Latest Results from RHIC + Progress on Determining q ^ L in RHI Collisions Using Di-Hadron Correlations

Abstract

Results from Relativistic Heavy Ion Collider Physics in 2018 and plans for the future at Brookhaven National Laboratory are presented.

1. Introduction

The Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory (BNL) is one of the two remaining operating hadron colliders in the world, and the first and only polarized p+p collider. BNL is located in the center of the roughly 200 km long maximum 40 km wide island (named Long Island), and appears on the map as the white circle which is the berm containing the Relativistic Heavy Ion Collider (RHIC). BNL is 100 km from New York City in a region which nurtures science with Columbia University and the Bronx High School of Science indicated (Figure 1). Perhaps more convincing is the list of the many Nobel Prize winners from New York City High School graduates (Figure 2) which does not yet include one of this years Nobel Prize winners in Physics, Arthur Ashkin who graduated from James Madison High school in 1940 and Columbia U. in 1947.

There also have been many discoveries and Nobel Prizes at BNL (Figure 3).

In particular, Leon Lederman, who made many discoveries at BNL (Figure 4), died this year (2018) at the age of 96. Leon was the most creative and productive high energy physics experimentalist of his generation as well as the physicist with the best jokes. He was also my PhD thesis Professor. For more details, see https://physicstoday.scitation.org/do/10.1063/PT.6.4.20181010a/full/.

2. Why RHIC Was Built: To Discover the Quark Gluon Plasma (QGP)

Figure 5 shows central collision particle production in the PHENIX and STAR detectors, which were the major detectors at RHIC.

At the startup of RHIC in the year 2000, there were two smaller more special purpose detectors PHOBOS and BRAHMS, as shown in Figure 6, which finished data taking in 2005.

2.1. The First Major RHIC Experiments

The two major experiments at RHIC were STAR (Figure 7), which is still operating, and PHENIX (Figure 8), which finished data taking at the end of the 2016 run.

2.2. The New Major RHIC Experiment sPHENIX

sPHENIX is a major improvement over PHENIX with a superconducting thin coil solenoid which was surplus from the BABAR experiment at SLAC and is now working at BNL and has reached its full field (Figure 9).

The design of the sPHENIX experiment is moving along well (Figure 10) with a notable addition of a hadron calorimeter based on the iron return yoke of the solenoid.

sPHENIX has been approved by the U. S. Department of Energy (DoE) as a Major Item of Equipment (MIE) with the schedule of critical decisions shown in Figure 11a, and the planned multi-year RHIC runs indicated in Figure 11b. The present sPHENIX collaboration and its evolution is shown in Figure 12.

2.3. Following RHIC in U.S. Nuclear Physics: the Electron Ion Collider (EIC)

The first BNL EIC design in 2014 is shown in Figure 13. The 2018 JLab and BNL EIC designs are shown in Figure 14 and Figure 15.

The two new designs of the JLab (JLEIC) and BNL (eRHIC) both satisfy the Temple committee cost estimate of $1.5B, but R&D of the novel first BNL design is not idle.

Research and Development (R&D) for an Improved Less Expensive BNL Machine Is Ongoing

BNL and Cornell are in the process of experiments studying an energy recovery linac ERL (Figure 16a). Figure 16b is the main Linac cryo module made from superconducting RF cavities. Figure 16c is a return loop made from fixed-field alternating-gradient (FFAG) optics made with permanent Halbach magnets to contain four beam energies in a single 70 mm-wide beam pipe, designed and prototyped at Brookhaven National Laboratory (BNL).

3. RHIC Future Run Plan (Figure 17) and and the Present RHIC Run in 2018 (Figure 18)

3.1. 2018 RHIC Run Is ${}_{40}$Zr${}^{96}$ + ${}_{40}$Zr${}^{96}$ and ${}_{44}$Ru${}^{96}$ + ${}_{44}$Ru${}^{96}$, Why?

