2 Feynman rules, decay widths and cross sections

Transcription

1 2 Feynman rules, decay widths and cross sections 2.1 Feynman rules Normalization In non-relativistic quantum mechanics, wave functions of free particles are normalized so that there is one particle in a box of volume V, d 3 x ψ( x) 2 = 1. (2.1) Thus plane waves are normalized as V ψ p ( x) = 1 V e i p x. (2.2) They form a complete, orthonormal set, p p = δ p,p. (2.3) Although this normalization is not Lorentz invariant, it is convenient to start with this box normalization for the initial and final states also in a relativistic framework and to perform the transition to covariant continuum normalization in the final formula for decay widths and cross sections. Note that a) the propagators describing virtual particles are already covariant, b) S-matrix elements are not independent of the chosen normalization: Squared S-matrix elements depending on continuous variables are not probabilities, but probability densities. We use as relativistic normalization for all types of particles p p = 2E p V δ p, p. (2.4) In the continuum limit p V (2π) 3 d 3 p (2.5) corresponding to one state per phase space volume d 3 xd 3 p = h 3 and p p = 2E p (2π) 3 δ( p p ). (2.6) Thus the relation between matrix elements in non-relativistic and relativistic normalization for a transition from n n particles is given by M fi = (2E i V ) i=1 1/2 n (2E f V ) 1/2 M box fi. (2.7) 9

2 2 Feynman rules, decay widths and cross sections Feynman rules We associate to external particles wave functions without normalization and exp(±ikx) factors as follows u(p 1 ) v(p 4 ) ε (r) µ (k 1 ) ε (r) ν (k 2 ) v(p 2 ) ū(p 3 ) time Spin sums are given by u a (p,s)ū b (p,s) = (p/ + m) ab (2.8) s v a (p,s) v b (p,s) = (p/ m) ab. (2.9) s Polarizations sums of massless spin-1 bosons (as photons or gluons) are given (in Feynman gauge) by 2 r=1 of spin-1 bosons with mass M > 0 (as W ± and Z) are by ε (r) µ (k)ε(r) ν (k) = g µν, (2.10) 3 r=1 The propagator of virtual particles are ε (r) µ (k)ε(r) ν (k) = g µν + k µ k ν /M 2. (2.11) 10

3 2.2 Decay widths and cross sections µ ν p is F (p) = i p/+m p 2 m 2 +iε µ ν k id F,µν (k) = i(g µν+k µ k ν /M 2 ) k 2 M 2 +iε µ ν k id F,µν (k) = ig µν k 2 +iε 2.2 Decay widths and cross sections Decay widths We split the scattering operator S into a diagonal part and the transition operator T, S = 1 + it. Taking matrix elements using the box normalization, we obtain S fi = δ fi + (2π) 4 δ (4) (P i P f )M box fi (2.12) where we set also T fi = (2π) 4 δ (4) (P i P f )M fi. Squaring for i f using (2π) 4 δ (4) (0) = V T and multiplying with the density of final states, Eq. (2.5), in the continuum limit gives as differential transition probability dw fi = (2π) 4 δ (4) (P i P f )V T M box fi 2 The decay rate dγ is the transition probability per time, dγ fi = dw fi T = (2π) 4 δ (4) (P i P f )V M box fi 2 V d 3 p f (2π) 3. (2.13) V d 3 p f (2π) 3. (2.14) Going over to relativistic normalization eliminates the volume factors V, dγ fi = (2π) 4 δ (4) 1 (P i P f ) M fi 2 d 3 p f 2E i 2E f (2π) 3. (2.15) Introducing the n-particle phase space volume the decay rate becomes dφ (n) = (2π) 4 δ (4) (P i P f ) d 3 p f 2E f (2π) 3, (2.16) dγ fi = 1 2E i M fi 2 dφ (n). (2.17) Since both M fi 2 and the phase space dφ (n) are Lorentz invariant, the decay rate Γ 1/E i = 1/(γ i m i ) shows explicitly the time dilation effect. 11

