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DYNAMICAL APPROACH TO PAIR

arXiv:hep-ph/9209212v1 4 Sep 1992LA-UR-92-2753DYNAMICAL APPROACH TO PAIRPRODUCTION FROM STRONG FIELDSFred CooperTheoretical Division T-8Los Alamos National LaboratoryMS B-285 Los Alamos, NM 87545 USALectures given at the NATO ADVANCED STUDY INSTITUTEParticle Production in Highly Excited MatterII Ciocco, Italy July 19928/27/92

1. INTRODUCTIONIn relativistic heavy-ion collisions one is hoping to produce conditions where energydensities are high enough so that a new state of matter– the quark-gluon plasma can beproduced.

This state of matter lasts for a short period of time following the collision andmay or may not be in equilibrium. Following this phase a transition to ordinary hadronicmatter takes place and many of the processes which occur during the quark-gluon plasmaphase might be masked by processes which occur in the hadronic phase.In order todetermine processes which might be signals of the quark gluon plasma one needs to knowthe dynamical evolution of the plasma.

This is because the particles that get producedduring that phase have to travel through a time evolving plasma. In order to study thisproblem one needs a different way of thinking about field theory.

Traditionally experimentsin Elementary Particle Physics are black box experiments where initial particles enter aregion, final particles exit the experimental region and all that is asked is how manyparticles of what type, energy, etc.enter various detectors.This type of experimentrequires only a covariant S-Matrix theory to predict the probabilities to be expected in thedetectors. However, if we want to know signatures of the quark gluon plasma,we actuallyneed to follow the time evolution of the plasma and fields produced following the heavyion collision.This requires a non-covariant real time formalism for the time evolutionof the quantum fields.In these talks we would first like to discuss various formalismsfor doing real time calculations in quantum field theory and then study in detail a verysimplified model of the production of the quark-gluon plasma– Schwinger’s mechanism forPair-production from strong “classical” gauge fields.

The value of doing a “first principles”calculation at this time, even if it is over-simplified, is multifold:(1) We can test the validity of existing semiclassical transport models of lepton productionfrom the quark-gluon plasma. We have already discovered that these models have tobe modified to correctly include Pauli blocking and Bose enhancement effects whichwere ignored.

(2) We can determine the effective hydrodynamics and show that certain kinematic as-sumptions automatically lead to flat rapidity distributions independent of the form ofthe equation of state. (3) We can determine the dynamical equation of state and in the next order in a systematiccalculation in powers of (1/N) we will be able to study whether equilibration will occurand calculate self consistently lepton pair production rates.1

First I would like to list the various approaches available to study real time processesin Quantum Field Theory.Each of these approaches needs an approximation schemeto reduce the number of degrees of freedom in order to make the problem numericallytractable. Three methods that my collaborators and I have studied in detail are:1- Functional Schrodinger Equation + variational approximations [1] [2][3]2- Truncated Heisenberg Equations – Large-N expansion or Mean-field approximationsto the Dyson equations [4] [5][6] [7].3- Schwinger’s closed time path - Path Integral Formalism in a large-N expansion[8][9][10][11].There also exists an alternative formalism related to the truncated Heisenberg equa-tions based on the Wigner Distribution function which has been discussed by Rafelski andhis collaborators [12].

Once we have chosen a method we have to decide how to specifythe initial data at t=0. In these different approaches we have to specify1- Initial position and width of say a Gaussian Wave Packet at t=0 in the Schrodingerpicture2- Number and pair densities etc.

in Heisenberg picture.3- Initial density matrix in the Path Integral Approach.In classical field theory, such as classical electrodynamics, the theory is finite and anysmooth initial configuration of the field is allowed for the initial value problem. When wehave a semiclassical field theory for the expectation value of the fields however, the initialdata can be reinterpreted in terms of the particle language and even a smooth initialconfiguration of the field might not be consistent with certain physical constraints such asthe initial state having finite number density at t=0 with respect to an adiabatic vacuum.

(This requirement is automatic for finite temperature field theory). Thus arbitrary initialdata may not be consistent with renormalizability.

This is discussed in detail in [3][4].We also have an additional new problem to face – how to perform renormalization in anon-covariant formulation of the field theory. To do this we isolate the divergences in anadiabatic (WKB) expansion of Green’s functions.

This method is similar to the techniqueof adiabatic regularization used by Parker and Fulling [13] in their study of semiclassicalgravity. The problem we will address in detail in these lectures is pair production of eitherBosons or Fermions from strong Classical Fields which are either functions of time t, orfluid proper time τ = (t2 −z2)1/2.

We will compare the results of the numerical simulationof this problem (for the degradation of the field, the particle spectra, etc.) with a semi-classical transport approach using a Schwinger-inspired source term [14][15][16].

We willalso discuss the effective hydrodynamics derived from the expectation value of the energymomentum tensor of the quantum theory.2

2. SUMMARY OF THE DIFFERENT STRATEGIES IN λϕ4 FIELD THE-ORYFor simplicity let us first study these different approaches to initial value problems inthe simplest case- λϕ4 field theory.a) Schrodinger Picture: In the Schrodinger picture the Initial State is described by awave functional at t=0.

For example a Gaussian wave functional is< ϕ|Ψ > = ψ[ϕ, t]= exp[−Zx,y[ϕ(x) −ˆϕ(x)][G−1(x, y)/4 −iΣ(x, y)][ϕ(y) −ˆϕ(y)]] (2.1)The time evolution is given by the Functional Schrodinger equation [1]:i∂ψ/∂t = HψH =Zd3x[−12δ2/δϕ2 + 12(∇ϕ)2 + V (ϕ)](2.2)This is a generalization of the usual Schrodinger equation:ψ(x) =< ψ|x >, x →ϕ(x, t);p = −iδ/δx →π = −iδ/δϕi∂ψ/∂t = Hψ; H = −∂2/∂x2 + V (x)(2.3)with initial condition:ψ(0) = exp[−α(x −xo)2]. (2.4)One might imagine solving (2.2) on a computer by introducing a lattice in d dimensionsand converting the functional derivatives into partial derivatives.

