Applications and Phenomenology QFT II - Weeks 3 & 4 1. Leptonic Decays of Hadrons: from π → 𝓂 ν to B → 𝓂 ν QFT in Hadron Decays. Decay Constants. Helicity Suppression in the SM. 2. On the Structure and Unitarity of the CKM Matrix The CKM Matrix. The GIM Mechanism. CP Violation. The Unitarity Triangle. 3. Introduction to the “Flavour Anomalies”: Semi-Leptonic Decays B → D (*) 𝓂 ν . The Spectator Model. Form Factors. Heavy Quark Symmetry. B → K (*) 𝓂 + 𝓂 - . FCNC. Aspects beyond tree level. Penguins. The OPE. 4. Introduction to Radiative Corrections: B → μ ν γ The (infrared) pole structure of gauge field theory amplitudes. Collinear and Infrared Safety. Peter Skands Monash University — 2020
Flavour-Changing Neutral Currents Now, we move on to: ๏ In the SM, only the W can change quark flavours u i → W + d j d i → W − u j • “Charged Current”: and • The photon, Higgs, and Z, all couple flavour-diagonally ๏ ➡ No tree-level FCNC in SM • FCNC = processes involving b → s b → d , , or c → u transitions. In the SM, this requires at least two W vertices. ๏ Recall: we saw an example when discussing the GIM mechanism: ๏ GIM suppression by CKM unitarity: μ − W - s ¯ ∑ V ij V † jk = δ jk ∑ ν μ K 0 u j W + E.g.: u , c , t μ + d V ud V * us + V cd V * cs ∼ cos θ C sin θ C − sin θ C cos θ C = 0 2 Peter Skands Monash University
Suppressed in SM ➡ Good probes for BSM ๏ Also called “Rare Decays” • Due to suppression, they have small Branching Fractions. • How rare is rare? Recall our K → μμ example; BR(K → μμ ) ~ 10 -8 . So you need to collect ~ one billion K decays to see ~ 10 of these. ๏ For comparison, the charged-current (tree-level W) decays we looked at in ๏ the last lecture have much larger branching ratios, e.g., BR(K → π e ν ) ~ 40% ๏ Since FCNC amplitudes are tiny in the SM, any additional contributions from new physics may be relatively easy to see The equivalent of K → μμ ๏ In B Sector: d , s → ℓ + ℓ − ( B 0 (why not B * ?) B 0 • Leptonic Decays: , d , s → ν ¯ ν ) b → s ℓ + ℓ − , b → d ℓ + ℓ − • Semi-Leptonic: , and b → s ( d ) γ , b → s ( d ) ν ¯ ν • Multi-hadronic: beyond the scope of this course. Our case study: B → K (*) ℓ + ℓ − 3 Peter Skands Monash University
Diagrams contributing to b → s ℓ + ℓ - transitions (same type as you drew “Box” Penguins? K + → π + ℓ + ℓ − for ) s This is t , c , u actually a b ℓ + strong ν ℓ ¯ penguin; W − can you ℓ − see why? + “Penguins” (EW penguins) s s J. Ellis W − t , c , u b b u , c , t ℓ + ℓ + W − γ γ * * / / Z 0 Z 0 ℓ − ℓ − ➡ This is going to get complicated … so let’s think first . + more … 4 Peter Skands Monash University
➡ Top Quark Domination 1: Exploit CKM Unitarity and m t ≫ m c ๏ All of these amplitudes involve (same type as you drew “Box” K + → π + ℓ + ℓ − for ) s GIM-type sums: t , c , u ℳ = V ub V * us ℳ u + V cb V * cs ℳ c + V tb V * ts ℳ t b ℓ + ν ℓ ¯ CKM Unitarity : V ub V * us = − V cb V * cs − V tb V * W − ts ℓ − = V cb V * cs ( ℳ c − ℳ u ) + V tb V * ts ( ℳ t − ℳ u ) ➡ Any quark-mass- + “Penguins” (EW penguins) s independent terms must cancel. t , c Whatever is left must be , u b m n m n proportional to and c t ℓ + W − ➨ Top quark dominates γ * / Z 0 ℓ − …. ℳ ∼ V tb V * ts ℳ t Keeping only terms ∝ m n t 5 Peter Skands Monash University
2: Exploit q 2 ≪ m W2 ➡ Low-Energy Effective Theory ๏ Construct effective vertices, with effective coefficients For example, we previously wrote tree-level W exchange as an effective ๏ coefficient , multiplying two V-A fermion currents. ∝ G F / 2 Recall: (and all the other processes we looked at so far) B → D ℓν ๏ Full EW Theory V cb q 2 = ( p B - p D ) 2 → ≪ m W2 “Low-energy effective theory” ¯ and vertices qqW ¯ ℓν W “Effective 4-FermionVertex” ℒ = − G F c γ ρ (1 − γ 5 ) b ] [ ¯ Effective 4-fermion Lagrangian: V cb [¯ ℓγ ρ (1 − γ 5 ) ν ℓ ] 2 Question: what is the mass dimension of a 4- Effective 4-Fermion Operator fermion operator? coupling (with V-A structure) 6 Peter Skands Monash University
