11institutetext: Department for Atomic Physics, Eotvos Lorand University,
1A Pazmany Peter Walk, H-1117 Budapest, Hungary 22institutetext: Department of
Computational Sciences, Wigner Research Centre for Physics,
29-33 Konkoly-Thege Mikloos Street, H-1121 Budapest, Hungary

# Scale dependence of the $q$ and $T$ parameters of the Tsallis distribution
in the process of jet fragmentation

Karoly Urmossy _email address:_<EMAIL_ADDRESS>Antal Jakovac 22

(Received: date / Revised version: date)

###### Abstract

The dependence of the $q$ and $T$ parameters of the Tsallis-distribution-
shaped fragmentation function (FF) on the fragmentation scale (found to be
equal to the jet mass) is calculated via the resummation of the branching
process of jet fragmentation in the leading-log appriximation (LLA) in the
$\phi^{3}$ theory. Jet and hadron spectra in electron-positron ($e^{+}e^{-}$)
annihilations with 2- and 3-jet final states are calculated using virtual
leading partons. It is found that jets, produced earlier in the branching
process, are more energetic, and the energy, angle and multiplicity
distributions of hadrons stemming from them are broader. It is also found that
replacing the LL resummation in the branching process by a single splitting
provides good approximation for the jet energy distribution in 2-jet events.
Furthermore, a micro-canonical statistical event generator is presented for
the event-by-event calculation of hadron momenta in $e^{+}e^{-}$
annihilations.

###### pacs:

13.87.FhFragmentation into hadrons and 12.40.EeStatistical models of strong
interactions 05.40.-a Fluctuation phenomena-statistical physics

##  1 Introduction

The spectra of hadrons produced in high-energy collisions are more-or-less
well described by various versions of phenomonological models based on the
cut-power law or Tsallis-distribution (TS),
$f_{TS}(E)\propto[1+(q-1)E/T]^{-1/(q-1)}$. $E$ is the energy of the hadron,
and $q$ measures the deviation from the Boltzmann-Gibbs (BG) distribution
($q=1$). The dependence of the $q$ and $T$ parameters on the features of the
collisions ($\sqrt{s}$, centrality, type and number of the produced final
state particles) has extensively been studied recently in electron-positron
($e^{+}e^{-}$) bib:Becattini –bib:UKee , positron-proton ($e^{+}p$)
bib:FHLiu10 –bib:UKdis , proton-proton ($pp$) bib:Trainor –bib:Zhangbu2 ,
proton-nucleus ($pA$) and deuteron-nucleus (dA) bib:FHLiu2 –bib:Zhangbu2 ,
nucleus-nucleus (AA) bib:FHLiu2 –bib:WCZhang7  collisions, and Drell-Yan
processes bib:FHLiu14 . The found tendencies as well as the quality of the
fits to measured data depend on the version of the model used. Some models
conjecture that quark-gluon plasma (QGP), an expanding and cooling thermal
source is created in the examined collision, and use the Cooper-Frye formula
to calculate the distribution of hadrons produced at break-up bib:FHLiu5
–bib:Zhangbu3 , bib:LLiu –bib:Zhangbu2 , bib:FHLiu7 –bib:UKaa , bib:FHLiu15
–bib:Zhangbu4 . These models provide a better agreement with measured data in
$AA$ collisions, especially in the interval, where the hadron’s transverse
momentum is smaller than its mass $p_{T}\lessapprox m_{h}$. However, in order
to describe both the spectra and the azymuthal anisotropy $v_{2}$ in AA
collisions at $\sqrt{s}\geq 200$ GeV, and $p_{T}$ up to 20 GeV/c, two-
component models bib:FHLiu13 –bib:Zhangbu4  are needed. Such models make out
hadron yields from those hadrons, which stem from the QGP (’soft’ component),
and those, which stem from jets (’hard’ component). It has been found that the
more central the collisions (the greater the hadron multiplicity), the larger
the ratio of the soft hadrons and the closer their TS to the BG distribution.
This suggests that in more central collisions, the QGP is closer to
equilibrium bib:WCZhang7 –bib:Zhangbu4 . The distribution of the hard yields,
which dominates the spectrum and $v_{2}$ for $p_{T}\gtrapprox$ 5 GeV/c, agrees
with spectra measured in peripheral AA, or pp collisions. Its $q$ parameter
increases with $\sqrt{s}$ and hadron multiplicity. Quark-coalescence models
bib:UKaa  suggest that the power of hadron spectra scales with the number of
constituent quarks, thus, the $q$ parameters of mesons and baryons obey the
relation $(q_{M}-1)/(q_{B}-1)=3/2$. Although, this precise ratio has not been
observed, it was found that $q_{M}>q_{B}$. Besides, the $T$ parameter was
found to depend linearly on the hadron mass. In case of $e^{+}e^{-}$
annihilations, hadron spectra are measured up to a range, where hadron
energies are comparable with $\sqrt{s}$, thus, micro-canoncal statistical
models are used bib:Becattini –bib:Begun3  for the description of
fragmentation functions ($FF$) and event shapes. The canonical statistical
version of the TS distribution fails to describe the spectra of jets bib:LNGao
; bib:FHLiu12  and direct photons bib:FHLiu1  for $p_{T}\gtrapprox
0.1\sqrt{s}$ in $pp$, $pA$ and $AA$ collsions.

From the theoretical point of view, the TS distribution has been obtained in
the canonical and micro-canoncal ensembles in case when the temperature
bib:Beck ; bib:Wilk1 , volume bib:Begun2 ; bib:Begun3  or multiplicity
bib:BiroNflukt  fluctuates according to specific patterns due to some external
reasons. The TS is also the steady solution of the Langevin and Fokker-Planck
equations with damping and noise terms depending linearly on the particle’s
energy bib:BiroJako . Besides, thermodynamic properties of the Linear Sigma
Model bib:Castano , the Nambu-Joana-Lasinio Model bib:YPZhao , a hadron gas in
the presence of magnetic field bib:Pradhan  along with general thermodynamic
relations bib:Ishihara  and a small $q-1$ expansion have been obtained in the
framework of the TS statistics. It is, however, important to point out that a
solid derivation of the TS distribution from first principles of the strong
interaction is still missing. Nevertheless, it has been shown in bib:CYWong
that the $p_{T}$ spectrum of a leading parton (jet) obtained from the leading
order QCD cross section of two-parton scatterings in $pp$ collisions,
approximately takes the form $d\sigma^{\,Jet}/dx\approx(1-x)^{a}/x^{4.5}$,
with $x=2p_{T}/\sqrt{s}$. The $(1-x)^{a}$ factor comes from the form of the
most commonly used parton distribution functions (PDF), and the $x^{-(4+1/2)}$
factor stems from the hard scattering of partons in the incoming protons and
the integration for the not measured other jet momenta. This result has
estimated the power of jet spectra in accordance with Tevatron and LHC
results, and it has also accounted for the rapid decrease of the distribution
for large $p_{T}$, where the jet energy becomes comparable with $\sqrt{s}$. In
the paper, it is also argued that the low-$p_{T}$ behaviour of jet spetra,
where multiple scattering and non-perturbative processes become important,
cannot be explained based on the single hard scattering of partons. These
arguments suggest that the $q$ parameter of the TS may be calculated in
perturbation theory (PT), but the $T$ parameter is of non-perturbative origin.

To avoid having to deal with the parton structure of the colliding particles,
in this paper, we examine $e^{+}e^{-}$ annihilations, and derive the scale
dependence of the $q$ and $T$ parameters of the TS shaped hadron spectrum in
fragmentation processes. Inspired by bib:Graz ; bib:p3dglp ; bib:CollinsMC ,
we use the $\phi^{3}$ model, which is the simplest asymptotically free quantum
field theory (QFT) mimicing the 3-gluon vertex of QCD. As perturbative methods
for the calculation of the fragmentation of an off-shell leading parton to
hadrons need a non-perturbative input, the form of the fragmentation function
(FF) at a low starting scale, which cannot yet be derived from first
principles, there is room for model building at this point. For this purpose,
we use the model bib:UKpp3D ; bib:UKdis  presented in Sec. 2, because it takes
into account the finitness of the total energy of the produced hadrons using
micro-canonical statistics, and it obtains the TS distribution making use of
the experimentally observed negative binomial (NBD) hadron multiplicity
fluctuations. Similarly to bib:p3dglp ; bib:QCDdglp , in Sec. 3, we start from
the Dyson-Schwinger equation (DSE) for the fragmentation of a single parton to
hadrons, we put momenta of daughter partons onto the mass-shell, and sum up
the leading-log (LL) terms via the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi
(DGLAP) equation bib:DGLAP . We obtain the dependence of the $q$ and $T$
parameters on the virtuality of the leading parton (scale) via solving the
DGLAP equation for the moments of the FF. In Sec. 4, we calculate jet and
hadron distributions in $e^{+}e^{-}$ annihilations with 2-, and 3-jet final
states using a Monte-Carlo (MC) event generator method based on the previously
obtained FF. We summarize the results in Sec. 5.

The approach presented in this paper mainly differs from most pQCD methods in
that it allows for the virtualities of the leading partons to be arbitrarily
large (within the boundaries set by energy-momentum conservation), whereas
usual parton model calculations use on-shell leading partons, as they rely on
factorisation theorems (FT). Consequently, we obtain broad distributions for
jet masses and for the angles between jet and hadron momenta, unlike in works
like bib:dEnterria1  which use a kinematic approximation in which, momenta of
leading and daughter partons are nearly collinear. FTs bib:fact ; bib:fact2
prove that if jets are highly boosted bunches of particles of low total
momentum squared ($M_{J}=\sqrt{P_{J}^{2}}\ll E_{J}$), and the angles between
jets are large, then, their contribution to cross sections can be expressed by
multiplicative factors (convoluted with the hard part of the process).
Interference terms coming from gluon exchange between jets or jets and soft
processes can either be incorporated into the jet factors via Wilson-lines, or
can be neglected, as they are suppressed by factors of $M_{J}/E_{J}\ll 1$.
However, in a portion of events, the basic conditions of FTs do not hold, as
jet masses are comparable to jet energies according to measurements
bib:MjetEE1 ; bib:MjetEE2 ; bib:MjetPP .

