Measurement of $\hat{q}$ in Relativistic Heavy Ion Collisions using di-hadron correlations

The azimuthal width of the di-hadron correlations in p$+$p collisons, beyond the fragmentation transverse momentum, $j_T$, is dominated by $k_T$, the so-called intrinsic transverse momentum of a parton in a nucleon, which can be measured. The predicted azimuthal broadening in A$+$A collisions should produce a larger $k_T$ than in p$+$p collisions. The present work introduces the observation that the $k_T$ measured in p$+$p collisions for di-hadrons with $p_{Tt}$ and $p_{Ta}$ must be reduced to compensate for the energy loss of both the trigger and away parent partons when comparing to the $k_T$ measured with the same di-hadron $p_{Tt}$ and $p_{Ta}$ in Au$+$Au collisions. This idea is applied to a recent STAR di-hadron measurement, with result $\langle{\hat{q}L}\rangle=2.1\pm 0.6$ GeV$^2$. This is more precise but in agreement with a theoretical calulation of $\langle{\hat{q}L}\rangle=14^{+42}_{-14}$ GeV$^2$ using the same data. Assuming a length $\langle{L}\rangle\approx 7$ fm for central Au$+$Au collisions the present result gives $\hat{q}=0.30\pm 0.09$ GeV$^2$/fm, in fair agreement with the JET collaboration result of $\hat{q}\approx 1.2\pm 0.3$ GeV$^2$/fm at initial time $\tau_0=0.6$ fm/c in Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV.


