gamma_{S}$ has no role and hence it is regarded as 1.
Therefore, the number of degrees of freedom (ndf = number of data points -
number of free parameters) in 1CFO, 1CFO+$\gamma_{S}$ and 2CFO schemes are 5,4
and 3 respectively. For canonical and strangeness canonical ensembles free
parameters are $T,V$ and $\gamma_{S}$. In Fig. 1 we compare the goodness of
fit in terms of $\chi^{2}/$ndf of different CFO schemes in different
multiplicity classes where the same is defined in terms of charge particle
multiplicity. Fitting quality is good for 1CFO+$\gamma_{S}$ at lowest $\langle
dN_{ch}/d\eta\rangle$ where $\chi^{2}/$ndf is approximately 3. The
$\chi^{2}/$ndf increases slightly with increasing $\langle
dN_{ch}/d\eta\rangle$ and remains within the range 5-6 in other multiplicity
classes. For the 1CFO scheme, $\chi^{2}/$ndf is approximately 8 at the lowest
$\langle dN_{ch}/d\eta\rangle$ and reaches the highest value ($\sim 12$) when
$\langle dN_{ch}/d\eta\rangle$ is within 3-6. The $\chi^{2}/$ndf then starts
decreasing with increasing $\langle dN_{ch}/d\eta\rangle$. Compared to
1CFO+$\gamma_{S}$ scheme, the fitting quality is bad in 1CFO scheme in all the
multiplicity classes. Variation of $\chi^{2}/$ndf with $\langle
dN_{ch}/d\eta\rangle$ for 2CFO scheme is almost similar to that of 1CFO. The
only difference is that fitting quality becomes comparable to that of
1CFO+$\gamma_{S}$ or even better at some multiplicity classes when $\langle
dN_{ch}/d\eta\rangle>10$.

####  III.1.1 Results of canonical ensemble

Along with the results of the grand canonical ensemble, we have also shown the
quality of fitting in the canonical ensemble (CE) in Fig. 1. At the lowest
$\langle dN_{ch}/d\eta\rangle$, $\chi^{2}/$ndf for CE is 15.59 which is much
larger than that of three CFO schemes in grand canonical ensembles. It then
decreases with increase of $\langle dN_{ch}/d\eta\rangle$. The $\chi^{2}/$ndf
for CE is greater than that of 1CFO+$\gamma_{s}$ scheme in the whole range of
multiplicities shown in this figure. Another significant difference is that
the extracted chemical freeze-out temperature in CE is too large compared to
those of GCE and QCD crossover temperature which is around 155 MeV. At the
lowest multiplicity, $T$ is around 194 MeV. Temperature decreases with
increasing multiplicity and it reaches 173 MeV at the highest multiplicity. We
have observed that $\phi$ meson is responsible for large $\chi^{2}/$ndf in CE.
If we exclude $\phi$ in the fitting, $\chi^{2}/$ndf in the lowest multiplicity
class goes down from 15.59 to 2.54. We compared our result of CE with the
previous work in Ref. Sharma  _et al._ (2019). It has been shown in Ref.
Sharma  _et al._ (2019) that CE gives the best result in the low multiplicity
region which seems to contradict our result. However, this is not the case as
different set of particles are used in both analysis. Particularly in Ref.
Sharma  _et al._ (2019) $K_{s},K^{*}$ and $\phi$ are excluded from the
fitting. Another difference is the inclusion of $\Omega$ which is not there in
our fitting. Already we have discussed the issue with the $\phi$ meson.
Further we checked the effect on our results by excluding $K_{s},K^{*}$ and
$\phi$. If we remove only $K^{*}$ and $\phi$, the $\chi^{2}/$ndf at lowest
$\langle dN_{ch}/d\eta\rangle$ goes down from 15.59 to 1.28. By excluding
$K^{*}$, $\phi$ and $K_{s}$ the $\chi^{2}/$ndf becomes even better (0.9).
Corresponding chemical freezeout temperatures in these two cases are 201 and
202 MeV respectively. So, although there is an improvement of $\chi^{2}/$ndf
chemical freeze-out temperature is still absurd. This indicates that the CE is
not suitable for LHC energy. In presence of $\phi$ meson, a large freezeout
temperature in CE was also reported in Ref. Vovchenko  _et al._ (2019). The
reason for the failure of CE is probably because the accepted rapidity region
which is much smaller than the total rapidity. The system is more like a grand
canonical rather than canonical ensemble. There is no reason that the baryon,
strangeness and electric charge will be exactly conserved at one unit of
rapidity region. It is argued in Ref. Vovchenko  _et al._ (2019) for the exact
conservation of these charges several units of rapidity interval around the
midrapidity is needed rather than a single unit. Because of these problems in
CE we have not considered CE in the rest of the paper.

