# Trace of a product of Dirac Matrices in a Fermion loop

• A
• RicardoMP
In summary, the conversation discusses the process of working out a quark loop diagram and the use of gamma matrix and trace identities to calculate the trace of a product of Dirac matrices. The conversation also mentions the use of a Mathematica program called Package-X to assist with the calculations.
RicardoMP
I'm working out the quark loop diagram and I've drawn it as follows:

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)##.

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.

Copy the following into a blank Mathematica notebook for editable code:
Trace-PackageX.nb:
Notebook[{

Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"<<", "X"}]], "Input",
CellLabel->"In[1]:=",ExpressionUUID->"d68acd69-9e6f-4b16-be40-73037f22b7d6"],

Cell[BoxData["\<\"\\!\$$\\*TemplateBox[List[\\\"\\\\\\\"Package-X v2.1.1, by \ Hiren H. Patel\\\\\\\\nFor more information, see the \\\\\\\"\\\", \ TemplateBox[List[\\\"\\\\\\\"guide\\\\\\\"\\\", \\\"paclet:X/guide/PackageX\\\ \"], \\\"HyperlinkPaclet\\\"]], \\\"RowDefault\\\"]\$$\"\>"], "Print",
CellLabel->
"During evaluation of \
In[1]:=",ExpressionUUID->"befefef7-2330-4488-a63e-82744427c722"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
RowBox[{"Spur", "[",
RowBox[{
SubscriptBox["\[Gamma]", "\[Mu]"], ",",
RowBox[{
RowBox[{
RowBox[{"(",
RowBox[{"k", "-",
RowBox[{"q", "/", "2"}]}], ")"}], ".", "\[Gamma]"}], "+",
RowBox[{"m1", " ", "\[DoubleStruckOne]"}]}], ",",
SubscriptBox["\[Gamma]", "\[Nu]"], ",",
RowBox[{
RowBox[{
RowBox[{"(",
RowBox[{"k", "+",
RowBox[{"q", "/", "2"}]}], ")"}], ".", "\[Gamma]"}], "+",
RowBox[{"m2", " ", "\[DoubleStruckOne]"}]}]}], "]"}]], "Input",
CellLabel->"In[2]:=",ExpressionUUID->"c1fd717f-9d65-4d47-8be3-7152b8684d0b"],

Cell[BoxData[
RowBox[{
RowBox[{"8", " ",
SubscriptBox["k", "\[Mu]"], " ",
SubscriptBox["k", "\[Nu]"]}], "-",
RowBox[{"2", " ",
SubscriptBox["q", "\[Mu]"], " ",
SubscriptBox["q", "\[Nu]"]}], "+",
RowBox[{"4", " ", "m1", " ", "m2", " ",
SubscriptBox["\[DoubleStruckG]",
RowBox[{"\[Mu]", ",", "\[Nu]"}]]}], "-",
RowBox[{"4", " ",
RowBox[{"k", ".", "k"}], " ",
SubscriptBox["\[DoubleStruckG]",
RowBox[{"\[Mu]", ",", "\[Nu]"}]]}], "+",
RowBox[{
RowBox[{"q", ".", "q"}], " ",
SubscriptBox["\[DoubleStruckG]",
RowBox[{"\[Mu]", ",", "\[Nu]"}]]}]}]], "Output",
CellLabel->"Out[2]=",ExpressionUUID->"c7080a80-df67-433a-9f81-e826e245e4c9"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
CellLabel->"In[3]:=",ExpressionUUID->"a01137ae-5895-4b17-bb2d-77e1b5d0a83c"],