Our approach resembles the recursive method presented in bib:CollinsMC  which
generates virtual daughter partons in each round, however, in two steps. In
the first step, it generates on-shell daughter partons using a fixed-order
cross sections, then in the second step, it generates daughter parton ’masses’
using the distribution obtained from the fixed-order cross section of on-shell
grand-daughter production. This procedure starts again in the next round,
until parton virtualities decrease to the order of hadron masses. This way,
every parton in a given generation of the cascade may be virtual, however, the
vector part of their momenta are generated according to cross sections
involving mother and daughter partons, while virtualities are obtaied from
cross sections involving daughter and grand-daughter ones. In our method on
the other hand, only the leading parton of a jet is virtual, as it produces
on-shell daughters in the fragmentation process, but both the vector part of
its momentum and its ’mass’ are generated from the same distribution obtained
from the LL resummation of the fragmentation process (not from fixed-order
graphs, as in bib:CollinsMC ).

##  2 A statistical model for the FF at starting scale

We use the minimalistic conjecture, that at the starting scale $M_{0}$, the
sole constraint required throughout the hadronisation process is the
conservation of energy-momentum. This way, hadrons stemming from the leading
parton of momentum $P=\left(\sqrt{M_{0}^{2}+\mathbf{P}^{2}},\mathbf{P}\right)$
form a micro-canonical ensemble. In the micro-canonical ensemble, the main
quantity, which determines the distribution of particles, is the phasespace,
which, for $n$ massless particles in $D$ dimensions is

$\Omega_{n}(P)\;=\;\prod_{i=1}^{n}\int\frac{d^{D-1}\mathbf{p}_{i}}{p_{i}^{0}}\,\delta^{D}\left(\sum_{j}p_{j}^{\mu}-P^{\mu}\right)\sim
M_{0}^{n(D-2)-D}\;.$ (1)

This result follows from dimensional analysis and Lorentz-invariance, but the
detailed calculation can be found in Appendix A. Consequently, the one-
particle distribution in the micro-canonical ensemble is

$d_{n}(x,M_{0})\;=\;\frac{\Omega_{n-1}(P-p)}{\Omega_{n}(P)}=\frac{\left(\frac{2}{M_{0}}\right)^{\omega^{\ast}}\Gamma\left(\frac{\omega^{\ast}n}{2}\right)}{\kappa_{D-1}\Gamma(\omega^{\ast}\textasciibreve)\Gamma\left[\frac{\omega^{\ast}(n-2)}{2}\right]}\left(1-x\right)^{\frac{(n-2)\omega^{\ast}}{2}-1}\;,$
(2)

with $x=2pP/P^{2}$, $\omega^{\ast}=D-2$ and $D>2$. For example, the properly
normalized distributions in $D=4$ and 6 dimensions are

$d_{n}(x,M_{0})\;=\;\left\\{{{\frac{(n-1)(n-2)}{\pi
M_{0}^{2}}(1-x)^{n-3}\,,\qquad\textrm{if}\;D=4}\atop{\frac{(2n-1)\dots(2n-4)}{\pi^{2}M_{0}^{4}}(1-x)^{2n-5}\,,\qquad\textrm{if}\;D=6\;.}}\right.$
(3)

In $D=4$ dimensions, for $n>3$, the probability of a particle to acquire the
maximal energy (which is $M_{0}/2$ in the co-moving frame) is zero, as
$d_{n}(1,M_{0})=0$. The $n=3$ case is exceptional, as $d_{3}(x,M_{0})$ is a
uniform distribution. In the $n=2$ case, $d_{2}(x,M_{0})\sim\delta(1-x)$, and
energy-momentum conservation cannot be satisfied for $n<2$. Interestingly, the
same variable $x$, which is used in parton model calculations, emerges as the
natural variable of the micro-canonical ensemble.

As measured fragmentation functions are averaged over the fluctuation of
hadron multiplicity, we make use of the experimental observation bib:NjetPP ;
bib:UKee  that the multiplicity of hadrons in a jet fluctuates according to
the negative-binomial distribution (NBD)

$\mathcal{P}_{n}\;=\;{{n+r-1}\choose{n}}\bar{p}^{n}(1-\bar{p})^{r}\;,$ (4)

with parameters $\bar{p},r$, expectation value $\bar{n}=r\bar{p}/(1-\bar{p})$
and scattering $\sigma^{2}=r\bar{p}/(1-\bar{p})^{2}$. Thus, we use the one-
particle micro-canonical distribution averaged over multiplicity fluctuations
as fragmentation function at starting scale $M_{0}$:

$\displaystyle d\left(x,M_{0}^{2}\right)$ $\displaystyle\;=\;$
$\displaystyle\sum_{n}\mathcal{P}_{n}\,n\,d_{n}(x,M_{0})=$ (5)
$\displaystyle=$
$\displaystyle\frac{2\left(\frac{2}{M_{0}}\right)^{\omega^{\ast}}(1-\bar{p})^{r}}{\kappa_{D-1}\Gamma(\omega^{\ast}+1)}\left(-\frac{\partial}{\partial
x}\right)^{\omega^{\ast}+1}\sum_{n}{{n+r-1}\choose{n}}\bar{p}^{n}(1-x)^{n\omega^{\ast}/2}$
$\displaystyle=$
$\displaystyle\frac{2\left(\frac{2}{M_{0}}\right)^{\omega^{\ast}}(1-\bar{p})^{r}}{\kappa_{D-1}\Gamma(\omega^{\ast}+1)}\left(-\frac{\partial}{\partial
x}\right)^{\omega^{\ast}+1}\left[1-\bar{p}(1-x)^{\omega^{\ast}/2}\right]^{-r}\;.$

The evaluation of this formula in $D=4$ dimensions results in the Tsallis
distribution

$d(x,M_{0}^{2})\;=\;\left(\frac{\bar{p}}{1-\bar{p}}\right)^{3}\frac{r(r+1)(r+2)}{\pi
M_{0}^{2}}\left(1+\frac{\bar{p}}{1-\bar{p}}x\right)^{-(r+3)}\,,$ (6)

in which, the parameters of the multiplicity distribution expressed with the
usual $q$ and $\tau$ parameters are

$r+3\;=\;\frac{1}{q-1}\;,\qquad\frac{\bar{p}}{1-\bar{p}}=\frac{q-1}{\tau}\;.$
(7)

The usual tempereature is $T=\tau M_{0}/2$.

Note that, when calculating Eq. (6), the $n=2$ term $d_{2}(x,M_{0})$ was left
out of the sum in Eq. (5). Besides, the TS distribution does not go to zero at
$x=1$, due to the uniform distribution $d_{3}(x,M_{0})$ which was included in
the sum in Eq. (5). If we leave out the $n=3$ term from the sum, we arrive at
a modified version of the TS distribution

$d^{\ast}(x,M_{0}^{2})\;=\;\left(\frac{\bar{p}}{1-\bar{p}}\right)^{3}\frac{r(r+1)(r+2)}{\pi
M_{0}^{2}}\left[\left(1+\frac{\bar{p}}{1-\bar{p}}x\right)^{-(r+3)}-\left(1+\frac{\bar{p}}{1-\bar{p}}\right)^{-(r+3)}\right]\,,$
(8)

which obeys $d^{\ast}(1,M_{0}^{2})=0$, thus, it decreases more rapidly than
the TS function for $x\geq 0.1$ providing a better description of hadron
spectra in $e^{+}e^{-}$ annihilations, and jet spectra in $pp$ collisions.

##  3 Scale evolution of the FF

Figure 1: Graphs contributing to the fragmentation of a virtual parton of
momentum $P$ into a measured hadron of momentum $p$ plus anything at order
$g^{4}$ in the $\phi^{3}$ theory.

To obtain $D(p,P)$, the probability of a parton of momentum $P$ producing a
hadron of momentum $p$ (the amputated cut graph blob in Fig. 1.a) at higher
scales $P^{2}\geq M^{2}_{0}$, we use the DSE at $\mathcal{O}(g^{4})$:

$\displaystyle D(p,P)$ $\displaystyle\;=\;$ $\displaystyle
g^{2}\int\frac{d^{D}k}{(2\pi)^{D}}\,\frac{D(p,k)\bar{D}(P-k)}{k^{4}(P-k)^{4}}\left[1+g^{2}\int\frac{d^{D}q}{(2\pi)^{D}}\left\\{\frac{i\,n_{c}}{q^{2}(q-k)^{2}(P-q)^{2}}\right.\right.+$
(9)
$\displaystyle+\left.\frac{i\,n_{d}}{k^{2}q^{2}(q-k)^{2}}+\frac{i\,n_{e}}{(P-k)^{2}q^{2}(P-q-k)^{2}}\right\\}+\left.\bar{n}_{c}\left(\delta
Z_{g}+\frac{3}{2}\delta Z_{3}\right)-(\bar{n}_{d}+\bar{n}_{e})\delta
Z_{3}\right]$ $\displaystyle+$ $\displaystyle
g^{4}\int\frac{d^{D}k}{(2\pi)^{D}}\int\frac{d^{D}q}{(2\pi)^{D}}\,\frac{D(p,k)\bar{D}(q-k)\bar{D}(P-q)}{k^{4}(q-k)^{4}(P-q)^{4}}\times$
$\displaystyle\qquad\qquad\qquad\times\left\\{\frac{n_{f}}{q^{4}}+\frac{n_{g}}{(P-k)^{4}}+\frac{n_{h}}{q^{2}(P-k)^{2}}+\frac{n_{i}}{q^{2}(P-q+k)^{2}}\right\\}\;.$

Factors like
$\bar{D}(k)=\int\frac{d^{D-1}\mathbf{p}^{\prime}}{p^{\prime}_{0}}D(p^{\prime},k)$
arise when we integrate out the momenta of un-measured hadrons stemming from
other jets. The combinatorial factors $n_{c},\dots,n_{i}$ multiply terms
depicted in Fig. 1.c–i.

