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- Consider π (t=1 and t0=1). The only possible combinaon is π − = ud In general, it is possible to find several linearly independent components corresponding to the same t and t0. The appropriate combinaon is given by rules. Furthermore, the wave funcon must be ansymmetric among the . This problem is similar to that of a two- wave funcon! 1 1 0 − uu dd π = τ − π = ( − ) T=1 triplet: 2 2 π + = − ud

What about the symmetric combinaon? 1 T=0 singlet: η0 = uu + dd 2 ( )

To produce heavier mesons we have to introduce excitaons in the quark- anquark system or invoke s- and other more massive quarks The Eighold Way is a term coined by Murray Gell-Mann for a theory organizing and mesons into octets (alluding to the Noble Eighold Path of Buddhism). The Eighold Way is a consequence of flavor . Since the strong nuclear affects quarks the same way regardless of their flavor, replacing one flavor of quark with another in a should not alter its very much. Mathemacally, this replacement may be described by elements of the SU(3) group. The octets and other arrangements are representaons of this group.

The Dharma wheel (represents the Noble Eighold Path) The lightest strange mesons are or K-mesons. Since s-quark has zero isospin, kaons come in two doublets with t=1/2: {K + (us ), K 0 (ds )}, {K − (us), K 0 (ds)}

Y = A + S +C +B +T 1 Q = −t0 + Y 2 the SU(3) symmetry limit is met for massless u,d,s quarks π − (ud)+ p(uud) → K 0 (ds )+ Λ(uds) is conserved! Pseudoscalar mesons ! J total ! ! ! ! ! ! ! J = ℓ + S, S = s + s ℓ orbital angular momentum q q ! S total

S can be either 0 or 1. The mesons with the relative zero orbital angular momentum are lower in . For the pion, S=0, hence J=0. Consequently, are “scalar” . But what about their ? The parity of the pion is a product of intrinsic parities of the quark (+1), antiquark (-1) and the parity of the spatial wave function is 1. Hence, the pion has negative parity: it is a pseudoscalar . With (u,d,s) quarks, one can construct 9 pseudoscalar mesons (recall our earlier discussion about the number of !): 9 (nonet)=8 (octet)+1 (singlet) Members of the octet transform into each other under rotaons in flavor space (SU(3) group!). The remaining meson, η0, forms a 1-dim irrep. (us ) 1 497.61 π 0 = uu − dd (ds ) 2 ( ) 1 493.68 η = uu + dd − 2 ss 8 6 ( ) 1 η = uu + dd + ss 547.86 134.98 (ud ) 0 3 ( ) (du )

139.57 957.78 In reality, since the SU3 (flavor) symmetry is not exact one, the observed mesons are:

(su ) (sd ) η = η8 cosϑ +η0 sinϑ η' = −η8 sinϑ +η0 cosϑ

ϑ - Cabibbo angle, ~11o for pseudoscalar mesons

are in MeV/c2 The eta was discovered in pion-nucleon collisions at the (LBNL) in 1961 at a me when the proposal of the Eighold Way was leading to predicons and discoveries of new parcles.

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(Q=0)

Hmmm… Those are not CP eigenstates

Long Short

69% 31% The main decay modes of KS are:

… and both decays conserve CP. What about KL ? Three-body decay, very slow!

These decays are called semileptonic decays, producing one meson and two . They account for about

67% of KL decays compared to 33% for the 3π mode. 1m

−11 T(KS)=(8.954±0.004)×10 s

−8 T(KL)=(5.116±0.021)×10 s

Cronin & Fitch experiment, 1964 17 m beamline; KS should not be observable more than a few cenmeters down the beam line

Given the disparity of the lifemes of the two kaon species, you expect to see only the long-lived version at the end of the beam tube, but they found about 1 in 500 decays to be 2-pion decays, characterisc of the short-lived species.