Quenching of high-pT hadrons: a non-energy-loss scenario

A parton produced with a high transverse momentum in a hard collision is regenerating its color field, intensively radiating gluons and losing energy. This process cannot last long, if it ends up with production of a leading hadron carrying the main fraction z_h of the initial parton momentum. So energy conservation imposes severe constraints on the length scale of production of a single hadron with high pT. As a result, the main reason for hadron quenching observed in heavy ion collisions, is not energy loss, but attenuation of the produced colorless dipole in the created dense medium. The latter mechanism, calculated with the path-integral method, explains well the observed suppression of light hadrons and the elliptic flow in a wide range of energies, from the lowest energy of RHIC up to LHC, and in a wide range of transverse momenta. The values of the transport coefficient extracted from data range within 1-2 GeV^2/fm, dependent on energy, and agree well with the theoretical expectations.


Intoduction
Single hadrons produced with high transverse momenta in heavy ion collisions at high energies of RHIC [1,2] and LHC [3][4][5] turn out to be appreciably suppressed compared with pp collisions. While there is a consensus about the source of the suppression, which is final-state interaction with the co-moving medium, created in the collision, the mechanisms of the interaction is still under debate.
The popular interpretation of the observed suppression of high-p T hadrons is the loss of energy by a parton propagating through the medium created in the collision. The perturbative radiative energy loss is caused by the "wiggling" of the parton trajectory due to multiple interactions in the medium. Every time, when the parton gets a kick from a scattering in the medium, a new portion of its color field is shaken off. The loss of energy induced by multiple interactions is naturally related to the broadening of the parton transverse (relative to its trajectory, i.e. to p T ) momentum k T [6], a e-mail: boris.kopeliovich@usm.cl EPJ Web of Conferences whereq(l) is the rate of broadening ∆k 2 T , which may vary with l along the parton trajectory, Although it is natural to expect that dissipation of energy by the parton in a medium should suppress production of leading hadrons, realization of this idea in detail raises many questions. In particular, one usually assumes that energy loss results in a shift in the argument of the fragmentation function, z h ⇒ z H + ∆z h . This could be true, if hadronization of the parton started outside the medium. However it starts right away after the hard collision, and the main part of gluon radiation occurs on a short distance (see section 2).
Another assumption, which the energy loss scenario relies upon and which has never been justified, is that the path length of the parton in the medium is always longer than the span of the medium. So the colorless hadronic state, which does not radiate energy any more and is eventually detected, is produced outside the medium. The validity of this assumption should be investigated and the path length available for hadronization should be evaluated. This problem was debated in [7] on a more certain situation of semi-inclusive deep-inelastic scattering, as well as within dynamical models of hadronization [8,9] providing solid constraints on the above assumption.
Besides, theoretical arguments, some experimental data is also difficult to explain within the energy loss scenario. In particular, production of hadrons in semi-inclusive deeply inelastic scattering (SIDIS) offer a rigorous test for in-medium hadronization model in much more certain environment than in heavy ion collisions. Indeed in SIDIS off nuclei the medium density and geometry are well known and time independent; the fractional momentum z h (the argument of the fragmentation function) of the detected hadron is measured. Measurements performed in the HERMES experiment [10,11] well confirmed the predictions [12] made five years prior the measurements, within a model, which include evaluation of the hadronization length, which was found rather short. The comparison with data shown in Fig. 1 [7] demonstrate a good agreement.  [7,12] z hdependence of the nuclear suppression factor in inclusive electroproduction of pions with HERMES data [10,11]. Solid and dashed curves show the results with included or neglected energy loss corrections, respectively. On the other hand the attempts to explain the same results of HERMES within the energy loss scenario were not successful. An example of comparison the model [13] with data [11] depicted in Fig. 2 show that the model fails to explain the data at large z h > 0.5, which dominates high-p T hadron production in heavy ion collision (see Fig. 5). Even adjustment of the transport coefficientq (actually well known for the cold nuclear matter [14,15]) did not help.
Thus, the assumption of a long hadronization length should be checked thoroughly. The main space-time scales of the in-medium hadronization process are indicated schematically in Fig. 3. We call production length l p the distance at which color neutralization occurs and a colorless "pre-hadron" (a colorless state, which does not have any certain mass) is produced and starts developing a wave function. The latter process is characterized by the formation length, which usually is rather long, [7,17]. In what follows we concentrate on the production length scale l p , which is the only path available for energy loss.
Notice that the question whether the hadronization ends up within or without the medium, might have no certain answer. This is a typical quantum-mechanical uncertainty: the production amplitudes with different values of l p interfere. Such interference was evaluated in [18] for an example of a SIDIS process and found to be a considerable effect. However, an extension of those results to a hot medium is a challenge, so what follows we neglect the interference and employ the standard semi-classical space-time pattern of the production process.

