# Applicability of transverse mass scaling in hadronic collisions at the LHC

Lucas Altenkämper  UiB, Bergen, Norway  Friederike Bock  CERN, Geneva,
Switzerland  Constantin Loizides  ORNL, Oak Ridge, USA  Nicolas Schmidt  ORNL,
Oak Ridge, USA

###### Abstract

We present a study on the applicability of transverse mass scaling for
identified particle spectra in proton-proton collisions at $\sqrt{s}=7$ TeV
based on data taken by the ALICE experiment at the LHC. The measured yields
are parametrized and compared to estimates obtained from a generalized
transverse mass scaling approach applied to different reference particle
spectra. It is found that generalized transverse mass scaling is not able to
describe the measured spectra over the full range in transverse momentum. At
low $p_{\text{T}}$, deviations of $20$% or more are obtained, in particular if
pions are used as reference particles. A better scaling performance is
obtained, when kaons are used as reference particles. At high $p_{\text{T}}$
all tested spectra with the possible exception of the charged kaons exhibit a
scaling behavior. Investigating the feed-down contributions from resonance
decays to the charged pion yields reveals, that using them as reference a
general scaling may not be achievable. Our findings imply that for precision
measurements of direct photon and di-electron spectra at low transverse
momentum one should measure the relevant hadronic background, instead of
relying on $m_{\text{T}}$ scaling for its estimate.

identified particle spectra, transverse mass scaling, particle ratios

##  I Introduction

Already as early as 1965, Hagedorn proposed that the transverse mass spectra
of hadrons in pp collisions should be governed by a universal scaling law,
based on a statistical-thermodynamical approach for hadron production
Hagedorn:1965st . The spectra, expressed as a function of the transverse mass
$m_{\text{T}}=\sqrt{p_{\text{T}}^{2}+m_{0}^{2}}$ (where $p_{\text{T}}$ is the
transverse momentum and $m_{0}$ the rest mass), should follow an exponential
distribution with a universal inverse slope parameter. This parameter was
originally identified as the highest possible temperature for hadronic matter.
Indeed, it was found that the $m_{\text{T}}$ spectra of different particle
species (e.g. $\pi^{-}$, $\overline{\text{K}}^{0}$, $\Lambda$, $\rho^{0}$ and
$\omega$) produced in hadronic collisions at center-of-mass energies of
$8$–$16$ GeV are reasonably well described by an exponential or Bose-Einstein
distribution with the inverse slope parameter being independent of the
particle mass Deutschmann:1974ne ; Bartke:1976zj . Subsequently, the effect
was called “$m_{\text{T}}$-scaling” Gatoff:1992cv .

Furthermore, it was also shown that the data from pp collisions over a range
of about $6$ to $63$ GeV can be parametrized with a generalized function of
the transverse mass Alper:1974rw ; Bourquin:1976fe . The generalized function,
however, assumes a factorization of the rapidity ($y$) and $m_{\text{T}}$
dependence and is in contrast to the statistical-thermodynamical approach not
of purely exponential form. A phenomenological scaling law was also observed
at the CERN SPS by the WA80 collaboration in S-S and S-Au collisions at
center-of-mass energies of $200$ GeV per nucleon pair Albrecht:1995ug . The
transverse mass spectra of the $\eta$ and $\pi^{0}$ mesons were found to be
identical up to a constant factor in both systems. However, in contrast to the
production at lower collision energies the spectra were best described by a
power law, which differs from the universal scaling originally proposed by
Hagedorn. The approximate $m_{\text{T}}$-scaling was further tested for mesons
and baryons by the STAR collaboration in pp collisions at $\sqrt{s}=200$ GeV
Abelev:2006cs , and a difference between mesons and baryons was observed for
the first time with the meson spectra appearing to be systematically harder
than the baryon spectra for transverse masses above about $2$ GeV. The
difference between the exact $m_{\text{T}}$-scaling, as originally proposed by
Hagedorn, and the more general but approximate form was also investigated by
the PHENIX collaboration in pp collisions at $\sqrt{s}=200$ GeV Adare:2010fe .
The spectra were found to obey the scaling relation separately for mesons and
baryons, but in accordance with Abelev:2006cs  no scaling relation in between
mesons and baryons was observed. Additional tests based on RHIC data were
performed in LABEL:Khandai:2012xx for different collision systems and energies
using a modified Hagedorn parameterization (which is close to an exponential
at low $p_{\text{T}}$ and an exact power law at high $p_{\text{T}}$). It was
found that the $m_{\text{T}}$ spectra of mesons ($\phi$, K± and K${}^{0}_{S}$)
and baryons ($\Lambda$, $\Delta$, $\Xi$ and $\Omega$) could be described
separately by applying the approximate transverse mass scaling relation of the
pion or proton, respectively, in pp and d-Au collisions at $\sqrt{s}=200$ GeV.
However, a disagreement was observed for $\phi$ mesons and kaons in pp
collisions at $\sqrt{s}=900$ GeV, indicating that the generalized scaling
breaks down at higher collision energies.

