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Lecture 12: Electroweak

Lecture 12: Electroweak. Kaon Regeneration & Oscillation The Mass of the W The Massless Photon & Broken Symmetry The Higgs Mixing and the Weinberg Angle The Mass of the Z Z Decay. Useful Sections in Martin & Shaw:. Chapter 9, Chapter 10. Regeneration.

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Lecture 12: Electroweak

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  1. Lecture 12: Electroweak • Kaon Regeneration & Oscillation • The Mass of the W • The Massless Photon & Broken Symmetry • The Higgs • Mixing and the Weinberg Angle • The Mass of the Z • Z Decay Useful Sections in Martin & Shaw: Chapter 9, Chapter 10

  2. Regeneration Ko, Kostates of definite strangeness KL KL + KS strong interaction with matter picks out Ko & Ko which then re-mix So what are kaons???  that depends... who wants to know?! K1o, K2ostates of definite CP KSo, KLostates of definite lifetime

  3. Regeneration Ko, Kostates of definite strangeness ''Regeneration" KL KL + KS strong interaction with matter picks out Ko & Ko which then re-mix So what are kaons???  that depends... who wants to know?! K1o, K2ostates of definite CP KSo, KLostates of definite lifetime

  4. Strangeness Oscillation K2o = 1/2 ( KoKo ) K1o = 1/2 ( Ko + Ko ) Ko = 1/2 ( K1oK2o ) AK(t) = 1/2 ( AS(t)  AL(t) ) Strangeness Oscillation: Amplitudes for decaying states KSo and KLo as a function of time are AS(t) = AS(0) exp(imSt) exp(St/2)Sℏ/S AL(t) = AL(0) exp(imLt) exp(Lt/2)Lℏ/L or Ko = 1/2 ( K1o + K2o ) ≃ 1/2 ( KSo + KLo ) ≃ 1/2 ( KSoKLo ) AK(t) = 1/2 ( AS(t) + AL(t) )

  5. Strangeness Oscillation Intensities I(Ko) = 1/2 [AS(t) AL(t)][AS*(t) AL*(t)] = 1/4 {exp(St) + exp(Lt) 2 exp[(S+L)t/2] cosmt } Ko Ko Thus, if we start with a pure Ko beam at t=0, the intensity at time t will be (setting AS(0) = AL(0) = 1) I(Ko) = 1/2 [AS(t) + AL(t)][AS*(t) + AL*(t)] = 1/4 {exp(St) + exp(Lt) + 2 exp[(S+L)t/2] cosmt } and similarly, wherem  mLmS = 3.49x1012 MeV (m/m ≃ 7x1015)

  6. K, W and Z A B C D E F G H I J L M N O P Q R S T U V X Y

  7. Weak Coupling & the W Mass M =  42  / GF CERN, 1983 MW = 80 GeV !!  M ~ 100 GeV Recall that the ''matrix element" for scattering from a Yukawa potential is fVo = g2/(q2+M2) In the Fermi theory of decay, this is what essentially becomes GF or, more precisely, GF/2 = g2/(q2+M2) = 4W/(q2+M2)  GF2 and the relatively small value of GF characterizes the fact that the weak interaction is so weak We can get this small value either by making W small or by making M large  UNIFICATION !! So what if we construct things so W =  ??? Assuming M ≫ q2 ,  = 1/137 GF = 105 GeV2

  8. u u d hadrons e, ,  W- u u d e-, -, - hadrons Stochastic Cooling Electron Cooling p p

  9. Electroweak Interlude A Brief Theoretical Interlude (electroweak theory... at pace!!)

  10. Weak Isospin ( )L ( )L ( )L ( )L ( )L ( )L e e     u d c s t b e e W+ But how can this be the ''same" force when the W’s are charged and the photon certainly isn’t !? Is there a way we can ''bind up" the W’s along with a neutral exchange particle to form a ''triplet" state (i.e. like the pions) ?? Well, like with the pions, we seem to have a sort of ''Weak" Isospin since the weak force appears to see the following left-handed doublets as essentially two different spin states: IW(3) =  1/2(like p-n symmetry) Thus, in the process The W+ must carry away +1 units of IW(3) so let’s symbolically denote W+ and, similarly, W If IW = 1 for the W’s then, similar to the o, there is also a neutral state: Wo 1/2 ( ) (which completes the triplet)

  11. The Higgs  Higgs Mechanism There is, however, another orthogonal state: 1/2 ( ) If we ascribe this to the photon, then perhaps we might expect to see weak ''neutral currents" associated with the exchange of a Wo with a similar mass to the W so we’d have a nice ''single package" which describes EM and weak forces! Hold on... any simple symmetry is obviously very badly broken  the photon is massless and the W’s are certainly not! The photon is also blind to weak isospin and also couples to right-handed leptons & quarks as well Assume the symmetry was initially perfect and all states were massless Then postulate that there exists some overall (non-zero) ''field" which couples to particles and gives them additional virtual loop diagrams : (kind of like an ''aether" which produces a sort of ''drag") but in the limit of zero momentum transfer (rest mass), so represent as

  12. Mixing: the Photon & Z Further suppose that this field is blind to weak isospin and, thus, allows for it’s violation. This would allow the neutral weak isospin states to mix  like with the mesons(the W are charged and cannot mix) We will call the ''pure," unmixed statesWoand And we will call the physical, mixed statesZoand

