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19.1 Show that

- Put (19.65) into H of (19.40) and derive the terms of (19.69).

- I have hit upon a desperate remedy to save the 'exchange theorem' of statistics.

- The sum of the energies of the electron and neutron is constant.

- I am indispensable here in Z"urich because of a ball on the night of 6/7 December, so I can't go to T"ubingen.

- We arrived at an important part of the Lagrangian of the Standard Model at the end of the previous chapter.
- We could discuss the empirical consequences with little mo tivatio n and an d p ro ceed.
- An approach that is economical would assume a lot of knowledge of weak interaction.

- The full theory at energies well below the W+- ( 8 0 GeV) an d Z 0 ( 90 GeV) is still a useful approximation for many purposes.
- In order to understand the focus of much ongoing research, historical data needs to be carefully interpreted.

- The decaying nucleus emitted a light, spin- 1, which he called the 'neutron'.

- The problem with the continuous e- spectrum and what he called the 'wrong' statistics of the 14N and 6Li nuclei were both attractive to him.
- The problem was taking 14N for definiteness.
- One needs 14 protons and seven electrons for the known charge of seven if the nucleus was composed of only electrons and protons.
- The total nuclear spin is implied by this.
- If the usual'spin-statistics' connection were to hold, the spin of the nitrogen nucleus should be an integer.
- The second part of Pauli's hypothesis was overtaken by the discovery of the real neutron byChadwick in the late 19th century.

- At the Solvay Conference in 1933, Pauli restated his hypothesis, using the name 'neutrino' which had been suggested by Fermi.

- The analogy with electromagnetism was used a lot by Fermi.
- The whole process took place at a single spacetime point, like the emission of a photon in QED.
- The pair was charged as well.

- When calculating transition probabilities in first-order theory.

- The observed selection rules in some nuclear transitions could not be accounted for by the forms.
- The energy release is very small in nuclear decays.
- In this limit, the interactions imply that the nucleon spins cannot 'flip'.
- It is necessary for some o th er in teraction to be present.

- We will discuss other combinations soon.
- It is important to note that the in ter actio n m st alway s b e L o r en tz invar iant.

- We will have more to say on the matters in sectio n 20.6.
- The data might have the same form, but it was incomplete and apparently contradictory.

- The breakthrough came in the year 1956, when Lee and Yang suggested that parity wasn't always the same in weak decays.
- Parity violation was found to be a feature of weak interactions after it was looked for properly.

- The temperature was cooled to 0.01 K.

- To see this, we need to remember the definition of the twovectors.

- By extension, any spin is an axis.

- Consider how the distribution would be described in a transformed coordinate system.

- The two are indistinguishable from the other coordinate system equivalents we have studied.
- Under three-dimensional rotation and Lorentz transformation.
- This is an operational consequence.

- There is a modification that needs to be made to account for this result.

- The coordinate system can either be reflected or not.

- Parity is said to be a different kind of transformation.

- To accommodate parity violation, we have to use both polar and axes.
- Section 12.3.2 has 5 matrix already introduced.

- The pseudoscalar is an axis, while the spatial part is an axis.

- The structure of the weak interaction is unlocked by this.
- It was established after many years of careful experiments and false trails.

- In our present situation, th e h ad r o n ic tr a n sitio n is actu a lly o ccu r r in g.

- It is important to work with a different representation of the Dirac matrices.
- We are all in s ection 20.
- There is a light m odifi cation.

- We might suggest a look at sectio n 12.3.2 and chapter 17.

- To see the consequences of this, we need the forms of the Dirac spinors in this new representation.

- The Standard Model hasspinors entering into weak interactions.

- The equations (20.51) and (20.52) are very important.
- Thespinor will enter.

- It is related to the parity violation built into the V.

- The two resu lts were derived from the convenient representatio n (20.38) for the Dirac matrices.
- This can be owned by usin g g eneral helicity projectio.

- In his original letter, Pauli suggested that the mass of the neutrino might be the same as the electron m ass.
- After the d iscovery of parity violation, the result could be explained by the assumption that the neutrinos were massless particles.
- There are two component 'Weyl' equations.

- The neutrino could be blamed for the violation.
- In this model, the neutrinos have one definite licity.
- A theory leads to the same conclusion as we have seen.

- The questio n o f wheth er th ere is made between the two.

- We follow the same steps as those in section 20.3 for parity.

- The anti-particle transformation is the nearest we can get to a particle.
- This can only be done in formalism.

- The field should beconjugate.

- We h ave in tr o d u ced th e id e a o f p ar ticle- a n ti- p a r ticle co n ju g a tio It is also a good sy mmetry in weak teractions.