Originally, the phenomenological scaling law was observed directly on
$m_{\text{T}}$ differential yields. However, it is more practical to formulate
the scaling as a function of $p_{\text{T}}$, by rewriting the functional form
of the parametrization, that is used as a basis for the scaling, in terms of
$m_{\text{T}}$ rather than $p_{\text{T}}$ through the substitution

$\displaystyle p_{\text{T}}\rightarrow\sqrt{m_{\text{T}}^{2}-m_{0}^{2}}\,.$
(1)

In this way, since the corresponding invariant yields are equal,

$\displaystyle\frac{1}{p_{\text{T}}}\frac{\text{d}^{2}N}{\text{d}p_{\text{T}}\text{d}y}=\frac{1}{m_{\text{T}}}\frac{\text{d}^{2}N}{\text{d}m_{\text{T}}\text{d}y}\,,$
(2)

the scaling can be expressed in terms of transverse momentum where it must be
ensured that both spectra are evaluated at the same transverse mass. Hence, at
midrapidity, the invariant $p_{\text{T}}$ spectrum of a particle of type $X$,
$f^{\text{inv.}}_{\rm X}(p_{\text{T}})$, can be obtained by scaling the
parametrization of the invariant transverse momentum spectrum of another
particle $R$, $f^{\text{inv.}}_{\rm R}(p_{\text{T}})$, used as the reference,
by

$\displaystyle f^{\text{inv.}}_{\rm X}(p_{\text{T,X}})$ $\displaystyle=C^{\rm
X}_{\rm R}\times f^{\text{inv.}}_{\rm
R}\left(\sqrt{m_{\text{T,X}}^{2}-m_{0,{\rm R}}^{2}}\right)$
$\displaystyle=C^{\rm X}_{\rm R}\times f^{\text{inv.}}_{\rm
R}\left(\sqrt{m_{0,{\rm X}}^{2}+p_{\text{T,X}}^{2}-m_{0,{\rm R}}^{2}}\right),$
(3)

where the substitution from Eq. 1 and the evaluation of both spectra at the
same transverse mass,

$\displaystyle p_{\text{T,X}}^{2}+m_{0,{\rm
X}}^{2}=p_{\text{T,R}}^{2}+m_{0,{\rm R}}^{2},$ (4)

were applied to the initial (reference) spectrum. The parameter $C^{\rm
X}_{\rm R}$ denotes a constant offset between the spectra of the two particle
species. It can in principle be obtained from the data if the two particle
spectra exhibit a similar power law at high $p_{\text{T}}$. In the following
we will drop the explicit notation $C_{\rm R}^{\rm X}$ and denote the constant
scaling parameter simply with $C$.

The scaling relation for invariant, $p_{\text{T}}$-differential spectra can be
translated into a relation for non-invariant spectra by multiplying both sides
by their respective $p_{\text{T}}$,

$\displaystyle f_{\rm X}(p_{\text{T,X}})=C$
$\displaystyle\times\frac{p_{\text{T,X}}}{\sqrt{m_{0,{\rm
X}}^{2}+p_{\text{T,X}}^{2}-m_{0,{\rm R}}^{2}}}$ $\displaystyle\times f_{\rm
R}\left(\sqrt{m_{0,{\rm X}}^{2}+p_{\text{T,X}}^{2}-m_{0,{\rm R}}^{2}}\right),$
(5)

where the notation “inv.” was omitted to denote the change to non-invariant
spectra. Formulating approximate $m_{\text{T}}$-scaling in this way is
operationally similar to the generalized approach discussed in
LABEL:SchaffnerBielich:2001qj.

Collective effects, such as radial flow known to be present in heavy-ion, but
also in p-A and perhaps even in pp collisions Loizides:2016tew , should modify
the scaling relation to some degree. This can be demonstrated by assuming a
simple model of flow,

$\displaystyle p_{\text{T}}\rightarrow
p_{\text{T}}^{\prime}=p_{\text{T}}+\beta m_{0},$ (6)

where the particles initially produced with a transverse momentum
$p_{\text{T}}$ are exposed to a boost by the radially expanding system with
flow velocity $\beta$. Such a modification would introduce an additional mass
dependence that affects different species to a different degree, in particular
at low $p_{\text{T}}$. At sufficiently high $p_{\text{T}}$ any influence from
such effects would however diminish.

Although transverse mass scaling can not be expected to hold in general, it is
often applied to $p_{\text{T}}$ spectra in situations, where parametrizations
of particle yields are required for which no measurements are available.
Estimates of hadronic backgrounds in direct photon Adare:2012yt ; Adare:2012vn
; Afanasiev:2012dg ; Adam:2015lda  or di-electron measurements Adare:2009qk ;
Adamczyk:2015lme  are prominent examples where transverse mass scaling is
employed. Furthermore, in such measurements, the parametrizations are usually
extrapolated to lower and higher $p_{\text{T}}$.

In the following we will test the generalized $m_{\text{T}}$-scaling law on
the basis of $p_{\text{T}}$ differential yields of identified hadrons measured
by the ALICE experiment in pp collisions at $\sqrt{s}=7$ TeV. For the purpose
of this article, we focus on mesons, but the same study could be done for
baryons as several different species were measured. At first, we present in
Sec. II a large part of the available data and their parameterizations,
followed by a discussion of the resonance feed-down contribution to the
charged pion spectrum. Then in Sec. III the measured yields are compared to
the predictions obtained from applying the generalized $m_{\text{T}}$-scaling
relation to the parametrizations of different spectra. Furthermore, they are
compared to the particle ratios using the light mesons as reference particles.
We conclude in Sec. IV with a summary of our findings.

