# Observation of the dead cone effect in charm and bottom quark jets and its
QCD explanation

Stefan Kluth  Wolfgang Ochs  Max-Planck-Institut für Physik, Föhringer Ring 6,
80805 München, Germany  Redamy Perez Ramos  DRII-IPSA, Bis, 63 Boulevard de
Brandebourg, 94200 Ivry-sur-Seine, France,
Laboratoire de Physique Théorique et Hautes Energies (LPTHE), UMR 7589,
Sorbonne Université et CNRS, 4 place Jussieu, 75252 Paris Cedex 05, France

###### Abstract

The production of a heavy quark is accompanied by gluon bremsstrahlung which
is suppressed at small angles $\Theta\lesssim M_{Q}/E$ for mass $M_{Q}$ and
high energy $E$ according to perturbative Quantum Chromo Dynamics (QCD) (“dead
cone effect”). As particles at small angles typically have large momenta, the
heavy quark mass also causes a suppression of high momentum particles. In this
paper, we studied this effect in c- and b-quark events using data from Z boson
decays in $e^{+}e^{-}$ annihilation. The heavy quark fragmentation function
for charged particles is reconstructed in the momentum fraction variable $x$
or $\xi=\ln(1/x)$ by removing the decays of the heavy quark hadrons. Indeed,
we find an increasing suppression of particles with rising $x$ down to a
fraction of $\lesssim 1/10$ for particles with $x\gtrsim 0.2$ in b-quark and
$x\gtrsim 0.4$ in c-quark jets in comparison to light quark fragmentation. The
sensitivity to the dead cone effect in the present momentum analysis is
considerably increased in comparison to the recently presented angular
analysis. This amount of suppression and the differences between c- and
b-quark fragmentation are in good quantitative agreement with the expectations
based on perturbative QCD within the Modified Leading Logarithmic
Approximation (MLLA) in the central kinematic region. The data also support a
two parameter description in the MLLA of these phenomena (“Limiting
Spectrum”). The sensitivity of these measurements to the heavy quark mass is
investigated.

††preprint: MPP-2023-52

##  I Introduction

The dead cone effect is a prediction of QCD, the theory of strong interactions
within the Standard Model of particle physics. It originates from the
radiation pattern off a heavy quark as obtained in perturbation theory [1, 2].
For an energetic heavy quark $Q$ of mass $M_{Q}$ and energy $E_{Q}$ such that
$E_{Q}/M_{Q}\gg 1$, the gluon emission probability for small emission angle
$\Theta$ and low energy $\omega$, can be written as

$d\sigma_{Q\to
Q+g}\simeq\frac{\alpha_{s}}{\pi}C_{F}\frac{\Theta^{2}d\Theta^{2}}{(\Theta^{2}+\Theta^{2}_{0})^{2}}\frac{d\omega}{\omega},$
(1)

with angular cut-off $\Theta_{0}=M_{Q}/E_{Q}$; $\alpha_{s}$ denotes the strong
coupling constant and $C_{F}$ the QCD colour factor at the branching vertex
$Q\to Q+g$. Therefore, for smaller emission angles $\Theta<\Theta_{0}$, gluon
radiation is suppressed and vanishes in the forward direction such that the
region with the gluon depopulated cone around the flight direction of the
heavy quark $Q$ is called “dead cone”. For large emission angles
$\Theta\gg\Theta_{0}$, the gluon radiation pattern becomes identical to that
of a light quark jet, and the same statement holds for the internal angular
ordered structure of secondary gluon subjets.

In the early studies, as a first consequence of the dead cone effect, a
reduction of the full particle multiplicity in the heavy quark jet has been
predicted. This effect has indeed been observed in [3] and in the subsequent
update [4] with results nearby the QCD expectation. Only recently, a more
direct observation of the dead cone effect has been achieved by the ALICE
collaboration [5], which has presented results on the differential angular
structure of charm-quark and inclusive jets from proton-proton collisions at
$\sqrt{s}=13$ TeV at the Large Hadron Collider (LHC). A relative suppression
of small angle particle emission is observed in the heavy quark jet in
agreement with Monte Carlo Event Generators (MCEG) combining the hard
interactions of the partons from the protons with a QCD parton shower and a
hadronisation model. Already before, some preliminary angular studies of the
dead cone effect have been presented using data on charm-jets from HERA [6]
and data on b-jets from LEP [7].

Multi-parton final states in quark and gluon jets can be calculated
perturbatively from subsequent parton branchings with running coupling
$\alpha_{s}$ down to the transverse momentum cut-off $Q_{0}$, within the
probabilistic parton shower picture based on angular ordering [8, 9, 10, 11].
Some insight can be gained by first considering the Double Logarithmic
Approximation (DLA), which accounts for leading collinear and soft
singularities as in eq. (1) for $\Theta_{0}=0$. A more accurate description is
achieved in the MLLA if single logarithmic terms are included (relative order
${\cal O}(\sqrt{\alpha_{s}})$). In the comparison with experiments, the
hypothesis of “Local Parton Hadron Duality” (LPHD) [12] has turned out to be
quite successful in many applications. In this scheme, perturbative QCD
predictions on sufficiently inclusive observables for final state partons are
in close correspondence with these observables for hadrons, where $Q_{0}$ may
achieve low values, even down to the mass scale $\Lambda$ of QCD (“Limiting
Spectrum”).

The single inclusive gluon spectrum for a quark or gluon jet in the variable
$\xi=\ln(1/x)$, with $x=E_{g}/E$, has been computed as function of the primary
energy $E$ in MLLA [12, 13, 14]. The main feature is a Gaussian-like peak in
the $\xi$ spectrum, the so-called “hump-backed plateau” resulting from the
coherent emission of soft gluons in the cascade. Predictions for the $\xi$
spectrum and the primary energy dependence of the Gaussian parameters agree
well with $e^{+}e^{-}$ annihilation data, see e.g. [15] for a review and [16]
for a recent analysis. Generally, only charged particles are studied
experimentally since the resolution for direction and momentum measurements is
improved with respect to neutral particles, in particular for low momentum
particles dominating the hadronic final states. For an overview of the
perturbative QCD approach on parton shower evolution, approximations and some
applications to multiparticle production, see e.g. [11, 17].

In this paper the dead cone effect is studied for the first time by exploring
the internal momentum structure of heavy and light-quark jets, which can be
directly compared with the early QCD predictions within the MLLA-LPHD approach
[1, 2]. In jets particles with large momenta are emitted on average at small
angles and particles with small momenta at larger angles. Therefore, the dead
cone effect at small angles corresponds to a suppression of large momentum
gluons. An advantage of studying momentum spectra is that their determination
does not require a jet axis definition which may cause important systematic
uncertainties.