One then quickly realizesthat the number of degrees of freedom in equation (2.2) is rather overwhelming. To controlthis problem one uses variational trial wave functionals which become “exact” in the large-N limit– namely Gaussians.

The equations of motion for the variational parameters canbe obtained from Dirac’s variational principle [17]:Γ =Zdt < Ψ|i∂/∂t −H|Ψ >(2.5)δΓ = 0 →Schrodinger’s equation:i∂/∂t −H|Ψ >= 0(2.6)3

In the ϕ representation one can choose a Gaussian trial wave functional:< ϕ|Ψv >= ψv[ϕ, t] = exp[−Zx,y[ϕ(x) −ˆϕ(x, t)][G−1(x, y, t)/4 −iΣ(x, y, t)][ϕ(y) −ˆϕ(y, t)]] + iˆπ(x, t)[ϕ(x) −ˆϕ(x, t)](2.7)where the variational parameters have the meaning:ˆϕ(x, t) =< Ψv|ϕ|Ψv >; ˆπ(x, t) =< Ψv| −iδ/δϕ|Ψv >G(x, y, t) =< Ψv|ϕ(x)ϕ(y)|Ψv > −ˆϕ(x, t) ˆϕ(y, t)(2.8)Then the effective action for the trial wave functional isΓ( ˆϕ, ˆπ, G, Σ) =Zdt < Ψv|i∂/∂t −H|Ψv >=Zdtdx[π(x, t)∂ϕ(x, t)/∂t +ZdtdxdyΣ(x, y)∂G(x, y, t)/∂t−Zdt < H >(2.9)where< H >=Zdx{π2/2+2ΣGΣ+G−1/8+1/2(∇ϕ)2−1/2∇2G+1/2V ′′[ϕ]G+1/8V ′′′′[ϕ]G2}.< H > is a constant of the motion and is a first integral of the motion. For λϕ4 fieldtheory we get the following equations of motion:˙π(x, t) = ∇2ϕ −∂< V > /∂ϕ;˙ϕ(x, t) = π˙G(x, t) = 2Zdz[Σ(x, z)G(z, x) + G(x, z)Σ(z, x)]˙Σ(x, t) = −2Zdz[Σ(x, z)Σ(z, x) + G−2/8+ [12∇2x −∂< V > /∂G]δ3(x −y)(2.10)If there is translational invariance and ϕ=0 we obtain a second order differentialequation for G(k,t), the Fourier transform of G(x,t):2 ¨G(k, t)G(k, t) −˙G2(k, t) + 4Γ(k, t)G2(k, t) −1 = 0Γ(k, t) = k2 + m2(t); m2(t) = µ2 + 12λZ[dk]G(k, t)(2.11)4

This approximation is called the time-dependent Hartree-Fock Approximation and isequivalent to the leading term in a 1/N expansion of the field theory [3]. To understandthis trial wave function let us look at a simple quantum mechanics problem- the harmonicoscillator with a gaussian initial state.

Harmonic oscillator: V(x) =1/2 m x2,Initial conditions:Ψ(x, 0) = [2πG(0)]−1/2 exp{−x2/[4G(0)]}q(0) =< x >= 0(2.12)For the harmonic oscillator a Gaussian remains Gaussian as time evolves so thatΨ(x, t) = (2πG(t))−1/2 exp{−x2[G−1(t)/4 −iΣ(t)]}(2.13)We find that the conserved Energy can be written in terms of G as follows:E =< H >= ˙G2/8G + Gm2/2 + G−1/8 = ˙G2/8G + V [g](2.14)We plot V[g] in fig 1. From fig.

1 we see that the ground state is G = 1/(2m). If att=0, G0 = 1/(2M) ; m ̸ = M thenG(t) = 1/2(G0 + G1) + 1/2(G0 −G1) cos(2m(t −to))(2.15)Thus the width oscillates with frequency 2m between G0 and G1.

Generalizing tofree field theory (which is just independent harmonic oscillators) we have instead for eachmode of momentum k:< H(k) >= ˙G2/8G + (k2 + m2)G/2 + G−1/8(2.16)This leads to the same result for G(k,t) as for G(t) with m→ωk = (k2+m2)1/2. However infield theory, unlike quantum mechanics, an arbitrary initial Gaussian state is not necessarilya physically valid choice since it might correspond to an infinite particle density or energydensity when compared to the adiabatic vacuum.

Thus the particle interpretation impliesthat one needs to restrict the large k behavior of G(k) at t=0 to be a physically allowedinitial state with finite particle number, energy density etc.Otherwise one gets extraunwanted infinities in loops.b) Heisenberg Picture: Green’s function approach5

In problems where there is spatial homogeneity one has a Fourier decomposition fora charged field ϕ in terms of mode functions fk(t) which depend only on the time and theusual creation and annihilation operators a and b which satisfy the canonical commutationrelations:Φ(x, t) =Z[dk][fk(t)akeikx + f ∗k (t)b+k e−ikx][ak, a+k′] = [bk, b+k′] = (2π)3δ3(k −k′)(2.17)The initial state |i > is totally specified by specifying at t=0 the matrix elements of a andb:< i|a+k ak|i > = (2π)dδd(k −k′)n+(k)< i|bkak|i > = (2π)dδd(k + k′)F(k)etc. (2.18)The equation for the expectation value of the equation of motion is:< i|(−+ m2)ϕ + λ(ϕ+ϕ)ϕ|i >= 0(2.19)We see from these equations that we also need to solve the equation of motion for< i|λ(ϕ+ϕ)ϕ|i >.In general we get a Heirarchy of Green’s function equations- The BBGKY heirarchy.To make practical progress we need a truncation scheme which allows us to solve the lowestorder problem and then systematically calculate corrections.

