Problem set 6 or Quantum Field Theory course 2018.03.13. Toics covered Scattering cross-section and decay rate Yukawa theory and Yukawa otential Scattering in external electromagnetic ield, Rutherord ormula QED e e, e e + and e + e + scattering and Coulomb interaction Recommended reading Peskin Schroeder: An introduction to quantum ield theory Sections 4.3-4.8 Problem 6.1 Decay rate and scattering cross section The ininitesimal scattering cross section o two incoming articles A and B with our-momenta A, B is exressed as ( 1 d 3 ) ( 1 M dσ = ( 2E A 2E B v A v B (2π) 3 A, B { }) 2 (2π) 4 δ (4) A + B ), (1) 2E where index runs over all inal states and M is the invariant transition amlitude. To obtain the inal result or the cross section it is necessary to erorm the hase sace integrals to eliminate the delta unctions. In the lecture, we derived the ollowing exression or a 2 2 scattering rocess in the centre-o-mass rame: ( ) dσ 1 1 = dω 2E A 2E B v A v B (2π) 2 M 2, (2) 4E C M C M where E C M = E 1 + E 2 is the total energy. (a) Equation (1) can be slightly modiied to describe the decay o an unstable article in its rest rame: ( dγ = 1 d 3 ) ( 1 M ( 2m A (2π) 3 A { }) 2 (2π) 4 δ (4) A ), (3) 2E where A = (m A,0,0,0) T. Following the derivation o the cross section, erorm the hase scae integrals to obtain a ormula or the case o a decay to two articles in the inal state: ( ) dγ dω C M 1 1 = Θ(M 2m) 2m A (2π) 2 M 2, (4) 4E C M where Θ(x) is the Heaviside ste unction and Ω is the solid angle or the direction o the inal article with momentum 1. Hint: Note that one o the integrals (say, over 2 ) is trivial due to the "satial" art o the Dirac delta and the remaining art (energy conservation) can be used to erorm the radial integral. The ollowing identity might be useul: d = E d E. 1
(b) The model discussed in Problem 5.4 exhibits such decays. Calculate the inverse lietime Γ o a heavy Φ article using the above equation (we recall that the result o Problem 5.4(d) is M = µ). Derive the ollowing result: µ 2 Γ = Θ(M 2m) 32Mπ which shows exlicitly that there is no decay or M < 2m. 1 4m2 M 2 (5) (c) Calculate the ull scattering cross-section σ o two light φ articles with momenta A,B into two light ones with momenta 1,2 in the same model using Eq. (2) and erorming the inal state hase sace integral. Use the result o Problem 5.4.(e): [ ] M = µ 2 1 ( A + B ) 2 M 2 + 1 ( A 1 ) 2 M 2 + 1 ( A 2 ) 2 M 2, (6) and erorm the comutation in COM rame. Show that the cross section decays as s 3 when s = ( A + B ) 2. Remark: by simle dimensional analysis high-energy cross-sections u to nth order o erturbation theory should decay as µ 2n s 1+nd, where d is energy dimension o couling µ. For d 0 the theory is called (suer)renormalizable. Problem 6.2 Wick s theorem and Yukawa theory The simliied version o Yukawa s theory consists o two ields: a real scalar Φ(x) and a Dirac ermion Ψ(x) with lane-wave exansions Φ(x) = Ψ(x) = d 3 k (2π) 3 1 2Ek [c k e i kx + c k ei kx], (7) d 3 k 1 (2π) 3 2Ek s=1,2 [ a sk u s (k)e i kx + b sk v s (k)e i kx]. (8) The Hamiltonian oerator governing the time-evolution is H = H D + H KG + d 3 xg ΨΨΦ, (9) where H D is describing a ree Dirac ield and H KG is the Klein Gordon Hamiltonian o the real scalar ield. We intend to treat the limit o small couling using Wick s theorem. (a) Calculate the normalization actor N o the interacting vacuum (c. Problem 5.4) u to second order in g. Recall that contraction between ermionic ields only involves airing o Ψ with Ψ and also that reordering leading to ermionic airings must use anticommutation relation o the ields. (b) Contraction with incoming scalar articles was exressed in Problem 5.4(c) Φ(x) q Φ = Φ(+) (x) q Φ = e i qx 0, (10) Do the same calculation involving incoming ermions and antiermions: Ψ ξ (x) q, s,n Ψ ξ (x) q, s,n = Ψ (+) ξ (x) q, s,n = u s ξ (q)e i qx 0, (11) = Ψ (+) ξ (x) q, s,n = v s ξ (q)e i qx 0, (12) where n and n reer to article secies and the s is a sin label. Similar relations or outgoing articles can be derived by taking the adjoint. Note that due to comlex conjugation an outgoing antiermion must be contracted with Ψ, not Ψ. 2
Figure 1: Feynman rules or Yukawa theory. Dashed lines denote scalar roagators, solid lines are ermionic. (c) Use these results to comute to lowest order the amlitude o the rocess where a ermion and its antiarticle annihilate into a boson ( Φ q T ex i g where sin indices are droed or brevity. d 4 xψ(x)ψ(x)φ(x)) 1,n; 2,n, (13) (d) Calculate the scattering amlitude o two ermions to leading order in the interaction,n;k,n ( T ex i g d 4 xψ(x)ψ(x)φ(x)),n;k,n. (14) As there are two ermions both in the initial and inal states, a convention must be seciied or the ordering o their creation and annihilation oerators. Let us ix,n;k,n a a k 0,,n;k,n 0 ak a. (15) Hint: watch out or signs coming rom exchanging ermionic ields! Problem 6.3 Feynman rules or Yukawa theory, Yukawa otential 1 (igure rom Pe- The Feynman rules or Yukawa theory in momentum sace are shown in Fig. skin&schroeder). (a) Draw the lowest order Feynman diagrams or the rocesses (13) and (14). Note that in the latter case there are two diagrams. What is their relative sign? (b) Now the amlitude or rocess (14) can be directly written in momentum sace. Derive the ormula or the invariant amlitude im. 3
