Jet tomography of quark gluon plasma

PLENARY TALKS

Jet tomography of quark gluon plasma

Xin-Nian Wang

Nuclear Science Division, MS 70R0319, Lawrence Berkeley National Laboratory, Berkeley, CA 94720

ABSTRACT

Recent experimental measurements of high p_T hadron spectra and jet correlation at RHIC are analyzed within a parton model which incoporates initial jet production and final propagation in heavy-ion collisions. The suppre-sion of single hadron spectra, back-to-back correlation, their centrality dependence and azimuthal anisotropy point to a dense matter with an initial parton density about 30 times of that in a cold heavy nucleus.

1 Introduction

In high-energy heavy-ion collisions, a dense medium of quarks and gluons is expected to be produced and possibly a quark-gluon plasma is formed. One important step in establishing evidence of QGP formation is to charaterize the properties of the dense medium produced, for example, the parton and energy density and color confinement, among many other charateristics. Conventionally, one can study the properties of a medium via scattering experiments with particle beams. In deeply inelastic scattering (DIS) experiments, for example, leptons scatter off the nucleon medium via photon exchange with quarks. The response function or the correlation function of the electromagnetic currents,

is a direct measurement of the quark distributions in a nucleon or nucleus. Such experiments have provided unprecedented information about partonic structure of nucleons and nucleus and confirmed the prediction of QCD evolution[1].

For dynamic systems such as that produced in heavy-ion collisions, one can no longer use the technique of scattering with a beam of particles because of the transient nature of the matter. The lifetime of the system is very short, on the order of a few fm/c. The initial spatial size is only the size of the heaviest nuclei, about 6 fm in diameter in the transverse dimension. The system expands very rapidly both in the longitudinal and transverse direction. These characteristics make it impossible to use external probes to study the properties of the produced dense matter in high-energy heavy-ion collisions. Fortunately, one can prove that the thermal average of the above correlation function gives the photon emission rate from the evolving system. The emission rate depends mainly on the local temperature or the parton density while the total yield also depends the whole evolution history of the system. Therefore, a strongly interacting system can reveal its properties and dynamics through photon and dilepton emission. One can further study the resonance properties of the emitted virtual photons and their medium modification. The screening of strong interaction in a color deconfined medium should lead to dissociation of the binding states and thus the quarkonia suppression [2]. Such color screening is a result of strong interaction between quarks and gluons at high density and temperature. The same interaction will also cause attenuation of fast and energetic partons propagating through the medium. Such an effect is the underlying physics of the jet quenching [3] phenomenon and jet tomography technique for studying properties of dense matter in high-energy heavy-ion collisions.

Jet quenching as a probe of the dense matter in heavy-ion collisions, takes advantage of the hard processes of jet production in high-energy heavy-ion collisions. Similar to the technology of computed tomography (CT), study of these energetic particles, their initial production and interaction with the dense medium, can yield critical information about the properties of the matter that is otherwise difficult to access through soft hadrons from the hadronization of the bulk medium. Though relatively rare with small cross sections, the jet production rate can be calculated perturbative in QCD and agrees well with experimental measurements in high-energy pp() collisions. A critical component of the jet tomography is then to understand the jet attenuation through dense matter as it propagates through the medium.

2 Modified Fragmentation Function

A direct manifest of jet quenching is the modification of the fragmentation function of the produced parton, D_{a ®h}(z, m²) which can be measured directly. This modification can be directly translated into the energy loss of the leading parton.

To demonstrate medium modified fragmentation function and parton energy loss, one can study deeply inelastic scattering (DIS) eA [4, 5, 6]. Here, we consider the semi-inclusive processes, e(L₁) + A(p) ® e(L₂) + h(_h) + X, where L₁ and L₂ are the four-momenta of the incoming and the outgoing leptons, and _h is the observed hadron momentum. The differential cross section for the semi-inclusive process can be expressed as

where p = [p⁺, 0, 0_^] is the momentum per nucleon in the nucleus, q = L₂ – L₁ = [–Q²/2q^–, q^–, 0_^] the momentum transfer, s = (p + L₁)² and a_EM is the electromagnetic (EM) coupling constant. L_{m n} is the leptonic tensor while W_{m n} is the semi-inclusive hadronic tensor.

