# Elastic, Inelastic, and Path Length Fluctuations in Jet Tomography

###### Abstract

Jet quenching theory using perturbative QCD is extended to include (1) elastic as well as (2) inelastic parton energy losses and (3) jet path length fluctuations. The extended theory is applied to non-photonic single electron production in central Au+Au collisions at AGeV. The three effects combine to significantly reduce the discrepancy between theory and current data without violating the global entropy bounds from multiplicity and elliptic flow data. We also check for consistency with the pion suppression data out to 20 GeV. Fluctuations of the jet path lengths in realisitic geometry and the difference between the widths of fluctuations of elastic and inelastic energy loss are essential to take into account.

###### pacs:

12.38.Mh; 24.85.+p; 25.75.-qLight quark and gluon jet quenching observed via suppression phenix_pi0 in Cu+Cu and Au+Au collisions at AGeV at the Relativistic Heavy Ion Collider (RHIC) has been remarkably consistent thus far with predictions Gyulassy:2003mc -Vitev:2002pf . However, recent non-photonic single electron data Adler:2005xv ; Abelev:2006db ; Adare:2006hc ; elecQM05_STAR ; Adare:2006nq (which present an indirect probe of heavy quark energy loss) have significantly challenged the underlying assumptions of the jet tomography theory (see Djordjevic:2005db ). A much larger suppression of electrons than predicted was observed in the GeV region (see Fig. 1). These data falsify the assumption that heavy quark quenching is dominated by radiative energy loss when the bulk QCD matter parton density is constrained by the observed rapidity density of produced hadrons.

The observed “perfect fluidity” WhitePapers ; BNLfluid of the sQGP at long wavelengths ( GeV) provides direct evidence for highly nonperturbative bulk dynamics Gyulassy:2004zy ; Hirano:2005wx . Due to asymptotic freedom, a breakdown of perfect fluidity and nonperturbative effects are expected at several times greater than the mean thermal energy, GeV. Prior to these electron data, pQCD based jet quenching theory provided increasingly reliable predictions above GeV winter ; WhitePapers for the nuclear modification of light parton jets Gyulassy:2003mc -Vitev:2002pf . However, the non-photonic single electron data however raise the question of whether the novel nonperturbative physics of the strongly interacting Quark Gluon Plasma (sQGP) Gyulassy:2004zy produced at RHIC could persist down to much smaller wavelengths than previously expected. This question is also of pragmatic importance because high jets can be utilized as calibrated “external” tomographic probes of the bulk sQGP matter only if their dynamics can be predicted reliably.

The upper band of Fig. 1 shows that the predictions from Djordjevic:2005db considerably underestimate the electron nuclear modification of data even out to GeV. This discrepancy points to one or more of (1) missing perturbative QCD physics, (2) incomplete understanding of the initial heavy quark production and/or (3) novel non-perturbative mechanisms affecting partonic physics out to GeV. We note that GeV (single non-photonic) electrons originate in our calculations from the fragmentation and decay of both charm and bottom quarks with transverse momenta GeV (see Fig. 5 in Djordjevic:2005db ).

Possibility (3) is of course the most radical and would imply the persistence of non-perturbative physics in the sQGP down to extremely short wavelengths. Processes can be postulated to improve the fit to the data Rapp:2005at , but at the price of losing theoretical control of the tomographic information from jet quenching data. DGVW Djordjevic:2005db showed that by arbitrarily increasing the initial sQGP densities to unphysical , the non-photonic electrons from heavy quarks can be artificially suppressed to . Thus, to approach the electron data, conventional radiative energy loss requires either a violation of bulk entropy bounds or nonperturbatively large extrapolations of the theory. Even by ignoring the bottom contribution, Ref. Armesto:2005iq found that a similarly excessive transport coefficient Baier:2002tc , GeV/fm, was necessary to approach the level of suppression of electrons in the data.