To determine whether the separation of charges in the flow, $v_2$, of ${\pi }^{+}$ and ${\pi }^{-}$ shown in Figure 19 is due to a new phenomenon called the Chiral Magnetic Effect (Figure 20a), the 2018 measurements are made with collisions of Zr+Zr and Ru+Ru, which have the same number of nucleons but different electric charges (Figure 20b). If the effect is larger in Ru+Ru with stronger charge and magnetic field compared to Zr+Zr with the same number of nucleons, it would indicate that the charge asymmetry is a magnetic effect, possibly the Chiral Magnetic Effect.

3.2. Vorticity: An Application of Particle Physics to the QGP

It was observed at FERMILAB [1] that forward $\mathsf{\Lambda }$ were polarized in p+Be collisions, where the proton in the $\mathsf{\Lambda }\to p+{\pi }^{-}$ decay is emitted along the spin direction of the $\mathsf{\Lambda }$. In the A+A collision (Figure 21a), the forward going beam fragments are deflected outwards so that the event plane and the angular momentum ${\widehat{J}}_{sys}$ of the QGP formed can be determined. STAR claims that the $\mathsf{\Lambda }$ polarization, ${\overline{\mathcal{P}}}_{\mathsf{\Lambda }}$, is parallel to the angular momentum ${\widehat{J}}_{sys}$ of the QGP everywhere so that the vorticity $\omega =k_{B}T({\overline{\mathcal{P}}}_{\mathsf{\Lambda }}+{\overline{\mathcal{P}}}_{\overline{\mathsf{\Lambda }}})/\hslash$ can be calculated, a good exercise for the reader to see if you can get the $\omega \sim {10}^{22}/s$ which is ${10}^5$ times larger than any other fluid [2]. Another interesting thing to note is that the largest vorticity is at $\sqrt{s_{{}_{NN}}}=7.6-19$ GeV where the CERN fixed target experiments measure. Does this mean that their fluid (with minimal if any QGP) is also perfect?

STAR team receives secretary’s achievement award for vorticity in 2018 (Figure 22).

4. The Search for the Quark Gluon Plasma at RHIC

High energy nucleus–nucleus collisions provide the means of creating nuclear matter in conditions of extreme temperature and density, the Quark Gluon Plasma QGP (Figure 23). At large energy or baryon density, a phase transition is expected from a state of nucleons containing confined quarks and gluons to a state of “deconfined” (from their individual nucleons) quarks and gluons covering a volume that is many units of the confinement length.

4.1. Anisotropic (Elliptical) Transverse Flow—An Interesting Complication in all A+A Collisions (Figure 24)

Figure 25 shows that Elliptical flow ($v_2$) exists in all A+A collisions measured. At very low $\sqrt{s_{{}_{NN}}}$, the main effect is from nuclei bouncing off each other and breaking into fragments. The negative $v_2$ at larger $\sqrt{s_{{}_{NN}}}$ is produced by the effective “squeeze-out” (in the y direction) of the produced particles by slow moving minimally Lorentz-contracted spectators, which block the particles emitted in the reaction plane. With increasing $\sqrt{s_{{}_{NN}}}$, the spectators move faster and become more contracted so the blocking stops and positive $v_2$ returns.

4.2. Flow Also Exists in Small Systems and Is Sensitive to the Initial Geometry

Figure 26 shows that flow exists in small p+Au, d+Au, ${}^3$He+Au systems with preliminary sensitivity of $v_3$ to the initial geometry. Figure 27 (Top) shows that $v_2$ is about the same in all three systems but $v_3$ is much larger in ${}^3$He+Au, clearly indicating the sensitivity of flow to the initial geometry of the collision. Figure 27 (Bottom) shows that there is mass ordering in the flow which is strong evidence for hydrodynamics in these small systems. The solid red and dashed blue lines represent hydrodynamic predictions. These hydrodynamical models, which include the formation of a short-lived QGP droplet, provide the best simultaneous description of the measurements, strong evidence for the QGP in small systems.

4.2.1. It Takes Two Color Strings for Collectivity—Nagle, J.; et al. [6]

This is an answer to the interesting question of the minimal conditions for collectivity in small systems.

For the case of e${}^{+}$e${}^{-}$ collisions in Figure 28 utilizing the AAMPT framework and a single color string, the results indicate only a modest number of parton–parton scatterings and no observable collectivity signal.