In the parton model with the collinear factorization approximation, the leading-twist contribution to the semi-inclusive cross section can be factorized into a product of parton distributions, parton fragmentation functions and the hard partonic cross section. Including all leading log radiative corrections, the lowest order contribution from a single hard g^*+ q scattering can be written as

where (x, p, q) is the hard part of the process in leading order, the momentum fraction carried by the hadron is defined as z_h = /q^– and x_B = Q²/2p⁺q^– is the Bjorken variable. and m² are the factorization scales for the initial quark distributions in a nucleus and the fragmentation functions D_{q® h}(z_h, m²), respectively.

In a nuclear medium, the propagating quark in DIS will experience additional scatterings with other partons from the nucleus. The rescatterings may induce additional gluon radiation and cause the leading quark to lose energy. Such induced gluon radiations will effectively give rise to additional terms in the evolution equation leading to the modification of the fragmentation functions in a medium. These are the so-called higher-twist corrections since they involve higher-twist parton matrix elements and are power-suppressed. We will consider those contributions that involve two-parton correlations from two different nucleons inside the nucleus.

One can apply the generalized factorization to these multiple scattering processes[7]. In this approximation, the double scattering contribution to radiative correction can be calculated and the effective modified fragmentation function is

where D_{q® h}(z_h, m²) and D_{g® h}(z_h, m²)are the leading-twist fragmentation functions. The modified splitting functions are given as

Here, the fractional momentum is defined as x_L = /2p⁺q^–z(1 – z) and x = x_B = Q²/2p⁺q^– is the Bjorken variable. The twist-four parton matrix elements of the nucleus,

has a dipole-like structure which is a result of Landau-Pomerachuck-Migdal (LPM) interference in gluon bremsstrahlung. Here the intrinsic transverse momentum is neglected. In the limit of collinear radiation (x_L ® 0) or when the formation time of the gluon radiation, t_fº 1/x_Lp⁺, is much larger than the nuclear size, the destructive interference leads to the LPM interference effect.

Using the factorization approximation[4, 5, 7, 8], we can relate the twist-four parton matrix elements of the nucleus to the twist-two parton distributions of nucleons and the nucleus,

where is considered a constant. One can identify 1/x_Lp⁺ = 2q^–z(1 – z)/ as the formation time of the emitted gluons. When it becomes comparable or larger than the nuclear size, the above matrix element vanishes, demonstrating a typical LPM interference effect.

Since the LPM interference suppresses gluon radiation whose formation time is larger than the nuclear size MR_A/p⁺ in our chosen frame, should then have a minimum value of ~ Q²/MR_A ~ Q²/A^1/3. Here M is the nucleon mass. Therefore, the leading higher-twist contribution is proportional to a_sR_A/ ~ a_sR_A²/Q² due to double scattering and depends quadratically on the nuclear size R_A.

With the assumption of the factorized form of the twist-4 nuclear parton matrices, there is only one free parameter (Q²) which represents quark-gluon correlation strength inside nuclei. Once it is fixed, one can predict the z, energy and nuclear dependence of the medium modification of the fragmentation function. Shown in Fig. 1 are the calculated nuclear modification factor of the fragmentation functions for ¹⁴N and ⁸⁴Kr targets as compared to the recent HERMES data[9]. The predicted shape of the z-dependence agrees well with the experimental data. A remarkable feature of the prediction is the quadratic A^2/3 nuclear size dependence, which is verified for the first time by an experiment. By fitting the overall suppression for one nuclear target, we obtain the only parameter in our calculation, (Q²) = 0.0060 GeV² with a_s(Q²) = 0.33 at Q²» 3 GeV². The predicted n-dependence also agrees with the experimental data [22].