If we were only interested in the partonic part of the branching process,
$D(p,P)$ would denote the distribution of a parton of momentum $p$ created by
the leading parton of momentum $P$. In that case, the $p-$integrated cut blobs
would be replaced by the cut propagator:
$\bar{D}(k)/k^{4}\rightarrow\delta(k^{2})$ at leading order, and we would
arrive at the problem presented in bib:p3dglp .

Let us parametrize the fragmentation function $D$ for later model building
purposes as a product

$D(p,P)\;=\;P^{10-D}\rho\left(P^{2}\right)d\left(\frac{2pP}{P^{2}},P^{2}\right)$
(10)

of a “virtuality (or jet mass) distribution” $\rho\left(P^{2}\right)$
normalized as $\int dP^{2}\rho\left(P^{2}\right)=1$, and a “hadron momentum
distribution” $d\left(x,P^{2}\right)$ normalized as
$\int\frac{d^{D-1}p}{p_{0}}d\left(\frac{2pP}{P^{2}},P^{2}\right)=\bar{n}(P^{2})$,
being the average hadron multiplicity in the jet. The $P^{10-D}$ factor
renders the mass-dimension $[D(p,P)]=10-2D$ of the FF, and also removes the
poles coming from the propagators on both sides of the cut blobs, as we work
in $D=6$ dimensions.

Now, let us simplify the DSE via making the approximation of pushing the
virtualities of the daughter partons (cut blobs) down to a negligible value
$m_{0}^{2}\ll P^{2}$. It is interesting that the phasespace of a massless
hadron of momentum $p$ stemming from a leading parton of momentum
$k=(k_{0},\mathbf{k})=(k_{0},k,\mathbf{k}_{T})$ is an ellipsoid of center
$\mathbf{k}/2$, length $k_{0}$ and width $m=\sqrt{k^{2}}$. This can be seen
from the condition defining the boundary of the phasespace, which is the
requirement that the phasespace of the rest of the hadrons stemming from the
same jet $\Omega_{n-1}(k-p)$ needs to be positive. According to Eq. (1), it
requires that $(k-p)^{2}\geq 0$, or equivalently, $2pk/k^{2}\leq 1$. For
instance, using the parametrisation $P=(M,0,\mathbf{0})$,
$k=(k_{0},k,\mathbf{0})$ and
$p=\left(\sqrt{p_{\parallel}^{2}+p_{\perp}^{2}},p_{\parallel},\mathbf{p}_{\perp}\right)$,
we arrive at the equation of an ellipsoid:
$\left(\dfrac{p_{\perp}}{m/2}\right)^{2}+\left(\dfrac{p_{\parallel}-k/2}{k_{0}/2}\right)^{2}\leq
1$. If we push the virtuality of the hadronizing parton down to a negligible
value $m\rightarrow m_{0}\approx 0$, the phasespace for the daughter hadrons
shrinks to a one-dimensional interval as $p_{T}\rightarrow 0$,
$p\rightarrow(p_{\parallel},p_{\parallel},\mathbf{0})$ with
$p_{\parallel}\in[0,k_{0}]$. Besides, the first argument of $d(p,k)$
simplifies as
$\dfrac{2pk}{k^{2}}\rightarrow\dfrac{2p_{\parallel}}{k_{0}+k}\rightarrow\dfrac{p_{\parallel}}{k}=\dfrac{x}{z}$,
with $x=\dfrac{2pP}{P^{2}}=\dfrac{2p_{\parallel}}{M}$ and
$z=\dfrac{2kP}{P^{2}}=\dfrac{2k}{M}$. We may carry out the above procedure in
the SDE via setting

$\rho(\dots)\rightarrow(2\pi)\delta(\dots),\quad\textrm{and}\quad
d\left(\frac{2pk}{k^{2}},k^{2}\right)\rightarrow(2\pi)^{D-2}\delta^{D-2}(\mathbf{k}_{T})\,d_{0}\left(\frac{x}{z},m_{0}^{2}\right)\,,$
(11)

where $\mathbf{k}_{T}\mathbf{p}=0$. This way, and SDE becomes

$\displaystyle P^{2}D\left(x,P^{2}\right)$ $\displaystyle\;=\;$
$\displaystyle\frac{g^{2}}{2}\,d_{0}(x,m_{0}^{2})\;+\;\frac{g^{4}}{2}\int\frac{dz}{z}\,d_{0}\left(\frac{x}{z},m_{0}^{2}\right)A(z,P^{2})\;,\;\;\textrm{with}$
$\displaystyle A(z,P^{2})$ $\displaystyle\;=\;$
$\displaystyle\delta(1-z)\left[\int\frac{d^{D}q}{(2\pi)^{D}}\left\\{\frac{i\,n_{c}}{q^{2}(q-k)^{2}(P-q)^{2}}+\frac{i\,n_{d}}{m_{0}^{2}q^{2}(q-k)^{2}}\right.\right.+$
$\displaystyle\qquad\qquad\quad+\;\left.\left.\frac{i\,n_{e}}{m_{0}^{2}q^{2}(P-q-k)^{2}}\right\\}+\frac{\bar{n}_{c}}{g^{2}}\left(\delta
Z_{g}+\frac{3}{2}\delta
Z_{3}\right)-\frac{\bar{n}_{d}+\bar{n}_{e}}{g^{2}}\delta Z_{3}\right]$
$\displaystyle+$
$\displaystyle\frac{P^{2}}{2\pi}\int\frac{d^{D}q}{(2\pi)^{D}}(2\pi)\delta\left[(q-k)^{2}\right](2\pi)\delta\left[(P-q)^{2}\right]\;\times$
$\displaystyle\qquad\qquad\qquad\times\;\left\\{\frac{n_{f}}{q^{4}}+\frac{n_{g}}{(P-k)^{4}}+\frac{n_{h}}{q^{2}(P-k)^{2}}+\frac{n_{i}}{q^{2}(P-q+k)^{2}}\right\\}\;.$

Terms in $A(z,P^{2})$ are calculated in Appendix B.

In order to keep only the LL terms, and to eliminate the collinear divergence
from $A(z,P^{2})$, we differentiate Eq. (LABEL:evol4) with respect to $t=\ln
P^{2}$ to obtain

$\partial_{t}\mathcal{D}\left(x,P^{2}\right)\;=\;\frac{g^{4}}{2}\int\frac{dz}{z}\,d_{0}\left(\frac{x}{z},m_{0}^{2}\right)\partial_{t}A(z,P^{2})$
(13)

for the dimensionless function $\mathcal{D}(x,P^{2})=P^{2}D(x,P^{2})$.
(Derivations of the coupling $g$ results in terms of $\mathcal{O}(g^{3})$, and
thus are neglected.) As
$\mathcal{D}=\dfrac{g^{2}}{2}d_{0}+\mathcal{O}(g^{4})$, we arrive at the DGLAP
equation:

$\partial_{t}\mathcal{D}\left(x,P^{2}\right)\;=\;g^{2}\int\frac{dz}{z}\,\mathcal{D}\left(\frac{x}{z},P^{2}\right)\Pi(z,P^{2})\;,$
(14)

with splitting function (SF)

$\Pi(z)\;=\;\frac{\partial}{\partial\ln
P^{2}}A(z,P^{2})\;=\;\frac{n_{f}}{(4\pi)^{3}}\frac{1-z}{z^{2}}-\frac{n_{c}}{2(4\pi)^{3}}\delta(1-z)\;.$
(15)

Note that the SF is proportional to the distribution of daughter partons in
the LL approximation at $\mathcal{O}(g^{4})$:
$p_{0}{\dfrac{dN}{d^{5}p}}^{DP}\sim\Pi(z,P^{2})$, thus,
${\dfrac{dN}{dz}}^{DP}\sim z^{3}\Pi(z)=az(1-z)+b\,\delta(1-z)$, which is of
the form of the SF used in bib:Graz .

