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

TL;DR

This paper measures the transport coefficient  in relativistic heavy ion collisions using di-hadron correlations, accounting for energy loss effects, and compares results with theoretical predictions and previous experimental findings.

Contribution

It introduces a method to correct the measured  for energy loss effects in di-hadron correlations, providing a more precise estimate of  in Au+Au collisions.

Findings

 in Au+Au collisions estimated as 0.30 1.09 GeV^2/fm.

Results are consistent with previous theoretical calculations.

The method improves the precision of  measurements in heavy ion collisions.

Abstract

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.

Tables

Table 1. Table 1: Tabulations for q ^ ^ 𝑞 \hat{q} –STAR π 0 superscript 𝜋 0 \pi^{0} -h  STARPLB760

STAR PLB760
$\sqrt{s_{{}_{NN}}}=200$
Reaction	GeV/c	GeV/c	(GeV/c)2
p$+$p	14.71	1.72	$0.263\pm 0.113$
p$+$p	14.71	3.75	$0.576\pm 0.167$
Au$+$Au 0-12%	14.71	1.72	$0.547\pm 0.163$
Au$+$Au 0-12%	14.71	3.75	$0.851\pm 0.203$
p$+$p comp	14.71	1.72	$0.263\pm 0.113$
p$+$p comp	14.71	3.75	$0.576\pm 0.167$
Reaction	GeV/c	GeV/c	GeV2
Au$+$Au 0-12%	$2.28\pm 0.35$	$1.006\pm 0.18$	$8.41\pm 2.66$
Au$+$Au 0-12%	$1.42\pm 0.22$	$1.076\pm 0.18$	$1.71\pm 0.67$

Table 2. Table 2: Tabulations for q ^ ^ 𝑞 \hat{q} –STAR π 0 superscript 𝜋 0 \pi^{0} -h  STARPLB760

Reaction	GeV/c	GeV/c	GeV/c
p$+$p	14.71	1.72	$2.34\pm 0.34$
p$+$p	14.71	3.75	$2.51\pm 0.31$

Equations

$$\frac{-dE}{dx}\simeq {\alpha }_s⟨q^2(L)⟩={\alpha }_s{\mu }^{2}L/{\lambda }_{\mathrm{mfp}}={\alpha }_s\hat{q}L,$$

$$⟨\hat{q}L⟩/2={⟨k_T^2⟩}_{AA}-{⟨k{{}^{\prime }}_T^2⟩}_{pp}$$

$$\sqrt{⟨k_T^2⟩}=\frac{{\hat{x}}_h}{⟨z_t⟩}\sqrt{\frac{⟨p_{\mathrm{out}}^2⟩-(1+x_h^2)⟨j_T^2⟩/2}{x_h^2}}.$$

$${\frac{d{P}_{\pi }}{d{x}_E}|}_{p_{T_t}}=N(n-1)\frac{1}{{\hat{x}}_h}\frac{1}{(1+\frac{x_E}{{\hat{x}}_h}{)}^n},$$

$${\chi }^2=\left[\sum _{i=1}^n\frac{(y_i+{ϵ}_b{\sigma }_{b_i}-y_i^{\mathrm{fit}}{)}^2}{{\tilde{\sigma }}_i^2}\right]+{ϵ}_b^2,$$

$$⟨\hat{q}L⟩/2={\left[\frac{{\hat{x}}_h}{⟨z_t⟩}\right]}^2\left[\frac{{⟨p_{\mathrm{out}}^2⟩}_{AA}-{⟨p_{\mathrm{out}}^2⟩}_{pp}}{x_h^2}\right].$$

Full text

I Introduction

In the original BDMPSZ formalism BDMPSNPB484 ; BSZARNPS50 , 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:

[TABLE]

where $\mu$, is the mean momentum transfer per collision, and the transport coefficient $\hat{q}=\mu^{2}/\lambda_{\rm mfp}$ is the 4-momentum-transfer-squared to the medium per mean free path, $\lambda_{\rm mfp}$. Additionally, the accumulated momentum-squared, $\left<p^{2}_{\perp W}\right>$ transverse to a parton traversing a length $L$ in the medium is well approximated by BDMPSNPB484 $\left<p^{2}_{\perp W}\right>\approx\langle{q^{2}(L)}\rangle=\hat{q}\,L$. This results in a direct snd simple relationship between the parton energy loss (Eq. 1) and the di-jet azimuthal broadening, $\left<p^{2}_{\perp W}\right>/2$, 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 CCHK77 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, $\left<p^{2}_{T}\right>_{\rm pair}=2\left<k^{2}_{T}\right>$, where $\left<k_{T}\right>$ is the average ‘intrinsic’ transverse momentum of a quark or gluon in a nucleon as defined by Feynman, Field and Fox FFF . Again, only the component of $\left<p^{2}_{T}\right>_{\rm pair}$ perpendicular to the di-jet axis leads to acoplanarity. Thus in an A$+$A collision the relationship in Eq. 2 should hold:

[TABLE]

for azimuthal correlations of a trigger particle with $p_{Tt}$ and away-side particles with $p_{Ta}$. It is important to note the ′ in $\left<k{{}^{\prime}}_{T}^{2}\right>_{pp}$, introduced here, which indicates 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 $k_{T}$ calculated with the same di-hadron $p_{Tt}$ and $p_{Ta}$ in Au$+$Au collisions.

Many experiments at RHIC, including recent experiments with di-hadron STARPLB760 , jet-hadron STARPRL112 and di-jet JacobsNPA956 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 STARPLB760 in terms of the out of plane component, $p_{\rm out}$ 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.

II How information about the initial partons can be derived from two-particle correlations.

We shall calculate $\left<k_{T}^{2}\right>$ from p$+$p and Au$+$Au di-hadron measurements with the same trigger particle transverse momentum, ${p}_{Tt}$, away-side ${p}_{Ta}$ and $x_{h}=p_{Ta}/p_{Tt}$. The di-hadrons are assumed to be fragments of jets with transverse momenta $\hat{p}_{Tt}$ and $\hat{p}_{Ta}$ with ratio $\hat{x}_{h}=\hat{p}_{Ta}/\hat{p}_{Tt}$, where ${z_{t}}\simeq p_{Tt}/\hat{p}_{Tt}$ 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 ppg029 , which we write in a slightly different form in Eq. 3:

[TABLE]

Here $p_{\rm out}\equiv p_{Ta}\sin\Delta\phi$ (see Fig. 1) and we have taken $\left<j_{T\rm a\phi}^{2}\right>=\left<j_{T\rm t\phi}^{2}\right>=\left<j_{T}^{2}\right>/{2}$. The variable $x_{h}$ (which STAR calls $z_{T}$) is used as an approximation of the variable $x_{E}=x_{h}\cos\phi$ of the original terminology from the CERN ISR where $k_{T}$ was discovered and measured 40 years ago Darriulat76 ; CCHK77 ; FFF ; CCORPLB97 .

II.1 Bjorken parent-child relation and ‘trigger-bias’ JacobPLC48

If the fragmentation function of the jet is a function only of the fragmentation variable $z$ and not of the jet $\hat{p}$, then the single particle cross section has the same power law shape, $d^{3}\sigma/2\pi p_{T}dp_{T}dy\propto p_{T}^{-n}$, as the parent jet cross section.

Furthermore, large values of $\left<z_{t}\right>=p_{Tt}/\hat{p}_{Tt}$ dominate the single-particle cross section (e.g. $\pi^{0}$) used as the trigger for the di-hadron (e.g. $\pi^{0}$-h) measurement. This is called trigger-bias but is valid also for the simple single-particle measurements. Calculations of $\left<z_{t}\right>$ vs. $p_{Tt}$ for $\pi^{0}$ at \mbox{\sqrt{s_{{}_{NN}}}}=200 GeV are given in Ref. ppg089 .

II.2 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 $\pi^{0}$ $p_{T}$ spectra with 5<p_{T}\ \raise 1.29167pt\hbox{<\kern-7.5pt\raise-4.73611pt\hbox{\sim}}\ 20 GeV/c all follow the same power law with $n\approx 8.10\pm 0.05$ ppg080 . From the Bj parent-child relation, the energy loss of the trigger jet is found by measuring $\delta p_{T}/p_{T}^{pp}$, the shift in the $\pi^{0}$ spectra in Au$+$Au at a given $p_{T}$ from the $\left<T_{AA}\right>$ corrected p$+$p cross section (Fig. 2) ppg133 . The small dropoff of $\delta p_{T}/p_{T}^{pp}$ for $p_{T}\geq 14$ GeV/c indicates a small increase of $n$ with increasing $p_{T}$.

