The Cabibbo-Kobayashi-Maskawa (CKM) matrix Charge-raising current J µ W = ( ν e ν µ ν τ )γ µ (1 γ 5 ) V = A u L Ad L e µ τ + (ū c t)γ µ (1 γ 5 )V Mismatch between weak and quark masses, and between A u,d L, which relate weak and mass eigenstates d s b V = unitary 3 angles and 1 phase after removing unobservable overall phases of fields V V = V V = I P59 Spring, 013 1
Magnitudes of elements (PDG) V = V ud V us V ub V cd V cs V cb V td V ts V tb V ij = 0.974 0.6 0.0036 0.6 0.973 0.04 0.0087 0.041 0.9991 V ud from superallowed β decay; V us from K l3 V cd,cs from c decay and from c production (dimuons) in νn V cb,ub from b c and b u decay rates (theory uncertainties) V td,ts,tb from B 0 B 0 mixing, t b, unitarity P59 Spring, 013
Various exact parametrizations consistent with unitarity V = 1 0 0 0 c 3 s 3 c 13 0 s 13 e iδ 0 1 0 c 1 s 1 0 s 1 c 1 0 0 s 3 c 3 s 13 e iδ 0 c 13 0 0 1 where c ij cos θ ij, s ij sin θ ij s 1 sin θ C 0.3, s 13 s 3 s 1 ; δ = CP phase Want to test unitarity V V = V V = I. Apparent violation would signal new physics. ( V V ) = V 11 ud + V us + V ub = 1? (Universality) ( V V ) 31 = V ub V ud + V cb V cd + V tb V td = 0? Unitarity triangle: each complex number in ( V V ) is a vector in 31 complex plane. Should sum to zero, i.e., form triangle. P59 Spring, 013 3
V ub V α ud V tb V td γ β Vcb V cd ( ρ, η) α ρ + i η 1 ρ i η γ β (0, 0) 1 (1, 0) The unitarity triangle ( ρ, η) α ρ + i η Vcb V cd 1 ρ i η γ β (0, 0) 1 (1, 0) Divide each side by (well-measured) Overconstrain by redundant tests Sides by (CP -conserving) rates, e.g., b u decay rate, B 0 B 0 mixing Angles by CP violating effects CP area (Jarlskog invariant): J = Im (V us V cb V ub V cs ) A λ 6 η Deviation: extra family, fermion mixing, new sources of CP -violation (e.g., SUSY loops), new FCNC interactions (extended Higgs, Z ) P59 Spring, 013 4
The Wolfenstein parametrization. sin θ C 0.6 Empirical expansion in λ Simple to remember and work with V = 1 λ / λ Aλ 3 (ρ iη) λ 1 λ / Aλ + O(λ 4 ) Aλ 3 (1 ρ iη) Aλ 1 A 0.81 from b c decay rate; (ρ, η) real, expected O(1) Vertices of triangle at (0, 0), (1, 0), and ( ρ, η), where ρ = ρ(1 λ /), η = η(1 λ /) Each constraint determines allowed region in ρ, η plane P59 Spring, 013 5
1.5 1.0 excluded area has CL > 0.95 sin '! excluded at CL > 0.95 #m d & #m s V ub determined by rate for exclusive or inclusive b ulν (theory uncertainties) ) 0.5 0.0-0.5 $ K " V ub V ub & % SL! " ' " #m d ( V ub V ud ) /( V cb V cd ) = ρ + η radius of circle centered at (0, 0) -1.0 CKM f i t t e r ICHEP 10! $ K sol. w/ cos ' < 0 (excl. at CL > 0.95) -1.5-1.0-0.5 0.0 0.5 1.0 1.5.0 (CKMFITTER: http://ckmfitter.inp3.fr/) ( P59 Spring, 013 6
K 0 K 0 mixing Strong interactions conserve strangeness (S K = 1, S K = S Λ = 1) K 0 = d s K 0 = ds CP K 0 = η K K 0 CP K 0 = η K K0 K 0 and K 0 are states relevant to strong interaction processes, e.g., π p K 0 Λ allowed, but π p K 0 Λ ( Λ = sdu ) Choose quark (or meson) phases so that η K = 1 P59 Spring, 013 7
s u, c, t d K 0 W W K 0 d u, c, t s However, WCC can violate strangeness and lead to K 0 K 0 mixing at second order M ( ) MK 0 M 1 M1 M K 0 s W d K 0 u, c, t u, c, t K 0 d W s M K 0 = M K 0 by CP T tiny M 1 important Need GIM m c 1.5 GeV predicted Typeset by FoilTEX 1 Severe constraint on tree-level Z, FCNC Higgs families real M 1 eigenstates K 0 1, = K0 K 0 K S,L m = m K m K1 = M 1 G F 6π m c V cd V cs f K m KB K 3.5 10 6 ev (regeneration) P59 Spring, 013 8