However, a simple extension to two color strings (Figure 29), which represent a simplified geometry in p+p collisions, predicts finite long-range two-particle correlations (known as the ridge) and a strong $v_2$ with respect to the initial parton geometry.

4.2.2. A Fundamental Point about QCD and the String Tension

Unlike an electric or magnetic field between two sources which spreads over all space, in QCD as proposed by Kogut and Susskind [7] the color flux lines connecting two quarks or a $q-\overline{q}$ pair as in Figure 28 are constrained in a thin tube-like region because of the three-gluon coupling. Furthermore, if the field contained a constant amount of color-field energy stored per unit length, this would provide a linearly rising confining potential between the $q-q$ or $q-\overline{q}$ pair.

This led to the Cornell string-like confining potential [8], which combined the Coulomb $1/r$ dependence at short distances from vector-gluon exchange with QCD coupling constant ${\alpha }_s\left(Q^2\right)$, and a linearly rising string-like potential, with string-tension $\sigma$,

$$V\left(r\right)=-\frac{{\alpha }_s}{r}+\sigma r$$

(1)

which provided confinement at large distances (Equation (1)). Particles are produced by the string breaking (fragmentation).

4.3. The Latest Discovery Claims “Flow” in Small Systems Is From the QGP How Did We Find the QGP in the First Place?

4.3.1. $J/\psi$ Suppression, 1986

In 1986, T. Matsui and H. Satz [9] said that due to the Debye screening of the color potential in a QGP, charmonium production would be suppressed since the c-$\overline{\mathrm{c}}$ could not bind. With increasing temperature, T, in analogy to increasing $Q^2$, the strong coupling constant ${\alpha }_s\left(T\right)$ becomes smaller, reducing the binding energy, and the string tension, $\sigma \left(T\right)$, becomes smaller, increasing the confining radius, effectively screening the potential [10]

$$V\left(r\right)=-\frac{4}{3}\frac{{\alpha }_s}{r}+\sigma r\to -\frac{4}{3}\frac{{\alpha }_s}{r}{e}^{-{\mu }_{D}r}+\sigma \frac{(1-e^{-{\mu }_{D}r})}{{\mu }_D}$$

(2)

where ${\mu }_D={\mu }_D\left(T\right)=1/r_D$ is the Debye screening mass. For $r<1/{\mu }_D$, a quark feels the full color charge, but, for $r>1/{\mu }_D$, the quark is free of the potential and the string tension, effectively deconfined. The properties of the QGP cannot be calculated in QCD perturbation theory but only in Lattice QCD Calculations [11].

$J/\psi$ suppression eventually didn’t work because the free c and $\overline{\mathrm{c}}$ quarks recombined to make $J/\psi$’s [12]. See Alice publication [13].

4.3.2. Jet Quenching by Coherent LPM Radiative Energy Loss of a Parton in the QGP, 1997

In 1997, Baier, Dokshitzer, Mueller, Peigne, Schiff and Zakharov (BDMPSZ) [14] said that the energy loss from coherent Landau–Pomeranchuk–Migdal (LPM) radiation for hard-scattered partons exiting the QGP would result in an attenuation of the jet energy and a broadening of the jets (Figure 30).

As a parton from hard-scattering in the A+B collision exits through the medium, it can radiate a gluon; and both continue traversing the medium. It is important to understand that “Only the gluons radiated outside the cone defining the jet contribute to the energy loss”. In the angular ordering of QCD [15], the angular cone of any further emission will be restricted to be less than that of the previous emission and will end the energy loss once inside the jet cone. This does not work in the QGP so no energy loss occurs only when all gluons emitted by a parton are inside the jet cone. In addition to other issues, this means that defining the jet cone is a big issue—so watch out for so-called trimming.

