Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Electroweak interaction Gauge group: SU ( 2 ) × U ( 1 ) Y � � � � u j ν j doublets: Q j und L j = L L L = d j ℓ j L L j = 1 , 2 , 3 labels the generation. � t L � � ν eL � Examples: Q 3 , L 1 = L = b L e L singlets: u j R , d j R and e j R .
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Electroweak interaction Gauge group: SU ( 2 ) × U ( 1 ) Y � � � � u j ν j doublets: Q j und L j = L L L = d j ℓ j L L j = 1 , 2 , 3 labels the generation. � t L � � ν eL � Examples: Q 3 , L 1 = L = b L e L singlets: u j R , d j R and e j R . Important: Only left-handed fields couple to the W boson.
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary How many interactions does the Standard Model comprise?
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary How many interactions does the Standard Model comprise? Five! • three gauge interactions
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary How many interactions does the Standard Model comprise? Five! • three gauge interactions • Yukawa interaction of Higgs with quarks and leptons
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary How many interactions does the Standard Model comprise? Five! • three gauge interactions • Yukawa interaction of Higgs with quarks and leptons • Higgs self-interaction
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Yukawa interaction � � G + Higgs doublet H = with v = 174 GeV. v + h 0 + iG 0 √ 2 � � v + h 0 − iG 0 √ H = Charge-conjugate doublet: � 2 − G −
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Yukawa interaction � � G + Higgs doublet H = with v = 174 GeV. v + h 0 + iG 0 √ 2 � � v + h 0 − iG 0 √ H = Charge-conjugate doublet: � 2 − G − H Yukawa lagrangian: jk Q j jk Q j jk L j − L Y = Y d L H d k R + Y u H u k R + Y l L H e k L � R + h.c. Here neutrinos are (still) massless. The Yukawa matrices Y f are arbitrary complex 3 × 3 matrices.
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Yukawa interaction � � G + Higgs doublet H = with v = 174 GeV. v + h 0 + iG 0 √ 2 � � v + h 0 − iG 0 √ H = Charge-conjugate doublet: � 2 − G − H Yukawa lagrangian: jk Q j jk Q j jk L j − L Y = Y d L H d k R + Y u H u k R + Y l L H e k L � R + h.c. Here neutrinos are (still) massless. The Yukawa matrices Y f are arbitrary complex 3 × 3 matrices. The mass matrices M f = Y f v are not diagonal! u j L , R , d j ⇒ L , R do not describe physical quarks! We must find a basis in which Y f is diagonal!
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Any matrix can be diagonalised by a bi-unitary transformation. Start with y u 0 0 Y u = S † Q Y u S u Y u = y c and y u , c , t ≥ 0 � with � 0 0 y t 0 0 This can be achieved via Q j u j L = S Q jk Q k ′ R = S u jk u k ′ L , R with unitary 3 × 3 matrices S Q , S u . This transformation leaves L gauge invariant (“flavour-blindness of the gauge interactions”)!
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Next diagonalise Y d : y d 0 0 Y d = V † S † Q Y d S d Y d = y s and y d , s , b ≥ 0 � with � 0 0 y b 0 0 with unitary 3 × 3 matrices V , S d . Via d j R = S d jk d k ′ R we leave L gauge unchanged, while Y d H d R + Q L � − L quark Y u � = Q L V � H u R + h.c. Y
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Next diagonalise Y d : y d 0 0 Y d = V † S † Q Y d S d Y d = y s and y d , s , b ≥ 0 � with � 0 0 y b 0 0 with unitary 3 × 3 matrices V , S d . Via d j R = S d jk d k ′ R we leave L gauge unchanged, while Y d H d R + Q L � − L quark Y u � = Q L V � H u R + h.c. Y To diagonalise M d = V � Y d v transform d j L = V jk d k ′ L This breaks up the SU ( 2 ) doublet Q L .