In the large N expansion thelowest order approximation leads to a factorization< i|(ϕ+ϕ)ϕ|i > =< i|(ϕ+ϕ)|i >< i|ϕ|i >= G(x, x; t) < i|ϕ|i >(2.20)where the fourier transform G(k,t) of G(x-x’; t) obeys the same equation as the width ofthe Gaussian wave function in the Schrodinger equation in the Hartree approximation.2 ¨G(k, t)G(k, t) −˙G2(k, t) + 4Γ(k, t)G2(k, t) −1 = 0Γ(k, t) = k2 + m2(t); m2(t) = µ2 + 12λZ[dk]G(k, t)G(x, x; t) =Z[dk]G(k, t)(2.21)6

Thus the large-N expansion (Hartree approximation, mean field approximation) truncatesthe hierarchy of coupled Green’s function equations making it necessary to only solve thecoupled one and two-point Green’s function equations.In these mean field equations the problem reduces to an external field problem in thatthe quantum field ϕ obeys the equation:(−+ m2(t))ϕ = 0(2.22)Because we have an external field problem with spatial homogeneity: the mode functionsf(t) in (2.17) obey:(∂20 + ω2)f = 0; ω2 = k2 + m2(t)(2.23)The canonical commutation relations lead to a constraint on the mode functions:fk ˙f ∗k −f ∗k ˙fk = i(2.24)which is automatically satified by the WKB form ansatz:fk(t) = [2ΩK(t)]−1/2 exp[−iyk(t)]˙yk(t) = Ωk(t)(2.25)which lead to the equationΩ2k(t) + ¨Ωk/(2Ωk) −34( ˙Ωk/Ωk)2 = ω2k(t). (2.26)At t=0 one has in general for the initial state:< i|a+k ak|i >= (2π)dδd(k −k′)n+(k)< i|bkak|i >= (2π)dδd(k −k′)F(k)For an adiabatic vacuum: n(k) =F(k) = 0, and the initial conditions on ΩareΩ(k, t = 0) = ω(k, t = 0); ˙Ω(k, t = 0) = ˙ω(k, t = 0).

(2.27)This formalism, however is perfectly general and one could take any initial state withan integrable phase space particle density n(k) and pair density F(k). As a particular7

choice one could have chosen at t=0 an equilibrium configuration of pions described by atemperature kT = β−1n(k) = 1/(exp[βE(k)] −1)(2.28)c) Path Integral Approach: Closed time-path formalismThe only formalism that allows a systematic approach to initial value problems is theclosed time-path approach of J. Schwinger[8]which was further elaborated by Keldysh[9]andput into a Path Integral framework by Chou, Su, Hao and Yu[10]. This Path Integral ap-proach allows standard Path Integral approximation schemes such as the large N approx-imation as well as ensuring causality for the Green’s functions for initial value problems[18].

The starting point for determining the Green’s functions of the initial value problemis the generating Functional:Z[J+, J−, ρ] =< i|T ∗(exp{−ZiJ−ϕ−})|out >< out|T(expZiJ+ϕ+)|i >(2.29)This can be written as the product of an ordinary Path integral times a complexconjugate one or as a matrix Path integral.Z[J+, J−, ρ] =Zdϕ+dϕ−< ϕ+, i|ρ|ϕ−, i > exp i[(S[ϕ+] + J+ϕ+) −(S∗[ϕ−] + J−ϕ−)]=Zdϕα exp i(S[ϕα] + Jαϕα) < ϕ1, i|ρ|ϕ2, i >(2.30)where < ϕ+i|ρ|ϕ−, i > is the density matrix defining the initial state.This leads to the following matrix Green’s functions [11]:G++ = δ2 ln Z/δJ+δJ+|j=0 =< T(ϕ(x1), ϕ(x2) >G−−= δ2 ln Z/δJ−δJ−|j=0 =< T ∗(ϕ(x1), ϕ(x2) >G+−= δ2 ln Z/δJ+δJ−|j=0 =< ϕ(x2), ϕ(x1) >G−+ = δ2 ln Z/δJ−δJ+|j=0 =< ϕ(x1), ϕ(x2) >(2.31)The matrix Green’s function structure insures causality. In this approach it is easyto generate a 1/N expansion in analogy with ordinary field theory.

The diagrams are thesame as in the usual 1/N expansion, except the Green’s functions are the matrix Green’sfunctions described above.If in lowest order in (1/N) we have an external field problemas described above, one can directly use the mode solutions of the previous methods todetermine the lowest order matrix Green’s function of eq. (2.31).

This obviates the needto discuss the initial density matrix of the theory, since it is these Green’s functions whichthen enter the diagrams of the higher order calculations.8

3. MAIN EXPANSION IDEA: FLAVOR SU(N)In many problems one of the fields can be treated classically to first approximation–pair production in Strong Electric or Gravitational fields.

This makes the lowest orderproblem an external field problem. One way to generate a systematic expansion whoselowest order is an external field problem is by introducing N copies of the original problemand expanding in Flavor SU(N).

This is most easily done in the Path Integral formalism.For the initial value problem one would use the matrix Green’s functions discussed above.Having an extra large parameter N allows an evaluation of the Path integral by Laplace’smethod (or the method of Steepest Descent). To obtain the large N expansion one realizesthat if there are N flavors the loops carry an extra N. Rescaling the fields then display anoverall factor of N in the effective action which includes the loops.

