Lecture 16 V2. October 24, 2017

Transcription

1 Lecture 16 V2 October 24, 2017 Recap: gamma matrices Recap: pion decay properties Unifying the weak and electromagnetic interactions Ø Recap: QED Lagrangian for U Q (1) gauge symmetry Ø Introduction of hypercharge and U Y (1) gauge symmetry Ø The EWK interaction from U Y (1)xSU L (2) gauge symmetry (See Lecture 17 for a better derivation of this) Some history 1

2 Class Plans Date Lecture Essay presentation (15 minutes) Oct Oct Minyu Lepton flavor universality tests Oct (HW3 due) Erin Neutrino-less double beta decay Nov Tianyu Hydro model of QGP Nov Long Detection of axions Nov Baran Neutrino mass from tritium decay Nov Sourav The strong CP problem Nov Michael Composite Higgs models Nov Ping Family/flavor symmetry in the SM Nov. 23 Thanksgiving break Nov Nov (last class) 2

3 Review of Pauli and gamma matrices Pauli matrices: apple apple I = = 1 0 i j + j i =2 ij j = 1,2,3 2 = apple 0 i i 0 3 = apple i = i Dirac or Weyl (chiral) matrices: In both cases i = apple 0 i i 0 µ + µ =2g µ µ =0, 1, 2, 3 apple I 0 Dirac 0 = 0 I apple 0 I Weyl 0 = I 0 In both cases 5 = i Dirac 5 = Weyl 5 = apple 0 I I 0 apple I 0 0 I I am using the Dirac representation (as in the handout) 3

4 Spin 0! l + l l 2 p P L u 1 = 1 2 E + m 6 4 l p 1 E+m 0 1+ p E+m l l forbidden by angular momentum conservation allowed by angular momentum conservation suppressed by P L projection of + ½ helicity state = u 1 5 =) suppression factor by [1 p E+m ] Matrix element suppression of electron decay/muon decay = 1 p p 1 E+m e E+m µ = m 1+ e m µ mµ m 1+ m µ m =8.5x10 3 and T (!e e 2 T (!µ µ (8.5x10 3 ) 2 = 7x

5 EWK unification (one generation, all masses = 0) 5

6 General Plan We have developed separate QED and trial weak theories by demanding invariance of the interaction (the Lagrangian) under the local gauge transformations U Q (1) and SU L (2). So let s start with these Lagrangians and attempt to connect the EM and weak forces therefore developing a unified electroweak theory. This will involve introducing four gauge vector fields: B µ and W iµ where i = 1,2,3. l The fields B µ and W iµ will then be related to the real force fields describing the electromagnetic and weak interactions: A µ (the photon), W +, W -, and Z o bosons. In the process of developing the theory, the EM and weak coupling will both be expressed in terms of one coupling strength α e. 6

7 Recap: the QED Lagrangian using U Q (1) gauge invarinace L QED First review the QED Lagrangian. Include in it both members of a generic weak interaction doublet. apple apple apple 1 u e = = or d e 2 Set the fermion mass = 0 as required for the parity-violating weak interaction. Require invariance under U Q (1) = exp[ ie (x)q] to get L QED Here Q i = Q i i with the electric charge of i =eq i. I choose to use the notation Q i for the fractional charge instead of f i. As usual e = p 4 e 7

8 Recap: the QED Lagrangian using U Q (1) gauge invariance L QED As derived in L9, the QED Lagrangian is: L QED =- 1 4 F µ F µ + i(~c) i µ i -e i µ Q i i A µ The first two terms are the kinetic energy of the the fermions and the boson (photon) field A µ. The last term is the one of interest, specifying the interaction of the fermion fields ψ i with the vector field A µ. What ever is done to obtain electroweak unification must preserve this result as it is the most precisely tested part of the SM. 8

