Unconventional mechanisms of heavy quark fragmentation

Heavy and light quarks produced in high-$p_T$ partonic collisions radiate differently. Heavy quarks regenerate their color field, stripped-off in the hard reaction, much faster than the light ones and radiate a significantly smaller fraction of the initial quark energy. This peculiar feature of heavy-quark jets leads to a specific shape of the fragmentation functions observed in $e^+e^-$ annihilation. Differently from light flavors, the heavy quark fragmentation function strongly peaks at large fractional momentum $z$, i.e. the produced heavy-light mesons, $B$ or $D$, carry the main fraction of the jet momentum. This is a clear evidence of the dead-cone effect, and of a short production time of a heavy-light mesons. Contrary to propagation of a small $q\bar q$ dipole, which survives in the medium due to color transparency, a heavy-light $Q\bar q$ dipole promptly expands to a large size. Such a big dipole has no chance to remain intact in a dense medium produced in relativistic heavy ion collisions. On the other hand, a breakup of such a dipole does not affect much the production rate of $Q\bar q$ mesons, differently from the case of light $q\bar q$ meson production.


I. INTRODUCTION
High-p T parton scattering leads to formation of four cones of gluon radiation: (i)-(ii) backward-forward jets formed by the color field of the colliding partons shaken off in in the hard collision; (iii)-(iv) the scattered partons carry no field up to transverse momenta k T < p T .These partons are regenerating the lost color field via gluon radiation forming the up-down jets, as is illustrated in Fig. 1 The radiation process is ordered in time or path length according to [1], Here x is the fractional light-cone momentum of the radiated gluon; k T is its transverse momentum relative to the initial quark direction.The radiated gluons subsequently hadronize forming a jet of hadrons.For heavy quarks the second term in the denominator play important role leading to the so called dead-cone effect [2].
In terms of the Fock state representation all radiated gluons pre-exist in the initial bare parton, and are liberated on mass shell successively in accordance with their coherence length/time Eq. (1.1).First are radiated gluons with small longitudinal and large transverse momenta.

A. Radiational energy loss in vacuum
How much energy is radiated over the path length L? Only gluons with radiation length l c < L contribute [3], where ω is the gluon energy; the soft cut-off parameter λ = 0.2 GeV.The perturbative radiation spectrum reads, ( We see that radiation by light and heavy quarks behave quite differently at small k T : (i) Light quarks: [2,3].Heavy quarks radiate less energy compared with the light ones.They promptly restore their color field and stop radiating.The amount of radiated energy for light and heavy flavors is depicted in Fig. 2 vs radiation length for different jet energies.We see that heavy quarks radiate only a small fraction 10-20% of their initial momentum.In particular, this explains the unusual shape of the experimentally observed fragmentation function D b/B (z) of b-quarks, presented in Fig. 3 [4] (and similar for charm [5]).Indeed, most of B-mesons carry a large fraction z ∼ 80%, of the b-quark momentum.
We conclude that such a specific shape of the fragmentation function of heavy quarks is a direct manifestation of the dead-cone effect.

II. PRODUCTION LENGTH
The process of gluon radiation by a heavy quark Q ends up with color neutralization by a light antiquark and production of a Qq dipole.As far as we are able to calculate the radiated fraction of the light-cone momentum (e.g. for b-quark) ∆p b + (L)/p b + , the production length L p distribution W (L p ) can be extracted directly from data on D b/B (z), The results for the differential distribution dW/dL p are depicted in Fig. 4 at several values of momenta p T .Remarkably, the mean value of L p is extremely short and shrinks with rising p T .This sounds counter-intuitive, however, the process has maximal hard scale allowed by the kinematics p T = E c.m. /2.The production length L p turns out to be much shorter than the confinement radius, indicating that the fragmentation mechanism is pure perturbative.At L = L p , a small-size dipole bq is produced, with no certain mass, but with a certain radius.It is to be projected on the B-meson wave function, giving Ψ B (0) (compare with [6]).

