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18 Chiral Symmetry Breaking

- The existence of nucleon parity doublets, which are not observed, is one of the simplest consequences of a symmetry.

- The appearance of massless (Goldstone) bosons, one for every symmetry not respected by the vacuum, should be an immediate physical consequence if th is so.

- Rather than being massless, 0 will emerge as 'anomalously light'.
- This is how they appear, particularly with respect to the octet of mesons, which differ only in q-q spin alignment.
- It is perhaps not a coincidence that there exists such an entity.
- The Goldstone state is in the form of a pion'.

- It would not be enough for the hypothesis to be accepted if this was the only observable consequence of breaking symmetry.

- Two circumstances greatly increase the phenomenological implications of the idea.

- There are some remarkable connections between weak and strong-in teractio n p arameters.
- The subject of anomalies in sectio n 18.5 will be troduced by us.

- It would take us far from our main focus to pursue these in terestin g avenues.
- In this chapter, we hope to conv in the reader.

- There will be symmetry and a massless boson.

- There are features of the analogy on both sides that need qualification.
- The appropriate generalization (Nambu and Jona-Lasinio 1961b) has to be understood because the particle symmetry we want to interpret is SU(2)f 5 not U(1)5.
- The SU(2)f 5 symmetry is not gauged.

- The quantum field theory vacuum is similar to the ground state.
- According to Nambu's analogy, the vacuum for a massive Dirac particle is to be seen as a group of correlated pairs of massive fermions.
- The members of a pair must have the same spin since the vacuum does not carry linear or angular momentum.
- They have opposite chirality, that's what this means.
- The expression for the Nambu vacuum is similar to the expression for the ground state.

- The helicity is equal to 5 eigenvalue.

- Operators carrying non-zero chirality can have non-vanishing vacuum expectation values.

- The nucleon mass was spontaneously generated.
- The four-fermion type was taken to be the fermion-fermion interaction.
- It was possible to show that a massless bound state appeared in the -ff channel when the gap equation had a non-trivial solution.
- The Goldberger-Treiman relation was derived and a number of other applications were suggested.

- For various processes, the amplitudes for the emission of a single soft pion could be calculated.

- The N-J- L theory of strong interactions was an illustrative example of this, but not a complete theory of strong interactions.
- It was widely accepted as that theory in 1973.
- The 'fermions in question' are quarks and interactions between th em ar e g lu o n ex ch an g es.

- A small explicit quark mass term in the Lagrangian is held to be responsible for the non-zero pion mass, which is interpreted as being generated spontaneously.
- Let's now look at two-flavour QCD.

- Symmetry does not correspond to any known law and there are no ear-massless iso scalar 0- mesons.
- It can be understood as one of the Goldstone bosons, as well as the group SU(3)f 5, which is the s quark.

- The path integrals leads to violations of symmetry.
- The one with which we are concerned is symmetry.

- A simple trick reduces the complexity of the equation.

- The symmetry group of the combined symmetries is called SU( 2 )f L x SU( 2 )f R.

- There is a simple interpretation of the decoupling effected by 18.
- 27.

- L in troduc sected inio n 12.3.2, whist I p roject out the arts of any fermion field.

- L equals 0.

- There is a llows that you can see to see im m e d.

- It can be argued that the larger sy mmetry SU(3)LxSU( 3 ) R is appropriate to three massless flavours.
- This is an issu e that cannot be settled within theory.
- Nu m e rical so lu tio n s o f QCD o n a lattice.

- It's similar to changing on an external field in th e f er r o m a g n e tic p r o b lem.
- SU( 2 )LxSU( 2 ) R is preserved at 18.41 and 18.42.

- This suggests that the vacuum will align in a way to break the isospin.

- The phenomenological implications of spontaneously broken sy mmetry are discussed.

- We know that this current has a non-zero matrix.

- 2 is zero, i.e.
- If th e p is assless.

- The Cabibbo angle is C.

- The matrix element has to be a 4-vector.

- The invariant amplitude for the process is the quantity in brackets.

- We commented on the physics before moving on.

- The pion spin is zero.

- 2 times greater than the r ate to electrons, a resu lt which agrees with experiment, and grossly contradictin g the naive expectation that the rate with the larger release should domin ate.
- This is one of the main indications for the 'V - A' structure of 18.47, as we all see in sectio n 2 0.

- To be ex tracted is 93 MeV.

- The spinors are written in flavour and Dirac space.

- The pion is the propagator of a massless particle.

- The well-known Goldberger-Treiman (G-T) relation was copyrighted in 2004.
- The value of the effective pion-nucleoncoupling constant is only 5% below the experimental value.

- These techniques can be used to calculate 'Goldstone' p ion.
- The in terested reader was referred to by us.

