Extracting the temperature dependence in high- p ⊥ particle energy loss
Stefan Stojku, Bojana Ilic, Marko Djordjevic, Magdalena Djordjevic
EExtracting the temperature dependence in high- p ⊥ particleenergy loss Stefan Stojku , Bojana Ilic , Marko Djordjevic and Magdalena Djordjevic Institute of Physics Belgrade, University of Belgrade, Belgrade, Serbia Faculty of Biology, University of Belgrade, Belgrade, Serbia (Dated: July 16, 2020)The suppression of high- p ⊥ particles is one of the main signatures of parton energyloss during its passing through the QGP medium, and is reasonably reproduced bydifferent theoretical models. However, a decisive test of the reliability of a certainenergy loss mechanism, apart from its path-length, is its temperature dependence.Despite its importance and comprehensive dedicated studies, this issue is still await-ing for more stringent constraints. To this end, we here propose a novel observable toextract temperature dependence exponent of high- p ⊥ particle’s energy loss, based on R AA . More importantly, by combining analytical arguments, full-fledged numericalcalculations and comparison with experimental data, we argue that this observable ishighly suited for testing (and rejecting) the long-standing ∆ E/E ∝ L T paradigm.The anticipated significant reduction of experimental errors will allow direct extrac-tion of temperature dependence, by considering different centrality pair in A + A collisions (irrespective of the nucleus size) in high- p ⊥ region. Overall, our resultsimply that this observable, which reflects the underlying energy loss mechanism, isvery important to distinguish between different theoretical models. I. INTRODUCTION
The main goal of ultra-relativistic heavy ion program [1–4] at RHIC and LHC is inferringthe features of the created novel form of matter − Quark-Gluon Plasma (QGP) [5, 6], whichprovides an insight into the nature of the hottest and densest known medium. Energyloss of rare high- p ⊥ partons traversing the medium is considered to be one of the crucialprobes [7] of the medium properties, which also had a decisive role in QGP discovery [8].Comparison of predictions stemming from different energy loss models with experimental a r X i v : . [ nu c l - t h ] J u l data, tests our understanding of the mechanisms underlying the jet-medium interactions,thereby illuminating the QGP properties. Within this, an important goal presents a searchfor adequate observables for distinguishing the energy loss mechanisms.Connected to this, it is well-known that the temperature ( T ) dependence of the energyloss predictions is directly related to the underlying energy loss mechanisms, e.g., pQCD radiative energy loss (BDMPS and ASW [9–11]; GLV [12]; AMY [13]; HT [14]; and someof their extensions [15–19]) is typically considered to have cubic T dependence ( T , stem-ming from entropy, or energy density dependence), while collisional energy loss [7, 20–22]is generally considered to be proportional to T . Additionally, AdS/CFT-motivated jet-energy loss models [23, 24] display even quartic ( T ) dependence on temperature. Thedifferent functional dependence on T found in these models are the results of: consideredenergy loss mechanism (elastic or inelastic); different treatment of the QCD medium: fi-nite or infinite size; inclusion or omission of finite temperature effects (i.e., application oftemperature-modified, or vacuum-like propagators). Therefore, assessing the accurate tem-perature dependence is important for disentangling relevant effects for adequate descriptionof leading parton energy loss, and consequently for understanding the QGP properties.For a comprehensive study on temperature (and path-length) dependence of differentenergy loss models we refer the reader to [17]. However, even this systematic study couldn’tsingle out T dependence, as the attempt to simultaneously describe high- p ⊥ R AA and v data within these models requires some more rigorous physical justifications. Moreover, thecurrent error bars at the RHIC and the LHC are still too large to resolve between differentenergy loss models. Having this in mind, we here propose a novel observable to extract thescaling of high- p ⊥ particle’s energy loss on temperature. We expect that this observable willallow direct extraction of T dependence from the data in the upcoming high-luminosity 3 rd run at the LHC, where the error-bars are expected to notably decrease.We also propose high- p ⊥ h ± as the most suitable probe for this study, as the experimen-tal data for h ± R AA are more abundant and with smaller error-bars, compared to heavierhadrons