Examples:λϕ4 : χ = ϕ2Z =ZdχZdϕ exp[−Z(∂µϕ)2 + λχϕ2 −λχ2 + µ2ϕ + Jϕ + Sχ]ϕ →ϕi, i = 1, 2, · · ·N; λ →λ/N; ϕi →N 1/2ϕi; χ →Nχ, λχ →λχ.Integrating over ϕ we obtain:Z =Zdχ exp{−N[χ2 + 12Tr ln G−1 −jGj]}=Zdχ exp{−NSeff(χ)}G−1 = [−+ µ2 + λχ]δ(x −y)(3.1)Evaluating the Path Integral at the Saddle point, δSeff(χ)/δχ = 0 leads to the selfconsistent external field problem[−+ µ2 + λχ]ϕ = 0; χ = ϕ2 + G(xx)(3.3)In QED we obtain an external field problem by integrating out the fermions (whichhave now N flavors to give an extra N to the determinant) and then rescaling the fields todisplay the overall factor of N in the effective Action: QED:Z =ZdAµZdΨdΨ exp[Zdx{−14F 2 + Ψ(iγ∂−e ̸ A + m)Ψ} + Ψη + ηΨ]Ψ →Ψi; e →e/√N, A →A√N(3.3)9

Integrate out the N species of fermionsZdAµ exp{−NSeff(Aµ)}Seff(Aµ) =Zdx14F 2 + Tr ln(S−1(x, y; A)) + ηS(x, y; A)η]S−1(x, y; A) = (iγ∂−e ̸ A(x) + m)δ(x −y)(3.4)Evaluating the Path Integral at the saddle point, δSeff(Aµ)/δAµ = 0 leads to theexternal field problem:(iγ∂−e ̸ A + m)Ψ = 0(3.6)where A is an external field,Ψ is a quantum field. We also obtain the semiclassical MaxwellEquation:∂µF µλ =< jλ >= e < ΨγλΨ > .

(3.7)In all these problems one has in leading order in 1/N a straightforward problem of aquantum field theory in a background field which allows a normal mode decomposition interms of the solutions of the classical field equations. Renormalization can be carried outby an adiabatic expansion of the mode equation[13].

The effect of quantum fluctuationsabout the semiclassical field can be systematically taken into account by calculating thefluctuations about the leading stationary phase point in the Path Integral order by orderin the 1/N expansion.4. PARTICLE PRODUCTION IN THE CENTRAL RAPIDITY REGION INHEAVY ION COLLISIONSA popular picture of high-energy heavy ion collisions begins with the creation of a fluxtube containing a strong color electric field[19].

The field energy is converted into particlesas qq pairs and gluons which are created by tunnelling- the so-called Schwinger mechanism[20][21][22]. The particle production can be modeled as an inside-outside cascade which issymmetric under longitudinal boosts and thus produces a plateau in the particle rapiditydistribution.

The boost invariant dynamics, in a hydrodynamical picture gets translatedinto energy densities (such as E2 ) being functions of the proper time. We take this asan initial condition on the fields in an initial value problem based on this pair-productionmechanism.

First let us look at the case where the electric field is a function of real timet, treating later the more realistic case where E= E(τ); τ = (t2 −z2)1/2. Thus we first10

consider particle production from a spatially uniform electric field such as that producedbetween two parallel plates. This is an idealized model of a flux tube for QCD.

The problemof pair production from a constant Electric field (ignoring the back reaction) was studiedby J. Schwinger in 1951 [20]. The physics is as follows: One imagines an electron boundby a potential well of order |V0| ≈2m and submitted to an additional electric potentialeEx (as shown in fig.

2 ). The ionization probability is proportional to the WKB barrierpenetration factor:exp[−2Z Vo/eodx{2m(Vo −|eE|x)}1/2] = exp(−43m2/|eE|)(4.1)A direct calculation due to Schwinger from first principles using the effective actionin an arbitrary constant electric field (ignoring the back reaction) gives insteadw = [αE2/(2π2)]Σ∞n=1(−1)n+1n2exp(−nπm2/|eE|).

(4.2)This equation tells us that pair production is exponentially suppressed unless eE≥πm2. So we expect (as we find in fig.

3) that there is a crossover value of E wherethe time it takes for E to first reach zero (remember there are plasma oscillations) isrelatively short. Schwinger’s result only applies when we can ignore dynamical photons(as well as back reaction)and is related to the lowest order in 1/N calculation where theelectric field is treated as a classical object.

Schwinger’s analytical result was subsequentlyused as source term for an approximate transport theory [14], [15], [16]approach to theback reaction connected with pair production which we will later compare with our exactnumerical results.We will choose the electric field in the z direction and choose a particularly simplegauge:→E = E(t)ˆk;→A = A(t)ˆk; E(t) = −dA/dt(4.3)To maintain spatial homogeneity we have from Maxwell’s equation:∇.E = ρ(4.4)that the plasma of produced particles must be neutral. In scalar QED, the equation forthe quantum field ϕ is−(∂α −ieAα)(∂α −ieAα)Φ + µ2Φ = 0(4.5)11

and for the electromagnetic field:∂αF βα =< C{−ie(Φ+∂βΦ −Φ∂βΦ+) −2e2AβΦ+Φ} >(4.6)where C denotes charge symmetrization with respect to Φ+ and Φ. For our constraints onthe field E and our choice of gauge we get:−dE/dt =< jZ >= eZ[dk](kZ −eA(t))G(k, t)(4.7a)where G(k, t) = [< ϕ†ϕ + ϕϕ† > −2ϕ∗ϕ]F T(4.7b)For QED we have instead the field equation:[iγ∂−e̸ A(t) −m]Ψ(x, t) = 0(4.8)and the semiclassical Maxwell equation:−dE/dt =< jZ >= 12e < i|[Ψ(x, t), γ3Ψ(x, t)]|i >(4.9)The fact that the external field is independent of space (spatial homogeneity) meansthat one has a simple normal mode expansion of the fields just as in λϕ4 field theorydescribed earlier.For Scalar QED we haveΦ(x, t) =Z[dk][fk(t)akeikx + f ∗k (t)b+k e−ikx][∂2o + ω2k(t)]fk(t) = 0ω2k(t) = [k −eA(t)]2 + µ2 + k2⊥(4.10)Repeating the arguments of (2.22 - 2.25) we again obtain for the generalized frequencyΩk(t):Ω2k(t) + ¨Ωk/(2Ωk) −34( ˙Ωk/Ωk)2 = ω2k(t).