##  II Available data and feed-down correction

A large part of the available identified and $p_{\text{T}}$ differential
particle yields measured by the ALICE experiment in proton-proton collisions
at $\sqrt{s}=7$ TeV are summarized in Fig. 2, where statistical uncertainties
are displayed as error bars and systematic uncertainties as open boxes. The
particles shown are $\pi^{\pm}$ Adam:2016dau , $\pi^{0}$ Abelev:2012cn ,
$\eta$ Abelev:2012cn , $\phi$ Abelev:2012hy , $\text{K}^{\pm}$ Adam:2016dau ,
$\text{K}^{*}(892)^{0}$ Abelev:2012hy , $\text{p}/\bar{\text{p}}$ Adam:2016dau
, $\Omega^{-}/\overline{\Omega}^{+}$ Abelev:2012jp ,
$\Xi^{-}/\overline{\Xi}^{+}$ Abelev:2014qqa  and
$\Sigma(1385)^{\pm}/\overline{\Sigma}(1385)^{\pm}$ Abelev:2014qqa . In the
case of charged particles, the average of all contributing yields (e.g.
$\pi^{\pm}=(\pi^{+}+\pi^{-})/2$) is used synonymously for the corresponding
species throughout this article. Furthermore, parametrizations fitted to the
different yields are also shown in Fig. 2. The ratios between the data points
and the fits are presented in Fig. 2. Systematic uncertainties were not
included in the fit and hence are not shown. Except for $\text{K}^{\pm}$, the
fits typically describe the data to better than $10$%. The fit parameters are
summarized in Tab. 1 and are explained below.

Hagedorn (Eq. 7) | Meas. range ($\text{GeV}/c$) | Fit range ($\text{GeV}/c$) | $A$ | $a$ | $b$ | $p_{0}$ | $n$ |  |  | Ref.
---|---|---|---|---|---|---|---|---|---|---
$\pi^{0}$ | $0.3<p_{\text{T}}<25.0$ | $0.3<p_{\text{T}}<16.0$ | $85.9$ | $0.48$ | $-0.008$ | $0.59$ | $6.10$ |  |  | Abelev:2012cn
$\pi^{\pm}$ | $0.1<p_{\text{T}}<20.0$ | $0.2<p_{\text{T}}<20.0$ | $98.1$ | $0.377$ | $-0.074$ | $0.556$ | $6.04$ |  |  | Adam:2016dau
$\text{K}^{\pm}$ | $0.2<p_{\text{T}}<20.0$ | $0.4<p_{\text{T}}<20.0$ | $2.17$ | $0.854$ | $0.00$ | $0.808$ | $5.59$ |  |  | Adam:2016dau
$\text{K}^{*}(892)^{0}$ | $0.0<p_{\text{T}}<\hskip 4.0pt6.0$ | $0.0<p_{\text{T}}<\hskip 4.0pt6.0$ | $0.303$ | $0.79$ | $0.00$ | $1.07$ | $5.37$ |  |  | Abelev:2012hy
$\text{p}/\overline{\text{p}}$ | $0.3<p_{\text{T}}<20.0$ | $0.4<p_{\text{T}}<20.0$ | $0.604$ | $0.602$ | $0.00$ | $1.210$ | $6.86$ |  |  | Adam:2016dau
$\Omega^{-}/\overline{\Omega}^{+}$ | $0.8<p_{\text{T}}<\hskip 4.0pt5.0$ | $0.8<p_{\text{T}}<\hskip 4.0pt5.0$ | $0.0014$ | $0.15$ | $0.00$ | $4.34$ | $15.2$ |  |  | Abelev:2012jp
$\Xi^{-}/\overline{\Xi}^{+}$ | $0.6<p_{\text{T}}<\hskip 4.0pt8.5$ | $0.6<p_{\text{T}}<\hskip 4.0pt8.5$ | $0.012$ | $0.56$ | $0.00$ | $1.75$ | $7.1$ |  |  | Abelev:2014qqa
$\Sigma(1385)^{\pm}/\overline{\Sigma}(1385)^{\pm}$ | $0.7<p_{\text{T}}<\hskip 4.0pt6.0$ | $0.7<p_{\text{T}}<\hskip 4.0pt6.0$ | $0.01$ | $0.76$ | $0.00$ | $1.6$ | $6.5$ |  |  | Abelev:2014qqa
$\omega(782)$ | $2.0<p_{\text{T}}<17.0$ | $2.0<p_{\text{T}}<17.0$ | $0.036$ | $0.707$ | $0.00$ | $2.01$ | $6.00$ |  |  |  Peresunko:2012tt *
|  |  |  |  |  |  |  |  |  |
Tsallis (Eq. 8) |  |  | $\text{d}N/\text{d}y$ | $n$ | $T$ |  |  |  |  |
$\phi$ | $0.4<p_{\text{T}}<\hskip 4.0pt6.0$ | $0.4<p_{\text{T}}<\hskip 4.0pt6.0$ | $0.197$ | $6.85$ | $0.277$ |  |  |  |  | Abelev:2012hy
|  |  |  |  |  |  |  |  |  |
Empirical ratio (Eq. 9) |  |  | $A$ | $\beta$ | $T$ | $N$ | $B$ | $p_{0}$ | $n$ |
$\eta/\pi^{0}$ | $0.4<p_{\text{T}}<15.0$ | $0.4<p_{\text{T}}<\hskip 4.0pt8.0$ | $0.55$ |  $-2.2$e5 | $1.56$ | $39.7$ | $-0.013$ | $0.87$ | $0.49$ | Abelev:2012cn
$\rho^{0}/\pi^{\pm}$ | $0.5<p_{\text{T}}<12.0$ | $0.5<p_{\text{T}}<12.0$ | 0.85 | -3.0e5 | 1.56 | 62.3 | -0.013 | 2.41 | 3.16 |  Riabov:2017jig **
Table 1: Summary of fit parameters obtained for the particle spectra shown in
Figs. 2 and 2. Fit parameter uncertainties are not given since the
parameterizations are just used to approximate the data without implying any
physical interpretation. (*) For the $\omega(782)$ meson, the preliminary
measurement Peresunko:2012tt  has been used for the parameterization. (**) The
$\rho^{0}$ and $\rho^{\pm}$ spectra were obtained by multiplying the
parameterization of the preliminary $\rho^{0}/\pi^{\pm}$ from pp
$\sqrt{s}=2.76$ TeV Riabov:2017jig  with the given charged pion
parameterization.

