\thechapter Introduction

Hydrodynamic Description of Dilepton Production

Abstract

The first part of this thesis focuses on the production of thermal dileptons from a hadronic gas at finite temperature. The rates are calculated by an expansion in Pion density and constrained by broken chiral symmetry and vacuum correlation functions, many of which have been measured by experiment. We focus on emission processes having two Pions in the final state.

Next, follows a separate discussion on viscous hydrodynamics and its effect on spectra and elliptic flow. A non-central hydrodynamic model of Au-Au collisions in 2+1 dimensions is simulated. Off-equilibrium corrections to the distribution can bring about large changes in the differential elliptic flow, especially at higher . Also discussed is the shear viscous correction to dilepton production in a quark-gluon plasma (QGP) emanating from qq̄ annihilation in the Born approximation. It is argued that a thermal description is reliable for invariant masses less than . Shear viscosity leads to qualitative differences in dilepton spectrum, which could be used to extract information on the thermalization time, viscosity to entropy ratio and possibly the thermalization mechanism in heavy-ion collisions.

Finally, the dilepton rates used in this work are integrated over the space-time evolution of the collision region and compared to the recent results from the NA60 experiment at CERN and the PHENIX experiment at RHIC. The role played by chiral symmetry restoration in the hadronic phase and viscosity in the QGP phase is discussed.

August 2008\programPhysics\directorIsmail ZahedProfessor, Department of Physics and Astronomy\chairmanAlexandre AbanovProfessor, Department of Physics and Astronomy\fstmemberAxel DreesProfessor, Department of Physics and Astronomy\outmemberDmitri KharzeevSenior Scientist, Brookhaven National Laboratory

\makeapproval{dedication}

To my family.

List of Figures:
Acknowledgements.
First and foremost, I must thank Ismail Zahed. Without his patience and guidance this thesis would not have been possible. I am also grateful to Derek Teaney who also advised me on many of the topics in this thesis and played a role throughout. Both Derek and Ismail, trained me thoroughly and played an intricate role in my development as a scientist. I would also like to thank Gerry Brown and Edward Shuryak for their encouragement and support throughout my PhD. I am also grateful to have been a member of the Stony Brook graduate student community. The high quality of graduate course work at Stony Brook was critical for my scientific development and intellectually enlightening. I was also fortunate enough to study as an undergraduate at the The Cooper Union where I received the strong foundation necessary for any academic endeavour. This was only possible because of Peter Cooper’s conviction that “education should be as free as water or air.” My fellow graduate classmates have also been a driving force in my studies. I would like to thank Clint Young and Shu Lin for being backboards in which to bounce ideas from. Finally, I am forever indebted to my wife Shaughnessy for introducing me to a life outside of Physics.

Chapter \thechapter Introduction

1 Basics of Quantum Chromodynamics

Quantum chromodynamics (QCD) is believed to be the theory which describes the strong interactions of quarks and gluons which are found in hadrons. The dynamics of both quarks and gluons are dictated by the QCD Lagrangian [[1]]

(1)

where we have used the following notation

(2)

and is the gluon field having color index and is a quark field having flavor index and color index .

Unfortunately there is no calculational scheme which works well for all energies. For large momentum transfers asymptotic freedom states that the coupling constant becomes small, therefore allowing for a perturbative treatment. However, as the momentum transfer decreases the coupling becomes larger binding the quarks and gluons into hadrons. This leads to the property of confinement, whereby the force required to separate two quarks increases with their relative distance.

Since perturbative calculations are only allowable at high energies due to asymptotic freedom non-perturbative methods have been developed in order to gain insight into QCD. One of the more well established non-perturbative approaches is lattice QCD. Only recently have first principal calculations by lattice QCD in the strong coupling regime become available, albeit with limitations. Since the metric in lattice QCD is Euclidean the calculation is limited to static properties. For example, it becomes very difficult, if not impossible, to calculate scattering amplitudes or transport coefficients.

There have also been many non-perturbative methods developed based on effective theories. In order for these theories to represent nature they should contain the same symmetries of QCD. QCD with massless quarks and flavors has an exact global flavor symmetry called chiral symmetry. This symmetry is spontaneously broken generating three (for flavor) Goldstone bosons called the Pions. In the real world the quarks are massive and electromagnetism is present, so the flavor symmetry is only approximate, leading to pseudo-Goldstone bosons having a small mass which can be calculated in the framework of chiral perturbation theory. This explicit breaking gives rise to the partially conserved axial current (PCAC) hypothesis, with the axial-vector current.