We can quantify the modification of the fragmentation by the quark energy loss which is defined as the momentum fraction carried by the radiated gluon,

where x_m = m²/2p⁺q^–z(1 – z) = x_B/z(1–z) if we choose the factorization scale as m² = Q². When x_A << x_B << 1 we can estimate the leading quark energy loss roughly as

Since x_A = 1/MR_A, the energy loss áDz_gñ thus depends quadratically on the nuclear size.

In the rest frame of the nucleus, p⁺ = m_N, q^– = n, and x_B º Q²/2p⁺q^– = Q²/2m_Nn. One can get the averaged total energy loss as DE = náDz_gñ » (Q²)(Q²)m_N(C_A/N_c)3 ln(1/2x_B). With the determined value of , áx_Bñ » 0.124 in the HERMES experiment[9] and the average distance áL_Añ = R_A for the assumed Gaussian nuclear distribution, one gets the quark energy loss dE/dL » 0.5 GeV/fm inside a Au nucleus.

3 Jet Quenching in Hot Medium at RHIC

To extend our study of modified fragmentation functions to jets in heavy-ion collisions, we can assume » m² (the Debye screening mass) and a gluon density profile r(y) = (t₀/t)q(R_A – y)r₀ for a 1-dimensional expanding system. Since the initial jet production rate is independent of the final gluon density which can be related to the parton-gluon scattering cross section[10] [a_sx_TG(x_T) ~ m²s_g], one has then

where t_f = 2Ez(1 – z)/ is the gluon formation time. One can recover the form of energy loss in a thin plasma obtained in the opacity expansion approach[11],

Keeping only the dominant contribution and assuming s_g» C_a2p/m² (C_a=1 for qg and 9/4 for gg scattering), one obtains the averaged energy loss,

Neglecting the logarithmic dependence on t, the averaged energy loss in a 1-dimensional expanding system can be expressed as » (dE₀/dL) (2t₀/R_A), where dE₀/dL µ r₀R_A is the energy loss in a static medium with the same gluon density r₀ as in a 1-d expanding system at time t₀. Because of the expansion, the averaged energy loss ádE/dLñ_1d is suppressed as compared to the static case and does not depend linearly on the system size.

In order to calculate the effects of parton energy loss on the attenuation pattern of high p_T partons in nuclear collisions, we use a simpler effective modified fragmentation function[12, 13],

where , are the rescaled momentum fractions. This effective model is found to reproduce the pQCD result from Eq.(4) very well, but only when Dz = DE_c/E is set to be Dz » 0.6 áz_gñ. Therefore the actual averaged parton energy loss should be DE/E = 1.6Dz with Dz extracted from the effective model. The factor 1.6 is mainly caused by the unitarity correction effect in the pQCD calculation.

Since gluons are bosons, there should also be stimulated gluon emission and absorption by the propagating parton because of the presence of thermal gluons in the hot medium. Such detailed balance is crucial for parton thermalization and should also be important for calculating the energy loss of an energetic parton in a hot medium[14]. Taking into account such detailed balance in gluon emission, one can then get the asymptotic behavior of the effective energy loss in the opacity expansion framework [14],

where the first term is from the induced bremsstralung and the second term is due to gluon absorption in detailed balance which effectively reduce the total parton energy loss in the medium. Numerical calculations show that the effect of the gluon absorption is small and can be neglected for partons with very high energy. However, the thermal absorption reduces the effective parton energy loss by about 30-10% for intermediate values of parton energy. This will increase the energy dependence of the effective parton energy loss in the intermediate energy region. One can parameterize such energy dependence as,

The threshold is the consequence of gluon absorption that competes with radiation that effectively shuts off the energy loss. The parameter m is set to be 1 GeV in the calculation.