Figure 1: Identified, $p_{\text{T}}$ differential particle yields measured by
ALICE in pp collisions at $\sqrt{s}=7$ TeV and corresponding parametrizations.
Statistical uncertainties are shown as error bars, systematic uncertainties
are shown as open boxes. The shown data are $\pi^{\pm}$ Adam:2016dau ,
$\pi^{0}$ Abelev:2012cn , $\eta$ Abelev:2012cn , $\phi$ Abelev:2012hy ,
$\text{K}^{\pm}$ Adam:2016dau , $\text{K}^{*}(892)^{0}$ Abelev:2012hy ,
$\text{p}/\bar{\text{p}}$ Adam:2016dau , $\Omega^{-}/\overline{\Omega}^{+}$
Abelev:2012jp , $\Xi^{-}/\overline{\Xi}^{+}$ Abelev:2014qqa  and
$\Sigma(1385)^{\pm}/\overline{\Sigma}(1385)^{\pm}$ Abelev:2014qqa . The lines
are fits to the data summarized in Tab. 1.

Figure 2: Ratios of measured yields to the respective parametrizations and
reduced $\chi^{2}$ values of the fits. In the case of the $\eta$ meson, the
reduced $\chi^{2}$ value of the corresponding $\eta/\pi^{0}$ ratio fit is
given. Only statistical uncertainties are shown. For more info see caption of
Fig. 2.

All particle yields, except those of the $\eta$ and $\phi$ mesons, were
parametrized with a modified Hagedorn function Khandai:2012xx ; Adare:2009qk ,
multiplied with $p_{\text{T}}$ to account for the spectra being non-invariant:

$\displaystyle\frac{\text{d}^{2}N}{\text{d}y\text{d}p_{\text{T}}}=p_{\text{T}}\times
A\times\left(\exp{\left(ap_{\text{T}}+bp_{\text{T}}^{2}\right)}+\frac{p_{\text{T}}}{p_{0}}\right)^{-n}.$
(7)

This functional form approaches an exponential at low $p_{\text{T}}$ and a
power law at larger transverse momenta and describes the measured spectra very
well over the full measured range in $p_{\text{T}}$. In the case of the
$\text{K}^{\pm}$ meson, a step at $p_{\text{T}}\approx 6$ GeV/$c$ is observed,
which can be traced back to significantly different analysis techniques used
for the lower and higher parts of the measured spectrum Adam:2016dau .

The yield of the $\phi$ meson was best described by a Tsallis function
Cleymans:2012ya  multiplied with $p_{\text{T}}$, which can be written as

$\displaystyle\frac{\text{d}^{2}N}{\text{d}y\text{d}p_{\text{T}}}=p_{\text{T}}\times\frac{\text{d}N}{\text{d}y}\times\frac{(n-1)(n-2)}{nT(nT+m(n-2))}\left(1+\frac{m_{\text{T}}-m}{nT}\right)^{-n}.$
(8)

A different approach for the fitting procedure was chosen for the $\eta$
yield. In order to reliably describe the high $p_{\text{T}}$ part of the
spectrum, the ratio to the neutral pion yield, shown in Fig. 4, was fitted
first with an empirical Klaus  function

$\displaystyle\frac{\eta}{\pi^{0}}(p_{\text{T}})=\frac{A\cdot\exp\left(\frac{\beta
p_{\text{T}}-m_{\text{T}}^{\eta}}{T\sqrt{1-\beta^{2}}}\right)+N\cdot
B\cdot\left(1+\left(\frac{p_{\text{T}}}{p_{0}}\right)^{2}\right)^{-n}}{\exp\left(\frac{\beta
p_{\text{T}}-m_{\text{T}}^{\pi^{0}}}{T\sqrt{1-\beta^{2}}}\right)+B\cdot\left(1+\left(\frac{p_{\text{T}}}{p_{0}}\right)^{2}\right)^{-n}}.$
(9)