- We n to a n o th e r ex a m p le o f a p h e n o m en o lo g ical m o d e l ex h ib itin g spontaneously broken axial iso sp in sy mmetry.
- It will be a bit more complicated than the model of f sectio n 17.4 which is similar to it in that the breaking is done by hand.
- It embodies many of the previous results.

- The first term is the same as the second.

- The constants of the motion are obeyed by IOP Publishing.
- The specific realizatio n o f the operato rs in terms of the fields in the model under consideration are not held by these results.

- The first term is the same as in 18.20.

- There is an 'Internal' space involving the field components.
- We can imagine the four directions as 1, 2, 3 and 4.
- The rotation in the 14, 24 and 34 directions are similar to spacetime in special relativity.

- This is the Copyright 2004 IOP Publishing Ltd co m b in atio n is v isib le.

- ' R' tr a n sf o r m atio n law f o r th e m e so n field s inth is m o d e l.

- A normal mass is represented by 2 in (18.72).

- In m a ssle s, b e in g th e th r e e G oldstone modes 'perpendicular' to th e o n e selected.

- The model can be modified to include a finite mass.

- Other features of low-energy interactions among pions and nucleons are described in 2 terms.

- Th e m ain p h en o m en o lo gical p r o b lem with.

- The result of imposing is quite remarkable.
- We should consider just the meson sector.

- There are a number of comments.

- The Lagrangian could still be useful, even in loops, if one worked at energies below the scale set by the dimensions.
- We expect the theory to only work for energies close to the threshold.
- This is a general feature that is very important.

- The meson sector of the'spontaneously broken' model does not include derivatives.

- The model fails to account for prominent phenomena as the energy scale rises.

- In his'vector meson dominance' theory, Sakurai stressed the change in hadronic dynamics.
- It is not yet possible to get it from QCD.

- A good representation of nonstrange mesonic dynamics can be found below 1 GeV.

- In our discussions of symmetries so far, there is one result which we have never queried.
- Noether's theorem is referred to in section 12.3.1.
- There is a corresponding conserved current for every continuous symmetry of a Lagrangian.
- We didn't investigate whether quantum corrections might violate the classical law.
- This can happen and when it does the afflicted current is said to be anomaly.
- General analysis shows that there are anomalies in renormalizable theories of fermions.

- The one-pion intermediate state contribution is considered.

- One way of understanding how anomalies arise is through consideration of the renormalization process, which is necessary once we get beyond the classical approximation.
- It was possible to find alternative regularizations which respected the current.

- Bell and Jackiw pointed out the model.
- The non-perturbative perspective is provided by Peskin and Schroeder.

- We won't attempt an extended discussion of this technical subject.
- We want to alert the reader to the existence of these anomalies, to indicate how they arise in one simple model, and to explain why, in some cases, they are to be welcomed, while in others they must be eliminated.

- This is because, once again, when the symmetry is broken, the pion state is connected to the vacuum.

- The on-shell point is 2.

- There is a problem with the hypothesis of spontaneously broken symmetry.

- I n su ch a situ atio, it is h e lp f u l to c o n sid er a d etailed calcu latio n p er f o r m ed with in a specific model.
- The model was considered by Bell and Jackiw and also by Adler.

- I re 18.

- I re 18.

- I n p r in cip le, va r io U s p o ssib ilities.

- No regularization can be found.

- The details are given in Itzykson and Zuber.

- The usual field strength is tenso rs.

- There is a violation at the quantum level of a symmetry of the classical Lagrangian.
- A very small ectio would resu mably b e very small.

- The triangles used a f ermion with charg e. 2 for each quark contributed.
- There are two quarks of SU(2)xSU( 2 ).

- Unless there is three replicas of each quark, the agreement with experiment is not valid.

- We will have a serious problem if we have an anomalies in a current with a local symmetry.
- anomalies in currents coupled to gauge field can't be lerated.
- If the theory is satisfactory at the quantum level, all anomalies must be canceled.
- The observation that the anomaly is independent of the mass of the circulating fermion is what makes this possible.
- It is possible to cancel between quark and lepton 'triangles' in the weak interaction.
- The complete cancellation of anomalies occurs in the G SW theory.

- Anomaly cancellation is a constraint on theories.

- The reasons include: However, we also k n ow f r o m sectio n 2.5 th at weak in ter actio n s are short-ranged.
- At first glance, it seems that th is seem to be weak in ter actio n s.

T h e d e tailed a p p licatio n to th e electr oweak theo r y will b e m ad e in ch ap ter 2 2

- The classical argument for why a gauge field quantum cannot have mass should be noted.

I re 19 I on-f er on-f er on-f er and on-f er on-f er and on-f er and on-f er and on-f er and on-f er and on-f er and

- The reasoning for the non-Abelian case is the same as the Sim ilar reasoning.
- We may have to settle for a theory that is not a gauge theory.