Radiative energy loss in vacuum
First of all, one should discriminate between vacuum and medium-induced radiative energy loss. High-p T partons radiate gluons and dissipate energy even in vacuum, and the corresponding rate of energy loss may considerably exceed the medium-induced value Eq. (1), because the former is caused by a hard collision.

Regeneration of the color field of a high-p T parton
High-p T scattering of partons leads to an intensive gluon radiation in forward-backward directions, which is related to the initial color field of the partons, shaken off due to the strong acceleration caused by the hard collision. The Weitzäcker-Williams gluons accompanying the colliding partons do not survive the hard interaction and lose coherence up to transverse frequencies k ∼ < p T . Therefore, the produced high-p T parton is lacking this part of the field and starts regenerating it via radiation of a new cone of gluons, which are aligned along the new direction. One can explicitly see the two cones of radiation in the Born approximation calculated in [19]. This process lasts a long time proportional to the jet energy (E ≈ p T ), since the coherence length (or time) l g of gluon radiation depends on the gluon fractional light-cone momentum x and its transverse momentum k relative to the jet axis as (to be concrete we assume that the jet is initiated by a quark), Here M qg is the invariant mass of the recoil quark and radiated gluon. One can trace how much energy is lost over the path length L via gluons which have lost coherence, i.e. were radiated, during this time interval, where Q ∼ p T is the initial quark virtuality; the infra-red cutoff is fixed at Λ = 0.2 GeV. The radiation spectrum reads A few examples of fractional vacuum energy loss by a quark vs distance from the hard collision is depicted in Fig. 4 for initial energies 10, 20, 100, 200 GeV (compare with heavy flavors in [20]). The rate of energy dissipation is considerable and energy conservation may become an issue for a long path length, if one wants to produce a leading hadron. Indeed, the production rate of high-p T hadrons comes from a convolution of the the parton distributions in the colliding hadrons (which suppresses large fractional momenta x, i.e. high p T ), with the transverse momentum distribution in the hard parton collisions (also suppresses large p T ), and with the fragmentation function D(z h ) of the produced parton. The latter has maximum at small z h ≪ 1, which, however, is strongly suppressed by the convolution, pushing the maximum towards large values of z h . Numerical results of the convolution for the mean value z h [21,22] are depicted in Fig. 5, separately for quark and gluon jets (upper and bottom solid curves) and at different energies, √ s = 200, 2760 and 7000 GeV. We see that the lower the collision energy is, the larger is z h , especially at high p T , because the parton k T distribution gets steeper. In the energy range of the LHC the magnitude of z h practically saturates as function of √ s and p T , and becomes indistinguishable for quark and gluonic jets.
It is worth emphasizing the difference between inclusive production of a high-p T hadron and a high-p T jet. In the former case, according to the above consideration, the detected hadron carries the main fractional light-cone momentum z h of the jet, which it originates from. In the latter case, if only the whole jet transverse momentum p jet T is required to be large, and no other constraints are imposed, the fractional momenta of hadrons in the jet are typically very low, so energy conservation energy conservation does not imply any severe constraints on the hadronization time scale, which rises with p T and may be long.

How long does it take to produce a hadron?
Apparently, production of a hadron with fractional momentum z h becomes impossible after the parton initiating the jet, radiates a substantial fraction of its initial energy, ∆E/E > 1 − z h . Thus, energy conservation imposes an upper bound for the production length l p . Figures 4 and 5 show that such a maximal value of l p is rather short. Remarkably, its value is nearly independent of p T . This might seem to be in contradiction with the Lorentz factor, which should lead to l p ∝ p T . However, the intensity of gluon radiation and the rate of energy loss also rise, approximately as p 2 T , what leads to an opposite effect of l p reduction.
More precisely the p T dependence of l p can be derived within a dynamic model for hadronization. It was done in Ref. [9,21] employing the model of perturbative hadronization [8]. Some numerical results are plotted in Fig. 6 for fragmentation of quarks and gluons by solid and dashed curves respectively. We see that the mean production length is rather short and slowly decreases with p T . The production length for gluon jets is shorter, because of a more intensive vacuum energy loss and a stronger Sudakov suppression, which leads to a reduction of l p .
The production length in Fig. 6 demonstrates a trend to decrease with p T , which is in variance with the naive expectation of a rise due to the Lorentz factor. As was explained above, this happens due to the growing virtuality and radiative dissipation of energy.
Notice that we took into consideration so far only the dissipation of energy in vacuum. Apparently, adding the medium induced energy loss one can only enhance the energy deficit and make the production length even shorter.