This function contains two separate contributions from soft and hard
processes, where $B$ is a relative normalization between the soft and hard
part of the parametrization and $N$ is the constant ratio between the two
particle species which is approached at high $p_{\text{T}}$. The soft part of
the spectra is described by a blast wave inspired function Schnedermann:1993ws
that depends on the radial flow velocity $\beta$ and the kinetic freeze-out
temperature $T$. While this Ansatz was originally intended to describe the
production in a system where a hot medium is produced, it also yields a good
description of the data in proton-proton collisions. The fit to the
$\eta/\pi^{0}$ ratio is compared with the data in Fig. 4. At high
$p_{\text{T}}$, the fit approaches a constant ratio of
$R_{\eta/\pi^{0}}\approx 0.55$, which is larger than the global average of
$R_{\eta/\pi^{0}}=0.459\pm 0.013$ at LHC energies Acharya:2017tlv . However,
the fit is only intended to serve as the best possible parametrization of the
available data points within the measured range and should not be taken as a
quantitative estimate of the $\eta/\pi^{0}$ ratio at high $p_{\text{T}}$. The
parametrization of the $\eta$ yield is then obtained by multiplying the
parametrization of the ratio with the parametrization of the neutral pion
yield. It should be noted that the reduced $\chi^{2}$ value for the $\eta$
meson quoted in Fig. 2 refers to the fit of the particle ratio shown in Eq. 9.

Although typically neutral pions are used as a basis for transverse mass
scaling of mesons, we use the charged pions since they are measured with a
better precision and to lower $p_{\text{T}}$ as can be seen in Fig. 2. The
direct comparison of the corresponding neutral and charged pion spectra for
$\sqrt{s}=7$ TeV Abelev:2012cn ; Abelev:2014laa  and $\sqrt{s}=2.76$ TeV
Acharya:2017hyu ; Abelev:2014laa  is shown in Fig. 4. The statistical and
systematic uncertainties are shown as error bars and open boxes, respectively.
As can be seen, the charged and neutral pions spectra agree within the
systematic and statistical uncertainties, in particular when considering that
part of the systematic uncertainties is correlated in $p_{\text{T}}$. The
difference at low $p_{\text{T}}$ is expected from isospin violating decays,
and the ratios are consistent with predictions from PYTHIA Sjostrand:2014zea .

Figure 3: The $\eta/\pi^{0}$ ratio measured in pp collisions at $\sqrt{s}=7$
TeV Abelev:2012cn , the corresponding parametrization and the ratio between
the data and fit. Statistical uncertainties are shown as error bars,
systematic uncertainties are shown as open boxes.

Figure 4: The $\pi^{0}/\pi^{\pm}$ ratio measured in pp collisions at
$\sqrt{s}=7$ TeV Abelev:2012cn ; Abelev:2014laa  and $\sqrt{s}=2.76$ TeV
Acharya:2017hyu ; Abelev:2014laa . Statistical uncertainties are shown as
error bars, systematic uncertainties are shown as open boxes. The ratios are
compared to calculations using PYTHIA 8.2, Monash 2013 tune Sjostrand:2014zea
.

The general transverse mass scaling relation as represented in Eq. 5 is
defined for primary particles. However, there is a significant contribution to
the measured yields from particle decays, especially to light particles as
$\pi^{\pm}$ (or $\pi^{0}$). For example, in the case of the $\pi^{\pm}$
measurement, contributions from weak decays (so-called secondaries) are
subtracted from the measured spectrum Adam:2016dau  in accordance with the
definition of primary particles that is used within ALICE ALICE-
PUBLIC-2017-005 .
Nevertheless, this does not remove all possible feed-down contributions from
resonance decays. In order to estimate the effect of feed-down on the
applicability of the scaling relation, the charged pion yield was calculated
for some of those contributions. This was done with a particle decay
simulation using the PYTHIA 6.4 “particle-decayer” algorithm Sjostrand:2006za
. The largest sources of charged pions are $\rho^{0,\pm}$, $\omega$, $\eta$,
$\eta^{\prime}$, $\phi$, $\text{K}^{*}(892)^{0}$ as well as the various
$\Delta$, $\Xi$, $\Omega$ and $\Sigma$ resonances. In pp collisions at
$\sqrt{s}=7$ TeV the $\eta$ Abelev:2012cn , $\phi$ Abelev:2012hy ,
$\text{K}^{*}(892)^{0}$ Abelev:2012hy  and $\Sigma(1385)^{\pm}$ Abelev:2014qqa
$p_{\text{T}}$-differential cross sections have been measured and a
preliminary $\omega$ yield Peresunko:2012tt  above $4$ GeV/$c$ is available.
For the $\rho$ meson no yield measurements in pp collisions at $\sqrt{s}=7$
TeV have been performed, yet. Thus, the preliminary $\rho^{0}/\pi^{\pm}$ for
pp collisions at $\sqrt{s}=2.76$ TeV Riabov:2017jig  has been parametrized,
assuming no collision energy dependence of the particle ratio. Afterwards,
this has been multiplied with the charged pion yield seen in pp collisions at
$\sqrt{s}=7$ TeV to obtain the $p_{\text{T}}$ differential $\rho$ yield at
this center-of-mass energy. The remaining particles were generated according
to the parametrized $p_{\text{T}}$-yields quoted in Tab. 1 uniformly in
rapidity and azimuthal angle with the ranges $[-1,1]$ and $[0,2\pi]$,
respectively. Charged pions from decays of the generated mother particles were
counted within the originally analyzed rapidity range of $[-0.8,0.8]$. The
contributions obtained from the decay simulation as well as the primary
charged pion spectrum, which has been generated according to the measured
spectrum, are shown in Fig. 5. It can be seen that the by far strongest
contribution among the considered sources arises from $\rho$ decays followed
by the $\omega$ decays, while the others are sub-dominant. Above transverse
momenta of $3$–$4$ GeV/$c$ all feed-down contributions appear to have a
similar slope, which is expected as all mother particles have a similar slope
as well. Below $3$ GeV/$c$, on the other hand, the shape of the different
contributions differs significantly and the total charged pion yield can be
understood as the sum of all of those contributions. However, the
uncertainties of the parameterizations of the mother particles are significant
below $1$ GeV/$c$ and thus the subtraction of the different feed-down
contributions has not been done. Additional contributions will arise from the
remaining baryonic resonances, which should have similar transverse momentum
distribution as the contribution from the $\Sigma(1385)^{\pm}$ decays, but are
not considered in this study.
Based on these observations it can be inferred, that the transverse mass
scaling based on the charged or neutral pion will break down at low
$p_{\text{T}}$, as the measured pion spectrum is not governed by just a single
(exponential) function of the pion mass. For pp collisions at center-of-mass
energies $8$–$16$ GeV, for which the initial scaling relation was observed,
heavy —in particular baryonic— resonance production is strongly suppressed and
thus the pion spectra from different sources are more similar. With increasing
center-of-mass energy, however, the contribution increases changing the shape
of the pion spectrum at low transverse momenta and thus the scaling only holds
at higher transverse momenta, where the spectra are dominated by the hard
particle production mechanisms. If a heavier reference particle is chosen for
the scaling the influence of the resonances decreases. However, the scaling
might also break down at lower $p_{\text{T}}$ than for the pions for the same
reason and a similar feed-down study should be performed for the corresponding
reference particles.

