Nuclear and Particle Physics - Lecture 16 Neutral kaon decays and oscillations

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1 1 Introction Nclear an Particle Phyic - Lectre 16 Netral kaon ecay an ocillation e have alreay een that the netral kaon will have em-leptonic an haronic ecay. However, they alo exhibit the phenomenon of mixing, in a very cloe analogy with the netrino. Netral kaon The cro-generation weak interaction relt in the non-nit Cabibbo matrix an thi allow the an to ocillate; the an tate can change into each other via a econ orer weak proce. ct ct ct ct Hence, nlike for the netrino cae, we nertan the interaction connecting them an o can preict the vale of the mixing angle (which for netrino we wrote a θ. The phyical tate with well-efine mae are calle S an L an thee are mixtre of an, jt a for the netrino. Again, there i a ma plitting between the tate, L S 15 m = 3.5 x 1 GeV Becae thi i a low energy, econ orer weak effect, the ma plitting i mall an i aron GeV/c, which i even le than for the netrino. The kaon cae ha imilaritie an ifference with netrino. The kaon ecay, o the overall ocillation i ampe with time by the exponential ecay contant. e know the interaction which connect the tate an o know (to a goo approximation what the mixing angle i; it trn ot to be cloe to 45. The kaon ecay in everal way an thee actally project ot ifferent tate o particlar ecay tell how mch of particlar tate there are a a fnction of time. For the netrino, only the weak interaction tate can be oberve. e will ame for the ret of thi calclation that CP i a conerve qantity, which i not exactly tre bt i a goo approximation. In thi approximation, CP commte with the Hamiltonian an o energy (or in the ret frame, ma eigentate are alo CP eigentate. Hence, thee mixe tate are tate of efinite CP. The an are J P = tate o we know that ˆP =, ˆP = 1

2 Thee two tate are antiparticle of each other o we ll aopt the convention that Ĉ = Reapplying the operator then mt get back to where we tarte, o o thi mean alo. Hence, Ĉ = Ĉ = Ĉ = Ĉ ˆP =, Ĉ ˆP = e nee to fin CP eigentate a thee are the ma eigentate in the limit of CP conerve, o conier the combination S = 1 ( +, L = 1 ( Applying the combine operator, thee give Ĉ ˆP S = + S, Ĉ ˆP L = L Hence, S an L are P, C an CP eigentate, i.e. they are each their own antiparticle an have J P C = an +, repectively, which give CP vale of +1 an 1, repectively. Thee form the phyical (i.e. ma eigentate. Thee are clearly eqivalent to the netrino mixe tate with θ = 45, o co θ = in θ = 1/. Note, we can alo invert the eqation to give = 1 ( S + L, = 1 ( S L 3 Netral kaon ecay One of the intereting thing abot thee tate i how they ecay. Like charge kaon, they can ecay emi-leptonically (meaning to l, ν l an pion or haronically (meaning to pion. The emileptonic ecay are ifferent for the an becae each only give lepton of particlar charge l + ν l π, l ν l π + Thi i obvio from the Feynman iagram for thee ecay + l - l ν l ν l Hence, emi-leptonic ecay project ot the an tate. Thee wol be expecte to have eqal partial with an ince the S an L are eqal mixtre of thee tate, they hol therefore ecay eqally to each charge type.