Bottom quark jets are very weakly quenched by radiative energy loss. Using the FONLL production cross-sections, their contribution significantly reduces the single electron suppression Djordjevic:2005db compared to that of the charm jets alone. The ratio is not sensitive to the scaling of all cross-sections by a constant. However, it is sensitive to any uncertainty in the relative contribution of charm and bottom jets to the electrons Armesto:2005mz . Recent data from STAR on electrons from p+p collisions Abelev:2006db may indicate an even larger uncertainty in the production than expected from FONLL. However, PHENIX p+p to electron data are compatible with the upper limit of FONLL predictions Cacciari:2005rk ; MNR , similar to the comparison between FONLL and Tevatron data.

The discrepancy between the ‘DGLV Rad only’ predictions and the data in Fig. 1 and recent work Mustafa:2004dr ; Dutt-Mazumder:2004xk ; Zapp:2005kt motivated us to revisit the assumption that pQCD elastic energy loss Bjorken:1982tu is negligible compared to radiative. In earlier studies, the elastic energy loss Bjorken:1982tu ; Thoma:1990fm ; Braaten:1991jj ; Wang:1994fx ; Mustafa:1997pm ; Lin:1997cn was found to be GeV/fm, which was erroneously considered to be small compared to the several GeV/fm expected from radiative energy loss. The apparent weakness of conventional pQCD collisional energy loss mechanisms was also supported by parton transport theory results Molnar:2001ux -Moore:2004tg , which showed that the typical thermal pQCD elastic cross section, mb, is too small to explain the differential elliptic flow at high GeV and also underestimates the high quenching of pions.

In contrast, Mustafa Mustafa:2004dr found that radiative and elastic average energy losses for heavy quarks were in fact comparable over a very wide kinematic range accessible at RHIC. In Fig. 2, we confirm Mustafa’s finding and extend it to the light quark sector as well. The fractional energy loss, , from DGLV radiative for quarks (solid curves; see also App. IB) is compared to TG Thoma:1990fm and BT Braaten:1991jj estimates of elastic (dashed curves; see also App. IA). For light quarks, the elastic energy loss decreases more rapidly with energy than radiative energy loss, but even at 20 GeV the elastic is only 50% smaller than the radiative.

From Fig. 2 we see that for GeV light and charm quark jets have elastic energy losses smaller but of the same order of magnitude as the inelastic losses. But due to the large mass effect Dead-cone -Zhang:2003wk ,Armesto:2005iq , both radiative and elastic energy losses remain significantly smaller for bottom quarks than for light and charm quarks, but the elastic energy loss can now be greater than inelastic up to GeV. We present both TG and BT as a measure of the theoretical uncertainties of the Coulomb log (see App IC for benchmark numerical examples). These are largest for the heaviest b quark. As they are not ultrarelativistic, the leading log approximation Thoma:1990fm ; Braaten:1991jj breaks down in the kinematic range accessible at RHIC. More rigorous computations of elastic energy loss Djordjevic:2006tw and numerical covariant transport techniques Molnar:2001ux can be used to reduce the theoretical uncertainties in the elastic energy loss effects.

Theoretical Framework.

The quenched spectra of partons, hadrons, and leptons are calculated
as in Djordjevic:2005db from the generic pQCD convolution

(1) | |||||

where denotes quarks and gluons. For charm and bottom, the initial quark spectrum, , is computed at next-to-leading order using the code from Cacciari:2005rk ; MNR ; for gluons and light quarks, the initial distributions are computed at leading order as in Vitev:2002pf . is the energy loss probability, is the fragmentation function of quark to hadron , and is the decay function of hadron into the observed single electron. We use the same mass and factorization scales as in Vogt and employ the CTEQ5M parton densities Lai:1999wy with no intrinsic . As in Vogt we neglect shadowing of the nuclear parton distribution in this application.

We assume that the final quenched energy, , is large enough that the Eikonal approximation can be employed. We also assume that in Au+Au collisions, the jet fragmentation function into hadrons is the same as in collisions. This is expected to be valid in the deconfined medium case, where hadronization of cannot occur until the quark emerges from the sQGP.

