Both W+ and W- exist because of charg e-raising and charge-lowerin g currents.
We are going to follow a more scenic route for the time being and accept that we have ordinary 'unsophisticated' massive particles, charged and un charged.
The IVB model relates to the current-current one.
I re 21.
The equation (22.29) of chapter 22 shows that this is a fundamental relation.
Out of all the apparent differences between the two, W is so larg e.
We investig ate wh e th er th e I VB m o d e l can d o any b.
We will take you at issu e a g in sectio.
The section heading indicates that matters will be fundamentally no better in the IVB model, but the demonstration demonstrates that.
As will be explained in sectio n 22.1.2, th e factors o f two have been chosen to be those that would actually appear in the unitary gauge.
The photon propagator was responsible for the fall with the QED cross-sectio n 21.12 and at least for the process.
The answer is no.
To calculate the total cross-section, we must combine the three states of polarization for each of the W's.
It is worth taking a closer look at this term.
Oth e r U n itar ity - v io latin g p r o cesses can easily b e inve n ted, a n d we h ave to conclude that the IVB model is no more fit to be called a model.
It was not a good enough cure that F was not dimensionless.
I re 21.
We turn in our distress to the QED.
The fact that there are two r ather than one is significant.
We need to sum over the photon polarization states in the cross-section.
This is not a trivial point.
Since the longitudinal W's caused the 'bad' high-energy behavior of the IVB model, the 'good' high-energy behavior of QED might have its origin in the absence of such states.
It requires the gauge field quanta to be massless.
If the local sy mmetry is broken, this peculiarity can arise.
Before we implement th at id, we need to be aware of the unitarity one.
There is an objection to the argument about unitarity violations.
The argument is conducted completely with the perturbatio n theory.
It's simply that p ertu rbatio n theory is so me.
F o r d e r p er tu r b atio n th e o r y a r e ir r e leva.
Another way of stating the results of the previous two sections is to say the current one is weak.
Chapters 1 0 and 11 of vo lu me 1 were given an elementary in troductio n. In particular, we d iscussed in some d etail, in sectio n 11.8, th e d ifficu lties th a t a r ise wh en o n e tr ies to d o h ig h e r- o F, also h a d d e n sio n ( m a ss)-2 The 'non-renormalizable' p roblem was essentially th at, as one approached the dangerous scale and needed to supply the values from the experiment.
I re 21.
I re 21.
Th is ex actly th e c o m p a r iso n we wer e m ak in g in the previous section, but now we have arrived at it from considerations.
The blame once again lies with the longitudinal p o lar izatio n states.
This problem can be avoided if L et U see h ow QED--a r en o r m alizab le theo r y.
Replacing the 1 T h e r eader would give us the leading high-energy behavior.
I re 21.
Four- poi nt e+ e- ve r t ex.
I re 21.
There is a question of renormalizing figure 21.7.
The particles in this process are virtual and not real.
3 f o r so m e th in g sim ilar in the case of one-loop diagrams.
W+W is translated into a figure of 21.8--and the r e is n o'c r o ssed'.
The introduction of a new vertex, figure 21.10, is not present in the original IVB theory.
If we include it, the theory is non-renormalizable, as in the current-current case.
The search for such mechanisms can be pushed to a su ccessf.
We have a more powerful principle.
The'spontaneously broken' gauge theory concept was developed in chapter 19.
This strongly suggests that these theories are renormalizable.
It was clear that it would be possible to make h igher- if Hooft's p roof th at th ey were to explode.
We now have all the pieces in place, and can introduce the G SW theory based on the local gauge symmetry of SU(2) x U(1).
In sectio n 2 0, 5 were recorded.
In the limit in which all the people are neglected.
When the photon inands are replaced by the photon momentum, the two amplitudes disappear.
It is now well established that the one originally proposed by Glashow, which was subsequently treated as a spontaneously broken gauge symmetry by Weinberg and Salam, produces a theory which is in.
We will not give a critical review of all the ex p erimen tal ev id en ce bu t.
Considering the transitions caused by these interactions is what gives an im p o r tant.
This is similar to discovering the m ultiplet structure of atomic levels and hence the representations of the rotation group, a prominent symmetry of the Sch r "odinger equation.
Between th e'weak m U ltip lets' we sh all b e consid ering and those asso ciated with symmetries which are not spontaneously.
In chapter 12 we saw how an unsymmetrical non-Abelian symmetry leads to a state of mind.
The result only holds if the vacuum is left invariant.
This is the situation in the theory.
The consequences of the weak symmetry group are accessible to experiment.
In section 20.10 we saw how weak transitions invo lv in g charg ed quarks suggested a doublet structure.
The simplest way to think about it is that there is a 'weak SU(2) g roup' involved.
We emphasize once more the weakness of iso sp in is d istin ct f r o m th e h a d r o n ic iso sp in o f ch ap ter 1 2
The left-handed components of the field enter as a consequence of the V - A structure.
