Department of Physics

Propagator analysis (particle spectrum) for one of Yun-Cherng’s torsion theories

W. E. V. Barker

The idea here is to recover the massive and massless propagator structures for a given theory. We also want to show that the unitarity conditions are independent of Gauss-Bonnet.

Introduction

This notebook is just an introduction to the “particle spectrum” calculation. The steps which you will focus on in the first part of the project are given an orange background.

The HiGGS and PSALTer packages

Load the HiGGS environment.

CloseKernels[];
<< xAct`HiGGS`;

The theory we are studying

Here is the theory under investigation.

Lagrangian = T[i, g, h] (Bet1 PT1[-i, -g, -h, a, c, d] +
       Bet2 PT2[-i, -g, -h, a, c, d] +
       Bet3 PT3[-i, -g, -h, a, c, d]) T[-a, -c, -d] + R[i, j, g, h] (Alp1 PR1[-i, -j, -g, -h, a, b, c, d] +
       Alp2 PR2[-i, -j, -g, -h, a, b, c, d] +
       Alp3 PR3[-i, -j, -g, -h, a, b, c, d] +
       Alp4 PR4[-i, -j, -g, -h, a, b, c, d] +
       Alp5 PR5[-i, -j, -g, -h, a, b, c, d] +
       Alp6 PR6[-i, -j, -g, -h, a, b, c, d]) R[-a, -b, -c, -d] - (1/2) Alp0 G[a, c] G[b, d] R[-a, -b, -c, -d] + AlpGB (R[a, b, -a, -b] R[i, j, -i, -j] - 4 R[q, -i, -q, -j] R[p, j, -p, i] + R[-i, -j, -k, -l] R[k, l, i, j]) /. {Bet1 -> 0, Alp1 -> -2 Alp6, Alp2 -> 2 Alp6, Alp3 -> 0, Alp4 -> -Alp6, Alp0 -> 0};
%/.PActivate;
%//ToNewCanonical

Linearise the theory on the background

Now we take that theory, which was expressed in terms of field strength tensors, and we expand it to second order in perturbed gauge fields around the vacuum.

ToOrderA=MakeRule[{A[-a,-b,-c],Prt A[-a,-b,-c]},MetricOn->All,ContractMetrics->True];
ToOrderF=MakeRule[{F[-a,-b],Prt F[-a,-b]},MetricOn->All,ContractMetrics->True];
ToOrder=Join[ToOrderA,ToOrderF];

Lagrangian/.ExpandStrengths/.ToF;
%/.ToOrder;
Series[%,{Prt,0,2}]//Normal;
%/.Prt->1;
%/.PActivate;
%//ToNewCanonical;
SecondOrderLagrangian=%//ToNewCanonical

Back to the PSALTer package for evaluation

PSALTer only provides one function, which does the whole of the propagator analysis in one go. Currently named in honour of Yun-Cherng, but that will have to change!

YunCherngLin[SecondOrderLagrangian,F,A];

Hopefully that last line above tells us about the massless polarizations in the theory. For more details, we can discuss.