NLO Scattering in ϕ 4 Theory Finite System Size Corrections Self-Consistency

Previously a 2 → 2 NLO scattering amplitude for massive ϕ 4 theory with periodic boundary conditions was derived to investigate finite system size effects. This utilized new techniques and results, and as such benefited from self-consistency checks. Here we revisit the self-consistency of the derived amplitude, in order to more rigorously establish the self-consistency. Special attention is given to potentially pathological sums, which sets up for numerical evaluation investigating the finite system effects on quantities of interest, such as the effective coupling.


Introduction
In heavy-ion collisions [1,2] there is an apparent formation of Quark Gluon Plasma (QGP) [3], where the correlations between the outgoing low-momentum particles appear to be well described by nearly inviscid relativistic hydrodynamics.This calculation uses an Equation of State (EoS) provided by a lattice QCD calculation that is extrapolated to infinite system size [4].
It is currently unclear what happens in QCD just above the transition temperature T = 150 MeV [5].There is strong evidence of a second order phase transition, but the nature of the new phase is unknown.While for large temperatures asymptotic freedom requires that the phase is essentially an ideal gas, the more complicated behaviour discussed above is found at relatively small (and experimentally accessible) temperatures.It is therefore necessary to understand how reliably the behaviour found in the finite systems (such as heavy-ion or parton collisions) can be extrapolated to effectively infinite systems, such as the QGP found in the ∼ 0.000001 seconds after Big Bang.Understanding and modeling this phase is therefore not only of interest in high energy physics but also in cosmology, in particular in studies of the early universe.
The dependence of the low viscosity on the lattice QCD calculation of the EoS brings the underlying assumptions of the calculation under scrutiny.A possibly erroneous assumption to be investigated is that heavy-ion collisions can be well approximated as infinite sized systems [6].Indeed quenched lattice QCD calculations have shown significant possible corrections dependent on the size of the system [7].An analytic derivation of the finite size effects on the equation of state (or equivalently the trace anomaly) is therefore sought.
In order to perform such calculations new techniques, results and understanding will be necessary.To this end, the NLO 2 → 2 scattering amplitude for massive ϕ 4 theory with periodic boundary conditions has been derived with new, potentially useful, techniques and results.In order to understand the validity an applicability of such techniques self-consistency is sought.Potential pathologies related to future numerical evaluation is of great interest.

Finite system size ϕ 4 theory
We consider the massive scalar in an n + 1 dimensional spacetime where we enforce periodic spatial boundary conditions, where the i th dimension has size [−πL i , πL i ].In order to calculate the NLO 2 → 2 scattering amplitude we define (−iλ) 2 iV (p 2 ) ≡ as usual [8], which gives us the scattering amplitude ) in terms of the Mandelstam variables s, t, u and the renormalized loop "integral".The renormalized has been calculated [9] using several new techniques.Most notably is the newly proposed denominator regularization which was used with modified MS renormalization.Also utilized was a newly derived analytic continuation of the so called generalized Epstein zeta function.In order to take the finite system size effects captured in this amplitude seriously, and to motivate the overall correctness of the techniques and results that go into it, we would now like to perform some rigorous self-consistency tests.

Infinite dimension size limit
If we consider Equation 1 in the L 3 → ∞ limit, we need to interchange the limit and the infinite sum over k 3 which we can show is supported by the dominated convergence theorem.We first note that for k 3 ̸ = 0 and Now since Re( We can then rigorously see that by taking a given L i → ∞ we can destroy terms in the sum with k i ̸ = 0, reducing the effective dimension of the sum.Furthermore we can also see that taking all three L 1 , L 2 , L 3 → ∞ the second term in Equation 1 completely disappears, leaving ∆ 2 , which is exactly the infinite system result derived in [8].

Unitarity
Showing that the optical theorem σ tot = 2 Im(M) holds is equivalent to showing that unitarity is respected by the scattering amplitude M = λ 1 + λ V (s) + V (t) + V (u) .We can start by computing the NLO total cross-section with n finite sized dimensions and m = 3 − n infinite dimensions: Here we took the usual expression for the total cross-section and for the finite dimensions we simply had to replace the integrals with sums over a lattice Λ * n with lattice spacings of L −1 i in the i th finite dimension, and a lattice determinant of det(Λ * n ) = L −1 i .We also had to replace the momentum conserving Dirac delta functions with momentum conserving Kronecker delta functions (weighted by the lattice determinants inverse).We can then integrate out the p 2 integral and sum the ⃗ k 2 sum to get Here we can now introduce Q = 1 − 4m 2 s for convenience, to finally obtain Where θ is the usual Heavyside step function and is the solid angle of a (2 − n)dimensional sphere.Now in order to calculate 2 Im(M), we can first note that for the t, u channels, Re(∆ 2 ) > 0, trivially showing Im(V (t)) = Im(V (u)) = 0. We therefore have 2 Im(M) = 2λ 2 Im(V (s)), and note that Re(∆ 2 ) < 0 for Where Λ n is the lattice dual of Λ * n .Now by considering the natural log of a negative real number, and that Im K 0 (ix + ε) = − π 2 J 0 (x) + O(ε), which can easily be found by writing K 0 in terms of the Hankel function of the first kind: The Poisson summation formula over lattices is given by ⃗ where F ( ⃗ k) = dm f ( ⃗ j)e −2πi ⃗ k• ⃗ j is the Fourier transform.Since the function we are considering is radial, we can instead use the equivalent radial form [10] Since this is equal to Equation 4, we have shown that the amplitude respects the optical theorem.

Conclusion
Using the dominated convergence theorem we have shown that the previously derived 2 → 2 NLO scattering amplitude for massive ϕ 4 theory with periodic boundary conditions has the expected infinite system size limit.We have further shown that using the Vitali convergence theorem and the Poisson summation formula, we can show that the optical theorem holds for the investigated amplitude.This shows that the novel methods and results employed in the derivation of said amplitude respected unitarity.