Propagation of color dipoles in a dense medium
As was discussed above, the intensive gluon radiation following the hard parton interaction, must stop in a short while (see Fig. 6) in order to be able to fragment into a hadron with large z h (see Fig. 5) and to obey the energy conservation constraints. The gluon radiation can stop only by means of color neutralization and production of a colorless dipole. In vacuum (e.g. in pp collision) the matrix element for a hadron production contains a direct projection of the initial distribution amplitude of the dipole separation to the hadron wave function, where Ψ h and Ψ in are the light-cone wave function of the hadron and the distribution amplitude of the produced dipole, respectively. We use the mixed representation of transverse dipole separation r and fractional light-cone momentum α carried by the quark.
In the case of production inside a medium, the produced dipole has to survive the propagation through the medium, i.e. to experience no inelastic collisions. The inelastic cross section depends on the dipole separation, which is fluctuating during the propagation. The effective way to solve this problem on a strict quantum-mechanical ground is the path integral method [17,22]. In the case of in-medium production the matrix element Eq. (6) is modified to, Here Gq q (l 1 , r 1 ; l 2 , r 2 ) is the Green function describing propagation of the dipole between longitudinal coordinates l 1 and l 2 with initial and final separations r 1 and r 2 respectively. It satisfies the twodimensional Schrödinger equation [23][24][25][26], The first term in square brackets plays role of kinetic energy in Schrödinger equation, while the imaginary part of the light-cone potential Vq q (l 2 , r 2 ) is responsible for absorption in the medium. The relation between the rate of broadening and the dipole cross section derived in [27], allows to present the imaginary part of the potential as [21,28], Thus, color transparency [29] controls dipole attenuation in a medium. The real part of the potential describes the nonperturbative interaction between q andq in the dipole [24,30]. However, it should not affect much the dipole evolution on the initial perturbative stage of development. Therefore we will treat theqq as free noninteracting partons.
Measurements of the suppression of high-p T hadrons in heavy ion collisions can provide precious information about the properties of the created hot matter. One can quantify those properties by the value of the transport coefficientq, which is proportional to the medium density. The latter is time dependent, and is assumed to dilute as ρ(t) = ρ 0 t 0 /t due to the longitudinal expansion of the produced medium. We adopt the popular parametrization [31] of the transport coefficient, which depends on impact parameter and path length (time) as, where n part ( b, τ) is the number of participants;q 0 corresponds to the maximum medium density produced at impact parameter τ = 0 in central collisions (b = 0) at the time t = t 0 = l 0 after the collision.
In what follows we treat the transport coefficientq 0 as an adjusted parameter.
International Conference on New Frontiers in Physics 2013

