We review recent calculations of the probability that a hard parton radiates
an additional energy fraction due to scattering in spatially extended matter,
and we discuss their application to the suppression of leading hadron spectra
in heavy ion collisions at collider energies.Comment: 10 pages, 5 figures, Talk given at Workshop on Strong and Electroweak
  Matter (SEWM 2002), Heidelberg, Germany, 2-5 Oct 200

Wiedemann, Urs Achim

English

arXiv

We review recent calculations of the probability that a hard parton radiates an additional energy fraction due to scattering in spatially extended matter, and we discuss their application to the suppression of leading hadron spectra in heavy ion collisions at collider energies.We review recent calculations of the probability that a hard parton radiates an additional energy fraction due to scattering in spatially extended matter, and we discuss their application to the suppression of leading hadron spectra in heavy ion collisions at collider energies.We review recent calculations of the probability that a hard parton radiates an additional energy fraction due to scattering in spatially extended matter, and we discuss their application to the suppression of leading hadron spectra in heavy ion collisions at collider energies.We review recent calculations of the probability that a hard parton radiates an additional energy fraction due to scattering in spatially extended matter, and we discuss their application to the suppression of leading hadron spectra in heavy ion collisions at collider energies.We review recent calculations of the probability that a hard parton radiates an additional energy fraction ΔE due to scattering in spatially extended matter, and we discuss its applications for the suppression of leading hadrons in heavy ion collisions at collider energies

CERN Document Server

arXiv:hep-ph/0302194v1  21 Feb 2003JET TOMOGRAPHY IN HEAVY ION COLLISIONSURS ACHIM WIEDEMANNCERN TH Division, CH-1211 Geneva 23E-mail: Urs.Wiedemann@cern.chWe review recent calculations of the probability that a hard parton radiates anadditional energy fraction ∆E due to scattering in spatially extended matter, andwe discuss their application to the suppression of leading hadron spectra in heavyion collisions at collider energies.1. IntroductionExperiments at the Relativistic Heavy Ion Collider RHIC in Brookhavenshow for the first time a significant quenching of high-p⊥ leading hadronspectra. In particular, transverse momentum spectra for neutral pion 1,2and charged hadron 1,3 are suppressed if compared to spectra in p+p col-lisions rescaled by the number of binary collisions. This suppression ismost pronounced (up to a factor ∼ 5) in central Au+Au collisions andsmoothly approaches the binary scaling case with decreasing centrality. Theazimuthal anisotropy v2(p⊥) of hadroproduction stays close to maximal upto the highest transverse momentum 4. Moreover, the disappearance ofback-to-back high-p⊥ hadron correlations 5 provides an additional indica-tion that final state medium effects play a decisive role in hadroproductionup to p⊥ ∼ 10 GeV.Parton energy loss has been proposed to account for the small nuclearmodification factor 6, the azimuthal anisotropy 7 and the disappearanceof dijets 8,9. Several studies (see references given in 10) indicate, how-ever, that in the kinematical regime relevant for RHIC (p⊥ < 12 GeV),p⊥-broadening, shadowing, formation time and possibly other effects con-tribute significantly to the high-p⊥ nuclear modification as well.Here, we discuss the status of parton energy loss calculations and theircomparison to data in the