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Next diagonalise Y d : y d 0 0 Y d = V † S † Q Y d S d Y d = y s and y d , s , b ≥ 0 � with � 0 0 y b 0 0 with unitary 3 × 3 matrices V , S d . Via d j R = S d jk d k ′ R we leave L gauge unchanged, while Y d H d R + Q L � − L quark Y u � = Q L V � H u R + h.c. Y To diagonalise M d = V � Y d v transform d j L = V jk d k ′ L This breaks up the SU ( 2 ) doublet Q L . ⇒ L gauge changes!
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary In the new “physical” basis M u = Y u v and M d = Y d v are diagonal. Also the neutral Higgs fields h 0 and G 0 have only ⇒ flavour-diagonal couplings!
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary In the new “physical” basis M u = Y u v and M d = Y d v are diagonal. Also the neutral Higgs fields h 0 and G 0 have only ⇒ flavour-diagonal couplings! The Yukawa couplings of the charged pseudo-Goldstone bosons G ± still involve V : Y d d R G + − d L V † � Y u u R G − + h.c. − L quark = u L V � Y
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary In the new “physical” basis M u = Y u v and M d = Y d v are diagonal. Also the neutral Higgs fields h 0 and G 0 have only ⇒ flavour-diagonal couplings! The Yukawa couplings of the charged pseudo-Goldstone bosons G ± still involve V : Y d d R G + − d L V † � Y u u R G − + h.c. − L quark = u L V � Y The transformation d j L = V jk d k ′ L changes the W-boson couplings in L gauge : � � L W = g 2 u L V γ µ d L W + + d L V † γ µ u L W − √ µ µ 2 The Z-boson couplings stay flavour-diagonal because of V † V = 1.
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary V is the Cabibbo-Kobayashi-Maskawa (CKM) matrix. V ud V us V ub V = V cd V cs V cb V td V ts V tb
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary V is the Cabibbo-Kobayashi-Maskawa (CKM) matrix. V ud V us V ub V = V cd V cs V cb V td V ts V tb Leptons: Only one Yukawa matrix Y l ; the mass matrix M l = Y l v of the charged leptons is diagonalised with L j L = S L jk L k ′ e k R = S e jk e k ′ L , R No lepton-flavour violation!
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary V is the Cabibbo-Kobayashi-Maskawa (CKM) matrix. V ud V us V ub V = V cd V cs V cb V td V ts V tb Leptons: Only one Yukawa matrix Y l ; the mass matrix M l = Y l v of the charged leptons is diagonalised with L j L = S L jk L k ′ e k R = S e jk e k ′ L , R No lepton-flavour violation! ⇒ Add a ν R to the SM to mimick the quark sector or add a Majorana mass term Y M LHH T L c . M The lepton mixing matrix is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix.
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Discrete symmetries x → − � x � Parity transformation P: Charge conjugation C: Exchange particles and antiparticles, e.g. e − ↔ e + t → − t Time reversal T:
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary C and P 1954/1955: CPT is a symmetry of every Lorentz-invariant quantum field theory.
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary C and P 1954/1955: CPT is a symmetry of every Lorentz-invariant quantum field theory. 1956/1957: P is not a symmetry of the microscopic laws of nature!
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary C and P 1954/1955: CPT is a symmetry of every Lorentz-invariant quantum field theory. 1956/1957: P is not a symmetry of the microscopic laws of nature! 1964: CP is not a symmetry of the microscopic laws of nature!
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary C and P 1954/1955: CPT is a symmetry of every Lorentz-invariant quantum field theory. 1956/1957: P is not a symmetry of the microscopic laws of nature! 1964: CP is not a symmetry of the microscopic laws of nature! ⇒ Also the T symmetry must be violated, there is a microscopic arrow of time!
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary K and M 1973: Explanation of CP violation by postu- lating a third fermion generation.