##  II Experimental determination of particle spectra in $b$ \- and $c$ \-
quark events

###  II.1 Reconstruction of heavy quark fragmentation functions

Experimental data are presented as function of the charged particle momenta
$p$, with $x_{p}=2p/W$ or $\xi_{p}=\ln(1/x_{p})$ at c.m.s. energy $W$. In
$e^{+}e^{-}$ annihilation, we refer to the fragmentation function

$\bar{D}(\xi,W)=\frac{1}{2}\frac{1}{\sigma_{tot}}\frac{d\sigma^{h}}{d\xi}(\xi,W)$
(2)

where $\bar{D}(\xi,W)=xD(x,W)$ with the inclusive $x$-distribution $D(x,W)$
for particles from both hemispheres. In order to obtain the heavy quark
fragmentation function $\bar{D}_{Q}^{ch}(\xi_{p},W)$, $Q=c,b$, we start from
the measured $\xi_{p}$-distribution of light hadrons in events tagged as
originating from $Z\rightarrow Q{\bar{Q}}$ decays. This distribution also
contains the charged hadrons from B-hadron or Charm-hadron decays and they
have to be subtracted. The $\xi_{p}$-distributions of charged B-hadron or
Charm-hadron decay products have not been measured separately111A measurement
could use the impact parameters of reconstructed tracks w.r.t. the primary
vertex to select the heavy hadron decay products. and we obtain them from a
MCEG program; the most recent version of Pythia8 [18, 19] is used for this
purpose with 100’000 events generated.

Figure 1: Distribution of charged hadrons in $\xi_{p}=\ln(1/x_{p})$ for full
$b\bar{b}$ events including B-hadron decays at 91.2 GeV: DELPHI and OPAL data
and prediction from the Pythia8 MCEG; B-hadron decay charged particle
$\xi_{p}$-distribution from Pythia8 with systematic error ($6.5\%$); the
b-quark fragmentation functions as derived from DELPHI and OPAL data after
subtraction of the B-hadron decay charged particle distribution rescaled to
the experimental decay multiplicity eq. (3) (left panel); corresponding
results for $c\bar{c}$ events: full spectrum by OPAL and from Pythia8, Charm-
hadron decay charged particle distribution from Pythia8 and the c-quark
fragmentation function (right panel).

The full $\xi_{p}$-distribution of events with B-hadrons including their
decays has been measured by several groups [20, 21, 22, 23] with good mutual
agreement. The results obtained222Only DELPHI and OPAL presented their data
for b- or c-quarks and uds-quarks in $\xi$. by DELPHI [20] and OPAL [21] are
shown in Fig. 1 together with the MCEG prediction for this distribution (left
panel). The predictions agree within $\sim$$10\%$ with the data. The measured
charged hadron multiplicities $N_{b}^{ch}=23.17\pm 0.35$ (DELPHI) and
$N_{b}^{ch}=23.16\pm 0.45$ (OPAL) compare to $N_{b}^{ch}=21.26\pm 0.02$, as
obtained with Pythia8. Furthermore in Fig. 1, the $\xi_{p}$-distribution of
the charged B-hadron decay particles is displayed which integrates up to the
multiplicity $n^{dec}_{b}=9.80$.

The parameters of the MCEG program are tuned to a large variety of data from
$e^{+}e^{-}$ and pp collisions [19]. Therefore, one cannot expect an optimal
agreement in all processes. Indeed, for the B-hadron decay multiplicity, an
experimental value has been determined with a rather small error [4]
(practically identical to the earlier result [3]) which differs from the MCEG
prediction:

$n^{dec}_{b}=9.80\ \ (\rm{Pythia8})\ \ \ \ \ \ n^{dec}_{b}=11.10\pm 0.18,\ \
(\rm{experiment}).$ (3)

The experimental result is based on the evaluation [24] of the measurements by
ALEPH, CDF, DELPHI, L3, OPAL and SLD for the single B-hadron decay
multiplicity $N_{b}=4.955\pm 0.062$ and includes contributions from $K^{0}$
and $\Lambda$ decays among others. This number for $n^{dec}_{b}$ exceeds the
one obtained by Pythia8 by $13\%$.

In our subsequent analysis, only the shape of the $\xi_{p}$-distribution is
taken from the MCEG simulation but its normalisation is scaled by $13\%$ to
obtain the experimental decay multiplicity eq. (3). We add a systematic error
of $6.5\%$ to all data points in order to allow for variations in a band of
the missing decay rate. Subtracting this rescaled B-hadron decay distribution
from the spectra of the full b-events by DELPHI and OPAL, the final b-quark
fragmentation function is derived and displayed in Fig. 1 (left panel) as
well. There is a good agreement between the results from both experiments. We
have compared our result for the B-hadron decay distribution with the results
obtained by the DELPHI collaboration [20] using the JETSET MCEG by adding the
contributions from $\pi,K,p$. The b-quark fragmentation function computed with
their result agrees with ours within the errors, lying for $\xi_{p}\gtrsim 3$
below our result at the edge of the error bars.

The corresponding results for $c\bar{c}$-events are shown in Fig. 1 (right
panel). The $\xi_{p}$-distribution for the full events including the Charm-
hadron decays as obtained by OPAL [21] are found in good agreement with the
Pythia8 results: the full multiplicity by OPAL $N_{c}^{ch}=21.55\pm 0.74$
compares with $N_{c}^{ch}=20.05$. Also shown is the $\xi_{p}$-distribution of
the charged decay products of the Charm-hadrons which integrates to the
multiplicity $n_{c}^{dec}=5.04$ and compares well with the observed decay
multiplicity, $n_{c}^{dec}=5.2\pm 0.3$ [4] such that no rescaling is applied.
Subtracting the distribution of decay products from the full distribution
finally yields the experimentally derived c-quark fragmentation function.

Figure 2: Fragmentation function in $\xi_{p}=\ln(1/x_{p})$ for the light uds-
quarks in comparison with experimentally derived $\xi_{p}$-distributions for
the b-quark and c-quark fragmentation (upper panels); ratio of the heavy quark
over the light quark fragmentation functions showing the strong suppression of
the heavy quark fragmentation for small $\xi_{p}$ (large momenta) by an order
of magnitude which constitutes the dead cone effect (lower panels).

###  II.2 Evidence for the dead cone effect in heavy quark events

Next we compare in Fig. 2 the light uds-quark $\xi_{p}$-distributions measured
by DELPHI and OPAL with the b-quark and c-quark $\xi_{p}$-distributions
derived in the last subsection by removing the charged heavy hadron decay
products. One observes the strong suppression of particle production in the
b-quark and c-quark fragmentation (upper panels) and in the ratios of both
distributions (lower panels). While the ratios approach unity for large
$\xi_{p}$ in the low momentum limit, the suppression progresses down to a
ratio of $\lesssim 1/10$ for small $\xi_{p}$ values (large momenta) before it
levels off. There is a notable difference between c- and b-quark
fragmentation: the ratio for the b-quark starts decreasing already at
$\xi_{p}\sim 5$ and falls down to $\sim 1/10$ near $\xi_{p}\sim 2$, whereas
for the c-quark, it starts decreasing later at $\xi_{p}\sim 3$ and falls to
zero near $\xi_{p}\sim 1$. This difference will be related to the different
quark masses $M_{Q}$ below.

The dead cone effect is established in this process with a high significance,
which is comfortably larger than five $\sigma$ for both b-quarks and c-quarks.
According to eq. (1), the dead cone effect is characterised by the full
suppression of the small angle fragmentation from the heavy quark. The results
in Fig. 2 show the efficiency of the momentum space analysis in reflecting the
almost complete suppression of heavy quark fragmentation in the corresponding
limit of large momenta. This suppression is stronger than the maximal
suppression by a factor of about $1/2$ observed in the angular analysis by
ALICE [5]. This may be related to the finite jet resolution and the difficulty
to define the gluon emission angle in this analysis.

