8.4 Particle-Field Interactions
8.4.2 Pair Production
The probability for a particle-pair annihilation process is given by a microscopic cross-section and the properties of the involved particlesŠ. Such a process removes particles from the particle ensemble and adds that amount of energy and momentum to the Ąeld, creating excitations and Ćuctuations on the Ąelds. In the inverse process, the pair production, removes energy and momentum from the Ąeld which efectively damps its Ćuctuations and creates new particles in the particle reservoir. The probability of this process can not be derived directly from the particlesŠ properties because scalar Ąelds have no particle-like properties.
In Chapter 5 and Section 7.4.4 the motivation and basic steps have been discussed. In the numerical implementation of the DSLAM model, the employed calculation steps are discussed now for the decay process of σ-Ąeld excitations
σ →qq .¯ (8.63)
For every point on the numerical ĄeldŠs ϕ-grid
(nx, ny, nz) (8.64)
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the mean Ąeld value of the Ąeld, its derivative, its energy and momentum has to be calculated
⟨ϕ(nx, ny, nz)⟩ ,
⟨ϕ˙(nx, ny, nz)⟩ , E(nx, ny, nz, ϕ,ϕ˙) , P(nx, ny, nz, ϕ,ϕ˙) .
(8.65)
The energy and momentum are calculated accordingly to equation (7.13) and (7.14). The Ąeld values can be taken directly from the numerical grid values with
⟨ϕ(nx, ny, nz)⟩=ϕ(nx, ny, nz) (8.66) or the mean Ąeld values can be extracted by a Gaussian convolution of the actual numerical values in analogy to the Gaussian smearing of the quark density in (8.46). In case of an interaction the Ąeld ϕis not only changed at the interaction pointp(nx, ny, nz) but also at its neighbor points due the Gaussian parametrization (see eq. (7.27)). Because an interaction changes a sub-volume of the system, this same volume can be used to calculate an average energy, momentum and mean Ąeld within this sub-volume. Both method turned out to work equally well, they only difer if the Ąeld ϕ has many excitations which are smaller than the size of the interaction volume.
In this case the non-convoluted algorithm will show a slightly larger pair-production rate until the high-frequency excitations are damped. In the case of thermal Ćuctuations, both methods behave equally.
After calculating⟨ϕ⟩, ⟨ϕ˙⟩,E andp for a Ąeld-cell, a distribution function has to be derived from this quantities. By assuming a boosted Boltzmann equation as a thermal distribution
fσ(p) = exp(−u·p T
)
, (8.67)
with the cell-velocityvis given by the cells energy and momentum:
⃗v= p⃗
E and u=γ
(1
⃗v )
. (8.68)
All free parameter of this distribution function are given by the temperatureT and the collective velocity given by the cell u. By integrating the distribution function over all momenta, the particle density of the system is given. However, in the DSLAM model the system should be able to deviate from thermal and chemical equilibrium. For every grid cell local thermal equilibrium has to be assumed, otherwise a local Ąt with a distribution function would not be possible. Still, the temperature can very strongly over the system volume for a small numerical grid.
Chapter 8 Numerical Implementation-Details of the DSLAM Model In the DSLAM model, the mean Ąeld value ⟨σ⟩ is used to derive the temperature by inverting the thermodynamic relation for the thermal equilibrium value of σ
⟨σ(T)⟩ → T(⟨σ⟩) . (8.69)
This is done by numerically calculating the thermodynamic relation and creating an inverse lookup interpolation, details are given in Section 8.8. The same method is used to derive the temperature dependent mass of the sigma quanta mσ with
mσ =mσ(⟨σ⟩) . (8.70)
To calculate a decay probability for Ąeld excitations, a sigma-particle densitynσ has to be derived.
To allow a deviation between the local ensemble temperature and the local particle density, the density is not derived from the coarse-grained temperature but from the local energy density.
nσ(T) =∫ d3p fσ(p, T) , (8.71) the energy density is given by
ϵσ(T) =∫ d3p ϵ(p) fσ(p, T) . (8.72) For a system with massless particles, these relations would have the form
n(T) = 2T3 , (8.73)
ϵ(T) = 3T4 , (8.74)
which leads to
n(ϵ) = 2ϵ
3Tσ . (8.75)
For massive particles this relation is n(ϵ) = ϵ
3Tσ −m2σTσK1(mσ/Tσ) , (8.76) with the modiĄed Bessel function of the second kind Kn.
After the parameter T, mσ, nσ and v are determined, a σ particle is sampled from fσ(p, T).
The particle-decay probability is given by the already in Chapter5 and Section7.4.4 discussed Breit-Wigner cross section
σ¯qq→σ(s) = σ¯ Γ2 (√
s−mσ)2+(12Γ)2 . (8.77)
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The decay event of this particle is sampled with Monte-Carlo techniques decay if Γ ∆t
Ntest∆V > ξ with ξ∈[0,1]. (8.78) In case of a decay, the energy and momentum
E =√m2σ+p2σ , P=pσ
(8.79)
are removed from the Ąeld. This is done in analogy to the already discussed method in Section 8.4.1. At the interaction point of the Ąeld-excitation decay, the energy density and momentum density has to be reduced accordingly to the values given by (8.79).
Removing energy and momentum from a Ąeld can be described in two steps. First, the kinetic energy of the Ąeld at the interaction point is reduced. This changes both the energy and momentum. To get the correct energy and momentum at the same time, an additional Gaussian wave packet is added in the second step to the Ąeld to correct for the missing or excessive momentum. The Gaussian parametrization for energy removing is therefore
δϕ(x,v) =♣v♣ ·exp (
−x2 2σ2
)
, (8.80)
δϕ(x,˙ v) =κϕ(x) exp˙ (
−x2 2σ2
)
+♣v♣exp (
−(x−∆tv)2 2σ2
) .
(8.81)
with the width σ of the Gaussian parametrization and the damping coeicient κ. For κ = 0 the complete energy is removed from the Ąeld, forκ = 1 none of the energy is removed. Note that in (8.81) the parametrization of the Gaussian wave packet is slightly diferent than in the parametrization for adding energy to a Ąeld. In (8.81)κ damps the Ąeld andv is the direction and strength of the ŞcorrectionŤ wave packet which is added to the system.
˜ v= v
♣v♣ (8.82)
However the actual velocity of the wave packet is normed to 1. This has a practical reason: the ŞcorrectionŤ wave packet which adds missing momentum removed by κ should interfere with the least possible impact. A wave packet with the maximal physical velocity needs the least packet-ŞheightŤ to add momentum to the system. In this case vis not directly associated with the amount of momentum which should be removed from the system but as a correction part in combination with κ.
Chapter 8 Numerical Implementation-Details of the DSLAM Model The four non-linear equations are solved with respect toκandv. In case of Ąnding a solution, the energy and momentum is removed from the system and theσ-particle is decayed to a particle-pair which is added to the particle-ensemble at the interaction point.
In the rest frame of the σ-particle the particles are created and their momentum is set to
⃗
p1=−⃗p2 . (8.83)
The spatial orientation is sampled isotropically and their momenta are boosted back into the laboratory-frame. More details of the decay kinematics can be found in Appendix D.