kinematical p⊥-range probed at RHIC.122. Medium-induced gluon radiation from a static mediumSeveral groups 11,12,13,14 calculated recently the modification of the ele-mentary splitting processes q → qg and g → gg due to multiple scattering.The inclusive energy distribution of gluon radiation off an in-medium pro-duced parton takes the form 13,15ωdIdω=αs CR(2π)2 ω22Re∫ ∞ξ0dyl∫ ∞yldy¯l∫d2u∫ χω0d2k e−ik⊥·u e−12∫∞y¯ldξ n(ξ) σ(u)× ∂∂y· ∂∂u∫ u=r(y¯l)y=0Dr exp[i∫ y¯lyldξω2(r˙2 − n(ξ)σ (r)i ω)], (1)where gluons of transverse momentum k⊥ < χω are included. The radia-tion of hard quarks or gluons differs by the Casimir factor CR = CF or CA,respectively. The properties of the medium enter eq. (1) by the product ofthe time-dependent density n(ξ) of scattering centers times the strength ofa single elastic scattering σ(r) = 2∫dq(2pi)2 |a(q)|2(1− eiq·r), where |a(q)|2denotes the elastic high-energy cross section of a single scatterer.For explicit calculations, one has to approximate the path integral in(1). This is done either by a saddle point approximation 12 obtained in themedium-induced soft multiple scattering approximation,n(ξ)σ(r) ≃ 12qˆ(ξ) r2 . (2)Here, the only medium-dependent quantity is the transport coefficient 16qˆ(ξ) which characterizes the transverse momentum squared µ2 transferredto the projectile per mean free path λ. Alternatively, one can proceed inthe opacity expansion which amounts to expanding the path integral in (1)in powers of the elastic scattering center 13,14K(r, yl; r¯, y¯l) = Dr exp[i∫ y¯lyldξω2(r˙2 − n(ξ)σ (r)i ω)](3)= K0(r, yl; r¯, y¯l)−z′∫zdξ n(ξ)∫dρK0(r, yl; ρ, ξ)σ(ρ)2K0(ρ, ξ; r¯, y¯l) + . . . .Fig.1 compares numerical results for the gluon energy distribution ob-tained in these two approximations. Qualitatively, this figure can be under-stood by estimating to what degree initial state gluons decohere from thepartonic projectile. Decoherence depends on the relative phase ϕ accumu-lated by the gluon due to scattering. This phase grows with the transversemomentum accumulated by the emitted gluon.3Figure 1. The gluon energy distribution in the multiple soft (LHS) and single hard(RHS) scattering approximation calculated for different values of the kinematical con-straints R, R¯ respectively. Fig. from Ref.10.For multiple soft scattering, we have 17ϕ =〈k2⊥2ω∆z〉∼ qˆ L2ωL =ωcω, (4)which defines the “characteristic gluon frequency” ωc =12 qˆ L2. Given thenumber Ncoh of scattering centers which add coherently in the gluon phase(4), one find k2⊥ ≃ Ncoh µ2. With the coherence time of the emitted gluon,tcoh ≃ ωk2⊥≃√ωqˆ and Ncoh =tcohλ =√ωµ2 λ , one estimates for the gluonenergy spectrum per unit pathlengthωdI(mult)dω dz≃ 1NcohωdI1 scattdω dz≃ αstcoh≃ αs√qˆω. (5)This 1/√ω-energy dependence agrees with the small-ω behaviour in Fig.1.It is cut off for ω > ωc where the phase (4) is smaller than unity and thereduced decoherence suppresses gluon emission.For single hard scattering (N = 1 opacity expansion) with momentumtransfer µ, decoherence occurs if the typical gluon formation time t¯coh =2ωµ2is smaller than the typical distance L between the production point of theparton and the position of the scatterer. The relevant phase isγ =Lt¯coh≡ ω¯cω, (6)4which defines the characteristic gluon energy ω¯c =12µ2 L. The gluon energyspectrum per unit pathlength can be estimated in terms of the coherencetime t¯coh,ωdIN=1dω dz≃ αst¯coh≃ αs µ2ω. (7)The full calculation in Fig.1 agrees with this 1/ω-dependence in the rangeω > ω¯c.In QCD, collinear gluons are hard and softer gluons tend to be emit-ted under larger angles. As a consequence, the limitations on transversemomentum phase