Standard Model of Particle Physics

Standard Model of Particle Physics Chris Sachrajda School of Physics and Astronomy University of Southampton Southampton SO17 1BJ UK SUSSP61, St Andrews August 8th 23rd 2006

Contents 1. Spontaneous Symmetry Breaking 2. Electroweak Theory 3. QCD 4. Flavourdynamics and Non-Perturbative QCD - I 5. Flavourdynamics and Non-Perturbative QCD - II

Lecture 5 Flavourdynamics and Non-Perturbative QCD II 1. Introduction 2. Selected Topics in Kaon Physics K 0 K 0 Mixing K ππ decays. 3. B 0 B 0 Mixing 4. B J/PsiK S Golden Mode.

PDG2006 Unitarity Triangle 1.5 excluded area has CL > 0.95 excluded at CL > 0.95 1 γ sin 2β m d ms & md 0.5 η 0 ε K V ub /V cb γ α β α -0.5 α ε K -1 γ sol. w/ cos 2β < 0 (excl. at CL > 0.95) -1.5-1 -0.5 0 0.5 1 1.5 2 ρ

K 0 K 0 Mixing d s d u,c,t s u,c,t u,c,t s d s u,c,t d The CP-eigenstates (K 1 and K 2 ) are linear combinations of the two strong-interaction eigenstates: K 1 = 1 2 ( K 0 + K 0 ) CP K 1 = K 1 and K 2 = 1 2 ( K 0 K 0 ) CP K 2 = K 2. I use the phase convention so that CP K 0 = K 0.

K 0 K 0 Mixing Cont. Because of the complex phase in the CKM-matrix, the physical states (the mass eigenstates) differ from K 1 and K 2 by a small admixture of the other state: K S = K 1 + ε K 2 (1 + ε 2 ) 1 2 and K L = K 2 + ε K 1 (1 + ε 2 ) 1 2, The parameter ε depends on the phase convention chosen for K 0 and K 0.

K 0 K 0 Mixing Cont. For K ππ and K πππ decays, the two pion states are CP-even and the three-pion states are CP-odd the dominant decays are: K S ππ and K L 3π. This is the reason why K L is much longer lived than K S. K L and K S are not CP-eigenstates, however K L 2π and K S 3π decays may occur. CP-violating decays which occur due to the fact that the mass eigenstates are not CP-eigenstates are called indirect CP-violating decays. A measure of the strength of indirect CP-violation is given by the physical parameter ε K defined by the ratio: ε K A(K L (ππ) I=0 ) A(K S (ππ) I=0 ) = (2.280 ± 0.013)10 3 e i π 4.

Directly CP-violating decays are those in which a CP-even (-odd) state decays into a CP-odd (-even) one: K L K 2 + εk 1. Direct (ε ) ππ ππ Indirect (ε K ) Consider the following contributions to K ππ decays: s d s u s uū d d I = 0, Complex (a) d I = 0, Real (b) ū d d d I = 0 or 2, Real (c) Thus direct CP-violation in kaon decays manifests itself as a non-zero relative phase between the I = 0 and I = 2 amplitudes. We also have strong phases, δ 0 and δ 2 which are independent of the form of the weak Hamiltonian.

K ππ Decays Cont. A(K 0 π + π ) = A(K 0 π 0 π 0 ) = 2 3 A 0 e iδ 0 1 + 3 A 2 e iδ 2 2 3 A 0 e iδ 0 1 2 3 A 2 e iδ 2. The parameter ε, which is used as a measure of CP-violation is defined by: ( ε = e iφ ImA2 ImA ) 0, 2 ReA 2 ReA 0 where ReA 2 ReA 0 and φ = π 2 + δ 2 δ 0 π 4. ε is manifestly zero if the phases of the I = 0 and I = 2 weak amplitudes are the same. The I = 1/2 rule puzzle - Why is 1 so large? ( 1 22.)