To calculate the modified high p_T spectra in A + A collisions, we use a LO pQCD model [15, 16],

with medium modified fragmentation funcitons given by Eq. (14) and the fragmentation functions in free space (z_c, Q²) are given by the BBK parameterization [17]. Here, z_c = p_T/p_Tc, y = y_c, s(ab ® cd) are elementary parton scattering cross sections and t_A(b) is the nuclear thickness function normalized to ò d²bt_A(b) = A. We will use a hard-sphere model of nuclear distribution in this paper. The K » 1.5 – 2 factor is used to account for higher order pQCD corrections.

The parton distributions per nucleon f_a/A(x_a, Q², r) inside the nucleus are assumed to be factorizable into the parton distributions in a free nucleon given by the MRSD–^¢ parameterization [18] and the impact-parameter dependent nuclear modification factor which will given by the new HIJING parameterization [19]. The initial transverse momentum distribution g_A(k_T, Q², b) is assumed to have a Gaussian form with a width that includes both an intrinsic part in a nucleon and nuclear broadening. This model has been fitted to the nuclear modification of the p_T spectra in p+A collisions at up to the Fermilab energy = 40 GeV. Shown in Fig. 2 are the first prediction made in 1998 [15] of the Cronin effect at RHIC for p + Au collisions at = 200 GeV as compared to the recent RHIC data. As one can see, the initial multiple scattering in nuclei can give some moderate Cronin enhancement of the high p_T spectra. Therefore, any suppression of the high p_T spectra in Au + Au collisions has to be caused by jet quenching.

We assume a 1-dimensional expanding medium with a gluon density r_g(t, r) that is proportional to the transverse profile of participant nucleons. According to Eq. 13, we will calculate impact-parameter dependence of the energy loss as

where DL(b, , f) is the distance a jet, produced at , has to travel along at an azimuthal angle f relative to the reaction plane in a collision with impact-parameter b. Here, r₀ is the averaged initial gluon density at t₀ in a central collision and ádE/dLñ_1d is the average parton energy loss over a distance R_A in a 1-d expanding medium with an initial uniform gluon density r₀. The corresponding energy loss in a static medium with a uniform gluon density r₀ over a distance R_A is [22] dE₀/dL = (R_A/2t₀)ádE/dLñ_1d. We will use the parameterization in Eq. (16) for the effective energy dependence of the parton quark energy loss.

Shown in Fig. 3 are the calculated nuclear modification factors R_AB(p_T) = /áN_binaryñ for hadron spectra (|y| < 0.5) in Au + Au collisions at = 200 GeV, as compared to experimental data [24, 23]. Here, áN_binaryñ = ò d²bd²rt_A(r)t_A(| – |). To fit the observed p⁰ suppression (solid lines) in the most central collisions, we have used m = 1.5 GeV, ₀ = 1.07 GeV/fm and l₀ = 1/(sr₀) = 0.3 fm. The hatched area (also in other figures in this paper) indicates a variation of ₀ = ±0.3 GeV/fm. The hatched boxes around R_AB = 1 represent experimental errors in overall normalization. Nuclear k_T broadening and parton shadowing together give a slight enhancement of hadron spectra at intermediate p_T = 2 – 4 GeV/c without parton energy loss.

The flat p_T dependence of the p⁰ suppression is a consequence of the strong energy dependence of the parton energy loss. The slight rise of R_AB at p_T < 4 GeV/c in the calculation is due to the detailed balance effect in the effective parton energy loss. In this region, one expects the fragmentation picture to gradually lose its validity and is taken over by other non-perturbative effects, especially for kaons and baryons. As a consequence, the (K + p)/p ratio in central Au + Au collisions is significantly larger than in peripheral Au + Au or p + p collisions. To take into account this effect, we add a nuclear dependent (proportional to áN_binaryñ) soft component to kaon and baryon fragmentation functions so that (K + p)/p » 2 at p_T ~ 3 GeV/c in the most central Au + Au collisions and approaches its p + p value at p_T > 5 GeV/c. The resultant suppression for total charged hadrons (dot-dashed) and the centrality dependence agree well with the STAR data. One can directly relate h^± and p⁰ suppression via the (K + p)/p ratio: = [1 + (K + p)/p]_AA/[1 + (K + p)/p]_pp. It is clear from the data that (K + p)/p becomes the same for Au + Au and p + p collisions at p_T > 5 GeV/c. To demonstrate the sensitivity to the parameterized parton energy loss in the intermediate p_T region, we also show in 0-5% centrality (dashed line) for m = 2.0 GeV and ₀ = 2.04 GeV/fm without the soft component.