- It does not.

- This is what we need to describe fermions which have a conserved quantum number and can be distinguished from anti-particles.
- This has been the case so far, since we only considered charged fermions.
- In the case of the neutral neutrinos, the situation is not so clear.

- In section 1.3.1 of volume 1, we gave a brief discussion of leptonic quantum numbers ('lepton flavours'), adopting a traditional approach in which the data are interpreted in terms of conserved quantum numbers carried by neutrinos, which serve to distinguish neutrinos from anti-neutrinos.
- In the light of what we have learned about the helicity properties of the V - A interaction, we must now examine the matter more closely.

- Davis was following a suggestion made by Pontecorvo in 1955.
- It was well established.
- It should also exist.
- The cross-section is small but using a large amount of target volume could compensate.

- The reaction (20.74) should be observed because decay are not distinguished by the weak interaction.
- Davis found no evidence for reaction at the expected level of cross-section, which could be seen as confirmation of the 'conserved electron number hypothesis'.

Another interpretation is1-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-6556

- Positive helicity is what it has.

- The property of the V is that it conserves helicity in the zero mass limit.
- The positive helicity lepton will be the e+ not the e-.
- The property of the V - A interaction, together with the small value of the neutrino mass, make it impossible to forbid (20.74).

- Carryin g a lepton number is equal to the e+.
- The carrier of the number + 1 is called the licity.

- We can also refer to the two states as sim p ly two d iff e n t h e licity states.
- As we have seen, the job is required just as much as the number rules.

- The conjugate of a neutrino state is not an anti-neutrino.
- It is like a photon, which is, of course, also its own anti-particle.

- The appendix P describes the quantum theory of Majorana fermions.

- Since the limit of strictly massless neutrinos has been reached, the distinction between the 'Dirac' and 'Majorana' possibilities is essentially a metaphysical one.
- The '-' label becomes redundant.

- I re 20. decay wi t hout emi ssi on of a neut r i no, a t est f or Maj or ana- t ype neut r i nos.

- The process A - A + e- + e- is where A is.

- Like the outgoing e- We can either make it easy or suppress it.
- The nucleus physics overlap factor is complicated.
- As y et, clear ev idence for this p rocess h has been obtained.

- There is evidence for a flavour quantum number distinguishing neutrinos which interact in association with one kind of charged lepton from those which interact in association with a different charged lepton.
- A number of observations combined to demonstrate that 'neutrino oscillations' occur, in which states of one flavour can acquire a component of another, as it propagates.
- The detour would be too large to continue with the details of the physics at this point.

- None of the known neutrino experiments is sensitive to the diff erence between the Majorana option and the dir ac option.

- It's fair to say that the Standard Model treats n eutrinos as Dirac p articles, and that's what we'll assume in the rest of this part.
- If neutrinos are Major ana p articles, the way they appear would suggest an origin other than the Standard Model.

- It is to these pairs that the 'V - A' structure applies, as already indicated in section 20.4.

- The argument is suppressed.

- 2 adds up all of the p roducts.

- When it is 20.83, the terms are used to describe any physical processes.

- The mass of F is 2.
- We will return to this part of Fermitype V - A theory in sectio n 21.4.

- We have talked about charged current processes so far.
- There is no way to pair the leptons in 20.
- 92 so as to cancel the number and have zero charge.
- In principle, neutral current contribution could be present.
- A new wave of experimental activity was generated in 1973.
- The first version of the Standard Model predicted their existence.
- The neutral current processes are mediated by the W+- bosons.

- In section 20.9 we will discuss neutral currents.

- It is time to calculate something after so much discussion.
- inverse muon decay is a pure charged current process.

- For example, the appropriate spin-averaged matrix element squared is 8.184.
- We have to average over initial electron states for unpolarized electrons and sum over the final muon polarization states.
- There is no averaging over the weak interaction between left-handed and massless neutrinos.
- There is no sum over the final neutrino helicities.
- Right-handed neutrinos contribute nothing to the cross-section.

- Let's look at it in some detail since it involves a couple of new features.

- In ap p e n d ix J o f vo lu m e 1 is a copyrighted work.

- This is a consequence of the p arity violation.

- It is easy to p e r f o r m th e calcu latio before sp ecializin g.

- The calculation of the purely leptonic p rocesses can be done in an analogous way.

- We discuss the elastic neutrino scattering case in sectio n 20.11, as well as the electron scattering case in chapter 9.
- We sh a ll r e tu r n to th e im p licatio n s o f th is in sectio n 2 0

- As in the case of the charged currents, the exact form of the neutral currentcouplings was determined through much detailed experimental work.
- Parametrizing the currents in a convenient way is what we are doing for the moment.

