CP VIOLATION Thomas Mannel Theoretical Physics I, Siegen University Les nabis School 2011
Outline of the course Lecture 1: Basics of CP and the Standard Model Lecture 2:
CP Violation in the Standard Model Lecture 2: Thomas Mannel Theoretische Physik I Universität Siegen Les Nabis School, Bad Honnef 1.-5. August 2011
Outline of Lecture 2 1 GIM Mechanism 2 Generalities 3 Quantum Mechanics of Flavour Oscillations Time-Dependent 4 Kaon CP Violation CP Violation in the B System Where do we go from here?
Introduction
Introduction GIM Mechanism What are the Phenomena induced by the CKM phases? The Cabbibo Submatrix is almost unitary: CP phenomena small in Kaons and charmed hadrons Rich CP phenomenology in bottom hadrons Strong suppression of CP effects also for top quarks
Glashow Illiopoulos Maiani Mechanism Not really related to CP, but it is relevant for CP pheneomenology Transitions between different kinds of Quarks
Historically: Why is Br(K π 0 l ν) >>> Br(K π ν ν) i.e. in general (FC)CC (charged currents) >>> FCNC (flavour changing neutral currents)? The SM does not have the colored arrows of the last panel at tree level! i.e. there are no interaction vertices at tree level of the form b sz 0, c uγ However, in a QFT this alone does not help: Terms can be induced as counterterms due to the necessity of renormalization
Look at loop diagrams: Example b sγ via a loop A(b sγ) = V ub V usf (m u ) + V cb V csf (m c ) + V tb V tsf (m t ) The individual contributions are indeed Log-Divergent with a mass independent coefficient! f (m q ) = ln(λ) + f fin (m q )
Summing everything: A(b sγ) = ln(λ) [V ub V us + V cb V cs + V tb V ts] + = finite expression This is due to CKM Unitarity GIM Mechanism = Unitarity of the CKM Matrix = No tree level FCNC s in the renormalized theory! All FCNC s are suppressed by a loop factor 1/(16π 2 ) All FCNC s are suppressed by factors (mu 2 i mu 2 j )/MW 2 for down-type quark FCNC s and by (md 2 i md 2 j )/MW 2 for up-type quark FCNC s
FCNC s are not so much suppressed in the down sector, since m 2 t /M 2 W O(1) FCNC s for Top and Charm are heavily suppressed m 2 b /M2 W 1
H eff for b decays at low scales Effective interaction: H eff = 4G F 2 λ CKM k Ĉk(Λ)O k (Λ) Tree Operators" O 1 = ( c L,i γ µ s L,j ) ( dl,j γ µ u L,i ), O 2 = ( c L,i γ µ s L,i ) ( dl,j γ µ u L,j ).
If two flavours are equal: QCD Penguin Operators O 3 = ( s L,i γ µ b L,i ) ( q L,j γ µ q L,j ), O 4 = ( s L,i γ µ b L,j ) O 5 = ( s L,i γ µ b L,i ) O 6 = ( s L,i γ µ b L,j ) q=u,d,s,c,b q=u,d,s,c,b q=u,d,s,c,b ( q L,j γ µ q L,i ), ( q R,j γ µ q R,j ), ( q R,j γ µ q R,i ). q=u,d,s,c,b
Electroweak Penguins: Replace the Gluon by a Z 0 or Photon: P 7 P 10 Rare (FCNC) Processes: O 7 = O 8 = e 16π m b( s 2 L,α σ µν b R,α )F µν g 16π m b( s 2 L,α Tαβσ a µν b R,α )G aµν O 9 = 1 2 ( s L γ µ b L )( lγ µ l) O 10 = 1 2 ( s L γ µ b L )( lγ µ γ 5 l)
Generalities
Generalities : Generalities How can a complex CKM matrix element produce an observable effect? CP violation: Interplay of weak and strong phases: A(B f ) = λ 1 a 1 + λ 2 a 2 λ 1, λ 2 : coupling constants with weak phases a 1, a 2 : hadronic matrix elements with strong phases CP Asymmetry A CP = Γ(B f ) Γ(B f ) Γ(B f ) + Γ(B f )
Generalities We have Γ(B f ) = λ 1 2 a 1 2 + λ 2 2 a 2 2 + 2 Re(λ 1 a 1 λ 2a 2) The CP image is obtained by λ i λ i, but a i a i due to CP invariance of strong interactions Γ(B f ) = λ 1 2 a 1 2 + λ 2 2 a 2 2 + 2 Re(λ 1a 1 λ 2 a 2) Thus we have (check this!) Γ(B f ) Γ(B f ) = 2 Re[(λ 1 a 1 λ 2a 2) (λ 1a 1 λ 2 a 2)] = 2 Im[λ 1 λ 2] Im[a 1 a 2]