- Consider figure 19.1, which shows some kind of fermion- fermion scattering, proceeding in fourth-order perturbation theory via the exchange of two massive vector bosons, which we will call X- p ar ticles.
- To calcu late th is am p litu d e and we can find it by following the 'Heuristic' r oute outlined in section 7.3.2.
- The propagator should be proportional to the quantity in the square brackets.

- We cannot just take the massless limit of the latter, as a massless particle seems to be a very different kind of thing.

- 5 couplin g will not affect the argument.
- We warned the reader that it was d imensionless.
- It is a case wh ere it does not.

- The theory is not normalizable.

- The answer is important.

- The effect of Equation (19.12) is to make sure that there are only three independent particles.
- Direction is longitudinal.

- Theories with massive charged vect or bosons are not normalizable, so we don't give further d etails here.

- We explained in sectio n 11.8 why relevant theories should be normalizable.

- The freedom to make gauge transformations seems to be only possible in a massless vector theory.

- A related point is th at, as sectio n 7.3.1 sh owed, free photons ex ist in only two polarizatio n states.

- In chapter 17 there is a natural generalization of the global symmetry breaking.
- By way of advance notice, the crucial formula is (19.75) for the propagator, which is to be compared with (19.22).

- Schwinger made the first serious challenge to the view that the photon should be massless.
- Anderson pointed out that several situations in solid state physics could be interpreted in terms of a massive electromagnetic field.

- Anderson had doubts about the hadronic application because gauge bosons can only acquire a mass if the sy mmetry is broken.
- The multiplet structure that we saw in chapter 17 would have a non-Abelian symmetry.
- Weinberg and Salam made the correct applicatio n of these ideas to the g eneration with the help of the weak force.
- There is no need to start with the non-Ab elian case.
- The non-relativistic Abelian is illustrated in the physics.
- The physics of superconductivity is the case.
- Anderson's presentation influenced our presentation.

- We will follow a less'microscopic' and somewhat more 'phenomenological' approach, which has a long history in theoretical studies of superconductivity and is, in some ways, closer to our eventual application in particle physics.

- The 'order p arameter' was a measure of the degree of ordering of a system below some transition temperature.

- The Ginzburg-Landau description could be derived from a certain type of theory.
- The work relates to static phenomena.
- We will follow a more qualitative approach for the time being.

- A situation similar to that in the Bogoliubov superfluid is what inspired this terminology.

- The variation of magnetic field is one of the characteristics encountered in a number of contexts.

Say, say, say, say, say, say, say, say, say

- The magnetic field will be'screened out' from penetrating further into the medium by this expression.

- When a magnetic field is applied to a system of charged particles, the magnetic effect of the resulting currents tends to cancel (or screen) the applied field.
- This is the cause of atomic diamagnetism on the atomic scale.
- The applied flux density in the interior cancels these.

- The order of magnitude for the thickness of the surface layer is correct and it is zero.

- An 'effective non zero photon mass' is an equation that is easy to interpret.

- Now co n sid er th e static ve r sio n o f.

- The photon is an equivalent mass.

- The answer to the puzzle about the change in the number of spin degrees of freedom in going from a massless to a massive gauge field is a field.
- There are many unanswered questions.

- This model is only G of 17.69 and 17.77.

- The rule says to add the Maxwell piece.

- It doesn't lend itself to a natural particle interpretation, which only appears after making a shift to the classical minimum.

- There is a huge difference between the global and local cases.

- This must mean that there is no more physical manifestations of the massless mode.
- The local case has never had an unexpected result.
- We will return to our desire to gauge away the longitudinal polarization states later.

- We started with four degrees of freedom.

- The analysis shows us two things.
- The current (19.46) provides a'screening' effect on the gauge field, which is a relativistic analogue of (19.26).

- We can go further.

- The Go ld stone field disappears as a massless d eg ree o f freedom, and reappears as the longitudinal part of the massive field.

- From our covariant treatment, we know that the components of the field must have the same mass.

- This is a high-frequency mode that is discussed in section 17.3.2.
- A fully gauge-invariant description of the electromagnetic properties of superconductors is obtained when this aspect of the dynamics of the electrons is included.

- We return to the equations.

- In superconductivity, the choice of gauge takes the Copyright 2004 IOP Publishing.
- The gauge is called the London gauge.
- In the next section, we will discuss a subtle change in the argument which applies in the case of real superconducto rs and leads to the phenomenon of flux quantization.

- On r e flectio n, co m e as a r elief, that the propagator is gauge dependent.
- 1 wo U ld b e c o m p letely

- I re 19 In t he hood of a FL ux fi l ament.

- While we have a number of relevant results assembled, it is convenient to include a discussion of flux quantization.

- The 'Confining' property of the physical model is in tr in ter est, th e p h e n o m en o n m a y a lso p r ov id.