The main difference between our previous calculation Djordjevic:2005db and the present one is the inclusion of two new physics components in the energy loss probability . First, is generalized to include for the first time both elastic and inelastic energy loss and their fluctuations. We note that Vitev Vitev:2003xu was the first to generalize the GLV formalism to include initial state elastic energy loss effects in d+Au. In this work, Eq. (2) extends the formalism to include final state elastic energy loss effects in .

The second major change relative to our previous applications is that we now take into account geometric path length fluctuations as follows:

(2) | |||||

Here

(3) |

is the locally determined effective path length of the jet given its initial production point and its initial azimuthal direction relative to the impact parameter plane . The geometric path averaging used here is similar to that used in Gyulassy:2000gk and by Eskola et al. Eskola:2004cr . However, the inclusion of elastic energy losses together with path fluctuations in more realistic geometries was not considered before.

We consider a diffuse Woods-Saxon nuclear density profile Hahn , which creates a participant transverse density, , computed using the Glauber profiles, , with inelastic cross section mb. The bulk sQGP transverse density is assumed to be proportional to this participant density, and its form is shown (for the slice) in Fig. 3 by the curve labeled . The distribution of initial hard jet production points, , is assumed on the other hand to be proportional to the binary collision density, . This is illustrated in Fig. 3 by the narrower curve labeled .

The combination of fluctuating DGLV radiative Djordjevic:2003zk with the new elastic energy losses and fluctuating path lengths (via the extra integrations) adds a high computational cost to the extended theory specified by Eqs. (1,2). In this first study with the extended theory, we explore the relative order of magnitude of the competing effects by combining two simpler approaches.

In approach I, we parameterize the heavy quark pQCD spectra by a simpler power law, , with a slowly varying logarithmic index . For the pure power law case, the partonic modification factor, , (prior to fragmentation) is greatly simplified. This enables us to perform the path length fluctuations numerically via

(4) |

where

(5) |

Both the mean and width of those fractional energy losses depend on the local path length. (See App ID for numerical illustrations of Eq.(5) for a fixed fm light quark case.)

We emphasize, however, that no externally specified a priori path length, , appears in Eq. (4); the path lengths are allowed to explore the whole geometry. Fig. 4 shows the broad distribution of lengths probed by hard partons in approach I.

In the second approach, we determine effective path lengths, , for each parton flavor, , by varying fixed predictions and comparing to approach I; see Fig. 4. In approach II, is calculated directly from Eq. (1) with in Eq. (2) replaced by the fixed approximation

(6) |

Here, jet quenching is performed via two independent branching processes in contrast to the additive convolution in Eq. (4). For small energy losses the two approaches are similar. They differ however in the case of long path lengths when the probability of complete stopping approaches unity. In the convolution method, the probability of is interpreted as complete stopping, whereas in the branching algorithm the long path length case is just highly suppressed. In both cases we take into account the small finite probability that the fractional energy loss due to fluctuations.

To illustrate the difference in approach II, consider the case of power law initial distributions as in Eq. (4). In this case

(7) | |||||

The branching implementation is seen via the product of two distinct factors in contrast to the one quenching factor in Eq. (4). For small both approaches obviously give rise to the same .

Due to the high computational cost in approach I, only the TG elastic is used for the heavy quarks and only BT for light quarks. The Coulomb log uncertainties are estimated only in approach II.

In both approaches, fluctuations of the radiative energy loss due to gluon number fluctuations are computed as discussed in detail in Ref. Djordjevic:2005db ; Djordjevic:2004nq . This involves using the DGLV generalization Djordjevic:2003zk of the GLV opacity expansion Gyulassy:2000er to heavy quarks. Bjorken longitudinal expansion is taken into account by evaluating the bulk density at an average time Djordjevic:2005db ; Djordjevic:2004nq . For elastic energy loss, the full fluctuation spectrum is approximated here by a Gaussian centered at the average energy loss with variance Moore:2004tg . In approach I the correct, numerically intensive integration through the Bjorken expanding medium provides . In approach II the approximation is again used; numerical comparisons show that for fm this reproduces the full calculation well. Finally, we note that we use the additional numerical simplification of keeping the strong coupling constant fixed at .