##  III MLLA expectations for heavy quark fragmentation functions

###  III.1 The MLLA QCD relations between light and heavy quarks jets

The dead cone effect has first been studied for total multiplicities of light
hadrons in QCD jets within the MLLA of perturbative QCD [1, 2] and the
predictions have been compared with data in [3, 4]. In the MLLA the
accompanying multiplicity in the production of a heavy quark pair
$N_{Q\bar{Q}}(W)$ at c.m.s. energy $W$ can be expressed in terms of the
multiplicity $N_{q\bar{q}}(W)$ in the light quark $q\bar{q}$ production
(q=u,d,s) with the multiplicity in the dead cone subtracted as

$N_{Q\bar{Q}}(W)=N_{q\bar{q}}(W)-N_{q\bar{q}}(\sqrt{e}M_{Q}),$ (4)

with the dead cone mass scale $W_{0}=\sqrt{e}M_{Q}$ ($e=\rm{exp}(1)$). The
directly observed full charged hadron multiplicity in heavy quark events
produced in $e^{+}e^{-}$ annihilation $N_{Q}^{ch}\equiv N^{ch}_{e^{+}e^{-}\to
Q\bar{Q}}$ can be written as

$N_{Q}^{ch}(W)=N_{Q\bar{Q}}^{ch}(W)+n_{Q}^{dec},$ (5)

i.e. as the sum of multiplicities accompanying the heavy quarks
$N_{Q\bar{Q}}^{ch}$ and the charged multiplicities from the decays of the two
heavy hadrons $n_{Q}^{dec}$.

As an important consequence of eq. (4), the difference between the observed
charged particle multiplicities in heavy and light quark events

$\delta_{Q\ell}=N_{Q}^{ch}(W)-N_{q\bar{q}}^{ch}(W)$ (6)

is predicted in the MLLA as

$\delta_{Q\ell}^{MLLA}=n_{Q}^{dec}-N_{q\bar{q}}(\sqrt{e}M_{Q}),$ (7)

such that this quantity is independent of the total energy $W$ and depends
only on the heavy quark mass $M_{Q}$. As reviewed in the dead cone analysis
[4], this difference is indeed found to be independent of the c.m.s. energy in
$e^{+}e^{-}$ annihilation to b-quarks up to LEP 2 energies within the
experimental uncertainties. An alternative model without the dead cone effect
and a pronounced energy dependence of $\delta_{b\ell}$ has been excluded at
high confidence level.

For our analysis of the inclusive spectra, the relation for multiplicities as
their integrals, will serve as an important cross check. For the b-quark with
the values $n_{b}^{dec}=11.10\pm 0.18$ and $W_{0}=8$ GeV with $N_{q\bar{q}}(8\
\rm{GeV})=6.7\pm 0.34$, the prediction is close to the experimental result [4]

$\delta_{b\ell}^{MLLA}=4.4\pm 0.4,\ \ \ \delta_{b\ell}^{exp}=3.14\pm 0.14,$
(8)

but a significant difference remains.

For the c-quark, with the values $n_{c}^{dec}=5.2\pm 0.3$, $W_{0}=2.7$ GeV and
$N_{q\bar{q}}(2.7\ \rm{GeV})=3.7\pm 0.3$, one finds the charged particle
multiplicity difference [4] as

$\delta_{c\ell}^{MLLA}=1.5\pm 0.4,\ \ \ \delta_{c\ell}^{exp}=1.0\pm 0.4,$ (9)

with consistent results between theory and experiment.

The spectrum $D_{Q}(x,E)$ of gluons with energy fraction333For the massless
partons $E=p$ in the calculations, in the comparison with light hadrons we
take $x=p_{h}/E$ $x=E_{g}/E$ at primary energy $E$ accompanying the
$Q{\bar{Q}}$ pair can be treated in a similar way to that of mean
multiplicities in eq. (4); for a review, see [25]. Some insight can be gained
by first considering the results for the leading double logarithmic
approximation (DLA). The difference between the heavy quark $D_{Q}(x,E)$ and
the light quark $D_{q}(x,E)$ spectra due to the dead cone effect comes from
the radiation of very energetic gluons at small angles $\Theta<\Theta_{0}$.
This radiation can be considered as resulting from a Lorentz boost by the
factor $\gamma=E/M_{Q}$ along the heavy quark direction from the corresponding
radiation at lower hardness $M_{Q}$. In the DLA, a simple formula in analogy
to the equation for multiplicities eq. (4) can be written as [2, 26]:

$D_{Q}(x,E)=D_{q}(x,E)-D_{q}(x,M_{Q}).$ (10)

Hence, the heavy quark fragmentation function is represented in terms of the
light quark fragmentation functions at the different energy scales $E$ and
$M_{Q}$. This equation cannot be strictly correct, since the $x$-distribution
at large $x$ decreases with rising energy because of scale breaking effects,
therefore $D_{q}(x,E)<D_{q}(x,M_{Q})$ and $D_{Q}(x,E)$ in eq. (10) would
become negative (see e.g. Fig. 8 below).

While the equation for multiplicities eq. (4) is derived systematically within
the MLLA, the corresponding analysis for the inclusive spectra in MLLA is not
yet available at the same rigor. An improved equation for the inclusive
$x$-spectra has been presented which reproduces the equation for
multiplicities in MLLA after integration over $x$ and avoids a negative
fragmentation function. The MLLA estimate has been reported as [2]

$\bar{D}_{Q}(x,W)=\bar{D}_{q}(x,W)-\bar{D}_{q}\left(\frac{x}{\langle
x_{Q}\rangle},\sqrt{e}M_{Q}\right),$ (11)

where $\bar{D}(x,W)=xD(x,W)$. This expression, after integration over the
variable $x$, reproduces eq. (4) for the multiplicities. For our comparison
with the heavy quark fragmentation function $\bar{D}_{Q}(\xi,W)$ with variable
$\xi$ as determined in the last section, we rewrite this relation as

$\bar{D}_{Q}(\xi,W)=\bar{D}_{q}(\xi,W)-\bar{D}_{q}(\xi-\xi_{Q},\sqrt{e}M_{Q}),$
(12)

with $\bar{D}(\xi,W)=xD(x,W)$ and $\xi_{Q}=\ln(1/\langle x_{Q}\rangle)$.
Again, as in the equation for multiplicities, the low energy scale is
$W_{0}=\sqrt{e}M_{Q}$. Furthermore, the mean momentum fraction ${\langle
x_{Q}\rangle}$ of the primary heavy quark $Q$ is introduced which reduces the
light particle energies to $x<{\langle x_{Q}\rangle}$. The shift of the
$\xi$-spectrum by $\xi_{Q}$ corresponds to an MLLA correction of ${\cal
O}(\sqrt{\alpha_{s}})$ as can be seen by a Taylor expansion of
$\bar{D}_{q}(\xi-\xi_{Q},\sqrt{e}M_{Q})$ in $\xi_{Q}$ at $\xi_{Q}=0$. The eq.
(12) represents an approximation that does not work well at small $\xi$ since
the shifted contribution $\bar{D}_{q}(\xi-\xi_{Q},W_{0})$ has to vanish for
$\xi<\xi_{Q}$. Comparisons of these predictions with experiment should take
these limitations into account.