III. FRAGMENTATION IN A DENSE MEDIUM A. Formation length of a Qq meson
The light antiquark in the B-meson carries a tiny fraction of its momentum, x ≈ m q /m Q , i.e. about 5%.The produced bq dipole has a small transverse separation, but it expands with a high speed, enhanced by 1/x, i.e. is an order of magnitude faster than symmetric qq or QQ dipoles.
where ⟨r 2 T ⟩ = 8/3⟨r 2 ch ⟩, and ⟨r 2 ch ⟩ B = 0.378 fm 2 as was evaluated in the potential model [7].The B meson is nearly as big as the pion, since its radius is controlled by the mass of the light antiquark.
According to (3.1) the dipole heavy-light Qq dipole separation promptly reaches the large hadronic size.This is confirmed by comparison data, for J/ψ detected in P b − P b nuclear collisions.Data demonstrate a color opacity for B-mesons (prompt production) and color transparency effect for J/ψ decaying to B (non-prompt production).The nuclear suppression factors R AA for these two channels are compared in Fig. 5 [8].While Eq. (3.1) describes the early, perturbative stage of the dipole expansion, the further evolution filters out the states with large relative phase shifts.The longest time takes discrimination between the two lightest hadrons, the ground state B and the first radial excitation B ′ , which concludes the formation process.Correspondingly, the full formation path length can be evaluated as, E.g. for the oscillatory potential m B ′ − m B = 0.6 GeV, so l f = 0.06 fm[p T /1 GeV] is extremely short for medium-large transverse momenta.

B. Attenuation of dipoles propagating in a dense medium
The mean free path of a Q − q meson in a hot medium characterizing by the transport coefficient (the rate of broadening) q, E.g. at q = 1 GeV 2 / fm λ B = 0.04 fm, so a formed B-meson breaks up in the medium nearly instantaneously.A b-quark propagating through the hot medium, easily picks up and loses accompanying light antiquarks without an essential reduction of its momentum.Meanwhile the b-quark keeps dissipating its energy with a rate, slightly enhanced by medium induced radiative energy loss [9] effects.Eventually the detected B-meson is produced in the dilute periphery of the medium.
The heavy quark keeps losing energy even inside a colorless Qq dipole sharing its momentum with the light quark, as is illustrated in Fig. 6 presenting a unitarity cut of a qq Reggeon, Thus, the heavy quark Q dissipates a part of its 6: Redistribution of energy inside the Qq dipole.The gluons radiated by Q are absorbed by q so the dipole energy remains unchanged.
energy on a long path from the hard collision point to the medium periphery.

C. Medium modified production rate
The cross section of a heavy-light meson M production in pp collisions can be presented in the factorized form, We replaced the b → B fragmentation function by the differential expression (2.1).The medium-modified L p distribution is given by, Here, for the sake of simplicity, we fixed the Qq dipole separation at the mean value.This approximation is rather accurate due to shortness of l f .Otherwise, one can calculate the attenuation factor in (3.6) exactly, applying the path integral technique [11,12].Eventually, the production rate of heavy-light mesons in AA collisions with impact parameter ⃗ s reads, The effective production length Lp in the medium turns out to be much longer than in vacuum, because the heavy-light meson is produced mainly at the medium periphery, long distance from the hard collision point.

D. Data analysis
Now we are in a position to calculate the nuclear ratio to be compared with data.Here and T A (s) is the nuclear thickness function.
The model cannot fully predict (as well as any other model) the nuclear ratio, because the medium density is not known, but is rather the goal of the research.We embedded this information into the broadening rate (transport coefficient) following the popular model [13] q(l, ⃗ s, ⃗ τ , ϕ) = q0 t 0 t where n part (⃗ s, ⃗ τ ) is the number of participants at transverse coordinates ⃗ s and ⃗ τ relative to the centers of the colliding nuclei.The falling time dependence, 1/t is due to longitudinal expansion of the produced medium.The time interval t 0 required for equilibrated medium production.We fixed it at the frequently used value t 0 = 1 fm.The only fitted parameter is q0 , which is the maximal value of the brodening rate (transport coefficient) at s = τ = 0 and t = t 0 .In fact, measurement of this parameter is our goal.Comparison with ATLAS [8] and CMS data [14] for B-meson production (non-prompt J/ψ) in lead-lead collisions at √ s = 5.02 TeV is presented in Fig. 7.We see FIG. 7: Nuclear ratio AA/pp for B-meson production in lead-lead collisions as function of pT (Left) and versus centrality (Right).Data are from ATLAS [8] and CMS [14].
that data are described pretty well, either for p T , or N part dependences.The adjusted parameter q0 ranges within q0 = 0.2 − 0.25 GeV 2 / fm.This magnitude is considerably smaller compared with the values usually measured for light quarks.See discussion below.We successfully described data on D-meson production as well, as is demonstrated in Fig. 8. Notice that c-quarks FIG.8: The same as in Fig. 7, but for production of D-mesons in experiments ALICE [15] and CMS [16].
radiate in vacuum more energy than b-quarks, while the effects of absorption of cq and bq dipoles in the medium are similar.Therefore D-mesons are suppressed in AA collisions more than B-mesons.R AA (p T ) for D-mesons steeply rises with p T due to color transparency.Since bq dipoles expand much faster than cq, no color transparency effects are seen in R AA (p T ) for B-mesons, as was demonstrated in the right pane of Fig. 5.
Interesting that the found broadening rate parameter for c-quarks q0 = 0.45 − 0.55 GeV 2 / fm, significantly exceeds the value q0 = 0.2 − 0.25 GeV 2 / fm we found for b-quarks, while is quite less than q0 ≈ 2 GeV 2 / fm for light quarks (see below).Such a hierarchy of broadening rates for different quark flavors might look puzzling, if q were a real transport coefficient in terms of statistical medium properties.It coincides with the rate of broadening [17] only within the Born approximation, i.e. single gluon exchange for an inelastic process.In reality, broadening is subject to strong higher-order corrections and usually considerably exceeds the Born approximation estimate.The rate of broadening reads [18,19], where g(x, µ 2 ) is the gluon density; ρ 2 is the medium density per unit of length.The characteristic scale of the process µ is related to the mean transverse momentum of the radiated gluons.For light quarks it is given by the non-perturbative effective gluon mass, m g ∼ 0.7 GeV [12,20].For heavy quarks gluon radiation is subject to the dead-cone effect and the scale is much larger µ 2 ≈ m 2 Q .This is why the rate of broadening for heavy quarks is significantly reduced.This is another manifestation of the dead-cone effect.
The left plot in Fig. 8 shows a considerable disagreement with data at small transverse momenta p T ≲ 10 GeV.While the measured R AA (p T ) is steeply falling with p T , our calculations predict a nearly constant value.Such kind of disagreement has been observed earlier for light quarks, as is displayed in Fig. 9 Apparently a bump at small p T is presented in R AA (p T ) for D-mesons as well, while our calculations in Fig. 8 disregard the hydrodynamic component.

