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
Semi-Leptonic Decays of Hadrons Now, we move on to: V ud V us V ub V cd V cs V cb V tb V td V ts 2 Peter Skands Monash University
Semi-Leptonic Decays of Hadrons V ud V us V ub V cd V cs V cb ๏ Simplifying factors: • These are all tree-level diagrams, in which one of the quarks acts as a pure “spectator”. • There is only one hadron in the the final state • Should be possible to write the amplitude as a lepton current interacting (via a virtual W) with the “active” quark, embedded in a “hadronic current” 3 Peter Skands Monash University
<latexit sha1_base64="OEL5eoqIOm9Ot3P0LdGm6TEQxms=">ACG3icbZDLSsNAFIYn9VbjLerSTbAIbixJqeimUtSF4KaCvUCTlslk0g6dXJiZCXte7jxVdy4UMSV4MK3cdJGqK0/DPx85xzOnN+JKOHCML6V3NLyupafl3d2Nza3tF29xo8jBnCdRTSkLUcyDElAa4LIihuRQxD36G46Qyu0nrzATNOwuBeDCNs+7AXEI8gKCTqaqXrjlEZIcuBLInHF5al3nZOKiM+A6yIpMj9RV2tYBSNifRFY2amADLVutqn5Yo9nEgEIWct0jEnYCmSCI4rFqxRxHEA1gD7elDaCPuZ1MbhvrR5K4uhcy+QKhT+jsRAJ9zoe+Izt9KPp8vpbC/2rtWHjndkKCKBY4QNFXkx1EepULpLGEaCDqWBiBH5Vx31IYNIyDhVGYI5f/KiaZSKZrl4elcuVC+zOPLgAByCY2CM1AFN6AG6gCBR/AMXsGb8qS8KO/Kx7Q1p2Qz+CPlK8fyX6grA=</latexit> Cabibbo Favoured vs Cabibbo Suppressed ๏ Which is Cabibbo Favoured vs Cabibbo Suppressed? And why. D 0 → π − ℓ + ν D 0 → K − ℓ + ν • D 0 = | c ¯ u > K − = | s ¯ u > V cs V cd π − = | d ¯ u > ๏ Which is CKM Favoured vs CKM Suppressed? And why. • B → D 𝓂 ν • B → π 𝓂 ν B − = | b ¯ u > ➠ Our case study. V cb • • V ub Has gotten attention recently, as part of the “flavour anomalies”. 4 Peter Skands Monash University
Starting Point for B → D ℓ ν : The Spectator Model ๏ Unlike B → ℓ ν , this is not an annihilation • Looks like a weak decay of the heavy quark , accompanied by a non- interacting spectator : V cb ๏ Suggests a simple starting point for semi-leptonic decays : • Assume the quark(s) which accompany the heavy quark play no role . ℒ = − G F c γ ρ (1 − γ 5 ) b ] [ ¯ V cb [¯ ℓγ ρ (1 − γ 5 ) ν ℓ ] 2 Can give some insights (e.g., lepton spectrum) but is not a precision tool. 5 Peter Skands Monash University
B → D ℓ ν with Hadronic Effects ๏ Can promote the spectator model's quark-level matrix element to a hadronic one by sandwiching it between initial and final hadronic states: ℳ = G F c γ ρ (1 − γ 5 ) b | B ( p B ) ⟩ [ ¯ V cb ⟨ D ( p D ) | ¯ ℓγ ρ (1 − γ 5 ) ν ℓ ] 2 Both B and D are pseudoscalars. To construct a vector, must use L=1 ⇒ negative parity ⇒ Axial part does not contribute. = G F c γ ρ b | B ( p B ) ⟩ [ ¯ V cb ⟨ D ( p D ) | ¯ ℓγ ρ ν ℓ ] 2 Unlike for pion decay, we have two (independent) momenta here, p B and p D ⇒ a priori two Lorentz-covariant combinations = G F V cb [ f + ( q 2 )( p B + p D ) ρ + f − ( q 2 )( p B − p D ) ρ ] [ ¯ ℓγ ρ ν ℓ ] 2 f + and f - are called Form Factors ν ) 2 = Momentum Transfer They depend on q 2 = ( p B − p D ) 2 = p 2 W = ( p ℓ + p ¯ 6 Peter Skands Monash University
(Alternative Parameterisation) We wrote: ℳ = G F V cb [ f + ( q 2 )( p B + p D ) ρ + f − ( q 2 )( p B − p D ) ρ ] [ ¯ ℓγ ρ ν ℓ ] 2 Another common parametrisation [Wirbel, Stech, Bauer, Z.Phys. C29 (1985) 637] is to write in terms of a “Transverse” F 0 and a “Longitudinal” F 1 form factor: V cb [ F 1 ( q 2 ) ( p B + p D − m 2 ] [ ¯ B − m 2 + F 0 ( q 2 ) m 2 B − m 2 ℳ = G F ρ q ) D D q ρ ℓγ ρ ν ℓ ] q 2 q 2 2 with F 1 (0) = F 0 (0) and q = p B - p D Thus: f + = F 1 Exercise: prove this f − = ( F 0 − F 1 )( m 2 B − m 2 D )/ q 2 Note: for decays involving vector mesons , polarisations ε μ ⇒ more form factors. 7 Peter Skands Monash University
Looks like we went from bad to worse? ๏ Our ignorance about non-perturbative physics is now cast as two whole functions. • How can we learn anything (precise) from this? • Frustrating when the process looks so simple … ๏ Let’s take a second look at the problem, physicist style: • The B meson is a heavy-light system; m b ~ 4 GeV ≫ Λ QCD (confinement scale ~ 200 MeV) ๏ Compton wavelength • of heavy 1 quark: r had ∼ Light-quark cloud. Λ QCD = (Complicated confinement stuff.) λ Q ∼ 1 ≪ r had m b • ➤ Large separation of scales! 8 Peter Skands Monash University