###  3.1 Solving the DGLAP equation

Introducing Mellin transforms
$\tilde{f}(\omega)=\int\limits_{0}^{1}dxx^{\omega-1}f(x)$, the DGLAP equation
Eq. (14) simplifies to

$\partial_{t}\tilde{\mathcal{D}}\left(\omega,P^{2}\right)\;=\;g^{2}\,\tilde{\mathcal{D}}\left(\omega,P^{2}\right)\tilde{\Pi}(\omega)\;,$
(16)

where the Mellin-transform of the SF is

$\tilde{\Pi}(\omega)\;=\;\frac{1}{(4\pi)^{3}}\left[\frac{n_{f}}{(\omega-1)(\omega-2)}-\frac{n_{c}}{2}\right]\;.$
(17)

The solution of the DGLAP equation is

$\displaystyle\tilde{\mathcal{D}}\left(\omega,P^{2}\right)$
$\displaystyle\;=\;$ $\displaystyle
e^{b(P^{2})\tilde{\Pi}(\omega)}\tilde{\mathcal{D}}\left(\omega,P_{0}^{2}\right),\quad\textrm{with}\quad
b(P^{2})=\int\limits_{\ln P^{2}_{0}}^{\ln
P^{2}}dt\,g^{2}=\frac{2}{\beta_{0}}\ln\left[\frac{\ln(P^{2}/\Lambda^{2})}{\ln(P_{0}^{2}/\Lambda^{2})}\right]$
(18) $\displaystyle\quad\textrm{and coupling}\quad
g^{2}=\frac{2}{\beta_{0}\ln(P^{2}/\Lambda^{2})}\;.$

If we substitute our form of the FF in Eq. (10) (and take into account that
$\mathcal{D}(x,P^{2})=P^{2}D(x,P^{2})$), we obtain that

$M^{12-D}\rho(M^{2})\tilde{d}(\omega,M^{2})=e^{b(P^{2})\tilde{\Pi}(\omega)}M_{0}^{12-D}\rho(M_{0}^{2})\tilde{d}(\omega,M_{0}^{2})\;,$
(19)

where $d(x,M_{0}^{2})$ is the hadron distribution at a low initial scale in
Eq. (5), for which, we have constructed the statistical model in Sec. 2. To
solve for both the jet mass distribution $\rho(M^{2})$ and the hadron
distribution $d(x,M^{2})$, we exploit the normalisation condition

$\bar{n}(M^{2})=\int\frac{d^{D-1}p}{p}d\left(\frac{2p}{M},M^{2}\right)=\left(\frac{M}{2}\right)^{\omega^{\ast}}\kappa_{D-1}\tilde{d}(\omega^{\ast},M^{2})\;,$
(20)

with $\omega^{\ast}=D-2$ and solid angle $\kappa_{D}=2\pi^{D/2}/\Gamma(D/2)$.
Consequently, when taking Eq. (19) at $\omega=\omega^{\ast}$, $\tilde{d}$
drops out, and we get the solution for $\rho$:

$\rho(M^{2})=\left(\frac{M_{0}}{M}\right)^{14-2D}\frac{\bar{n}_{0}}{\bar{n}}\rho(M_{0}^{2})e^{b(P^{2})\tilde{\Pi}(\omega^{\ast})}\;.$
(21)

Writing this back into Eq. (19), we obtain the hadron distribution in the jet:

$\tilde{d}(\omega,M^{2})=e^{b(M^{2})[\tilde{\Pi}(\omega)-\tilde{\Pi}(\omega^{\ast})]}\left(\frac{M_{0}}{M}\right)^{D-2}\frac{\bar{n}}{\bar{n}_{0}}\,\tilde{d}(\omega,M_{0}^{2})\;.$
(22)

Now, we can readily obtain the FF by substituting Eq. (21) and Eq. (22) into
Eq. (10):

$\tilde{D}(\omega,M^{2})=M_{0}^{10-D}\left(\frac{M_{0}}{M}\right)^{2}\rho(M_{0}^{2})\,\tilde{d}(\omega,M_{0}^{2})\,e^{b(M^{2})\tilde{\Pi}(\omega)}\;.$
(23)

As we have a model for $d(x,M_{0}^{2})$, we could use the inverse Mellin-
transform to calculate the FF, however, we will use a less complicated
approximation in the next section.

Finally, the $p-$integrated cut blobs being the factors, which jets contribute
to cross sections, if we do not measure the hadrons in them, take the form

$\displaystyle\bar{D}(M^{2})$ $\displaystyle=$
$\displaystyle\int\frac{d^{D-1}p}{p}D\left(\frac{2p}{M},M^{2}\right)=\kappa_{D-1}\left(\frac{M}{2}\right)^{D-2}D(\omega^{\ast},M^{2})$
(24) $\displaystyle=$ $\displaystyle
M^{D-4}M_{0}^{14-2D}\rho(M_{0}^{2})\,e^{b(M^{2})\tilde{\Pi}(\omega^{\ast})}\sim
M^{D-4}\ln^{2\tilde{\Pi}(\omega^{\ast})/\beta_{0}}(M^{2}/\Lambda^{2})\;.$

###  3.2 Scale evolution of the $q$ and $T$ parameters of the TS distribution

Although the TS function Eq. (6) is not an exact solution of the DGLAP
equation, along with the hadron multiplicity distribution Eq. (4), it
describes measured data of hadrons stemming from jets in $e^{+}e^{-}$ bib:UKee
and $pp$ bib:UKpp ; bib:UKpp3D  collisions at various energy scales. Based on
this, we conjecture that Eq. (5) is a reasonably good approximation of
$d(x,M^{2})$, the hadron distribution in a jet at any scale $M$. To determine
the scale dependence of the parameters $\bar{n}$ and $\sigma$ of the model,
let us insert
$d(\omega,M)=\sum\limits_{n}\mathcal{P}_{n}(M)\,n\,\tilde{d}_{n}(\omega,M)$
with

$\tilde{d}_{n}(\omega,M)\;=\;\frac{\left(\frac{2}{M}\right)^{\omega^{\ast}}\Gamma(\omega)\Gamma\left(\frac{\omega^{\ast}n}{2}\right)}{\kappa_{D-1}\Gamma(\omega^{\ast})\Gamma\left(\omega+\frac{\omega^{\ast}(n-2)}{2}\right)}$
(25)

into Eq. (22), to obtain

$\sum_{n}\mathcal{P}_{n}(M)\,n\,\tilde{d}_{n}(\omega,M)=e^{b(M^{2})[\tilde{\Pi}(\omega)-\tilde{\Pi}(\omega^{\ast})]}\left(\frac{M_{0}}{M}\right)^{\omega^{\ast}}\frac{\bar{n}}{\bar{n}_{0}}\,\sum_{n}\mathcal{P}_{n}(M_{0})\,n\,\tilde{d}_{n}(\omega,M_{0})\;.$
(26)

As the TS distribution is only an approximation of the exact solution of the
DGLAP equation, we cannot eliminate $\omega$ from this equation. Nevertheless,
we may require Eq. (26) to hold for a suitable set of fixed values of $\omega$
to force the scale evolution of certain moments of $x$ to be exact. In this
paper, we prescribe that Eq. (26) hold for
$\omega=\\{\omega^{\ast},\omega^{\ast}\pm 1\\}$ (which refers to momenta
$\left\langle\frac{1}{x}\right\rangle,\langle 1\rangle$, and $\langle
x\rangle$) and obtain

$\displaystyle\bar{n}(M)$ $\displaystyle\;=\;$
$\displaystyle\bar{n}_{0}\,e^{-b\,a_{+}}\;,$ $\displaystyle\sigma^{2}(M)$
$\displaystyle\;=\;$
$\displaystyle\bar{n}\left[\frac{2}{\omega^{\ast}}+e^{b\,a_{-}}\left(\frac{\sigma_{0}^{2}}{\bar{n}_{0}}+\bar{n}_{0}-\frac{2}{\omega^{\ast}}\right)\right]-\bar{n}^{2}\;,$
(27)

with $b=b(M^{2})$, $a_{\pm}=\tilde{\Pi}(\omega^{\ast}\pm
1)-\tilde{\Pi}(\omega^{\ast})$ and initial values
$\sigma(M_{0})=\sigma_{0},\bar{n}(M_{0})=\bar{n}_{0}$. Furthermore, we choose
the initial scale $M_{0}$ to be the scale, at which, the multiplicity
distribution becomes Poissonian ($\sigma^{2}_{0}=\bar{n}_{0}$), to get even
simpler results:

$\displaystyle\bar{n}(M)$ $\displaystyle\;=\;$
$\displaystyle\bar{n}_{0}\,e^{-b\,a_{+}}\;,$ $\displaystyle\sigma^{2}(M)$
$\displaystyle\;=\;$
$\displaystyle\bar{n}\left[\frac{2}{\omega^{\ast}}+e^{b\,a_{-}}\left(1+\bar{n}_{0}-\frac{2}{\omega^{\ast}}\right)\right]-\bar{n}^{2}\;.$
(28)

The obtained logarithmic growth of the mean hadron multiplicity with the
energy scale $\bar{n}\propto\ln^{a}(M)$ is in accordance with observations.
Consequently, the scale dependence of the standard parameters of the hadron
multiplicity distribution is

$\displaystyle p$ $\displaystyle\;=\;$ $\displaystyle
1-\frac{\bar{n}}{\sigma^{2}}=1-\frac{1}{\frac{2}{\omega^{\ast}}+e^{b\,a_{-}}\left(1+\bar{n}_{0}-\frac{2}{\omega^{\ast}}\right)-\bar{n}_{0}\,e^{-b\,a_{+}}}\;,$
$\displaystyle r$ $\displaystyle\;=\;$
$\displaystyle\frac{\bar{n}^{2}}{\sigma^{2}-\bar{n}}=\frac{\bar{n}_{0}\,e^{-b\,a_{+}}}{\frac{2}{\omega^{\ast}}+e^{b\,a_{-}}\left(1+\bar{n}_{0}-\frac{2}{\omega^{\ast}}\right)-\bar{n}_{0}\,e^{-b\,a_{+}}-1}\;.$
(29)

In $D=4$ dimensions, the FF takes the form of the TS distribution Eq. (6), and
the scale evolution of its parameters (using Eq. (7)) is

$\displaystyle q-1$ $\displaystyle\;=\;$
$\displaystyle\frac{\sigma^{2}-\bar{n}}{\bar{n}^{2}+3(\sigma^{2}-\bar{n})}=\frac{1-e^{-b\,(a_{+}+a_{-})}}{3-2e^{-b\,(a_{+}+a_{-})}}\;,$
$\displaystyle\tau$ $\displaystyle\;=\;$
$\displaystyle\frac{\bar{n}}{\bar{n}^{2}+3(\sigma^{2}-\bar{n})}=\frac{\tau_{0}}{3\,e^{b\,a_{-}}-2\,e^{-b\,a_{+}}}\;,$
(30)

having introduced $\tau_{0}=1/n_{0}$ in accordance with the equipartition
principle. This type of logarithmically rising tendency of $q$, and falling
tendency of $\tau$ has been observed in various high-energy collisions listed
in Sec. 1. Looking at the asymptotic behaviour, it is interesting to point out
that $q$ is bounded from above: $q\;\rightarrow\;\frac{4}{3}$.