####  III.1.2 Results of strangeness canonical ensemble

In Fig.1, the worst fitting is observed in case of SCE. The $\chi^{2}/$ndf is
$\sim 20$ at the lowest $\langle dN_{ch}/d\eta\rangle$. Similar to CE,
$\chi^{2}/$ndf for SCE decreases with increase of $\langle
dN_{ch}/d\eta\rangle$. The $\chi^{2}/$ndf for SCE is always larger than that
of 1CFO+$\gamma_{s}$ scheme. Again a relatively better $\chi^{2}/$ndf is shown
in Ref. Sharma  _et al._ (2019) but with the cost of giving up $K_{s},K^{*}$
and $\phi$ in the fitting. Since $\chi^{2}/$ndf in Fig.1 is bad in SCE, we
will not consider this CFO scheme in the rest of the paper.

Overall from Fig. 1, we can say that, except a few multiplicity classes,
1CFO+$\gamma_{S}$ scheme is better compared to other CFO schemes.

####  III.1.3 Extracted chemical freeze-out parameters and comparison between
data and model

Extracted chemical freeze-out parameters in different multiplicity classes are
listed in the table-1 where instead of $\langle dN_{ch}/d\eta\rangle$,
percentage of centrality is mentioned Acharya  _et al._ (2019). Note that the
$\langle dN_{ch}/d\eta\rangle$ decreases as the percentage of the centrality
increases. The $\langle dN_{ch}/d\eta\rangle$ dependence of chemical freeze-
out temperature, the radius of the fireball and the strangeness suppression
factor in three different CFO schemes are shown in Fig.2. At the lowest value
of $\langle dN_{ch}/d\eta\rangle$ temperature is around 152 MeV for
1CFO+$\gamma_{S}$ scheme. The temperature gradually increases with the
increase of $\langle dN_{ch}/d\eta\rangle$ and reaches around 164 MeV at the
highest $\langle dN_{ch}/d\eta\rangle$. Similar temperatures are obtained for
the 1CFO scheme as well. On the other hand for the 2CFO scheme, a clear
separation of freeze-out temperature of the non-strange and strange hadrons is
observed and the separation increases with increasing $\langle
dN_{ch}/d\eta\rangle$. The multiplicity dependence of freeze-out temperature
of strange hadrons in the 2CFO scheme is similar to the 1CFO and
1CFO+$\gamma_{S}$ schemes. However, no such multiplicity dependence of freeze-
out temperature is observed for the non-strange hadrons in this CFO scheme.
Not only that, freeze-out temperature of non-strange hadrons, which lies
around 145 MeV, is significantly (5-20 MeV) lower compared to that of strange
hadrons which suggests that the strange hadrons freeze earlier compared to
non-strange hadrons. Compared to different CFO schemes in the grand canonical
ensemble, the larger temperature is observed for the canonical ensemble in all
$\langle dN_{ch}/d\eta\rangle$. Although it is not shown still we would like
to mention here that the extracted temperature at lowest $\langle
dN_{ch}/d\eta\rangle$ for CE is around 195 MeV which is very large compared to
other CFO schemes and therefore seems unrealistic.