Experimentally the two parameters ε K (which, following standard conventions I rename from now on as ε, ε ε K ) and ε can be determined by measuring the ratios: η 00 A(K L π 0 π 0 ) A(K S π 0 π 0 ) ε 2ε η + A(K L π + π ) A(K S π + π ) ε + ε. Direct CP-violation is found to be considerably smaller than indirect violation. By measuring the decays and using η 00 2 ( ε ) η + 1 6Re +, ε The NA31 and E371 experiments have measured ε /ε, and the combined result is: ε /ε = (17.2 ± 1.8)10 4.

ε and the Unitarity Triangle We need to know the matrix element: K 0 H S=2 eff K 0. The form of the effective Hamiltonian is H S=2 eff = G2 F 16π 2 M2 W X O S=2 (µ) where X is a function of the CKM-matrix elements, with coefficients which can be calculated perturbatively and which depend on the (u,)c, t masses. The non-perturbative QCD corrections are contained in the matrix element: K 0 sγ µ (1 γ 5 )d sγ µ (1 γ 5 )d K 0 8 3 m2 K f 2 K B K(µ). Uncertainty in B K is a major restriction on the Unitarity Triangle analysis.

Recent Lattice Results for B K Recent summaries of the quenched value of B K include: B MS K (2GeV) = 0.58(4) S.Hashimoto (ICHEP 2004) B MS K (2GeV) = 0.58(3) C.Dawson (Lattice 2005). Dynamical computations of B K are underway by a number of collaborations, but so far the results are very preliminary. C.Dawson s guesstimate (from comparison of unquenched & quenched results at similar masses and lattice spacings) B MS K (2GeV) = 0.58(3)(6) C.Dawson (Lattice 2005). We need to wait until reliable dynamical results are available in the next year or two.

A precise determination of ε would fix the vertex A to lie on a hyperbola A ( ρ, η) C (0,0) B (1,0)

B 0 B 0 Mixing d,(s) b d,(s) t b t t b d,(s) In B 0 B 0 mixing, the top quark dominates and hence from the measured mass differences V td and V ts. b t d,(s) The non-perturbative QCD effects are contained in the matrix element of the B = 2 operator: O B=2 = bγ µ (1 γ 5 )d bγ µ (1 γ 5 )d 8 3 m2 B f 2 B B B(µ). The uncertainty in this matrix element dominates that in the final answer for V td. PDG2006 use m d = 0.507 ± 0.004 and take the lattice value f Bd ˆB Bd = (244 ± 11 ± 24)MeV to obtain V td = (7.4 ± 0.8) 10 3

B 0 B 0 Mixing Cont. The uncertainties are reduced in the lattice calculation of the ratio ξ = f B BBs s = 1.21 ± 0.04 +0.04 0.01 V td f Bd BBd V ts = 0.208+0.008 0.006, where the new Tevatron result of m s = (17.31 +0.33 0.18 ± 0.07) ps 1 has been used. From a comprehensive unitarity triangle analysis without using the lattice result for the B = 2 matrix element: G.Martinelli (for UTfit Collaboration) - Ringberg April 2006 CDF (2006) m S = (17.33 +.42.21 (stat) ± 0.07syst)ps 1 ξ = 1.15 ± 0.08 (V ub exclusive) V td 1 ρ i η. or ξ = 1.05 ± 0.10 (V ub inclusive) or ξ = 1.06 ± 0.09 (V ub combined)

The Golden Mode - B K S J/Ψ Mixing Induced CP-Violating Decays In order to study CP-violation we need to be sensitive to the weak phase interference. The strong interactions also generate phases, so, in general, we need to be able to control the hadronic effects. For the golden-mode B J/ΨK s this is possible to a great degree of accuracy precise determination of sin(2β). I will now review the theoretical background behind this statement. The two neutral mass-eigenstates are given by B L = 1 ) (p B 0 + q B 0 p 2 + q 2 and B H = 1 ) (p B 0 q B 0. p 2 + q 2 where p and q are complex parameters.

