Theoretical Particle Physics 1 Exercise Sheet 9

Prof. Thorsten Feldmann Handout: June 20th 2014 Philipp B¨oer Discussion of Homework: June 27th 2014

Homework for June 27th 2014

H20: Higgs Decay into Fermions So-called Yukawa interactions couple a scalar ﬁeld φ to a scalar fermion current of a Dirac ﬁeld ψf . The interaction Hamiltonian reads ¯ Hint(x) = yf ψf (x) ψf (x) φH (x) , (1) where yf is the Yukawa coupling constant. In the Standard Model of particle physics, for instance, a Yukawa interaction induces the decay of the Higgs boson into fermion-antifermion ¯ pairs, H → ff (as long as mH > 2mf ). [Remark: Also Yukawa interaction can be used to describe the strong nuclear force between nucleons (fermions), mediated by pions (spin-0 mesons).]

(a) Draw the Feynman diagram, that describes the leading order transition amplitude in a perturbation series in yf . Use the Feynman rules (in momentum space) to derive an expression for the decay amplitude iM(H → ff¯).

P 2 (b) Calculate the sum s |M| , where s are all possible spin orientations of the ﬁnal state fermions. Here use the relations (see exercise sheet 5) X X uα(p, s)¯uβ(p, s) = (p/ + mf )αβ , vα(p, s)¯vβ(p, s) = (p/ − mf )αβ , (2) s s to write the result as a trace of Dirac matrices. Then use

tr[a/] = 0 , tr[a//b] = 4 a · b , (3)

P 2 2 2 to calculate the appearing traces, and ﬁnally express s |M| in terms of mH and mf . (c) Use the result from the previous sheet (H19: two particle phase space) to calculate the partial decay width dΓ(H → ff¯). As an example calculate the Higgs decay width into ¯ ¯ bb-quark pairs, Γ(H → bb), for mH = 125 GeV, mb = 5 GeV and yb = 0.03. [Hint: You also have to sum over the three diﬀerent colour states of the b-quark.]

1 H21: Yukawa Potential

Consider the scattering amplitude for a 2 → 2 process kAkB → pApB of Dirac fermions mediated by a Yukawa interaction (see above). For simplicity assume the scattering of two distinguishable fermions (e.g. electrons and muons), such that only one diagram (“t-channel, 2 2 where t is the so-called Mandelstam variable t ≡ (pA − kA) = q ) contributes to the Born amplitude. The scattering amplitude can be written as

0 0 [¯uB(pB, sB)uB(kB, sB)][¯uA(pA, sA)uA(kA, sA)] iM(kAkB → pApB) = (−iyeyµ) 2 2 . (4) (pA − kA) − mφ Now investigate the non relativistic limit

kA ' (mA, kA) , pA ' (mA, pA) ,

kB ' (mB, kB) , pB ' (mB, pB) , (5) where terms of order |p|/m 1 have been neglected in the energy E.

(a) Show that in this limit

2 0 t ' −|p − kA| andu ¯A(pA, s )uA(kA, sA) ' 2m δ 0 etc. A A sAsA holds, and verify that the scattering amplitude is given by

−yeyµ iM ' −iV˜ (p − kA) 4memµ δ 0 δ 0 , V˜ (q) ≡ . A sAsA sB sB 2 2 |q| + mφ

(b) Calculate the Fourier transform of V˜ and compare the result to the general form of a Yukawa potential e−2r/r0 V (r) ∝ . r

How does the range of inﬂuence r0 depends on the mass of the mediated particle mφ? Calculate r0 in the case of

(i) exchange of pions in the strong nuclear force (mπ = 140 MeV),

(ii) exchange of a Higgs boson (mH = 125 GeV). (c) How does the situation looks like, if you interchange one particle (two particles) by the corresponding antiparticle(s)?