Since now we have the approximate form of $\tilde{d}(\omega,M^{2})$, we may
use it in the expression of the FF $\tilde{D}(\omega,M^{2})$ via only
substituting Eq. (21) into Eq. (10):

$\tilde{D}(\omega,M^{2})=M^{10-D}\left(\frac{M_{0}}{M}\right)^{14-2D}\frac{\bar{n}_{0}}{\bar{n}}\rho(M_{0}^{2})\,\tilde{d}(\omega,M^{2})\,e^{b(M^{2})\tilde{\Pi}(\omega^{\ast})}\sim\tilde{d}(\omega,M^{2})\;.$
(31)

From the point of view of the $\omega$ dependence, this formula differs from
Eq. (23) in the change of
$\tilde{d}(\omega,M_{0}^{2})\,e^{b(M^{2})\tilde{\Pi}(\omega)}\rightarrow\tilde{d}(\omega,M^{2})$.

Figure 2: The partonic part of 2–, and 3–jet final states in the
$e^{+}e^{-}\rightarrow hX$ process. We will refer to the process shown in
panel b as the “split”, while the one in panel c the “crossed” 3-jet event.
Figure 3: The cut blob in Fig. 2 denoting the fragmentation pocess of a
virtual leading parton of momentum $P$ fragmenting into hadrons, among which,
one has momentum $p$ in the LL approximation.

##  4 Hadron and jet distributions in $e^{+}e^{-}\rightarrow 2-3$ jet events

Similarly to bib:Graz ; bib:CollinsMC , we mimic the
$e^{+}e^{-}\rightarrow\gamma^{\ast}\rightarrow q\bar{q}$ process via
introducing the term $eA\phi^{2}$ into the Lagrangian, thus, coupling a
’scalar photon’ to the ‘scalar parton’ field of the $\phi^{3}$ model. When
calculating the distributions of jets stemming from $e^{+}e^{-}$ annihilations
with 2-, and 3-jet final states depicted in Fig. 2, we use the un-cut
propagators of the leading partons attached to the $\bar{D}(k_{i})$ cut blobs
given in Eq. (24). These cut blobs are the resummed LL terms of the process
(shown in Fig. 3) of a virtual leading parton emitting on-shell daughter
partons before fragmenting into hadrons when its virtuality reaches some low
scale $M_{0}$. As the cut blobs along with the parton legs of momenta $k_{i}$
on both their sides contribute a factor of $\bar{D}(k_{i}^{2})/k_{i}^{4}$, we
introduce the function

$F_{n}(k_{1},\dots,k_{n})\;=\;\prod_{i=1}^{n}\frac{\bar{D}(k_{i}^{2})}{k_{i}^{4}}\,\delta^{D}\left(\sum_{j=1}^{n}k_{j}-P\right)\propto\prod_{i}\frac{\ln^{\alpha}\left(\dfrac{k_{i}^{2}}{\Lambda^{2}}\right)}{k_{i}^{8-D}}\,\delta^{D}\left(\sum_{j}k_{j}-P\right)\;,$
(32)

with $\alpha=2\tilde{\Pi}(\omega^{\ast})/\beta_{0}$, and
$P=(\sqrt{s},\mathbf{0})$ being the total momentum of the incoming
$e^{+}e^{-}$ pair. This way, the momentum distribution of jets in a 2–jet
event reads

$\frac{1}{\sigma_{2jet}}\frac{d\sigma_{2jet}}{d^{D}k_{1}d^{D}k_{2}}\;=\;F_{2}(k_{1},k_{2})\;,$
(33)

and we can make out the 3–jet cases in panels b and c of Fig. 2 as

$\frac{1}{\sigma^{b}_{3jet}}\frac{d\sigma_{3jet}^{b}}{d^{D}k_{1}\dots
d^{D}k_{3}}=g^{2}\frac{F_{3}(k_{1},k_{2},k_{3})}{(k_{1}+k_{2})^{4}}\;,\qquad\frac{1}{\sigma^{c}_{3jet}}\frac{d\sigma_{3jet}^{c}}{d^{D}k_{1}\dots
d^{D}k_{3}}=g^{2}\frac{F_{3}(k_{1},k_{2},k_{3})}{(k_{1}+k_{2})^{2}(k_{2}+k_{3})^{2}}\;.$
(34)

As we do not focus on the renormalisation of the electric charge $e$, we have
omitted the term with the radiative correction to the photon-parton vertex in
the 2–jet case in Fig. 2.a.

In order to regularize the divergences originating from the poles of the
propagators of leading partons, we prescribe an extra condition $k_{i}^{2}\geq
m_{0}^{2}$ so that jet masses be larger than some low energy scale
$m_{0}>\Lambda$, being of the order of the proton mass. Such a requirement is
natural, as the jet mass cannot be smaller than the sum of the masses of the
hadrons it containes. This condition also eliminates the collinear divergence
from the 3-jet events, which would arise if leading partons were taken to be
on-shell. In models relying on factorisation, such divergences may be removed,
for example, by the subtraction of the configurations, in which, some jet
momenta are parallel bib:CollinsMC .

When calculating distributions of quantities involving jet momenta
numerically, we first generated jet momenta randomly according to the uniform
distribution in the phasespace, and accepted them, if they satisfied the
condition of $k_{i}^{2}\geq m_{0}^{2}$. Then, we filled the calculated
quantities of interest into histograms weighted by the corresponding jet
cross-sections in Eq. (33) and Eq. (34). To generate random jet momenta, we
exploited that a thermal ensemble readily provides particles with uniform
momentum distribution in the phasespace. In fact, given $n$ particles, having
an arbitrary set of initial momenta $k_{i}$ ($\sum k_{i}=P$), after at least
$10\,n$ pairwise “collisions”, they reach a thermal state. In these imaginary
collisions, the incoming particles exchange a random momentum, thus, the new
momentum of one of the pair has uniform distribution in the center-of-mass
(CM) frame:

$d\mathcal{P}(k)\;=\;dk^{0}\,d|\mathbf{k}|d\Omega\,|\mathbf{k}|^{D-2}\,\Theta(k_{0}-|\mathbf{k}|)\,\Theta\left(E^{CM}-|\mathbf{k}|-k_{0}\right)\,\Theta\left(\frac{E^{CM}}{2}-|\mathbf{k}|\right)\;.$
(35)

This method works also for a mixture of particles of various masses, in case
of which, analytic formulas (like Eq. (2), we have obtained for massless, on-
shell particles) for the direct generation of particle momenta one-by-one are
not available.

Having obtained the jet momenta $k_{i}$, the probability of a hadron stemming
from the $i^{th}$ jet to have momentum $p$ is given by
$d\left(\dfrac{2pk_{i}}{k_{i}^{2}},k_{i}^{2}\right)$ according to Eq. (31).
The functional form of $d$ is given in Eq. (5) (or Eq. (6) in $D=4$
dimensions), and the values of its parameters depend on the jet mass according
to Eq. (28)–(30). This way, the spectrum of hadrons stemming from the $i^{th}$
jet in an $n-$jet event is calculated as

$\frac{1}{\sigma^{h}_{n-jet}}p_{0}\frac{d\sigma^{h}_{n-jet}}{d^{D-1}\mathbf{p}}\;=\;\int\prod\limits_{j=1}^{n}d^{D}k_{j}\,\frac{1}{\sigma_{n-jet}}\frac{d\sigma_{n-jet}}{d^{D}k_{1}\dots
d^{D}k_{n}}\,d\left(\dfrac{2pk_{i}}{k_{i}^{2}},k_{i}^{2}\right)\;.$ (36)

Setting the $M_{0}$ starting scale of the DGLAP evolution of the FF to be
equal to the lower cut-off for the allowed jet masses, $M_{0}=m_{0}$, our
model has 4 parameters: the mean hadron multiplicity $\bar{n}_{0}$ at starting
scale $M_{0}$, along with the $\beta_{0}$ and $\Lambda$ parameters of the
coupling. We calibrated these parameters via fitting the calculated hadron
distribution in 2-jet events in $D=4$ dimensions to the spectrum of charged
hadrons measured at $\sqrt{s}=$ 200 GeV by the OPAL Collaboration bib:MjetEE1
, and obtained the following values: $\Lambda$ = 98.3 MeV, $\beta_{0}$ =
0.08107, $\bar{n}_{0}$ = 6.814, $M_{0}$ = 3.038 GeV/$c^{2}$. Fit results are
shown in the top-left panel of Fig. 4. As the $\phi^{3}$ model is
renormalizable in $D=6$ dimensions, we used the formulas for the coupling and
the SF $\tilde{\Pi}(\omega^{\ast})$ obtained in 6 dimensions, even when
calculating the 4-dimensional results. This inaccuracy is, however, of not
much relevance as the aim of this paper is to derive the scale dependence of
the parameters of the TS distribution in a high-energy process, and to show
the effect of using virtual leading partons even in the “hard” part of cross
sections.

Figure 4: Measured distributions of hadron energy (top-left) and heavy jet
mass (top-right) in $e^{+}e^{-}$ annihilations compared with our calculations
for 2-jet events based on the $\phi^{3}$ theory with virtual leading partons.
Bottom, our results for jet energy and momentum distributions in 4 and 6
dimensions.

In the top-right panel of Fig. 4, we compare our results on the distribution
of the mass of jets in 2-jet events in 4 and 6 dimensions with the available
experimental result, which is the distribution of the heavy jet mass (the
larger among the two total momentum squared of particles in the semispheres of
the phasespace). Although, OPAL data were taken at a slightly lower collision
energy ($\sqrt{s}=189$ GeV), our results are in accordance with it in the
high-jet mass range. For lower values, the heavy jet mass distribution
decreases rapidly, and we do not expect our result to describe it correctly.
Out of curiosity, we have also plotted a dataset measured at a lower energy
$\sqrt{s}=$ 91 GeV.