(c) Let us consider the scattering o distinguishable non-relativistic ermions (in this case, only the irst o the two diagrams contribute). Non-relativistically we can write the our-momenta as = (m,). (16) Use this exression to evaluate ( ) 2 and u s () in Weyl reresentation u to leading order. Show by calculating the sinor roducts ū s ()u s ( ) that the sin is conserved or each article during the scattering in the nonrelativistic limit. (d) To sum u, write the non-relativistic scattering amlitude or distinguishable articles: i g 2 im = 2 + m 2 2mδ ss 2mδ r r. (17) Φ This is to be comared with the non-relativistic Born aroximation or scattering: i T = iṽ (q)2πδ(e E ), (18) where Ṽ (q) is the Fourier transorm o the scattering otential and q =. Read o Ṽ (q) or the Yukawa interaction. Note: actors o 2m come rom relativistic normalization and can be droed rom comarison with Born s ormula. (e) Obtain the exression o the Yukawa otential in real sace by erorming an inverse Fourier transorm. Hint: use a comlex contour when calculating the radial art o this integral. () You ind that two ermions eel an attractive otential due to their interaction. Without doing any lengthy calculations, just by looking at the signs argue that the situation is the same or an antiermion air and also or a ermion-antiermion scattering. Problem 6.4 Rutherord scattering Hint: There is a minus sign between ū s ()u s ( ) and v s ()v s ( ), and it is also necessary to disentangle all airings roerly. Consider the scattering o a Dirac ermion on an external classical electromagnetic ield described by a vector otential A µ (x). The interaction Hamiltonian is H I = e d 3 xψ(x)γ µ Ψ(x)A µ (x) (19) where e is the electric charge. (a) Use Wick s theorem to show that to leading order, the transition amlitude is given by the ollowing exression: i T = i eū s ( )γ µ u s ()à µ (q), (20) where q = and à is the our-dimensional Fourier transorm o A. (b) Assume that the external ield is static so energy is conserved by such a rocess. Then we have i T = im (2π)δ(E E i ). (21) For this rocess (1) is modiied as dσ = 1 d 3 1 M (i 2E i v i (2π) 3 ) 2 (2π)δ(E i E ). (22) 2E where v i is the velocity o the incoming article. Perorm the radial art o the hase sace integral in to obtain the dierential cross-section dσ dω = 1 16π 2 M 2 i =. (23) Hint: utilize the Dirac delta to imose a constraint on the relation between i and. 4
Figure 2: Feynman rules or interactions involving the hoton. The metric tensor is denoted by g µν. (c) Now secialize the classical ield to the Coulomb ield o a nucleus: with A i = 0. Perorm a Fourier transorm to obtain A 0 (q). A 0 = Z e 4πr. (24) Taking the nonrelativistic limit o sinors in the Weyl reresentation, derive the Rutherord ormula dσ dω = Z 2 α 2 4m 2 v 4 sin 4 (θ/2), (25) where θ is the scattering angle and α = e 2 /4π is the ine structure constant. Problem 6.5 Scattering in QED and Coulomb interaction In this exercise we consider scattering amlitudes o electrons and ositrons. To handle such rocesses, two additional Feynman rules are needed or the hoton that mediates the interaction (see Fig. 2.). (a) Draw the diagrams o e e e e scattering u to second order in e. What is the corresonding scattering amlitude? (b) Consider only one o the two ossible diagrams: the one without exchanging external lines. The amlitude or this rocess is im = ( i e) 2 ū( )γ µ u() iη µν ( ) 2 ū(k )γ ν u(k). (26) Take the nonrelativistic limit o sinors in the Weyl reresentation. What is ū( )γ µ u() in this limit? Hint: erorm the calculation at = = 0 and kee the term that does not vanish. (c) Read o Ṽ (q) or the electromagnetic interaction by comaring the non-relativistic limit o (26) to exression (18). Perorm an inverse Fourier transorm to derive the Coulomb otential in real sace. What is its sign? Hint: it is convenient to regularise the Fourier integral by adding a small ositive term to the denominator thus reducing the integral to that ound in the Yukawa case. (d) Draw second-order diagrams or the rocesses: e e + e e + and e + e + e + e +. Write down the amlitude which involves no annihilation/external leg exchange, resectively. Comare their sign with the nonrelativistic limit o (26) to show that attraction only aears or the electronositron air. 5