No te at, as anticip involve ated for a spontaneously broken sy mmetry, th ese doublets all pairs of particles which are not mass degenerate.
This is a g e n e r a lizatio n to 3x 3 m ix in g o f th e 2x 2 GIM mixing introduced in section 20.10, and it will be discussed further in section 22.7.1.
For the time being, we ignore the mixing in the neutrino sectors, but return to it in section 22.7.
The two gauge fields associated with transitions between doublet members will have charge +-1 because the members of a weak isodoublet differ by one unit of charge.
The photon is massless and the W's must somehow acquire mass.
Schwinger arranged the th e th e th e th e th e th e th e th e th e th e th e th e th e th e th e th e th e th e No prediction of the W mass could be made.
The breakdown of a non-Abelian gauge must be the cause of the W mass, as we saw in sectio n 1 9.
There is an obvious suggestion to have the neutral member W0 of the SU2L act as a conduit for these currents.
The plan was to put the W mass in 'by hand'.
The attractive f eature o f including the photon has been lost.
A key contribu tio n was made by Glashow in 1961.
The structure is g roup.
The piece of mathematics we went through in section 19.6 is an important part of the Standard Model.
The main results of section 19.6 are reproduced here.
Weak Is ospi n and hyper charge.
The rules for the propagators can be read off from 22.8 and are in appendix Q.
There is no weak in ter actio n s co and a b asic assu mp.
We all see 5 terms.
We arrive at our assignments in table 22.1.
The table has 'R' components in it.
The original Standard Model took the neutrinos to be massless with no mixing.
We proceed in the massless n eutr inos approximation.
The raising and lowering operators are used for doublets.
I re 22.
The form we used in the current-current theory may be compared with W.
This is an important equation that gives a precise version of the qualitative relation.
There is no theory that can predict the value of the scale of symmetry breaking.
I n g en er al, th e ch arg e- ch an g in g p ar t o f.
Her m itian c o n ju g a te.
The phenomenological currents of the earlier model are exactly what L gauge th eory are.
The rules can be read off from 22.33
I re 22.
Z 0 is not a pure 'V - A'.
A n d h e n c e ex h ib it.
There are two more rules contained in (22.37) and (22.38).
W (22.46) is already suggested in chapter 19.
5 cancel from 21:45.
The matrix is the Cabibbo-Kobayashi-Maskawa matrix.
We are all in the same sectio n 2 2.
The sum will be over all the quark flavours.
The expressions are the same as given.
We note one important feature of the Standard Model currents.
In sectio n 1 8 an o m alies wer e d iscu ssed.
There is no explanation provided by the Standard Model.
We n o ted in sectio.
The predictions of the theory show the power of the underlying symmetry to tie together many unrelated quantitities, which are all determined in terms of only a few basic parameters.
Neutrino-electron graphs are related to Z0 exchange.
The width of the quark channels would be the same, apart from a factor of three for the different colour channels.
We neglected all fermion mass in making these estimates.
The GIM mechanism ensures that all flavourchanging terms are canceled.
The hadronization of the q-q channels has a branching ratio of 69.3%.
I re 22.
There was a scattering in sectio n 20.11.
Wherever possible, the lepton mass has been neglected.
I re 22.
The experimental fits to these predictions are reviewed by Commins and Bucksbaum.
The Standard Model parameters can be determined at the e+e- colliders.
30 years ago, the cross-section calculations were made.
The flu x o f scattered electrons were inelastically scattered.
A c lear sig n a l f o r p a r ity v io latio n and an asy mmetry b etween th e results.
The W+- and Z0 1983 are some of the main experimental evidence.
The sea quarks will be expected to contribute.
It is required that the QCD corrections to (22.81) be included.
The order is 1.5-2 at these energies.
I re 22.
On model a mpl I tude f or W+- or Z 0 pr oduct.
I re 22.
The total cross-section for p-p is about 70 MB at these energies, and hence (22.84) is 10 times smaller.
The rates could go up if the q-q modes of W and Z0 were used.
W and Z 0 would appear as slight shoulders on the edge of a very steep hill.
Ite th e u n favo, th e lep to n ic m o d e s prov id.
The signature for (22.87) is an isolated and back to back, e+ e- pair with an invariant mass of around 90 GeV.
The e+ e- pairs required come from the decay of a m assive slowly moving Z 0 The mass resolution was folded in.
The uncertainty in the absolute calibration of the calorimeter energy scale is reflected in the systematic error.
The agreement with (22.57) is good, but there is a suggestion that the tree-level prediction is on the low side.
I re 22.
I re 22.
The dotted, full and dashed lines are predictions of the Standard Model.
It is possible to use (22.90) as an important measure of such neutrinos.
It is possible to determine Z accurately.
The mass resolution of the -pp experiments was of the same order as the total expected Z0 width, so that (22.90) could not be used directly.
