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4 Standard Model Tree-Level Processes

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4 Standard Model Tree-Level Processes 4 STANDARD MODEL TREE-LEVEL PROCESSES This chapter describes in some detail the tree-level computations of some elementary standard model processes. Both leptons and quarks are treated on the same footing, ignoring for the most part that quarks do not exist as free states. Quantum corrections to these results are relegated to a later chapter. 4.1 Tree-Level Parameters In chapter 2 we described the classical standard model Lagrangian in terms of eighteen parameters: three gauge coupling constants, three charged lep- ton masses, six quark masses, three flavor-mixing angles, one CP-violating phase, the Higgs mass and the vacuum value of the Higgs field which is fixed by the masses of the W and Z gauge bosons. All these parameters have been measured to varying degrees of accuracy, except the Higgs mass for which there is an upper limit of 95 GeV (LEP, 1999). All parameters of the standard model run (some only crawl) with the scale at which they are measured. Lepton masses which hardly run with scale can be measured directly through kinematics. To distinguish the running from the physical fermion mass, we denote the latter in capital letters. The experimental values of the charged lepton masses are Me = :511 MeV ;Mµ = 105:66 MeV ;Mτ = 1777:05 MeV : (4.1) Since quarks are subject to the strong QCD force, they do not exist as asymptotic states. This makes the measurements of their masses at best indirect, and subject to qualifications. The masses of the three lightest quarks are extracted from the effective low energy chiral Lagrangian (see next chapter), with the range of values 1 2 STANDARD MODEL TREE-LEVEL PROCESSES Mu = 2 − 8 MeV ;Md = 5 − 15 MeV ;Ms = 100 − 300 MeV : (4.2) For heavier quarks, one may adopt a universal operational definition for their physical mass as say, half the energy needed for pair production of the quark-antiquark bound state. The other three quarks have masses which are all above the QCD scale, making their extraction from the data more direct, with values ranging over Mc = 1:0 − 1:6 GeV ;Mb = 4:1 − 4:5 GeV ;Mt = 175:5 ± 5:5 GeV : (4.3) The masses and widths of the W and Z bosons have been measured to remarkable accuracies MW = 80:37 ± 0:09 GeV ; ΓW = 2:06 ± 0:06 GeV ; (4.4) MZ = 91:187 ± 0:007 GeV ; ΓZ = 2:496 ± 0:0027 GeV : (4.5) The rest of the parameters appear in the Cabibbo-Kobayashi-Maskawa ma- trix, traditionally represented as 0 1 Vud Vus Vub U = @ Vcd Vcs Vcb A : (4.6) Vtu Vts Vtb Not all of its nine matrix elements have been measured directly, but all can be inferred from the unitarity relations X ∗ VkiVkj = δij ; i; j = u; c; t : (4.7) k=d;s;b Since the rotation group is generated by the three family Gell-Mann matrices lambdab2 ; λ5 ; λ7, we can write the CKM matrix `ala Euler U = eia4λ4 eia5λ5 eia7λ7 eia2λ2 ; (4.8) where the CP-violating phase is set along λ4. Following L. Wolfenstein, we express its parameters as a power series in the Cabibbo angle, 2 3 3 a2 = λ ; a7 = Aλ ; a5 = Aρλ ; a4 = −Aηλ ; (4.9) 4.2 W-decay 3 where λ is the Cabibbo angle. The first three parametrize the rotations, and Aη denotes the CP violating phase. In matrix form, 0 λ2 3 1 1 − 2 λ Aλ (ρ − iη) U = λ2 2 ; (4.10) @ −λ 1 − 2 Aλ A Aλ3(1 − ρ − iη) −Aλ2 1 where A; ρ, and η are of order one. Their (1999) values λ ≈ :2205 ± :0018 ; A ≈ 0:81 ± 0:06 ; (4.11) are obtained from direct measurements of Vus and Vcb, respectively. From the branching ratio of B-meson decay into charm and up quarks, we get p ρ2 + η2 = 0:36 ± 0:09 : (4.12) The extraction of the phase from data is more indirect, as it necessarily involves loops with the third family, since CP-violation in B-decay has not yet been observed. One can at best constrain its value, either by deducing Vtd from B − B mixing, or K − K mixing. One can quote the central values ρ ≈ 0:05 ; η ≈ 0:35 : (4.13) Finally we note that since the CKM matrix is unitary, the first column times the complex conjugate of the third column must vanish, yielding Aλ3(ρ + iη) − Aλ3 + Aλ3(1 − ρ − iη) = 0 : (4.14) With this parametrization, this is obviously tautological, but we may view the three factors, ρ + iη, −1, and 1 − ρ − iη as the sides of a triangle in the complex (ρ, η) plane, called the unitarity triangle. Its sides and its angles (CP-violation) can be independently determined from experiments. If after these measurements, the triangle does not close, there must be sources of CP-violation beyond the standard model. 