for all centrality classes, where this is also expected to hold in the future. Therefore,in this paper, we concentrate on h ± in 5.02 TeV Pb + Pb collisions at the LHC, with the goalto elucidate this new observable, and test its robustness to medium evolution and collidingsystem size. By combining full-fledged numerical predictions and scaling arguments withinour DREENA [25, 26] framework, this new observable yields the value of temperature depen-dence exponent, which is in accordance with our previous estimate [27]. More importantly,we utilize this observable to question the long-standing ∆ E/E ∝ L T paradigm, used in awide-range of theoretical models [9–12, 14–19]. II. THEORETICAL FRAMEWORK
In this study, we use our state-of-the-art dynamical energy loss formalism [28–30], whichincludes several unique features in modeling jet-medium interactions: (1) The calculationswithin the finite temperature field theory and generalized Hard-Thermal-Loop approach [31](contrary to many models which apply vacuum-like propagators [9, 10, 12, 14]), so thatinfrared divergences are naturally regulated in a highly non-trivial manner; (2) Finite size ofcreated QGP; (3) The QCD medium consisting of dynamical (moving) as opposed to staticscattering centers, which allows the longitudinal momentum exchange with the mediumconstituents; (4) Both radiative [28, 29] and collisional [30] contributions are calculatedwithin the same theoretical framework; (5) The inclusion of finite parton’s mass [32], makingthe formalism applicable to both light and heavy flavor; (6) The generalization to a finitemagnetic mass [33], running coupling [34] and beyond the soft-gluon approximation [35] isperformed.The analytical expression for single gluon radiation spectrum reads [25, 28, 33, 34]: dN rad dxdτ = C ( G ) C R π x Z d q π d k π µ E ( T ) − µ M ( T )[ q + µ E ( T )][ q + µ M ( T )] T α s ( ET ) α s (cid:16) k + χ ( T ) x (cid:17) × (cid:20) − cos (cid:16) ( k + q ) + χ ( T ) xE + τ (cid:17)(cid:21) k + q )( k + q ) + χ ( T ) (cid:20) k + q ( k + q ) + χ ( T ) − kk + χ ( T ) (cid:21) , (1)where k and q denote transverse momenta of radiated and exchanged gluons, respectively, C ( G ) = 3, C R = 4 / C R = 3) for quark (gluon) jet, while µ E ( T ) and µ M ( T ) are elec-tric (Debye) and magnetic screening masses, respectively. Temperature dependent Debyemass [25, 36] is obtained by self-consistently solving Eq. (5) from Ref. [25]. α s is the(temperature dependent) running coupling [25, 34, 37], E is the initial jet energy, while χ ( T ) = M x + m g ( T ), where x is the longitudinal momentum fraction of the initial partoncarried away by the emitted gluon. M is the mass of the propagating parton, while thegluon mass is considered to be equal to its asymptotical mass m g = µ E / √ dE coll dτ = 2 C R πv α s ( ET ) α s ( µ E ( T )) Z ∞ n eq ( | ~ k | , T ) d | ~ k |× (cid:20) Z | ~ k | / (1+ v )0 d | ~ q | Z v | ~ q |− v | ~ q | ωdω + Z | ~ q | max | ~ k | / (1+ v ) d | ~ q | Z v | ~ q || ~ q |− | ~ k | ωdω (cid:21) (2) × (cid:20) | ∆ L ( q, T ) | (2 | ~ k | + ω ) − | ~ q | | ∆ T ( q, T ) | ( | ~ q | − ω )((2 | ~ k | + ω ) + | ~ q | )4 | ~ q | ( v | ~ q | − ω ) (cid:21) , where n eq ( | ~ k | , T ) = Ne | ~ k | /T − + N f e | ~ k | /T +1 is the equilibrium momentum distribution [21] includ-ing gluons, quarks and antiquarks. k is the 4-momentum of the incoming medium parton, v is velocity of the initial jet and q = ( ω, ~ q ) is the 4-momentum of the exchanged gluon. | ~ q | max is provided in Ref. [30], while ∆ T ( q, T ) and ∆ L ( q, T ) are effective transverse andlongitudinal gluon propagators given by Eqs. (3) and (4) from Ref. [25].Despite very complicated temperature dependence of Eqs. (1) and (2), in [27] it was ob-tained that our dynamical energy loss formalism [34] (which accommodates some of uniquejet-medium effects mentioned above) has an exceptional feature of near linear T dependence.That is, while T dependence for radiative energy loss is widely used [9–19], from Eq. (1)it is evident that this simplified relation is reproduced with approximations of using vac-uum gluon propagators (leading to the absence of m g ( T ) from χ expression) and neglectingrunning coupling. It is straightforward to show that in that case leading T dependence is: ∆ E rad E ∝ µ E T ∝ T ( µ E ∝ T ). However, Eq. (1) clearly demonstrates that a more realistic T dependence is far from cubic, where in [27] it was shown that asymptotic T dependenceof our full radiative energy loss is between linear and quadratic.Additionally, commonly overlooked (due to being smaller compared to radiative at high- p ⊥ ) collisional energy loss, must not be neglected in suppression predictions [39]. Moreover,widely-used dominant T dependence of collisional energy loss [7, 20–22] can also be shownto be a consequence of: i) using tree-level diagrams, and consequently introducing artificialcut-offs to non-physically regulate ultraviolet (and infrared) divergencies (e.g., in [7]) in thehard momentum transfer sector [21]; or ii) considering only soft momentum exchange [20].That is, it is straightforward to show that Eq. (2) recovers leading T dependence from [20]if: 1) only soft gluon sector is considered, with upper limit of integration artificially setto | ~ q | max ; 2) only forward emission is accounted for ( ω > T dependence of our collisional energy loss (Eq. (2)) reduces not to commonly considered quadratic, but rathernearly linear dependence for asymptotically large p ⊥ . Therefore, a state-of-the-art energyloss model leads to a much slower growth of the energy loss with temperature compared tocommon paradigm, where the widely assumed faster growth can be reproduced only throughquite drastic simplifying assumptions.Since the goal of this paper is the extraction of the temperature dependence exponent ofthe energy loss, this study will furthermore provide an opportunity to test our dynamicalenergy loss formalism on more basic level. III. NUMERICAL FRAMEWORK
In this paper, the predictions are generated within our fully optimized DREENA [25, 26]numerical framework, comprising: i) Initial parton momentum distribution [40]; ii)
Energyloss probability based on our dynamical energy loss formalism [28–30] (discussed in theprevious section), which includes multi-gluon [41] and path-length fluctuations [42]. Thepath-length fluctuations are calculated according to the procedure provided in [43], (seealso [26]); and iii)
Fragmentation functions [44].In generating numerical predictions for comparison with 5.02 TeV Pb + Pb data fordifferent centrality classes, we use no fitting parameters , i.e., all the parameters correspondto standard literature values, and for their values we refer reader to [26].In the first part of our study, the average temperature for each centrality class is obtainedaccording to the procedure outlined in Refs. [26, 45]. Similarly, initial temperature ( T )for each centrality, in a part of this study where we test the sensitivity of the obtainedconclusions to the medium evolution, is estimated in accordance with [25]. IV. RESULTS AND DISCUSSION
In this section, we first address the choice of the suitable observable for extracting energyloss temperature dependence. For this purpose, an observable which is sensitive only tothe details of jet-medium interactions (to facilitate extraction of T dependence), ratherthan the subtleties of medium evolution (to avoid unnecessary complications and ensurerobustness), would be optimal. R AA has such features, since it was previously reportedthat it is very sensitive to energy loss effects [39] and the average medium properties, i.e.,average temperature, while being practically insensitive to the details of medium evolution(in distinction to v ) [25, 26, 46–48]. Therefore, it is plausible that the appropriate observableshould be closely related to R AA .Our theoretical and numerical approaches described above (where the dynamical en-ergy loss explicitly depends on T ), are implemented in a fully optimized DREENA frame-work [25, 26], which makes it suitable for this study. To more easily interpret the obtainedresults, we start from constant T medium, i.e., DREENA-C [26] and continue toward evolv-ing medium case, i.e., DREENA-B framework [25]. We here exploit that DREENA-C andDREENA-B are analytically trackable, allowing to derive appropriate scaling behavior. Toadditionally test the obtained results, we will then use our DREENA-A (A stands for Adap-tive) framework, which employs full 3+1D hydrodynamics evolution [49].With the intention of extracting simple functional dependence on T (of the otherwiseanalytically and numerically quite complex dependence of the fractional energy loss, seeEqs. (1, 2)), we first provide the scaling arguments. These scaling (analytical) argumentswill then be followed by a full-fledged numerical analysis. Namely, in [25, 26, 41, 48] it wasshown that, at very large values of transverse momentum p ⊥ and/or in peripheral collisions,the following estimates can be made:∆ E/E ≈ ηT a L b ,R AA ≈ − ξT a L b , (3)where η denotes a proportionality factor, depending on initial parton transverse momentumand its flavor, while ξ = ( n − η/