Numerical Results: Parton Level

In Fig. 5, we show the quenching pattern of from the second approach
for a “typical” path length scale fm, similar to that used in previous calculations Djordjevic:2005db .
The curves show , prior to hadronization, for
. The dashed curves show the quenching arising from only the
DGLV radiative energy loss. The solid curves show the full results
after including TG elastic as well as DGLV radiative energy loss. Adding
elastic energy loss is seen to increase the quenching of all flavors for fixed
path length. Note especially the strong nonlinear
increase of the gluon suppression and the factor
increase of the bottom suppression. The curious switch of
the and the quenching reflects the
extra valence (smaller index ) contribution to light quarks.

Fig. 5 emphasizes the unavoidable result of using a fixed, “typical” path length scale, , in jet tomography: the pion and single electron quenching can never be similar. If pions were produced only by light quarks and electrons only by charm, then we would expect comparable quenching for both. However, contributions from highly quenched gluons decrease the pion while weakly quenched bottom quarks increase the electron . Therefore, in the fixed length scenario, we expect a noticeable difference between pion and single electron suppression patterns.

The solid curves of Fig. 3 labeled by the parton flavor show the relative transverse coordinate density of surviving jets defined by

(8) |

is given by the initial transverse production distribution times the quenching factor from that position in direction with final momentum . The case shown is for a GeV jet produced initially at and moving in the direction along the positive x axis. The quenching is determined by the participant bulk matter along its path , and varies with because the local path length changes according to Eq. (3).

What is most striking in Fig. 3 is the hierarchy of -dependent length scales. No single, representative path length can account for the distribution of all flavors. In general heavier flavors are less biased toward the surface (in direction ) than lighter flavors since the energy loss decreases with the parton mass. Gluons are more surface biased than light quarks due to their color Casimir enhanced energy loss. In addition, note the surprising reversal of the and suppressions, also seen in Fig. 5. Fig. 2 shows that the energy loss of is somewhat less than for ; however, the higher power index, , of relative to – as predicted by pQCD and due to the valence component of – compensates by amplifying its quenching.

However, none of the distributions can be categorized as surface emission. The characteristic widths of these distributions range from fm. We show below that such a large dynamic range of path length fluctuations is essential for consistent reproduction of both electron and pion data.

We turn next to Figs. 6 and 7, which show the interplay between the dynamical geometry seen in Fig. 3 and the elastic-enhanced quenching of partons. In Figs. 6 and 7 the solid green curves labeled “DGLV+TG/BT: Full Geometry” are the results using approach I based on Eq. (4). The curves labeled TG and BT are from approach II based on Eq. (7). The effective fixed in II were taken to match approximately the green curves in which full path length fluctuations are taken into account. This procedure is not exact because of the different numerical approximations involved, but the trends are well reproduced. The are determined only to fm accuracy, as this suffices for our purposes here. We show the comparison between approaches I and II for heavy quarks in Fig. 6 using and fm and for gluons and light quarks in Fig. 7 using and fm; see Fig. 4 for a visual comparison of the input length distributions used. This hierarchy of -dependent length scales is in accord with that expected from Fig. 3.

Note that in comparison to the fixed fm case in Fig. 5, geometric fluctuations reduce gluon jet quenching in Fig. 7 by a factor . Nevertheless, even with path length fluctuations the gluons are still quenched by a factor of 10 when elastic energy loss is included in addition to radiative.

