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.
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)