9 The hypercharge Lagrangian using U Y (1) gauge invariance L Y Introduce a generalized U Y (1) gauge symmetry based on an operator Y that will be called hyper-charge. U Y (1) = exp[ i g0 2 (x)y] The weak doublet member Li is an eigenstates of Y =) Y Li = Y i Li where Y i (not bold) is the eigenvalue for the state Li. Introduce a massless vector field B µ 0 that interacts with the weak doublet fermions ψ i in a manner analogous to the development of QED using the U Q (1) gauge symmetry. Why do this? The reason is that it introduces a flexibility that will allow the unification of QED and the weak interaction, namely U Y (1) x SU L (2) invariance. What exactly the hypercharge operator Y is will emerge from this process. 9

10 The hypercharge Lagrangian using U Y (1) gauge invariance L Y Obtain the Lagrangian for hyper-charge by copying that obtained for QED with Q à Y/2 and e à g. L Y =- 1 4 B µ B µ + i(~c) i µ i - g 0 i µ Y 2 i B 0 µ As for QED, first terms are the kinetic energy of the the fermions and (abelian) boson field field B µ 0. The last term is the one of interest, specifying the interaction of the fermion fields ψ i with the vector field B µ0. ( Looking ahead: the field B µ 0 will turn out to be a mixture of the electromagnetic vector field A µ and the real neutral electroweak boson field Z µ.) 10

11 The weak interaction Lagrangian using SU L (2) gauge invariance Look back into L14 and 15 for the SU L (2) invariant weak Lagrangian. It can be written as: L weak =- 1 4 W aµ W µ a + i(~c) i µ@ µ i - g w j µ [T al ] jk k W aµ Two simplify the expression, and anticipate the the EWK unification, I define T al = T a [ 1 2 (1 5)] with as usual T 1 = 1 2 apple T 2 = 1 2 apple 0 i i 0 T 3 = 1 2 apple and the [T a ] jk just the jk-th element of the 2x2 T a matrix. The term of interest here is the last one expressing the interaction between the weak doublet fermion fields ψ i with the three vector field W aµ. 11

12 The weak interaction Lagrangian using SU L (2) gauge invariance As discussed in L14, the form of the interaction terms can be simplified by introducing the charged and neutral vector fields: W ± µ =(W 1µ iw 2µ )/ p 2 W 0 µ = W 3µ Here W µ ± are the charged vector fields for the weakly interacting W + and W - bosons. The neutral field is related to the both the neutral weak boson Z and the photon field. Exactly how this occurs is obtained from the EWK unification. 12

13 The weak interaction Lagrangian using SU L (2) gauge invariance Expressing the interaction terms in the weak Lagrangian on page 13 in terms of simplifies to: W ± µ and W 0 µ g w p2 1 µ 2L W + µ g w p2 2 µ 1L W µ g w i µ [T 3L ] ii i W 0 µ The [T 3L ] ii are eigenvalues of the operator T 3L acting on the electroweak doublet members ψ i. Show that they are: [T 3L ] ii T 3Li =+ 1 2 for 1L and 0 for 1R = 1 2 for 2L and 0 for 2R 13

14 Unification with U Y (1) x SU L (2) gauge invariance 14

15 The electroweak Lagrangian Using U Y (1) x SU L (2) gauge invariance Collecting all the above together, the trial EWK Lagrangian is: L EWK =- 1 4 W aµ W a µ B µ B µ + i(~c) i µ@ µ i g w p2 1 µ 2L W + µ g w p2 2 µ 1L W µ - g w i - g 0 i µ T 3Li i W 0 µ µ Y i 2 il B 0 µ 2 charged fields 2 neutral fields A fundamental requirement is that this must be able to reproduce the predictions for QED obtained from U Q (1) invariance. That is the electromagnetic interaction must be: -e i µ Q i il A µ Imposing this requirement leads to the SM (massless) EWK theory. 15