In the bottom panels of Fig. 4, we compare the calculated distributions of the
energy and the vector part of jet momenta in 2-jet events in 4 and 6
dimensions. As expected, these distributions have a sharp peak at
$\sqrt{s}/2$, where the distributions of on-shell particles would have a
$\delta$-peak. Besides, the 6-dimensional distributions are broader, due to
the lower power of jet virtualities in the denominator of Eq. (32).

Figure 5: Energy, momentum, mass and rapidity distributions of jets in 2–, and
3–jet final states stemming from $e^{+}e^{-}$ annihilations calculated at
$\sqrt{s}=$ 200 GeV in $D=6$ dimensions. Colors show which graph refers to
which jet in Fig. 2.

Fig. 5 shows distributions of jet energies, momenta, masses and rapidities
($\eta^{J}=\ln\left(\frac{E^{J}+P^{J}_{z}}{E^{J}-P^{J}_{z}}\right)$, where
$P^{J}_{z}$ is the component of $P^{J}$, which is parallel to the beam axis)
calculated in 6 dimensions. Color encoding shows which histogram refers to
which jet in Fig. 2, where the Feynmann graphs of the corresponding 2-, and
3-jet processes are depicted. It is trivial that distributions of the first
two jets of momenta $k_{1}$ and $k_{2}$, colored in red in Fig. 2.b (which we
will refer to as the “split” process) coincide. So do the distributions of the
first and third jets of momenta $k_{1}$ and $k_{3}$, colored in blue in Fig.
2.c (which we will call the “crossed” process). However, it is interesting
that the second jet in the crossed process has the same distributions as do
the first two jets in the split process. Besides, a clear hierarchy among the
mean jet energies, momenta, and masses are visible. This seems to be in
connection with which “generation” the leading parton of a jet is produced. In
Fig. 2, we see that green jets belong to the $1^{st}$ generation, as their
leading partons were created in the first splitting. Consequently, their
average energy $\bar{E}^{J}$ is the largest. The magenta jet also belongs to
the $1^{st}$ generation, but it stems from a 3-jet event, where the total
energy is distributed among more jets then in the 2-jet case, thus, the
$\bar{E}^{J}$ of the magenta jet is somewhat smaller. The $\bar{E}^{J}$ of red
jets is the smallest, because they belong to the $2^{nd}$ generation, whereas
the $\bar{E}^{J}$ of the blue jets is in between those of the red and magenta
jets, as the blue jets come from the interference of a $1^{st}-$ and a
$2^{nd}-$generation jet.

Rapidity, as well as, anglular distributions are the same for all the jets due
to rotational symmetry.

Figure 6: Distributions of the angles between jet momenta (left). Distribution
of the $q$ (middle) and $\tau$ (right) parameters of the spectrum of hadrons
in jets. Colors show which graph refers to which jet in Fig. 2.

From the left panel of Fig. 6, we can see that in a split process, the angle
between the red jets $\theta_{12}$ and the one between the red and the magenta
jet $\theta_{13}$ are most likly to be ordered as
$\theta_{12}<\dfrac{\pi}{2}<\theta_{13}$, whereas jets in a crossed process
favour the three-pronged star configuration.

As jet masses fluctuate event-by-event in $e^{+}e^{-}$ annihilations, so do
the $q$ and $\tau$ parameters of hadron spectra in jets due to Eq. (30).
According to the middle and left panels of Fig. 6, $q$ and $\tau$ has sharp
distributions. Consequently, the TS distribution provides a good description
of the hadron spectrum, which is the sum of spectra in single jets averaged
over jet-momentum fluctuations. It is important to point out, that the scale
evolution of FFs can be examined even if the measured spectrum is available at
only a single value of $\sqrt{s}$, if we set the jet mass as fragmentation
scale.

Figure 7: Top, double differential distributions of the jet mass and energy.
Bottom, average jet mass vs. jet energy. Legend shows which graph refers to
which jet in the split and crossed events in Fig. 2.

As factorisation theorems are expected to work best when jet masses are small
compared to jet energies, we have plotted the double-differential distribution
$d\sigma^{J}/dE^{J}dM^{J}$ along with the dependence of the $\langle
M^{J}\rangle/E^{J}$ ratio as a function $E^{J}$ in Fig. 7 for the three types
of jets produced in our model in 3-jet events. As it can be seen, although,
the most probable events are those with small jet masses, the average jet mass
is never negligable compared to the jet energy. When one of the jets acquires
energy $E^{J}\geq\sqrt{s}/2$, the situation gets even worse, since in that
case, the energies as well as the momenta of the other jets are small.
Consequently, all jet momenta has to be small, thus, the mass of the jet with
large $E^{J}$ has to be large as well.

Figure 8: On- vs. off-shell jet energy distributions. Colors show which graph
refers to which jet in Fig. 2.

At this point, it is not surprising that jet energy distributions of our model
(using off-shell leading partons and LL resummation for the fragmentation
process) differ significantly from those, obtained using fixed-order cross
sections with on-shell leading partons, as can be seen in Fig. 8. The
distribution of the on-shell leading partons in 3-jet events are

$\displaystyle{\frac{d\sigma}{dx}}^{on-shell}$ $\displaystyle\;\sim\;$
$\displaystyle x(1-x)[1-A\ln(1-x)]\quad\textrm{for jet 1-2 and}$ (37)
$\displaystyle\;\sim\;$ $\displaystyle\frac{x^{3}}{1-x}\quad\textrm{for jet 3
in a {split} event;}$ $\displaystyle\;\sim\;$ $\displaystyle
x^{2}\quad\textrm{for jet 1-3 and}$ $\displaystyle\sim$ $\displaystyle
x(1-x)\quad\textrm{for jet 2 in a {crossed} event}.$

The Feynmann-graphs of the processes of the creation of these partons are
depicted in Fig. 1.f–i (if we replace the jet blobs by cut free propagators).
Consequently, the corresponding distributions are the terms, listed in Eq.
(LABEL:1p31) multiplied by the phasespace factor $x^{3}$ (which we have
already calculated when determining the SF). In the off-shell case, the
distribution of jet 1-2 in a split event coincides with that of jet 2 in a
crossed event. In the on-shell case, these distributions in Eq. (37) only
differ in the $\ln(1-x)$ term coming from the dimensional regularisation of
the collinear divergence resulting from the configuration when jet 1 and 2 are
parallel. A different regulatisation method might remove this term. According
to Figs. 7–8, the shape of the jet energy distributions obatained using on-,
and off-shell partons are close to each other only within limited intervals
around $E^{J}\approx\sqrt{s}/4\pm\sqrt{s}/8$, where $\langle
M^{J}\rangle/E^{J}\approx 0.3$. Outside of this region, the $\langle
M^{J}\rangle/E^{J}$ ratio is much higher, and the on-, and off-shell
distributions differ significantly.

Figure 9: Left, 2-jet event with off-shell leading partons, where the blobs
denote the fragmentation process resummed in the LL approximation. Right,
3-jet event with on-shell leading partons.

In the middle panel of Fig. 8, we can see that, although, the shape of the on-
shell distribution of jet 3 is not close to that of the off-shell one, it
approximates the off-shell distribution in 2-jet events. This observation
suggests that, having a pair of jets created in a splitting (as depicted in
Fig. 9), in order to get a good approximation for the energy distribution of
the jet of momentum $k$, it is enough to keep only the virtuality of the
leading parton of the other jet. Furthermore, it is also enough to keep only
the first splitting in the other jet, instead of resumming the whole ’parton
ladder’ in the LL approximation. This result supports the argument in
bib:CollinsMC  that, when we create a parton shower, if we neglect the
virtualities of partons of a given generation due to using on-shell cross
sections, our inaccuracy is compensated for when we create the next
generation.

Figure 10: Hadron energy (top-left), rapidity (top-right), angle (bottom-left)
and multiplicity (bottom-right) distributions. Colors show which graph refers
to which jet in Fig. 2.

The spectrum, rapidity and angular distributions of hadrons stemming from each
jet in 2-, and 3-jet events, shown in Fig. 10, can be obtained using Eq. (36).
Besides, the distribution of the $N_{h}$ number of hadrons in a given jet is

$\frac{1}{\sigma^{h}_{n-jet}}\frac{d\sigma^{h}_{n-jet}}{dN_{h}}\;=\;\int\prod\limits_{j=1}^{n}d^{D}k_{j}\,\frac{1}{\sigma_{n-jet}}\frac{d\sigma_{n-jet}}{d^{D}k_{1}\dots
d^{D}k_{n}}\,\mathcal{P}_{N_{h}}(k^{2})\;.$ (38)

The difference between these distributions are due to the hierarchy of the
average mass, energy and momentum of the different jets (see Fig. 5)
containing the hadrons. The more energetic the jet, the more high-energy
hadrons stem from it, as can be seen in the top-left panel. The larger the jet
mass, the wider the phasespace ellipsoid of the hadrons in the jet, thus, the
average angle between hadron momenta and the momentum of the mother jet is
larger as well, according to the bottom-left panel. The mean number of hadrons
in a jet grows monotonically with the jet mass in accordance with the bottom-
right panel and Eq. (28). Finally, the shape of hadronic pseudo-rapidity
($\eta=\ln\frac{p_{0}+p_{z}}{p_{0}-p_{z}}$) distributions shown in the top-
right panel can be derived from rotational invariance and dimensional
arguments.

