FeynCalc Tutorial 2. (Dated: November 7, 2016)

Transcription

1 FeynCalc Tutorial 2 (Dated: Novemer 7, 206) Last time we learned how to do Lorentz contractions with FeynCalc. We also did a simple calculation in scalar QED: two scalars annihilating into two photons at leading order (LO). In this tutorial we will first learn how to implement calculation with the spinors and gamma matrices. Then we are going to calculate the famous Compton scattering at LO. Let s first check whether you can load FeynCalc properly. Run the following command in your Mathematica noteook. If you get the similar output, you can start from section. I. If your FeynCalc cannot e loaded properly, we have a version for the class not requiring the installation, just ask. I. Review of the last tutorial Let s first test to see how much you have rememered from the last tutorial. Below are the three examples we had last time. Discuss with your teammates and make sure you understand the meaning of the FeynCalc functions and the results. Example : g µν p µ p ν with p 2 = m 2.

2 Example 2: (p k) µ (p k) µ (p k) 2 p 2 Example 3: ɛ αβµν g αβ II. Spinors and Gamma matrices A. Notations The procedure is the same as last tutorial. We first need to convert the spinor expressions into something FeynCalc can understand, then use some FeynCalc functions to do the necessary simplifications. The conversion rules for spinor calculations are summarized in the following tale. symol in FeynCalc Spinor u with mass m u(p, m), ū(p, m) Spinor[p,m] Spinor v with mass m v(p, m), v(p, m) Spinor[-p,m] Gamma Matrix γ µ, γ µ GA[µ] Gamma 5 γ 5, γ 5 GA[5] or GA5 Dirac Slash /p = γ p GS[p] Notice that there is an extra minus sign for spinor u and spinor v. It seems confusing that the ū and u share the same notation in FeynCalc. Here FeynCalc uses a trick: since in reality every Fermion chain starts with a arred spinor and ends with a unarred spinor, FeynCalc interprets the Spinor[p,m] at the eginning of a Fermion chain as ū, while interpreting the 2

3 Spinor[p,m] at the end of a Fermion chain as u. The case for v and v is the same. In summary, whether a Spinor is arred or unarred depends on its position in a Fermion chain. Now let s do a simple example, Example : ū(p 2, m)γ µ (/k + m)γ ν v(p, m). Since the γ-matrices are anti-commutative, the order of them is important. FeynCalc uses the dot multiplication. to keep the order. As a test, you can remove all the. in the input aove, and see what the output looks like. Missing dots in Fermion chains is a very common mistake in real calculation, so please e very careful and doule check the input every time you have a Fermion chain. B. DiracSimplify DiracSimplify[expr] simplifies products of γ-matrices in expr and expands noncommutative products. Repeated Lorentz indices are contracted. Here are some examples. Example 5: simplify γ µ γ µ. Example 6: simplify γ µ (/k + m)γ µ. Example 7: simplify /kγ µ /k. Check whether these results are the same as your expectation. Now try to simplify γ µ /k/qγ µ and γ µ /k/p/qγ µ and compare the results with your classmates. DiracSimplify[expr] also applies the Dirac equation. for example 3

4 Example 8: simplify ū(p 2, m) ( /p 2 /p + m) u(p, m). The result is proportional to m! (Why?) C. Tr The trace of γ-matrices can e easily calculated with Tr. For example Example 9: calculate Tr[γ α γ µ γ β γ ν ]. Example 0: calculate Tr[γ α γ µ γ β γ ν γ 5 ]. Do these two outputs make sense to you? You can try the trace with more γ-matrices Example : calculate Tr[γ α γ µ γ β γ ν γ λ ] and Tr[γ α γ µ γ β γ ν γ λ γ 5 ]. Both of them equal to zero! (Why?) You are encouraged to calculate the trace with 6 and 8 γ-matrices. Count how many terms are in the outputs. III. Project: Compton scattering Compton scattering is the process e + γ e + γ. At LO, there are two Feynman diagrams,

5 p k p k p2 k2 p2 k2 The cross section can e expressed as ( a ) ( ) σ = 2s dπ 2 e,γ spin M 2, () where dπ 2 is the two-ody phase space including (2π) δ () (p + p 2 k k 2 ), the factor (/) spin averages (sums) the spin of the initial (final) state electron and photon. The matrix element can e written as e,γ spin M 2 = = e,γ spin e spin (M a µν + M µν ) (M µ ν a + M µ ν ) ɛ µ (p 2 )ɛ µ (p 2)ɛ ν(k 2 )ɛ ν (k 2 ) (M a µν + M µν ) (M µ ν a + M µ ν ) ( g µµ )( g νν ), (2) where M a and M are for the two Feynman diagrams, respectively. In the second step of Eq. (2), we have summed over the polarization of the two photons. There are four terms in the sum in Eq. (2). Next I will show how to use FeynCalc to calculate one of them: e spin (M µν a ) (M µ ν a ) ( g µµ )( g νν ). (3) You need to add the contriution of the diagram () to get the full result in Eq. (2). From the Feynman rule of QED, we have M a µν = ū s (k )( ieγ ν ) i( /p + /p 2 + m) (p + p 2 ) 2 m 2 ( ieγµ )u s (p ) = ( ie 2 ) (p + p 2 ) 2 m ū s(k 2 )γ ν ( /p + /p 2 + m)γ µ u s (p ), () where s and s are the spins of the final and initial electrons, respectively. Now let s pass the calculation to FeynCalc. First we need to define the scalar products 5

6 where m is the electron mass. complex conjugate Next we type Eq. () in Mathematica and calculate its Then we can use FeynCalc function FermionSpinSum to sum over the spins of the initial and final electrons. Notice that the output of FermionSpinSum is a trace (why?). Although it appears as tr in the output, the internal function is DiracTrace, as one can check y looking at its standard form. 6

7 We can do the trace y changing DiracTrace to Tr, which is explained in susection II C. where we have used momentum conservation k 2 = p + p 2 k and identity s + t + u = 2m 2. The factor / is the factor in Eq. (3). So far, we have calculated the result for Eq. (3). It is your turn to write down the Feynman expression of diagram () and get the full result in Eq. (2). Compare your result with Eq. (3.9) on Matthew Schwartz s ook to see whether the two results agree. (Hint: you need to replace p2 and p2 in Eq. (3.9) into SP[p,p2] and SP[p,k2] to compare the two results.) 7