16 The electroweak Lagrangian using U Y (1) x SU L (2) gauge invariance The key is to realize that the two massless neutral fields W o µ and Bo µ can mix with each other to produce two new fields. With great perception, we will call the new fields A µ and Z µ. A general mixing that preserves normalization introduces one apple apple apple parameter: Zµ cos w sin = w W 0 µ A µ sin w cos w Bµ 0 In SM jargon, θ w is called the Weinberg angle. At this point a free parameter. The procedure is to replace the fields W o µ and Bo µ by A µ and Z µ and then adjust θ w is to recover the predictions of QED. W 0 µ = cos w Z µ + sin w A µ B 0 µ = sin w Z µ + cos w A µ 16

17 The electroweak Lagrangian using U Y (1) x SU L (2) gauge invariance Substitute W o µ and Bo µ in terms of A µ and Z µ into the neutral current terms in on page 17. The result is: i i L EWK µ [ g 0 Y cos i w 2 + g wsin w T 3Li ] i A µ µ [-g 0 Y sin i w 2 + g wcos w T 3Li ] i Z µ where everything in brackets acts only on ψ il states. Compare this to the the QED prediction for the interaction between charged fermions and the field A µ. -e i µ Q i il A µ Equate the two expressions for the current coupling to A µ for Li. eq i = g 0 Y cos i w 2 + g wsin w T 3Li 17

18 The electroweak Lagrangian using U Y (1) x SU L (2) gauge invariance Here is the critical step that produces the EWK unification. Obtain the equality by substitution: eq i = g 0 cos w Y i 2 + g wsin w T 3Li g 0 = e/cos w and g w = e/sin w This now defines the hypercharge operator that was introduced with the U Y (1) symmetry. Q i = T 3Li + Y i /2 For left-handed fermion states This is called the Gell-Mann - Nishijima relation. 18

19 The electroweak Lagrangian using U Y (1) x SU L (2) gauge invariance Since eigenvalues Q i and T 3Li are known for each particle in the weak doublets, then Y i can be calculated for each left-handed member of the weak doublets. Q i = T 3Li + Y i /2 Q T T 3L Y ν el 0 1/2 +1/2-1 e L -1 1/2-1/2-1 u L +2/3 1/2 +1/2 +1/3 d L -1/3 1/2-1/2 +1/3 19

20 The electroweak Lagrangian using U Y (1) x SU L (2) gauge invariance The structure of the EWK interactions is determined by the numbers on the previous page, emerging from the requirement of U Y (1)xSU L (2) symmetry, plus two measured parameters: the electromagnetic coupling strength e = p 4 e = the Weinberg angle usually quoted as sin 2 W = The SM predictions are completely defined for the interactions of the fermions in the two weak doublets apple apple apple 1 u e = or 2 d e with the photon and the W +, W - and Z 0 weak bosons. 20

21 The electroweak interaction using U Y (1) x SU L (2) gauge invariance Collecting all terms (see Lecture 17 for the final steps in obtaining this result): - g w 2 p 2 - g w 2 p 2 -e i 2 µ (1 5 ) 1 W + µ 1 µ (1 5 ) 2 W µ µ Q i i A µ V-A parity violating charged weak currents parity conserving neutral (QED) current µ [T 3Li x w Q i ] i Z µ -g z i parity violating neutral weak currents where x w sin 2 w = (at M z ), w = 28.7 o e= p 4 e = (at M z with = 1/128) g w = e/sin w = g z g 0 /sin w = e/(sin w cos w ) =

22 Discovery of weak neutral currents This postulated EWK theory was known before there was an observation of a W or Z boson. One of the unique predictions was that neutrinos should scatter off electrons: µ + e! µ + e should occur from the neutral current exchange of a Z boson. µ µ This was observed using a neutrino beam sent into a large heavy-liquid bubble chamber called Gargamelle in 1973, providing the first (indirect) evidence for a neutral weakly interacting gauge boson. e Z e 22

23 Discovery of the W and Z bosons The W and Z bosons were directly observed as particles in This was done at CERN using the Super-Proton Synchrotron that could collide protons and antiprotons at a cm energy of 540 GeV. 23

24 Next Lecture: Replay of EWK unification derivation Example calculations 24