Assume CP is conserved CP violation CP K 0 = K 0 CP K 0 = K 0 CP K 0 1, = CP K0 K 0 = ±K 0 1, CP π + π = + π + π CP π 0 π 0 = + π 0 π 0 for J = 0 CP 3π 3π for J = Q = 0 Thus, K S K 0 1 π (τ S = 8.9 10 11 s); K L K 0 3π (τ L = 5.1 10 8 s) Fitch-Cronin (1964): observed K L π with B(K L π) 3 10 3 CP P59 Spring, 013 9
Regeneration Produce pure K 0 or K 0 state at t = 0 ψ(0) = K 0 = 1 ( K S + K L ) Ignoring decays, at proper time τ = d/βγ (distance travelled d): ψ(τ ) = 1 [ K S e im sτ + K L e im Lτ ] = [ ( ) ( ) mk τ cos K 0 mk τ ] i sin K 0 e im K 0 τ Probability sin m K τ of oscillation into K 0 P59 Spring, 013 10
Oscillation time: π/ m K 1. 10 9 s τ KS = Γ 1 S ψ(τ ) = 1 [ K S e imsτ e Γ sτ }{{} K S π = 8.9 10 11 s im + K L e Lτ e ] Γ L τ }{{} K L 3π Pure K L beam for τ KS τ τ KL ; 3π CP violation Place thin target (regenerator) at distance d 1 : K 0 K 0 and K 0 K 0 with different amplitudes a, b. For τ 1 τ KS : ψ(τ 1 ) = K L = K0 + K 0 ψ(τ ) }{{} emerging = a K 0 + b K 0 = a b K S + a + b K }{{} L regenerated P59 Spring, 013 11
For small τ 1 (non-negligible K S in ψ(τ 1 ) ) (λ exp(i m K τ Γ S τ /)): ψ(τ ) [(a + b)λ + (a b)] K S +[(a b)λ + (a + b)] K L For b 0: K S ψ(τ ) [ ] 1 + e Γ Sτ + e Γ Sτ / cos ( m K τ ) Measure m K by varying parameters P59 Spring, 013 1
CP violation and phases CP violation requires complex observable phases Some phases unobservable: only overall phase of amplitude; can absorb in phase of state Need interference between amplitudes, e.g., diagrams involving all three families Phase in CKM can generate Im M 1 0. Effect small due to small mixings even if CKM phase large s u, c, t d K 0 W W K 0 d u, c, t s s W d K 0 u, c, t u, c, t K 0 d W s Typeset by FoilTEX P59 Spring, 013 13
Toy example H eff = M K 0 K 0 K 0 }{{} K 0, K 0 mass + M 1 ( e iϕ K 0 K 0 + e iϕ K 0 K 0) }{{} complex mixing mass (indirect) + a ( e iβ K 0 + e iβ K 0) ππ }{{} complex decay amplitude (direct) Absorb e iϕ by K 0 e iϕ K 0 and K 0 e iϕ K 0 H decay eff ( = a e i(β ϕ) K 0 + e i(β ϕ) K 0 ) ππ = a cos(β ϕ) K0 K 0 }{{ } K S Observable CP by phase difference β ϕ +i sin(β ϕ) K0 + K 0 } {{ } K L ππ P59 Spring, 013 14
Actual K L,S system; amplitude ratios η + A(K L π + π ) A(K S π + π ) = ɛ + ɛ η 00 A(K L π 0 π 0 ) A(K S π 0 π 0 ) = ɛ ɛ Difference because two isospin amplitudes for π, I = 0 (large) and I = (small) ɛ from mixing (indirect) and CP in I = 0 amplitude (small) ɛ from difference in I = 0 and I = phases (direct suppressed) Experiment: ɛ =.9(1) 10 3 Re(ɛ /ɛ) = 1.65(6) 10 3 (CERN, FNAL) P59 Spring, 013 15
Physical states CP parameters K L = p K 0 + q K 0 = ɛ K0 1 + K0 1 + ɛ K S = p K 0 q K 0 = K0 1 + ɛ K0 1 + ɛ p q 1 + ɛ 1 ɛ ɛ = ɛ }{{} mixing ɛ = i e i(δ δ 0 ) }{{} strong + i ImA 0 ReA 0 }{{} I=0 decay ReA ReA 0 }{{} 0.045 [ ImA ImA ] 0 } ReA {{ ReA 0 } I=0, mismatch P59 Spring, 013 16
Box diagrams ɛ box Im(V ts V td)f (m t, m c, angles, QCD) η [(1 ρ)f 1 + F ] s u, c, t d K 0 W W K 0 d u, c, t s Hyperbola in ( ρ, η) plane 1.5 1.0 excluded area has CL > 0.95! sin ' excluded at CL > 0.95 #m d & #m s s W d K 0 u, c, t u, c, t K 0 d W s 0.5 $ K " #m d ) 0.0-0.5 " V ub V ub & % SL! ' " Typeset by FoilTEX -1.0 CKM f i t t e r ICHEP 10! $ K sol. w/ cos ' < 0 (excl. at CL > 0.95) -1.5-1.0-0.5 0.0 0.5 1.0 1.5.0 ( P59 Spring, 013 17
ɛ from direct CP violation (i.e., in decay amplitudes) Believed dominated by penguin diagrams (A 0 enhanced) Re(ɛ /ɛ) = 1.65(6) 10 3 η Roughly consistent with SM, but large theory uncertainty s W u, c, t d G(Z, γ) d d P59 Spring, 013 18