- This is the most general possible p arametrization.

- The various neutral current processes have relative amplitudes.

- The weak mixing angle is W.

- The following reactions can be used for experimental measurement.
- There are formulas for cross-sections.
- The experiments are reviewed and discussed in Commins and Bucks 1983 and Winter 2000.
- The reader must know that modern precision measurements are sensitive to higher-order (loop) corrections, which must now be included in comparing the full G SW theory with experiment.
- Confirmation of the theory was already emerging in the late 1970s and early 1980s, before the actual discovery of the W+- and Z0 bosons.
- It is interesting to note that the presence of V interactions in the neutral current processes may suggest that there is a link between the two.
- As we will see, W.

- The phenomenon of flavour change in weak hadronic processes is familiar to quarks.
- Cabibbo took the first step towards the modern theory of quark currents.

- A great many processes are described in the effective interaction.

- The idea that the 'total current' should be the sum of a hadronic and a leptonic part is already familiar from electromagnetism.

- The strangeness and the charge increase by one unit when there is a change in the quark.

- The inverse mass squared is F's dimensions.
- Such a correction could be significant if accurate data existed.
- The present limit on the branching is less than 10%, and the experiment placed very tight limits on the non-existence of (20.140).

- The (c,s) pair could be involved in a second quark current.

- The form (20.142) was already suggested by two people.

- GIM speculated that the non-renormalizability could be overcome if the weak interactions were described by a triplet (W+, W-, W0) of gauge bosons.

- 3 wo U ld b e m ed iated b y W0 would correspond to neutral current transitions for quarks.
- The correct sy mmetry is the SU(2) x U(1) of Glashow, which was invoked by GIM.

- There is one feature that is worth nothing.

- There are no neutral flavour-changing currents in this model, which will be extended to three flavours in sections 22.3 and 22.7.1.

- Gaillard and Lee performed a one-loop calculation of the KL - KS mass difference in the G SW model as extended b y GIM to quarks.
- 5 GeV for the charm quark mass, a result spectacularly confirmed by the discovery of the c-c states in charmonium.

- There is a calculation within the framework of the 'current-current' model involving neutrinos and quarks.
- The parton model introduced in chapter 9 will be used to calculate cross-sections for deep inelastic neutrino scattering.

- I re 20.

- The str ated in fig U r e 2 0 is illu str ated.
- We are used to the idea that such p rocesses are, in fact, the W+.

- This scaling can be interpreted in the same way as scatter in g f r o m partons.

- Three term changes sign.

- T h e cr o ss- sectio can b wr itten in th e f o r m.

- The parton-level subprocesses are where we are now.

- A simple helicity argument shows that this follows from the V - A nature of the current.

- There is no momenta in the momentum-space matrix element in our current interaction.

- The current-currentcoupling's point-like nature can be inferred from the lack of orbital angular momentum.
- The initial and final helicities add to zero and backward scattering is allowed.

- The results of the quark parton are followed by many simple and striking predictions.

- I re 20.

- 2 0 is the fig.

- The second figure is in agreement with the first figure, which shows that the majority of the nucleon momentum is carried by charged partons.

- The model's success inelastic scattering was confirmed as early as 1975 by parton m odel.

- QCD is 250 + 50 MeV.

- It's simple since it's not coupled to the distribution.

- I am 1 for the protons.

- 40 was in agreement with the expected value.

- There are 2 functions for electron and neutrino scattering.

- The mean squared charg and the d quarks in the nucleon are related.

- Further confirmation of the quark parton picture is given by the agreement.

- The simple p arto n model has been mentioned several times.
- In the context of deep inelastic charged lepton scattering, the full machinery introduced in chapter 15 can be used.

- I re 20.
- 2 r a ngi ng bet with 10 and 1000 G e V 2.

- I re 20.
- Our-fermion non-leptoni has a weak transition.

- The interaction is responsible for the transitions at the quark level.

- The figure is schematic since there are strong QCD interactions which are responsible for binding the three-quark systems into baryons and the q-q system into a meson.

- I re 20.
- 0 usi ng t he pr ocess of 20.
- 10, w he addition of t wo'spect ator' quarks.

- One universal function is the 'Isgur- Wise form factor', which is a large number of hadronic form factors.

- Leader and Predazzi are references for a review of this important area.

- Lpe to n m a sses is called F.

- The 'V - A current-current' phenomenon of weak in teractions was developed in the p receding chapter.
- The account of a wide range of data gives a remarkably accurate account, so one might wonder why it shouldn't be considered a fully-fledged theory.
- One good reason for wanting to do this would be in order to carry out calculations beyond the lowest point.
- Modern high-energy experiments require higher-order calculations.

