Determination of the Quark Content of Scalar Mesons Using Hydrodynamical Flow in Heavy Ion Collisions
DDetermination of the Quark Content of Scalar Mesons UsingHydrodynamical Flow in Heavy Ion Collisions
M. Wussow and N. Grau ∗ Department of Physics, Augustana College, Sioux Falls, SD 57197 (Dated: November 8, 2018)
Abstract
We study the possibility of determining the quark content of the scalar mesons a (980) and f (980) through their hydrodynamical flow signature in relativistic heavy ion collisions. Utilizingthe constituent quark scaling of hydrodynamic flow, we find that the tetraquark a (980) or f (980)mesons will have a v of 0.38 at transverse momentum of 6 GeV/c in 20-60% central Au+Aucollisions at √ s NN = 200 GeV. The feasibility of measuring a (980) → π η and f (980) → π + π − into the PHENIX and STAR detectors at the Relativistic Heavy Ion Collider (RHIC) is alsodiscussed. Even though the mid-rapidity cross sections for these mesons at high- p T are non-negligible, their broad mass range will make them difficult to detect in both p + p and Au+Aucollisions. ∗ [email protected] a r X i v : . [ h e p - ph ] A ug . INTRODUCTION Nearly all hadrons that have been measured can be described as a color singlet stateof either three quarks or anti-quarks or a quark-anti-quark pair. However, the theory ofQuantum Chromodynamics (QCD) allows more exotic structures to exist, like tetraquarks( q ¯ qq ¯ q ) [1], pentaquarks ( qqqq ¯ q ) [2] or multi-gluon states (glueballs) [3]. Several unexpectedcharm states have been discovered recently. For example, the X (3872) [4] and Z + (4330) [5],have not satisfactorily been explained as two-quark states but could be tetraquark states[6, 7].Not only could exotic quark states exist at the higher masses, but several well-establishedlow mass resonances are candidates for multiquark states. In Ref. [1], Jaffe showed that thelowest mass scalar q ¯ qq ¯ q states form a nonet, several of which describe established particles.High-statistics measurements from radiative φ decays to the f (980) [8] and a (980) [9]indicate that these are consistent with four-quark states.Assigning valence quark content for a particle is based on the determination of its quan-tum numbers. It would be extremely useful to have a more direct measurement of theirvalence quark structure. It appears that data from relativistic heavy ion collisions from theRelativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory have yielded adynamical observable that is sensitive to the number of valence quarks of a hadron.In collisions of gold nuclei at √ s NN = 200 GeV, a quark-gluon plasma is formed andbehaves as a strongly-interacting perfect fluid [10, 11]. In collisions with a non-zero impactparameter, the overlap region forms an almond-like shape. This is depicted in Fig. 1. Becauseof thermodynamic pressure gradients in this region, particle production is not azimuthallysymmetric. More particles are emitted along the reaction plane, the plane resulting fromthe beam and impact parameter axes, than perpendicular to it. This azimuthal asymmetryis characterized by the coefficients of a Fourier expansion of the invariant yield [12]1 p T d Ndp T dϕdy = 12 πp T d Ndp T dy × (cid:34) (cid:88) n v n ( p T , y ) cos ( nϕ ) (cid:35) , (1)where p T , y , and ϕ are the particle’s momentum transverse to the beam direction, rapidiy,and angle with respect to the reaction plane, respectively. The v n ( p T , y ) encode the strengthof the azimuthal asymmetry. 2 ReactionPlane In PlaneOut ofPlane h (cid:160)
FIG. 1. A beam view of a relativistic heavy ion collision. The impact parameter and the beam axis(into the page) form the reaction plane. The shaded, almond-shaped overlap region is the sourceof particle production, which is asymmetric in ϕ . When fluctuations in the initial energy and entropy density of the overlap region aresmall, symmetry requires only even Fourier coefficients to be non-zero. The dominant termin the series is v . Both the PHENIX and STAR experiments have measured a large numberof identified baryon and meson v as a function of p T [13–16]. These data follow a universaltrend in v /n q vs. KE T /n q (see Fig. 2 and 3). Here, n q is the number of valence quarks ofthe particle and KE T = m T − m = (cid:113) p T + m − m is the transverse kinetic energy of theparticle. Since this data fits light-flavor (up, down, and strange) baryons and mesons for KE T /n q < ∼ v is sensitive to the number of valence quarksof that particle.Such a result can be explained quantitatively by the recombination model of hadroniza-tion [17]. When the quark-gluon plasma cools, hadronization must take place. The recom-bination model allows for quarks nearby in phase space to cluster and, with appropriatequantum numbers, form hadrons. The