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary K and M 1973: Explanation of CP violation by postu- lating a third fermion generation. Makoto Kobayashi and Toshihide Maskawa: CP Violation in the Renormalizable Theory of Weak Interaction , Prog.Theor.Phys.49:652-657,1973,
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary K and M 1973: Explanation of CP violation by postu- lating a third fermion generation. Makoto Kobayashi and Toshihide Maskawa: CP Violation in the Renormalizable Theory of Weak Interaction , Prog.Theor.Phys.49:652-657,1973, ∼ 7300 citations (ranks 3rd in elementary particle physics). Nobel Prize 2008.
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary K and M 1973: Explanation of CP violation by postu- lating a third fermion generation. Makoto Kobayashi and Toshihide Maskawa: CP Violation in the Renormalizable Theory of Weak Interaction , Prog.Theor.Phys.49:652-657,1973, ∼ 7300 citations (ranks 3rd in elementary particle physics). Nobel Prize 2008.
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Strong interaction The QCD lagrangian permits a term which violates P , CP , and T , but experimentally the corresponding coefficient θ is found to be smaller than 10 − 11 .
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Strong interaction The QCD lagrangian permits a term which violates P , CP , and T , but experimentally the corresponding coefficient θ is found to be smaller than 10 − 11 . ⇒ The strong interaction essentially respects C , P , and therefore T , � � � � � � H strong , P H strong , C H strong , T = = = 0 We can assign C and P quantum numbers, which ⇒ can be + 1 or − 1, to hadrons.
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Strong interaction The QCD lagrangian permits a term which violates P , CP , and T , but experimentally the corresponding coefficient θ is found to be smaller than 10 − 11 . ⇒ The strong interaction essentially respects C , P , and therefore T , � � � � � � H strong , P H strong , C H strong , T = = = 0 We can assign C and P quantum numbers, which ⇒ can be + 1 or − 1, to hadrons. Example: A π 0 meson has P = − 1 and C = + 1. A π + has P = − 1, but is no eigenstate of C , because C | π + � = | π − � .
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Strong interaction The QCD lagrangian permits a term which violates P , CP , and T , but experimentally the corresponding coefficient θ is found to be smaller than 10 − 11 . ⇒ The strong interaction essentially respects C , P , and therefore T , � � � � � � H strong , P H strong , C H strong , T = = = 0 We can assign C and P quantum numbers, which ⇒ can be + 1 or − 1, to hadrons. Example: A π 0 meson has P = − 1 and C = + 1. A π + has P = − 1, but is no eigenstate of C , because C | π + � = | π − � . Also QED respects C , P , and T .
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Parity violation 1956: θ − τ puzzle : A seemingly degenerate pair ( θ, τ ) of two mesons with P = + 1 and P = − 1, weakly decaying as P = + 1 “ θ ” → ππ P = − 1 “ τ ” → πππ
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Parity violation 1956: θ − τ puzzle : A seemingly degenerate pair ( θ, τ ) of two mesons with P = + 1 and P = − 1, weakly decaying as P = + 1 “ θ ” → ππ P = − 1 “ τ ” → πππ Explanation by Lee and Yang: “ θ ” and “ τ ” are the same particle, instead the weak interaction violates parity.
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Parity violation 1956: θ − τ puzzle : A seemingly degenerate pair ( θ, τ ) of two mesons with P = + 1 and P = − 1, weakly decaying as P = + 1 “ θ ” → ππ P = − 1 “ τ ” → πππ Explanation by Lee and Yang: “ θ ” and “ τ ” are the same particle, instead the weak interaction violates parity. K + = “ θ ” = “ τ ” .
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Maximal P violation In the SM only left-handed fields feel the charged weak interaction, no couplings of the W-boson to u j R , d j R , and e j R .
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Early monograph on parity violation:
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Early monograph on parity violation: Lewis Carroll: Alice through the looking glass
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Maximal parity violation
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Maximal parity violation
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Charge conjugation C maps left-handed (particle) fields on right-handed (antiparticle) fields and vice versa: C → ψ C ψ C L ≡ ( ψ C ) R is right-handed. ψ L ← L , where ⇒ The weak interaction also violates C!