It is important to note that the same value of $n$ for the $\pi^{0}$ 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 $\left<z_{t}\right>$ for p$+$p and Au$+$Au measurements may differ slightly because the maximum possible parton energy \mbox{\sqrt{s_{{}_{NN}}}}/2 is reduced by the energy loss. The effect on $\left<z_{t}\right>$ from p$+$p to Au$+$Au was estimated by increasing $p_{Tt}$ in the calculation of $\left<z_{t}\right>$ in p$+$p collisions ppg089 by the largest $\delta p_{T}/p_{T}^{pp}=0.20$ for centrality 0-10% (Fig.2) with result for $p_{Tt}=7.8$ GeV/c, $\left<z_{t}\right>=0.63\pm 0.07$, and for $p_{Tt}=7.8/0.80=8.78$ GeV/c, $\left<z_{t}\right>=0.66\pm 0.06$. Since the difference for the largest $\delta p_{T}/p_{T}^{pp}=0.20$ is considerably less than the error in the calculation, we shall use the measured or calculated $\left<z_{t}\right>$ in p$+$p also for Au$+$Au with the same $p_{Tt}$.

II.3 The away particles from a hadron-trigger do not measure the fragmentation function ppg029

It was generally assumed, as implied by Feynman, Field and Fox in 1977 FFF , that the $p_{Ta}$ (or $x_{E}$, or $x_{h}$) distribution of away-side hadrons from a single hadron trigger with $p_{Tt}$, corrected for $\left<z_{t}\right>$, would be the same as that from a jet-trigger and would measure the away-jet fragmentation function as it does for direct photon triggers ppg095 . However, attempts to try this at RHIC led to the discovery ppg029 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, $\hat{x}_{h}=\hat{p}_{Ta}/\hat{p}_{Tt}$, Eq. 4

[TABLE]

with the value of $n=8.10$ ($\pm 0.05$) fixed as determined in Ref. ppg080 , where $n$ is the power-law of the inclusive $\pi^{0}$ spectrum and is observed to be the same in p$+$p and Au+Au collisions in the $p_{T_{t}}$ range of interest as noted in section II.2 above.

III How to apply this information to find $\hat{q}$ from p$+$p and Au$+$Au di-hadron measurements

A recent STAR $\pi^{0}$+h di-hadron measurement in p$+$p and Au$+$Au collision at $\sqrt{s_{{}_{NN}}}$ =200 GeV STARPLB760 is used to measure $\left<\hat{q}L\right>$ by calculating $k_{T}$ in each case as in Eq. 2. For a di-jet produced in a hard scattering, the initial $\hat{p}_{Tt}$ and $\hat{p}_{Ta}$ will both be reduced by energy loss in the medium to become $\hat{p}{{}^{\prime}}_{Tt}$ and $\hat{p}{{}^{\prime}}_{Ta}$ that will be measured by the di-hadron correlations with $p_{Tt}$ and $p_{Ta}$ 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 $\left<k_{T}^{2}\right>$ in the original collision should not change unless the medium induces multiple scattering from $\hat{q}$. Thus, without $\hat{q}$ and assuming the same fragmentation transverse momentum $\left<j_{T}^{2}\right>$ in the original jets and those that have lost energy, the $p_{\rm out}$ between the away hadron with $p_{Ta}$ and the trigger hadron with $p_{Tt}$ will not change (Fig. 1), but the $\left<k{{}^{\prime}}_{T}^{2}\right>$ will be reduced according to Eq. 3 because the ratio of the away to the trigger jets $\hat{x}{{}^{\prime}}_{h}=\hat{p}{{}^{\prime}}_{Ta}/\hat{p}{{}^{\prime}}_{Tt}$ will be reduced. Thus the calculation of $k{{}^{\prime}}_{T}$ from the di-hadron p$+$p measurement to compare with Au$+$Au measurement with the same di-hadron trigger $p_{Tt}$ and $p_{Ta}$ must use the values of $\hat{x}_{h}$, and $\left<z_{T}\right>$ from the Au$+$Au measurement to compensate for the energy lost by the original dijet in p$+$p collisions.