4 2 Feynman rules, decay widths and cross sections Two-particle decays We evaluate the two particle phase space dφ (2) in the rest frame of the decaying particle, dφ (2) = (2π) 4 δ(m E 1 E 2 ) δ (3) ( p 1 + p 2 ) d 3 p 1 2E 1 (2π) 3 d 3 p 2 2E 2 (2π) 3 (2.18) We perform the integration over d 3 p 1 using the momentum delta function. In the resulting expression, dφ (2) = 1 1 (2π) 2 δ(m E 1 E 2 ) d 3 p 2, (2.19) 4 E 1 is now a function of p 2, E 2 1 = p2 2 +m2 1. Introducing spherical coordinates, d3 p 2 = dωp 2 2 dp 2, dφ (2) = 1 16π 2dΩ and evaluating the delta function with M E 1 E 2 = M x and gives 0 δ(m E 1 E 2 ) p2 2 dp 2, (2.20) dp 2 dx = p 2x (2.21) dφ (2) = p 2 16π 2 dω. M (2.22) where p 2 2 = p2 cms = 1 [ M 2 4M 2 (m 1 + m 2 ) 2][ M 2 (m 1 m 2 ) 2] (2.23) equals the cms momentum of the two final state particles. Three-particle decays The three particle phase space dφ (3) is dφ (3) = (2π) 4 δ (4) (P p 1 + p 2 + p 3 ) d 3 p 1 2E 1 (2π) 3 d 3 p 2 2E 2 (2π) 3 d 3 p 3 2E 3 (2π) 3 (2.24) We can use again the momentum delta function to perform the integration over d 3 p 3, dφ (3) = 1 (2π) 5 δ(m E 1 E 2 E 3 ) d3 p 1 d 3 p 2 8 E 3, (2.25) To proceed we have to know the dependence of the matrix element on the integration variables. If there is no preferred direction (either for scalar particles or spin averaged), we obtain dφ (3) = = 1 4πp 2 1 dp 1 2πd cos ϑp 2 2 dp 2 8(2π) 5 δ(m E 1 E 2 E 3 ) E 3 1 (p 1 dp 1 ) (p 1 p 2 d cos ϑ)(p 2 dp 2 ) 32π 3 δ(m E 1 E 2 E 3 ) (2.26) E 3 We rewrite next the momentum integrals as energy integrals. Energy-momentum relation E 2 i = m2 i + p2 i gives E ide i = p i dp i for i = 1,2. Furthermore, E 2 3 = ( p 1 + p 2 ) 2 + m 2 3 = p p p 1 p 2 cos ϑ + m 2 3 (2.27) 12

5 2.2 Decay widths and cross sections and thus E 3 de 3 = p 1 p 2 d cos ϑ for fixed p 1, p 2. Performing the angular integral, we obtain and finally dφ (3) = 1 32π 3 de 1dE 2 de 3 δ(m E 1 E 2 E 3 ) (2.28) dφ (3) = 1 32π 3 de 1dE 2. (2.29) The last step is only valid, if the argument of the delta function is non-zero. Thus the remaining task is to determine the boundary of the integration domain. We introduce the invariant mass of the pair (i,j) m 2 23 = (p p 1 ) 2 = (p 2 + p 3 ) 2 = M 2 m 2 1 2ME 1 (2.30) m 2 13 = (p p 2 ) 2 = (p 1 + p 3 ) 2 = M 2 m 2 2 2ME 2 (2.31) m 2 12 = (p p 3 ) 2 = (p 1 + p 2 ) 2 = M 2 m 2 3 2ME 3. (2.32) where the last column is valid in the rest frame of the decaying particle with mass M. With E 1 +E 2 +E 3 = M one finds m m2 13 +m2 12 = M2 +m 2 1 +m2 2 +m2 3. Therefore only two out of the three variables are independent. Let s choose m 2 23 and m2 13 as integrations variables, with m 2 23 as the outer one. Then maximal value of m2 23 follows from E 1 = m 1 in Eq. (2.30) as m 2 23 (M m 1) 2, the minimal one (choosing the cm frame of pair 23) from m 2 23 = m2 2 + m (E 2E 3 p 2 + p 3 ) 2 = M 2 m 2 1 2ME 1 (m 2 + m 3 ) 2 (2.33) Combined we have (m 2 + m 3 ) 2 m 2 23 (M m 1) 2. (2.34) For given value of m 2 23, we have now to determine the allowed range of m2 13. Inserting energy and momentum conservation into E3 2 = p2 3 + m2 3, we obtain (M E 1 E 2 ) 2 = m p2 1 + p p 1 p 2 (2.35) The extrema correspond to p 1 p 2 = ±p 1 p 2 = ± (E1 2 m2 1 )(E2 2 m2 2 ) (2.36) Inserting this into (2.35) and eliminating E 1/2 via and gives the desired border values. A plot of E 1 = M2 + m 2 1 m2 23 2M E 2 = M2 + m 2 2 m2 13 2M (2.37) (2.38) dγ de 1 de 2 M fi 2 (2.39) informs us directly about the absolut value of M fi 2. If the decay proceeds via a resonance with mass m R, the number of events along m 2 ij = m2 R is strongly enhanced. 13