Generalities To have a CP asymmetry we need a strong phase difference aside from a complex coupling Where can we get a phase difference from? From two different strong amplitudes with different strong phases From time dependence induced by flavour mixing... Strong phases are notoriously difficult to compute
Quantum Mechanics of Flavour Oscillations Time-Dependent
Mixing: Flavour Oscillations Quantum Mechanics of Flavour Oscillations Time-Dependent The Standard model allows for F = 2 transitions: A( B = 2, D = 2) carries the weak phase 2β A( B = 2, S = 2) has only small weak phase δγ A( S = 2, D = 2) carries the weak phase 2β C = 2 is heavily GIM suppressed
Quantum Mechanics of Flavour Oscillations Time-Dependent Quantum Mechanics of Flavour Oscillations Time Dependent superpositions of neutral meson states: M(t) = a(t) M 0 + ā(t) M 0 with the Schrödinger equation: i d ( ) ( ) ( ) a(t) H11 H = 12 a(t) dt ā(t) H 21 H 22 ā(t) ( a(t) = H ā(t) M 12 and M 21 originate from Box Diagramms CPT enforces H 11 = H 22 H is not hermitean: H = [M i2 ] Γ with hermitean M and Γ )
Quantum Mechanics of Flavour Oscillations Time-Dependent In the rest frame of the meson (M 1 M 0 and M 2 M 0 ) [M i2 ] Γ = m (0) M δ ij+ 1 2m H ij n M i H weak n n H weak M j E +, n + iɛ m (0) M Non-hermitean problem: The decay of the mesons leads out of the 2-dim. Hilbert space. Use 1 ω + iɛ = P to get the absorptive piece Γ ( ) 1 iπδ(ω) ω
Quantum Mechanics of Flavour Oscillations Time-Dependent Thus the general form of M i 2 Γ is ( M11 (i/2)γ 11 M 12 (i/2)γ 12 M 12 (i/2)γ 12 M 11 (i/2)γ 11 with complex A, p and q. Eigenstates: M short = ) ( A p 2 = q 2 A 1 ( p M0 q M 0 ) p 2 + q 2 ) M long = 1 ( p M0 + q M 0 ) p 2 + q 2 If CP were conserved, we had p = q and the CP eigenstates would be eigenstates of H This happens for real couplings, however, the CKM matrix elements are complex
Quantum Mechanics of Flavour Oscillations Time-Dependent Although non hermitean, we still can diagonalize Eigenvalues m short (i/2)γ short m long (i/2)γ long with 2pq = (m long m short ) i 2 (Γ long Γ short ) ( = 2 M 12 i ) ( 2 Γ 12 M 12 i ) 2 Γ 12 and p q = M12 1 2 Γ 12 M 12 1Γ 2 12
Quantum Mechanics of Flavour Oscillations Time-Dependent with and g ± = 1 2 Solution of the Schrödinger Equation (Check this) ( ) ( ) a(t) a(0) = R(t) a(t) a(0) ( R(t) = g + (t) (p/q)g (t) ) (q/p)g (t) g + (t) ( [ exp im long t 1 ] [ 2 Γ longt ± exp im short t 1 ]) 2 Γ shortt,
Quantum Mechanics of Flavour Oscillations Time-Dependent E.g. a pure M 0 state at t = 0 evolves as M(t) = g + (t) M 0 + p q g (t) M 0 For negligible lifetime differences this simplifies to ( ) ( ) 1 cos R(t) = e imt (1/2)Γt 2 m t q 1 p i sin 2 m t q ( ) ( ) 1 1 p i sin 2 m t cos 2 m t with M = (m long + m short )/2 and m = m long m short The Time Evolution generates a Phase difference
Quantum Mechanics of Flavour Oscillations Time-Dependent Time Dependent Consider a state f common to both M 0 and M 0 Then there are two ways how to end up in this state: 1 We have A(M 0 f ) 2 We have A(M 0 M 0 ) and subsequently A( M 0 f ) These are indistinguishable and hence can interfer However, A(M 0 M 0 ) is time dependent. Thus: Time Dependent CP Asymmetry a M f CP (t) = Γ(M 0(t) f ) Γ( M 0 (t) f ) Γ(M 0 (t) f ) Γ( M 0 (t) f )
Quantum Mechanics of Flavour Oscillations Time-Dependent After some algebra, using the results above with a M f CP (t) = C M f cos( m t) S M f sin( m t) cosh( Γ t/2) + D M f sinh( Γ t/2) C M f = 1 λ 2 1 + λ 2 S M f = 2 Im λ 1 + λ 2 D M f = 2 Re λ 1 + λ 2 and Note that λ = p q A( M 0 f ) A(M 0 f ) C 2 M f + S 2 M f + D 2 M f = 1 or C 2 M f + S 2 M f 1
Kaon CP Violation CP Violation in the B System Where do we go from here?