Figure 5: Primary charged pion yields (black) in pp collisions at $\sqrt{s}=7$
TeV , according to the primary particle definition in ALICE-PUBLIC-2017-005 ,
generated according to the charged pion parameterization given in Tab. 1.
Furthermore, the different feed-down contribution from a particle decay
simulation are shown per source. The uncertainties given are related to the
statistics of the decay particle simulation and the uncertainties on the
parameterization of the sources are not taken into account.

##  III Results

The general transverse mass scaling relation according to Eq. 5 was applied to
the yield parametrizations of different reference particles: $\pi^{\pm}$,
$\eta$, $\phi$ and $\text{K}^{\pm}$, in order to evaluate the applicability of
the scaling. The corresponding scaling parameter $C$ from Eq. 5 has been
estimated by fitting the respective particle ratios at high transverse
momenta, where the $p_{\text{T}}$ threshold has been chosen such, that the
asymptotic value has been reached. For the ratios of the $\phi$ meson to the
different reference particles the plateau values in pp collisions at
$\sqrt{s}=7$ TeV are not yet reached, thus the corresponding ratios in pp
collisions at $\sqrt{s}=2.76$ TeV have been fitted, as they extend further in
$p_{\text{T}}$. This assumes that the $\phi$-related ratios at high
$p_{\text{T}}$ do not depend strongly on the center-of-mass energy at LHC
energies. The corresponding particle ratios with the various reference
particles can be found in Figs. 7–9, where the obtained high $p_{\text{T}}$
values are displayed as blue band including the corresponding uncertainties.
Furthermore, the particle ratios of the $\eta$ and $\phi$ meson with respect
to the neutral pion are displayed in Fig. 9 for completeness, as these are
usually used for the validation of the $m_{\text{T}}$-scaling procedure for
the direct photon and low mass di-electron analyses. The obtained scaling
parameters are listed in Tab. 2.

| Scaled
---|---
Basis | $\eta$ | $\phi$ | $(\text{K}^{+}+\text{K}^{-})/2$
$\pi^{0}$ | $0.482\pm 0.025$ | $0.199\pm 0.013$ | -
$(\pi^{+}+\pi^{-})/2$ | $0.482\pm 0.030$ | $0.196\pm 0.013$ | $0.503\pm 0.004$
$\eta$ | - | $0.344\pm 0.016$ | $1.113\pm 0.031$
$\phi$ | $2.90\pm 0.13$ | - | $2.54\pm 0.17$
$(\text{K}^{+}+\text{K}^{-})/2$ | $0.898\pm 0.025$ | $0.393\pm 0.026$ | -
Table 2: Constant scaling factors ($C$ in the scaling relation, Eq. 5)
obtained from the ratio fits for different combinations of reference and
scaled particle types for pp collisions at $\sqrt{s}=7$ TeV at high transverse
momenta. For the $\phi$ meson ratios the high $p_{\text{T}}$ points for pp
collisions at $\sqrt{s}=2.76$ TeV have been considered in the fits.

Figure 6: The $\eta/\pi^{\pm}$ (left) and K${}^{\pm}/\pi^{\pm}$ ratio (right)
measured in pp collisions at $\sqrt{s}=7$ TeV together with the ratios using
the different particles as basis for the scaling of the $\eta$ meson and
charged kaon and the parameterization of the charged pion, respectively. The
fitted high $p_{\text{T}}$ constant $C$ is reported as well as a blue band in
the $p_{\text{T}}$ region, where it was obtained. For comparison the
corresponding particle ratios measured in pp collisions at $\sqrt{s}=2.76$ TeV
are displayed as well Acharya:2017hyu ; Abelev:2014laa .