I. INTRODUCTION
In the original BDMPSZ formalism [1, 2], the energy loss of an outgoing parton, −dE/dx, per unit length (x) of a medium with total length L, is proportional to the 4-momentum transfer-squared, q 2 , and takes the form: −dE dx α s q 2 (L) = α s µ 2 L/λ mfp = α sq L , where µ, is the mean momentum transfer per collision, and the transport coefficientq = µ 2 /λ mfp is the 4-momentum-transfer-squared to the medium per mean free path, λ mfp . Additionally, the accumulated momentum-squared, p 2 ⊥W transverse to a parton traversing a length L in the medium is well approximated by [1] p 2 ⊥W ≈ q 2 (L) =q L. This results in a direct snd simple relationship between the parton energy loss (Eq. 1) and the di-jet azimuthal broadening, p 2 ⊥W /2, because only one of the components of the accumulated momentum transverse to the outgoing parton is in the scattering plane, the other being along the beam axis for mid-rapidity di-jets.
It has long been established [3] that even in p+p collisions, or in the initial hard-scattered parton pair in A+A collisions, the mid-rapidity di-jets from hard-scattering are not back-to-back in azimuth but are acollinear with a net transverse momentum, p 2 T pair = 2 k 2 T , where k T is the average 'intrinsic' transverse momentum of a quark or gluon in a nucleon as defined by Feynman, Field and Fox [4]. Again, only the component of p 2 T pair perpendicular to the di-jet axis leads to acoplanarity. Thus in an A+A collision the relationship in Eq. 2 should hold: for azimuthal correlations of a trigger particle with p T t and away-side particles with p T a . It is important to note the in k 2 T pp , introduced here, which indicates that the k T measured in p+p collisions for di-hadrons with p T t and p T a must be reduced to compensate for the energy loss of both the trigger and away parent partons when comparing to the k T calculated with the same di-hadron p T t and p T a in Au+Au collisions. Many experiments at RHIC, including recent experiments with di-hadron [5], jet-hadron [6] and di-jet [7] azimuthal correlations have searched for azimuthal broadening in Au+Au collisions compared to p+p collisions but have not found a significant difference in the azimuthal angular Gaussian width of the away-peaks. Here we shall reexamine the STAR di-hadron measurement [5] in terms of the out of plane component, p out rather than the azimuthal angular width, taking account of the energy lost by the original parton-pair in Au+Au collisions when comparing to the p+p measurement. We shall calculate k 2 T from p+p and Au+Au dihadron measurements with the same trigger particle transverse momentum, p T t , away-side p T a and x h = arXiv:1702.00840v3 [nucl-ex] 25 Jul 2017 p T a /p T t . The di-hadrons are assumed to be fragments of jets with transverse momentap T t andp T a with ratiô x h =p T a /p T t , where z t p T t /p T t is the fragmentation variable, the fraction of momentum of the trigger particle in the trigger jet, and j T is the jet fragmentation transverse momentum. The standard equation at RHIC comes from PHENIX [8], which we write in a slightly different form in Eq. 3: ( Here p out ≡ p T a sin ∆φ (see Fig. 1) and we have taken The variable x h (which STAR calls z T ) is used as an approximation of the variable x E = x h cos φ of the original terminology from the CERN ISR where k T was discovered and measured 40 years ago [3,4,9,10].
FIG. 1. Azimuthal projection of di-jet with trigger particle pT t and associated away-side particle pT a, and the azimuthal components j T φ of the fragmentation transverse momentum. The initial state kT of a parton in each nucleon is shown schematically: one vertical which gives an azimuthal decorrelation of the jets and one horizontal which changes the transverse momentum of the trigger jet.
A. Bjorken parent-child relation and 'trigger-bias' [11] If the fragmentation function of the jet is a function only of the fragmentation variable z and not of the jetp, then the single particle cross section has the same power law shape, d 3 σ/2πp T dp T dy ∝ p −n T , as the parent jet cross section.
Furthermore, large values of z t = p T t /p T t dominate the single-particle cross section (e.g. π 0 ) used as the trigger for the di-hadron (e.g. π 0 -h) measurement. This is called trigger-bias but is valid also for the simple singleparticle measurements. Calculations of z t vs. p T t for π 0 at √ s N N = 200 GeV are given in Ref. [12].
B. The energy loss of the trigger jet from p+p to Au+Au can be measured.
At RHIC, in p+p and Au+Au collisions as a function of centrality the π 0 p T spectra with 5 < p T < ∼ 20 GeV/c all follow the same power law with n ≈ 8.10±0.05 [13]. From the Bj parent-child relation, the energy loss of the trigger jet is found by measuring δp T /p pp T , the shift in the π 0 spectra in Au+Au at a given p T from the T AA corrected p+p cross section (Fig. 2) [14]. The small dropoff of δp T /p pp T for p T ≥ 14 GeV/c indicates a small increase of n with increasing p T . FIG. 2. p pp T dependence of δpT /p pp T of π 0 from p+p to Au+Au for the centralities indicated at √ s N N =200 GeV [14].
It is important to note that the same value of n for the π 0 spectra in p+p and Au+Au collisions implies the same value of n for the original parton in p+p and the one that has lost energy in Au+Au. However z t for p+p and Au+Au measurements may differ slightly because the maximum possible parton energy √ s N N /2 is reduced by the energy loss. The effect on z t from p+p to Au+Au was estimated by increasing p T t in the calculation of z t in p+p collisions [12] by the largest δp T /p pp T = 0.20 for centrality 0-10% ( Fig.2) with result for p T t = 7.8 GeV/c, z t = 0.63 ± 0.07, and for p T t = 7.8/0.80 = 8.78 GeV/c, z t = 0.66 ± 0.06. Since the difference for the largest δp T /p pp T = 0.20 is considerably less than the error in the calculation, we shall use the measured or calculated z t in p+p also for Au+Au with the same p T t .
C. The away particles from a hadron-trigger do not measure the fragmentation function [8] It was generally assumed, as implied by Feynman, Field and Fox in 1977 [4], that the p T a (or x E , or x h ) distribution of away-side hadrons from a single hadron trigger with p T t , corrected for z t , would be the same as that from a jet-trigger and would measure the awayjet fragmentation function as it does for direct photon triggers [15]. However, attempts to try this at RHIC led to the discovery [8] that the x E distribution does not measure the fragmentation function. The good news was that it measured the ratio of the away jet to the trigger jet transverse momenta,x h =p T a /p T t , Eq. 4 with the value of n = 8.10 (±0.05) fixed as determined in Ref. [13], where n is the power-law of the inclusive π 0 spectrum and is observed to be the same in p+p and Au+Au collisions in the p Tt range of interest as noted in section II B above.

III. HOW TO APPLY THIS INFORMATION TO FINDq FROM P+P AND AU+AU DI-HADRON MEASUREMENTS
A recent STAR π 0 +h di-hadron measurement in p+p and Au+Au collision at √ s N N =200 GeV [5] is used to measure qL by calculating k T in each case as in Eq. 2. For a di-jet produced in a hard scattering, the initial p T t andp T a will both be reduced by energy loss in the medium to becomep T t andp T a that will be measured by the di-hadron correlations with p T t and p T a in Au+Au collisions. As both jets from the initial di-jet lose energy in the medium, the azimuthal angle between the di-jets from the k 2 T in the original collision should not change unless the medium induces multiple scattering fromq. Thus, withoutq and assuming the same fragmentation transverse momentum j 2 T in the original jets and those that have lost energy, the p out between the away hadron with p T a and the trigger hadron with p T t will not change ( Fig. 1), but the k 2 T will be reduced according to Eq. 3 because the ratio of the away to the trigger jetsx h = p T a /p T t will be reduced. Thus the calculation of k T from the di-hadron p+p measurement to compare with Au+Au measurement with the same di-hadron trigger p T t and p T a must use the values ofx h , and z T from the Au+Au measurement to compensate for the energy lost by the original dijet in p+p collisions.

