Exercise 7: Rho Meson Goals: Computing vector meson two-point function Concepts: Gamma matrices Directory: \$HOME/examples/07-rho How to: \$ cd \$HOME/examples/07-rho \$ make \$ submit rho /scratch/data/beta6_gauge_8x8x8x16 0.1493

In this exercise we compute a vector meson two point function and introduce γ matrices. QDP has a hardwired prepresentation of the Clifford algebra basis that is used by all routines in the library. The user can access the γ-matrices by using a small integer. All products of γ-matrices can be expresses as a product of the following form: γ1aγ2bγ3cγ4d where each of a, b, c and d is either 0 or 1. Then the index of this particular product is the number [dcba]2. Since Sρ = Trt(Pγ1PTγ1), and P=γ5P+γ5, one computes Sρ as follows (this is rho.c). Since γ-matrices do not commute, we need to be mindful on which side of the propagator they are, see lines 37 and 38)

 ` 12 print_rho(QDP_DiracPropagator *prop, int t0) 13 { 14 int i, nt, gamma; 15 QLA_Complex *corr; 16 QDP_DiracPropagator *prop_gamma, *gamma_prop_gamma;... 29 /* allocate temporary propagators */ 30 prop_gamma = QDP_create_P(); 31 gamma_prop_gamma = QDP_create_P(); 32 33 /* gamma_1 gamma_5 = g_2 g_3 g_4 -> 1110_2 = 15 - 1 */ 34 gamma = 15 - 1; 35 36 /* multiply prop by GAMMA on both sides */ 37 QDP_P_eq_P_times_gamma(prop_gamma, prop, gamma, QDP_all); 38 QDP_P_eq_gamma_times_P(gamma_prop_gamma, prop_gamma, gamma, QDP_all); 39 40 /* do sum[trace(prop^+ GAMMA prop GAMMA)] over each timeslice */ 41 QDP_c_eq_P_dot_P_multi(corr, prop, gamma_prop_gamma, timeslices, nt);`
The propagator now is complex. We print only the real part of it:
 ` 43 /* print result */ 44 printf0("BEGIN RHO\n"); 45 for(i=0; i

### Problems

• What will happen if you use QDP_P_eq_P_times_gamma() instead of QDP_P_eq_gamma_times_P() on line 38?
• Modify the program to compute all 16 γ-matrices. How many different states are there?
• Run the program with different values of the hopping parameter. How does mρ depend on κ?
• Why there is a minus sign in front of QLA_real() on line 47?