first model of recombination explained the measuredenhancement of the p/π + ratio in Au+Au collisions compared to p+p collisions. Because ofthe steeply falling momentum spectrum of quarks, the ability for three quarks of approxi-mately equal momentum combining to form a proton is more favorable than two combiningto form a meson with the same final momentum as the proton.3n such a model, the quark v will become the hadron v based on the combination ofthe quarks that take place. The initial suggested scaling of v was [18] v ( p T ) ≈ n q v q ( p T /n q ) (2)However, detailed measurements of v showed this scaling was incomplete and the KE T scaling was found to work better experimentally [15]. A more detailed recombination model,where one considers resonance-like scattering in the pre-hadronic state, which conservesenergy and momentum in detail, is able to reproduce the observed KE T scaling [19–21].With the discovery of constituent quark scaling of hadron v at RHIC, we would like tosearch for evidence of particles with exotic quark structures. In this paper we present thestudy of the flow of light scalar mesons, f (980) and a (980), at RHIC. If the recombinationmodel of hadronization is responsible for the translation of v of quarks to v of hadrons atRHIC, then these mesons will follow the observed scaling of v and their number of valencequarks will then be known. A definitive measure of a tetraquark state would be an importantresult for QCD. II. FLOW OF SCALAR MESONS
Using data it is possible to determine an expectation for the v signature of the a (980)and f (980). Fig. 2 shows the compilation of PHENIX data for v of identified hadronsin 20-60% central Au+Au collisions at √ s NN = 200 GeV [13, 14]. The v /n q is plottedas a function of KE T = m T − m . Up to KE T /n q of approximately 1 GeV, all of thedata follow the same trend. The PHENIX π data were published in 10% centrality bins.They were combined for the figure using an eccentricity weighted average of the appropriatecentrality bins. The eccentricity was determined by a Glauber Monte Carlo of the Au+Aucollisions [22]. We fit a 4 th -order polynomial to the combined pion data: π ± for KE T /n q < π for KE T /n q > v from STAR in minimum bias (0-80%) central Au+Aucollisions at √ s NN = 200 GeV [16]. A fit to the K S data, which exists over the broadestrange in KE T /n q , is also shown as the dashed line.From here it is straight-forward to undo the scaling and plot the v as a function of p T byknowing the mass and the number of valence quarks of the particle of interest. This is shown4 (GeV) q /n T KE q / n v PHENIX Au+Au 20-60% (cid:47) Kp (cid:113) (cid:47) FIG. 2. (Color online) The combined PHENIX v data [13, 14] for identified π ± (open circles), π (green closed circles), K ± (red squares), p (¯ p ) (blue triangles), and φ (magenta inverted triangles)in 20-60% central Au+Au collisions at √ s NN = 200 GeV. The black line is a fit to the combinedpion data. (GeV) q /n T KE q / n v (cid:47)(cid:82) S0 K (cid:85) FIG. 3. (Color online) The combined STAR v data [16] for identified protons (green open circles),Λ (open squares), Ξ (red crosses), π ± (magenta circles), and K S (blue squares) in 0-80% Au+Aucollisions at √ s NN = 200 GeV. The dashed line is a fit to the kaon data. in Fig. 4 for 20-60% collisions and Fig. 5 for 0-80% collisions for a particle of mass 980 MeVand consisting of either two (dashed line) or four quarks (dot-dashed line). There is a verylarge separation between these two curves. It is rougly consistent with the pocket formulaEq. 2, where the 4-quark v curve is roughly twice the 2-quark v , especially at intermediate p T where the recombination model dominates the particle production. In the mid-centralcollisions (20-60%), the 4-quark v at 6 GeV is 2.6 times larger than the 2-quark v andreaches a value of 0.38, a v that dwarfs all other v measurements from light hadrons. Even5 (GeV) T p v /f a 2-quark /f a (cid:47) FIG. 4. (Color online) The extrapolated v of the a (980) or f (980) in 20-60% Au+Au collisionsas a function of transverse momentum if they are either 2-quark states (dashed blue line) or 4-quarkstates (red dot-dashed line). For comparison the PHENIX pion v is given as the solid (black) line. (GeV) T p v Min. Bias Au+Au 4-quark /f a 2-quark /f a S0 K FIG. 5. (Color online) The extrapolated v of the a (980) or f (980) in minimum bias (0-80%)Au+Au collisions as a function of transverse momentum if they are either 2-quark states (dashedblue line) or 4-quark states (red dot-dashed line). For comparison the STAR K S v is given as thesolid (black) line. in minimum bias collisions the 4-quark v reaches a value of 0.30 at 6 GeV and is 2.5 timeshigher than the 2-quark v .We note that, in both the minimum bias and in mid-central collisions, the 4-quark v peaks between p T of 5-6 GeV or KE T /n q = 1-1.2 GeV. Therefore, the entire curve is withinthe measured KE T scaling of v at RHIC. We expect the 2-quark v to follow the trend ofthe π or the K S and decrease as a function of p T , widening the gap between the expected v signatures. 6 II. FEASIBILITY OF MEASURING SCALAR MESONS AT RHIC