Figure 7: The $\eta/$K± (left) and $\phi/\pi^{\pm}$ ratio (right) measured in
pp collisions at $\sqrt{s}=7$ TeV together with the ratios using the different
particles as basis for the scaling of the $\eta$ and $\phi$ meson and the
parameterizations of the charged kaon and pion, respectively. For comparison
the corresponding particle ratios measured in pp collisions at $\sqrt{s}=2.76$
TeV are displayed as well Acharya:2017hyu ; Abelev:2014laa ; Adam:2017zbf .
The fitted high $p_{\text{T}}$ constant $C$ is reported as well as a blue band
in the $p_{\text{T}}$ region, where it was obtained. The scaling constant for
the $\phi/\pi^{\pm}$ is obtained solely on the data obtained for pp collisions
at $\sqrt{s}=2.76$ TeV , as the plateau value has not been reached for the
results at $7$ TeV.

Figure 8: The $\phi/$K± (left) and $\phi/\eta$ ratio (right) measured in pp
collisions at $\sqrt{s}=7$ TeV together with the ratios using the different
particles as basis for the scaling of the $\phi$ meson and the
parameterizations of the charged kaon and $\eta$ meson, respectively. For
comparison the corresponding particle ratios measured in pp collisions at
$\sqrt{s}=2.76$ TeV are displayed as well Acharya:2017hyu ; Abelev:2014laa ;
Adam:2017zbf . The fitted high $p_{\text{T}}$ constant $C$ is reported as well
as a blue band in the $p_{\text{T}}$ region, where it was obtained. Both
scaling constants mainly rely on the data obtained for pp collisions at
$\sqrt{s}=2.76$ TeV, as the plateau value has not been reached for the results
at $7$ TeV.

Figure 9: The $\eta/\pi^{0}$ (left) and $\phi/\pi^{0}$ ratio (right) measured
in pp collisions at $\sqrt{s}=7$ TeV together with the ratios using the
different particles as basis for the scaling of the $\eta$ and $\phi$ meson,
respectively, and the parameterization of the neutral pions. For comparison
the corresponding particle ratios measured in pp collisions at $\sqrt{s}=2.76$
TeV and $8$ TeV are displayed as well Acharya:2017hyu ; Adam:2017zbf ;
Acharya:2017tlv . The fitted high $p_{\text{T}}$ constant $C$ is reported as
well as a blue band in the $p_{\text{T}}$ region, where it was obtained. For
the $\phi/\pi^{0}$ the scaling constant was obtained based on the data
measured at $\sqrt{s}=2.76$ TeV, as the plateau value has not been reached for
the results at $7$ TeV.

As the particle rest masses of the $\eta$ meson and charged kaon are fairly
similar the scaling factor is very close to unity for these combinations, and
the $p_{\text{T}}$ threshold for the high momentum fit can be significantly
lowered from about $4$ GeV/$c$, as for most of the other combinations, to $1$
GeV/$c$. In the case of the scaling of the $\eta$ and K± from the $\phi$
meson, large uncertainties on the scaling factors are observed. This is due to
the large statistical uncertainties associated to the $\eta$ and $\phi$
spectra (see Fig. 9) in combination with the strong restriction on the fit
range due to the large mass of the $\phi$ meson. It should also be noted that
some of the factors listed in Tab. 2 are larger than one, which is due to the
fact that in those cases the reference particle is heavier than the particle
that it is scaled to. This poses a limit for the scaled function since
$\sqrt{m_{0,{\rm X}}^{2}+p_{\text{T,X}}^{2}-m_{0,{\rm R}}^{2}}$ becomes
imaginary below a certain $p_{\text{T}}$. In these cases a lower limit of
$p_{\text{T}}\geq\sqrt{m_{0,{\rm R}}^{2}-m_{0,{\rm X}}^{2}}$ was used
generally, i.e. also for the extrapolated scaled function. However, this limit
does not affect the scaling procedure itself but places only a restriction on
the $p_{\text{T}}$ range of the extrapolated function obtained from the
scaling procedure.

Figure 10: Ratio of the measured $\eta$ yield (points) and its fit (lines) to
the scaled parametrization obtained from different reference particles. The
$p_{\text{T}}$ thresholds for the obtained scaling constants ($C$) are shown
as vertical red line and reflect the point at which the ratio between scaled
and reference particles is approximately flat.

Figure 11: Ratio of the measured $\phi$ yield (points) and its fit (lines) to
the scaled parametrization obtained from different reference particles. The
$p_{\text{T}}$ thresholds for the obtained scaling constants ($C$) are shown
as vertical red line and reflect the point at which the ratio between scaled
and reference particles is approximately flat.

Figure 12: Ratio of the measured charged kaon yield (points) and its fit
(lines) to the scaled parametrization obtained from different reference
particles. The $p_{\text{T}}$ thresholds for the obtained scaling constants
($C$) are shown as vertical red line and reflect the point at which the ratio
between scaled and reference particles is approximately flat.