Heavy Quark Symmetry Soft gluons exchanged between the heavy quark and the light constituent cloud can only resolve distances much larger than λ Q ~ 1/ m Q ๏ ➤ In limit m Q → ∞ , the light degrees of freedom: • Are blind to the flavour (mass) and spin of the heavy quark. • Experience only the colour field of the heavy quark (which extends over distances large compared with 1/m Q ) ๏ ➤ If we swap out the heavy quark Q by one with a different mass and/or spin, the light cloud would be the same. • ⇒ Relations between B, D, B * , and D * , and between Λ b and Λ c . • For finite m Q , these relations are only approximate. Deviations from exact heavy-quark symmetry: “ symmetry breaking corrections” ๏ Can be organised systematically in powers of α s ( m Q ) (perturbative) and 1/ m Q (non- ๏ perturbative) in a formalism called HQET (heavy-quark effective theory) . 9 Peter Skands Monash University
Physics of heavy-quark symmetry Isgur & Wise , Phys. Lett. B 232 (1989) 113; Phys. Lett. B 237 (1990) 527 ๏ Before we consider decays, consider just elastic scattering* of a B meson Induced by giving a kick to the b quark at time t 0 : *Elastic Scattering: means B meson does not break up. • Before t 0 : light degrees of freedom orbit around the heavy quark, which acts as a static source of colour. • On average, b quark and B meson have same velocity, v . Illustrations and physics arguments inspired by the BaBar Physics Book. • At t 0 : instantaneously replace colour source by one moving at velocity v ’ (possibly with a different spin). • • After t 0 : If v=v’ (spectator limit), nothing happens; light degrees of freedom have no way of knowing anything changed. • But if v ≠ v’, the light cloud will need to be rearranged (sped up), to form a new B meson moving at velocity v’. ➤ Form-factor suppression. (Large Δ v ⇒ elastic transition less likely.) ๏ 10 Peter Skands Monash University
Elastic Form Factor of a Heavy Meson (Isgur-Wise Function) Isgur & Wise , Phys. Lett. B 232 (1989) 113; Phys. Lett. B 237 (1990) 527 ๏ In limit m b → ∞ , form factor can only depend on the difference between v and v’: • Lorentz invariance ☛ use the relative boost between the rest frames of the initial- and final-state mesons. v μ = p μ v ′ � μ = p ′ � μ Using and the relative boost is γ = v ⋅ v ′ � ≥ 1 m b m b Exercise: prove this • ➤ In this limit, a dimensionless probability amplitude ξ ( γ ) describes the transition amplitude. ( ξ is called the Isgur-Wise function.) • ➤ The hadronic matrix element can be written as: ⟨ ¯ B ( p ) ⟩ = ξ ( γ )( p + p ′ � ) μ B ( p ′ � ) | ¯ b p ′ � γ μ b p | ¯ ξ is the elastic form factor of a heavy meson. Only depends on γ = v.v’ , not m B . Constraint: at γ =1 (zero momentum transfer), current conservation ⇒ ξ (1)=1 Question: why is ξ (1)=1 intuitive? 11 Peter Skands Monash University
Implications ๏ Using heavy-quark symmetry, we can replace the b quark in the final-state meson by a c quark: ⟨ ¯ c v ′ � γ μ b v | ¯ B ( v ) ⟩ = m B m C ξ ( v ⋅ v ′ � ) ( v + v ′ � ) μ D ( v ′ � ) | ¯ Writing it terms of velocities, v and v’, instead of momenta Same Isgur-Wise functions! (This corresponds to the field definitions in HQET) ๏ Compare with the general expression from before: ℳ = G F V cb [ f + ( q 2 )( p B + p D ) ρ + f − ( q 2 )( p B − p D ) ρ ] [ ¯ ℓγ ρ ν ℓ ] 2 ⇒ the functions f + and f - are not independent. Both are related to ξ . Assignment Problem 3: derive expressions for f + ( ξ ) and f - ( ξ ) 12 Peter Skands Monash University
The Partial Widths ๏ In the limit that m b ,m c ≫ Λ QCD , the differential semileptonic decay rates become: … in terms of the “recoil variable” w = v ⋅ v ′ � ¯ (Similar expressions can be derived for semi-leptonic Λ b → Λ c ℓ ¯ ν or B → D ** ℓ ¯ ν Different clouds so different Isgur-Wise functions .) ξ • Reminder: corrections from finite m Q (breaking of heavy quark symmetry). α n Perturbative: order s ( m Q ) ๏ ( Λ QCD / m Q ) n Non-perturbative: order analysed in HQET (effective QFT with ๏ velocity-dependent Q fields, expansion in powers of 1/ m Q starting from m Q → ∞ ) 13 Peter Skands Monash University
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