Results at the energies of LHC
Now we are in a position to calculate the suppression factor R AB (b, p T ) for high-p T hadrons produced in nuclear A-B collision with impact parameter b. The suppression orrurs due to the difference between the matrix elements (6) and (7), so we get, where ; φ is the azimuthal angle between the dipole trajectory and reaction plane (impact vector b).
We also included medium-induced radiative energy loss during the short path from l = l 0 ∼ 1 fm to l = l p (if l p > l 0 ), where the parton experiences multiple interactions, which induce extra radiation of gluons and additional loss of energy [6], Although this is a small correction, it is included in the calculations by making a proper shift of the variable z h in the fragmentation function.
The results of calculations [22] for charge hadron suppression factor R AA (p T ) are plotted in Fig. 7 in comparison with data from the ALICE [3] and CMS [4,5] experiments at √ s = 2.76 TeV and different centralities indicated in the plot. The dashed lines are calculated with the path-integral expression, Eq. (11), calculated with the space-and time dependent transport coefficient Eq. (10), where the parameterq 0 = 2 GeV 2 /fm, which controls the normalization of R AA , was adjusted to data. Notice that this parameter is independent of p T and centrality. It affects the normalization, but not the shape of the p T dependence. The solid curves also include the effects of initial state interactions (ISI) and energy conservation in nuclear collisions [32,33], as is described below in Sect. 5.
As far as hadrons propagated over a longer path are suppressed more, naturally the azimuthal angle distribution of the produced hadrons correlates with the geometry of the collisions and gains an asymmetry. Data for such an asymmetry, characterized by the parameter v 2 = cos(2φ) , provide an alternative sensitive way to test the model for suppression. The asymmetry can be calculated in a way similar to Eq. (11), Here we neglected the initial dipole size r 1 ∼ 1/p T ≈ 0 at l 1 = 0. The results of calculations [22] are compared in Fig. 8 with data from the ALICE [34] and CMS [35] experiments at √ s = 2.76 TeV and different centralities. The calculated asymmetry in the perturbative regime is rather small and is falling monotonically with p T in agreement with data.  (11). The solid curves also include the effects of initial state interactions [32,33], as is described in Sect. 5.
Notice that an azimuthal asymmetry appears for any mechanism of suppression. One may wonder if a successful description of the cross section (Fig. 7) should automatically lead to a good agreement with v 2 (p T ), i.e. whether data for v 2 suggest a complementary test of the model? The answer is positive, v 2 data provide a more stringent examination of the model. Indeed, while the suppression of the cross section depends on the accumulated suppression along the hadron path in the medium, azimuthal asymmetry is sensitive to the path length distribution.

Large x T , towards the kinematic bound
Multiple interactions of the projectile hadron and its debris propagating through the nucleus lead to a dissipation of energy. Important observation made in [32] (see also [36]) is that the resulting loss of energy is proportional to the energy of the projectile hadron, therefore the related effects do not disappear at very high energies. An easy and intuitive way to understand this is the Fock state representation for the projectile hadron wave function. Those sates are the fluctuations of this hadron "frozen" by Lorentz time dilation. The interaction with the target modifies the weights of the Fock states, some interact stronger, some weaker. An example is the light-cone wave function of a transversely polarized photon [37]. In vacuum it is overwhelmed byqq Fock states with vanishingly small separation (this is why the normalization of the wave function is ultraviolet divergent). However, those small size fluctuations have a vanishingly small interaction cross section, and the photoabsorption cross section turns out to be finite. In each Fock component the hadron momentum is shared by the constituents, and the momentum distribution depends on their multiplicity: the more constituents the Fock state contains, the smaller is the mean fractional momentum per a constituent parton. Higher Fock components interact with a nuclear target stronger and gain larger weight factors compared to low Fock states. Thus, the x distribution in the projectile hadron is softer on a nuclear than on a proton targets.
In the case of a hard reaction on a nucleus, such softening of the momentum distribution of a projectile parton can be viewed as an effective loss of energy of the leading parton due to initial state multiple interactions: the mean energy of the leading parton on a nuclear target decreases compared the hard reaction on a proton target. Such a reduction of the fractional momentum of the leading parton is apparently independent of the initial hadron energy. Thus, the effective loss of energy is proportional to the initial energy.
Notice that this is different from energy loss by a single parton propagating through a medium and experiencing induced gluon radiation. In this case the mean fractional energy carried the radiated gluons vanishes at large initial energies E as ∆E/E ∝ 1/E [6,38,39]. Initial state energy loss is a minor effect at high energies and mid rapidities. However, it may essentially suppress the cross section upon approaching the kinematic bound, either Correspondingly, the proper variable, which controls this effect is ξ = The magnitude of suppression was evaluated in [32,33]. It was found that each of multiple interactions (treated within the Glauber approximation) in the nucleus supplies a suppression factor U(ξ) ≈ 1 − ξ. Summing up over the multiple ISI interactions in pA collision with impact parameter b one arrives at a new parton distribution function in the projectile proton compared with pp collisions, Here σ e f f is the effective hadronic cross section controlling multiple interactions. It is reduced by Gribov inelastic shadowing, which makes the nuclear medium more transparent. The effective cross section was evaluated in [32,40] at about σ e f f ≈ 20 mb. The normalization factor C in Eq. (14) is fixed by the Gottfried sum rule.
With the parton distribution functions Eq. (14) modified by ISI energy loss one achieves a good parameter-free description of available data from the BRAHMAS [41] and STAR [42] experiments at forward rapidities in dA collisions large x L [32,33].
The ISI energy-loss should also be important at large p T , in particular in the RHIC energy range, where x T in data reaches values of 0.2 − 0.3. Notice that the real values of x T , essential for energy conservation, are about twice larger,x T = x T /z h , so it reaches values of 0.5 at RHIC, and about 0.4 at LHC. The measured nuclear modification of pions produced at high p T in dAu collisions at √ s = 200 GeV [43] indeed demonstrates a significant suppression, as one can see in Fig. 9. With the same modification factor Eq. (14), which was successful at forward rapidities, a good agreement is achieved at large p T as well, as is demonstrated in Fig. 9.   [44] (triangles) and [45] (squares).
The effects of ISI energy loss also affect the p T dependence of the nuclear suppression in heavy ion collisions. These effects are calculated similarly, using the modified parton distribution functions Eq. (14) for nucleons in both colliding nuclei. The resulting additional suppression significantly reduces R AA (p T ) at the energy of RHIC. This is demonstrated in Fig. 10 on the example of central gold-gold collisions at √ s = 200 GeV. Since parameter q 0 is expected to vary with energy, it was readjusted and found to be q 0 (RHIC) = 1.6 GeV 2 /fm, less than in collisions at the LHC 1 . Data on centrality dependence of R AA (p T ), presented in Fig. 10, is also well explained.
One can access larger values of x T either by increasing p T and a fixed energy, or by reducing the collision energy keeping unchanged the p T range. So the data [46] at lower energies √ s = 62 GeV depicted in Fig. 11 play important role for study of the energy loss effects. Indeed, the data demonstrate an unusual falling p T dependence of R AA (p T ) predicted with the suppression factor Eq. (14). 1 Notice that q 0 is also A-dependent.
International Conference on New Frontiers in Physics 2013 On the contrary, at much higher energies of LHC one would not expect any sizable effects of energy loss. Nevertheless, even at such high energies one can reach x T sufficiently large for ISI energy loss effects to show up. The results depicted by solid curves in Fig. 7 include such energy loss corrections, which cause leveling off and even fall of R AA (p T ) at p T 100 GeV.