(4.11)where now ω is given by (3.10) Spatial homogeneity requires translational invariance,W(x −x′, t, t′) =Z[dk]W(k, t, t′)eik(x−x′).12

This in turn requires that< a+k ak > = (2π)dδd(k −k′)n+(k)< b+k bk > = (2π)dδd(k −k′)n−(k);< bkak > = (2π)dδd(k + k′)F(k)(4.12)Thus we obtain for G(k,t)G(k; t) = Ω−1(k, t){1 + n+(k) + n−(k) + 2F(k) cos[2yk(t)]}(4.13)This is the most general form of the propagator that one would use in the diagrams ofthe 1/N expansion, where n and F are the particle and pair phase space densities at t=0.These parameters also totally specify (in leading order in 1/N) the density matrix at t=0.To solve the field theory in leading order in 1/N (ignoring questions of renormalizationto be discussed below) one solves the second order differential equation for each modefunction Ωk(t), determines G(k,t) and then solves the back reaction equation:−dE/dt = eZ[dk](kZ −eA(t))G(k, t)(4.14)For QED one has to deal with the spinor structure:ψ(x, t) =Z[dk][uks(t)bkeikx + v−ks(t)d†−ke−ikx](4.15)If we choose a basis where γ0γ3 is diagonal:γ0γ3χs = λsχs, s = 1, 2 →λ = 1; s = 3, 4 →λ = −1χ†rχs = 2δrs(4.16)Then the spinors u and v obey the equation{γ0∂t + iγ3π + iγ⊥p⊥+ m}uks(t)vks(t)= 0(4.17)Squaring the Dirac equation by letting:uksvks= {−γ0∂t −iγ3π −iγ⊥p⊥+ m}χsf +k (t)χsf −−k(t)(4.18)13

we find that the mode functions f now obey:[∂20 + ω2k(t) −iλs ˙π]fk(t) = 0,ω2k(t) = π2 + p2⊥+ µ2π = k −eA(4.19)If the operators ak and bk obey the usual anticommutation relations:{aks, a†k′s} = {bks, a†k′s} = (2π)3δ3(k −k′)δss′(4.20)the fk are constrained to satisfyω2f ∗αf β + ˙f ∗α ˙f β + iλsπ[f ∗α ˙f β −˙f ∗αf β] = δαβ/2(4.21)Parametrizing the positive and negative frequency solutions:f±(t) = N± expZ t0g±(τ)dτ,(4.22)we find:g+ = −[λs ˙π + ˙Ω]/2Ω−iΩ(4.23)where the generalized frequencies, Ωk(t) now satisfies the equation:Ω2k(t) + ¨Ωk/(2Ωk) −34( ˙Ωk/Ωk)2 −˙π2/(4Ω2) −λs ˙π ˙Ω/Ω2(4.24)Ignoring renormalization, the solution of QED is obtained by simutaneously solvingfor these modes and also for E(t) which is obtained from the Maxwell equation:dE/dt = 2eΣ4s=1Z[dk](p2⊥+ m2)λs|f +ks(t)|2(4.25)5. RENORMALIZATIONThe equations of the previous section as they stand are not finite in the continuumsince the sum over modes in (4.14) and (4.25) contains a divergence related to the renor-malization of the charge (as well as the wave function) resulting from the charged particleloops in the definition of the current.Let us first look at Scalar QED where the back-reaction equation is:14

−dE/dt =< j >= eZ[dk](kz −eA(t))Ω−1[1 + N(k)...](5.1)We first see that N(k) has to fall fast enough at large k to not lead to any furtherdivergences– this is equivalent to the condition that the initial number density ρ is finite.The integral of Ω−1 contains a divergence proportional to dE/dt which renormalizes thecharge (as well as the field E). To isolate this divergence one makes an adiabatic expansionof the equation for the generalized frequencies Ωk.

That is, we imagine that the timederivatives are small d/dt→ǫ d/dt :ǫ2[¨Ωk/(2Ωk) −34( ˙Ωk/Ωk)2] = ω2k(t) −Ω2k(t)(5.2)and we then expand in powers of ǫ1/Ωk = 1/ωk[1 + ǫ2{¨ωk/4ωk −38( ˙ωk/ωk)2} + 0(ǫ4ω−4k )](5.3)We see that terms with higher derivatives are associated with more convergence factorsof 1/k so that one only has to consider the first two terms in the adiabatic expansion toisolate the divergences which are interpreted as the standard charge renormalization. Thelog divergence comes from the term¨ωk = e(dE/dt)(k −eA)ω−1(5.4)After integrating over k this leads to a term of the form:e2δe2dE/dt; δe2 = 112Z[dk]ω−3k= π(0)(5.5)where π(0) is the usual vacuum polarization at q2=0.

Subtracting this term from bothsides of eq. (5.1) we obtain:−edE/dt(1 + e2π(0)) = e2[Z[dk](kz −eA(t))G −eπ(0)dE/dt].

(5.6)The Ward identity tells us that eE = eRER; and the renormalized charge is determinedbye2R = e2/(1 + e2π(0))(5.7)so the explicity mode by mode finite renormalized equation is−dER/dt = eRZ[dk](k −eA(t))[Ω−1 −ω−1 −e2R(k −eA(t))(dE/dt)ω−5/4](5.8)15

For QED one gets instead after charge renormalization:dER/dt = 2eRΣ4s=1Z[dk][(p2⊥+ m2)λs|f +ks|2 −e2RdER/dt ω−3(5.9)6. HEAVY ION COLLISIONS AND BOOST INVARIANT DYNAMICSIn e+ e−annihilation, hadronic collisions and in heavy-ion collisions particle produc-tion in the central rapidity region can be modeled as an inside-outside cascade which issymmetric under longitudinal boosts which leads to a plateau in the particle rapidity dis-tributions.

This boost invariance also emerges dynamically in Landau’s hydrodynamicalmodel [23] and forms an essential kinematic ingredient in the analyses of Cooper, Frye andSchonberg [24] as well as Bjorken[25]. It was recognized by Cooper and collaborators andfurther elaborated by Bjorken that in a hydrodynamical framework scale invariant initialconditions :v = z/t, ǫ(x, t) →ǫ(τ), τ 2 = t2 −z2(6.1)would automatically lead to flat rapidity distributions.