IV Calculation of ${\left<\hat{q}L\right>}$ from the STAR measurement STARPLB760 .

IV.1 Determine $\left<p^{2}_{\rm out}\right>$ from the $\pi^{0}$-h correlation function

This is accomplished by fitting the $\pi^{0}$-h correlation functions for $12<p_{Tt}<20$ GeV/c STARPLB760 to a gaussian in $\sin{\Delta\phi}$ for the away-side, $\pi/2\leq\Delta\phi\leq 3\pi/2$ ppg029 ; ppg089 ; ppg195 ; and a gaussian in $\Delta\phi$ for the trigger side $-\pi/2\leq\Delta\phi\leq\pi/2$ (Fig. 3). The results for Au$+$Au 0-12% centrality are $\left<p^{2}_{\rm out}\right>=0.547\pm 0.163$ (GeV/c)2, $\chi^{2}/{\rm dof}=33/21$ for $1.2<p_{Ta}<3$ GeV/c and $\left<p^{2}_{\rm out}\right>=0.851\pm 0.203$ (GeV/c)2, $\chi^{2}/{\rm dof}=23/21$ for $3<p_{Ta}<5$ GeV/c. The same fits to the p$+$p measurements with the same $p_{Tt}$ and $p_{Ta}$ yielded $\left<p^{2}_{\rm out}\right>=0.263\pm 0.113$ (GeV/c)2, $\chi^{2}/{\rm dof}=186/21$ for $1.2<p_{Ta}<3$ GeV/c and $\left<p^{2}_{\rm out}\right>=0.576\pm 0.167$ (GeV/c)2, $\chi^{2}/{\rm dof}=137/21$ for $3<p_{Ta}<5$ GeV/c where the errors have been corrected up by $\sqrt{\chi^{2}/{\rm dof}}$.

IV.2 Determine $\hat{x}_{h}=\hat{p}_{Ta}/\hat{p}_{Tt}$

This is done by a fit of Eq. 4 to the STAR measurements of what they call the away-side $z_{T}$ distribution STARPLB760 (called the $x_{h}$ or $x_{E}$ distribution here) for $12<p_{Tt}<20$ GeV/c in p$+$p and Au$+$Au 0-12% centrality collisions (Fig. 4). The fit ppg079 takes account of the statistical and correlated systematic errors, $\sigma_{i}$ and $\sigma_{b_{i}}$, for each data point with $dP/dx_{E}=y_{i}$ :

[TABLE]

where ${\tilde{\sigma}}_{i}$ is the statistical error, $\sigma_{i}$, scaled by the shift in $y_{i}$ such that the fractional error remains unchanged: $\tilde{\sigma}_{i}=\sigma_{i}\left(1+\epsilon_{b}\sigma_{b_{i}}/{y_{i}}\right)$, where $\epsilon_{b}$ is to be fit.

The fit worked very well with a result for Au$+$Au of $\hat{x}_{h}=0.36\pm 0.05$ with $\chi^{2}/\rm{dof}=38.8/5$ where the error has been corrected upward by $\sqrt{\chi^{2}/\rm{dof}}$. This is consistent with the value $\hat{x}_{h}=0.48\pm 0.10$ for $9\leq p_{Tt}\leq 12$ GeV/c from a PHENIX measurement ppg106 ; MJTGrenoble (see Fig. 4).

The value of $\hat{x}_{h}$ for the p$+$p measurement, although not needed for determining $\left<\hat{q}L\right>$ in the present method, was determined for the STAR p$+$p data with fitted result $\hat{x}^{pp}_{h}=0.84\pm 0.04$ which is in decent agreement with the result $\hat{x}^{pp}_{h}=0.73\pm 0.04$ for $9\leq p_{Tt}\leq 12$ GeV/c from the PHENIX measurement (Fig. 4).

IV.3 Determine $\left<z_{t}\right>$

This was the easiest part of the calculation because STAR STARPLB760 had determined that $\left<z_{t}\right>=0.80\pm 0.05$ in their p$+$p collisions for $\pi^{0}$ with $12<p_{Tt}<20$ GeV/c.