In order to establish that a measurement of the v can be made, we must get a sense ofthe signal strength. Important in v measurements of resonances is the ability to measurethe signal and the background v . In this section, we estimate the signal-to-background in p + p and Au+Au collisions to determine the feasibility of the measurement.In Au+Au collisions there will be at least three production sources: from radiative φ decays, “prompt” production from fragmentation, and production from recombination. Wetake the approach of estimating the first two components of these production mechanismsin p + p collisions. From this we can evaluate if the measurement in heavy ion collisions isfeasible.Specifically, we are interested in the cross section of these mesons at high p T , i.e. at6 GeV where there is maximal difference between the 2-quark and 4-quark v . We focuson the RHIC experiments where these mesons would be measured. The a (980) → π η decay would be possible in PHENIX where their calorimeter has measured both π [23] and η [24] inclusive production in p + p and heavy ion collisions. The f (980) → π + π − wouldbe a favorable decay for STAR, which has particle identification capabilities over a largekinematic range [25].To estimate the cross section, we first study the radiative φ decays. PHENIX has mea-sured the cross section of φ and several other neutral mesons in p+p collisions at √ s = 200GeV [26]. The spectral shape for all mesons is well described with a Tsallis distribution [27], E d σdp = 12 π dσdy ( n − n − nT + m ( n − nT + m ) × ( nT + m T nT + m ) − n (3)where dσ/dy is the integrated cross section of the particle production at midrapidity. Wemodeled the radiative φ → a (980) γ decay using the measured spectral shape from PHENIX: dσ/dy = 0.42 mb, n = 10, and T = 120 MeV. With the measured branching fraction fromKLOE (7 × − ), the dσ/dy for 6 GeV a (980) production into the PHENIX pseudorapidityacceptance from φ decays is 9 nb. Requiring that the η daughter decays to 2 photons, whichis the most efficient method to measure η mesons in PHENIX, reduces the cross section to3 nb. If, instead, the φ would radiatively decay into an f (980), based on the measuredbranching fraction from KLOE (10 − ), dσ/dy for 6 GeV f (980) into STAR would be 58 nb.7f you require the f (980) to decay into two pions, this reduces the cross section to 19 nb.These cross sections, therefore, represent the lower bounds on the production cross sectionfor these scalar mesons into PHENIX and STAR, respectively.There should be an additional prompt production of a (980) and f (980) mesons fromfragmentation. This has not been measured at collider energies. We estimate the promptcross section the following way. If the mesons are two-quark state, because their mass andquark content are similar to that of the η (cid:48) , their prompt production will be similar. PHENIXhas measured the η (cid:48) production cross section in p+p collisions [26]. Its spectral shape is alsogiven by the Tsallis distribution, Eq. 3. We use the PHENIX parametrization, dσ/dy = 0.7mb, n = 10, and T = 120 MeV, which yields a dσ/dy = 87 nb at 6 GeV. Requiring the η from the a (980) decay into two photons reduces the cross section to 34 nb. The f (980)decaying into two pions reduces the cross section to 29 nb.For a lower bound estimate of the prompt cross section, we use the following argument.If the mesons are 4-quark states, their production compared to 2-quark fragmentation wouldgo approximately as the ( p/π + ) . This is because this ratio encodes the difference in thestring fragmenting into di-quark-anti-di-quark compared to quark-anti-quark pairs. In p + p collisions, this ratio is approximately 0.1. So we would expect that a lower estimate of theprompt cross section would be a factor of 100 less than the 2-quark production. In this case,the cross section for 6 GeV a (980) into PHENIX and 6 GeV f (980) into STAR is 0.34 nband 0.29 nb, respectively. It is interesting to note that because the η (cid:48) cross section at 6 GeVis the same order of magnitude as the cross section from φ decays, the lower bound valuesare significantly less than the radiative φ production. Therefore, the φ decays would reallyrepresent an absolute lower bound for their production in p + p collisions.To find the signal-to-background ratio, we generated PYTHIA p+p events at √ s = 200GeV with Tune A [28]. For the a (980) background all π - η