IV. CALCULATION OF qL FROM THE STAR
MEASUREMENT [5].

B. Determinex h =pT a/pT t
This is done by a fit of Eq. 4 to the STAR measurements of what they call the away-side z T distribution [5] (called the x h or x E distribution here) for 12 < p T t < 20 GeV/c in p+p and Au+Au 0-12% centrality collisions (Fig. 4). The fit [17] takes account of the statistical and correlated systematic errors, σ i and σ bi , for each data point with dP/dx E = y i : whereσ i is the statistical error, σ i , scaled by the shift in y i such that the fractional error remains unchanged: The fit worked very well with a result for Au+Au of x h = 0.36 ± 0.05 with χ 2 /dof = 38.8/5 where the error has been corrected upward by χ 2 /dof. This is consistent with the valuex h = 0.48 ± 0.10 for 9 ≤ p T t ≤ 12 GeV/c from a PHENIX measurement [18,19] (see Fig. 4).
The value ofx h for the p+p measurement, although not needed for determining qL in the present method, was determined for the STAR p+p data with fitted result x pp h = 0.84 ± 0.04 which is in decent agreement with the resultx pp h = 0.73 ± 0.04 for 9 ≤ p T t ≤ 12 GeV/c from the PHENIX measurement (Fig. 4).

C. Determine zt
This was the easiest part of the calculation because STAR [5] had determined that z t = 0.80 ± 0.05 in their p+p collisions for π 0 with 12 < p T t < 20 GeV/c.
The p 2 out values from the fits to the correlation functions in p+p and Au+Au plus the results forx AA h = 0.36 ± 0.05, z t = 0.80 ± 0.05 above are used to calculate k 2 T using Eq. 3 with the value j T 2 = 0.62 ± 0.04 GeV/c [8,12] for both p+p and Au+Au. Equation 6 is used for qL /2. The results are given in Table I.
For completeness, the results for k 2 T pp with the p+p valuesx pp h = 0.84 ± 0.04, z t = 0.80 ± 0.05 are given in Table II. For the 12 < p T t < 20 ( p T t = 14.71) GeV/c, 1.2 < p T a < 3 ( p T a = 1.72) GeV/c bin, the result of qL = 8.41 ± 2.66 GeV 2 agrees with the Ref. [20] result, qL = 14 +42 −14 GeV 2 , but is not consistent with zero because of the much smaller error. The result for the 3 < p T a < 5 ( p T a = 3.75) GeV/c bin, qL = 1.71±0.67 GeV 2 , is at the edge of agreement, 2.4 σ below the value in the lower p T a bin, but also 2.6 σ from zero. If the different p T a ranges do not change the original di-jet configuration, then the value of qL should be equal in both ranges and can be weighted averaged with a result of qL = 2.11 ± 0.64 GeV 2 . Taking a guess for L in an Au+Au central collision as 7 fm, half the diameter of an Au nucleus, the result would beq = 1.20 ± 0.38 GeV 2 /fm for the lowest p T a bin,q = 0.24 ± 0.096 GeV 2 /fm for the higher p T a bin, with weighted averageq = 0.30 ± 0.09 GeV 2 /fm. These results are close to or lower than the result of the JET collaboration [21]q = 1.2±0.3 GeV 2 /fm at τ 0 = 0.6 fm/c.
The new method presented here gives results for qL comparable with the theoretical calculations noted [20,21] but is more straightforward and transparent for experimentalists. This is possibly the first experimental evidence for the predicted di-jet azimuthal broadening [1, 2]. It is noteworthy that the value ofx AA h = p T a /p T t ≈ 0.4 combined with the 20% loss ofp T t for the trigger jet (Fig. 2), which is surface biased [22], implies that the away jet has lost ≈ 3 times more energy than the trigger jet and thus traveled a longer distance so spent a longer time in the QGP. This may affect [23] the value of q used for comparison from the JET collaboration which used only single (trigger) hadrons for their calculation.
It is important to emphasize that the calculated values of qL are proportional to the square of the value ofx h derived from the measured away-side z T (i.e. x E ) distribution using Eq. 4. Although in the literature for more than a decade in a well-cited paper [8] and referenced in an important QCD Resource Letter [24], Eq. 4 has neither been verified nor falsified by a measurement of di-jet correlations with a di-hadron trigger. Future measurements at RHIC [25,26] will be able to do this and thus greatly improve the understanding of di-jet and di-hadron azimuthal broadening.