IV. SUMMARY
Our observations and results can be summarized as follows.
• Heavy and light quarks originated from hard collisions radiate differently.The former is subject to the deadcone effect, suppressing radiation of low-k T gluons.Consequently heavy quarks regenerate their color field much faster than light ones and radiate a significantly smaller fraction of the initial energy.The heavier is a quark, the less it radiates.
• The fragmentation function usually depends on two variables D M/q (z, Q 2 ), fractional light-cone momentum of produced meson, and the scale Q 2 .However, we consider here the case of "maximal" scale, when the jet energy FIG.9: Suppression factor RAA(pT ) for lead-lead collisions at √ s = 2.76 TeV vs centrality.The dashed and dotted lines are calculated within the pQCD [21] and hydrodynamic [22] mechanisms, respectively.The solid lines represent both mechanisms summed up.Data for RAA are from the ALICE [23] and CMS [24,25] experiments.and the hard scale coincide.This happens e.g. in e + e − annihilation, or high-p T jet production at Feynman x F = 0.
• The dead-cone effect suppressing bremsstrahlung of heavy quarks, explains the unusual shape of the fragmentation function of heavy quarks D M/Q (z), observed at LEP and SLAC.It peaks at large fractional momentum z, i.e. the produced heavy-light mesons, B or D, carry the main fraction of the jet momentum.On the contrary, the fragmentation function of light quarks is falling steadily with z towards z = 1.
• Differently from propagation of a small q − q dipole, which survives in the medium due to color transparency, a Q − q dipole promptly expands to a large transverse size, controlled by the small mass of the light quark.Such a big dipole has no chance to remain intact in a hot medium.On the other hand, a breakup of such a dipole hardly affects the production rate of Q − q mesons.
• We successfully described data on p T and centrality dependence of the production rate of B and D mesons in heavy ion collisions.The only unavoidable parameter of such analyses is the broadening rate (usually called transport coefficient) of the quark in the medium.Its maximal value q0 was found 0.2 − 0.25 GeV 2 /f m, 0.4 − 0.45 GeV 2 /f m and 2 GeV 2 /f m for b, c and light quarks respectively.Such hierarchy of the broadening rates is related to the same dead-cone effect.Suppression of bremsstrahlung leads to a considerable reduction of broadening.

FIG. 1 :
FIG. 1: High-pT collision in the c.m. frame of two partons, which leads to production of four jets: (i)-(ii) soft color field shaken off in the collision; (iii)-(iv) transverse cones of gluons radiated due to regeneration of the stripped off color field.

2 FIG. 2 :FIG. 3 :
FIG. 2: Radiational energy loss in vacuum by light (u,d), c and b quarks, depicted by blue, red and green curves respectively.Radiated energy ∆E is plotted as function of path length for different jet energies.

FIG. 4 :
FIG. 4:The Lp-distribution of B-mesons produced with different pT in pp collisions.