  13. Masses and Couplings GW GW M2W = W + W GW GG GW GW M2W = W + W +  GG GW GG GG M2 =  +   + Wo M2Wo = GW2 Wo + GW GG M2 = GG2 + GW GG W Think about mathematically introducing this Higgs coupling by applying some ''mass-squared" operator to the initial states (since mass always enters as the square in the propagator) where the right-most terms represent the weak isospin - violating terms Assume couplings to W’s are all the same (GW) but coupling to  may be different (GG ) MW2 = GW2 For the W the mass would then simply be given by (where G2 contains the coupling plus a few other factors) For the latter 2 equations, we can think of M2 as an operator which yields the mass-squared, M2 , for the coupled state:

  14. Massless Photon / Massive Z GW GG (M2-GG2)  = Wo GW2 GG2 (M2-GG2) M2Wo = GW2 Wo + Wo Thus, associate M2 = 0 and MZ2 =GW2 + GG2 From the second of these: Substituting into the first: M4 M2 GG2 = M2 GW2 GW2 GG2 + GW2 GG2 M2 ( M2 GG2 GW2) = 0  M2 = 0orM2 = GW2 + GG2 Note also that MZW

  15. Weinberg Angle & Z Mass W ''Weinberg Angle" tan W = GG / GW ''anomaly condition" ''unification condition" Ql + 3Qq = 0 (leptons) (quarks) MZ = MW/cosW We can parameterize the  as a mixture of Wo and  as follows:  sinWWo cosW M2 = M2 ( sinW Wo cosW) = 0 Thus, applying M2 : 0 = ( GG2+ GW GGW ) sinW  (GW2Wo GW GG cosW GW GGsinW GW2 cosW = 0 Coefficient of Wo GG2sinW  GW GGcosW = 0 Coefficient of  MZ2/MW2 = (GW2 + GG2)/GW2 = 1/cos2W is satisfied separately for each generation

  16. Neutral Current Event Neutral Current Event (Gargamelle Bubble Chamber, CERN, 1973)   p

  17. Z Discovery  MZ = 91 GeV (predicted) Z e+ e MZ = 91 GeV(observed!!) From comparing neutral and charged current rates  sin2W = 0.226 MW = 80 GeV

  18. Flavour-changing neutral currents ( ) ( ) u d cosqC + s sinqC u d´ pre-ABBA weak doublet = = (d cosqC + s sinqC) (d cosqC + s sinqC) u u Z0 Z0 uu + (dd cos2qC + ss sin2qC) + (sd + ds ) sinqC cosqC DS = 0 DS = 1 While we’re here... So, consider the coupling to the Z0 : + Probability ∝ product of wave functions: “Flavour-Changing Neutral Currents”  never seen!

  19. GIM mechanism ( ) ( ) ( ) ( ) c s cosqC- d sinqC u d cosqC + s sinqC u d´ c s´ = = (d cosqC + s sinqC) (d cosqC + s sinqC) u u Z0 Z0 + (s cosqC- d sinqC) (s cosqC- d sinqC) c c Z0 Z0 + + uu + cc + (dd+ss)cos2qC + (ss+dd) sin2qC) + (sd + ds - sd - sd) sinqC cosqC DS = 0 DS = 1 Postulate 2 doublets: (Glashow, Iliopolis & Maiaini: “GIM” mechanism) &

  20. Resonant Cross Section formation ''rate" of initial state Blam ! prob for decay to particular final state given the total number of available states ( ) dP dN Transition Rate W = 0 ( ) ( ) dP dE dE dN  0 = dP 1 f dE 2 (E-E0)2 + 2/4 = ( ) ( ) 1 1 V q2 dq d (2)3 dE dN dE = But recall that W = vB / V ( ) 1 V q2 2 v 0f q2 (E-E0)2 + 2/4  = (recall  = ℏ/)

  21. Relativistic Treatment But this is non-relativistic! From considering scattering from a Yukawa potential (which followed from the relativistic Klein-Gordon equation) we found the ''propagator" 1/(q2 + M2) So consider the diagram: Under a fully relativistic treatment, q is the 4-momentum transfer and, if we sit in the rest frame of the intermediate state, q2 = p2 E2 = E2 Also note that, for a decaying state, the intermediate mass takes on an imaginary component M  M  i /2 since ~ exp(iE0t) = exp(iMt)  exp{i(Mi/2)}t = exp(iMt) exp(t/2)

  22. Relativistic Breit-Wigner 1 (Mi/2)2 E2 1 M2 /4  iM E2 1 M2  iM E2 = ≃ (in the limit ≪ M) ( )( ) 1 M2  iM E2 1 (E2  M2)2 + M2 1 M2  iM E2 = ~ compare 0f q2 (E-E0)2 + 2/4 and we’d expect something like  M20f E2 (E2-M2)2 + M22  = ~  M20f E2 (E2  M2)2 + M22 = CM CM Thus, the propagator goes like And the cross section will be proportional to the square of the propagator : so, roughly, /2  M In fact, a full relativistic treatment yields

  23. The Z Resonance [ ] MZ2 eeX E2 (E2  MZ2)2 + MZ22 (e+e X)= CM CM eeX  BreeBrX= Thus, for the production of Z0 near resonance and the subsequent decay to some final state ''X" : since ee can be related by time-reversal to ee Peak of resonance  MZ Height of resonance  product of branching ratios

  24. Z Decay: Generation Limit ''Invisible modes" Neutrinos !! (limit for light, ''active" neutrinos) Results: hadrons= 1.741  0.006 GeV MZ = 91.188  0.002 GeV Z = 2.495  0.003 GeV ll= 0.0838  0.0003 GeV ≠ 2.495 !! 1.741 + (3 x 0.0838) = 1.9924 So what’s left ???

  25. End To Generation Game An End To The Generation Game ??? (not necessarily a bad thing!)

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