If ${\langle x_{Q}\rangle}$ is taken from experiment, the heavy quark
fragmentation function at c.m.s. energy $W$ can be obtained by the relation
eq. (12) from the light quark fragmentation functions at energies $W$ and
$W_{0}$ in absolute normalisation.

As numerical values of these parameters, we take for b-quarks, $W_{0}=8.0$
GeV444This value corresponds to a b-quark pole mass $M_{b}=4.85\pm 0.15$ which
is consistent with the most recent world average pole mass $M_{b}=4.78\pm
0.06$ [27] [4] and the experimental evaluation $\langle x_{b}\rangle=0.7092\pm
0.0025$ [28]. For c-quarks, we use $W_{0}=2.7$ GeV [4] and the experimental
value $\langle x_{c}\rangle=0.495\pm 0.006$ [29]. This yields the shift
parameters

$\xi_{c}=0.70,\ \ \ \xi_{b}=0.36.$ (13)

These numbers are also consistent with the results in [17], based on
calculations for the heavy quark $x$-spectra in [30].

###  III.2 Experimental test of the MLLA relation between light and heavy
quark jets

At first, we probe the MLLA expectation eq. (12) by inserting for
$\bar{D}(\xi,W)$ the experimentally observed distributions in
$\xi_{p}=\ln{1/x_{p}}$ at the respective energies $W$. At the low energy
$W_{0}=2.7$ GeV for the c-quark fragmentation, we insert the
$\xi_{p}$-distribution data obtained by the BES collaboration at the nearby
energy 2.6 GeV [31]. There are no data nearby $W_{0}=8.0$ GeV for the b-quark
fragmentation and therefore, we obtain the corresponding
$\xi_{p}$-distribution from the interpolation between two neighbouring
energies, also a correction for charm production has been applied (see App.
A). The $\xi_{p}$-distribution at $W_{0}=8.0$ GeV, so obtained and shifted by
$\xi_{b}=0.36$ according to eq. (12), i.e. $\bar{D}_{q}(\xi-\xi_{b},W_{0})$,
is shown in Fig. 3 (left panel) as a dashed line; also shown are the data for
$\bar{D}_{q}(\xi,W)$ at $W=91.2$ GeV, both referring to uds-quark events. From
their difference, according to eq. (12), one obtains the MLLA prediction for
the b-quark distributions $\bar{D}_{b}(\xi_{p},W)$ where the error bars shown
include the systematic errors.

Figure 3: Fragmentation functions in $\xi_{p}=\ln(1/x_{p})$ for uds-quarks at
91.2 GeV and at $W_{0}=8.0$ GeV by interpolation between neighbouring
energies, with correction for charm decays (see App. A) and with shift
$\xi_{b}=0.36$, see eq. (13). Subtracting these distributions as in eq. (12),
yields the MLLA prediction for the b-quark fragmentation function to be
compared with the experimentally derived one on linear and logarithmic scales
(left panels); corresponding results for c-quark fragmentation: at 91.2 GeV
the OPAL uds-quark data; at $W_{0}\sim 2.6$ GeV the data by BES; the MLLA
prediction for the c-quark fragmentation function to be compared with the
experimentally derived c-quark fragmentation function (right panels).

This MLLA-expected distribution for the b-quark is now compared in Fig. 3 to
the experimentally derived b-quark fragmentation functions using DELPHI and
OPAL data (see Fig. 1 (left panel) as discussed in the last section). Again we
point to the strong suppression of the b-quark fragmentation function which
becomes almost complete for the high momentum particles with $\xi_{p}\lesssim
2$. The MLLA expectations match with the experimentally derived b-quark
fragmentation data at a quantitative level in the region around the peak of
the 8 GeV distribution between $\xi_{p}=1.5$ and $\xi_{p}=3.2$. There are
deviations at small $\xi_{p}$ (large momenta) at a low level of particle
density. A larger deviation occurs in the region above $\xi_{p}=3$, which
corresponds to the ultrasoft particles with momenta $p\lesssim
Q_{0}\sim\Lambda$ at the hadronic mass scale. This region is outside the range
of validity of the perturbative approach.

Now we turn to the results on the c-quark events in the right panel of Fig. 3.
By subtracting the low energy BES $\xi_{p}$-distribution at $W_{0}=2.6$ GeV,
after a shift by $\xi_{c}=0.7$, from the uds-quark distribution at $W=91.2$
GeV, one obtains the MLLA-predicted c-quark fragmentation function. The
experimentally derived and the predicted $\xi_{p}$-distributions for c-quark
fragmentation are compatible with each other within the larger errors over the
full considered region supporting the MLLA ansatz. There is no direct evidence
for an excess multiplicity at large $\xi_{p}$ as seen in b-quark fragmentation
but the errors are larger.

The different behaviour of the c-quark and b-quark fragmentation functions is
caused by the different behaviour of the $\xi_{p}$-distributions at the low
energies $W_{0}(M_{Q})$, i.e. at 2.6 and 8.0 GeV, respectively. In this way,
the essential features of the dead cone effect are explained by the
subtraction of particles with the $\xi_{p}$-distribution at the respective
mass scale $M_{Q}$ from the full particle ensemble at energy $W$.

We also observe in Fig. 3, that the light and heavy quark spectra approach
each other for large $\xi_{p}$ as theoretically expected, since the soft
particles are mainly emitted at large angles and are thereby insensitive to
the cut-off $\Theta_{0}=M_{Q}/E_{Q}$, i.e. the dead cone effect.

It is noted, that our results on the shapes of observed uds-quark and MLLA-
expected b-quark distributions in Fig. 3 qualitatively agree with the
expectations presented in the original publication [1].

Figure 4: Fragmentation function in $\xi_{p}=\ln(1/x_{p})$ for uds-quarks at
91.2 GeV together with the MLLA Limiting Spectrum distributions; at
$W_{0}=8.0$ GeV the Limiting Spectrum at this energy including the correction
for charm decays (normalisation $K_{ch}=1.43$, App. A) and with shift
$\xi_{b}=0.36$, see eq. (13). Subtracting these distributions as in eq. (12)
yields the MLLA prediction for the b-quark fragmentation function to be
compared with the experimentally derived one (left panels); the corresponding
results for c-quark fragmentation (right panels).

