Measuring CP Violation

Particle Physics Dr Victoria Martin, Spring Semester 2012 Lecture 15: Measuring CP Violation

!Mixing and decays of kaons !CP violation in K0 and B0 !CKM Matrix revisited !CP violation experiments! !The unitarity triangle

From Last Friday: Summary

•Parity P and Charge Conjugation C are maximally violated in weak interactions due to vector ! axial vector structure of interaction vertex. •Conserved in strong and electromagnetic interactions. •The combined symmetry CP describes the difference between matter and anti-matter •almost a good symmetry in the weak interactions. •CPT symmetry must be conserved… it’s one of the foundations of QM and field theory! •CP violation observed in K0 decays and mixing.

2 Neutral Meson Mixing •Second order weak interactions can mix long-lived neutral mesons with their antiparticles: 0 0 0 0 0 0 0 0 0 "K ( s ̅ d ) , D ( c ̅ u ) , B (b ̅ d) , Bs (b ̅ s) K # K̅ D # D̅ B # B̅ Bs # Bs̅ • e.g. take the neutral kaons K0 K̅0 as an example: 0 0 P K0 = K0 P K = K K̅0 K0 | −| 0 | 0 −| CP K0 = K CP K = K0 | −| | −| • The CP eigenstates are: 1 0 0 K1 = K K CP =+1 | √2 | −| 1 0 0 K2 = K + K CP = 1 | √2 | | − • Decay eigenstates are (approximately) K1 and K2 - not the same as the flavour eigenstates K0 and K0̅ • Two common decay modes are 2! and 3! • !0!0 and !+!" have CP = +1 • !0!0!0 and !+!"!0 have CP = "1 3

Neutral Kaons continued • CP is almost conserved in the decay of the neutral kaons. • Decay K$!! has large phase space ⇒ quick decay, travels ~ cm before decay • named “K-short” or KS with %S = 0.09 ns • Decay K$!!! has small phase space ⇒ slow decay, travels ~ 10 m before decay • “K-long” or KL with %L = 51 ns

• After a neutral kaon is produced, at some time, t, later it will be described 0 by: ψ(t)=a(t) K0 + b(t) K | | • The evolution of the kaon state described by mass & decay matrices: M, & ∂ψ(t) i i = Hˆ ψ(t)=(Mˆ Γˆ)ψ(t) ∂t − 2 i i i ˆ ˆ MK 2 ΓK ∆mK 2 ∆ΓK M Γ = − i − i − 2 (∆mK)∗ (∆ΓK)∗ MK ΓK − 2 − 2 "12 "10 "1 • Mass difference 'mK = mS " mL = 3.52(1) x10 MeV = 0.53 x 10 s is oscillation frequency

4 Neutral Kaons with CP Violation •The CP eigenstates are K1 and K2 1 0 0 K1 = K K CP =+1 | √2 | −| 1 0 0 K2 = K + K CP = 1 | √2 | | − •The decay KS and KL are not quite identical to the CP eigenstates; in terms of a (small) parameter !:

1 0 K = (1 + ) K0 (1 ) K | S N | − − | 1 0 K = (1 + ) K0 + (1 ) K | L N | − | 0 0 •Note, both KS and KL contain more K (matter) than K̅ (antimatter). The decay states contain both CP = +1 and CP = "1: CP is violated. 1 K = ( K K ) | S N | 1− | 2 1 K = ( K + K ) | L N | 2 | 1 •! is measured to be |! | ~ 2 # 10"3, the amount of indirect CP violation 5

CP Violation in K$!!

• KL ~ K2 + ! K1 mainly CP = "1 plus a little CP = +1 • It can decay into both $$$ (CP = "1) and $$ (CP = +1)

• Measured rate of KL % $$ is slightly larger than can be accommodated by ! • A small amount (!’) of the K2 in KL decays directly to $$. • Known as direct CP violation.

• It took 40 year’s of effort to measure these effects!