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Charge conjugation C maps left-handed (particle) fields on right-handed (antiparticle) fields and vice versa: C → ψ C ψ C L ≡ ( ψ C ) R is right-handed. ψ L ← L , where ⇒ The weak interaction also violates C! But: Nothing prevents CP and T from being good symmetries. . .
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Charge conjugation C maps left-handed (particle) fields on right-handed (antiparticle) fields and vice versa: C → ψ C ψ C L ≡ ( ψ C ) R is right-handed. ψ L ← L , where ⇒ The weak interaction also violates C! But: Nothing prevents CP and T from being good symmetries. . . . . . except experiment!
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary CP violation Neutral K mesons: K long and K short (linear combinations of K and K ). Dominant decay channels: K long → πππ CP = − 1 K short → ππ CP = + 1
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary CP violation Neutral K mesons: K long and K short (linear combinations of K and K ). Dominant decay channels: K long → πππ CP = − 1 K short → ππ CP = + 1 1964: Christenson, Cronin, Fitch and Turlay observe K long → ππ and therefore discover CP violation. ǫ K ≡ � ( ππ ) I = 0 | H weak | K long � � ( ππ ) I = 0 | H weak | K short � = ( 2 . 229 ± 0 . 010 ) · 10 − 3 e i 0 . 97 π/ 4 .
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary CP violation in the SM Example: W coupling to b and u : � � L W = g 2 V ub u L γ µ b L W + ub b L γ µ u L W − + V ∗ √ µ µ 2
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary CP violation in the SM Example: W coupling to b and u : � � L W = g 2 V ub u L γ µ b L W + ub b L γ µ u L W − + V ∗ √ µ µ 2 u L γ µ b L CP → − b L γ µ u L − CP transformation CP W + → − W − µ − µ
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary CP violation in the SM Example: W coupling to b and u : � � L W = g 2 V ub u L γ µ b L W + ub b L γ µ u L W − + V ∗ √ µ µ 2 u L γ µ b L CP → − b L γ µ u L − CP transformation CP W + → − W − µ − µ � � g 2 Hence V ub b L γ µ u L W − ub u L γ µ b L W + CP L W + V ∗ − → √ µ µ 2 Is CP violated?
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary CP violation in the SM Example: W coupling to b and u : � � L W = g 2 V ub u L γ µ b L W + ub b L γ µ u L W − + V ∗ √ µ µ 2 u L γ µ b L CP → − b L γ µ u L − CP transformation CP W + → − W − µ − µ � � g 2 Hence V ub b L γ µ u L W − ub u L γ µ b L W + CP L W + V ∗ − → √ µ µ 2 Is CP violated? Not yet. . . Rephasing b L → e i φ b L , u L → e i φ ′ u L amounts to � � g 2 CP + reph . V ub e i ( φ ′ − φ ) b L γ µ u L W − ub e i ( φ − φ ′ ) u L γ µ b L W + L W µ + V ∗ √ − → , µ 2 so that we can achieve V ub e i ( φ ′ − φ ) = V ∗ ub .
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Alternatively we could have used the rephasing to render V ub real from the beginning. Observation by Kobayashi and Maskawa: A unitary n × n matrix has n ( n + 1 ) phases. In an n -generation 2 SM one can eliminate 2 n − 1 phases from V by rephasing the quark fields. The remaining ( n − 1 )( n − 2 ) phases are 2 physical, CP-violating parameters of the theory!
