( ) ( ) K t K t t H (27.1) In this basis the Hamiltonian is a 2x2 matrix with the form, H (27.3) = M i Γ,

Transcription

1 Physics 557 Lecture 7 The Neutral ao System: Techical Details I this Lecture we wat to make coectio to some of the more formal aspects of the behavior of the eutral kao system We will study the time depedece of the system i Hamiltoia otatio (ie this system ca be usefully aalyzed as a -state Q system t t i = t ( t ( t H (7 where i pricipleh icludes all relevat iteractios eg the strog electroweak ad Yukawa iteractios H = HStrog HYukawa HE H (7 I this basis the Hamiltoia is a x matrix with the form H (73 = i Γ where ad Γ are agai x matrices Due to the decays the full Hamiltoia is o loger Hermitia but the idividual compoets ad Γ are Hermitia Cosider first the limit withh = Sice are eigestates of strageess which is coserved by all iteractios except the weak iteractios we have Strog Yukawa E = H H H (74 Thus i this limit strageess is coserved ad there are o decays H H (75 where due to CPT the diagoal terms are equal = So what chages as we tur o the weak iteractios? Now the eutral kaos ca both decay ad mix Lecture 7 Physics 557 Witer 6

2 If we assume that CPT is still a good symmetry as it is i the Stadard odel the diagoal elemets are still equal ad we have iγ iγ = * * i i Γ Γ H (76 If we also focus iitially o the limit where CP (which exchages coserved this matrix must be symmetric ad we have * * = = = Γ = Γ = Γ Γ = Γ * * is (77 ie both ad Γ are real The resultig eigevalues of the CP coservig problem are i Γ i ( Γ Γ = i Γ i Γ Γ where we have chose the sigs to match the defiitios of the previous lecture The rotatio matrix that performs the diagoalizatio is (78 U = (79 so that we have as oted above UU U U = Γ Γ Γ = Γ Γ (7 The correspodig eigestates are as defied i the previous lecture Lecture 7 Physics 557 Witer 6

3 t t t = U = ( t ( t ( t CP = ± (7 The relevat dimesioless parameters to describe the resultig mixig ad oscillatio which we discussed i the last lecture are typically chose to be Γ Γ Γ Γ x = = y = = Γ Γ Γ Γ Γ Γ (7 For the case of kaos we have Γ 997 Γ = ± Γ Γ Γ Γ y ev x 945 (73 The first umber tells us that the first eigestate decays much more rapidly tha the secod while the secod results says that the oscillatio time is comparable to the shorter of the two decay times Thus as we oted i the previous lecture we essetially observe oly a sigle oscillatio The domiat physics is the decay of the first eigestate The correspodig umbers for the D ad B systems are System x y ± 5 77 ± 8 D D B B 77± 8 < % From these umbers we coclude that we will ot be able to observe mixig i the D system but should see both mixig ad real oscillatios i the B system So where do these umbers come from? Cosider first just the compoet ad keep terms through secod order i the weak iteractio Lecture 7 3 Physics 557 Witer 6

4 = H = H H H E E (74 As oted above H is ivariat uder CPT i the stadard model ad it follows that H = H H = CPT H CPT - * H = H = (75 I the last expressio the mius sig follows from our choice of the CP phase of the eutral kaos ad the prime o remids us that the spis are all flipped due to T But the equatio above icludes a sum over all states So through secod order the diagoal terms are still idetical Thus = ad both are real sice is Hermitia as oted above Agai sice is Hermitia we also have real off-diagoal cotributios = = * = H H H E (76 SiceH has S the first order off-diagoal term vaishes ad we have oly the secod order term Thus to this order we ca express the mass splittig as H H = E (77 I a -geeratio world with o CP violatio the relevat itermediate states i the laguage of the elemetary degrees of freedom are either W s or the various Lecture 7 4 Physics 557 Witer 6