- Our discussion of superconductivity has dealt with one class of superconducto rs, called type I.
- The type of superconducto rs that allow thin filam ents of flux is called the II superconducto rs.
- The material is not superconducting and the field is high.

- The variation is shown in the figure.

- The result is that the flux through the line is quantized.

- The equation (19.59) can be integrated around any closed loop in the type-II superconductor.

- There is a situation in which a magnetic monopole is placed in a superconductor.

- In one way or another, this is the basic model which underlies many theoretical attempts to understand confinement.
- A model of a meson is provided by the monopole-anti monopole pair in a type-II vacuum.
- As the distance between the pair increases, so does the energy of the filament, since it can't move in a straight line.
- This is the kind of linearly rising potential energy required by hadron spectroscopy.
- The configuration is stable because there is no way for the flux to leak away.

- One will want particles carrying non-zero values of the colour quantum numbers to be confined.
- The U(1) case has quantum numbers that are similar to electric charge.
- We imagine that magnetism and electricity will be different in all of the foregoing.
- When monopoles are present, the equations have a symmetry.
- The coherent state formed by condensation of fermion pairs was the main feature of the ground state.
- Magnetic monopoles would have to be introduced by hand.
- In the case of non-Abelian gauge field theories, the solution of the magnetic monopole type occurs.
- This circumstance can happen in a grand unified theory with SU(3)c and a residual U(1)em.
- Dirac pointed out that the existence of just one monopole implies that the charge is quantized.

- The group structure of SU(3) is different from that of U(1) models, and we don't want to be restricted just to static solutions.
- Other schemes are also possible if the real QCD vacuum is formed as a coherent plasma of monopoles with confinement of electric charges and flux.
- The confinement problem has a non-perturbative nature.

- This gauge enabled us to transform away from the phase degree of freedom.

- It is certain that th is not right, as we can see a count of rees of freedom.
- The consequence of the unwanted degree of freedom is subtle, but it is basic to all gauge theories and already appeared in the photon case.

I re 19

I re 19

- The m omentum 4 -vector is the only quan tity that can pair to the vector index.

- 2 is a ssle.

- The inverse we needed for the photon propagator in (7.88) does not exist.

- I re 19 For m al summat, I ser I es I n fi gur e 19.

- A clever way to deal with this in the present was suggested by 't Hooft.
- This condition is covariant and it reduces freedom by one.

- It can be shown that there is a consistent set of Feynman rules for this gauge and the theory is renormalizable thanks to many cancellation of divergences.

- We will treat (19.72) as a classical field relation if we don't enter into the full details of quantization.

- The field itself is owned by IOP Publishing.
- It is easy to enter in loop calculations for tree-diagram calculations.

- The remarks about the limit for the 'naive' massive vector boson propagato r in sectio n 19.1 may be taken safely.

- The non-Abelian SU(2) case is the one that is relevant to the electroweak theory.
- The general non-Ab elian case was treated by Kibble.

- The discussion of the breaking of a local non-Abelian sy mmetry will be limited to the particular case.
- This is only the model studied in sectio n 17.6.

- We have to decide how to choose the non-zero vacuum expectation value.
- We don't want a'superconducting' massive photon to emerge from the theory, as the physical vacuum is not a superconductor.
- We don't want to give a vacuum value to a charged field.
- We want it to behave as a weak superconducto r, generatin g mass for W+- and Z0

- In section 17.6 we have already considered 2.

- In order to see the physical particle spectrum, we need to consider the oscillations about (19.83).

- The W and Z particles have propagators in the gauge.

- In the Ab e lian case, an d th is will.

- The ghost interactions in the non-Abelian sector are discussed in section 13.5.3.
- In appendix B of Cheng and Li, there are the complete Feynman rules.

- The model introduced here is actually the 'Higgs sector' of the Standard Model.
- The W+- and Z0 gauge bosons can be given mass.
- This seems to be an ingenious and even elegant way to arrive at a renormalizable theory.
- One may wonder if this is purely phenomenological, similar to the superconducto r.

- The Standard Model uses the Higgs sector to generate mass for all the fermions.

- We saw in ch ap ter 1 8 h that it is likely a t th e str o n g QCD in teractions.
- Even as far as the quarks are concerned, we saw that the Lagrangian mass was required to give a finite mass to the pion.
- It seems unavo id able for both quarks and leptons.
- The genes of fermio n mass would be similar to the genes of th e g eneratio n o f th e e n e rg y g ap.
- Technicolour models are generically known as 'technicolour models' and they have been studied tensively.
- The theories are alr ead y tig h tly co n str ain e d b y the precision electroweak experiments, and meetin g these constraints seems to require elaborate k inds.
- A new strongly interacting sector could be explored in the next generation of colliders.
- Such ideas are beyond the scope of the p resent vo lume.
- Chapter 22 will explain the model o f sectio n 18.3.

- The last part of the book deals with weak interactions.

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