• Measured parameters are decay rates ratios: + iφ+ Γ(KL π π−) η+ = η+ e − = → + = + Γ(KS π π−) − | −| → 0 0 iφ00 Γ(KL π π ) η00 = η00 e = Γ(K →π0π0) = 2 | | S→ − • Experimental measurements: |(| = 2.23(1) ) 10"3 |(’/(| = 1.65(26) ) 10"3 • Phases: *+" = *00 = 43.4(7)° • Note the observed amount of CP violation in these experiments is very small. 6 CP and T Violation in KL $ ! " #

CP violation is also observed in the semileptonic decay • + KL$! " # ! + "" stands for e or µ W "! "K0 can only decay as K0$!$ "+ # s̅ u ̅ "K̅0 can only decay as K̅0$!+ "$ # ̅ d d 0 0 • KL = 1/N [ (1+() K + (1"() K̅ ] !!

W! ̅ • Measure asymmetry in KL decay rates: "! + + s u Γ(KL π− ν) Γ(KL π −ν¯) δ = → − → d̅ d̅ Γ(K π +ν)+Γ(K π+ ν¯) L → − L → − (1 + )2 (1 )2 δ = − − =2 e () (1 + )2 + (1 )2 R − •Measured to be ( = 3.27(12) ) 10"3 •CP violation due to the mixing of the CP eigenstates

General Formalism for CP Violation

•Define light and heavy mass states ML and MH of neutral mesons: 0 0 0 0 2 2 ML = p M + q M MH = p M q M q + p =1 •Associated| | parameters: | | | − | | | | | ∆m = m m ∆Γ = Γ Γ Γ =(Γ + Γ )/2 H − L H − L H L Amplitudes for decays of the flavour eigenstates M0 and M̅0 • Don’t to a CP eigenstate f are A and A ̅ memorise these equations!

dΓ 0 Γt 2 q 2 2 q 2 (M f)=e− [( A + A ) cosh(∆Γt/2) + ( A A ) cos(∆mt) dt → | | |p | | | − |p | q q +2 e( A∗A) sinh(∆Γt/2) 2 m( A∗A) sin(∆mt)] R p − I p

dΓ 0 Γt p 2 2 p 2 2 (M f)=e− [( A + A ) cosh(∆Γt/2) + ( A A ) cos(∆mt) dt → | q | | | | q | − | | q p +2 e( AA∗) sinh(∆Γt/2) + 2 m( AA∗) sin(∆mt)] R p I q •In practice measure oscillations with frequency %mt; amplitudes of sin and cos terms used to extract parameters p, q, A and A ̅ 8 Types of CP Violation 1.Direct CP violation in decay amplitudes A =1 A • Can occur in both charged and neutral particle decays • Gives a cos(&mt) term in neutral meson decays

q 2.CP violation in neutral meson mixing =1 p 4 4 • Asymmetry in semileptonic decays 'SL = (1 "|q/p| )/(1+|q/p| ) q A 3.CP violation due to interference mixing and decay m =0 I p A • Gives a sin(&mt) term in neutral meson decays

0 All three of these types been observed in K decays ((’ , +SL , ( ) and first and last in B0 decays

•NB: a complex phase between q/p and A/A̅ leads to CP violation

Cabibbo-Kobayashi-Maskawa Matrix

• Mass eigenstates and weak eigenstates of quarks are not identical. "Decay properties measure mass eigenstates with a definite lifetime and decay width "The weak force acts on the weak eigenstates. • Weak eigenstates are admixture of mass eigenstates, conventionally described using CKM matrix a mixture of the down-type quarks:

d Vud Vus Vub d weak mass s = V V V s eigenstates    cd cs cb    eigenstates b Vtd Vts Vtb b       † • The CKM matrix is unitary, VCKM VCKM = 1 implies nine “unitarity relations”

Vud∗ Vcd∗ Vtd∗ Vud Vus Vub 100 V ∗ V ∗ V ∗ V V V = 010  us cs ts   cd cs cb    Vub∗ Vcb∗ Vtb∗ Vtd Vts Vtb 001 • The most frequently discussed  is (1st row ! 3rd column):   VudVub∗ + VtdVtb∗ + VcdVcb∗ =0

10 The Wolfenstein Parameterisation

• An expansion of the CKM matrix in powers of , = Vus = 0.22 1 λ2/2 λ Aλ3(ρ iη) − 2 −2 4 VCKM = λ 1 λ /2 Aλ + (λ )  Aλ3(1− ρ iη) −Aλ2 1  O − − −   • Parameterisation reflects almost diagonal nature of CKM matrix: "The diagonal elements Vud, Vcs, Vtb are close to 1

"Elements Vus, Vcd ~ , are equal 2 "Elements Vcb, Vts ~ , are equal 3 "Elements Vub, Vtd ~ , are very small

• Diagonal structure means down quark mass eigenstate is almost equal to down quark weak eigenstate • similarly for strange and bottom mass eigenstates • Note that the complex phase & only appears in the very small elements, and is thus hard to measure.