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Alternatively we could have used the rephasing to render V ub real from the beginning. Observation by Kobayashi and Maskawa: A unitary n × n matrix has n ( n + 1 ) phases. In an n -generation 2 SM one can eliminate 2 n − 1 phases from V by rephasing the quark fields. The remaining ( n − 1 )( n − 2 ) phases are 2 physical, CP-violating parameters of the theory! ( n − 1 )( n − 2 ) n 2 1 0 2 0 3 1 Kobayashi-Maskawa phase δ KM 4 3
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary CKM metrology The Cabibbo-Kobayashi-Maskawa (CKM) matrix V ud V us V ub V = V cd V cs V cb V td V ts V tb involves 4 parameters: 3 angles and the KM phase δ KM . Best way to parametrise V : Wolfenstein expansion
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Expand the CKM matrix V in V us ≃ λ = 0 . 2246: A λ 3 � � 1 − λ 2 1 + λ 2 ( ρ − i η ) λ V ud V us V ub 2 2 V cd V cs V cb ≃ − λ − iA 2 λ 5 η 1 − λ 2 A λ 2 2 V td V ts V tb A λ 3 ( 1 − ρ − i η ) − A λ 2 − iA λ 4 η 1 with the Wolfenstein parameters λ , A , ρ , η CP violation ⇔ η � = 0
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Expand the CKM matrix V in V us ≃ λ = 0 . 2246: A λ 3 � � 1 − λ 2 1 + λ 2 ( ρ − i η ) λ V ud V us V ub 2 2 V cd V cs V cb ≃ − λ − iA 2 λ 5 η 1 − λ 2 A λ 2 2 V td V ts V tb A λ 3 ( 1 − ρ − i η ) − A λ 2 − iA λ 4 η 1 with the Wolfenstein parameters λ , A , ρ , η CP violation ⇔ η � = 0 Unitarity triangle: Exact definition: A=( ρ,η) − V ∗ ub V ud α ρ + i η = V ∗ cb V cd ρ+ i η 1−ρ− i η � � V ∗ ub V ud � � � e i γ � � = � V ∗ cb V cd γ β C=(0,0) B=(1,0)
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary In the SM the flavour violation only occurs in the couplings of µ and G ± to fermions. W ± ⇒ At tree-level flavour-changes only occur in charged- current processes.
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary In the SM the flavour violation only occurs in the couplings of µ and G ± to fermions. W ± d s b b u u u W W W W ⇒ At tree-level flavour-changes only occur in charged- current processes. � � � � � � � � ` ` ` ` ` ` ` ` Semileptonic decays: determining | V ud | | V us | | V cb | | V ub | .
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Flavour-changing neutral current (FCNC) processes Examples: b u,c,t s s b W t s u,c,t b B s − B s mixing penguin diagram
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Flavour-changing neutral current (FCNC) processes Examples: b u,c,t s s b W t s u,c,t b B s − B s mixing penguin diagram FCNC processes are the only possibility to gain information on V td and V ts . However: FCNC processes are highly sensitive to physics beyond the SM.
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Flavour-changing neutral current (FCNC) processes Examples: b u,c,t s s b W t s u,c,t b B s − B s mixing penguin diagram FCNC processes are the only possibility to gain information on V td and V ts . However: FCNC processes are highly sensitive to physics beyond the SM. In principle can determine all parameters λ , A , ρ , η from tree-level processes. View FCNC processes as new physics analysers ⇒ rather than ways to measure V td and V ts .
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary B − B mixing basics Consider B q − B q mixing with q = d or b u,c,t q q = s : A meson identified (“tagged”) as a B q at time t = 0 is described by | B q ( t ) � . q u,c,t b
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary B − B mixing basics Consider B q − B q mixing with q = d or b u,c,t q q = s : A meson identified (“tagged”) as a B q at time t = 0 is described by | B q ( t ) � . q u,c,t b For t > 0: | B q ( t ) � = � B q | B q ( t ) �| B q � + � B q | B q ( t ) �| B q � + . . . , with “. . . ” denoting the states into which B q ( t ) can decay.
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary B − B mixing basics Consider B q − B q mixing with q = d or b u,c,t q q = s : A meson identified (“tagged”) as a B q at time t = 0 is described by | B q ( t ) � . q u,c,t b For t > 0: | B q ( t ) � = � B q | B q ( t ) �| B q � + � B q | B q ( t ) �| B q � + . . . , with “. . . ” denoting the states into which B q ( t ) can decay. Analogously: | B q ( t ) � is the ket of a meson tagged as a B q at time t = 0.
