1. P780.02 Spring 2003 L7Richard Kass Neutral Kaons and CP violation Since CP is not conserved in neutral kaon decay it makes more sense to use mass (or lifetime) eigenstates rather than |k 1 > and |k 2 >: There can be two types of CP violation in k L decay: indirect (“mixing”): k L  because of its k 1 component direct: k L  because the amplitude for k 2 allows k 2  It turns out that both types of CP violation are present and indirect >> direct! Long lifetime state with  L  5x10 -8 sec. Short lifetime state with  S  9x10 -11 sec.  is a (small) complex number that allows for CP violation through mixing. Review: The strong interaction eigenstates (with definite strangeness) are: If S=strangeness operator then: They are particle and anti-particle and by the CPT theorem have the same mass. Experimentally we find:

2. P780.02 Spring 2003 L7Richard Kass Neutral Kaons and CP violation The k L and k S states are not CP (or S) eigenstates since: In fact these states are not orthogonal! If CP violation is due to mixing (“indirect”only) then the amplitude for k L  is: While the amplitude for k L  is: From experiments we find that |  | = 2.3x10 -3. However, the standard model predicts a small amount of direct CP violation too!

3. P780.02 Spring 2003 L7Richard Kass Neutral Kaons and CP violation The standard model predicts that the quantities  +- and  00 should differ very slightly as a result of direct CP violation (CP violation in the amplitude). CP violation is now described by two complex parameters,  and , with  related to direct CP violation. The standard model estimates Re(  /  ) to be (4- 30)x10 -4 ! Experimentally what is measured is the ratio of branching ratios: After many years of trying (starting in 1970’s) and some controversial experiments, a non-zero value of Re(  /  ) has been recently been measured (2 different experiments): Re(  /  )=17.2  1.8x10 -4 At this point, the measurement is more precise than the theoretical calculation! Calculating Re(  /  ) is presently one of the most challenging HEP theory projects.

4. P780.02 Spring 2003 L7Richard Kass Neutral Kaons and Strangeness Oscillations Here we want to consider how the neutral kaon state evolves with time. This will allow us to measure the K L, K S mass difference and the phases of the CP violation parameters:  +- and  oo. How a two particle quantum system evolves in time is covered in many texts including: Particle Physics, Martin and Shaw section 10.2.5 Introduction to High Energy Physics, Perkins Introduction to Nuclear and Particle Physics, Das and Ferbel The Feynman Lectures, Vol III, section 11-5. Lectures on Quantum Mechanics, Baym, Ch 2. Note: Feynman's lectures were written in about 1963, before there was good experimental data on this topic! For example, CP violation would not be discovered for another year. While the following derivation is for the neutral kaon system it is also applicable to B-meson oscillations and neutrino oscillations. First, consider the case where CP is conserved. The CP eigenstates are |K 1 > and |K 2 > and they are solutions to the time dependent Schrodinger equation:

5. P780.02 Spring 2003 L7Richard Kass Neutral Kaons and Strangeness Oscillations Here H eff is a phenomenological Hamiltonian that describes the system. Since our particles can decay, H eff is not a hermitian operator! Since there are two states in this problem it is customary to describe H eff by a 2x2 matrix (e.g. see Das and Ferbel Chapter XII) and |K 1 > and |K 2 > by column vectors. The H eff matrix is written in terms of the masses (m 1, m 2 ) and lifetimes (  1,  2 ) of the two states: The eigenvalues and eigenvectors of H eff are given by: f(t) contains the time dependence with: In this representation the states are orthogonal to each other: = =0. The solutions to the Hamiltonian are:

6. P780.02 Spring 2003 L7Richard Kass Neutral Kaons and Strangeness Oscillations Consider an experiment which produces a beam of pure K 0 ’s (at t=0) using the strong interaction,  - p  K 0. From the previous lecture we saw that we can express a K 0 in terms of a mixture of K 1 and K 2 states (and visa versa): Using the time dependent solutions for K 1 and K 2 we can find the time dependent solution for K 0 : The amplitude for finding a K 0 in the beam at a later time (t) is given by:

7. P780.02 Spring 2003 L7Richard Kass Neutral Kaons and Strangeness Oscillations The probability to find a K 0 in the beam at a later time is given by: To make the units come out right in the cos term substitute (m 2 -m 1 )c 2 for (m 2 -m 1 ). The third term in the above equation is an interference term and it causes an oscillation that depends on the mass difference between k 1 and k 2.

8. P780.02 Spring 2003 L7Richard Kass Neutral Kaons and Strangeness Oscillations We could also calculate, using the same procedure, the probability that a beam initially consisting of K 0 ’s contains K 0 ’s at a later time: Note: the sign of the interference term is now “-”. Since the strangeness of a K 0 differs from a K 0 ’s the strangeness content of the beam is changing as a function of time. This phenomena is called strangeness oscillations. More generally we call this phenomena flavor oscillations since it also occurs with B-mesons (b quark oscillations) and neutrinos (e.g. e   ). We can measure the strangeness content of a beam as a function of time (distance) by putting some material in the beam and counting the number of strong interactions that have S=+1 in the final state vs. the number with S=-1. S= +1 S= -1 Strangeness oscillations for an initially pure K 0 beam. A value of (m 2 -m 1 )  1 =0.5 is used.

9. P780.02 Spring 2003 L7Richard Kass Neutral Kaons, Flavor Oscillations, & CP Violation CP violation requires modifying H eff to include additional complex parameters. The details of the derivation are given in many texts, e.g. Weak Interactions of Leptons and Quarks by Commins and Bucksbaum. From an experimentalist's point of view a good quantity to measure is the yield (as a function of proper time, time measured in the rest frame of the K) of  +  - decays from a beam that is initially K 0. Since we are measuring the sum of the square of the amplitude |K L  +  - > +|K S  +  - > there will be an interference term in the number of  +  - decays/time (  I(t)). The yield of  +  - decays is given by: Thus by measuring this yield we gain information on the mass difference as well as the CP violation parameters  +- and  +-.  sec x 10 -10 5 15 Event rate for  +  - decays as a function of proper time. The best fit requires interference between the K L and K S amplitudes. m 2 -m 1 =3.491  0.009x10 -6 eV |  +- | =(2.29  0.01)x10 -3  +- =(43.7  0.6) 0  S =0.893x10 -10 sec  L =0.517x10 -7 sec