- O n e o f th e p illar s o f th e Stan d a rd Mo d e l, wa s f o r m. T h e se h a d to d o, m a in ly, with cer tain in, if viewed as a 'th eory'.
- We lieve th at th e G SW and we are currently up to currently.
- There is an idea of what is at stake in regard to experiments relating to those parts of the G SW theory which have not been established.

- It is worth emphasizing once again a more positive motivation for a gauge theory of weak interactions.
- In the previous chapter, it was noted that the universality structure was also between different types of lepton.
- The generalization of this property in the non-Abelian theories of chapter 13 is very similar to the universality property of QED.
- A gauge theory would provide a framework.

- I re 21.

- We b eg in b y sh owin g wh y th is.

- Since the troubles we all find occur at h ig h energ ies, we can simplify the expressions by neglecting the lepton mass.
- In th is lim it the invar ian t am p litu d e is.

- Take a look at a partial wave analysis of this process.

- The CM energy rises as the Copyright falls.

- The current-current model has a 'unitarity disease'.

- When all particles carry spin, a careful partial wave analysis is needed to fill in all the details.
- We will sketch the conclusions of the analysis.

- The reader may recall a similar argument made in section 11.8, which led to an estimate of the 'dangerous' energy scale.
- The discussion referred to a hypothetical four-fermion interaction without the V -A structure and was concerned with renormalization.

- Unitarity and renormalizability are related.

- We are led to search for a theory of weak interactions if we accept the clue from QED.
- Pressing the analogy with QED will allow us to see how it might happen.
- The model was motivated by the currents of QED.
- In QED, the currents interact directly with each other, whereas in Fermi, they only interact indirectly.

- I re 21.

- We are led to think that weak interactions are caused by the exchange of IVB's.

- 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.
- 1989 Hollik; 1990 for reviews.

- The scheme is defined by (22.170).

- The strength of neutral current processes is compared to charged current processes.

- There are surprising features in (22.173) and (22.174).

- We shall return to this after discussing (22.173) if we considered the typical divergence of a scalar particle in a loop.
- The sensitivity is only a function of time.

- The fermion mass in the numerator is square.

- Consider the contribution from the longitudinal components of the W's.
- The W+ and Z0 allowed three of the four Higgs components to become massive.
- The'swallowed' Higgs fields to fermions are determined by the same Yukawa couplings that were used to generate the fermion mass.

- We have not yet discussed the H symmetry of the assumed minimal Higgs sector.

- It is a natural consequence of having the symmetry broken by an SU(2)L doublet Higgs field or any number of doublets.

- There may be some extra symmetry in (22.8) which is special to the doublet structure.

- W is 1.

- The requirement of a massless photon is a consequence of the global SU(2) symmetry of the interactions and the vacuum under.

- The component is global SU (2).

- The numerator has 2 in it.
- This question can now be answered.

- The symmetry is broken by the quark mass difference.

- The gauge interactions of the quarks obey the same symmetry as the transformation on the L components, except that it is global.
- The R components are decoupled from the gauge dynamics and we are free to make the transformation if we want.

- I re 22.

- H would act like a c.

- We will r e tu r n to th is p a r ticu lar d e tail.

- Even without a Higgs contribu tio n h owever, it turns out th at th e electroweak th eo r y is r e n o r m a lizab le.

- After all, be so dramatic.

- The sensitivity of the St andard Model is only logarith mic.

- The 2 s.d.
- has been ruled out by searches that rule out a Higgs mass less than 120 GeV.

- H 115 GeV.

- The screening was shown to be a consequence of the isospin SU(2) symmetry we have just discussed.

- The situation was different with the top quark.

- The W and Z particles were discovered in 1983, but the data was not very sensitive to virtual effects.

- H is 60 GeV.

- This is a triumph for both theory and experiment.
- The quantum fluctuations of a new particle could pin down its mass so precisely.
- Nature has made use of a renormalizable, spontaneously broken, non-Abelian gauge theory.

- The feature of the'real' top events is noteworthy.

- The real process is much faster than W. Consider the production of a pair of t-ts.
- The strong interactions which should eventually 'hadronize' them will not play a role until they are separate.
- If they travel close to the speed of light, they can only travel up to 16 m before they die.
- T's tend to decay before they experience the QCD interactions.
- This fast decay of the t quark means that there will be no t-t 'toponium'.

- The discovery of the missing particle in the Standard Model is of paramount importance now that the t quark is real.

- We should end this chapter with a review of the Higgs physics.

- The particles in this world are massless.