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Schr¨ odinger equation: � � � � � � � B q | B q ( t ) � M q − i Γ q � B q | B q ( t ) � i d = dt � B q | B q ( t ) � � B q | B q ( t ) � 2 with the 2 × 2 mass and decay matrices M q = M q † and Γ q = Γ q † . � � � B q | B q ( t ) � obeys the same Schr¨ odinger equation. � B q | B q ( t ) �
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Schr¨ odinger equation: � � � � � � � B q | B q ( t ) � M q − i Γ q � B q | B q ( t ) � i d = dt � B q | B q ( t ) � � B q | B q ( t ) � 2 with the 2 × 2 mass and decay matrices M q = M q † and Γ q = Γ q † . � � � B q | B q ( t ) � obeys the same Schr¨ odinger equation. � B q | B q ( t ) � 3 physical quantities in B q − B q mixing: � � − M q � � � � � M q � Γ q � , � , 12 φ q ≡ arg Γ q 12 12 12
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Diagonalise M q − i Γ q 2 to find the two mass eigenstates: | B L � p | B q � + q | B q � . Lighter eigenstate: = | B H � p | B q � − q | B q � Heavier eigenstate: = with masses M q L , H and widths Γ q L , H . Further | p | 2 + | q | 2 = 1.
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Diagonalise M q − i Γ q 2 to find the two mass eigenstates: | B L � p | B q � + q | B q � . Lighter eigenstate: = | B H � p | B q � − q | B q � Heavier eigenstate: = with masses M q L , H and widths Γ q L , H . Further | p | 2 + | q | 2 = 1. Relation of ∆ m q and ∆Γ q to | M q 12 | , | Γ q 12 | and φ q : M H − M L ≃ 2 | M q ∆ m q = 12 | , Γ L − Γ H ≃ 2 | Γ q ∆Γ q = 12 | cos φ q
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary b u,c,t q M q 12 stems from the dispersive (real) part of the box diagram, internal t . Γ q 12 stems from the absorpive (imag- inary) part of the box diagram, inter- nal c , u . q u,c,t b
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary odinger equation to find the desired B q − B q Solve the Schr¨ oscillations: � � e − Γ q t cosh ∆Γ q t |� B q | B q ( t ) �| 2 = |� B q | B q ( t ) �| 2 + cos (∆ m q t ) = 2 2 � � e − Γ q t cosh ∆Γ q t |� B q | B q ( t ) �| 2 ≃ |� B q | B q ( t ) �| 2 − cos (∆ m q t ) ≃ 2 2 with Γ q ≡ Γ q L + Γ q H 2
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Time-dependent decay rate: d N ( B q ( t ) → f ) Γ( B q ( t ) → f ) = 1 , N B d t where d N ( B q ( t ) → f ) is the number of B q ( t ) → f decays within the time interval [ t , t + d t ] . N B is the number of B q ’s present at time t = 0.
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Time-dependent decay rate: d N ( B q ( t ) → f ) Γ( B q ( t ) → f ) = 1 , N B d t where d N ( B q ( t ) → f ) is the number of B q ( t ) → f decays within the time interval [ t , t + d t ] . N B is the number of B q ’s present at time t = 0. With | f � ≡ CP | f � define the time-dependent CP asymmetry: a f ( t ) = Γ( B q ( t ) → f ) − Γ( B q ( t ) → f ) Γ( B q ( t ) → f ) + Γ( B q ( t ) → f )
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Example 1: B d → J /ψ K S | f � = −| f � (CP-odd eigenstate) ⇒ � � B d B d b c b c J/ψ J/ψ c d d c s s K S K S a J /ψ K S ( t ) ≃ − sin ( 2 β ) sin (∆ m d t ) , � � − V cd V ∗ cb where β = arg V td V ∗ tb
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Example 2: B s → ( J /ψφ ) L = 0 | f � = | f � (CP-even eigenstate) ⇒ � � B s B s b c b c J/ψ J/ψ c s s c s s φ φ sin ( 2 β s ) sin (∆ m s t ) a ( J /ψφ ) L = 0 ( t ) = − cosh (∆Γ s t / 2 ) − cos ( 2 β s ) sinh (∆Γ s t / 2 ) , � � − V ts V ∗ tb ≃ λ 2 η where β s = arg V cs V ∗ cb
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary The Wolfenstein parameters λ and A are well determined from the semileptonic decays K → πℓ + ν ℓ and B → X c ℓ + ν ℓ , ℓ = e , µ .
