FeynCalc Tutorial 2 (Dated: November 7, 2016) 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 notebook. If you get the similar output, you can start from section. I. If your FeynCalc cannot be loaded properly, we have a version for the class not requir- ing the installation, just ask. I. Review of the last tutorial Let's first test to see how much you have remembered 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. µ ν 2 2 • Example 1: gµνp p with p = m . 1 µ 2 2 • Example 2: (p − k) (p − k)µ − (p − k) − p αβµν • Example 3: gαβ II. Spinors and Gamma matrices A. Notations The procedure is the same as last tutorial. We first need to convert the spinor expres- sions 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 table. symbol in FeynCalc Spinor u with mass m u(p; m),u ¯(p; m) Spinor[p,m] Spinor v with mass m v(p; m),v ¯(p; m) Spinor[-p,m] µ Gamma Matrix γ , γµ GA[µ] 5 Gamma 5 γ , γ5 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 theu ¯ and u share the same notation in FeynCalc. Here FeynCalc uses a trick: since in reality every Fermion chain starts with a barred spinor and ends with a unbarred spinor, FeynCalc interprets the Spinor[p,m] at the beginning of a Fermion chain asu ¯, while interpreting the 2 Spinor[p,m] at the end of a Fermion chain as u. The case forv ¯ and v is the same. In summary, whether a Spinor is barred or unbarred depends on its position in a Fermion chain. Now let's do a simple example, µ ν • Example 4:u ¯(p2; m)γ (k= + m)γ v(p1; 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 above, and see what the output looks like. Missing dots in Fermion chains is a very common mistake in real calculation, so please be very careful and double check the input every time you have a Fermion chain. B. DiracSimplify DiracSimplify[expr] simplifies products of γ-matrices in expr and expands non- commutative products. Repeated Lorentz indices are contracted. Here are some examples. µ • Example 5: simplify γ γµ. µ • Example 6: simplify γ (k= + m)γµ. • Example 7: simplify kγ= µk=. µ Check whether these results are the same as your expectation. Now try to simplify γ k==qγµ µ and γ k=p==qγµ and compare the results with your classmates. DiracSimplify[expr] also applies the Dirac equation. for example 3 • Example 8: simplifyu ¯(p2; m)(p=2 − p=1 + m) u(p1; m). The result is proportional to m! (Why?) C. Tr The trace of γ-matrices can be easily calculated with Tr. For example • Example 9: calculate Tr[γαγµγβγν]. • Example 10: calculate Tr[γαγµγβγνγ5]. Do these two outputs make sense to you? You can try the trace with more γ-matrices • Example 11: calculate Tr[γαγµγβγνγλ] and Tr[γαγµγβγνγλγ5]. 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, 4 p1 k1 p1 k1 k2 p2 k2 p2 ( a ) ( b ) The cross section can be expressed as 1 Z 1 X σ = dΠ jMj2; (1) 2s 2 4 e−,γ spin 4 (4) where dΠ2 is the two-body phase space including (2π) δ (p1 + p2 − k1 − k2), the factor P (1=4) spin averages (sums) the spin of the initial (final) state electron and photon. The matrix element can be written as 1 X 2 1 X µν µν µ0ν0 µ0ν0 ∗ ∗ ∗ jMj = (M + M ) × (M + M ) × (p ) 0 (p ) (k ) 0 (k ) 4 4 a b a b µ 2 µ 2 ν 2 ν 2 e−,γ spin e−,γ spin 1 X µν µν µ0ν0 µ0ν0 ∗ = (M + M ) × (M + M ) × (−g 0 )(−g 0 ); 4 a b a b µµ νν e−spin (2) where Ma and Mb 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: 1 X µν µ0ν0 ∗ (M ) × (M ) × (−g 0 )(−g 0 ): 4 a a µµ νν (3) e−spin You need to add the contribution of the diagram (b) to get the full result in Eq. (2). From the Feynman rule of QED, we have µν ν i(p=1 + p=2 + m) µ M =u ¯ (k )(−ieγ ) (−ieγ )u 0 (p ) a s 1 (p + p )2 − m2 s 1 1 2 (4) 2 1 ν µ 0 = (−ie ) 2 2 × u¯s(k1)γ (p=1 + p=2 + m)γ us (p1); (p1 + p2) − m where s and s0 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 where m is the electron mass. Next we type Eq. (4) in Mathematica and calculate its complex conjugate 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 by looking at its standard form. 6 We can do the trace by changing DiracTrace to Tr, which is explained in subsection II C. 2 where we have used momentum conservation k2 = p1 + p2 − k1 and identity s + t + u = 2m . The factor 1=4 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 (b) and get the full result in Eq. (2). Compare your result with Eq. (13.119) on Matthew Schwartz's book to see whether the two results agree. (Hint: you need to replace p12 and p24 in Eq. (13.119) into SP[p1,p2] and SP[p1,k2] to compare the two results.) 7.