2, where n is the steepness of a power law fit to theinitial transverse momentum distribution, i.e., dσ/dp ⊥ ∝ p − n ⊥ . T and L denote the averagetemperature (of the QCD medium) along the jet path and the average path length traversedby the energetic parton. The scaling factors for temperature and path-length energy lossdependence are denoted as a and b , respectively.We next formulate the following quantity R TAA , with the goal to isolate the temperaturedependence: R TAA = 1 − R AA − R refAA , (4)which presents (1 − R AA ) ratio for a pair of two different centrality classes. The centralityclass that corresponds to R refAA (i.e., the quantity in the denominator) is denoted as the refer-ent centrality, and is always lower (corresponding to more central collision) than centrality - - - - - - - - ln ( T / T ref ) l n ( L / L r e f ) FIG. 1: ln (
L/L ref ) vs. ln ( T /T ref ) in 5.02 TeV Pb+Pb collisions at the LHC for various cen-trality pairs. The referent centralities (for quantities in denominators) acquire one of the followingvalues: (5 − , − , − , − , − − in the numerator. We term this new quantity, given by Eq. (4), as a temperature dependentsuppression ratio ( R TAA ), which we will further elucidate below.Namely, by using Eq. (3), it is straightforward to isolate average T and average path-length dependence of R TAA : R TAA = 1 − R AA − R refAA ≈ ξT a L b ξT aref L bref = TT ref ! a · LL ref ! b , (5)which in logarithmic form reads:ln( R TAA ) = ln − R AA − R refAA ! ≈ a ln TT ref ! + b ln LL ref ! . (6)However, the remaining dependence of the newly defined quantity on the path length isundesired for the purpose of this study. So, in order to make use of the previous equation,we first test how the two terms on the right-hand side of Eq. (6) are related. To this end,in Fig. 1 we plot ln( L/L ref ) against ln(
T /T ref ) for several combinations of centralities, asdenoted in the caption of Fig. 1. For a particular centrality class, L and T are calculatedby averaging over path-length distributions given in [26].Conveniently, Fig. 1 shows a linear dependence ln( L/L ref ) ≈ k ln( T /T ref ), with k ≈ . R TAA ) ≈ ( a + kb ) ln TT ref ! , (7)so that with f = a + kb : R TAA ≈ (cid:18) TT ref (cid:19) f , (8)where this simple form facilitates extraction of a .In Eq. (8), R TAA depends solely on T and effectively temperature dependence exponent a (as k and b [48] are known), which justifies the use of ”temperature-sensitive” term withthis new quantity. Therefore, here we propose R TAA , given by Eq. (4), as a new observable,which is highly suitable for the purpose of this study.The proposed extraction method is the following: We use our full-fledged DREENA-C numerical procedure to generate predictions for R AA and thereby for the left-hand sideof Eq. (8). Calculation of average T is already outlined in the previous section. We willgenerate the predictions with full-fledged procedure, where we expect asymptotic scalingbehavior (given by Eq. (8)) to be valid at high p ⊥ ≈
100 GeV. Having in mind that valuesof k and b parameters have been extracted earlier, the temperature dependence exponent a in very high- p ⊥ limit can then be estimated from slope ( f ) of a ln( R TAA ) vs. ln( T /T ref )linear fit, done for a variety of centrality pairs.However, before embarking on this task, we first verify whether our predictions of R TAA fordifferent centrality classes, based on the full-fledged DREENA-C framework, are consistentwith the available experimental data. In Fig. 2 we compare our R TAA vs. p ⊥ predictionsfor charged hadrons with corresponding 5.02 TeV Pb + Pb LHC data from ALICE [50],CMS [51] and ATLAS [52], for different centrality pairs as indicated in the upper-left cornerof each plot. Despite the large error bars, for all centrality pairs we observe consistencybetween our DREENA-C predictions and experimental data, in p ⊥ region where our formal-ism is applicable ( p ⊥ (cid:38)
10 GeV). Moreover, we also notice the flattening of each curve withincreasing p ⊥ ( ∼ R TAA and ln(
T /T ref ), which we test in Fig. 3. Note thatall quantities throughout the paper are determined at p ⊥ = 100 GeV, and by calculating - % - % - % - % - % - % R AA T - % - % - % - % - % - % - % - % p ⊥ ( GeV ) - % - % - % - % FIG. 2:
Charged hadron R TAA for different pairs of centrality classes as a function of p ⊥ . The predictions generated within our full-fledged suppression numerical procedure DREENA-C [26] (black curves with corresponding gray bands) are compared with ALICE [50] (red triangles),CMS [51] (blue squares) and ATLAS [52] (green circles) data. The lower (upper) boundary of eachband corresponds to µ M /µ E = 0 . µ M /µ E = 0 . R TAA for various centrality pairs (see figure captions) within full-fledged DREENA procedure.Remarkably, from Fig. 3, we observe that ln( R TAA ) and ln(