space translate into a depletion of the infrared regionof the gluon energy distribution, seen in Fig.1. The estimates (5) and (7)determine the dependence on the characteristic gluon energies ωc (ω¯c) only.They do not include the constraint on transverse phase space. The latteris determined by the parameters 18Rχ =12qˆ χ2 L3 = χ2 ωc L , R¯χ =12χ2 µ2 L2 = χ2 ω¯c L . (8)Clearly, keeping ωc (ω¯c) fixed and taking R (R¯) to infinity amounts to thelimit of infinite pathlength. In this limit, the projectile can accumulatean arbitrarily large medium-induced transverse momentum and the limiton transverse momentum phase space is removed. For a medium of finitesize, however, gluon emission at angles Θ2c ≃ 〈k2⊥〉medω2 is suppressed sincethe emitted gluons are sensitive to the kinematical constraint k⊥ ≤ O(ω).In the multiple soft scattering approximation, this translates intoΘ2c ≃√ωqˆω2≃(ωωc)−3/21R∼ 1 =⇒ ωˆωc∝(1R)2/3. (9)The position of the maximum of ω dI(mult)dω as a function of R is consistentwith this dependence on ωˆ, see Fig. 1.For the single hard scattering approximation, the corresponding esti-mate for the infrared cut-off ωˆ due to transverse momentum phase spaceconstraints readsΘ2c ≃µ2ωˆ2≃( ω¯cωˆ)2 1R¯∼ 1 =⇒ ωˆω¯c∝ 1√R¯. (10)The position of the maximum of ω dIN=1dω in Fig. 1 changes ∝ 1√R¯ , in accor-dance with this estimate. We thus have a semi-quantitative understandingof how phase space constraints deplete the non-perturbative soft region ofthe medium-induced gluon energy distribution. This suppression of the5non-perturbative small-ω contributions helps to make the calculation ofmedium-induced energy loss perturbatively stable.In Ref.10, we compare the single hard and multiple soft scattering ap-proximations of (1) in detail. In general, one finds that the gluon energydistribution is significantly harder in the single hard scattering approxima-tion. For example, the average energy loss ∆E =∫dω ω dIdω receives in themultiple soft scattering approximation a dominant contribution from theregion ω < ωc. In contrast, the dominant contribution comes from the hardregion ω > ω¯c in the single hard scattering approximation.Remarkably, although both approximations emphasize different kine-matical regions, the observable results are quantitatively comparable ifcomparable sets of model parameters are used. To relate the model param-eters in both approximations, one observes that up to logarithmic accuracyµ2 n0 L ≃ qˆ L, where n0L determines the average number of scatteringcenters within the in-medium pathlength L. This impliesR ≃ (n0L) R¯ , ωc ≃ (n0L) ω¯c . (11)In the multiple soft scattering approximation, one uses the transport co-efficient qˆ to characterize the average transverse momentum squared perunit pathlength. In the opacity expansion, one specifies not only this infor-mation, but additionally the average number of scattering centers in whichthis momentum transfer takes place. As a consequence, the opacity expan-sion has one additional model parameter n0L. Numerically, we find thatfor the choice n0L = 3 in (11), the energy density distribution, quench-ing weight and even the angular dependence of the average energy loss arequantitatively comparable 10. We found deviations from this relation forthose quantities for which due to kinematical constraints only the regionω < ωc contributes in both approximations, see below. This is in particularthe case for the comparison to RHIC data in Fig.4, where ω < E < ωc.3. Quenching WeightsIf gluons are emitted independently by a hard parton, then the probabilityP (∆E) that this parton loses an additional energy fraction ∆E can becalcualted from the normalized sum of the emission probabilities for anarbitrary number of n gluons which carry away a total energy ∆E:P (∆E) =∞∑n=01n![n∏i=1∫dωidI(ωi)dω]δ(∆E −n∑i=1ωi)exp[−∫dωdIdω].