1.4 Violation in the Standard Model 21 1.4 Violation in the Standard Model 21 VtdVtb The Unitarity_ Triangle ` V V VtdVtb VudVub∗ + VtdVtb∗ + VcdVcb∗ =0ud ub _ a V V • Forms a triangle in the complex plane: cd cb ` V V (a)ud ub * a V V VcdVcb : cd cb • Dividing through by A d _ (a) VudVub VtdVtb |VcdVcb| |VcdVcb|

A d a ` 0 VudVub _ V V l td tb 0 1 |VcdVcb| |V V | 7–92 cd cb (b) 7204A5 This unitarity triangle is often use to present measurements of CP violation • Figure 1-2. The rescaled Unitarity Triangle, all sides divided by . in B-meson decay. a VudVub∗ VtdVtb∗ ` The rescaled Unitarity Triangle (Fig. 1-2)0 is derived from (1.82) by (a) choosing a phase convention • Lengths and angles of the triangle are: V V l V V such that is real, and (b) dividing0 the lengthscd ofcb∗ all sides by cd ;cb∗ (a) aligns one1 side of the triangle with the real axis, and (b) makes the length of this side 1. The form of the triangle 7–92 (b) 7204A5 is unchanged.VtdV ∗ Two vertices of the rescaled UnitarityVcdV Triangle∗ are thus fixed at (0,0) andV (1,0).udV The∗ α arg coordinatestb of the remainingβ vertexarg are denoted by cb . It is customaryγ thesearg days to expressub the ≡ −CKM-matrixVudVub∗ in terms ofFigure four≡ Wolfenstein 1-2. The− parametersV rescaledtdVtb∗ Unitaritywith Triangle,≡ all sides− V dividedcdplayingVcb∗ by . the role of an expansion parameter andrepresenting the -violating phase [27]: • Triangle has a finite area only if relative complex phase between CKM elements The rescaled Unitarity Triangle (Fig. 1-2) is derived from (1.82) by (a)(1.83) choosing a phase convention 12 such that is real, and (b) dividing the lengths of all sides by ; (a) aligns one side of the triangle with the real axis, and (b) makes the length of this side 1. The form of the triangle is small, and for each element in , the expansion parameter is actually . Hence it is sufficient to keepis unchanged. only the first few Two terms vertices in this expansion. of the rescaled The relation Unitarity between Triangle the parameters are thus of (1.78)fixed at (0,0) and (1,0). The and (1.83)coordinates is given by of the remaining vertex are denoted by . It is customary these days to express the CKM-matrix in terms of four Wolfenstein parameters with playing (1.84) the role of an expansion parameter and representing the -violating phase [27]: This specifies the higher order terms in (1.83).

REPORT OF THE BABAR PHYSICS WORKSHOP (1.83)

is small, and for each element in , the expansion parameter is actually . Hence it is sufﬁcient to keep only the ﬁrst few terms in this expansion. The relation between the parameters of (1.78) and (1.83) is given by

(1.84)

This speciﬁes the higher order terms in (1.83).

REPORT OF THE BABAR PHYSICS WORKSHOP B-Physics Measurements • Several dedicated experiments designed specifically to investigate B-hadrons. • BaBar and Belle: e+e$$ Υ(4s) $ B0 B̅0 to measure the decays of b and c quarks, e.g. Vcb and Vub 0 0 0 • LHCb experiment at LHC looks B , Bs , D mesons produced in pp collisions.