In the context of transport or fieldtheory modelling of the heavy ion collision, after an initial time τ0, energy densities areexpected to be functions only of the fluid “proper time” τ. We therefore assume thatthe kinematics makes the electric field E only a function of the proper time τ.

For thiskinematical choice it is convenient to introduce new variables τ, η the fluid “proper time”and the fluid rapidity (when v=z/t) via :z = τ sinh η, t = τ cosh η. (6.2)This change of coordinates to (τ, η) from (t,z) can be accommodated by the usual formalismof curved space [26][27] (except the curvature here is zero).

One introduces the metric incurved spacegαβ = diag(−1, 0, 0, τ 2). (6.3)Maxwell’s equations(−g)−1/2∂β[(−g)−1/2F αβ] = jα(6.4)becomes for an electric field E(τ) in the z directionE(τ) = Fzt = Fητ/τ = −τ −1∂τA(τ)(6.5)−1/τ∂τ[1/τ∂τA(τ)] =< jη >(6.6)16

For Scalar Electrodynamics the equation for χ = √τϕ is(∂2τ + τ −2[(∂η −ieA(τ))2 + 1/4] −∂2x −∂2y + m2)χ = 0(6.7)The rescaled field χ has the same Fourier decomposition as φ had in flat space withthe mode functions f obeying[∂2τ + ω2k(τ)]fk(τ) = 0(6.8)however nowω2k(τ) = [k −eA(τ)]2/τ 2 + k2⊥+ µ2 + 1/(4τ 2)(6.9)so that the longitudinal momenta get suppressed at large τ. For fermions one has the addedcomplication that the covariant derivative now has a spin piece: (denote the Minkowskiindices with α, β the curvilinear coordinates with µν)∇µ = ∂µ + Γµ −ieAµΓµ = 12ΣαβV υα Vβυ;µΣαβ = 14[γα, γβ](6.10)and the vierbein represents the transformation to the Minkowski coordinates:gµν = V αµ V βν ηαβ;∼γµ = γαV µα(6.11)Maxwell’s equation becomes:−τ −1dE(τ)/dτ =< jη > = 12e < i|[Ψ,∼γηΨ]|i >= 12τ e < i|[Ψ†, γ0γ3Ψ]|i >(6.12)and the fermion mode functions now obey[∂2τ + ω2k(τ) −iλs ˙π]fk(τ) = 0(6.13)whereω2k(τ) = π2 + p2⊥+ µ2; π = (p −eA(τ))/τ(6.14)The divergences in Maxwell’s equation in curved space can be renormalized as beforeby an adiabatic expansion in the variable τ.

The details of this calculation are presentedin [28]:17

7. PARTICLE PRODUCTION RATES AND THE BOGOLIUBOV TRANS-FORMATIONThe wave functions of the first order adiabatic expansion eikxf 0k(t) wheref 0k(t) = (2ωk)−1/2 exp[−iZ t0ωk(t′)dt′](7.1)form an alternative basis for expanding the scalar fields and allows one to define an in-terpolating number density N(k,t) which becomes the true one as t→∞.

Expanding thefield in terms of f0k(t) we haveΦ(x, t) =Z[dk][a0k(t)eikxf 0k(t) + f ∗0−k(t)b0†k (t)e−ikx]where a0k(t →∞) = aoutketc. (7.2)In this expansion the creation and annihiliation operators are time dependent.

We alsohave our previous expansion in terms of the time independent operators a and b relatedto the initial state:Φ(x, t) =Z[dk][fk(t)akeikx + f ∗−k(t)b†ke−ikx]. (7.3)We recognize that ak and a0k(t) are related by a unitary transformation.

The Bogoli-ubov coefficients are defined bya0k(t) = α(k, t)ak + β∗(k, t)b†kb0k(t) = α(k, t)bk + β∗(k, t)a†k|α(k, t)|2 + |β(k, t)|2 = 1(7.4)The number of particles produced per unit volume is justV −1dN/dk =< t = 0|bout †kboutk+ aout †kaoutk |t = 0 >(7.5)The interpolating number density is defined in terms of the first order adiabatic op-erators:V −1dN(k, t)/d3k = ⟨t = 0|b0 †k (t)b0k(t) + a0 †k (t)a0k(t)|t = 0⟩= (1 + N(k))|β|2 + N(k)|α|2 + 2Re{αβF(k)}(7.6)18

For N=F=O (the adiabatic vacuum at t=0)V −1dN(k, t)/dk = |β|2 = (4ωkΩk)−1[(Ωk −ωk)2 + 14( ˙Ωk/Ωk −˙ωk/ωk)2](7.7)We see that adiabatic initial conditions (no particle production at t=0) areΩk(0) = ωk; ˙Ωk(0) = ˙ωk(0)(7.8)For fermions we have instead:V −1dN(k, t)/dk = Σsw(ω + λπ)(2ω)−1[ω2|f +|2 + | ˙f +|2 −iω(f ∗+∂0f + −f +∂0f ∗+)].Similar expressions exist for the boost invariant problem [28].8. TRANSPORT APPROACH TO MULTIPARTICLE PRODUCTIONA classical kinetic theory approach to the back-reaction problem as discussed in[14][15][16]introduces a phase space single particle distribution function f(x, p, t) in thepresence of a homogeneous electric field and with a phenomenological source term inspiredby Schwinger’s solution for the constant external field.df/dt = ∂f/∂t + eE(t)∂f/∂p= dN/dtdzdp= |eE(t)| ln[1 ± exp[−πm2⊥/|eE(t)|]]δ(p)(8.1)± stand for boson(+) or fermion case (-).

The right hand side of (8.1) is a naive useof Schwinger’s formula (valid when no particles are present and for constant fields with Ereplaced by E(t) {or E(τ)}. This approach was recently used to predict dilepton productionfrom the quark-gluon plasma [29].