The amplified role of elastic energy loss is due to its smaller width for fluctuations relative to radiative fluctuations. Even in moderately opaque media with , inelastic energy loss fluctuations are large because only a few, 2-3, extra gluons are radiated GLV_suppress . Thus, gluon number fluctuations, lead to a substantial reduction in the effect of radiative energy loss. On the other hand, elastic energy loss fluctuations are controlled by collision number fluctuations, , which are relatively small in comparison for a significant proportion of the length scales probed. Therefore, fluctuations of the elastic energy loss do not dilute the suppression of the nuclear modification factor as much as fluctuations. The increase in the sensitivity of the final quenching level to the opacity is a novel and useful byproduct of including the elastic channel; see Fig. 11 in Appendix D. The inclusion of elastic energy loss significantly reduces the fragility of pure radiative quenching Eskola:2004cr and therefore increases the sensitivity of jet quenching to the opacity of the bulk medium HWG .

Numerical Results: Pions and Electrons

We now return to Fig. 1 to discuss the consequence of including
elastic energy loss of and quarks on the electron spectrum. The inclusion of the collisional energy loss
significantly improves the comparison between theory and the single electron data. That is, the lower yellow band can reach below in spite of keeping , consistent with measured multiplicity, and
using a conservative . A large source of the uncertainty represented
by the lower yellow band is the modest
but poorly determined elastic energy loss, , of bottom quarks (see Fig. 2). There is additional uncertainty from the relative contributions to electrons from charm and bottom
jets. The dashed lines show an extreme version of this in which charm jets are the only source of electrons. If the charm to bottom ratio given by FONLL calculations is accurate, the current data suggests that even the combined radiative+elastic pQCD mechanism is not
sufficient to explain the single electron suppression.

As emphasized in Djordjevic:2005db , any proposed energy loss mechanisms must also be checked for consistency with the extensive pion quenching data phenix_pi0 , for which preliminary data now extend out to GeV. This challenge is seen clearly in Fig. 5, where for fixed fm, the addition of elastic energy loss would overpredict the quenching of pions. However, the simultaneous inclusion of path fluctuations leads to a decrease of the mean and , path lengths that partially offsets the increased energy loss. Therefore, the combined three effects considered here makes it possible to satisfy without violating the bulk entropy constraint and without violating the pion quenching constraint now observed out to 20 GeV; see Fig. 8. We note that the slow rise of with in the present calculation is due in part to the neglect of initial smearing that raises the low region and the EMC effect that lowers the high region (see Vitev:2002pf ).

Conclusions

The elastic component of the energy loss cannot be neglected when considering pQCD jet quenching.
While the results presented in this paper are encouraging, further
improvements of the jet quenching theory will be required before
stronger conclusions can be drawn.

From an experimental perspective, there is at present significant disagreement between measured p+p to electron baselines Abelev:2006db ; Adare:2006hc . In addition, direct measurement of spectra will be essential to deconvolute the different bottom and charm jet quark dynamics.

On the theoretical side, more work is needed to sort out coherence and correlation effects between elastic and inelastic processes that occur in a finite time and with multiple collisions. Classical electrodynamics calculations presented in Peigne:2005rk suggested that radiative and elastic processes could destructively interfere over lengths far longer than previously thought. As described in Adil:2006ei , a proper accounting of the current shows finite size effects persist out only to the expected lengths of order the screening scale, fm. Additionally, work on the quantum mechanical treatment of elastic energy loss in a finite medium Djordjevic:2006tw ; Wang:2006qr also concluded that finite size effects on remain small except in peripheral collisions.

There are several other open problems that require further study. The inclusion of all the initial state effects from Vitev:2002pf will be needed to fully check the consistency of the pion predictions with the data. Only an approximate fluctuation spectrum for elastic energy loss has been included here; still needed is an examination of the effect of the full fluctuation spectrum.

The radiative and elastic energy losses depend sensitively on the coupling, and . Future calculations will have to relax the current fixed approximation. In Peshier:2006hi , the running of the coupling is seen to increase the magnitude of the elastic energy loss and alter the energy dependence. More complete calculations of both radiative and elastic energy losses will involve integrals that probe momentum scales that are certainly nonperturbative. Therefore it will be important to study the irreducible uncertainty associated with the different maximum cutoff prescriptions commonly used.