- In particle physics, a lot of effort has gone into examining various analogous 'dynamical sy mmetry b reaking' th eories.
- There is a field ex ists with a p o ten tial.
- They are put in 'by hand' via Yukawa-like couplings to the Hig g's field.

- I am as to b e ad m itted th at th is p ar t o f th e Stan d ar d Mo d el ap p ear s to b e th least satisfacto r y.
- The Yukawalike fermion couplings are both unconstrained and different in magnitude.
- All of these are renormalizable, but they are not calculable and have to be taken from experiment.

- A commonly held view is that the 'Higgs Sector' is on a somewhat different footing than the rest of it.
- It could be seen as more of a 'phenomenology' than a 'theory'.
- We may be able to mentio n a point in this connection.
- The term would usually be just the mass term of the field.
- The matter is more delicate in the Higgs case.

- The scale of 'new physics' is very high.

- There is nothing like a dramatic 'fine-tuning' problem when it comes to mass corrections.

- There is a good reason for this in the case of the electron mass.
- To contain only logarith ms of the cut-off, self-energ y corrections must be proportional to their mass.

- Thehierarchy problem is a problem.
- One would really like to understand far better if we stress that we ar e d ealin g h er e with an a so lu tely cr.

- The scale at which the Standard Model ceases to be a calculable theory will be shown in the next section.
- The kind of physics that might enter at energies of a few TeV is suggested.
- For example, 'technicolour' models think that the Higgs field is a result of some new heavy air in g id.
- A seco n d p o ssib ility is a protective sy mmetry that can be put alongside fermions in super multiplets.
- The third possibility is that of large extra dimensions.

- We must return to these fascinating ideas because they take us well beyond our subject.

- H is by way of orientation.

- It is not possible to be arbitrarily larg e.

- The essential point is correct, though this is an oversimplified argument.
- The theory is non-perturbative at a lower scale.

- H can't be too large.
- We have considered violations of unitarity by the lowest order d iagrams before.
- The high-energy behavior can be seen between different lowestorder diagrams.
- They are related to their renormalizability.
- There is a process in which two W's scatter from each other.
- An unexpected result can be understood at first sight.

- The scattering of longitudinal W's is the result of the scattering of the three Goldstone bosons.

- H gets bigger than a certain value.
- It is reminiscent of the original situation with the four-fermion current- current interaction itself.
- This could be a clue that we need to replace the Higgs theory.
- The line of reasoning seems to imply that the Higgs boson will be found at a mass well below 1 TeV, or that some weak interactions will become strong with new physical consequences.
- The construction of the LHC was motivated by this 'no lose' situation.

- The Standard Model predicts some aspects of the Higgs production and decay processes.

- This excludes a lot of possibilities in decay and production.

- The cross-section is the same for pp and p-p colliders.

- I re 22.
- I am on the associ at I on W or Z.

- I re 22.
- I am on a t-t pair.

- A p -p c o llid er g ives a so m ewh at larg er cross-section than a pp collider.

- It will have to be detected via its decays.
- 140 GeV decays to fermion-anti-fermion pairs and b-b has the largest branching ratio.

- 500 GeV.
- The width of the state will become comparable to its mass as part of the strong interaction regime discussed earlier.

- I re 22.

- The larger branching ratios of the quark jets are due to the decay of the vector bosons.
- Final states containing hadronic jets will have to contend with hadron collides.
- hope for discovery is likely.

- The physics runs of the LHC are scheduled to start in 2007.

- This crucial energy regime will be explored with high precision thanks to the collider.

- The part of (22.23) which has the form (22.35) can be identified.

- If you want to verify (22.56), use the vertex.

- To derive, insert (22.7) into (22.125).

- The neutral current part is diagonal in the'mass' basis.

- The gauge in which it is real has 1 in it.

- To verify, use (22.181).

- If you want to verify, use the Higgs Couplings given in appendix Q.

- The law of combination is not commutative.

- An infinite group is a set that does not have a finite number of elements.

- The inverse is the usual matrix inverse.

- Although matrix multiplication is not commutative in general, it happens to be for certain matrices.
- The way the four matrices are combined is the same as the way the four numbers are combined.

- We are concerned with various kinds of coordinate transformations, not only spacetime ones but also internal ones.
- The elements of a group are specified by three real parameters.
- There are two ways to define the axis of the rotation.
- Three of the real parameters are for 3D rotation and three are for pure velocity transformations.

- The values of eight real parameters are used to specify the matrices of SU(3).

- If we are given the form of the group elements in the neighbourhood of any one element, we can'move out' from that neighbourhood to other nearby elements using the mathematical procedure known as 'analytic continuation'.