5 combiatios of( u u ( u c ( c u ( c c as suggested i the followig figures (where with the choice above time rus from left to right but this caot matter with T coserved For ow let us forget the issues of color ad the kao wave fuctio (ie cofiemet issues ad cosider the perturbative form of the box diagrams exhibited above Appropriately iterpreted (ie igorig extra quark-atiquark pairs these are the squares of the quark diagrams for the decay processes discussed i the previous lecture The sum over ow becomes a cotiuous itegral over the mometum k flowig aroud the box Formally the cotributio to the real part arises from the pricipal value of the itegral while the cotributio to the imagiary part Γ comes from the discotiuity (imagiary part arisig from real physical itermediate states (ie the states that the ca actually decay ito First cosider the u u itermediate state The matrix elemet correspodig to the secod figure has the form (recall that there are 4 flavor chagig vertices ad a i factor for the loop 4 g g d k i siθ i cosθ i uu C C u s u d ( π σ σ k m u g k k udγ ( γ 5 i γ ρ ( γ 5 us i k mu k W αρ α ρ k m u g k k W vdγ α ( γ 5 i γ σ ( γ 5 v s i k mu k W 4 W (78 where we have made the further approximatio of settig all the exteral mometa to zero This is a reasoable approximatio sice they will be kept small compared to W by a appropriate wave fuctio for the Notice that due to the usual properties of the helicity projectio operators we ca igore the terms with the quark mass i the umerator Oly the k terms will cotribute We ext otice that the itegral appears to be quadratically diverget i the UV This troublig feature is Lecture 7 5 Physics 557 Witer 6

6 fixed whe we iclude the GI mechaism ie sum over the various possible quarks i the itermediate state I our model world with -geeratios this yields the followig factor arisig from the mixig agles ad fermio propagators uu cu uc cc si θ cos θ si θ cos θ si θ cos θ C C C C C C ( k m ( k mu ( k m u c ( k mc ( mc mu θc θc ( k mu ( k mc = si cos (79 The itegral is ow well behaved i the UV (ad the IR I fact if the quarks were degeerate i mass (m u = m c the cotributio of these box diagrams would vaish (A similar GI cacellatio also occurs whe we iclude the full 3-geeratio structure of the C mixig matrix as we will discuss below To leadig order i σ the small expasio parameter mc W we ca focus o the just the g g αρ terms from the W propagators We ow have si θ cos θ 6 udγ µ ( 5 us v ρ ν γ γ ρ γ dγ γ γ ( γ 5 v s Iµν 4 C C uu cu uc cc qq ig mc mu (7 So fially the itegral has the form I µν = 4 µ ν d k k k (7 4 ( π ( k mc ( k mu ( k W I the limit W mc mu oe ca show (see the Appedix that this itegral has the value (ote that the itegral must be proportioal to g µν sice that is the oly possible tesor µν µν g I 4 π i m Thus we have (7 64 W c Lecture 7 6 Physics 557 Witer 6

7 qq g 4 si θ cos θ C C c π W ρ ν u u γ γ γ γ v γ γ γ γ v d ν ρ 5 s d 5 s m (73 We ca further simplify this expressio with the followig useful spior idetity (where we iclude a related idetity for completeess ote the order of idices - you are ecouraged to check these ρ ν ν { γ γνγ ρ ( γ 5 }{ γ γ γ ( γ 5 } = 4{ γν ( γ 5 }{ γ ( γ 5 } ρ ν ν { γ ργνγ ( γ 5 }{ γ γ γ ( γ 5 } = { γν ( γ 5 }{ γ ( γ 5 } 6 (74 Thus we obtai a factor of 4 ad remove 4 of the 6 gamma matrices With a appropriate Fierz trasformatio we ca put the cotributio of the secod box diagram the WW itermediate state i a idetical form ASIDE: Fierz trasformatios are discussed for example i Chapter 3 of Peski ad Schroeder especially exercise 36 I geeral they allow us to rearrage the order of the spiors i matrix elemets of iterest The characteristic structure is A B AB C D ( uγ u ( u3γ u4 = CCD ( uγ u4 ( u3γ u A Γ = 5 µ { γ γ } CD (75 Naively addig the two (idetical cotributios leads to aother factor of two However if we ow iclude the color quatum umber for the quarks ad the fact that the icomig ad outgoigqq states are color siglets (the quark ad atiquark have the same color ad the states are properly ormalized we fid that the WW box has weight but theqq box has weight of oly /3 for a total factor of 4/3 rather tha ASIDE: To see this color factor thik of the color siglet meso wave fuctio as havig two color idices oe for the quark ad oe for the atiquark The expected form is ψ δ ab ab 3 = (76 Lecture 7 7 Physics 557 Witer 6