Effective vertices for b → s ℓ + ℓ - (same type as you drew Apply same idea to FCNC processes . “Box” K + → π + ℓ + ℓ − for ) s “Integrate out” the short-distance propagators, leaving only operators b ℓ + for the external states: O i with some effective coefficients, C i ℓ − (which now in general will contain integrals over whatever loops contribute to them in the full theory) + “Penguins” (EW penguins) s s b b ℓ + ℓ + ℓ − ℓ − (Re) classify all possible low-energy operators in terms of Lorentz (+ colour) structure Inami & Lim, Progr. Theor. Phys. 65 (1981) 297 7 Peter Skands Monash University
The Operator Product Expansion For a textbook, see e.g., Donoghue, Golowich, Holstein, “Dynamics of the SM”, Cambridge, 1992 For a review, see e.g., Buchalla, Buras, Lautenbacher, Rev. Mod. Phys. 68 (1996) 1125 ๏ Effective Lagrangian for b → s transitions ℒ = − G F ts ∑ • = sum over effective vertices V tb V * C k 𝒫 k • with overall G F & CKM factor, 2 k Q: why only t? • and operators coefficients 𝒫 k × C k “Wilson Coefficients” In general, we need to do some loop integrals to compute them. Operators directly responsible for semi-leptonic decays: s s γ μ (1 − γ 5 ) b ] [ ¯ 𝒫 ℓ 9 V = [¯ ℓγ μ ℓ ] b ℓ + 𝒫 9 V + 𝒫 10 A s γ μ (1 − γ 5 ) b ] [ ¯ 𝒫 ℓ 10 A = [¯ ℓγ 5 γ μ ℓ ] ℓ − (+QED Magnetic Penguin) s b e s σ μν (1 + γ 5 ) b ] F μν 𝒫 7 γ 𝒫 7 γ = 8 π 2 m b [¯ σ μν = − i γ 4 [ γ μ , γ ν ] 1 Warning: I have not been particularly systematic about 2 (1 − γ 5 ) vs (1 − γ 5 ) in these slides. 8 Peter Skands Monash University
(Non-Leptonic Operators) (i,j=1,2,3 and a=1,…,8 are SU(3) C indices ; indicate colour structure ) W exchange / Charged-Current: s Exercise : consider tree-level diagrams b for W exchange between two quark s i γ μ (1 − γ 5 ) c i ] [¯ 𝒫 1 = [¯ c j γ μ (1 − γ 5 ) b j ] 𝒫 1 , 𝒫 2 c ¯ Note: some authors swap currents and justify why the (LO) Wilson these, e.g. s i γ μ (1 − γ 5 ) c j ] [¯ 𝒫 2 = [¯ c j γ μ (1 − γ 5 ) b i ] c Buchalla et al. coefficients are C 1 = 1 and C 2 = 0. Strong/QCD Penguins (Sum over q=u,d,s,c,b) 2 Lorentz s Why not t? s i γ μ (1 − γ 5 ) b i ] [¯ 𝒫 3 = [¯ q j γ μ (1 − γ 5 ) q j ] structures & 2 b s i γ μ (1 − γ 5 ) b j ] [¯ 𝒫 4 = [¯ q j γ μ (1 − γ 5 ) q i ] 𝒫 3 − 𝒫 6 q ¯ possible colour s i γ μ (1 − γ 5 ) b i ] [¯ 𝒫 5 = [¯ q j γ μ (1 + γ 5 ) q j ] structures q s i γ μ (1 − γ 5 ) b j ] [¯ 𝒫 6 = [¯ q j γ μ (1 + γ 5 ) q i ] s b g s m b s i σ μν (1 + γ 5 ) T a ij b j ] G a 𝒫 8 G = 8 π 2 [¯ 𝒫 8 G μν g Electroweak Penguins (Sum over q=u,d,s,c,b) 3 e q s i γ μ (1 − γ 5 ) b i ] [¯ 𝒫 7 = 2 [¯ q j γ μ (1 + γ 5 ) q j ] s 2 Lorentz 3 e q s i γ μ (1 − γ 5 ) b j ] [¯ 𝒫 8 = 2 [¯ q j γ μ (1 + γ 5 ) q i ] structures & 2 b 𝒫 3 − 𝒫 6 q ¯ 3 e q possible colour s i γ μ (1 − γ 5 ) b i ] [¯ 𝒫 9 = 2 [¯ q j γ μ (1 − γ 5 ) q j ] structures q 3 e q s i γ μ (1 − γ 5 ) b j ] [¯ 𝒫 10 = 2 [¯ q j γ μ (1 − γ 5 ) q i ] 9 Peter Skands Monash University
Renormalisation & Running Wilson Coefficients ๏ At tree level, C 1 = 1 and all other C i = 0 (they all involve loops) • Not good enough. (Among other things, FCNC would be absent!) ๏ At loop level, we must discuss renormalisation • In this part of the course, we focus on applications; not formalism • Suffice it to say that, just as we can do a tree-level comparison between the full theory (EW SM with full W propagators) and the effective theory, to see that and the C 1 = 1 other C i are zero at tree level, we can do the same kind of comparison at loop level. • This procedure - determining the coefficients of the effective theory from those of the full theory - is called matching and is a general aspect of deriving any effective theory by “integrating out” degrees of freedom from a more complete one. ๏ Two aspects are especially important to know. At loop level: • We do the matching a specific value of the renormalisation scale , characteristic of the degrees of freedom being integrated out , here μ match = m W . • This determines the values of the Wilson coefficients at that scale , C i ( m W ) . • We must then “run” those coefficients to a scale characteristic of the physical process at hand , in our case μ R = m b . In general, C i ( m b ) ≠ C i ( m W ) . 10 Peter Skands Monash University
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