Acknowledgments: Valuable input from Azfar Adil on parton spectra and discussions with Xin Dong, Barbara Jacak, John Harris, Peter Lévai, Denes Molnar, Thomas Ullrich, Ivan Vitev, Xin-Nian Wang, and Nu Xu are gratefully acknowledged. This work is supported by the Director, Office of Science, Office of High Energy and Nuclear Physics, Division of Nuclear Physics, of the U.S. Department of Energy under Grants No. DE-FG02-93ER40764. M. D. acknowledges support by the U.S. Department of Energy under Grants No. DE-FG02-01ER41190 during the final parts of this work.

## I Appendix

### i.1 Collisional Energy Loss

The leading logrithmic expression for the elastic energy loss of a jet with color Casimir in an ideal quark-gluon plasma with active quark flavors and temperature, , is given by Bjorken:1982tu

(9) |

where the Coulomb log is controlled by the ratio that involves relevant minimum and maximum momentum transfers or impact parameters. For scattering in an assumed ultrarelativistic gas of partons the jet velocity dependence is

(10) |

Estimates for differ below asymptotic energies and are given in Bjorken:1982tu , Thoma:1990fm , and Braaten:1991jj that we denote by Bj, TG, and BT respectively:

with the crossover between and being taken at for numerical computation.

### i.2 DGLV Radiative Energy Loss

For completeness we also record the DGLV formula for radiative energy loss used in our calculations. We neglect finite kinematic limits on the momentum transfer integral, and perform the finite and azimuthal integrations analytically. The mean fractional radiative energy loss can be then evaluated numerically from the expression

(12) | |||||

where

(13) | |||||

(14) | |||||

(15) |

We employ the asymptotic 1-loop transverse gluon mass . The A,B,C functions denote

(16) | |||||

(17) | |||||

(18) | |||||

where the abbreviated symbols denote

(19) | |||||

(20) | |||||

(21) | |||||

(22) | |||||

(23) |

with .

### i.3 Benchmark Numerical Examples

In this section we record numerical benchmark cases of both the elastic and radiative mean energy loss to illustrate the above formulas. Consider a uniform Bjorken cylinder with density

(24) |

We assume fm. The temperature evolves as

where is the number of active quark flavors. The effective static approximation simulates the effect of Bjorken expansion by evaluating at , where is the jet path length to the the cylinder surface. The gluon density is computed from , and the density of quarks plus antiquarks is . The Debye mass squared is . In Table I the results for are given for a charm jet of energy .

Radiative | Collisional | |||
---|---|---|---|---|

(GeV) | DGLV | Bj | TG | BT |

10 | 0.2111 | 0.2022 | 0.1594 | 0.1596 |

11 | 0.2126 | 0.1894 | 0.1506 | 0.1552 |

12 | 0.2129 | 0.1782 | 0.1430 | 0.1621 |

13 | 0.2123 | 0.1683 | 0.1358 | 0.1530 |

14 | 0.2110 | 0.1596 | 0.1294 | 0.1450 |

15 | 0.2093 | 0.1518 | 0.1237 | 0.1379 |

### i.4 Energy Loss Fluctuation Spectrum

This section illustrates the fluctuation spectra of induced gluon number and the distribution of fluctuating energy loss for a specific case of a 15 GeV up quark jet with path length fm. Fig. 9 shows the first order induced gluon number distribution for this case. Fig. 10 shows the fractional radiative energy loss distribution taking into account Poisson fluctuations of the gluon number computed as in GLV_suppress . The finite probability, , of radiating zero gluons contributes a that is not shown. There is also a finite probablity, , of complete stopping with .

The width of the elastic energy fluctuations seen in Fig. 11 is significantly smaller than the radiative width. The narrowing of the convoluted elastic plus radiative distributions significantly reduces the distortion effects due to fluctuations. Because of the steep fall off of the initial unquenched parton spectra, the smaller width of the elastic energy fluctuations considerably amplifies the quenching effect due to collisional energy loss in comparison to the larger but much broader radiative contribution. In terms of an effective renormalization as discussed in GLV_suppress , is closer to unity than the renormalization characteristic of pure radiative energy loss distributions.

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