- The real p arameters are the transformations.

- It is chosen for convenience, f o r exam p le thos e o f S O.

- The two groups are the same one with the same structu re constants.

- They are not the same for large transformations.

- Under SO(4) transformations, 2 is left invariant.

- The six generators have separated into two sets of three, each set obeying the algebra of SO (3) and the other set, in this form.

- SU2xSU2 is the name of the algebra (M. 43)-(M.45).

- The appendix D of volume 1 contains 2.

- We can think of infinitesimal pure velocity transformations as similar to ordinary infinitesimal 3D rotation.

- There is a minus sign on the right-hand side.

- In all pairs of indices,'s are anti-symmetric.

- The Lie algebra of the group is satisfied by these g enerators.
- The ey have the same algebra.

- There is a method for getting matrix representations of Lie algebras.

- This is the same as the number of independent mutually commuting generators.

- One particular representation of the generators can be obtained by considering the general form of a matrix in the group which is infinitesimally close to the unit element.

- The SU(3) algebra is satisfying.

- The coefficients on the right-hand side are not related to the SO(3) structure constants.

- As required,'s satisfy the group algebra.

- It is remarkable that 10 x 10 matrices are the same as the rotation matrices of SO (3).

- Consideration of matrix representations of the Lorentz group gives insight into the equations of quantum mechanics.

- Consider the infinitesimal Lorentz transformation.

- Lie groups can be represented by matrices.
- The group is non-compact.

- The same as (M.43)-(M.45) is the Copyright 2004 IOP Publishing.
- The same is true in a general finitedimensional representation.
- They behave like two independent people.

- The second part shows how a spinor transforms.

- Let's consider boosting.

- The rough problem 4.15 will be recognized by the reader, with a sign change for the in fin itesimal velocity parameters in (M.101) and (M.102) as compared with (4.151) and (4.15).
- 0 becomes a (0, 1/2) object.

- The two groups are the same.

- The groups are close to each other.

- It is possible that the groups are fully isomorphic because of the infinitesimal matrix transformations.

- The parameters used to characterize elements o f SO( 3 ) a n d SU( 2 ) were re-considering.

- We are going to ex amin e in more detail now because of the real parameters.

- We want to know if it is one-to-one.

- The whole sphere is the upper hemisphere's SU2 and SO3 parameters.

- It's a homomorphism.

- Let's return to the correspondence between SU and SO.

- The identity matrix of SU (2) is not clearly defined by our theorem.
- In the Copyright 2004, the SU(2) matrices cannot be said to represent rotation.

- Up to a sign, SU (2) matrices provide a representation of SO (3).

- The groups are isomorphic if wefactor out this sign.
- A more precise way of saying this is given in Jones.

- First, we consider (N.1).

- 4 can be exposed and dealt with by a renormalization scheme like we did with the cut-off procedure.
- The difference between the simple cut-off regularization we used in chapter 10 and the one we used in chapter 11 is that the gauge invariance is preserved.

- It may include in the subtraction certain finite terms as well.

- One can show straightforwardly that the gauge-non-invariant part of (11.18)--i.e.
- using these results.
- There is a piece in braces.
- The renormalization programme can be carried out with the help of the technique of regularization.

- Classical functions are regarded as the e fields.
- There are elements of time-ordered products of bosonic operators that could be represented.
- We must think in terms of 'classical' anti-commuting variables when we represent fermionic operators by path integrals.

- The necessary mathematics was developed by Grassmann and applied to quantum amplitudes by Berezin.

- Grassmann numbers can be added and subtracted in a variety of ways.
- We need to integrate over Grassmann numbers for our application.
- As with ordinary numbers and functions, integration would be an inverse of differentiation.
- Let's start with differentiation.

- We must approach Grassmann integ ratio via an inve r.

- The path integral formalism uses this property to make manipulations similar to those in section 16.4 but with Grassmann numbers.

- We need a convention about the order in which the integrals are to be performed.

- Since our application will be to Dirac fields, which are complex-valued, we need to introduce complex Grassmann numbers, which are built out of real and imaginary parts in the usual way.

- Under complex conjugation, O.14 is consistent.

- The path integral formalism requires some Gaussian integrals over Grassmann variables, which we are ready to evaluate.

- When written in 'discretized' form, this result isficient to establish the assertion m ad e in section 16.4 concerning the integral.

- Gaussian integrals over complex Grassmann variables are proportional to the inverse of the determinant.

- They are equivalent.

- The reader can interpret this as a finite-dimensional determinant.

- There is a Copyright 2004 by IOP Publishing.