4.4. BDMPSZ: The Cone, the Energy Loss, Azimuthal Broadening, Is the QGP Signature

The energy loss of the outgoing parton, $-dE/dx$, per unit length (x) of a medium with total length L, is proportional to the total four-momentum transfer-squared, $q^2\left(L\right)$, and takes the form:

$$\frac{-dE}{dx}\simeq {\alpha }_s\langle q^2\left(L\right)\rangle ={\alpha }_s{\mu }^{2}L/{\lambda }_{\mathrm{mfp}}={\alpha }_s\widehat{q}L$$

where $\mu$, is the mean momentum transfer per collision, and the transport coefficient $\widehat{q}={\mu }^2/{\lambda }_{\mathrm{mfp}}$ is the four-momentum-transfer-squared to the medium per mean free path, ${\lambda }_{\mathrm{mfp}}$.

Additionally, the accumulated momentum-squared, $〈p_{\perp W}^2〉$ transverse to a parton traversing a length L in the medium is well approximated by

$$〈p_{\perp W}^2〉\approx \langle q^2\left(L\right)\rangle =\widehat{q}L.$$

5. Jet Quenching at RHIC, the Discovery of the QGP

The energy loss of an outgoing parton with color charged fully exposed in a medium with a large density of similarly exposed color charges (i.e., a QGP) from Landau–Pomeranchuk–Migdal (LPM) coherent radiation of gluons was predicted in QCD by BDMPSZ [14].

Hard scattered partons (Figure 31a) lose energy going through the medium so that there are fewer partons or jet fragments at a given $p_T$. The ratio of the measured semi-inclusive yield of, for example, pions in a given A+A centrality class divided by the semi-inclusive yield in a p+p collision times the number of A+A collisions $〈N_{\mathrm{coll}}〉$ in the centrality-class is given by the nuclear modification factor, $R_{AA}$ (Figure 31b), which equals 1 for no energy loss.

PHENIX discovered jet quenching of hadrons at RHIC in 2001 [16] (Figure 32). Pions at large $p_T>2$ GeV/c are suppressed in Au+Au at $\sqrt{s_{{}_{NN}}}$ =130 GeV compared to the enhancement found at the CERN SpS at $\sqrt{s_{{}_{NN}}}$ =17 GeV. This is the first regular publication from a RHIC experiment to reach 1000 citations.

5.1. Status of $R_{AA}$ in Au+Au at $\sqrt{s_{{}_{NN}}}$ = 200 GeV

Figure 33 shows the suppression of all identified hadrons, as well as $e^{\pm }$ from c and b quark decay, with $p_T>2$ GeV/c measured by PHENIX until 2013. One exception is the enhancement of protons for $2<p_T<4$ GeV/c, which are then suppressed at larger $p_T$. Particle Identification is crucial for these measurements since all particles behave differently. The only particle that shows no-suppression is the direct single $\gamma$ (from the QCD reaction $g+q\to \gamma +q$) which shows that the medium produced at RHIC is the strongly interacting QGP since $\gamma$ rays only interact electromagnetically.

5.2. Recent Measurements to Test the Second BDMPSZ Prediction

(1) The energy loss of the outgoing parton, $-dE/dx$, per unit length (x) of a medium with total length L, is proportional to the total four-momentum transfer-squared, $q^2\left(L\right)$, and takes the form:

$$\frac{-dE}{dx}\simeq {\alpha }_s\langle q^2\left(L\right)\rangle ={\alpha }_s{\mu }^{2}L/{\lambda }_{\mathrm{mfp}}={\alpha }_s\widehat{q}L$$

where $\mu$, is the mean momentum transfer per collision, and the transport coefficient $\widehat{q}={\mu }^2/{\lambda }_{\mathrm{mfp}}$ is the four-momentum-transfer-squared to the medium per mean free path, ${\lambda }_{\mathrm{mfp}}$.

(2) Additionally, the accumulated momentum-squared, $〈p_{\perp W}^2〉$ transverse to a parton traversing a length L in the medium is well approximated by

$$〈p_{\perp W}^2〉\approx \langle q^2\left(L\right)\rangle =\widehat{q}L〈\widehat{q}L〉={〈k_T^2〉}_{AA}-{〈k{{}^{\prime }}_T^2〉}_{pp}.$$

(3)

Although only the component of $〈p_{\perp W}^2〉$⊥ to the scattering plane affects $k_T$ (Figure 34), the azimuthal broadening of the di-jet is caused by the random sum of the azimuthal components $〈p_{\perp W}^2〉/2$ from each outgoing di-jet or $〈p_{\perp W}^2〉=\widehat{q}L$.

