Trace of a product of Dirac Matrices in a Fermion loop

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Discussion Overview

The discussion revolves around the calculation of a quark loop diagram in quantum field theory, specifically focusing on the trace of a product of Dirac matrices within a fermion loop. Participants explore the mathematical formulation and explicit representation of Dirac indices and matrix components in the context of loop calculations.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents a detailed expression for the quark loop diagram involving Dirac matrices and asks how to explicitly write the Dirac indices and slashed momenta.
  • Another participant questions the necessity of writing out the indices explicitly, suggesting that using gamma matrix and trace relations is more straightforward for computing the trace.
  • A third participant points out that useful formulae for traces can be found in Peskin & Schroeder, indicating that these can be derived from the anticommutation relations of the Dirac matrices.
  • One participant clarifies their intent to use trace identities for calculations while wanting to show the explicit indices to confirm the numerator is indeed a trace.
  • Another participant provides the formula for the matrix product and trace, explaining how to compute the trace using the Einstein summation convention.
  • A participant mentions a Mathematica program called Package-X that can assist with Dirac traces and loop calculations, suggesting it as a tool for verification after manual calculations.

Areas of Agreement / Disagreement

Participants express differing views on the necessity and method of explicitly writing out Dirac indices and matrix components. While some advocate for using established trace relations, others emphasize the importance of clarity in representation. The discussion remains unresolved regarding the best approach to take.

Contextual Notes

Participants reference specific mathematical techniques and identities related to Dirac matrices and traces, but there is no consensus on the optimal method for expressing these in the context of the quark loop calculation.

RicardoMP
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I'm working out the quark loop diagram and I've drawn it as follows:

quark loop.jpg


where the greek letters are the Lorentz and Dirac indices for the gluon and quark respectively and the other letters are color indices.
For this diagram I've written:

$$i\Pi^{\mu\nu}=\int_{}^{}\frac{d^4k}{(2\pi)^4}.[ig(t_a)_{ij}\gamma^\mu].[\frac{i[(\not\! k-\frac{\not{q}}{2})+m_1]}{(k-\frac{q}{2})^2-m_1+i\epsilon}\delta_{il}].[ig(t_b)_{kl}\gamma^\nu].[\frac{i[(\not\! k+\frac{\not{q}}{2})+m_2]}{(k+\frac{q}{2})^2-m_2+i\epsilon}\delta_{kj}]$$

In Peskin & Schroeder's Introduction to QFT it is said: "a closed fermion loop always gives a factor of -1 and the trace of a product of Dirac matrices". In short, I can write the above expression as:

$$i\Pi^{\mu\nu}=g^2T_F\delta{ab}\int_{}^{}\frac{d^4k}{(2\pi)^4}\frac{tr[\gamma^\mu.((\not\! k-\frac{\not{q}}{2})+m_1).\gamma^\nu.((\not\! k+\frac{\not{q}}{2})+m_2)]}{((k-\frac{q}{2})^2-m_1+i\epsilon).((k+\frac{q}{2})^2-m_2+i\epsilon)}$$

where ##T_F## is the Dynkin index in the fundamental representation I get after contracting the generators and deltas.
What I'm having a hard time with is on how to explicitly write the Dirac indices in each gamma matrix and slashed momenta in order to get something of the sort, e.g, ##(matrix)_{ii}=tr(matrix)##.
 
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What is tr(AB) in terms of the components of A and B?

Also, I am unsure why you would want to write this out explicitly. It is far (far!) easier to use gamma matrix and trace relations to compute the trace.
 
There are some useful formulae for such traces in the appendix of Peskin&Schroeder. You can derive them by using the anticommutation relations of the Dirac matrices.
 
Orodruin said:
What is tr(AB) in terms of the components of A and B?

Also, I am unsure why you would want to write this out explicitly. It is far (far!) easier to use gamma matrix and trace relations to compute the trace.
I might have not been clear, I'm sorry. I do want to use the trace identities in order to do the calculations. I just wanted to write out the indices explicitly so I show clearly that the numerator is indeed a trace.
 
First the matrix product is (Einstein summation convention used)
$$(AB)_{ij}=A_{ik} B_{kj}$$
Then the trace is the sum of the diagonal elements. In the Ricci calculus you just have to set ##i=j## in the above formula (implying summation over ##i## then of course):
$$\mathrm{Tr}(AB)=A_{ik} B_{ki}.$$
 
I have written a Mathematica program called Package-X that can help you with your Dirac traces and loop calculation. I suggest after completing the calculation by hand, you check it with the output of Package-X.

Here is what the calculation might look like in your Mathematica notebook.

Trace-PackageX.png


Copy the following into a blank Mathematica notebook for editable code:
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