- 0, f o r ex am p le, wh is its own.
- A n eutral scalar field has only one field degree of freedom, whereas a charged scalar field has two field degrees of freedom.
- The two degrees of freedom correspond to the states that are physically distinct.
- In sectio n 7.2, there is an additional doublin g o f the number o f d eg rees of freedom to four in all, corresponding to particles and anti-particles.
- F o r n e U tr a l f er m io n s su ch as n e U tr in o s.

- I'm clear, a t fir st sig ht, wh e r e ther e is r o m f o r su c h a p o ssibility.
- We begin by rethinking the wave equations for spin- 1 particles.

- One way of approaching the number of degrees of freedom is within the framework of quantum-mechanical wave equations.

- The p aragraph contains equations.

- The first results were given in 1957.

- It is assumed that it is the right operation in the present case as well.

- The Majorana condition can be seen in the form (P.21).

- When we try to describe this theory in terms of a Lagrangian, there is a problem.

- We seem unable to form the required Lagrangian because 2 is anti-symmetric.

- We should now consider the quantum field case.

- Th ese r esu lts sh ow at, a s c laim ed in sectio n 2 2.
- The field transforms in the same way as the original field.

- In this quantum field formalism, the mass term can be considered at the end of the previous section.
- Other types of mass term are the same.

- Theory in this sector is serving.

- There is a clear discussion in Bilenky.

- The mixing parameters can be absorbed with 2.
- D and one are very small.
- The 2 of (P.40) is much lighter than the 1 of (P.40).

- The SU(3) colour degree of freedom is not explicitly written.

- It is important to remember that the rules given are only adequate for tree-diagram calculations.

- It is convenient to use the U-gauge Feynman rules in which there are no physical particles.

- The talk is at the 20th Int.

- Gauge Theories in Particle Physics: A Practical Introduction, Third Edition, Vol. 2 CONTENTS PREFACE TO VOLUME 2 OF THE THIRD EDITION Acknowledgments

- Part 5: Non Abelian Symmetries Chapter 12: Global Non-Abelian Symmetries 12.1 The flavour symmetry SU(2)f 12.1.1 The nucleon isospin doublet and the group SU(2) 12.1.2 Larger (higher-dimensional) multiplets of SU(2) in nuclear physics 12.1.3 Isospin in particle physics 12.2 Flavour SU(3)f 12.3 Non-Abelian global symmetries in Lagrangian quantum field theory 12.3.1 SU(2)f and SU(3)f 12.3.2 Chiral symmetry Problems Chapter 13: Local Non-Abelian (Gauge) Symmetries 13.1 Local SU(2) symmetry: the covariant derivative and interactions with matter 13.2 Covariant derivatives and coordinate transformations 13.3 Geometrical curvature and the gauge field strength tensor 13.4 Local SU(3) symmetry 13.5 Local non-Abelian symmetries in Lagrangian quantum field theory 13.5.1 Local SU(2) and SU(3) Lagrangians 13.5.2 Gauge field self-interactions 13.5.3 Quantizing non-Abelian gauge fields Problems

- Part 6: QCD and the Renormalization Group Chapter 14: QCD I: Introduction and Tree-Graph Predictions 14.1 The colour degree of freedom 14.2 The dynamics of colour 14.2.1 Colour as an SU(3) group 14.2.2 Global SU(3)c invariance and 'scalar gluons' 14.2.3 Local SU(3)c invariance: the QCD Lagrangian 14.3 Hard scattering processes and QCD tree graphs 14.3.1 Two-jet events in -pp collisions 14.3.2 Three-jet events 14.4 Three-jet events in e+e- annihilation Problems Chapter 15: QCD II: Asymptotic Freedom, The Renormalization Group and Scaling Violations in Deep Inelastic Scattering 15.1 QCD corrections to the parton model prediction for 15.2 The renormalization group and related ideas 15.2.1 Where do the large logs come from? 15.2.2 Changing the renormalization scale 15.2.3 The renormalization group equation and large -q2 behaviour in QED 15.3 Back to QCD: asymptotic freedom 15.4 A more general form of the RGE: anomalous dimensions and running masses 15.5 Some technicalities 15.6 Hadrons revisited 15.7 QCD corrections to the parton model predictions for deep inelastic scattering: scaling violations 15.7.1 Uncancelled mass singularities 15.7.2 Factorization and the DGLAP equation 15.7.3 Comparison with experiment Problems Chapter 16: Lattice Field Theory and the Renormalization Group Revisited 16.1 Introduction 16.2 Discretization 16.3 Gauge invariance on the lattice 16.4 Representation of quantum amplitudes 16.5 Connection with statistical mechanics 16.6 Renormalization and the renormalization group on the lattice 16.6.1 Introduction 16.6.2 The one-dimensional Ising model 16.6.3 Further developments and some connections with particle physics 16.7 Numerical calculations Problems