###  III.3 Description of $\xi_{p}$-distributions by the Limiting Spectrum

In this subsection we will probe the MLLA suggested prediction eq. (12) by
inserting, for the description of inclusive $\xi_{p}$-distributions, the
analytical MLLA results for the $\xi$-distribution of gluons. In a particular
approach, the transverse momentum cutoff is taken at its minimum value
$Q_{0}=\Lambda$ and one obtains the so-called “Limiting Spectrum”, which can
be written in terms of an integral representation of the confluent
hypergeometric function [11]. In this application of the MLLA framework, one
assumes that the parton cascade evolves down to the hadron mass scale
$Q_{0}=\Lambda$ and represents the hadron cascade in an average sense (“Local
Parton Hadron Duality” (LPHD) [12]). As the bulk of particles inside the
Gaussian hump have only small momenta of a few GeV, the number of active
flavours in the calculation of the coupling $\alpha_{s}$ is usually taken as
$n_{f}=3$ [11, 32, 33]. The only remaining parameters in this approach are the
QCD scale $\Lambda$ and an overall normalisation factor $K_{ch}$. Previous
fits to the charged particle spectra for all flavours yielded values, see e.g.
[32], with $\Lambda=250$ MeV and $K_{ch}=1.28$ at c.m.s. energy $W=91.2$ GeV.
Distributions at lower energies down to 14 GeV could be fitted with the same
$\Lambda$ but for $K_{ch}$ increased by 14$\%$. This energy dependence is
interpreted as an effect from higher order corrections to the MLLA.

The Limiting Spectrum function will now be used for the description of the
$\xi_{p}$ spectra in this analysis. The $\xi_{p}$-distribution for b-quark
fragmentation at 8 GeV is obtained by using the same parameters as for the
neighbouring energies and applying a correction for charm quark production
(see App. A). This spectrum at 8 GeV is shown in Fig. 4 (left panel) as the
dashed line, also shown is the corresponding fit to the uds-data at $W=91.2$
GeV. The parameters of the fit are obtained as $\Lambda=275$ MeV in this
energy range, $K_{ch}=1.28$ at $W=91.2$ GeV and $K_{ch}=1.52$ at the lower
energies. From these two Limiting Spectrum distributions one obtains the MLLA
based predictions for the b-quark fragmentation function using eq. (12) shown
as the short dashed line. This prediction is in quantitative agreement with
the experimentally derived b-quark spectrum (see Sec. II.1) in the region
around the maximum of the spectrum at 8 GeV for $\xi_{p}\gtrsim 1.8$, with a
mild disagreement below that value where errors are large and the relation eq.
(12) is only approximately valid (at large x). The predictions end at the
kinematic limit of the Limiting Spectrum $\xi_{0}=\xi_{max}^{lim}+\xi_{b}$ =
3.05 with $\xi_{max}^{lim}=\rm{ln}(W/(2Q_{0}))$.

The same procedure is followed with the c-quark fragmentation in the right
panel of Fig. 4. At the low energy of 2.6 GeV both parameters have to be
adjusted and are reported by BES [31] as $\Lambda=342$ MeV and $K_{ch}=1.52$.
In the narrow available region in $\xi_{p}$ the predictions at the energies
2.6 and 91.2 GeV are nearby but somewhat below the data.

Figure 5: Ratio of the heavy b-quark over the light uds-quark fragmentation
functions (left panel) and the corresponding ratio for c-quark (right panel)
together with the MLLA expectations based on comparison with experimental data
and with Limiting Spectrum distributions.

In Fig. 5 we show again the ratio of heavy quark over light quark
fragmentation functions in comparison with the MLLA prediction eq. (12) using
data input and Limiting Spectrum fits. These ratios are correctly reproduced
in the central region $1\lesssim\xi_{p}\lesssim 3$ for b-quarks and in
$1\lesssim\xi_{p}\lesssim 2$ for c-quarks. Below $\xi_{p}\sim 1$ ($x\gtrsim
0.4$) the Limiting Spectrum ratios are rising again, because of a mismatch in
the lower limit in $\xi_{p}$ for the limiting spectrum at 91.2 GeV and the
shifted one at the low energy $W_{0}$ (i.e. at 2.6 or 8 GeV), therefore we
excluded those results from the figure (see also next section). The amount of
suppression from the dead cone effect is correctly reproduced for both heavy
quark fragmentation processes.

The figures 4, 5 and 8 show the good overall description of the
$\xi_{p}$-distributions for both light and heavy quarks within the very
compact MLLA-LPHD and Limiting Spectrum ($Q_{0}=\Lambda$) approach in terms of
only two parameters, the QCD scale $\Lambda$ and the slowly moving
normalisation parameter $K_{ch}$. For the very low energy around 2 GeV also a
change of $\Lambda$ from 275 to 340 MeV is required. These small variations
reflect the relevance of higher order corrections beyond MLLA.

###  III.4 Behaviour of fragmentation functions near kinematic limits

In the figures 3 and 4, it is demonstrated that the MLLA expectation eq. (12)
quantitatively predicts the suppression of particle production in the central
region around the maximum of the $\xi_{p}$-distribution at the lower mass
scale $W_{0}$. For the b-quark, however, there is a major surplus of particles
at large $\xi_{p}$ beyond expectation and a smaller excess at small $\xi_{p}$.
To quantify these effects more clearly, we investigate the difference between
the heavy and light quark fragmentation functions at $W=91.2$ GeV which,
according to the MLLA expectation eq. (12), should just yield the expected
light quark $\xi_{p}$-distribution at the lower mass scale $W_{0}$. This will
clarify how the predicted $\xi_{p}$\- distribution at the low energy $W_{0}$
deviates from the observed one.

Figure 6: Expected $\xi_{p}$-Fragmentation Function (FF) at $W_{0}=8.0$ GeV
constructed according to MLLA eq. (12) as difference of
$\xi_{p}$-distributions for uds-quark and b-quark jets with DELPHI and OPAL
data as input; compared in absolute normalisation with interpolated Distorted
Gaussian (DG) at 8 GeV with shift $\xi_{b}=0.36$ and the same DG distribution
convoluted with the leading particle $x_{b}$-spectrum of the heavy b-quark
(left panel), the corresponding results for the expected
$\xi_{p}$-Fragmentation Function at $W_{0}=2.7$ GeV from uds-quark and c-quark
jets in comparison with BES data (right panel).

These differences are shown in Fig. 6 for the b-quark (left panel) and c-quark
fragmentation (right panel). The MLLA expected $\xi_{p}$-distributions are
compared with the experimental $\xi_{p}$-distribution: for the b-quark at 8
GeV with the distribution obtained by interpolation with corresponding
systematic errors (dashed line, see sec. III.2), for the c-quark with the
observed distribution at 2.6 GeV by the BES collaboration. The differences
between the MLLA expected and experimental $\xi_{p}$-distributions are shown
in the lower part of Fig. 6.

This figure clearly shows that the difference between heavy and light quark
fragmentation, in its main features, can just be related to the low energy
“hump backed plateau” which changes with the MLLA mass scale
$W_{0}=\sqrt{e}M_{Q}$. When the energy $W_{0}$ is increased from 2.6 to 8.0
GeV, the $\xi_{p}$-distribution shifts to a higher mean value $\bar{\xi}_{p}$
with larger width and increasing height in agreement with the behaviour known
from experiment in absolute terms. This result not only explains the limits of
their ratios in Fig. 5 for small and large $\xi_{p}$ with $R\sim 0$ and $R=1$,
but also the behaviour in between.