The middle panel of Fig.2 shows the multiplicity dependence of fireball radius
at the freeze-out. As expected, $R$ increases with increasing $\langle
dN_{ch}/d\eta\rangle$ in all the CFO schemes in the grand canonical ensemble.
Similar to the temperature, freeze-out radii are also close to each other in
1CFO and 1CFO+$\gamma_{S}$ schemes. In these two CFO schemes, $R$ is around
1.3 fm at the lower $\langle dN_{ch}/d\eta\rangle$ and reaches upto $\sim$2.3
fm at the highest $\langle dN_{ch}/d\eta\rangle$. Again in the 2CFO scheme,
two separate freeze-out radii is observed for strange and non-strange hadrons.
Extracted size of the fireball for non-strange hadrons is larger than that of
strange hadrons in all multiplicity classes.

The multiplicity dependence of the strangeness suppression factor is shown in
the lower panel of Fig.2. At the lowest $\langle dN_{ch}/d\eta\rangle$, the
$\gamma_{S}$ is around 0.72 which is far away from the equilibrium value 1.
With the increase of $\langle dN_{ch}/d\eta\rangle$, the $\gamma_{S}$
gradually increases towards its equilibrium value and reaches up to 0.89 at
highest $\langle dN_{ch}/d\eta\rangle$. It indicates that full strangeness
equilibration is not achieved at lower values of $\langle
dN_{ch}/d\eta\rangle$. However, it is moving towards the equilibrium when
$\langle dN_{ch}/d\eta\rangle$ is relatively larger. Therefore, it is expected
that the CFO scheme which deals with the non-equilibrium situation of the
strange hadron is the best suited when $\langle dN_{ch}/d\eta\rangle$ is
small. For this reason, in the previous figure (Fig. 1), better fitting
quality (i.e, lower $\chi^{2}$/ndf) at lower $\langle dN_{ch}/d\eta\rangle$
was observed in 1CFO+$\gamma_{S}$ scheme compared to other 1CFO and 2CFO
schemes where full equilibration was assumed.

We have observed that the extracted freeze-out temperature and $\gamma_{s}$ in
1CFO+$\gamma_{s}$ scheme in this present work is in good agreement with those
of Ref. Sharma  _et al._ (2019) despite having the difference in the fitted
particle list mentioned previously.

Figure 2: (Color online) The multiplicity dependence of chemical freeze-out
parameters (temperature $T$, radius $R$ and strangeness non-equilibrium factor
$\gamma_{S}$) obtained in three different freeze-out schemes. Figure 3: (Color
online) The comparison between data Acharya  _et al._ (2019) and model Wheaton
and Cleymans (2009) for p+p collisions at $\sqrt{s}$ = 7 TeV in three
different freeze-out schemes: 1CFO (left column), 1CFO$+\gamma_{S}$ (middle
column) and 2CFO (right column). The deviation of the data from the model (Eq.
8) for each freeze-out scheme are also shown.

Now we will discuss the comparison between experimental data of hadrons and
model estimation. For the rest of the paper, we will consider CFO schemes in
grand canonical ensemble only since in Fig. 1 we have already seen that
fitting quality is much better in grand canonical ensemble combined to CE or
SCE. In the upper panels of Fig. 3, we show the comparison between
experimentally measured hadronic yields and the best-fitted model calculation
in three CFO schemes: 1CFO (left column), 1CFO$+\gamma_{S}$ (middle column)
and 2CFO (right column). For the illustration purpose, we choose only three
multiplicity bins: 0-1 %, 20-30 % and 70-100 %. Experimental data is indicated
by solid points while a line symbol is used to show the model estimation. The
lower panel of this figure shows the deviation which is defined as

$\sigma=\frac{\text{Data - Model}}{\text{Error of Data}}.$ (8)