From the values of $R_{AA}$ observed at RHIC (after 12 years), the JET Collaboration [17] has found that $\widehat{q}=1.2\pm 0.3$ GeV${}^2$/fm at RHIC, $1.9\pm 0.6$ at LHC at an initial time ${\tau }_0=0.6$ fm/c; however, nobody has yet measured the azimuthal broadening predicted. Before proceeding, one has to know the meaning of $k_T$ defined by Feynman, Field and Fox [18] as the transverse momentum of a parton in a nucleon (Figure 34).

5.2.1. The Key New Idea of ${〈k{{}^{\prime }}_T^2〉}_{pp}$ Instead of ${〈k_T^2〉}_{pp}$ in Equation (3)

The di-hadron correlations of $p_{Ta}$ with $p_{Tt}$ (Figure 34) are measured in p+p and Au+Au collisions. The parent jets in the original Au+Au collision as measured in p+p will both lose energy passing through the medium but the azimuthal angle between the jets should not change unless the medium induces multiple scattering from $\widehat{q}$. Thus, the calculation of $k{{}^{\prime }}_T$ from the di-hadron p+p measurement to compare with Au+Au measurements with the same di-hadron $p_{Tt}$ and $p_{Ta}$ must use the value of ${\widehat{x}}_h$ and $〈z_t〉$ of the parent jets in the A+A collision. The variables are $x_h\equiv p_{Ta}/p_{Tt},{\widehat{x}}_h\equiv {\widehat{p}}_{Ta}/{\widehat{p}}_{Tt},〈z_t〉\equiv p_{Tt}/{\widehat{p}}_{Tt}$, where, e.g., $p_{Tt}$ is the trigger particle transverse momentum and ${\widehat{p}}_{Tt}$ means the trigger jet transverse momentum.

The same values of ${\widehat{x}}_h$ and $〈z_t〉$ in Au+Au and p+p give the cool result [20]:

$$〈\widehat{q}L〉={\left[\frac{{\widehat{x}}_h}{〈z_t〉}\right]}^2\left[\frac{{〈p_{\mathrm{out}}^2〉}_{AA}-{〈p_{\mathrm{out}}^2〉}_{pp}}{x_h^2}\right]$$

(4)

For di-jet measurements, the formula is even simpler:

(i) $x_h\equiv {\widehat{x}}_h$ because the trigger and away “particles” are the jets; (ii) $〈z_t〉\equiv 1$ because the trigger “particle” is the entire jet not a fragment of the jet; and (iii) $〈p_{\mathrm{out}}^2〉={\widehat{p}}_{Ta}^2{sin}^2(\pi -\Delta \varphi )$. This reduces the formula for di-jets to:

$$〈\widehat{q}L〉=\left[{〈p_{\mathrm{out}}^2〉}_{AA}-{〈p_{\mathrm{out}}^2〉}_{pp}\right]={\widehat{p}}_{Ta}^2\left[{〈{sin}^2(\pi -\Delta \varphi )〉}_{AA}-{〈{sin}^2(\pi -\Delta \varphi )〉}_{pp}\right]$$

(5)

5.2.2. A Test of Equation (5) for $〈\widehat{q}L〉$

Al Mueller et al. [21] gave a prediction for the azimuthal broadening of di-jet angular correlations for 35 GeV jets at RHIC (Figure 35).

To check my Equation (5), I measured the half width at half maximum (HWHM), which equals $1.175\sigma$ for a Gaussian, for each curve in Figure 35, and calculated ${(\sigma \times 35)}^2$ to get $〈p_{\mathrm{out}}^2〉$ for each $\widehat{q}L$, and used Equation (5) to get 9.6 GeV${}^2$ and 21.5 GeV${}^2$, respectively, for the 8 GeV${}^2$ and 20 GeV${}^2$ plots. This is an excellent result considering that I had to measure the HWHMs in Figure 35 with a pencil and ruler.