2 Heavy Ion Collisions

Chiral symmetry which is spontaneously broken in the QCD vacuum is partially restored at finite temperature and/or density. In addition, as the temperature is increased from zero, it is thermodynamically favorable for there to be a phase transition from a resonance gas of hadronic bound states to a quark gluon plasma. One of the goals of the heavy ion collision program is to produce a quark-gluon plasma and study its properties. It has already been accepted by many in the heavy ion community that a quark gluon plasma (QGP) has been triggered at RHIC consisting of a strongly interacting, low viscosity fluid.

In order to confirm these conclusions and quantify the properties of the QGP, a detailed study of heavy ion phenomenology is required. One of the main experimental observations that led to the conclusion of the low viscosity nature of the QGP is the large amount of collective flow of the produced particles and its interpretation as coming from a hydrodynamic expansion. Even though hydrodynamic behavior is able to explain a large amount of the available hadronic data it fails at a number of places, such as at high transverse momentum () and at forward rapidity. It is believed that the deviations from ideal hydrodynamic behavior could be explained by dissipative effects.

In order to quantify these assertions, viscous relativistic hydrodynamic simulations have to be developed. The first order Navier-Stokes theory is plagued with difficulties (e.g. the parabolic nature of the equations permit acausal signal propagation). In order to correct for this unsatisfactory behavior a number of second order theories have been developed. At this point in time there is still not a consensus in the heavy-ion community on which theory is appropriate to use. A full study of viscous hydrodynamics is not only imperative for making quantitative predictions on the properties of the matter produced at RHIC but also helps in our theoretical understanding of kinetic theory results. The importance of having a theoretical understanding of viscous relativistic hydrodynamics is not limited to heavy-ion collisions but is also necessary, for example, in cosmological simulations of the early universe.

A second interesting phenomenological tool to study heavy ion collisions is electromagnetic probes. In contrast to hadronic observables which interact strongly throughout the entire evolution of the heavy ion collision, electromagnetic probes leave the medium without further interaction and therefore carry direct information on the time evolution of the system [[2]]. This is in contrast to hadronic observables which thermalize after the collision and thus provide information only on the late stages of the evolution.

In theory, the electromagnetic spectral function of the quark-gluon plasma could be extracted from thermal photon and dilepton emission, which would in turn permit one to learn about its properties (e.g. transport coefficients, presence of bound states, etc.) In practice, however, this is not possible since the QGP dilepton yields are quenched by hadronic emission. Therefore, in order to probe the QGP phase, there must be a solid theoretical understanding of the hadronic emission processes.

There is a long history of experimental dilepton measurements [[3]] which we don’t attempt to summarize here. In regard to dilepton measurements from heavy-ion collisions there were three experiments prior to the recent results from NA60 and PHENIX. These three past experiments were the NA45, HELIOS-3 and NA38/50, which were all performed at the CERN SPS collider, and focused respectively on low, intermediate and high mass dileptons. All three experiments found an enhancement in the dilepton yields above expected hadronic sources (which is comprised of a cocktail designed to describe the measured dilepton spectra in p-p and p-Be collisions). The quality of data however was limited. Dilepton measurements in general are much more difficult then measuring hadronic observables. Not only is there a large background which must be rejected but the cross sections involved are also relatively small. These two facts together require high luminosity experiments in order to collect precision data.

Let us discuss the low mass enhancement found at NA45. For central S-Au and Pb-Au collisions NA45 found an enhancement by as much as a factor of 3-5 above known sources in the emission of di-electrons with invariant masses GeV. A number of theoretical explanations were given for this phenomenon including melting of the due to chiral symmetry restoration [[4], [5], [6], [7], [8]]. The statistics were unfortunately too poor to confirm these predictions. The upgrade from NA50 to NA60 consisted of a new vertex tracker, which now allows track matching in both coordinate and momentum space. This leads to a considerable improvement in statistics and should allow one to discern between different theoretical approaches.

Recently, the PHENIX experiment has also measured di-electron invariant mass spectra and found an enhancement by a factor as large as 7-8 above the cocktail for the most central collisions. At first glance it appears that this result may be inconsistent with the measurements by NA60. However, one must remember that the resultant yields must first be folded through the complicated detector acceptance which is specific to either PHENIX or NA60.