BaBar at SLAC California, 1999 - 2008

Belle at KEK, LHCb at LHC Japan 1999-2010 2009 - 20.. p

p 13

CP Violation in B0 decays

• At BaBar and Belle e+e$$ Υ(4s) $ B0 B̅0 coherent state of B0 and B̅0 • B0 = bd,̅ B̅0 = bd̅ • Υ(4s) is bb ̅ state with m > m(B0) + m(B̅0)

0 • At time t1 identity one meson as B through its decay (known as flavour tagging) 0 • At time t1 the other meson must be B̅ 0 0 • The B̅ meson evolves and may oscillate into B , decays at later time t2

Oscillations occur with +i2' e phase from CKM a frequency 'md over matrix elements a time 't = t2-t1 involved in mixing B0 # B̅0 Measure sin('md't) and cos('md't) amplitudes for B0 and B̅0 flavour tags

Next few slides show measurements made in the B-meson sector, the details of these measurements are non-examinable.

14 CP Violation in 0 0 BaBar B % J/( K 200 a) B 0 tags J/( KS B 0 tags CP = +1 Evts. / 0.4 ps

-5 0 5 0.5 b)

Raw asym. -0.5

Interference between mixing and -5 0 5 b $ c decay amplitude gives CP c) B0 tags J/( K asymmetries with Im(,) - 0 100 L B 0 tags CP = "1

A(t) = S sin('md't) + C cos('md't) Evts. / 0.4 ps

-5 0 5 0.5 S = 2Im (,)/(1+|,|2) = sin2. d) C = (1"|,|2)/(1+|,|2) 0

Raw asym. -0.5 BaBar measurement: -5 0 5 S = 0.655 ± 0.024 't (ps)

CP Violation in B $ !!, //

• Measurements of B $ !!, B $ // decays constrain the phase angle ( • Measure ACP (t) = S sin('md't) + C cos('md't) due to interference between mixing and b $ u decay

d ̅ + + + + / , ! }) , ! } d ̅ B0

0 B }/", !$ }/", !$

16 0 0 B and Bs Mixing

• Measurements of the mixing B0 # B̅0 |Vtb| |Vtd| and Bs # Bs̅ used to determine CKM matrix elements • Particularly |Vtd| and |Vts| which are not well known. |Vtd| |Vtb|

Bs mixing "1 Measured 'ms = 17.77 ± 0.12 ps Oscillation frequency 'md = 0.51/ps Lifetime %(b) = 1.5ps "1 Ratio &md/&ms gives LHCb: 'ms = 17.63 ± 0.11 ps |Vtd/Vts| = 0.209±0.006

“CKM Fit”

•All measurements including results from BaBar, Belle, LHCb used to determine lengths and sides of unitarity triangle. •Determines best values for 0 and / parameters in Wolfenstein parameterisation. •Current measurements indicate it is a closed triangle - consistent with only small CP violation.

18 From Lecture 2: CP Violation in Charm Decays

• Charge-Parity Symmetry examines the difference between matter and anti-matter • LHCb experiment at the LHC looked for difference in the decays of an anti-matter pair of D-mesons: • D0 = cu;̅ D0̅ = cu̅ • ) is the decay rate, how quickly this decay takes place (units of energy) 0 + 0 + 0 + 0 + Γ(D K K−) Γ(D K K−) Γ(D π π−) Γ(D π π−) CP = → − 0 → → − 0 → A Γ(D0 K+K )+Γ(D K+K ) − Γ(D0 π+π )+Γ(D π+π ) → − → − → − → − • LHCb measured: =[ 0.82 0.21(stat) 0.11(syst)]% ACP − ± ± • Clear different between behaviour of D0 and D0̅ !

http://lhcb-public.web.cern.ch/lhcb-public/ 19

Summary

• The CP symmetry describes the difference between matter and anti-matter - almost a good symmetry in the weak interactions. 0 0 0 0 • Small amounts of CP violation observed in K B D Bs through decays and mixing. • Three types of CP violation: 1.Direct CP violation in decay amplitudes 2.CP violation in neutral meson mixing 3.Indirect CP violation due to interference of mixing and decay. • CP violation is accommodated in the Standard Model through a complex phase in the CKM matrix. • The unitarity triangle of the CKM matrix is used to understand observation of the CP violation, and see if measurements are consistent. • The amount of CP observed in the Standard Model not enough to explain the matter ! anti-matter asymmetry of the universe.

• Friday: neutrino oscillations!