3 The itation i very ifferent for the haronic ecay, where either two or three pion are kinematically poible. Both the an give the ame final tate qark o they cannot be itingihe from their ecay. Thee will then interfere an we nee to think in term of the S an L irectly. e aw in previo lectre that the parity of a pion i 1 an that of an orbital anglar momentm tate i ( 1 L. Hence a tate of n pion ha parity P = ( 1 n ( 1 L. For pinle kaon ecay L =, o two pion have P = +1 an three pion have P = 1. hat abot C? The π i J P C = +, o Ĉ nπ = (C π n nπ = + nπ For π + π in L =, Ĉ exchange them; to wap them back give another ( 1L = +1 factor, o Ĉ π + π = + π + π alo. Finally, for π + π π, thi i imilar to the above bt with an extra π which give an extra C π = +1, o for all cae of two or three pion Hence, combining the two operation, then Ĉ nπ = + nπ Ĉ ˆP π = + π, Ĉ ˆP 3π = 3π e are aming CP i conerve; hence the only ecay allowe are S ππ, L πππ Hence, the haronic ecay project ot the S an L tate. Becae the ma of the three pion i cloe to that of the kaon, the phae available for the two pion ecay i mch bigger an the partial with are therefore very ifferent Γ( S ππ = GeV, Γ( L πππ = GeV In fact, the two pion ecay ominate the S ecay an o the emi-leptonic ecay are well below 1%. In contrat, the three pion ecay of the L are comparable with the emi-leptonic ecay. Thi alo relt in the S lifetime being mch horter than the L lifetime τ S = , τ L = which are ifferent by a factor of aron 6. Hence a pre S wol ecay away mch qicker than a pre L beam. However, whichever way we make kaon, we cannot make pre S or L beam. e alway make them via trong or EM interaction, ch a π p Λ 3

4 where the Λ efinitely contain an qark o the mt have forme a, not a. Hence, when proce, the kaon are initially in either a or a tate. e nee to look at what happen with time. For a imple ecaying particle, the wavefnction in it ret frame goe a o that ψ e imc t/ h e Γt/ h = f(t ψ e Γt/ h = e t/τ a reqire. An initial can be ecompoe a ψ( = = 1 ( S + L The S an L are the tate with efinite mae an with, o the tate at a later time i ψ(t = 1 (f S (t S + f L (t L Conier the ecay to pion. At time t, the intenity of S in the beam i imply f S(t = e Γ1t/ h / an imilarly for L, o we ee half the proce particle ecay rapily to two pion an ome fraction of the ret ecay lowly to three pion. The ecay to lepton are trickier; we nee to project ot the an tate a thee ecay to the ifferent emi-leptonic moe. Thi i more like what we i for netrino. Hence, we write ψ(t = 1 [ ( f S (t + ( + f L (t ] = 1 [ (f S (t + f L (t + (f S (t f L (t ] Therefore, the intenity of, an hence the l + ν l π ecay, i given by 1 4 f S(t + f L (t = 1 [ ] f S (t + f L (t + f S (t f L (t + f L (t f S (t 4 = 1 [ ] e Γ1t/ h + e Γt/ h + e (Γ 1+Γ t/ h co( mc t/ h 4 4

5 where m = m m 1. The intenity to an hence the l ν l π + ecay, i imilar bt the ign of the coine term change. The interference change a a fnction of time becae the e imt term are lightly ifferent (by m an o the eparate part lowly rift in an ot of phae. The rate of each of thee two ecay actally go p an own with time. The al meare i the aymmetry, ince efficiencie, etc., cancel. Thi allow m to be meare. Effectively, we are comparing it with Γ rather than m, o thi i often expree a x = m Γ 1 =.4738 ±.9 5

6 4 CP violation The amption we mae that CP i a ymmetry i actally not tre, becae the CM matrix i not prely real. The obvio place to look for ch a effect i for the ecay S πππ an L ππ. The former i har to ee a the S i alway proce with L an the three pion ecay i har to ee given the large backgron from the CP conerving L ecay. The ame wol be tre for the L ππ ecay, except that the S lifetime i mch horter. Hence, by waiting long enogh, all the S meon ecay away. It i then traightforwar to ty the ecay of the L an inee, ecay to two pion are een with a branching fraction of.%. Thi in fact wa the firt obervation which emontrate CP violation back in In aition, CP violation alo mean the rate of L l + ν l π, L l ν l π + are not exactly eqal. The aymmetry plot above oe not in fact go exactly to zero for long time, bt ha a vale.3%. Thi mean a L i more likely to ecay to an l+ than an l ; a clear example of a matter-antimatter ifference. Sch ifference, which arie becae of CP violation, are thoght to be the reaon why the Univere ha more matter than antimatter. Since then, the only other ytem which ha hown CP violation i the B ytem, which alo ocillate like the. CP violation in B ecay wa only firt oberve in 1. 6