Summary
Summarizing, for the process of inclusive high-p T hadron production we performed evaluation of the production length l p available for gluon radiation and energy loss, and found it to be rather short in many instances. As a result, the main reason for the observed suppression in heavy ion collisions is not induced energy loss, but attenuation of early produced color dipoles propagating through a dense absorptive matter. Having no free parameters, except the medium density characterized by the transport coefficientq, we reached a good agreement with data on nuclear suppression and azimuthal elliptic flow in a wide range of energy, from the lowest energies of RHIC up to LHC, and in a wide range of transverse momenta, from p T = 5-7 GeV up to 100 GeV. The region of smaller p T apparently is dominated by hydrodynamic mechanisms of hadron production. The recent attempt [50] to unify hydrodynamics with perturbative QCD calculations, presented above, was successful, the whole range of p − T was well described with the same medium temperature.
Our analysis led to quite reasonable values of the parameter characterizing the hot medium, q 0 = 1.2, 1.6 and 2 GeV 2 /fm at √ s = 62, 200 and 2760 GeV respectively. This is close to the expected magnitude q 0 ∼ 1 GeV 2 /fm [6], as well as to the value extracted from data on J/Ψ suppression [28,47]. The latter is an alternative probe for the created hot matter, and different probes obviously must result in the same properties of the probed medium. The pure energy loss scenario did not pass this important test, it leads to a magnitude of the transport coefficient [48], which is an order of magnitude larger than expected [6].
Another important test of the energy loss scenario would be a direct measurement of the transport coefficient, i.e. broadening of a jet propagating through the hot medium. On the contrary to this expectation no broadening was found in back-to-back photon-jet azimuthal correlation [49].
Concluding, few words of precotion to avoid further confusions: (i) One should clearly distinguish between vacuum and medium-induced radiative energy loss. The former is much more intensive and is the main reason for shortness of l p .
(ii) One should also discriminate between the terms "jet quenching" and "hadron quenching". The latter is what was considered in this paper, a hadron detected inclusively with a high p T carries the main fraction z h of the accompanying jet. This is why intensive vacuum energy loss and energy conservation impose severe constraints on the value of l p . If, however, the whole jet has a large p T , while none of hadrons in the jet are not forced to have large z h , the hadronization length is subject to the usual Lorentz time dilation and is long.