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
Class Plans Date Lecture Essay presentation (15 minutes) Oct. 24 16 Oct. 26 17 Minyu Lepton flavor universality tests Oct. 31 18 (HW3 due) Erin Neutrino-less double beta decay Nov. 2 19 Tianyu Hydro model of QGP Nov. 7 20 Long Detection of axions Nov. 9 21 Baran Neutrino mass from tritium decay Nov. 14 22 Sourav The strong CP problem Nov. 16 23 Michael Composite Higgs models Nov. 21 24 Ping Family/flavor symmetry in the SM Nov. 23 Thanksgiving break Nov. 28 25 Nov. 30 26 (last class) 2
Review of Pauli and gamma matrices Pauli matrices: apple apple 1 0 0 1 I = 0 1 1 = 1 0 i j + j i =2 ij j = 1,2,3 2 = apple 0 i i 0 3 = apple 1 0 0 1 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 0 1 2 3 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
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 0 3 7 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 = 7x10 5 2 4
EWK unification (one generation, all masses = 0) 5
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
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
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 @ 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
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
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 @ 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
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 0 1 1 0 T 2 = 1 2 apple 0 i i 0 T 3 = 1 2 apple 1 0 0 1 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
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
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
Unification with U Y (1) x SU L (2) gauge invariance 14
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 - 1 4 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
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
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
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
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
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 = 0.303 the Weinberg angle usually quoted as sin 2 W = 0.231 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
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 = 0.2313 (at M z ), w = 28.7 o e= p 4 e = 0.313 (at M z with = 1/128) g w = e/sin w = 0.651 g z g 0 /sin w = e/(sin w cos w ) = 0.743 21
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
Discovery of the W and Z bosons The W and Z bosons were directly observed as particles in 1983. This was done at CERN using the Super-Proton Synchrotron that could collide protons and antiprotons at a cm energy of 540 GeV. 23
Next Lecture: Replay of EWK unification derivation Example calculations 24