Cell[BoxData[
FormBox[
RowBox[{
RowBox[{"8", " ",
SuperscriptBox["k", "\[Mu]"], " ",
SuperscriptBox["k", "\[Nu]"]}], "-",
RowBox[{"4", " ",
SuperscriptBox["k", "2"], " ",
SuperscriptBox["\[ScriptG]",
RowBox[{"\[Mu]", "\[InvisibleComma]", "\[Nu]"}]]}], "+",
RowBox[{"4", " ", "m1", " ", "m2", " ",
SuperscriptBox["\[ScriptG]",
RowBox[{"\[Mu]", "\[InvisibleComma]", "\[Nu]"}]]}], "-",
RowBox[{"2", " ",
SuperscriptBox["q", "\[Mu]"], " ",
SuperscriptBox["q", "\[Nu]"]}], "+",
RowBox[{
SuperscriptBox["q", "2"], " ",
SuperscriptBox["\[ScriptG]",
RowBox[{"\[Mu]", "\[InvisibleComma]", "\[Nu]"}]]}]}],
CellLabel->
3f50f76001f2"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
RowBox[{
RowBox[{
RowBox[{"LoopIntegrate", "[",
RowBox[{
RowBox[{"Spur", "[",
RowBox[{
SubscriptBox["\[Gamma]", "\[Mu]"], ",",
RowBox[{
RowBox[{
RowBox[{"(",
RowBox[{"k", "-",
RowBox[{"q", "/", "2"}]}], ")"}], ".", "\[Gamma]"}], "+",
RowBox[{"m1", " ", "\[DoubleStruckOne]"}]}], ",",
SubscriptBox["\[Gamma]", "\[Nu]"], ",",
RowBox[{
RowBox[{
RowBox[{"(",
RowBox[{"k", "+",
RowBox[{"q", "/", "2"}]}], ")"}], ".", "\[Gamma]"}], "+",
RowBox[{"m2", " ", "\[DoubleStruckOne]"}]}]}], "]"}], ",", "k", ",",
RowBox[{"{",
RowBox[{
RowBox[{"k", "-",
RowBox[{"q", "/", "2"}]}], ",", "m1"}], "}"}], ",",
RowBox[{"{",
RowBox[{
RowBox[{"k", "+",
RowBox[{"q", "/", "2"}]}], ",", "m2"}], "}"}]}], "]"}], "/.",
RowBox[{
RowBox[{"m2", "|", "m1"}], "\[Rule]", "m"}]}], "//",
"LoopRefine"}]], "Input",
CellLabel->"In[4]:=",ExpressionUUID->"26c2c3b2-c443-45d6-985b-d7d5004495c0"],

Cell[BoxData[
RowBox[{
RowBox[{
RowBox[{"(",
RowBox[{
RowBox[{"-",
FractionBox[
RowBox[{"4", " ",
RowBox[{"DiscB", "[",
RowBox[{
RowBox[{"q", ".", "q"}], ",", "m", ",", "m"}], "]"}], " ",
RowBox[{"(",
RowBox[{
RowBox[{"2", " ",
SuperscriptBox["m", "2"]}], "+",
RowBox[{"q", ".", "q"}]}], ")"}]}],
RowBox[{"3", " ",
RowBox[{"q", ".", "q"}]}]]}], "-",
FractionBox[
RowBox[{"4", " ",
RowBox[{"(",
RowBox[{
RowBox[{"12", " ",
SuperscriptBox["m", "2"]}], "+",
RowBox[{"5", " ",
RowBox[{"q", ".", "q"}]}]}], ")"}]}],
RowBox[{"9", " ",
RowBox[{"q", ".", "q"}]}]], "-",
RowBox[{
FractionBox["4", "3"], " ",
RowBox[{"(",
RowBox[{
FractionBox["1", "\[Epsilon]"], "+",
RowBox[{"Log", "[",
FractionBox[
SuperscriptBox["\[Micro]", "2"],
SuperscriptBox["m", "2"]], "]"}]}], ")"}]}]}], ")"}], " ",
SubscriptBox["q", "\[Mu]"], " ",
SubscriptBox["q", "\[Nu]"]}], "+",
RowBox[{
RowBox[{"(",
RowBox[{
RowBox[{
FractionBox["4", "3"], " ",
RowBox[{"DiscB", "[",
RowBox[{
RowBox[{"q", ".", "q"}], ",", "m", ",", "m"}], "]"}], " ",
RowBox[{"(",
RowBox[{
RowBox[{"2", " ",
SuperscriptBox["m", "2"]}], "+",
RowBox[{"q", ".", "q"}]}], ")"}]}], "+",
RowBox[{
FractionBox["4", "9"], " ",
RowBox[{"(",
RowBox[{
RowBox[{"12", " ",
SuperscriptBox["m", "2"]}], "+",
RowBox[{"5", " ",
RowBox[{"q", ".", "q"}]}]}], ")"}]}], "+",
RowBox[{
FractionBox["4", "3"], " ",
RowBox[{"q", ".", "q"}], " ",
RowBox[{"(",
RowBox[{
FractionBox["1", "\[Epsilon]"], "+",
RowBox[{"Log", "[",
FractionBox[
SuperscriptBox["\[Micro]", "2"],
SuperscriptBox["m", "2"]], "]"}]}], ")"}]}]}], ")"}], " ",
SubscriptBox["\[DoubleStruckG]",
RowBox[{"\[Mu]", ",", "\[Nu]"}]]}]}]], "Output",
CellLabel->"Out[4]=",ExpressionUUID->"b84b5d35-33bc-4be1-97fe-7f6dcc187173"]
}, Open  ]]
},
WindowSize->{676, 876},
WindowMargins->{{Automatic, 626}, {Automatic, 156}},
FrontEndVersion->"12.0 for Mac OS X x86 (64-bit) (April 8, 2019)",
StyleDefinitions->"Default.nb"
]`