###  4.1 A micro-canonical event generator

We obtained the hadron distributions in the previous section via generating
all the hadrons in the jets using the micro-canonical model presented in Sec.
2. That is, after having generated the jet momenta, we used the NBD
multiplicity distribution in Eq. (4) (the parameters of which depend on the
jet mass according to Eq. (28)) to obtain the numbers of hadrons in the jets.
After that, we generated the momenta of hadrons in each jet according to the
micro-canonical ensemble using the random collisional method, described in the
paragraph before Eq. (35). As in our approximation, hadrons are on-shell and
massless, we replaced Eq. (35) with

$d\mathcal{P}(p)\;=\;d|\mathbf{p}|d\Omega\,|\mathbf{p}|^{D-2}\,\Theta\left(\frac{E^{CM}}{2}-|\mathbf{p}|\right)$
(39)

for the generation of the momentum of one of the outgoing particles in the CM
frame in each imaginary collision. This way, the one-particle distribution in
each jet of $n$ hadrons becames of the form of Eq. (3), and the $n$-averaged
distribution becames Eq. (5).

Since the results presented in the previous section only involve $d(x,k^{2})$,
the single-hadron distribution in a jet averaged over multiplicity
fluctuations (the scale evolution of which, we have derived from first
principles), actually, we used the statistical model only to obtain the form
of the FF at an initial scale $M_{0}\approx$ 3 GeV (which value we obtained
from fitting the model to measured data). However, the micro-canonical
fragmentation model could, in principle, be used to obtain multi-particle
observables as well, since it is very simple to generate momenta of all the
particles in all jets event-by-event within its scope. Although, this way, we
would discard the microscopic dynamics of the branching process of parton
production within jets, as all measurables are derived solely from the
phasespace of particles in a statistical ensemble.

##  5 Summary

  * •

We have summarized the status of the application of the Tsallis (TS)
distribution in theoretical and experimental high-energy physics.

  * •

We have resummed the fragmentation processes of a virtual leading parton
(initiating the jet) emitting on-shell daughter partons in the leading-log
approximation (LLA) in the $\phi^{3}$ theory. We have found that the
fragmentation scale is the virtuality of the leading parton, which is equal to
the jet mass $M_{J}$, and calculated the $M_{J}$-dependence of the $q$ and $T$
parameters (Eq. (28)) of a TS-shaped fragmentation function (FF).

  * •

Unlike in approaches based on factorisation theorems, in this paper, we have
calculated the energy $E_{J}$, momentum $|\mathbf{P}_{J}|$ and mass $M_{J}$
distributions of jets produced in $e^{+}e^{-}$ annihilations with 2– and 3–jet
final states using virtual leading partons attached to the previously derived
TS-shaped FFs. The results show (Fig. 5) that there is a hierarchy among the
$\langle E_{J}\rangle$, $\langle|\mathbf{P}_{J}|\rangle$ and $\langle
M_{J}\rangle$ of the jets depending on which generation the leading parton of
the jet was produced at. Besides, we have found that the energy distribution
of a jet, obtained using virtual leading partons and LL resummation in a 2-jet
event, was well approximated by the distribution of an on-shell leading
parton, whose jet pair contained only a single splitting (Fig. 8-9).

  * •

Furthermore, when calculating hadronic distributions, we have found that the
larger the jet mass, the more, and also more energetic hadrons stem from it.
The angle between the momenta of hadrons and that of the jet is larger too
(Fig. 10).

  * •

We have developed an event generator to obtain the momenta of all hadrons
produced in jets event-by-event, using the FF based on micro-canonical
statistics and superimposed NBD hadron multiplicity fluctuations. We have
derived that the mean multiplicity of hadrons in a jet depends on the jet mass
as $\bar{n}\sim\ln^{a}(M_{J})$ (see Eq. (26)).

##  6 Appendix

##  Appendix A Phasespace of $n$ massless particles

$\Omega_{n}(P)\;=\;\prod_{i=1}^{n}\int\frac{d^{D-1}\mathbf{p}_{i}}{p_{i}^{0}}\,\delta^{D}\left(\sum_{j}p_{j}^{\mu}-P^{\mu}\right)\sim\int
d^{D}s\,e^{-is_{\mu}P^{\mu}}\varphi^{n}(s)\;,$ (40)

where the Fourier-transform of the single-particle phasespace $\varphi(s)$ can
be evaluated in the frame, in which, $s=(\sigma,\mathbf{0})$ (with
$\sigma^{2}=s_{0}^{2}-\mathbf{s}^{2}$), and $p=(p,\mathbf{p})$:

$\varphi(\sigma)\;=\;\int\frac{d^{D-1}\mathbf{p}}{p^{0}}\,e^{is_{\mu}p^{\mu}}\sim\int
dp\,p^{D-3}e^{i\sigma p}\sim\frac{1}{(i\sigma)^{D-2}}\;.$ (41)

We may evaluate the inverse Fourier-transform in Eq. (1) in the frame, where
$P=(M_{0},\mathbf{0})$, thus,

$\Omega_{n}(P)\;\sim\;\int d^{D-1}\mathbf{s}\int
ds_{0}\,\frac{e^{-is_{0}M_{0}}}{[(s_{0}+|\mathbf{s}|)(s_{0}-|\mathbf{s}|)]^{n(D-2)/2}}\;.$
(42)

As we have poles at $s_{0}=\pm|\mathbf{s}|$, we use Cauchy’s formula
$\oint\frac{dzf(z)}{(z-z_{0})^{n}}\sim f^{(n-1)}(z_{0})$, and arrive at terms
of the form of

$\displaystyle\Omega_{n}(P)$ $\displaystyle\;\sim\;$
$\displaystyle\sum_{j}A^{\pm}_{j}\int
d^{D-1}\mathbf{s}\,\left(\frac{\partial}{\partial
s_{0}}\right)^{j}e^{-is_{0}M_{0}}\left(\frac{\partial}{\partial
s_{0}}\right)^{n(D-2)/2-1-j}\left.\frac{1}{(s_{0}\pm|\mathbf{s}|)^{n(D-2)/2}}\right|_{s_{0}=\pm|\mathbf{s}|}$
(43) $\displaystyle\sim\sum_{j}A^{\pm}_{j}\,M_{0}^{j}\int
ds\,s^{D-1-n(D-2)+j}\,e^{-isM_{0}}\sim M_{0}^{n(D-2)-D}\;.$

The actual values of the constant factors $A^{\pm}_{j}$ multiplying the terms
coming from the poles at $s_{0}=\pm|\mathbf{s}|$, are of no importance from
the point of view of the particle distributions.

##  Appendix B Calculation of the splitting function

Via introducing the renormalized field and coupling $g=Z_{g}g_{r}$ and
$\phi=Z_{3}^{1/2}\phi_{r}$, along with $Z_{3}=1+\delta Z_{3}$ and
$Z_{g}=1+\delta Z_{g}$, we arrive at the renormalised Lagrangian

$\displaystyle\mathcal{L}$ $\displaystyle\;=\;$
$\displaystyle\frac{1}{2}(\partial_{\mu}\phi_{r})^{2}+(Z_{3}-1)\frac{1}{2}(\partial_{\mu}\phi_{r})^{2}+\frac{g_{r}}{3!}\phi_{r}^{3}+(Z_{g}Z_{3}^{3/2}-1)\frac{g_{r}}{3!}\phi_{r}^{3}$
(44) $\displaystyle=$
$\displaystyle\frac{1}{2}(\partial_{\mu}\phi_{r})^{2}+\delta
Z_{3}\frac{1}{2}(\partial_{\mu}\phi_{r})^{2}+\frac{g_{r}}{3!}\phi_{r}^{3}+(\delta
Z_{g}+\frac{3}{2}\delta Z_{3})\frac{g_{r}}{3!}\phi_{r}^{3}\;.$

As $\delta Z_{3}$ and $\delta Z_{g}$ are of $\mathcal{O}(g^{2})$, terms
proportional to them come as perturbative corrections. This way, a propagator
of momentum $p$ is $i/(p^{2}+i\epsilon)$, a vertex is $-ig_{r}$, the counter
terms are $ip^{2}\delta Z_{3}$ and $-ig_{r}(\delta Z_{g}+\frac{3}{2}\delta
Z_{3})$, and a cut propagator is $2\pi\delta(p^{2})$. Propagators and vertices
on the right of the cuts are complex conjugates.

We may write $A(z,P^{2})=\delta(1-z)A_{1}(z,P^{2})+A_{2}(z,P^{2})$ in Eq.
(LABEL:evol4). When calculating $A_{1}(z,P^{2})$, we use the identity
$\prod\limits_{i}\dfrac{1}{A_{i}^{\alpha_{i}}}=\dfrac{\Gamma(\alpha)}{\prod\limits_{i}\Gamma(\alpha_{i})}\prod\limits_{i}\int\limits_{0}^{1}\dfrac{d\xi_{i}\,\xi_{i}^{\alpha_{i}-1}\delta(1-\alpha)}{(\sum\xi_{i}A_{i})^{\alpha}}$,
with $\alpha=\sum\alpha_{i}$, thus,

$\displaystyle A_{1}(z,P^{2})$ $\displaystyle\;=\;$
$\displaystyle\int\limits_{0}^{1}d\xi_{1}\int\frac{d^{D}q}{(2\pi)^{D}}\left\\{\int\limits_{0}^{1-\xi_{1}}d\xi_{2}\frac{2i\,n_{c}}{\left[(1-\xi_{1}-\xi_{2})q^{2}+\xi_{1}(q-\hat{k})^{2}+\xi_{2}(P-q)^{2}\right]^{3}}\right.\;+$
(45)
$\displaystyle\quad+\;\left.\frac{i\,n_{d}}{m_{0}^{2}\left[(1-\xi_{1})q^{2}+\xi_{1}(q-\hat{k})^{2}\right]^{2}}+\frac{i\,n_{e}}{m_{0}^{2}\left[(1-\xi_{1})q^{2}+\xi_{1}(P-q-\hat{k})^{2}\right]^{2}}\right\\}$
$\displaystyle+$ $\displaystyle\frac{\bar{n}_{c}}{g^{2}}\left(\delta
Z_{g}+\frac{3}{2}\delta
Z_{3}\right)-\frac{\bar{n}_{d}+\bar{n}_{e}}{g^{2}}\delta Z_{3}\;.$