Kaon CP Violation CP Violation in the B System Where do we go from here? A few words on Kaon CP violation Definition of CP violating quantities η + = π+ π H W K L π + π H W K S η 00 = π0 π 0 H W K L π 0 π 0 H W K S These quantities are definitively non-zero: η + = (2.275 ± 0.019) 10 3 η 00 = (2.285 ± 0.019) 10 3 Define: η + = ɛ + ɛ and η 00 = ɛ 2ɛ
Kaon CP Violation CP Violation in the B System Where do we go from here? ɛ is channel independent, is a property of the K L produced from K K oscillations ɛ channel dependent, direct CP violation ɛ 0 has been established! Important as a matter of principle, is predicted by the CKM picture However, it is very hard to make a quantitative prediction
Kaon CP Violation CP Violation in the B System Where do we go from here? CP Violation in the B System Start out simple: Charged B meson decay A CP (B + f ) = Γ(B+ f ) Γ(B f ) Γ(B + f ) + Γ(B f ) There have to be two contributions with different strong and weak phases: Tree Contribution Penguin Contribution
Kaon CP Violation CP Violation in the B System Where do we go from here? Disentangle the two contributions by looking at different channels π K + H eff B 0 = (P + T ) π + K 0 H eff B + = P The direct CP asymmetry becomes: A CP (B d π K + ) = 2r sin γ sin δ 1 + r 2 2r cos γ cos δ with P T = r eiδ and γ CKM Phase
Kaon CP Violation CP Violation in the B System Where do we go from here? B d J/ΨK s and the determination of β (Bigi, Sanda) The gold plated mode Practically no direct CP violation: C B J/ψK = 0 b c cs Penguin carries the same weak phase b uūs Penguin strongly CKM suppressed λ = p/q = exp(2iβ) and so λ = 1 As in all B 0 decays: The lifetime-difference can be neglected. Time dependent CP Asymmetry: A (J/ΨKs) CP (t) = sin(2β) sin( m t) B J/ΨK good laboratory for new physics (Fleischer, M.)
Kaon CP Violation CP Violation in the B System Where do we go from here? B d φk s Penguin process b s ss: Mediated by either tree matrix elements of penguin operators or penguin contraction of tree operators Not as clean: C (φks) = O(10%) S (φks) = sin(2β) + O(10%)
B ππ GIM Mechanism Kaon CP Violation CP Violation in the B System Where do we go from here? Interplay between Tree and Penguin Tree Vub V ud λ 3 Penguin Vtb V td λ 3 Expect sizable direct CP violation
Kaon CP Violation CP Violation in the B System Where do we go from here? Roadmap of Bottom Decays (R. Fleischer)
Kaon CP Violation CP Violation in the B System Where do we go from here? Attempt of a B CP Summary... It all depends on P/T which is hard to calculate Some modes are gold plated due to strongly CKM supressed P/T s B 0 J/ψK s measures β B s J/ψφ measures the B s mixing phase δγ In the meantime β from B 0 J/ψK s is a super-precise measurement We need to worry about P/T here! It will nor be easy to pin down small new CP violation in nonletonic decays, be it two or more-body... but you should take it a a challenge.
Kaon CP Violation CP Violation in the B System Where do we go from here???? Many Open Questions??? Flavour and CP is not understood: 22 (out of 27) free Parameters of the SM originate from the Yukawa Sector (including Lepton Mixing) Why is the CKM Matrix hierarchical? Why is CKM so different from the PMNS? Why are the quark masses (except the top mass) so small compared with the electroweak VEV? Why do we have three families? There is no guiding principle for a theory of flavour Why is CP Violation in Flavour-diagonal Processes not observed? Where is the missing CP violation to explain the matter-antimatter asymmetry of the Universe?