(12)6The summation over arbitrarily many gluon emissions in (12) can be per-formed analytically by Laplace transformation 17,18,19. In general, onefinds a discrete and a continuous part,P (∆E) = p0 δ(∆E) + p(∆E) . (13)The discrete weight p0 may be viewed as the probability that no additionalgluon is emitted due to in-medium scattering and hence no medium-inducedenergy loss occurs. For finite in-medium pathlength, there is always a finiteprobability p0 that the projectile is not affected by the medium. For infinitein-medium pathlength, however, one finds limR→∞ p0 = 0.Figure 2. The continuous part of the quenching weight (13), calculated in the multiplesoft scattering limit for a hard quark (upper row) or hard gluon (lower row). Fig. fromRef.10.In Fig. 2, we show the continuous part of the quenching weight (13) inthe multiple soft scattering approximation. The parton energy loss ∆E is7generally seen to increase with increasing characteristic gluon energy ωc orincreasing factorR. This implies that ∆E grows with increasing momentumtransfer qˆ from the medium, and with increasing in-medium pathlength L,as naively expected.To understand the negative contributions of the quenching weight forsmall valuesR, one recalls that the gluon energy distribution ω dIdω calculatedin (1) is only the medium-induced modification of a radiation pattern dI(vac)dωwhich occurs in the absence of a medium,ωdI(tot)dω= ωdI(vac)dω+ ωdIdω. (14)By writing the corresponding probabilities (11) in Mellin space, one showsthatP (tot)(∆E) =∫ ∞0dE¯ P (∆E − E¯)P (vac)(E¯) . (15)The probability P (tot)(∆E) is normalized to unity and it is positive definite.In contrast, the medium-induced modification of this probability, P (∆E),is a generalized probability. It can take negative values for some range in∆E, as long as its normalization is unity,∫ ∞0dE¯ P (E¯) = p0 +∫ ∞0dE¯ p(E¯) = 1 . (16)The discrete weight p0 is found to coincide with this normalization condi-tion. A qualitatively similar behaviour of the quenching weights is foundin the single hard scattering approximation.4. Angular dependence of radiation probabilityThe quenching weights discussed above allow to calculate the average en-ergy loss outside an opening angle Θ,〈∆E〉(Θ) =∫dω ωdI>Θdω(ωc, R = ωcL)=∫dE¯ E¯[P (E¯, ωc, R = ωcL)− P (E¯, ωc, Rχ = χ2ωcL)].(17)This calculation is straightforward since a finite emission angle Θ can betaken into account simply by reducing the kinematical constraint k⊥ <ω in (1) to k⊥ < χω, χ = sinΘ. In Fig.3, we compare the resultingangular radiation patterns obtained in the single hard and multiple softscattering approximation. In agreement with the discussion following eq.8(11) above, we find that for comparable sets of model parameters ωc, Rand ω¯c, R¯, n0 L respectively, the multiple soft and single hard scatteringapproximations lead to a comparable angular dependence of 〈∆E〉(Θ) forΘ > 10◦. For smaller angles, deviations persist which one can understandquantitatively 10 and which should be regarded as an intrinsic uncertaintyof this calculation.Figure 3. The average energy loss (17) radiated outside an angle Θ as calculated in themultiple soft (lower three lines) and single hard (upper three lines) scattering approxi-mation. Fig. from Ref.10.5. ApplicationThe quenching weight P (∆E) determines the quenching factor 17Q(p⊥) =∫d∆E P (∆E)(dσvac(p⊥ +∆E)/dp2⊥dσvac(p⊥)/dp2⊥), (18)which determines the reduction of transverse momentum leading