Horizontal lines at $\pm 1$ are drawn to indicate $\pm 1\sigma$ deviation.
Deviations are less than $3\sigma$ for all the hadrons except pion in the 1CFO
scheme. In 1CFO+$\gamma_{S}$ scheme, significant improvement is observed
particularly in the 70-100 % centrality class where the modulus of deviation
is less than $2\sigma$ for all the hadrons including pion. On the other hand,
for all the centrality classes descriptions of $\pi,K,K^{0}_{s}$ and $p$ are
very good (modulus of the deviation is less than 1$\sigma$) in the 2CFO
scheme. Although, it is not good for strange hadrons like $K^{\star},\Lambda$
and $\Xi$ in 70-100 % centrality class. These hadrons are described better in
the 1CFO+$\gamma_{S}$ scheme. Note that a separate freeze-out surface is used
for the hadrons containing the strange quark in the 2CFO scheme; still, its
performance is not better than the 1CFO+$\gamma_{S}$ scheme. The description
of both non-strange and strange hadrons in 70-100 % centrality is better in
1CFO+$\gamma_{S}$ scheme compared to the other two CFO schemes. The
$\chi^{2}/$ndf is also smaller in 1CFO+$\gamma_{S}$ scheme (see the Fig. 1).
Therefore, we can say that 1CFO+$\gamma_{S}$ is the most appropriate freeze-
out scenario for 70-100 % centrality class. The same is true for 20-30%
multiplicity classes as well. As we move toward higher multiplicity class,
improvement is observed in 2CFO scheme. In 0-1% multiplicity class,
description of $\pi,K,K^{0}_{s}$ and $p$ is better in 2CFO scheme compared to
1CFO and 1CFO+$\gamma_{S}$ schemes. As a result, $\chi^{2}/$ndf is slightly
small in case of 2CFO scheme compared to other two CFO schemes in this
multiplicity class.

###  III.2 Comparison of different freeze-out scenarios at LHC

Figure 4: (Color online) $\chi^{2}/ndf$ in three different CFO schemes in
grand canonical ensemble as a function of average charged particle
multiplicity in p+p collisions at $\sqrt{s}=7$ TeV and compared with the same
in p+Pb and Pb+Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 Abelev  _et al._
(2014b); Adam  _et al._ (2016a, b) and 2.76 TeV Abelev  _et al._ (2013a, b,
2014a, 2015), respectively.  Figure 5: (Color online) Extracted freeze-out
parameters in three different CFO schemes in grand canonical ensemble as a
function $\langle dN_{ch}/d\eta\rangle$ considering three different collision
system p+p at $\sqrt{s}$ = 7 TeV, p+Pb at $\sqrt{s_{\rm NN}}$ = 5.02 and Pb+Pb
collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV.  Figure 6: (Color online)
Product of extracted volume and temperature cube as a function of average
charged particle multiplicity in p+p collisions at $\sqrt{s}=7$ TeV and
compared with the same in p+Pb and Pb+Pb collisions at $\sqrt{s_{\rm NN}}$ =
5.02 and 2.76 TeV Chatterjee  _et al._ (2017), respectively for the best
freeze-out scheme.