4.2 W-decay As a first example of a tree-level application of the standard model, we calculate the decay rate of the W -boson into an electron and its associated antineutrino, 4 STANDARD MODEL TREE-LEVEL PROCESSES − − W ! e +ν ¯e : The decay proceeds at tree-level through the interaction ig2 + −µ Lint = p W (x)J (x) ; (4.15) 2 µ using only the electron part of the charged current. This process is repre- sented by the Feynman diagram p W e e k q It corresponds to the matrix element Z ig2 4 −(λ) + µ (a) (b) Tfi = p d xhW (k)jW (x)¯νe(x)γ L e(x)je (p)ν ¯ (q)i ; (4.16) 2 µ where we have rewritten the current in terms of four-component Dirac fields to facilitate the computations to come; L is the projection operator onto left-handed states 1 + γ L = 5 : (4.17) 2 Assuming that the plane wave states are normalized in a box of volume V , we have −ik·x −(λ) + 1 e ∗(λ) hW (k)jWµ (x)j0i = p p µ (k) ; (4.18) V 2EW (λ) where µ denote the plane wave solutions to the massive Klein-Gordon equation with polarization λ = 1; 2; 3; they obey µ (λ) k µ (k) = 0 : (4.19) 4.2 W-decay 5 Similarly, the fermion fields acting on the plane wave states yield 1 1 h0je(x)je(a)(p)i = p p eip·xu(a)(p) ; (4.20) V 2Ee (b) 1 1 iq·x (b) h0jν¯e(x)jν¯e (q)i = p p e v¯ (q) ; (4.21) V 2Eνe where a; b are the fermion polarizations which run over two values. The four-component covariant plane wave states, u(a) and v(a), are normalized to (a) (b) (a) (b) ab u¯ (p)u (p) = 2meδab ; v¯ (p)v (p) = −2mνe δ ; (4.22) where the bar denotes the Pauli adjoint. Putting it all together, and inte- grating over x, we obtain the matrix element ig (2π)4 δ(4)(k − p − q) T = 2 ∗(λ)(k)¯v(a)(q)γµLu(b)(p) : (4.23) fi 3=2 p µ 4V EW EeEνe Its absolute square gives the transition probability over all of space and time. One of the two δ-functions (times (2π)4) is interpreted as the volume of space-time, VT . The differential decay rate is the transition probability per unit of time, multiplied by the number of final states; with our normalization of the plane wave states, for each final state with three momentum ~p, the number of available states is V d3p ; (4.24) (2π)3 leading to the differential decay rate 2 3 3 g2 (4) d p d q dΓ = 2 δ (k − p − q) × 16(2π) EW Ee Eνe ∗(λ) (λ) (a) µ (b) (b) ρ (a) µ (k)ρ (k)¯v (q)γ Lu (p)¯u (p)γ Lv (q) :(4.25) The sums over the neutrino and electron polarizations, using X (a) (a) X (a) (a) v (q)¯v (q) = q= −mνe ; u (p)¯u (p) = p= +me ; (4.26) a a yield the trace 6 STANDARD MODEL TREE-LEVEL PROCESSES Tr (γµL[p= +me]γρL[q= −mνe ]) = Tr(γµ p=γρL q=) ; (4.27) since LγρL = 0. To obtain the total decay rate, we average over the three initial polarizations of the W -boson, using X kµkρ ∗(λ)(k) (λ)(k) = −(g − ) : (4.28) µ ρ µρ M 2 λ W Substituting in the differential decay rate, and dividing by the number of polarization states of the W , we obtain 2 (4) 3 3 µ ρ −g2δ (k − p − q) d p d q µρ k k dΓ = 2 (g − 2 )Tr(γµp/γρLq=) : (4.29) 48(2π) EW Ee Eνe MW To evaluate the trace, we use the identity σ γµγνγρ = gµνγρ + gνργµ − gµργν + iµνρσγ5γ ; (4.30) where 0123 = 1, with the result Tr(γµγνγργσL) = 2(gµνgρσ + gµσgρν − gµρgσν + iµνρσ) : (4.31) Then, by using the kinematical relations (q · p ≡ q0p0 − ~q · ~p) 1 p · k = m2 + p · q ; q · k = p · q ; p · q = (M 2 − m2) ; (4.32) e 2 W e we find, in the rest frame of the W -boson, 2 2 2 3 3 g2MW me me (4) d p d q dΓ = 2 1 − 2 1 + 2 δ (k − p − q) : (4.33) 24(2π) MW 2MW Ee Eνe Use of the generic phase space integral formula Z d3p d3q I ≡ δ(4)(k − p − q) ; p0 q0 2π p = [k2 − (m − m )2][k2 − (m + m )2] ; (4.34) k2 1 2 1 2 where 4.2 W-decay 7 q q 2 2 q0 = m1 + ~q · ~q; p0 = m2 + ~p · ~p: Setting m1 = 0 and m2 = me, it yields the final result 3 2 2 2 − − GF MW me me Γ(W ! e +ν ¯e) = p 1 − 2 1 + 2 ; (4.35) 6π 2 MW 2MW written in terms of the Fermi coupling constant, GF , 2 GF g2 p = 2 ; (4.36) 2 8MW which parametrizes the strength of the current-current interaction in the effective Hamiltonian GF + −µ Hint = 4 p J J : (4.37) 2 µ The factor of 4 is to account for the L in the charged current (Fermi did not envisage Parity violation).
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