A potential problem with replacing constant E by E(t)is that in the field theory simulations E(t) is rapidly varying in time. A more seriousproblem is that once particles are produced, Schwinger’s derivation, which was for particleproduction from the vacuum, is no longer valid.

This however can be fixed up by thefollowing arguement. Once particles are present there is an additional quantum mechanicaleffect due to statistics– Bose enhancement or Pauli Suppression.

For the external field19

problem one always has a normal mode decomposition at each time t. Thus the creationand annihilation operators at different t are again connected by a unitary transformation:b(k, t + ∆t) = α(t + ∆t)b(k, t) + β(t + ∆t)d†(k, t)|α|2 + |β|2 = 1; |b†b| = n+; |a†a| = n−; n+ = n−= n(8.2)Thereforen(t + ∆t) = n(t) + 2|β(t + ∆t)|2{1 ± n}(8.2)or∆n/∆t = 2|β|2{1 ± n}/∆t(8.3)where the +(-) is Bose enhancement (Pauli suppression). The Pauli suppression ensuresn(k) ≤1 for fermions.

Thus to include this effect we will modify the right hand side of(8.1) by multiplying by (1± 2 f (p,t)). This modified transport eqaution, as we will showbelow gives much better agreement with the field theory calculation.

One can solve theViasov equation using the method of characteristics. From dp/dt = eE and f(p,0)=0 oneobtains:f(p, t) = Σi ln[1 ± exp[−πm2/|eE(ti)|]](8.4)where the ti are determined fromp + eA(t) + eA(ti) = 0; ti < t(8.5)The back reaction equation is nowd2A/dt2 = jcond + jpoljcond = 2eZ[dp]pf(p, t)/(p2 + m2)1/2jpol = 2/EZ[dp](p2 + m2)1/2d3N/dtdxdp(8.7)wheredN/dtdxdp =(1 ± 2f(p, t))|eE(t)| ln[1 ± exp[−πm2⊥/|eE(t)|]]δ(p)A similar expression holds in boost invariant dynamics as discussed in [28].

The transportapproach with the enhancement (suppression) factor gives reasonable agreement with thedirect numerical solution of the field theory (in lowest order in 1/N) as long as we coarsegrain the field theory result in momentum bins.20

9. HYDRODYNAMIC CONSIDERATIONS: ENERGY FLOWFrom a hydrodynamical point of view, flat rapidity distributions seen in multiparticleproduction in p-p as well as A-p and A-A collisions are a result of the hydrodynamicsbeing in a scaling regime for the longitudinal flow.That is for v=z/t (no size scale in the longitudinal dimension) the light cone variablesτ, η:z = τ sinh η; t = τ cosh η(9.1)become the fluid proper time τ = t(1 −v2)1/2 and fluid rapidity:η = 1/2 ln[(t −x)/(t + x)] ⇒1/2 ln[(1 −v)/(1 + v)] = α(9.2)In the rest frame (comoving frame) of a perfect relativistic fluid the stress tensor hasthe form:Tµν = diagonal (ǫ, p, p, p)(9.3)Boosting by the relativistic fluid velocity four vector uµ(x, t) one has:Tµν = (ǫ + p)uµuν −pgµν(9.4)Letting u0 = cosh α; u3 = sinh α, we have when v = zt that η = α, the fluid rapidity.If one has an effective equation of state p = p(ǫ) (which happens if both p and ǫ arefunctions of the single variable (τ) as well as for the case of local thermal equilibrium)then one can formally define temperature and pressure as follows:ǫ + p = Ts; dǫ = Tds; lns =Zdǫ/(ǫ + p)(9.5)Then the equation:uµ∂νTµν = 0becomes:∂ν(s(τ)uν) = 0(9.6)Which in 1 + 1 dimensions becomesds/dτ + s/τ = 0 or sτ = constant(9.7)The assumption of hydrodynamical models is that the initial energy density for theflow can be related to the center of mass energy and a given volume (say of a Lorentz21

contracted disk of matter). It is also assumed that the flow of energy is unaffected by thehadronization process and that the fluid rapidity can be identified in the out regime withparticle rapidity.

Thus after hadronization the number of pions found in a bin of fluidrapidity can be obtained from the energy in a bin of rapidity by dividing by the energyof a single pion having that rapidty. That is one assumes that when the comoving energydensity become of the order of one pion/(compton wave length)[23][24]we are in the outregime.

This determines a surface defined byǫ(τf) = mπ/Vπ(9.8)On that surface of constant τdN/dη = 1/(mπu0)dE/dη = 1/(mπ cosh α)ZT 0µdσµ/dηdσµ = 4πa2(dz, −dt) = 4πa2τf(cosh η, −sinh η)dN/dη = 4πa2/mπ[(ǫ + p) cosh α cosh(η −α)p cosh η]/ cosh α = 4πa2ǫ(τf)/mπ (9.9)which shows that when η = α one gets a flat distribution in fluid rapidity.An extraassumption is needed to identify fluid rapidity α with particle rapidity y = 1/2 ln[(Eπ +pπ)/(Eπ −pπ)], where pπ is the longitudinal momentum of the pion.What I would like to show next is that in a field theory calculation based on theSchwinger mechanism if we make the kinematical assumption that the electric field Eis just a function of τ we obtain a flat rapidity distribution. We can also prove that thedistribution in fluid rapidity is the same as the distribution in particle rapidity.

We will alsoobtain renormalized expressions for ǫ(τ) and p(τ) (non-equilibrium dynamical equation ofstate).In the pair production problem we have shown that the interpolating phase spacenumber density is given by the Bogoliubov function (7.7) :dN/dηdkηdk⊥dx⊥= |β(kη, k⊥, τ)|2(9.10)we are interested in transforming from dηdkη to dz dy where y is the particle rapidityy = 1/2 ln[(Eπ + kzπ)/(Eπ −kzπ)].One can show that the transformation from (η, τ) to (z,t) is a canonical one (in thesense of Poisson brackets {η, kη} = {τ, Ω} = 1) with canonical momentumkη = −Ez + tp = −τm⊥sinh(η −y)Ω= (Et −pz)/τ) = m⊥cosh(η −y)(9.11)22

The phase space is unchanged by this change of variable thusd6N/(dηdkηdk⊥dx⊥) = d6N/dzdkzdk⊥dx⊥= JdN 6/dzdydk⊥dx⊥(9.12)where J−1 = ∂kη/∂y∂η/∂z. At fixed τ one can show that |J| = dz/dkη which leads todesired result, assumed by Landau that along the breakup surface τ = constant:dN/dy = dN/dη.