In order to support these interesting experimental programs it is necessary to generate realistic dilepton predictions. In this direction we use a comprehensive set of rates for dilepton production taking into account the symmetries of QCD (e.g. broken chiral symmetry) at finite temperature and density integrated over the space-time history of relativistic viscous hydrodynamic simulations of the collision.

3 Outline of this thesis

This thesis is separated into a number of self contained parts. However, the last section on heavy ion phenomenology will rely on all the material presented throughout.

The first part of this work focuses on the work done with my advisor, Ismail Zahed. In chapter two, we discuss the dilepton emission rates from a hadronic gas in thermal and chemical equilibrium. The rates take into account broken chiral symmetry in a consistent manner and rely on experimental data as input. The rates are treated in a density expansion and the effects of one and two pions in the final state are explored. The new work consisted of evaluating the rates to second order in pion density, which include all hadronic processes involving two pions in the final state. This work is currently unpublished.

Chapter three is a separate discussion on viscous hydrodynamic simulations. The work was done under the auspices of Derek Teaney and was published in [[9]]. We examine how shear viscosity changes the ideal hydrodynamic evolution and the effect it has on differential transverse momentum and elliptic flow spectra.

Chapter four goes back to dilepton production, this time from a quark gluon plasma out of kinetic equilibrium. In collaboration with Shu Lin [[10]] we consider how shear viscosity modifies the leading order born dilepton production rates.

In chapter five all the pieces are put together. The dilepton rates are integrated over the space time evolution presented in chapter three. Most of the work is done in kinetic equilibrium and was published in [[11], [12], [13]] and was done in collaboration with Ismail Zahed and with help from Derek Teaney regarding the hydrodynamic evolution and equation of state. A final section in chapter five discusses the role of shear viscosity on dilepton emission from both the QGP and hadronic phases.

Chapter \thechapter “Master Equation”Approach to Dilepton Production

4 Introduction

It can be shown [[112]] that to lowest order in electromagnetic interactions and to all orders in strong coupling the differential rate for dilepton pair production can be expressed in terms of the correlation function of the hadronic electromagnetic current.

When lepton mass is ignored the rate is given by

(3)

where

(4)

In the above equations is the time-like four-momentum of the lepton pair, is the hadronic part of the electromagnetic current and stands for the thermal averaging at a temperature .

In general, there are two ways in which the above thermal structure function, , can be evaluated. The first is by kinetic theory. By inserting a complete set of states for each incoming component of the thermal density matrix the above equations can be shown to agree with relativistic kinetic theory reaction by reaction. This method of evaluation is not only cumbersome but also relies on many approximations, such as the choice of Lagrangian and coupling constants.

A second approach, which is used here, is to relate the thermal structure function directly to spectral functions as was first done by Z. Huang [[15]]. From the spectral representation and symmetry the thermal structure function can be related to the absorptive part of the time ordered function

(5)

where

(6)

Let us now evaluate not reaction by reaction as done in kinetic theory but instead in a low temperature expansion as was first done by Dey, Eletsky and Ioffe [[16]]. At low enough temperature the heat bath will be dominated by pions and therefore one can keep the first term in the expansion of the trace in the thermal averaging. We quote the result and leave the details for the appendix. To leading order in one finds

(7)

where and are the axial-vector and vector correlators. This is an example of how the dilepton production rate at finite temperature can be determined from measurable experimental data at zero temperature. The above result shows that the vector and axial-vector spectral densities mix at finite temperature. Chiral symmetry makes the statement that . To leading order in temperature this occurs when MeV.

Even though the above result was restricted to zero momentum pions it is general in the sense that it was derived from current algebra and PCAC alone. As the number of soft-pion fields emitted or absorbed grows the current algebra formulation becomes increasingly difficult. For this reason Weinberg [[17]] developed a method of calculating current algebra results using an effective Lagrangian formulation at tree level. By renormalizing the tree level results one could obtain corrections to the soft-pion theorems. A one loop calculation can still only describe data up to about 200 MeV above threshold. Two loop calculations are intractable since over 100 new low energy constants appear.

A program that extends chiral symmetry consistently into the resonance region without the soft-pion restriction is discussed in the next section.