We will now present a broader picture of the multiplicity dependence of
freeze-out scenarios at the LHC. Here we will consider collision systems p+Pb
and Pb+Pb along with p+p. In Fig.4, we have shown the multiplicity dependence
of least $\chi^{2}$/ndf. Here we have taken some data from the previously
published results of p+Pb and Pb+Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02
and 2.76 TeV, respectively Chatterjee  _et al._ (2017) combined with the
present results of p+p collision at $\sqrt{s}=7$ TeV. The left panel shows
$\chi^{2}/ndf$ in three different CFO schemes in grand canonical ensemble as a
function of $\langle dN_{ch}/d\eta\rangle$. Different marker style is used for
different CFO schemes, while different colour is used for different collision
system. For the p+p collision, the result is already discussed in Fig. 1. For
p+Pb collision, lowest $\chi^{2}$/ndf is observed in 1CFO+$\gamma_{s}$ scheme
when $dN_{ch}/d\eta\rangle<30$. Above this 1CFO scheme performs better. While
for Pb+Pb collision, 2CFO gives best result when $\langle
dN_{ch}/d\eta\rangle>100$. Below this multiplicity, 1CFO performs well. The
variation of $\langle dN_{ch}/d\eta\rangle$ at LHC is more than three orders
of magnitude. Based on the applicability of different CFO schemes, we can
divide the whole range of $\langle dN_{ch}/d\eta\rangle$ into three regions:
low ($\langle dN_{ch}/d\eta\rangle<30$), intermediate ($30<\langle
dN_{ch}/d\eta\rangle<60$) and high ($\langle dN_{ch}/d\eta\rangle>100$)
multiplicities. Note that presently no data is available within $60<\langle
dN_{ch}/d\eta\rangle<100$. All p+p collisions and p+Pb collisions where
$\langle dN_{ch}/d\eta\rangle<30$ are best described by 1CFO+$\gamma_{S}$
scheme which is indicated by plus marker. It indicates that strange hadrons
containing strange quark are out of equilibrium when $\langle
dN_{ch}/d\eta\rangle<30$. In small systems, the fireball lifetime is expected
to be small. As a result, interaction among constituents is not sufficient
enough to achieve equilibration. The equilibrium CFO schemes are good only if
$\langle dN_{ch}/d\eta\rangle$ is approximately greater than 30. For the
intermediate range of multiplicity ($30<\langle dN_{ch}/d\eta\rangle<60$) 1CFO
scheme has least $\chi^{2}$/ndf which is shown by star marker. High
multiplicity p+Pb and low multiplicity Pb+Pb collisions fall in this category.
The Pb+Pb collisions with $\langle dN_{ch}/d\eta\rangle>100$ can be best
described by the 2CFO scheme which is shown by a circular marker.

In three panels of Fig.5 we show the multiplicity dependence of freeze-out
temperature (top panel), the volume of the fireball (middle panel) and the
$\gamma_{S}$ (bottom panel) at the LHC using three different CFO schemes in
grand canonical ensemble. freeze-out parameters of p+Pb and Pb+Pb collisions
at $\sqrt{s_{\rm NN}}$ = 5.02 and 2.76 TeV, respectively are taken from
Chatterjee  _et al._ (2017). We have already discussed the results of p+p
collision at $\sqrt{s}$ = 7 TeV in Fig. 2. One can see that compared to p+p
collision the freeze-out temperatures in different CFO schemes for p+Pb
collision are much closer to each other and lie around 155 MeV. Further, no
multiplicity dependence of temperature is observed in the case of p+Pb
collision. On the other hand for Pb+Pb collision, the freeze-out temperature
is around 155 MeV for 1CFO and 1CFO+$\gamma_{s}$ schemes while for the 2CFO
scheme, the temperature is slightly higher than 155 MeV for strange hadrons
and slightly lower for non-strange hadrons. Here also no significant
multiplicity dependence is observed. Now let us discuss the best-fitted
results of freeze-out temperature. For the low and intermediate regions of
multiplicity ($\langle dN_{ch}/d\eta\rangle<60$), best-fitted (for which
$\chi^{2}/ndf$ is minimum) freeze-out temperature lies within 155-165 MeV. In
the high multiplicity region ($\langle dN_{ch}/d\eta\rangle>100$) freeze-out
temperature of the hadrons containing strange quark also lies within this
range. However, for non-strange hadrons, the temperature is approximately 15
MeV less. It means non-strange hadrons freeze-out later than the strange
hadrons when $\langle dN_{ch}/d\eta\rangle>100$. The volume of the fireball at
the freeze-out increases almost linearly with the multiplicity. Volumes for
1CFO and 1CFO+$\gamma_{s}$ are almost same in all $\langle
dN_{ch}/d\eta\rangle$. In the 2CFO scheme, freeze-out volume of strange and
non-strange are different and it is larger for the non-strange hadrons since
they freeze-out later. Overall we observe a smooth variation of volume from
one collision system to another. With the increase of multiplicity, the
$\gamma_{S}$ increases towards unity and for Pb+Pb collision $\gamma_{S}$ is
greater than one. Values of $\gamma_{S}$ for different data sets lie in a
line. Here also we observe a smooth variation from one system to another. Note
the best fit result suggest that the $\gamma_{S}$ is only needed for the low
multiplicity events. Above $\langle dN_{ch}/d\eta\rangle\approx 30$ an
equilibration CFO schemes, either 1CFO or 2CFO can be used.