5.2.3. How to Calculate $\widehat{q}L$ with Equation (4) from Di-Hadron Measurements

The determination of the required quantities is well known to older PHENIXians who have read Ref. [19] or my book [22] as outlined below:

(A) $〈z_t〉$ is calculated from the Bjorken parent–child relation and “trigger bias” [23] (cf. Ref. [24]).

(B) The energy loss of the trigger jet from p+p to Au+Au can be measured by the shift in the $p_T$ spectra [25].

(C) ${\widehat{x}}_h$, the ratio of the away-jet to the trigger jet transverse momenta can be measured by the away particle $p_{Ta}$ distribution for a given trigger particle $p_{Tt}$ taking $x_E=x_{h}cos\Delta \varphi \approx x_h=p_{Ta}/p_{Tt}$ [19]:

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

(6)

5.2.4. Example: ${\widehat{x}}_h$ from Fits to the PHENIX Data from Ref. [26]

The fits in Figure 36 work very well, with excellent ${\chi }^2$/dof. However, it is important to notice that the dashed curve in Au+Au does not fit the data as well as the solid red curve which is the sum of Equation (6) with free parameters + a second term with the form of Equation (6) but with the ${\widehat{x}}_h$ fixed at the p+p value. It is also important to note that the solid red curve between the highest Au+Au data points is notably parallel to the p+p curve. A possible explanation is that, in this region, which is at a fraction $\approx 1$% of the $dP/d{x}_E$ distribution, the highest $p_{Ta}$ fragments are from jets that do not lose energy in the QGP.

5.2.5. Results from STAR ${\pi }^0-h$ and $\gamma -h$ Correlations [27]

Figure 37 is a table of results of my published calculation [20] of $〈\widehat{q}L〉$ from the STAR data. The errors on the STAR $〈\widehat{q}L〉$ here (with the *) are much larger than stated in my published calculation because I made a trivial mistake, which is corrected here. In addition, the new values of $〈\widehat{q}L〉$ reflect that Equation (4) defines $〈\widehat{q}L〉$ not $〈\widehat{q}L〉/2.$

5.3. Some $〈\widehat{q}L〉$ Results from PHENIX [26]

The away widths from PHENIX ${\pi }^0-h$ correlations [26] are shown in Figure 38 with the calculated $\widehat{q}L$ values for ${\pi }^0-h$$\sqrt{s_{{}_{NN}}}$ = 200 GeV, 20–60% centrality, $5<p_{Tt}<7$ GeV/c shown in Figure 39 and $7<p_{Tt}<9$ GeV/c in Figure 40.

5.4. Conclusions

It appears that the method works and gives consistent results for all the $\widehat{q}L$ calculations shown (Figure 37, Figure 39 and Figure 40). In the lowest $p_{Ta}\sim 1.5$ GeV/c bin, the results are all consistent with the JET collaboration [17] result, $\widehat{q}=1.2\pm 0.3$ GeV${}^2$/fm or $\widehat{q}L=8.4\pm 2.1$ GeV${}^2$ for $L=7$ fm, the radius of an Au nucleus. However, for $p_{Ta}>2.0$ GeV/c, all the results are consistent with $\widehat{q}L=0$. Personally, I think that this is where the first gluon emitted in the medium was inside the jet cone, so that all further emissions were also inside the jet cone due to the angular ordering of QCD so that there is no evident suppression; or that jets with fragments with $p_T\ge 3$ GeV/c, which are distributed narrowly about the jet axis, are not strongly affected by the medium [28]. I think that this also agrees with the observation in Figure 36 that two or three orders of magnitude down in the $x_E=p_{Ta}/p_{Tt}$ distributions the A+A best fit is parallel to the p+p measurement, which means that these A+A fragments are from jets that have not lost energy. This is consistent with all the $I_{AA}=x_E^{AA}/x_E^{pp}=(p_{Ta}^{AA}/p_{Ta}^{pp}){|}_{p_{Tt}}$ distributions ever measured (e.g., Figure 41 and Figure 42), which decrease with increasing $p_{Ta}$ until $p_{Ta}\approx 3$ GeV/c and then remain constant because the A+A and p+p distributions are parallel due to no jet energy loss for fragments in this range.