Substituting $\tilde{q}=q-\xi_{1}k-\xi_{2}P$ and
$L_{1}=-P^{2}\xi_{2}(1-\xi_{2})$ in the first term, $\tilde{q}=q-\xi_{1}k$ in
the second term and $\tilde{q}=q-\xi_{1}(P-k)$ in the third term, we get

$\displaystyle g^{2}A_{1}(z,P^{2})$ $\displaystyle\;=\;$ $\displaystyle
i\,g^{2}\int\limits_{0}^{1}d\xi_{1}\int\frac{d^{D}\tilde{q}}{(2\pi)^{D}}\left\\{\int\limits_{0}^{1-\xi_{1}}d\xi_{2}\frac{2n_{c}}{(\tilde{q}^{2}-L_{1})^{3}}\;+\;\frac{n_{d}+n_{e}}{m_{0}^{2}\,\tilde{q}^{4}}\right\\}+$
(46) $\displaystyle+\;\bar{n}_{c}\left(\delta Z_{g}+\frac{3}{2}\delta
Z_{3}\right)-(\bar{n}_{d}+\bar{n}_{e})\delta Z_{3}\;.$

Substituting $\tilde{q}=(iq_{E},\mathbf{q}_{E})$ (Wick-rotation) gives

$\displaystyle g^{2}A_{1}(z,P^{2})$ $\displaystyle\;=\;$ $\displaystyle
g^{2}\frac{\kappa_{D}}{(2\pi)^{D}}\int\limits_{0}^{1}d\xi_{1}\int\limits_{0}^{\infty}dq_{E}\,q_{E}^{D-1}\left\\{\int\limits_{0}^{1-\xi_{1}}d\xi_{2}\frac{2n_{c}}{(q^{2}_{E}+L_{1})^{3}}\;-\;\frac{n_{d}+n_{e}}{m_{0}^{2}\,\tilde{q}_{E}^{4}}e^{-\epsilon
q_{E}/m_{0}}\right\\}+$ (47) $\displaystyle+\;\bar{n}_{c}\left(\delta
Z_{g}+\frac{3}{2}\delta Z_{3}\right)-(\bar{n}_{d}+\bar{n}_{e})\delta Z_{3}\;.$

Using
$\int\limits_{0}^{\infty}\dfrac{dx\,x^{a-1}}{(x+1)^{a+b}}=\Gamma(a)\Gamma(b)/\Gamma(a+b)$,
$\Gamma(\epsilon)=1/\epsilon-\gamma_{E}$ and setting $D=6-2\epsilon$ along
with $g\rightarrow g\mu^{\epsilon}$, we obtain

$\displaystyle g^{2}A_{1}(z,P^{2})$ $\displaystyle\;=\;$
$\displaystyle\frac{g^{2}n_{c}}{2(4\pi)^{3}}\left(\frac{1}{\epsilon}-\ln\frac{-P^{2}}{\mu^{2}}-\gamma_{E}+\ln
4\pi-2\int\limits_{0}^{1}d\xi_{1}\int\limits_{0}^{1-\xi_{1}}d\xi_{2}\ln\xi_{2}(1-\xi_{2})\right)\;-$
(48)
$\displaystyle-\frac{g^{2}}{2(4\pi)^{3}}(n_{d}+n_{e})\frac{1}{\epsilon^{2}}+\;\bar{n}_{c}\left(\delta
Z_{g}+\frac{3}{2}\delta Z_{3}\right)-(\bar{n}_{d}+\bar{n}_{e})\delta Z_{3}\;.$

If we remove the divergences along with the $P^{2}$ and $\mu$ independent
constants by the counter terms, we are left with

$A_{1}(z,P^{2})\;=\;-\frac{n_{c}}{2(4\pi)^{3}}\ln\frac{P^{2}}{\mu^{2}}$ (49)

For the calculation of the second line of $A(z,P^{2})$ in Eq. (LABEL:evol4),
we parametrize momenta as $P=(M,0,\mathbf{0})$, $k=(Mz/2,Mz/2,\mathbf{0})$ and
$q=\alpha P+\beta k+q_{T}=(\alpha M+\beta Mz/2,\beta Mz/2,\mathbf{q}_{T})$.
This way, the integration measure becomes
$d^{D}q=(M^{2}z/2)d\alpha\,d\beta\,d^{D-2}\mathbf{q}_{T}$. Besides,
$2Pq=M^{2}(2\alpha+\beta z)$, $2kq=M^{2}\alpha z$ and
$q^{2}=M^{2}\alpha(\alpha+\beta z)-\mathbf{q}_{T}^{2}$. Furthermore, due to
the $\delta\left[(q-k)^{2}\right]$ term, $q^{2}=2kq=M^{2}\alpha z$. This way,
the second line of Eq. (LABEL:evol4) becomes

$\displaystyle g^{2}A_{2}(z,P^{2})$ $\displaystyle\;=\;$ $\displaystyle
g^{2}\frac{z\kappa_{D-2}}{4(2\pi)^{D+1}}\int d\alpha\int d\beta\int
dq_{T}^{2}\,(q_{T}^{2})^{D/2-2}\;\times$
$\displaystyle\times\;(2\pi)\delta\left[M^{2}\alpha(\alpha+\beta
z-z)-q_{T}^{2}\right](2\pi)\delta\left[M^{2}(1+\alpha z-2\alpha-\beta
z)\right]\;\times$
$\displaystyle\times\;\left\\{\frac{n_{f}}{\alpha^{2}z^{2}}+\frac{n_{g}}{(1-z)^{2}}+\frac{n_{h}}{\alpha
z(1-z)}+\frac{n_{i}}{\alpha z(1-2\alpha+z-\beta z)}\right\\}\;.$

Using the integral for $\beta$ to eliminate the second $\delta$ function gives

$\displaystyle g^{2}A_{2}(z,P^{2})$ $\displaystyle\;=\;$
$\displaystyle\frac{g^{2}\kappa_{D-2}}{4(2\pi)^{D-1}M^{2}}\int d\alpha\int
dq_{T}^{2}\,(q_{T}^{2})^{D/2-2}\delta\left[M^{2}\alpha(1-\alpha)(1-z)-q_{T}^{2}\right]\;\times$
$\displaystyle\times\;\left\\{\frac{n_{f}}{\alpha^{2}z^{2}}+\frac{n_{g}}{(1-z)^{2}}+\frac{n_{h}}{\alpha
z(1-z)}+\frac{n_{i}}{\alpha(1-\alpha)z^{2}}\right\\}$ $\displaystyle\;=\;$
$\displaystyle\frac{g^{2}\kappa_{D-2}M^{D-6}(1-z)^{D/2-2}}{4(2\pi)^{D-1}}\int
d\alpha\,[\alpha(1-\alpha)]^{D/2-2}\;\times$
$\displaystyle\times\;\left\\{\frac{n_{f}}{\alpha^{2}z^{2}}+\frac{n_{g}}{(1-z)^{2}}+\frac{n_{h}}{\alpha
z(1-z)}+\frac{n_{i}}{\alpha(1-\alpha)z^{2}}\right\\}$

Using the identity
$\int\limits_{0}^{1}d\alpha\alpha^{a-1}(1-\alpha)^{b-1}=\Gamma(a)\Gamma(b)/\Gamma(a+b)$
and $D=6-2\epsilon$ dimensions, where the coupling acquires dimension
$g\rightarrow g\mu^{\epsilon}$, the solid angle is
$\kappa_{D}=2\pi^{D/2}/\Gamma(D/2)$ and
$\Gamma(-\epsilon)=-1/\epsilon-\gamma_{E}$, we obtain

$\displaystyle g^{2}A_{2}(z,P^{2})$ $\displaystyle\;=\;$
$\displaystyle\frac{g^{2}(1-z)}{(4\pi)^{3}}\left(\frac{4\pi\mu}{M^{2}(1-z)}\right)^{\epsilon}\left\\{\frac{n_{f}}{z^{2}}\left(-\frac{1}{\epsilon}-\gamma_{E}\right)+\frac{n_{g}}{6(1-z)^{2}}+\frac{n_{h}}{2z(1-z)}+\frac{n_{i}}{z^{2}}\right\\}$
(52) $\displaystyle\;=\;$
$\displaystyle\frac{g^{2}}{(4\pi)^{3}}\left\\{-\left[\frac{1}{\epsilon}+\gamma_{E}+\ln\left(\frac{4\pi\mu}{M^{2}(1-z)}\right)\right]\frac{n_{f}(1-z)}{z^{2}}\;+\right.$
$\displaystyle\qquad\qquad+\left.\frac{n_{g}}{6(1-z)}+\frac{n_{h}}{2z}+\frac{n_{i}(1-z)}{z^{2}}\right\\}\;.$

We made use of $\Gamma(-1+\epsilon)=-1/\epsilon+\gamma_{E}-1$,
$\Gamma(-\epsilon)=-1/\epsilon-\gamma_{E}$. Note that the $1/\epsilon$ term
being the collinear divergence, cannot be eliminated via renormalisation,
however, it drops out of the splitting function, which is

$\Pi(z)\;=\;\frac{\partial}{\partial\ln
P^{2}}A(z,P^{2})\;=\;\frac{n_{f}}{(4\pi)^{3}}\frac{1-z}{z^{2}}-\frac{n_{c}}{2(4\pi)^{3}}\delta(1-z)\;.$
(53)

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