In the same LO pQCD parton model, one can also calculate di-hadron spectra,

for two back-to-back hadrons from independent fragmentation of the back-to-back jets. Let us assume hadron h₁ is a triggered hadron with p_T1 = . One can define a hadron-triggered fragmentation function (FF) as the back-to-back correlation with respect to the triggered hadron:

similarly to the direct-photon triggered FF [12, 13] in g-jet events. Here, z_T = p_T/ and integration over |y_1,2| < Dy is implied. In a simple parton model, the two jets should be exactly back-to-back. The initial parton transverse momentum distribution in our model will give rise to a Gaussian-like angular distribution. In addition, we also take into account transverse momentum smearing within a jet using a Gaussian distribution with a width of ák_^ñ = 0.6 GeV/c. Hadrons from the soft component are assumed to be uncorrelated.

Shown in Fig. 4 are the calculated back-to-back correlations for charged hadrons in Au+Au collisions as compared to the STAR data [25]. The same energy loss that is used to calculate single hadron suppression and azimuthal anisotropy can also describe well the observed away-side hadron suppression and its centrality dependence. In the data, a background B(p_T)[1 + (p_T) cos(2Df)] from uncorrelated hadrons and azimuthal anisotropy has been subtracted. The value of v₂(p_T) is measured independently while B(p_T) is determined by fitting the observed correlation in the region 0.75 < |f| < 2.24 rad [25].

With both the single spectra and dihadron spectra, the extracted average energy loss in this model calculation for a 10 GeV quark in the expanding medium is ádE/dLñ_1d» 0.85 ± 0.24 GeV/fm, which is equivalent to dE₀/dL » 13.8 ± 3.9 GeV/fm in a static and uniform medium over a distance R_A = 6.5 fm. This value is about a factor of 2 larger than a previous estimate [22] because of the variation of gluon density along the propagation path and the more precise RHIC data considered .

4 Summary

In summary, with the recent measurements of high p_T hadron spectra and jet correlations in d + Au collisions at RHIC, the observed jet quenching in Au + Au collisions has been established as a consequence of final state interaction between jets and the produced dense medium. The collective body of data: suppression of high p_T spectra and back-to-back jet correlation, high p_T anisotropy and centrality dependences of the observables, indicate that the cause of the jet quenching is due to parton energy loss rather than final state absorption of hadrons.

A simultaneous phenomenological study of the suppression of hadron spectra and back-to-back correlations, and high-p_T azimuthal anisotropy in high-energy heavy-ion collisions within a single LO pQCD parton model incorporating current theoretical understanding of parton energy loss can describe the experimental data of Au + Au collisions very well. With HIJING (EKS) parton shadowing, the extracted average energy loss for a 10 GeV quark in the expanding medium is ádE/dLñ_1d» 0.85(0.99) ± 0.24 GeV/fm, which is equivalent to dE₀/dL » 13.8(16.1) ±3.9 GeV/fm in a static and uniform medium over a distance R_A = 6.5 fm. This value is about a factor of 2 larger than a previous estimate [22] because of the variation of gluon density along the propagation path and the more precise RHIC data considered here .

Acknowledgement

This work is supported by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Divisions of Nuclear Physics, of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098

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Received on 12 December, 2003

Publication Dates

• Publication in this collection
  28 Jan 2005
• Date of issue
  Dec 2004

History

• Received
  12 Dec 2003