Instead of multiplicity, we can separate three zones in terms of dimension-
less quality $VT^{3}$ as well since it is proportional to the $\langle
dN_{ch}/d\eta\rangle$ and this is presented in Fig. 6. This figure shows that
the 1CFO+$\gamma_{S}$ scheme is the best for the small freeze-out system where
$VT^{3}<50$. For intermediate system size, where $50<VT^{3}<100$, 1CFO
performs well whereas for a large system, where $VT^{3}>100$, the 2CFO scheme
is more suitable. We observe that the best fit value of $VT^{3}$ increases
almost linearly with the multiplicity even though different CFO scheme is
suitable for the different zone of multiplicity. Further, Fig.6 shows that our
treatment in the grand canonical ensemble is justified since $VT^{3}>1$ in all
the multiplicity classes for all the collision systems Kraus  _et al._ (2009)

##  IV Summary

We have studied the transverse momentum integrated hadron yields in different
multiplicity classes of p+p collisions at $\sqrt{s}=7$ TeV using a thermal
model (THERMUS) and extracted the CFO parameters. Analysis has been done for
different CFO schemes $e.g.,$ 1CFO, 1CFO+$\gamma_{S}$, 2CFO in the grand
canonical ensemble. The analysis is also done for canonical and strangeness
canonical ensemble. The charge conservation is enforced across only a single
unit of rapidity within which experimental data are available. Within this
limitation it has been observed that for most of the multiplicity classes
1CFO+$\gamma_{S}$ scheme best describe the data. The value of $\gamma_{S}$
varies between 0.72 to 0.89 which indicates that hadrons containing strange
quark are out-of-equilibrium at the freeze-out in all multiplicity classes.
With the increase of multiplicity the chemical freeze-out temperature
increases from 152 MeV to 164 MeV. The fireball size also increases with
increases of multiplicity and the radius varies between 1.38 - 2.3 fm.

We have also tried to understand the applicability of different CFO schemes at
LHC considering two other collision systems p+Pb and Pb+Pb at $\sqrt{s_{\rm
NN}}=$ 5.02 and 2.76 TeV respectively. We observed a multiplicity (or freeze-
out volume) dependence of CFO schemes instead of colliding ion and energy
dependence. The 1CFO+$\gamma_{S}$ scheme is suitable to describe the hadronic
yield when multiplicity is less than 30 whereas 2CFO perform the best when
multiplicity is larger than 100. In the intermediate region
$30<dN_{ch}/d\eta<60$ 1CFO gives the best description of the data. The same
conclusion can be drawn in terms of freeze-out volume as well. The
1CFO+$\gamma_{S}$, 1CFO and 2CFO schemes best description of the data when the
dimensionless quantity $VT^{3}$ satisfies the conditions $VT^{3}<50$,
$50<VT^{3}<100$ and $VT^{3}>100$ respectively.

##  V Acknowledgement

AKD and RS acknowledge the support of XII th plan project no.
12-R$\&$D-NIS-5.11-0300 of Govt. of India. BM acknowledges financial support
from J C Bose National Fellowship of DST Govt. of India. SS acknowledges
financial support from the Polish National Agency for Academic Exchange
through Ulam Scholarship with the agreement no: PPN/ULM/2019/1/00093/U/00001.

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