For the b-quark, the interpolated distribution at 8 GeV approaches quite
closely the data in the central region $1\lesssim\xi_{p}\lesssim 3$, but falls
somewhat below the expectations for the very small $\xi_{p}\lesssim 1$
($x_{p}\gtrsim 0.4$) and there is a considerable and very significant excess
over the expectation in the large $\xi_{p}\gtrsim 3$ region. For the c-quark
fragmentation there is a good agreement but errors become large for the larger
$\xi_{p}$.

Figure 7: Expected $\xi_{p}$-Fragmentation Function (FF) at $W_{0}=8.0$ GeV
constructed according to MLLA eq. (12) as difference of
$\xi_{p}$-Fragmentation Functions for uds-quarks and b-quarks; compared in
absolute normalisation with interpolated Limiting Spectrum at 8.0 GeV with
shift $\xi_{b}=0.36$, see eq. (13) and the same Limiting Spectrum convoluted
with the leading particle $x_{b}$-spectrum of the heavy b-quark (left panel);
the corresponding results for c-quarks with Limiting Spectrum calculations
using parameters by BES and $\xi_{c}=0.6$, see eq. (13) (right panel).

We also compare these experimental results with the MLLA Limiting Spectrum
distributions in Fig. 7. The agreement for b-quarks is rather satisfactory up
to the region close to the upper limit at $p\sim\Lambda$ i.e. at $\xi\sim 3$,
but there are deviations at the very small $\xi_{p}\lesssim 1$ ($x_{p}\gtrsim
0.4$). For c-quarks the agreement in the central region around $\xi_{p}\sim 2$
is satisfactory but there are considerable differences at small $\xi_{p}$.
These deviations at small $\xi_{p}$ are a consequence of the approximate form
of the eq. (12) in which the $\xi_{p}$-distribution is shifted to
$\xi\gtrsim\xi_{Q}$, i.e. to the values 0.36 and 0.70 for b- and c-quarks,
respectively. Therefore the approximate form of eq. (12) works best in the
central region around the maximum, away from limits $\xi_{p}=0$ and
$\xi_{p}=\xi_{\rm max}^{\rm lim}=\ln(W/(2\Lambda))$.

It appears that the width of the observed distribution in b-quark
fragmentation is larger than the expected one. One possible explanation could
be that the momentum fluctuations of the heavy quark are larger than
anticipated in the MLLA formula eq. (11) where a fixed energy loss $\langle
x_{Q}\rangle$ is assumed. More generally, one could consider the emitting
heavy quark with a distribution $F_{Q}(x_{Q})$ or

$\bar{F}_{Q}(\xi_{Q})=x_{Q}F_{Q}(x_{Q}),\ \ \xi_{Q}=\rm{ln}(1/x_{Q}).$ (14)

Then the distribution of the final partons is obtained from the convolution
integral

$\bar{D}_{q}(\xi_{p},W_{0})=\int_{0}^{\xi_{p}}\
d\xi_{p}^{\prime}\bar{F}_{Q}(\xi_{p}^{\prime})\bar{D}_{q}(\xi_{p}-\xi_{p}^{\prime},W_{0}),\
\ W_{0}=8\ \text{GeV},$ (15)

which may also fill the region $\xi_{p}<\xi_{Q}$. With
$F_{b}(x_{b})=\delta(x_{b}-\langle x_{b}\rangle)$, the eq. (11) is restored.
If one takes, as an exercise, for the momentum spectrum of the heavy quark
$F_{b}(x_{b})$, the distribution as experimentally measured (DELPHI [28]), the
second curve (short dashed) in Fig. 6 and Fig. 7 from the convolution is
obtained, which is broader than the experimental $\xi_{p}$-distribution for 8
GeV, as expected, but the effect is rather small and cannot explain the
observed deviations.

Finally, the MLLA prediction for b-quark fragmentation and the experimental
spectrum at 8 GeV are compared separately for the regions below and above
$\xi_{p}=3$. To this end, the difference between the expected and experimental
distributions at 8 GeV, see Fig. 6, is fitted to a polynomial function, and
the respective multiplicities $\Delta N=N(8\ \rm{GeV})^{MLLA}$ \- N(8 GeV)exp
are calculated from the integrals over the two $\xi_{p}$ regions for Distorted
Gaussian (Fig. 6) and the Limiting Spectrum distributions (Fig. 7). The
results are shown in Tab. 1.

Table 1: Integrals $\Delta N$ over the difference data shown in Fig. 6 and Fig. 7 (left panels) between expected and experimental $\xi_{p}$-distributions for b-quark fragmentation for different $\xi_{p}$ regions and two interpolating functions at 8 GeV. $\xi_{p}$-range |  Distorted Gaussian |  Limiting Spectrum
---|---|---
$\rm{all}\ \xi_{p}$ |  $\Delta N=1.52\pm 0.25$ |  $\Delta N=0.24\pm 0.15$
$\xi_{p}<3$ |  $\Delta N_{\rm low}=0.14\pm 0.16$ |  $\Delta N_{\rm low}=0.24\pm 0.15$
$\xi_{p}>3$ |  $\Delta N_{\rm high}=1.37\pm 0.16$ |  $-$

The first number $\Delta N$ in Tab. 1 is obtained by summing the data points
over the full $\xi_{p}$ range and it should agree with the results using the
published total multiplicity data instead. For DELPHI, one has N(8 GeV)MLLA =
$N_{uds}-N_{b}+n_{\rm dec}=(19.44\pm 0.34)-(23.17\pm 0.38)+(11.10\pm
0.18)=7.87\pm 0.54$ and with $N_{uds}(8\ \rm{GeV})^{exp}$ = $6.1\pm 0.3$, one
finds $\Delta$N =$1.8\pm 0.6$, which compares well with the value from our fit
$\Delta$N$=1.52\pm 0.25$.

As a main result for the b-quark fragmentation, it can be seen from Tab. 1
that the full multiplicity in the lower part of the $\xi_{p}$-spectrum (its
integral over $\xi_{p}<3$) agrees with the MLLA expectation for both
interpolating functions, i.e. $\Delta N=0$, but at large $\xi_{p}$ there is a
significant difference for which we have no direct explanation, but the effect
appears in the kinematic region $p\lesssim Q_{0}=\Lambda$ outside the validity
of the perturbative MLLA approach.

In this way, we also suggest a solution to the problem found in the previous
study of full multiplicities [4], where a moderate but significant discrepancy
between expected MLLA results and the experimental finding was noted, namely
$\delta_{b\ell}^{\rm MLLA}-\delta_{b\ell}^{\rm exp}=1.26\pm 0.42$ multiplicity
units, see eq. (8), corresponding to our result on the equivalent quantity
$\Delta N=1.52\pm 0.25$. The observed difference of these two numbers comes
from the slightly different determination of N(8 GeV). We now conclude that
the previously observed discrepancy in the MLLA multiplicity equation (4) can
be related to the contribution from the ultrasoft particles with $\xi_{p}>3$.
These are contributions from other sources outside the control of perturbation
theory for the heavy quark jets.

For the c-quark fragmentation the statistical errors in figures 3 and 6 are
too large to confirm or exclude such an ultrasoft anomaly. The result on
multiplicities $\delta_{c\ell}^{\rm MLLA}-\delta_{c\ell}^{\rm exp}=0.5\pm 0.6$
from Eq. (9) does not show any such effect either.