6 2 Feynman rules, decay widths and cross sections Cross sections We consider now the interaction of 2 particles in the rest system of either particle 1 or 2. For simplicity, we consider 2 uniform particle beams. They may produce n final state particles. The total number of such scatterings is dn v rel n 1 n 2 dv dt, (2.40) where n i is the density of particles of type i = 1,2 and v rel is their relative velocity. The proportionality constant has the dimension of an area and is called cross section. We define in the rest system of either particle 1 or 2 while we set in an arbitrary frame dn = σv rel n 1 n 2 dv dt, (2.41) dn = An 1 n 2 dv dt. (2.42) Since both dn and dv dt = d 4 x are Lorentz invariant, the expression An 1 n 2 has to be Lorentz invariant too. Since the densities transform as the expression n i = n i,0 γ = n i,0 E i m i (2.43) A E 1E 2 p 1 p 2 (2.44) is also Lorentz invariant. In the rest system of particle 1, it becomes Thus we found that A in an arbitrary frame is given by A p 1 p 2 = A = σv rel (2.45) A = σv rel p 1 p 2. (2.46) A more handy expression for A is obtained as follows: In the rest frame 1, we have m 2 p 1 p 2 = m 1 E 2 = m 1. (2.47) 1 vrel 2 Thus Next we define the flux factor v rel = I = 1 m2 1 m2 2 (p 1 p 2 ) 2. (2.48) (p 1 p 2 ) 2 m 2 1 m2 2. (2.49) Inserting I using (3.36) into (3.34), we obtain dn = σ I V (n 1V )(n 2 dv )dt. (2.50) 14

7 2.2 Decay widths and cross sections Here, we regrouped the terms to make clear that after integration the total number N of scattering events is proportional to the number N 1 = n 1 V and N 2 = n 2 dv of particles of type 1 and 2, respectively. The number N of scattering events per time and per particle pair 12 is however simply the transition probability per time, dn N 1 N 2 T = dσ I V = dw T (2.51) Inserting the expression for dw, we find dσ = E 1E 2 V 2 I (2π) 4 δ (4) (P i P f ) M box fi 2 V d 3 p f (2π) 3 (2.52) Changing from the box to the continuum normalization introduces a factor (2E 1 V ) 1 (2E 2 V ) 1 for the initial state and f (2E fv ) 1 for the final state. Thus the arbitrary normalization volume V cancels and we obtain dσ = 1 4I (2π)4 δ (4) (P i P f ) M fi 2 with the final state phase space dφ (n). d 3 p f 2E f (2π) 3 = 1 4I M fi 2 dφ (n) (2.53) 2 2 scattering The flux factor becomes in the cms I 2 = (p 1 p 2 ) 2 m 2 1 m2 2 = p2 cms (E 1 + E 2 ) 2. (2.54) or I = p cms s. Adding also the know expression for the 2-particle phase space gives dσ dω = 1 64π 2 s p cms p cms M fi 2 (2.55) Introduce for the scattering process Mandelstam variables s,t, and u, s = (p 1 + p 2 ) 2 = (p 2 + p 4 ) 2 (2.56) t = (p 1 p 3 ) 2 = (p 2 p 4 ) 2 (2.57) u = (p 1 p 4 ) 2 = (p 2 p 3 ) 2. (2.58) Since s + t + u = m m2 2 + m2 3 + m2 4, the scattering amplitude A depends only on two variables, e.g M(s,t). In the center of mass (cm) frame, p 1 = p 2, and we see that the Mandelstam variable s = m m E 1E 2 + 2p 2 cm = (E a + E b ) 2 (2.59) has the meaning of cm energy squared. The other two variables correspond to the fourmomentum exchanged between initial and final state particles. 15