Figures 12 to 12 show the comparison of the different parametrizations
obtained from the $m_{\text{T}}$-scaling approach formulated in Eq. 5 to the
parametrizations of the corresponding particle yields. The performance of the
scaling is quantified as the ratio between the measured yield (shown as
points) and the scaled parametrization obtained from different reference
particles, and additionally with the ratio of the fit (shown as lines) to the
scaled parametrization. The red vertical lines mark the different
$p_{\text{T}}$ thresholds, which were used to establish the scaling constant
and which depend on the masses of the particles, that were involved in the
scaling and the $p_{\rm T}$ reach of the data, while the horizontal lines mark
$\pm 20$% deviation of the ratio from unity. Below the threshold, we find that
the ratio generally differs significantly from the measured spectra, and hence
one can conclude that the scaling does not work for low transverse momenta,
i.e. in the region significantly lower than the threshold used to determine
the scaling. In particular, the $\eta$ meson and charged kaon can not be well
reproduced by scaling the charged pions, which is also visible in Figs. 7–9.
If a heavier reference particle is used, on the other hand, like the charged
kaon or the $\eta$ meson the scaling performs better and the spectra can be
reproduced down to lower transverse momenta. This is in particular the case
when scaling the $\eta$ meson from the charged kaon and vice versa due to
their similar mass. It can also be observed for the $\phi$ meson, if it is
scaled from the charged kaon or $\eta$ meson, respectively. This supports the
assumption that the feed-down from heavier resonances explicitly breaks the
scaling from the pions and that this has a smaller influence when using the
charged kaons or eta mesons as reference particle. However, it has to be
ensured that the high $p_{\text{T}}$ scaling factor $C$ can be obtained with
sufficient precision, which was not possible using only the data measured in
pp collisions at $\sqrt{s}=7$ TeV. Similar conclusions can be drawn from the
comparisons of the calculated and measured particle ratios in Figs. 7–9 in
case the desired particle has been obtained from the parametrization of the
charged kaon (red) or $\eta$ meson (blue) yields. If the charged kaon and
$\eta$ meson were obtained from the $\phi$ meson parameterization (green) the
description of the data is worse than for other combinations, which can be
explained by the imprecise $C$ values as well as the limited transverse
momentum reach of the respective yield measurement of the $\phi$ meson.
Additionally, it can be seen that for parameterizations with a similar $n$ of
the power-law the ratio flattens at higher momenta, while a residual slope can
be observed if different $n$ values are obtained for the parameterizations of
the particles divided in the ratio. This implies that the scaling relation can
only hold for particle spectra, which have a similar power-law behavior at
high transverse momenta, as predicted. It is reflected in the agreement
between the scaled curves and the actual parameterizations above the chosen
$p_{\text{T}}$ thresholds for those particles, which have a similar $n$
according to Tab. 1. In these cases the agreement between the data and the
parameterizations is usually better than $5-10$%. For the charged kaons, on
the hand, a strong $p_{\text{T}}$-dependent difference at high transverse
momenta is observed. We have verified that the scaling at high $p_{\text{T}}$
cannot be restored fully by removing the step observed in Fig. 2 through
shifting the low and high $p_{\text{T}}$ points down and up, respectively. For
pp collisions at $\sqrt{s}=2.76$ TeV, however, the ratio of
K${}^{\pm}/\pi^{\pm}$ Adam:2016dau  appears to be constant above $4$ GeV/$c$
and approaches a similar value as obtained for pp collisions at $\sqrt{s}=7$
TeV, which is shown in Fig. 7. Hence, it is likely that the apparent deviation
of the $\text{K}^{\pm}/\pi^{\pm}$ from $\approx 0.5$ is just driven by the two
highest $p_{\text{T}}$ points, but with the data one can also not rule out
that it is broken at high $p_{\text{T}}$ in pp collisions at 7 TeV.

##  IV Summary

We have presented a study on the applicability of transverse mass scaling for
identified particle spectra in proton-proton collisions at $\sqrt{s}=7$ TeV
based on data taken by the ALICE experiment at the LHC. The measured yields of
$\pi^{\pm}$, $\eta$, $\phi$ and K± mesons were parametrized and compared to
estimates obtained from a generalized transverse mass scaling approach applied
to different reference particle spectra. Generalized transverse mass scaling
is found not to be able to describe the measured spectra over the full range
in transverse momentum, especially not when pions are used as reference
particles. At low $p_{\text{T}}$, deviations of $20$% or more are observed. A
better scaling performance is obtained, when kaons are used as reference
particles. At high $p_{\text{T}}$ all spectra with the possible exception of
the charged kaons exhibit a scaling behavior. Investigating the feed-down
contributions from resonance decays to the charged pion yields reveals that a
general scaling may not be expected when using them as reference. A similar
study should be done for heavier reference particles, in particular for kaons,
which within the available statistics exhibit a potential to be used as
reference particle instead of pions. Our findings imply that for precision
measurements of direct photon and di-electron spectra at low transverse
momentum one should measure the relevant hadronic background, instead of
relying on $m_{\text{T}}$ scaling for its estimate.

## Acknowledgments

The work of L.A. is supported by the Research Council of Norway, Norway. The
works of C.L. and N.S. are supported by the U.S. Department of Energy, Office
of Science, Office of Nuclear Physics, under contract number DE-
AC05-00OR22725.

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