## What is a trace of a product of Dirac matrices in a Fermion loop?

A trace of a product of Dirac matrices in a Fermion loop is a mathematical calculation used in quantum field theory to describe the behavior of fermions, which are particles with half-integer spin. It involves taking the trace of a product of Dirac matrices, which are mathematical objects used to represent the behavior of fermions, in a loop-like structure. This calculation is important in understanding the behavior of fermions in various physical systems.

## Why is the trace of a product of Dirac matrices in a Fermion loop important?

The trace of a product of Dirac matrices in a Fermion loop is important because it allows us to calculate various physical quantities related to fermions, such as their scattering amplitudes and decay rates. These calculations are crucial in understanding the behavior of fermions in different physical systems, from subatomic particles to the early universe.

## How is the trace of a product of Dirac matrices in a Fermion loop calculated?

The trace of a product of Dirac matrices in a Fermion loop is calculated using the properties of Dirac matrices and the rules of matrix multiplication. The result is a complex number that represents the contribution of the fermion loop to the overall physical quantity being calculated.

## What are some applications of the trace of a product of Dirac matrices in a Fermion loop?

The trace of a product of Dirac matrices in a Fermion loop has many applications in theoretical physics, particularly in quantum field theory. It is used in the calculation of scattering amplitudes, decay rates, and other physical quantities related to fermions. It also has applications in cosmology, where it helps us understand the behavior of fermions in the early universe.

## Are there any limitations to using the trace of a product of Dirac matrices in a Fermion loop?

While the trace of a product of Dirac matrices in a Fermion loop is a powerful tool in theoretical physics, it does have its limitations. One limitation is that it can only be used to calculate physical quantities related to fermions, and not other types of particles. Additionally, the calculation can become very complicated and difficult to solve for systems with a large number of particles, making it challenging to apply in some scenarios.

• Quantum Physics
Replies
3
Views
760
• Quantum Physics
Replies
1
Views
717
• Quantum Physics
Replies
1
Views
1K
• Quantum Physics
Replies
1
Views
1K
• Quantum Physics
Replies
1
Views
821
• High Energy, Nuclear, Particle Physics
Replies
1
Views
1K
• Quantum Physics
Replies
4
Views
1K
• Quantum Physics
Replies
1
Views
1K
• Quantum Physics
Replies
0
Views
531
• Quantum Physics
Replies
24
Views
2K