T /T ref ) are indeed linearly related,which confirms the validity of our scaling arguments at high- p ⊥ and the proposed procedure.Linear fit to calculated points in Fig. 3 leads to the proportionality factor f = a + kb =0 - - - - - - - - ln ( T / T ref ) l n ( R AA T ) FIG. 3: ln ( R TAA ) vs. ln ( T /T ref ) relation. ln( R TAA ) and ln(
T /T ref ) are calculated from full-fledged DREENA-C framework [26], for h ± at p ⊥ = 100 GeV in 5.02 TeV Pb+Pb collisions at theLHC for different centrality pairs. The referent centrality values are: (10 − , − , − , − − . ∼
4. This small value of f would lead to k smaller than 1 if (commonly assumed) a = 3and b = 2 are used. Such k value seems however implausible, as it would require ( T /T ref )to change more slowly with centrality compared to (
L/L ref ).More importantly, the temperature exponent can now be extracted ( b ≈ . a ≈ .
2. This indicates that temperature dependence of energetic particleenergy loss (at very high p ⊥ ) is close to linear (see Eq. (3)), that is, certainly not quadratic orcubic, as commonly considered. This is in accordance with previously reported dependenceof fractional dynamical energy loss on T to be somewhere between linear and quadratic [27],and as opposed to commonly used pQCD estimate a = 3 for radiative [9–19] (or even a = 2for collisional [7, 20–22]) energy loss.The extraction of T dependence, together with previously estimated path-length depen-dence [48], within DREENA framework, allows utilizing this new observable R TAA in discrim-inating between energy loss models, with the aim of better understanding QGP properties.To this end, in Fig. 4, we i ) Test sensitivity of R TAA on different medium evolutions (con-stant temperature, 1D Bjorken [53] and full 3+1D hydrodynamics [49]). ii ) Compare theasymptote derived from this study (( T /T ref ) . · ( L/L ref ) . ), with commonly used estimate1 DREENA - CDREENA - BDREENA - A R AA T - % - % - % - % p ⊥ ( GeV ) R AA T - % - % p ⊥ ( GeV ) - % - % FIG. 4:
The discriminative power of R TAA quantity in resolving energy loss mech-anism.
Four panels in Fig. 2 are extended to include comparison of our asymptotic scal-ing behavior (
T /T ref ) . · ( L/L ref ) . (gray dashed horizontal line) with common assumption( T /T ref ) · ( L/L ref ) (gray dot-dashed horizontal line). The figure also shows comparison of R TAA sobtained by three different numerical frameworks: constant temperature DREENA-C (black curve),1D Bjorken expansion DREENA-B [25] (cyan curve) and full 3+1D hydrodynamics evolution [49]DREENA-A (magenta curve). The remaining labeling is the same as in Fig. 2. of (
T /T ref ) · ( L/L ref ) .Several conclusions can be drawn from Fig. 4: i ) With respect to different models of QGPexpansion, we see that, as expected, obtained R TAA results are similar, i.e., not very sensitiveto the details of the medium evolution. As in DREENA-C (and DREENA-B, see the nextsubsection) the temperature dependence can be analytically tracked (which is however notpossible in more complex DREENA-A), this result additionally confirms that DREENA-Cframework is suitable for the extraction of energy loss temperature dependence. ii ) Ideally,2 T dependence exponent could be directly extracted from experimental data, by fitting astraight line to very high- p ⊥ part ( ∼
100 GeV) of R TAA for practically any centrality pair.However, the fact that data from different experiments (ALICE, CMS and ATLAS) are notideally consistent, and that the error-bars are quite sizeable, currently prevents from suchdirect extraction. The error-bars in the upcoming high-luminosity 3 rd run at the LHC arehowever expected to significantly decrease, which would enable the direct extraction of theexponent a from the data. iii) From Fig. 4 it importantly follows that even the current largeexperimental uncertainties seem to indicate inadequacy of commonly exploited energy lossdependence ∝ T L , practically for all considered centrality pairs. This is consistent withour other results, which all indicate breaking of the long-standing paradigm of energy losstemperature and path-length dependence. Future increase in measurements precision couldprovide the confidence to this conclusion and resolve the exact form of these dependenciesfrom the data, through our proposed observable. This discriminative power of R TAA quantityhighlights its importance in understanding the underlying energy loss mechanisms in QGP.