5 Chiral Reduction Formula

The limitations of current algebra and chiral perturbation theory can be avoided by instead using an S matrix formalism. H. Yamagishi and I. Zahed [[18]] have derived a single equation (coined the “Master Equation”) that contains all of the low energy theorems of current algebra.

The starting point for this program is an action I with its kinetic part invariant under local that is gauged with external sources. Examples are two-flavor QCD or the nonlinear sigma model.

For two-flavor QCD the action is given as

(8)

where is the gluon field strength tensor defined by

(9)

By Noether’s theorem currents are defined by

(10)

where and . The currents must satisfy the Noether’s equations in the presence of sources (also known as the Veltman–Bell [[19]] equations):

(11)
(12)

In the above equations is the vector covariant derivative and we have used the notation that . Schwinger’s quantum mechanical action principal

(13)

along with the completeness of asymptotic states leads to the Peierls-Dyson formula [[20]]

(14)

The Veltman-Bell equations can now be recast into the following form

(15)
(16)

where we have defined

(18)
(19)

It can be shown that and are the generators of local .

So far we have not considered whether chiral symmetry is present or explicitly and/or spontaneously broken. The spontaneous breaking of chiral symmetry is expressed in terms of the following asymptotic condition on the axial-vector field

(20)

The above condition assumes the absence of any additional stable axial vector or pseudo-scalar resonances. For explicit chiral symmetry breaking we must also impose

(21)

In order simplify the incorporation of the above boundary conditions into the approach a modified action and modified S-matrix are introduced.

(22)
(23)

Also needed is a change of variables, . We now use as independent variables and as modified current densities defined analogously as

(24)

Under this new change of variables the Veltman-Bell equations LABEL:eq:VB can be integrated upon introduction of a retarded and advanced Green function yielding a relation between the pion field and the other currents. After some manipulation these relations can be written in the following form

With the above master equation in hand it is easy to see a strategy in order to generate Ward identities. After reducing out a pion from a scattering amplitude the commutator on the LHS can be replaced by an operator of many functional derivatives acting on the scattering matrix on the RHS. These variations on the matrix can be written as time ordered products by

(26)

where .

6 Leading Order Lepton Emission:

The rate of dilepton emission per unit four volume for particles in thermal equilibrium at a temperature is related to the thermal expectation value of the electromagnetic current-current correlation function [[21], [22]]. For massless leptons with momenta and , the rate per unit invariant momentum is given by:

(27)

where , is the temperature and

(28)

where is the hadronic part of the electromagnetic current, H is the hadronic Hamiltonian and is the free energy. The trace is over a complete set of hadron states.

In order to take into account leptons with mass the right-hand side of Eq. 27 is multiplied by

(29)

Even though there are various approaches to calculating production rates, they differ in the way in which the current-current correlation function in Eq. 27 is approximated and evaluated. The approach taken here is to use a chiral reduction formalism in order to reduce the current-current correlation function in 28 into a number of vacuum correlation functions which can be constrained to experimental annihilation, -decay, two-photon fusion reaction, and pion radiative decay experimental data.

For temperatures T the trace in Eq. (28) can be expanded in pion states. Keeping terms up to first order in pion density yields [[6]]

(30)

with the phase space factor

(31)

The first term in 30 is the transverse part of the isovector correlator which can be determined experimentally from electroproduction data and gives a result analogous to the resonant gas model. At low and intermediate invariant mass the spectrum is dominated by the MeV) and MeV).

The term linear in pion density (the second term in Eq. 30) can be related to experimentally measured quantities via the chiral reduction formulas [[23]]. It is shown in [[6]] that the dominant contribution comes solely from the part involving two-point correlators which yields:

where is the real part of the retarded pion propagator given by and is the transverse part of the iso-axial correlator . The spectral functions appearing in Eq. (LABEL:eq:lin_in_meson1) can be related to both annihilation as well as -decay data as was compiled in [[24]].

It can be seen in Fig. 1 that the term linear in pion density decreases the rates from the resonance gas contribution for the mass region above the two pion threshold. However below the two pion threshold the only contribution to the rates come from the terms in Eq. LABEL:eq:lin_in_meson1. This is because the axial spectral density is integrated over all momentum in the thermal averaging (Eq. 30), which weakens the factor in Eq. LABEL:eq:lin_in_meson1 allowing the term in Eq. 27 to dominate at low .