###  III.5 Sensitivity of experimental dead cone data to the heavy quark mass

In the last subsection we have compared the experimental data on the
difference between light and heavy quark fragmentation in figures 6 and 7 with
the expectation for the fragmentation function at the given low energy scale
$W_{0}=\sqrt{e}M_{Q}$ according to the MLLA estimate eq. (12). In turn, one
could treat the low energy scale $W_{0}$ in this equation as a free parameter
to be determined from the best fit of the $\xi_{p}$ spectrum in $e^{+}e^{-}$
annihilation to the experimental data in Fig. 6. We perform such a fit to the
data in the lower pad of Fig. 6 and determine a constant shift to the zero
line (corresponding to the $\xi_{p}$-distribution at $W_{0}=8.0$ GeV) in the
central region $1.6\lesssim\xi_{p}\lesssim 2.6$ avoiding contributions from
the excess multiplicity at large $\xi$. The maximum height of the inclusive
$\xi_{p}$ spectra $h_{\rm max}(W)$ as measured by BES [31] and TASSO [34]
collaborations is found by interpolation to rise by 0.2 units for an increase
of $W$ by 1 GeV near $W=8.0$ GeV. For the DELPHI and OPAL data we obtain the
shifts $\delta h_{\rm max}=-0.17\pm 0.13$ and $\delta h_{\rm max}=-0.14\pm
0.15$, respectively, or combined $\delta h_{\rm max}=-0.16\pm 0.10$ and
correspondingly for the low energy scale

$W_{0}^{\rm exp}=(7.2\pm 0.5)\ \rm{GeV}\ \rm{(DELPHI\ and\ OPAL)}.$ (16)

This is to be compared with the evaluation [4] $W_{0}^{\rm MLLA}=(8.0\pm 0.2)$
GeV, based on the b-quark pole-mass $M_{b}=4.85\pm 0.15$ GeV and the MLLA
correction $\sqrt{e}=1.65$ applied to the lowest order DLA result $W_{0}^{\rm
DLA}=M_{Q}$. It is remarkable, how close the MLLA prediction with its large
correction factor comes to the experimental data. If we would take the quark
mass $M_{b}$ itself as low energy scale we had to replace the Gaussian curve
for $W_{0}=8.0$ GeV in Fig. 6 by the curve at 4.8 GeV, displayed in Fig. 8,
which is clearly far away from the data in Fig. 6. The experimental
uncertainty of the low energy scale $W_{0}$ in (16) is about 7$\%$, but there
are also theoretical uncertainties from possible corrections beyond MLLA to
the correction factor for the energy scale $\sqrt{e}$ and from the
approximation of applying a constant shift in eq. (12). Therefore, although
the dead cone measurements at present cannot be competitive to the available
heavy quark mass determinations, they come close to the predicted value within
less than 10$\%$.

##  IV Conclusions

The dead cone effect predicted by perturbative QCD has been studied using data
taken at LEP on identified heavy b- and c-quark and light uds-quark
fragmentation. The dead cone effect for particle production at small angles to
the primary quark is also reflected in the production of large momenta, as is
typical for the jet structure. In the present study, QCD expectations for the
momentum spectra in heavy quark jets based on the MLLA [1, 2] are
investigated.

At first, we reconstruct the inclusive distributions of charged particle
momenta using the variable $\xi_{p}=\ln(1/x_{p})$ in b-quark and c-quark
events by correcting for B-hadron and Charm-hadron decays. In the comparison
of heavy and light quark fragmentation we observe a convergence of the spectra
for large $\xi_{p}$ ($x_{p}\to 0$) but a strong suppression of the
fragmentation functions of the heavy b- and c-quarks with respect to the one
of the light uds-quarks with decreasing $\xi_{p}\to 0$ (increasing $x\to 1$)
down to a fraction of $\lesssim 1/10$. This observed almost complete
suppression reflects the presence of the dead cone with a high significance
($\gg 5\sigma$). There is a characteristic difference between b- and c-quark
fragmentation, in that the decrease for the c-quark is shifted towards lower
$\xi_{p}$ as compared to the b-quark. It would be desirable to replace the
MCEG based subtraction of the charged heavy hadron decay products by an
experimental measurement in order to remove any residual model dependence.

The $\xi_{p}$-distributions derived from experimental data are then compared
directly with the QCD expectations within the MLLA following the hypothesis of
Local Parton Hadron Duality (LPHD). This QCD analysis provides a quantitative
explanation of the dead cone effect: the difference between the heavy and
light quark fragmentation functions in the variable $\xi_{p}$ at high c.m.s.
energy $W$ is just given by the $\xi_{p}$-fragmentation function at the lower
energy $W_{0}=\sqrt{e}M_{Q}$ with the heavy quark mass $M_{Q}$ (see eq. (11)).

The equation 11, an estimate within MLLA, is tested first with the
experimentally observed or derived $\xi_{p}$-distributions as input. For both
the b-quark and the c-quark fragmentation this equation is found to be well
supported in the central kinematic region around the peak of the
$\xi_{p}$-spectrum at scale $M_{Q}$ corresponding to the momentum range
$x\lesssim 0.4$ and $p\gtrsim\Lambda$. It explains quantitatively the
suppression of both fragmentation functions down to about $1/10$ of the one
for uds-quarks. The different suppression profiles of c- and b-quark
fragmentation are directly related to the different shapes of the $\xi_{p}$
spectra (the “hump-backed plateau”) at the respective c- and b-quark mass
scales $W_{0}$, i.e. at the c.m.s. energies 2.7 and 8.0 GeV respectively. In
the MLLA estimate eq. (11), the mean fractional momentum $\langle
x_{Q}\rangle$ of the heavy quark appears as additional (known) parameter. This
parameter is important for the successful quantitative description. The
interplay between heavy quark fragmentation and energy loss deserves further
attention.

The MLLA estimate eq. (11) has also been tested using as input the analytic
expressions for the $\xi$-distributions obtained within MLLA with the
simplification $Q_{0}=\Lambda$ for the $p_{T}$-cut off and the QCD scale, the
so called “Limiting Spectrum”. This formula describes to a reasonable
approximation the experimental $\xi_{p}$-spectra in the c.m.s. energy range
2-100 GeV for allowed momenta $p>Q_{0}=\Lambda$ in terms of only two
parameters, the QCD scale $\Lambda$ and the normalisation $K_{ch}$. These
parameters show a small variation with energy which hints towards
contributions beyond MLLA. In this way, a very compact representation of
momentum spectra with two parameters for light and heavy quark jets in a wide
kinematic region for not too large x ($x\lesssim 0.4$) and $p>\Lambda$ is
established.