A. Effects of medium evolution
While in Fig. 4 we showed that R TAA results are robust with respect to the mediumevolution, the analytical procedure for extracting temperature dependence is different inDREENA-C and DREENA-B frameworks. Comparing scaling factors extracted from thesetwo procedure, can be used to test reliability of the proposed procedure. In this subsection,we consequently utilize DREENA-B framework [25], where medium evolution is introducedthrough Bjorken 1D hydrodynamical expansion [53], i.e., there is the following functionaldependence of T on path-length: T = T · (cid:18) τ l (cid:19) / , (9)where T and τ = 0 . R TAA (given by Eq. (4))in evolving medium (for coupled T and l , where l stands for traversed path length) reads: R TAA = R L T a l b − dl R L ref ( T ref ) a ( l ref ) b − dl ref = T a τ a/ R L l b − l a/ dlT a ,ref τ a/ R L ref l ref ) b − ( l ref ) a/ dl ref = T T ,ref ! a · LL ref ! b − a , (10)3 - - - - - - - - ln ( T / T ) l n ( L / L r e f ) FIG. 5: ln (
L/L ref ) vs. ln ( T /T ,ref ) for various pairs of centralities in evolving medium. The assumed centrality pairs are the same as in Fig. 1. The red solid line corresponds to the linearfit to the values. where we used Eq. (9). Again, we assess whether there is a simple relation between loga-rithms of (now initial) temperature ratio and average path-length ratio for different centralitypairs. Similarly to constant T case, from Fig. 5 we infer linear dependence between thesetwo quantities, where slope coefficient now acquires the value κ ≈ .
3. Thus, we may write: LL ref = T T ,ref ! κ = ⇒ T T ,ref = LL ref ! /κ , (11)which ensures that the R TAA quantity has a very simple form, depending only on averagepath-length and exponents a, b and κ : R TAA = LL ref ! aκ + b − a . (12)If we substitute value of a ≈ . T medium case, previously esti-mated b ≈ . κ ≈ .
3, we arrive at the following estimate: R TAA = LL ref ! . = ⇒ ln( R TAA ) = 1 . · ln LL ref ! . (13)This equation is quite suitable for testing the robustness of the procedure for extracting theexponent a to inclusion of the evolving medium. Namely, value 1.93 in Eq. (13) stems fromcoefficient a , which is extracted from constant T medium case. On the other hand, if we plotln( R TAA ), generated by full-fledged DREENA-B calculations (i.e., in evolving medium) whichis fundamentally different from DREENA-C, against ln(
L/L ref ) for variety of centrality4 - - - - - - - - ln ( L / L ref ) l n ( R AA T ) FIG. 6:
Testing the validity of our procedure for temperature dependence extractionin the case of expanding QCD medium. ln( R TAA ) vs. ln( L/L ref ) for h ± at p ⊥ = 100 GeV fordifferent pairs of centrality classes is plotted. Suppression predictions are obtained from full-fledgedDREENA-B [25] calculations. Referent centrality values are: (5 − , − , − , − , − , − − pairs, again we observe a linear dependence (see Fig. 6). Furthermore, a linear fit to thevalues surprisingly yields the exact same slope coefficient value of 1.93.Consequently, the procedure of extracting temperature dependence exponent, introducedfirst in the case of constant T medium, is applicable to the expanding medium as well. Thedisplayed consistency of the results provide confidence to general applicability of the proce-dure presented in this paper (suggesting robustness to the applied model of bulk medium)and supports the reliability of the value of extracted T dependence exponent a ≈ . B. Effects of colliding system size
We below extend our analysis to smaller colliding systems in order to assess generality ofthe conclusions presented above. Smaller colliding systems, such as Xe + Xe, Kr + Kr, Ar+ Ar and O + O are important to gradually resolve the issue of QGP formation in smallsystems (such as pA), and (except Xe + Xe, which is already in a run) are expected to bea part of the future heavy-ion program at the LHC [56].As already discussed in [48], for this analysis within DREENA-C framework [26] (which5 Pb + PbXe + XeKr + KrAr + ArO + O p ⊥ ( GeV ) R AA T - % - % - % - % - % - % FIG. 7:
Dependence of R TAA on a system size as a function of p ⊥ . Predictions for h ± generated within full-fledged DREENA-C [26] suppression numerical procedure are compared fordifferent colliding systems: Pb+Pb, Xe + Xe, Kr + Kr, Ar + Ar, O + O (for lines specificationsee legend). For clarity, the results are shown only for three centrality pairs, as specified in plot,although checked for all available centrality classes. Magnetic to electric mass ratio is fixed to µ M /µ E = 0 . we employ here for simplicity, since the robustness of the procedure to evolving mediumwas demonstrated above) note that R AA depends on: i) initial high- p ⊥ parton distribution, ii) medium average T and iii) path-length distribution. For different colliding systems(probably at slightly different √ s NN = 5 .