Figure 1: (Color online) The total integrated dimuon rates from a pion gas at T=150 MeV. The curve labeled “Res. Gas” shows the analogue of the resonance gas contribution (the first term in Eq. 30). The curves labeled “Vector” and “Axial-Vector” show the contributions from the respective spectral functions in equation LABEL:eq:lin_in_meson1.

7 Second Order Lepton Emission:

7.1 Introduction

We now keep terms up to second order in pion density

(33)

with the phase space factor

(34)

The first two terms in the above density expansion were considered in the previous section. As long as the system is sufficiently dilute (i.e. ) the expansion should converge rather quickly as long as no new thresholds open up. What we will find is that the contribution feeds into the low mass and low region where the zero and first order corrections do not contribute. This will also enhance the real photon rate () at small energy.

7.2 Result

In this section we quote the full on-shell Ward identity for . We note that we correct a number of typographic errors from the result quoted in [[7]]. We also discuss in detail which terms are kept in the numerical calculations and argue which terms can be safely neglected. First let us quote the terms included in the analysis:

(35)
(36)

Equation 35 contains the pion-spin averaged forward scattering amplitude (). This quantity can be constrained from measured photon fusion data by crossing and is discussed in section 7.3. We have defined the term in equation 36 as

(37)

The pions in the above expression for can be reduced out via the chiral reduction formula. Since most of the strength will come from the vector and axial-vector spectral densities we keep these terms only.

(38)

There are additional terms however which we should discuss. Most can be argued away by resonance saturation. One term which we should quote which could appear at higher mass is the following four point function

but we neglect it further in this analysis since we want to focus on the region below the mass. Experimental information about this term could be extracted from scattering data [[1]].

Figure 2: Schematic representation of the chiral reduction of . The dashed line is a pion.

For completeness we now quote the remaining terms of . These are not included in our analysis since they can be argued to be small in the kinematic regions which we are interested in.

(39)
(40)
(43)
(44)
(45)

We now discuss why the above terms are neglected. First look at eq. 39. It is proportional to the principal value of the real part of the retarded pion propagator defined as,

(46)

For on shell pions this term is proportional to and therefore vanishes. Now look at eq. 40 where we have defined as:

Making use of the chiral reduction formula one can reduce the incoming pion with the result:

(48)

From reducing out the incoming pion we find that . The term only contributes when the pion from the heat bath has the kinematics specified by the delta function. Due to the small amount of phase space this term will be suppressed compared to the terms in 35 and 36. The same argument can be made for neglecting eq. 43.

The matrix elements appearing in terms 43, 43 and 43 are shown in figure 3 where we have defined the term in eq. 43 as

(49)
Figure 3: Diagrams of the matrix elements in eqns 43, 43 and 43 (left to right). The dashed line is a pion and the wavy line denotes a photon.

It turns out that these three contributions can be argued to be small. First note that eqn. 43 vanishes in the chiral limit. Furthermore, after reducing out the incoming pions the remaining vacuum spectral functions will mostly consist of and for which the resonance saturation is small. Equation 43 depends on which after chiral reduction mostly reduces to the three correlators; , and for which there is no s-channel cut through resonance saturation. Finally the matrix element in eq. 43 will mostly contribute in the four and six range and higher. Since we are focusing our attention near the threshold it is safe to say that the above processes can be neglected since the correlators contribute at higher mass. In addition these processes are Boltzmann suppressed in comparison to the resonance and one final state reactions.

The final expression, eq. 45, is a direct consequence of the way chiral symmetry is broken through . The term can be related directly to the scalar form factor through and therefore vanishes.

7.3 Pion Compton Scattering Amplitude

We consider the reaction and define the Mandelstam variables to be

(50)

Let us express the total Compton scattering matrix element as

(51)

The pion compton scattering amplitude in the born approximation is by now a textbook example [[25]]. The three Feynman diagrams of figure 4 contribute which evaluate for forward scattering (i.e. and ) to

(52)

where

(53)
Figure 4: Tree level contribution to the scattering amplitude.

However, since we are always below threshold the amplitude is always real and does not contribute to the imaginary part of the amplitude. To go further we make use of the master formula approach of the reaction [[18]]. The pion compton scattering process for real photons () was examined in [[26]]. The dominant contributions from the chiral reduction of the process is