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Metrology of the unitarity triangle: The apex ( ρ , η ) is currently constrained from the following experimental input: � • | V ub | ∝ ρ 2 + η 2 measured in B → πℓν ℓ , B → X u ℓν ℓ and B + → τ + ν τ .
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Metrology of the unitarity triangle: The apex ( ρ , η ) is currently constrained from the following experimental input: � • | V ub | ∝ ρ 2 + η 2 measured in B → πℓν ℓ , B → X u ℓν ℓ and B + → τ + ν τ . • γ extracted from B ± → ( ) DK ±
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Metrology of the unitarity triangle: The apex ( ρ , η ) is currently constrained from the following experimental input: � • | V ub | ∝ ρ 2 + η 2 measured in B → πℓν ℓ , B → X u ℓν ℓ and B + → τ + ν τ . • γ extracted from B ± → ( ) DK ± � • ∆ m d ∝ ( 1 − ρ ) 2 + η 2
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Metrology of the unitarity triangle: The apex ( ρ , η ) is currently constrained from the following experimental input: � • | V ub | ∝ ρ 2 + η 2 measured in B → πℓν ℓ , B → X u ℓν ℓ and B + → τ + ν τ . • γ extracted from B ± → ( ) DK ± � • ∆ m d ∝ ( 1 − ρ ) 2 + η 2 � • ∆ m d / ∆ m s ∝ ( 1 − ρ ) 2 + η 2
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Metrology of the unitarity triangle: The apex ( ρ , η ) is currently constrained from the following experimental input: � • | V ub | ∝ ρ 2 + η 2 measured in B → πℓν ℓ , B → X u ℓν ℓ and B + → τ + ν τ . • γ extracted from B ± → ( ) DK ± � • ∆ m d ∝ ( 1 − ρ ) 2 + η 2 � • ∆ m d / ∆ m s ∝ ( 1 − ρ ) 2 + η 2 • sin ( 2 β ) from a J /ψ K S ( t ) and other b → ccs decays
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Metrology of the unitarity triangle: The apex ( ρ , η ) is currently constrained from the following experimental input: � • | V ub | ∝ ρ 2 + η 2 measured in B → πℓν ℓ , B → X u ℓν ℓ and B + → τ + ν τ . • γ extracted from B ± → ( ) DK ± � • ∆ m d ∝ ( 1 − ρ ) 2 + η 2 � • ∆ m d / ∆ m s ∝ ( 1 − ρ ) 2 + η 2 • sin ( 2 β ) from a J /ψ K S ( t ) and other b → ccs decays • α determined from CP asymmetries in B → ππ , B → ρρ and B → ρπ decays.
Basics C,P ,T CKM new physics global analysis SUSY Rare decays Summary Metrology of the unitarity triangle: The apex ( ρ , η ) is currently constrained from the following experimental input: � • | V ub | ∝ ρ 2 + η 2 measured in B → πℓν ℓ , B → X u ℓν ℓ and B + → τ + ν τ . • γ extracted from B ± → ( ) DK ± � • ∆ m d ∝ ( 1 − ρ ) 2 + η 2 � • ∆ m d / ∆ m s ∝ ( 1 − ρ ) 2 + η 2 • sin ( 2 β ) from a J /ψ K S ( t ) and other b → ccs decays • α determined from CP asymmetries in B → ππ , B → ρρ and B → ρπ decays. • ǫ K (the measure of CP violation in K − K mixing), which defines a hyperbola in the ( ρ , η ) plane.
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