44 TeV compared to Pb + Pb system) we employthe same high- p ⊥ distributions, since in [27] it was shown that for almost twofold increaseof the collision energy (from 2.76 TeV to 5.02 TeV) the change in corresponding initialdistributions results in a negligible change (approximately 5%) in suppression.Regarding the average temperature, one should note that T is directly proportional tothe charged particle multiplicity, while inversely proportional to the size of the overlaparea and average medium size [26, 45, 48, 57], i.e., T ∝ ( dN ch /dηA ⊥ L ) / . The transition tosmaller colliding systems, for a certain fixed centrality class, leads to the following scaling: A ⊥ ∝ A / , L ∝ A / [58, 59] and dN ch /dη ∝ N part ∝ A [60, 61], where A denotes atomicmass. This leads to T ∼ ( AA A ) / ∼ const , that is, we expect that average temperaturedoes not change, when transitioning from large Pb + Pb to smaller systems, for a fixedcentrality class. Lastly, path-length distributions for smaller systems and each centralityclass are obtained in the same manner as for Pb+Pb [26], and are the same as in Pb + Pb6collisions up to a rescaling factor of A / .By denoting all quantities related to smaller systems with a tilde, with Pb + Pb quantitiesdenoted as before, it is straightforward to show that the temperature sensitive suppressionratio for smaller systems satisfies: e R TAA = 1 − e R AA − e R refAA ≈ e T a e L b e T aref e L bref ≈ T a L b T aref L bref · ( e A/A ) b/ ( e A/A ) b/ = 1 − R AA − R refAA = R TAA , (14)where we used: e T = T and e L/L = ( e A/A ) / .To validate equality of R TAA s for different system sizes, predicted by analytical scalingbehavior (Eq. (14)), in Fig. 7 we compare our full-fledged R TAA predictions for h ± in Pb +Pb system with those for smaller colliding systems. We observe that, practically irrespectiveof system size, R TAA exhibits the same asymptotical behavior at high- p ⊥ . This not onlyvalidates our scaling arguments, but also demonstrates the robustness of the new observable R TAA to system size. Consequently, since for fixed centrality range, T should remain thesame for all these colliding systems, we obtained that temperature dependence exponent a should be the same independently of considered colliding system (see Fig. 3). Therefore,the proposed procedure for extracting the temperature dependence of the energy loss is alsorobust to the collision system size. As a small exception, O + O system exhibits slightdeparture from the remaining systems at high- p ⊥ , which might be a consequence of the factthat this system is significantly smaller than other systems considered here. V. CONCLUSIONS AND OUTLOOK
One of the main signatures of high- p ⊥ particle’s energy loss, apart from its path-length, isits temperature dependence. Although extensive studies on both issues were performed, notuntil recently the path dependence resolution was suggested [48]. Here we proposed a newsimple observable for extracting temperature dependence of the energy loss, based on one ofthe most common jet quenching observable − the high- p ⊥ suppression. By combining full-fledged numerical calculations with asymptotic scaling behavior, we surprisingly obtainedthat temperature dependence is nearly linear, i.e., far from quadratic or cubic, as commonlyassumed. Further, we verified its robustness and reliability on colliding system size andevolving QGP medium. 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