This is a collection of the files I developed in order to perform and test the explicit calculation of the 4-point function of the energy-momentum tensor in the free conformal field theory of a scalar field in 4 spacetime dimensions.
Calculations of the 2 and 3 point functions are detailed as well, in order to provide a framework in which the reader can familiarize with the tools employed and with the ideas underpinning the calculations and their tests, if she/he is not yet familiar with them.
Some of the calculations can be quite time consuming and the numerical checks of the Ward identities for the 4 point function requires your computer to have least 25 GB of memory at its disposal in order not to crush.
Mathematica (version 10.1 or higher).
PLEASE NOTE:
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"functional_derivatives.nb" MUST be run once in order to be able to run the rest of the calculation, particularly "correlators_calculation.nb", which in turn MUST be run in order to generate the correlation functions which are checked by "Ward_identities.nb".
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The packages PackageX and CollierLink must be loaded on the machine where the notebooks are run.
Here follows a concise description of the scope and purpose of each notebook:
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The noteook "~tensor_bases/tensor_bases_generation.nb" GENERATES the 4 files "tensmom#rank##". As the name suggests, the tensors in each of these files span a complete basis of tensors which are rank ## products of the metric tensor and # independent momenta. They are needed to check the Ward identities for the 3 and 4 point correlation functions.
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"functional_derivatives.nb" generates the file "all_functional_derivatives". As detailed in the paper, our computation requires heavy use of tensor strutures: rank 2, 4 and 6 trace anomalies for the 2, 3 and 4 point functions, rank 4, 6 and 8 countertems for the very same correlation functions. All of them are obtained by functionally differentiating scalars consisting of algebraic combinations of the Riemann tensor, the Ricci tensor and the Ricci scalar. The notebook starts by explaining the simplest functional derivatives of the metric tensor and goes on all the way up to anomalies counterterms and interaction vertices of the scalar with the background gravitational field, introducing gradually more complex structures. The second part of the notebook checks that the counterterms and the anomalies, which are computed non perturbatively, obey all the constraints they are supposed to. This is needed to make us more confident about our explicit calculations of the Green functions, provided that their divergent parts match the counterterms (they do indeed) and that they pass the check of the trace Ward identies with the anomalies (they do as well).
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"correlators_calculation.nb" explicitly computes the 2, 3 and 4 point functions, checks that they match the counterterms computed in "functional_derivatives.nb" and stores them in 3 files: "T2_scalar", "T3_scalar" and "T4_scalar"
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The .jpg files in the "figures" folder are simply graphical representations of the diagrams computed in
"correlators_calculation.nb" and of the vertices computed in "functional_derivatives.nb", which are loaded in the same notebooks just above the line of code computing each of them. -
The notebook "ward_identities.nb" checks the Ward identities for all of our correlation functions; analytically for two and three-point, numerically for four-point.
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The very short notebook "vanishing_euler_ct.nb" proves that the counterterm obtained by the integrated Euler density vanishes identically in exactly four spacetime dimensions,as it must be, though not away from it, as happens in Dimensional Regularization.
Have fun and please write me if you encounter any issues !
Mirko Serino