Introduction Kaon Physics B-Physics Summary The 2 2 mass-matrix takes the form M i 2 = ( A p 2 q 2 A where A,p and q are complex parameters. ). Starting with a B 0 meson at time t = 0, its subsequent evolution is governed by the Schrödinger equation: ( ) B 0 phys (t) = g +(t) B 0 q + g (t) B 0, p where and M = (M H + M L )/2. [ g + (t) = exp [ g (t) = exp t 2 t 2 ] exp[ imt] cos ] exp[ imt] isin ( M t 2 ( M t Starting with a B 0 meson at t = 0, the time evolution is B 0 phys (t) = (p/q)g (t) B 0 + g + (t) B 0. 2 ), ),

Introduction Kaon Physics B-Physics Summary Decays of Neutral B-Mesons into CP-Eigenstates Let f CP be a CP-eigenstate and A,Ā be the amplitudes Defining we have A f CP H B 0 and Ā f CP H B 0. λ q p f CP H B 0 phys = A[g +(t) + λ g (t)] and f CP H B 0 phys = A p q [g (t) + λ g + (t)]. Ā A The time-dependent rates for initially pure B 0 or B 0 states to decay into the CP-eigenstate f CP at time t are given by: [ (B 0 phys (t) f CP) = A 2 e t 1 + λ 2 + 1 ] λ 2 cos( M t) Imλ sin( M t) 2 2 [ ( B 0 phys (t) f CP) = A 2 e t 1 + λ 2 1 ] λ 2 cos( M t) + Imλ sin( M t). 2 2

Introduction Kaon Physics B-Physics Summary The time-dependent asymmetry is defined as: A fcp (t) (B 0 phys (t) f CP) ( B 0 phys (t) f CP) (B 0 phys (t) f CP) + ( B 0 phys (t) f CP) = (1 λ 2 )cos( M t) 2Imλ sin( M t) 1 + λ 2. If q/p = 1 (which is the case if M) and Ā/A = 1 (examples of this will be presented below), then λ = 1 and the first term on the right-hand side above vanishes. The form of the amplitudes A and Ā is: A = i A i e iδ i e iφ i and Ā = A i e iδ i e iφ i i Sum is over all the contributions to the process; the A i are real; the δ i are the strong phases; the φ i are the phases from the CKM matrix.

A = i A i e iδ i e iφ i and Ā = A i e iδ i e iφ i i In the most favourable situation, all the contributions have a single CKM phase (φ D say) and Ā A = exp( 2iφ D). Since Thus 12 << M 12, q/p = M 12 /M 12 exp( 2iφ M ), and From the box diagrams: ( ) q = VtdV tb p B d Vtd V tb λ = exp( 2i(φ D + φ M )). Imλ = sin(2(φ D + φ M )). and ( ) q = VtsV tb p B s VtsV. tb

Consider processes in which the b-quark decays through the subprocess b d j u i ū i. The corresponding tree-level diagram is b u i d j for which B d J/ΨK S In this case Ā A = V ibvij Vib V. ij ū i λ(b J/ΨK S ) = V tdv tb V td V tb V cs V cd V csv cd V cb V cs V cb V cs = sin(2β) The first factor is (q/p) Bd ; the second factor is the analogous one for the final state kaon; the third factor is Ā/A, with u i = c and d j = s. Recall that ( β = arg V cdvcb ) V td Vtb.

There is also a small penguin contribution to this process: W b t s G. Phase is that of V tb V ts, which is is equal (to an excellent approximation) to that of V cb V cs. Thus we have a single weak phase and hence hadronic uncertainties are negligible in the determination of the sin(2β) from this process (golden mode). This is an (almost) ideal situation but one which is very rare. PDG 2006 average the results from BaBar and Belle and obtain In PDG 2000, sin(2β) = 0.78 ± 0.08. sin(2β) = 0.687 ± 0.032.

Summary and Conclusions In these lectures I have tried to remind you of the main elements of the Standard Model of Particle Physics and to describe some of the attempts to explore its limits in the quark sector. I did not have time to discuss the recent developments in the determination of α and γ from two-body B-decays. The observation of ν oscillations window on BSM. Flavour Physics will continue to be a powerful tool with which to unravel the structure of BSM physics.