In the kinematic region of small $\xi_{p}\lesssim 1$ ($x_{p}\gtrsim 0.4$), at
low values for the heavy quark fragmentation functions, violations of the MLLA
relation have been observed. This is to be expected in the present approximate
scheme using a shifted spectrum at the low energy $W_{0}$. Integrating the
fragmentation function of the b-quark over the important region
$\xi_{p}\lesssim 3$ ($p\gtrsim\Lambda$) yields the respective multiplicity
which is found in good agreement with the MLLA expectation. A large and
significant excess of particle production in b-quark fragmentation over these
expectations is observed for large $\xi_{p}\gtrsim 3$ which concerns the
region of very soft particle production with $p\lesssim\Lambda$. The total
excess multiplicity in this kinematic region corresponds quantitatively to the
excess over MLLA expectations already noted in the previous study of the full
multiplicity [4]. This discrepancy comes from a kinematic region outside the
validity of the perturbative approach. In case of the c-quark fragmentation no
such excess at large $\xi_{p}$ can be resolved within the larger errors and
there is a satisfactory agreement between prediction and experimental data for
not too large momenta ($\xi_{p}\gtrsim 1$).

We estimated a value for the low energy scale $W_{0}=(7.2\pm 0.5$) GeV of the
dead cone subtraction from the data which is consistent within the
uncertainties with the MLLA prediction $W_{0}=\sqrt{e}M_{b}=(8.0\pm 0.2)$ GeV.
By relating $W_{0}$ to the b-quark mass this shows the mass sensitivity of the
dead cone effect, which is, however, currently limited by uncertainties of the
MLLA predictions.

We would like to point out that our results could be of direct relevance to
the experimental task of identifying (tagging) jets originating from heavy
quarks. Traditional heavy quark jet tagging algorithms only use variables
derived from particles associated with the heavy hadron decay, see e.g. [35]
for a review of algorithms used at LEP. Recent developments using advanced
machine learning techniques (see e.g. [36, 37] and references therein) include
all objects associated with the jet and thus their improved performance
compared to traditional algorithms could be related at least partially to the
dead cone effect. This topic should be investigated further.

###### Acknowledgements.

We would like to thank Valery Khoze for his interest in this work and helpful
comments.

##  Appendix A Hadron $\xi_{p}$-distribution at $W=8$ GeV from interpolation

The $\xi_{p}$-distributions of hadrons at $W=8$ GeV are obtained by
interpolation between the neighbouring energies. Such data have been collected
by the BES [31] and TASSO [34] experiments at $W=2-5$ GeV and $14-44$ GeV,
respectively, and can be fitted by a Distorted Gaussian, acknowledged to be
well suited for QCD analysis [14, 16]:

$\bar{D}\left(\xi,W\right)=\frac{N}{\sigma\sqrt{2\pi}}\exp{\left[\frac{1}{8}k-\frac{1}{2}s\delta-\frac{1}{4}(2+k)\delta^{2}+\frac{1}{6}s\delta^{3}+\frac{1}{24}k\delta^{4}\right]},$
(17)

with $\delta=(\xi-\bar{\xi})/\sigma$ and the mean $\bar{\xi}$. The parameters
in eq. (17) are the mean multiplicity $N^{ch}$, mean value $\bar{\xi}$ or
maximum peak position $\xi_{max}=\bar{\xi}-\frac{1}{2}\sigma s$, width
$\sigma$, skewness $s$ and kurtosis $k$. These moments show only a smooth
dependence as function of $\ln W$ [16]. Therefore, the first five moments are
determined at $W=4.8$ and 14 GeV from a fit to the $\xi_{p}$ spectra, and
those at $W=8.0$ GeV by interpolation in $\ln W$, see Tab. 2.

Table 2: Moments of the Distorted Gaussian distribution (DG) for BES and TASSO data and interpolated DG at $W=8.0$ GeV. The errors for $N$, $\xi_{\rm max}$ and $\sigma$ are $1-4$%, $1-5$% and $3-10$%, respectively, while $s$ and $k$ are not well determined by the available data. Moments | $N$ | $\xi_{\rm max}$ | $\sigma$ | $s$ | $k$
---|---|---|---|---|---
BES (4.8 GeV) | $4.62$ | $1.81$ | $0.70$ | $0.72$ | $0.14$
Interp. DG (8 GeV) | $6.49$ | $2.05$ | $0.74$ | $0.37$ | $-0.20$
TASSO (14 GeV) | $8.73$ | $2.35$ | $0.80$ | $-0.05$ | $-0.60$
Figure 8: Distribution of charged hadrons in $\xi_{p}=\ln(1/x_{p})$ at c.m.s.
energies $W=4.8$ and 14 GeV together with Distorted Gaussian fits and the
interpolated spectrum at $W=8.0$ GeV with systematic errors (left panel); the
same data and the uds-quark data at $W=91.2$ GeV as compared to the MLLA
Limiting Spectrum and prediction for $W=8.0$ GeV using $\Lambda_{QCD}=275$
MeV, $K_{ch}=1.28$ at 91.2 GeV and $K_{ch}=1.52$ at lower energies (right
panel).

In Fig. 8, the results of the fits and the interpolated spectrum at $W=8.0$
GeV are displayed. An estimate of the errors is obtained by averaging the
relative errors of $\bar{D}(\xi,W)$ of the BES and TASSO data at their maxima
$\delta\bar{D}_{max}$ (6$\%$), and by increasing their values for smaller
$\bar{D}(\xi)$ as
$\delta\bar{D}(\xi)=\delta\bar{D}_{max}\sqrt{\bar{D}_{max}/\bar{D}(\xi)}$ so
as to account for statistical fluctuations.

At $W=8.0$ GeV, there is also some production of c-quarks, which contributes
to the multiplicity $N^{ch}=6.49\pm 0.19$ with $0.4\pm 0.2$ units according to
the estimate in [4], such that $N^{ch}_{uds}=6.1\pm 0.3$ is taken in the
following. In order to obtain the $\xi_{p}$-distribution for uds-quarks, the
distorted Gaussian fit obtained with parameters displayed in Tab. 2 has to be
corrected for charm particle decays. This small correction is approximated by
a global lowering of the $\xi_{p}$-distribution by 6$\%$ and a systematic
error of 3$\%$ is linearly added to the experimental error above.

In Fig. 8 (right panel), the uds-quark $\xi_{p}$-distribution data by DELPHI
[20] and OPAL [21] at $W=91.2$ GeV and the data by BES [31] at $W=4.8$ GeV and
TASSO [34] at $W=14$ GeV have been fitted with the Limiting Spectrum with the
same $\Lambda=275$ MeV, the normalisation was changed from $K_{ch}=1.28$ at
$W=91.2$ GeV to $K_{ch}=1.52$ at the lower energies. There is a good overall
description of the data by the “hump backed plateau” distribution rising and
broadening between 4.8 and 91.2 GeV with deviations of up to 10$\%$ from the
data, except for the high end of the distribution, where there is a kinematic
limit for the massless partons at $p=Q_{0}=275$ MeV. With these parameters,
the Limiting Spectrum at $W=8.0$ GeV, shown as dashed curve in Fig. 8 (right
panel), can be found. The maximum of this curve is consistent with the value
$dn/d\xi|_{max}(8\ \rm{GeV})=3.5\pm 0.1$ found by interpolating, as before,
the corresponding maxima in the BES [31] and the TASSO [34] data.

The uds-quark $\xi_{p}$-distribution at $W=8.0$ GeV is obtained again after
charm decays are corrected for by rescaling the $\xi_{p}$ spectrum by 6$\%$
using $K_{ch}=1.43$.

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