8 where the roecker delta fuctio sets the colors equal (except of course if the quark is red the ati-quark is redbar ad the /3 gives the correct ormalizatio for each possible color The box graph with the WW itermediate states correspods to the quark aihilatig with the atiquark from the same meso (iitial or fial state ad yields a color factor F = ψ ψ δ δ = δ δ = = (77 * WW cd ab cd ab cd ab a b c d 9 a b c d 9 a c I cotrast for theqq itermediate state the quark ad ati-quark color flows across the diagram ad coects the icomig state with the outgoig oe The correspodig factor looks like F = ψ ψ δ δ = δ δ δ δ = = (78 * qq cd ab ac bd ac bd cd ab a b c d a b c d a ie the color coectio rus all the way aroud through the wave fuctios ad allows oly oe free sum over colors If we iclude these color factors ad substitute for g i terms of G F ie g = 4 G F W we obtai si θ cos θ 6π u γ γ v γ γ v C C qq WW GF m c ν u d ν 5 s d 5 s (79 Fially we replace the spiors with the usual creatio /aihilatio operators ( s d ad express the effective Hamiltoia as a 4-fermio term si θ cos θ H 6π dγ γ s dγ γ s C C eff = qq WW GF m c ρ ρ 5 5 (73 Thus we ca write = H (73 eff Lecture 7 8 Physics 557 Witer 6

9 where the factor i the deomiator is to accout for the relativistic ormalizatio of the bra ad ket Now comes the hard part how do we evaluate this expectatio value? Oe of the goals of lattice QCD is to be able to accurately calculate just such quatities That umerical task is essetially doe but here we proceed aalytically usig the followig (iitially surprisig approximatio I the 4-fermio operator above we imagie isertig the uit operator as represeted by a sum over states = ad the approximate the sum by keepig just the vacuum! What a approximatio!? It s called the vacuum isertio approximatio Actually it is ot so crazy Due to the virtual W s i the box graphs we have bee studyig the 4 weak iteractio vertices are all withi a volume characterized by a radius r ~ W which is quite small o the scale over which hadroic wave fuctios vary ie r ~ m π So it is ot so crazy to evaluate the matrix elemet i terms of the probability to fid the quark ad ati-quark at the same poit ie the wave fuctio at the origi (squared just as we did for the leptoic decays which is just what the projectio oto the vacuum state does So we use the idetificatio ( γ ρ ρ dγ s i f q (73 5 with the decay costat defied as we did above (ad for the pio Without FCNC we caot directly measure this quatity i leptoic decays of the but we ca measure the correspodig quatity for the charged kaos The we ivoke the fact that the strog iteractios coserve strog isospi to justify usig the same umber ie the same wave fuctio for the eutral (ote that we have already icluded the correct color factors for the two forms of the box diagram Thus we ow have ρ dγ γ s dγ γ s = f m (733 ρ 5 5 ad fially GF mc f si cos m θc θ (734 c 3π Pluggig i umbers m = 498 ev si θ 49 G F = 66 ev f m π 7 ev C cos θc 95 mc 5 ev we fid Lecture 7 9 Physics 557 Witer 6

10 68 ev (735 which works (ie agrees with data at the factor of level! This formalism was i fact used by Gaillard ad Lee i 974 to estimate that the mass of the charmed quark was ~ 5 GeV before it was see I the (real world of 3 geeratios we should also iclude the cotributio of the top quark For this cotributio we make the followig replacemet i terms of the C mixig matrix elemets * * c θc θc c cs cd t ts td m si cos m V V m V V (736 The charm quark cotributio (ie the quatity above is of order ~ GeV while due to the strog suppressio withi the mixig matrix the top cotributio is of order ~ 3 GeV (Strictly speakig we should also keep the correctios of ordermt W ad higher that we igored for the charm quark These are ow of order or but will ot make up for the mixig suppressio To see the suppressio more explicitly ote that we ca characterize the magitudes ad sigs of the C matrix elemets (igorig the phase as " V " 3 3 (737 where the rows are labeled (dsb ad the colums (uct ad ~ Thus the * charm mixig factor is V V ~ 5 while the top quark factor is V V ts * 7 td ~ ~ 3 cs cd This large suppressio more tha compesates for factor of 4 x 4 i the mass ratio squared It also qualitatively explais why the CP violatig cotributio to which arises from the phase i the top term is a small effect If we keep the phases the mixig matrix looks crudely like Lecture 7 Physics 557 Witer 6

11 Lecture 7 Physics 557 Witer i i e V e δ δ (738 We will pursue the subject of CP violatio i the ext lecture