hadronspectra due to medium-induced energy loss. Alternatively, the quench-ing weight can be used to determine medium-modified fragmentation func-tions 20D(med)h/q (x,Q2) =∫ 10dǫ P (ǫ)11− ǫ Dh/q(x1− ǫ ,Q2) . (19)9Figure 4. The nuclear modification factor for pi0-production 2 compared to model cal-culations involving parton energy loss only. Curves present the quenching factor (18)and the suppression factor (20) obtained from medium-modified fragmentation functions.They are given in the limiting cases where all parent partons are either quarks (upperlines) or gluons (lower lines). Calculations in the multiple soft scattering approximationuse R = 2000, ωc = 67.5 GeV, corresponding to qˆ = 0.75GeV2fmand L = 6 fm. In thesingle hard scattering approximation, we use R¯ = R, ω¯c = ωc. Fig. from Ref. 10.From these, the reduction of leading transverse momentum spectra canbe calculated 10 by convoluting with the perturbative hard matrix elementswhich are approximately proportional to x6,Rff (p⊥) =x6maxD(med)h/q (xmax, p2⊥)x6maxDh/q(xmax, p2⊥)∣∣∣∣∣p⊥=xmax Eq. (20)In Fig.4, we compare the data on the suppression of the π0-spectrato suppression factors calculated from (18) and (20). We find that bothdefinitions of the suppression factors lead to quantitatively comparable re-10sults. Moreover, the single hard and multiple soft scattering approximationlead to quantitatively comparable results for suitable choices of the modelparameters. Remarkably, we find that in the presence of kinematical con-straints depleting the infrared region of ω dIdω in Fig.1, the p⊥-dependenceof the quenching factor flattens considerably. This feature is necessary tofind in Fig.4 a shallow p⊥-dependence which is consistent with the data.The application of the current calculations of parton energy loss todata below p⊥ < 10 GeV entails significant theoretical uncertainties. Onthe one hand, there are plausible competing physics effects which may berelevant at p⊥ < 10 GeV (see Ref.10 for further discussion). On the otherhand, the high-energy “eikonal” approximation used in the derivation of (1)becomes questionable if ∆E ∼ E which is the case in Fig.4. This shouldmotivate improved calculations of (1) for which the finite energy effects ofthe partonic projectile are taken into account.References1. K. Adcox et al. [PHENIX Collaboration], Phys. Rev. Lett. 88 (2002) 022301.2. S. Mioduszewski [PHENIX Collaboration], arXiv:nucl-ex/0210021.3. C. Adler et al., Phys. Rev. Lett. 89 (2002) 202301.4. C. Adler et al. [STAR Collaboration], Phys. Rev. Lett. 90 (2003) 032301.5. C. Adler et al. [STAR Collaboration], arXiv:nucl-ex/0210033.6. E. Wang and X. N. Wang, Phys. Rev. Lett. 89 (2002) 162301.7. M. Gyulassy, I. Vitev and X. N. Wang, Phys. Rev. Lett. 86 (2001) 2537.8. B. Muller, arXiv:nucl-th/0208038.9. T. Hirano and Y. Nara, arXiv:nucl-th/0301042.10. C. A. Salgado and U. A. Wiedemann, arXiv:hep-ph/0302184.11. R. Baier, Y. L. Dokshitzer, A. H. Mueller, S. Peigne and D. Schiff, Nucl.Phys. B 483 (1997) 291.12. B. G. Zakharov, JETP Lett. 63 (1996) 952.13. U. A. Wiedemann, Nucl. Phys. B 588 (2000) 303.14. M. Gyulassy, P. Levai and I. Vitev, Nucl. Phys. B 594 (2001) 371.15. U. A. Wiedemann, Nucl. Phys. A 690 (2001) 731.16. R. Baier, Y. L. Dokshitzer, A. H. Mueller, S. Peigne and D. Schiff, Nucl.Phys. B 484 (1997) 265.17. R. Baier, Y. L. Dokshitzer, A. H. Mueller and D. Schiff, JHEP 0109 (2001)033.18. C. A. Salgado and U. A. Wiedemann, Phys. Rev. Lett. 89 (2002) 092303.19. F. Arleo, JHEP 0211 (2002) 044.20. X. N. Wang, Z. Huang and I. Sarcevic, Phys. Rev. Lett. 77 (1996) 231.

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