The W+- is where we turn now.
As in the case of Z0 - e+e- decay, slow moving massive W's will emit isolated electrons with high energy.
calorimetry can be used to balance the energy of the electrons.
The following argument shows W. Consider the decay of a W at rest.
A maximum likelihood fit was used to find the most probable value.
I re 22.
The agreement b etween the experiments is good and the predictions are on the low side.
One renormalization sch e m e is one of the Radiative corrections that can be applied.
I re 22.
I'm on the beam and he's on the posi t r on.
We may say that the early discovery experiments were remarkably convective in their confirmation.
We are going to further aspects o f the th eory.
It is not invariant if L is su bject to a tr a n sf o r m atio n o f th e form.
The same is true for Majorana fermions.
This kind of explicit breaking of the gauge symmetry cannot be condoned.
0 is lo n g itu d in a l. We studied the unitarity violations in the lowest-order theory for the IVB model.
The cancellation feature is one aspect of the renormalizability of the theory.
We will eventually have a 'nonrenormalizable' problem on our hands, all over again, because the cancellation no longer occurs.
I re 22.
Even though the breakdown occurs at energies beyond those currently reachable, it would constitute a serious flaw in the theory.
There is a way to give fermion mass without introducing an explicit mass term in the Lagrangian.
The model shows how a fermion with a Yukawa-type coupling will generate a fermion mass.
In each term, the two doublets aredotted together so as to form an SU(2)L scalar.
The symmetry is preserved at the Lagrangian level if (22.104) is SU(2)L-invariant.
I re 22.
I am on a ph.
The reader will not be surprised to hear that this graph is what is required to cancel the 'bad' high-energy behavior found in (22.10).
The upper component of (22.11) has a p p ear.
It is possible to arrange for all the fermions, quarks and leptons to get the same'mechanism'.
The quarks will be looked at more closely in the next section.
It appears that we are dealing with a 'phenomenological model' once more.
h oweve r, th er e is a n o th e r p o ssib ility.
It is possible to make a Dirac-type mass term of the form.
The anti-Hermitian 2 x 2 matrix would disappear for classical fields.
Majorana neutrinos do not have a number.
The (1,1) operator cannot combine with the (1,01) operator to form a singlet.
We can't make a tree-level Majorana mass by the mechanism of Yukawacoupling to the Higgs field.
We could generate effective operators via loop corrections, similar to how we generated an effective operator in QED.
The operator would have to violate the standard model interactions if it is true.
It was not possible to generate an effective operator in the theory.
It could arise as a low-energy limit of a theory defined at a higher mass scale, as the current- current model is the low energy limit of the G SW one.
appendix P, sectio n P.2 contains further discussion of the neutrino mass.
We will not pursue these considerations beyond the Standard Model.
We need to generalize the discussion to the three- family case.
We have to consider what is the most general interaction between the Higgs field and the various fields.
If we abandon renormalizability, we might as well abandon the whole motivation for the 'gauge' concept.
The 2 appearing are non-normalizable and have a coupling with dimensions.
We can still manage with only one field.
Consider the gauge-invariant interaction part of the Lagrangian.
The CKM matrix is well known.
The CKM matrix has many independent parameters.
A matrix has 2x32 real parameters.
R has to change in the same way as mass terms.
This leads to a parametrizatio.
I re 22.
T he uni t a r i t y t r i a ngl e' r e pr esent
This is a triangle.
The original Cabibbo-GIM type is considered in section 20.10 to be a fundamental difference.
Thematrix must be real.
13 was stressed by Kobayashi and Maskawa.
The construction of B factories is influenced by the effects of the B0 system.
The fit is consistent with (22.153).
A similar analysis can be done in the leptonic sector.
See also Pontecorvo 1967.
These are the states that we would identify with the physical neutrino states.
It is clear that the mixing of flavours does take p lace, indicating that there are differences in the mass of the particles.
It is an open question if the particles are Dirac or Majo rana.
Global phase transformations can't be made on Majorana fields as they don't carry a number.
The phases were violated.
Because the mass differences of the neutrinos are so small, they can be observed to occur over distances.
Section 20.6 has a description of decay.
The reason why we need a renormalizable electroweak theory is because of such remarkable precision.
One can be around in more ways than one.
The unconstrained b y theory is a p arameter.
The presence of 'new physics' may be indicated by the analysis of small discrepancies between data and predictions.
The introduction to one-loop calculations in QED at the end of volume 1 may have given the reader a right to expect an exposition of loop corrections.
We want to talk about a few of the simpler and more important aspects of one-loop corrections.
Cut-off independent results from loop corrections in a renormalizable theory can be obtained by taking the values of certain parameters from the original experiment, according to a welldefined procedure.
2 bu t these relations are changed.
In practice, the renormalizatio n scheme is to be sp ecified at any finite order.