- Part 7: Spontaneously Broken Symmetry Chapter 17: Spontaneously Broken Global Symmetry 17.1 Introduction 17.2 The Fabri-Picasso theorem 17.3 Spontaneously broken symmetry in condensed matter physics 17.3.1 The ferromagnet 17.3.2 The Bogoliubov superfluid 17.4 Goldstone's theorem 17.5 Spontaneously broken global U(1) symmetry: the Goldstone model 17.6 Spontaneously broken global non-Abelian symmetry 17.7 The BCS superconducting ground state Problems Chapter 18: Chiral Symmetry Breaking 18.1 The Nambu analogy 18.1.1 Two flavour QCD and SU(2)f LxSU(2)fR 18.2 Pion decay and the Goldberger-Treiman relation 18.3 The linear and nonlinear s-models 18.4 Chiral anomalies Problems Chapter 19: Spontaneously Broken Local Symmetry 19.1 Massive and massless vector particles 19.2 The generation of 'photon mass' in a superconductor: the Meissner effect 19.3 Spontaneously broken local U(1) symmetry: the Abelian Higgs model 19.4 Flux quantization in a superconductor 19.5 't Hooft's gauges 19.6 Spontaneously broken local SU(2)xU(1) symmetry Problems

- Part 8: Weak Interactions and the Electroweak Theory Chapter 20: Introduction to the Phenomenology of Weak Interactions 20.1 Fermi's 'current-current' theory of nuclear b-decay and its 20.2 Parity violation in weak interactions 20.3 Parity transformation of Dirac wavefunctions and field operators 20.4 V - A theory: chirality and helicity 20.5 Charge conjugation for fermion wavefunctions and field operators 20.6 Lepton number 20.7 The universal current-current theory for weak interactions of leptons 20.8 Calculation of the cross-section for nm + e ->m + ne 20.9 Leptonic weak neutral currents 20.10 Quark weak currents 20.11 Deep inelastic neutrino scattering 20.12 Non-leptonic weak interactions Problems Chapter 21: Difficulties With the Current-Current and 'Naive' Intermediate Vector Boson Models 21.1 Violation of unitarity in the current-current model 21.2 The IVB model 21.3 Violation of unitarity bounds in the IVB model 21.4 The problem of non-renormalizability in weak interactions Problems Chapter 22: The Glashow-Salam-Weinberg Gauge Theory of Electroweak Interactions 22.1 Weak isospin and hypercharge: the SU(2) x U(1) group of the electroweak interactions: quantum number assignments andW and Z masses 22.2 The leptonic currents (massless neutrinos): relation to current-current model 22.3 The quark currents 22.4 Simple (tree-level) predictions 22.5 The discovery of the W+- and Z0 at the CERN p-p collider 22.5.1 Production cross-sections forWand Z in p-p colliders 22.5.2 Charge asymmetry inW+- decay 22.5.3 Discovery of the W+- and Z0 at the p-p collider and their properties 22.6 The fermion mass problem 22.7 Three-family mixing 22.7.1 Quark flavour mixing 22.7.2 Neutrino flavour mixing 22.8 Higher-order corrections 22.9 The top quark 22.10 The Higgs sector 22.10.1 Introduction 22.10.2 Theoretical considerations concerning mH 22.10.3 Higgs phenomenology Problems

- Appendix M: Group Theory M.1 Definition and simple examples M.2 Lie groups M.3 Generators of Lie groups M.4 Examples M.4.1 SO(3) and three-dimensional rotations M.4.2 SU(2) M.4.3 SO(4): The special orthogonal group in four dimensions M.4.4 The Lorentz group M.4.5 SU(3) M.5 Matrix representations of generators and of Lie groups M.6 The Lorentz group M.7 The relation between SU(2) and SO(3)

- Appendix N: Dimensional Regularization

- Appendix O: Grassmann Variables

- Appendix P: Majorana Fermions P.1 Spin- 1/2 wave equations P.2 Majorana quantum fields

- Appendix Q: Feynman Rules for Tree Graphs in QCD and the Electroweak Theory Q.1 QCD Q.1.1 External particles Quarks Gluons Q.1.2 Propagators Quark Gluon Q.1.3 Vertices Q.2 The electroweak theory Q.2.1 External particles Leptons and quarks Vector bosons Q.2.2 Propagators Leptons and quarks Higgs particle Q.2.3 Vertices Charged-current weak interactions Neutral-current weak interactions (no neutrino mixing) Vector boson couplings Higgs couplings

- References

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