IV.4 Calculate $\left<k_{T}^{2}\right>_{AA}$, $\left<k{{}^{\prime}}_{T}^{2}\right>_{pp}$, $\left<\hat{q}L\right>/2$

The $\left<p^{2}_{\rm out}\right>$ values from the fits to the correlation functions in p$+$p and Au$+$Au plus the results for $\hat{x}^{AA}_{h}=0.36\pm 0.05$, $\left<z_{t}\right>=0.80\pm 0.05$ above are used to calculate $\sqrt{\left<k_{T}^{2}\right>}$ using Eq. 3 with the value $\sqrt{\left<j_{T}\right>^{2}}=0.62\pm 0.04$ GeV/c ppg029 ; ppg089 for both p$+$p and Au$+$Au. Equation 6 is used for $\left<\hat{q}L\right>/2$. The results are given in Table 1.

[TABLE]

For completeness, the results for $\sqrt{\left<k_{T}^{2}\right>}_{pp}$ with the p$+$p values $\hat{x}^{pp}_{h}=0.84\pm 0.04$, $\left<z_{t}\right>=0.80\pm 0.05$ are given in Table 2.

V Discussion and Conclusion

For the $12<p_{Tt}<20$ ($\left<p_{Tt}\right>=14.71$) GeV/c, $1.2<p_{Ta}<3$ ($\left<p_{Ta}\right>=1.72$) GeV/c bin, the result of $\left<\hat{q}L\right>=8.41\pm 2.66$ GeV2 agrees with the Ref. ChenCCNU0616 result, $\left<\hat{q}L\right>=14^{+42}_{-14}$ GeV2, but is not consistent with zero because of the much smaller error. The result for the $3<p_{Ta}<5$ ($\left<p_{Ta}\right>=3.75$) GeV/c bin, $\left<\hat{q}L\right>=1.71\pm 0.67$ GeV2, is at the edge of agreement, 2.4 $\sigma$ below the value in the lower $p_{Ta}$ bin, but also 2.6 $\sigma$ from zero. If the different $p_{Ta}$ ranges do not change the original di-jet configuration, then the value of $\left<\hat{q}L\right>$ should be equal in both ranges and can be weighted averaged with a result of $\left<\hat{q}L\right>=2.11\pm 0.64$ GeV2. Taking a guess for $\left<L\right>$ in an Au$+$Au central collision as 7 fm, half the diameter of an Au nucleus, the result would be $\hat{q}=1.20\pm 0.38$ GeV2/fm for the lowest $p_{Ta}$ bin, $\hat{q}=0.24\pm 0.096$ GeV2/fm for the higher $p_{Ta}$ bin, with weighted average $\hat{q}=0.30\pm 0.09$ GeV2/fm. These results are close to or lower than the result of the JET collaboration JET2014 $\hat{q}=1.2\pm 0.3$ GeV2/fm at $\tau_{0}=0.6$ fm/c.

The new method presented here gives results for $\left<\hat{q}L\right>$ comparable with the theoretical calculations noted ChenCCNU0616 ; JET2014 but is more straightforward and transparent for experimentalists. This is possibly the first experimental evidence for the predicted di-jet azimuthal broadening BDMPSNPB484 ; BSZARNPS50 . It is noteworthy that the value of $\hat{x}^{AA}_{h}=\hat{p}_{Ta}/\hat{p}_{Tt}\approx 0.4$ combined with the 20% loss of $\hat{p}_{Tt}$ for the trigger jet (Fig. 2), which is surface biased ppg054 , implies that the away jet has lost $\approx 3$ times more energy than the trigger jet and thus traveled a longer distance so spent a longer time in the QGP. This may affect IvanV the value of $\hat{q}$ 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 $\left<\hat{q}L\right>$ are proportional to the square of the value of $\hat{x}_{h}$ 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 ppg029 and referenced in an important QCD Resource Letter QCDAMJPhys , Eq. 4 has neither been verified nor falsified by a measurement of di-jet correlations with a di-hadron trigger. Future measurements at RHIC MuellerPLB763 ; sPHENIX will be able to do this and thus greatly improve the understanding of di-jet and di-hadron azimuthal broadening.

Acknowledgements.

Research supported by U. S. Department of Energy, Contract No. DE-SC0012704.