B 0 B 0 mixing CLEO, LEP, Tevatron, BaBar (SLAC), BELLE (KEK), LHCb Strong interaction states B 0 d = d b B 0 d = db = CP B 0 d B 0 s = s b B 0 s = sb = CP B0 s Mixed by second order weak (box diagrams) B Hi = p i B 0 i + q i B 0 i, i = d, s B Li = p i B 0 i q i B 0 i where p i q i = 1 ɛ i 1+ ɛ i and ɛ i = iimm i 1 / m i 1 ( m i m Hi m Li = M i 1 ) P59 Spring, 013 19
Observe m i m Hi m Li = M i 1 by B0 B 0 oscillations (cf ν oscillations). eg., suppose initial state tagged as B 0 i ψ(0) = B 0 i = 1 p i ( BLi + B Hi ) ψ(τ ) 1 ( ) B Li e im L τ i + B Hi e im H i τ p i ( ) mi τ cos B 0 i + i sin ( mi τ ) qi p i B 0 i Measure m i by time dependence (or integral) of decays, using b cl ν l (from B 0 ); b cl + ν l (from B 0 ) m d = 0.507(5) ps 1 3.3 10 4 ev (1 ρ) + η m s = 17.77(1) ps 1 0.01 ev 1 (QCD correlated theory error reduced in m d / m s ) P59 Spring, 013 0
1.5 excluded at CL > 0.95 excluded area has CL > 0.95! 1.0 #m d & #m s sin ' 0.5 #m d $ K " ) 0.0-0.5 " V ub V ub & % SL! ' " -1.0 CKM f i t t e r ICHEP 10! $ K sol. w/ cos ' < 0 (excl. at CL > 0.95) -1.5-1.0-0.5 0.0 0.5 1.0 1.5.0 (CKMFITTER group: http://ckmfitter.inp3.fr/) ( P59 Spring, 013 1
The CP asymmetries in B decays Many CP asymmetries to determine angles of unitarity triangle, but QCD uncertainties Cleanest: breaking) B 0 ( B 0 ) f, where f CP eigenstate (up to CP CP f = ± f η f f e.g., f = J/ψK S, π + π, ρk S Let B 0 (τ ) be time-evolution of state tagged as B 0 at τ = 0, and similarly for B 0 (τ ) P59 Spring, 013
CP -odd asymmetry where a f (τ ) = Γ(B0 (τ ) f) Γ( B 0 (τ ) f) Γ(B 0 (τ ) f) + Γ( B 0 (τ ) f) = (1 λ ) cos( m τ ) Imλ sin( m τ ) (1 + λ ) λ = p q }{{} mixing Ā A }{{} decay Usually confused by sum over diagrams, with (CP -conserving) strong phases (final state interactions) and (CP -violating) CKM phases A = α A α e iδ α e iϕ α Ā = η f α A α e iδ α }{{} strong e iϕ α }{{} CKM P59 Spring, 013 3
Golden modes, B J/ψK S,L (only one significant diagram no strong interaction uncertainty) ψ c c W b s B 0 K 0 d d λ = ( V tb V td ) V tb V }{{ td } B B mixing ( V cs V cb ) V cs V }{{ cb } decay, 1 Imλ = ± sin β, λ = 1 BaBar, Belle: sin β = 0.681(5) P59 Spring, 013 4 Typeset by FoilTEX 1
1.5 excluded at CL > 0.95 excluded area has CL > 0.95! 1.0 #m d & #m s sin ' 0.5 #m d $ K " ) 0.0-0.5 " V ub V ub & % SL! ' " -1.0 CKM f i t t e r ICHEP 10! $ K sol. w/ cos ' < 0 (excl. at CL > 0.95) -1.5-1.0-0.5 0.0 0.5 1.0 1.5.0 (CKMFITTER group: http://ckmfitter.inp3.fr/) ( P59 Spring, 013 5
Many other modes studied, including charmless u π + d B π + π W Tree-level only: λ = sin α However, competition from significant penguin theory uncertainty (isospin) Penguin-dominated, b s ss, e.g., B φk S (hint of discrepancy in sin β?) SM loops; may be significant new physics from tree (FCNC Z, extended Higgs) or enhanced supersymmetric loops (large tan β) CDF, D0 B s asymmetries? Not LHCb B 0 B 0 b d W u, c, t b G d b Typeset by FoilTEX B 0 G d W u, c, t d u ū d ū d π + π s s s d π φ K 0 P59 Spring, 013 6
Stringent tests of CKM universality, unitarity triangle Also, new B s µ + µ CP violation well studied in K and B sector Recent extensions to D D (cū cu) system Tests electroweak loops, renormalization, QCD corrections Strong interaction challenges largely overcome Constraints on new physics However, additional CP violation required for baryogenesis P59 Spring, 013 7