pairs, each decaying to twophotons, were combined. Since this will be measured into the PHENIX acceptance, we alsoapplied p T cuts consistent with the PHENIX capabilities [23, 24]. We required π p T > η p T > | y | < a (980) would have a cross section similar tothe η (cid:48) , then a peak is barely visible in the invariant mass distribution of π − η pairs. If one8 (GeV) (cid:100) (cid:47) m ( nb / G e V ) (cid:100) (cid:47) / d m (cid:109) d (980) + PYTHIA aPYTHIA background (GeV) (cid:100) (cid:47) m ) s i gna l / ba ck g r ound ( x FIG. 6. The 6 GeV pair p T π η invariant mass distribution for background from PYTHIA (dashed)and the parametrized a (980) signal (solid) assuming the same cross section as the η (cid:48) . In both thesignal and the background, the π and η decay to two photons and are required to be within thePHENIX y acceptance. The inset is the signal-to-background ratio over the full mass range of the a (980). integrates over the full mass range of 0.7-1.0 GeV, the signal-to-background is 10 − . Theinset in Fig. 6 shows the signal-to-background as a function of invariant mass. It reaches amaximum of 3 × − near 980 MeV. It follows that if the φ decay is the dominant productionmechanism, the signal-to-background will be reduced. With the p T cuts above, the signalfrom φ decay reduces to 2 nb over the whole mass range and the signal-to-background wouldbe reduced to 5 × − . Even though the cross section is small but measurable with RHICluminosities, the fact that the resonance is spread over 300 MeV will make the measurementdifficult. Additional cuts would be required to improve the signal-to-background.With the same PYTHIA background, we studied the f (980) signal-to-background intothe STAR acceptance. We assume the following cuts on the pion daughters of the f (980)decay: p T > | y | <
1. Even though the STAR experiment has particle identificationcapabilities below this, a higher- p T cut will naturally reduce the background. With thesecuts the signal is reduced to 22 nb. The resulting signal and background π + π − invariantmass distributions are shown in Fig. 7. Even though the f (980) cross section into STARis comparable to the a (980) into PHENIX, the π + π − background is larger. An integratedsignal-to-background is 4 × − . The inset shows the signal-to-background over the entiremass range.With these upper estimates of the signal-to-background values in hand, we estimate9 (GeV) - (cid:47) + (cid:47) m b / G e V ) µ ( - (cid:47) + (cid:47) / d m (cid:109) d (980) + PYTHIA fPYTHIA background (GeV) - (cid:47) + (cid:47) m ) s i gna l / ba ck g r ound ( x FIG. 7. The 6 GeV pair p T π + π − invariant mass distribution for background from PYTHIA(dashed) and the parametrized f (980) signal (solid) assuming the same cross section as the η (cid:48) .In both the signal and the background, the π ± are required to be within the STAR y acceptance.The inset is the signal-to-background ratio over the full mass range of the f (980). what they will be in heavy ion collisions. We assume the production of high- p T particles, φ , π , η and π ± , will scale as the number of nucleon-nucleon collisions, N coll , in the event.Therefore, we would naively estimate that the signal, which will scale as N coll , divided bythe background, which scales as N coll , would be reduced by a factor of N coll . In mid-centralcollisions (20-60%), a Glauber model yields an N coll of 200. This would be the reduction inthe signal-to-background with no additional sources of these mesons in heavy ion collisions.There is an additional increase in the a (980) and f (980) signal from recombination. Wefollow the estimation technique in Ref. [17] to obtain an order-of-magnitude estimate of theenhanced yield of tetraquarks due to recombiation. In that paper, the yield of mesons fromrecombination is related to products of the quark and anti-quark Wigner functions. Fortetraquark states, the yield from recombination will be related to the products of two quarkand two anti-quark Wigner functions. Assuming that the Wigner functions are exponentialand independent of the quark or anti-quark internal quantum numbers, the a (980) /η (cid:48) willbe a ratio of sums over the valence quark flavor, color, and spin. Using the notation ofRef. [17] this is dN q ¯ qq ¯ q dN q ¯ q = (cid:80) a,b,c,d (cid:80) a,b (4)where a , b , c and d represent the internal quantum numbers of valence quarks in the meson.This ratio is on the order of 10. The estimate assumes that the η (cid:48) mesons will come predom-10nantly from recombination. At 6 GeV, where we would like to apply this ratio, this mightnot seem to be a good assumption. However, as seen in Fig. 4 and 5, the tetraquark mesonproduction mechanism will be dominated by recombination up to this point. We have