(9.13)Schwinger’s pair production mechanism leads to an Energy Momentum tensor whichis diagonal in the(τ, η, x⊥) coordinate system which is thus a comoving one. In that systemone has:T µν = diagonal {ǫ(τ), p∥(τ), p⊥(τ), p⊥(τ)}(9.14)We see in a 3 dimensional problem, the field theory in this approximation has twoseparate pressures, one in the longitudinal direction and one in the transverse directionand thus differs from the thermal equilibruim case.

However, for a one-dimensional flowwe have that the energy in a bin of fluid rapidity is just:dE/dη =ZT 0µdσµ = A⊥τ cosh ηǫ(τ)(9.15)which is just the (1 + 1) dimensional hydrodynamical result of (9.9). This result does notdepend on any assumptions of thermalization.In the field theory calculation the expectation value of the stress tensor must be renor-malized since the electric field undergoes charge renormalization.

We can also determinethe two pressures and the energy density as a function of τ. Explicitly we have in thefermion case.ǫ(τ) =< Tττ >= τΣsZ[dk]Rττ(k) + E2R/2(9.16)whereRττ(k) = 2(p2⊥+ m2)(g+0 |f +|2 −g−0 |f −|2) −ω −(p2⊥+ m2)(π + e ˙A)2/(8ω5τ 2)p∥(τ)τ 2 =< Tηη >= τΣsZ[dk]λsπRηη(k) −12E2Rτ 2(9.17)23

whereRηη(k) = 2|f +|2 −(2ω)−1(ω + λsπ)−1−λse ˙A/8ω5τ 2 −λse ˙E/8ω5 −λsπ/4ω5τ 2+ 5πλs(π + e ˙A)2/(16ω7τ 2)andp⊥(τ) =< Tyy >=< Txx >= (4τ)−1ΣsZ[dk]{p2⊥(p2⊥+ m2)−1Rττ −2λπp2⊥Rηη} + E2R/2. (9.18)Thus we are able to numerically determine the dynamical equation of state pi = pi(ǫ)as a function of τ.10.

DISCUSSION OF NUMERICAL RESULTSThe physical quantities that we determine numerically are the time evolution of E(t),A(t) , and j(t). We will display in the figures the plasma oscillations and the time scalefor field energy to be essentially transferred into pair production.

The other quantitiesof physical interest we determine are the spectra of produced particles dN/dk and thedynamical equation of state. For comparison we have also solved the phenomenologicaltransport theory with and without the quantum correction due to statistics (i.e.

PauliBlocking and Bose Enhancement). In making plots for the spatially homogeneous case weuse the dimensionless variables [5]: ˜E = (eE/m2) ˜A = eA/m; mt = τ When the Electricfield ˜E is > 1 then it is quite easy for pairs to be produced and in that regime the finalresult is independent of the initial data.

We can see the approach to the tunneling regimeby comparing in the regime. 5 < ˜E0 < 2 the behavior of E(τ).

This is shown in fig. 3 for˜E0 = .5, 1, 2.

Once ˜E0 > 2 the behavior of ˜E(τ) is only weakly dependent on ˜E0. Oncethe pairs are produced one sees that there are plasma oscillations superimposed on whichthe electric field degrades.

These figures are from early simulations for scalar QED in 1 +1 dimensions [5].In fig. 4 we show ˜A(t), ˜e(t), < j(t) > for ˜E0 = 2 for scalar QED in 1+1 dimensions[6].In fig.

5 we show ˜E and ˜j for scalar QED in 1+1 dimensions for ˜E0 = 4. We comparethe naive Vlasov approach (dashed line) and the improved Vlasov approach ( dot- dashedline) .

We notice that including Bose enhancement corrections is quite important. We also24

notice that ˜j max = 2 e ρc so that particles continue to be produced when ˜E is near amaximum. In fig.

6 We show the exact particle spectrum as well as the momentum spacesmoothed result which is compared to the Vlasov Equation. Here ˜E0 =1 and we havescalar QED in 1+1 dimensions.In fig.

7 we show the results for E and j for ˜E0 = 4 in QED in 1+1 dimension comparedto the uncorrected Vlasov equation. We notice the dismal agreement.

In fig. 8 we seethe same curves compared to the improved Vlasov equation.

In fig. 9 we show the exactspectrum of produced pairs for QED in 1+1 dimensions for ˜E0 = 4.

We notice that n(k) ≤1which expresses the Pauli Principle. In fig.

10 we compare the binned version of the fieldtheory result with both the Naive and Improved transport theory. Next we present recentresults for Scalar QED in 1+1 dimensions using boost invariant Kinematics.

In fig. 11 weplot E,A and j vs. u=log(τ) for E0(u0) = 4 in the boost invariant case where E is a functionof the proper time τ (not to be confused with the previous τ).

In fig. 12 we compare E(u)and j(u) with the boost invariant transport theory with and without Bose enhancement.Finally we present preliminary results[30] for scalar QED in 3+1 dimensions.

In fig. 13we show the time evolution for E(t) and j(t) in 3+1 dimensions and compare with theBoltzmann-Vlasov model with and without Bose-enhancement.11.

ACKNOWLEDGEMENTSThe work presented here was done in collaboration with Emil Mottola, Yuval Kluger,So-Young Pi, Ben Svetitsky, Judah Eisenberg, Paul Anderson, Barrett Rogers and M.Samiullah.25

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