alsoneglected the reduction of the η (cid:48) production due to energy loss [24]. Including this estimateof the increase in yield due to recombination, the signal-to-background will decrease by afactor of 20 compared to p + p collisions.One could expect that there is additional signal from π − η and π − π rescattering in thehadronic state, which would enhance the yield of these mesons and would mimic a 4-quark v signal. We have checked this contribution using UrQMD [29] which simulates both a (980)and f (980) resonance production in the hadronic state. We ran UrQMD in minimum biasmode and determined the absolute cross section for each resonance at 6 GeV. For this weused σ AuAu = 6.4 b [22]. For the a (980) we determined a mid-rapidity cross section requiringthe daughters to decay to photons and require the same kinematic cuts into PHENIX aswe described above. We find a cross section of of 0.9 µ b. This should be compared withthe N coll - and recombination-enhanced cross sections fromx the the p + p production. Fromabove the the φ and η (cid:48) bound the cross section in p + p into the PHENIX between 2-22 nb. InAu+Au from N coll and recombination we expect the production cross section to be 4-44 µ b.The lower bound is a factor of 4 above the hadronic-rescattering cross-section. Similarly, the f (980) 6 GeV cross section into STAR using the same kinematic cuts as before was foundto be 0.1 µ b. The cross-section bounds on the f (980) in p + p collisions is between 14-22nb. In Au+Au we would expect this cross section would be between 30-44 µ b. Therefore,for f (980) production, the hadronic-scattering component is negligible. For the a (980), itis a small component. IV. DISCUSSION
From the previous section it is clear that the measurement of these resonances in heavy ioncollisions will be difficult. Suitable background rejection must take place. An example wouldbe placing higher p T cuts on the daughters in order to reduce the combinatorial background.However, increasing the π cut to be greater than 2 GeV when considering a (980) decays inPHENIX, will only increase the signal-to-background by 30%. One might wish to search forthe radiative φ decays in heavy ion collisions. Requiring that the resonance combined with a11hoton be in the φ mass range will certainly reduce the background significantly. However,the mesons from the radiative decays would not have the flow signature of interest. Becauseof the small Q of the decay, the a (980) or f (980) would generally be aligned with thehigh- p T φ and exhibit the 2-quark v signature of the φ . Potentially, one could measureinclusive a (980) and f (980) and compare with those from φ decay since their v signatureswould be different. However, this would require a long running time to achieve the necessarystatistics for such a measurement.This idea could be extended to other narrower tetraquark candidates, for example the X (3872) → J/ψπ + π − . The advantage of measuring this state would be the narrow 3 MeVwidth [30]. The drawbacks for this measurement at RHIC is that the cross section is quitelow considering the rate that was observed at Fermilab in inclusive production [31, 32].Furthermore, because the charm quark is massive compared to light quarks, its v is likelynot the same as the light quarks [33, 34]. The charm quark v has not been measured directlyat RHIC, but only indirectly from the electrons from semi-leptonic decays [35, 36]. In thatmeasurement there is a mixture of bottom and charm sources [37, 38]. At increased RHICluminosity, however, it may be possible to measure this resonance. Such a measurementmight be more feasible at the Large Hadron Collider where the rate is higher.From the empirical evidence for constituent quark scaling, we expect a large v for theexotic scalar mesons a (980) and f (980) in √ s NN = 200 GeV Au+Au collisions at RHICif they are tetraquark states. Their v reaches a value of 0.38 in 20-60% collisions at 6 GeVand is a factor 2.6 higher than the 2-quark v at the same p T . The mid-rapidity cross sectionof these mesons in p + p collisions will be larger than 19 nb for the f (980) and 3 nb for the a (980), which results from their radiative φ decays. This is not a negligible cross section atRHIC. Because of the broad mass structure, the signal-to-background is at and below 0.001over the full 300 MeV mass range of the resonances. The signal-to-background will further bereduced in heavy ion collisions due to the increase in the underlying event multiplicity. Evenwith the enhanced production due to recombination, the signal-to-background will decreaseby a factor of 20. Therefore, without taking steps to increase the signal-to-background, themeasurement will be difficult. 12 CKNOWLEDGEMENTS
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