where the symmetry factor of | arises because the two gluons are identical and the factors of g^guu1 are from the gluon propagators (in the Feynman gauge). In computing the imaginary part of the diagrams, the two intermediate gluons are fixed on their mass-shell, so that if fci and are their four-momenta, we have k\ — 0 and k% = 0, just as for the original annihilation diagrams. The only difference between (15.69) and the correct unitarity relation is that the g^guu' term from the propagators includes the contribution from the longitudinal polarization states, the term in (15.68). Unitarity requires that the sum be over physical, transverse states only, and hence we require

The difference between (15.70) and (15.69) is the extent to which the imaginary parts of diagrams A-E violate unitarity and is

Note that each of the terms in this difference involves at least one factor of gL, which in turn would be zero if the annihilation amplitudes were gauge invariant. Because k* = k% = 0, it follows from Eq. (15.66) that kinhvMX = = eiMfc2„AC = 0

in the previous section, it would seem to be an accident. As it is, the unitarity requirement is one way to see why such gauges require ghosts, and their presence was anticipated well before their quantization rules were known. Having the rules, we can regard this demonstration of unitarity as a confirmation that the rules are correct and that the theory is physical.

Building on what has been learned, the standard Electroweak model, usually just referred to as the Standard Model, will be described next. This model was developed by Weinberg and Salam in the late 1960's [We 67, Sa 68] and is the first example of the modern unification of forces. In this model, the weak and electromagnetic forces are unified into a single force.

We will only describe one generation of the lepton sector of the Standard Model, but the other generations and the quark sectors are all very similar (for an elementary account, see [AL 73, La 81]). Briefly, the Standard Model can be described as a theory which has a local SU(2)xU( 1) gauge symmetry spontaneously broken by the vacuum. Hence it combines features of the SU(2) gauge group studied in Sec. 13.2 with spontaneous symmetry breaking studied in Sees. 13.6-13.8. There are two Lagrangians to be discussed. The first is the unbroken Lagrangian in which the gauge symmetry is manifest, and the second is the transformed Lagrangian expressed in terms of physical fields. The particles contained in each of these Lagrangians are summarized in Table 15.1, and they will be described in detail as we proceed.

We begin with the unbroken Lagrangian, which will be denoted by C. It is constructed from a left-handed doublet of Fermi particles, a right-handed fermi singlet, a doublet of complex scalars, and four gauge bosons: three, A^, associated with the SU(2) symmetry, and one, B^, associated with the t/(l) symmetry. All particles but the scalars are massless. The left-handed particles include the 75) projection operator, and we use the notation where v and e are neutrino and electron fields. The right-handed singlet has the form

Under the gauge group, the lepton and scalar doublets transform as in Eq. (13.16)

where v and e are neutrino and electron fields. The right-handed singlet has the form

where D is either doublet, and

512 QUANTUM CHROMODYNAMICS AND THE STANDARD MODEL In these expressions,

A = n A - n A Bfu; — — dvB^ , where AM = as in Eq. (13.25), and the Z)M are covariant derivatives, defined with the following properties:

where D'fl is the gauge transformed version of DM. With these defining properties, the covariant derivatives have a form analogous to (13.32), including terms for both the U(2) and U(l) groups, when indicated. For example,

where the positive sign in the term follows from the assignment Y = -1 to L. The gauge transformations for the gauge fields are

With all these definitions, the gauge invariance of the Lagrangian (15.83) follows almost trivially. The only term requiring a demonstration is the interaction term, where the invariance under SU(2) is again obvious, but invariance under C/(l) is a consequence of compatible hypercharge assignments,

We now arrange for this gauge symmetry to be spontaneously broken by making m2 negative, as we did in Sec. 13.6. Then the energy density is a minimum when

where v is the same constant first introduced in Sec. 13.6. We will choose the real part of the neutral component of 4> to be the one which breaks the symmetry, and

V92+9'2

Hence the Weinberg angle and the electric charge are given by the two original coupling constants g and g'. Since these were both undetermined before, we have no prediction as of yet.

The final task is to express the original Lagrangian L in terms of <po (v and rj) and the new gauge fields A^ (the photon) and ZM (the neutral vector boson). To make contact with the old phenomenology, we also define the charged vector boson field by

Hence

\ (nAl + T2Al) = A (n + ir2) (Al - iAl) + ± (n - in) {Alß + iAl)

are the weak isospin raising and lowering operators. From Eqs. (15.77) and (15.84) we see that the operator t± connects neutrino and electron fields as follows:

and hence describes the creation of one unit of charge through processes like e'-tv, i>e e+ 0 —> e+ + ue e~~ + ve —i• 0 .

Hence the identification of WM with positively charged bosons is confirmed, in agreement with our discussion in Sec. 9.10. The fields A® and BM will be replaced by Afl and Z^, where

516 QUANTUM CHROMODYNAMICS AND THE STANDARD MODEL Hence, the lepton term in the original Lagrangian (15.84) becomes

where, for convenience, a new combination of coupling constants has been introduced,

2V2 cos 6W y/2 sin 26W

We will regard e and 6W as the independent parameters and introduce gw for convenience only. Note that the Lagrangian density (15.107) includes the new interactions shown in Fig. 15.6.

Next, look at the Lagrangian for the scalars. Reducing the covariant derivative gives

Recalling that tj and ZfL are real fields, we see that

2 4cos2

We have the proper kinetic energy term for a scalar, called the Higgs, and interactions of the scalar with the gauge bosons. However, more significantly, we have mass terms for the bosons. The squares of these masses are the coefficients of the W^W1 term and the \ZtlZ'L terms. Hence

4 cos2 6w

Hence the mass ratio is completely determined by the Weinberg angle,

These masses have been measured in colliding beam experiments. Recent values are Mw = 80.22 ± 0.26 GeV and Mz = 91.173 ± 0.020 GeV. The average value of sin2 9W extracted from many experiments is 0.2325± 0.0008 [RP 92].

• the operator | |

|Z | |

at each vertex where a Z° weak boson |
e 1 e Fig. 15.6a |

with polarization /j, is emitted from or | |

absorbed by an electron. |

• the operator |
at each vertex where a Z° weak boson with polarization /i is emitted from or absorbed by a neutrino. |
v 1 V Fig. 15.6b |

• the operator | |

-igw cos0w7M (1 - 75) |
iw- |

at each vertex where a W+ weak boson |
e 1 v |

with polarization fx is emitted from or |
_I__ |

a W~ boson is absorbed by a neutrino, |
Fig. 15.6c |

converting it to an electron. Electron | |

to neutrino conversion is described by | |

the same factor. |

Fig. 15.6 Interactions and Feynman rules for the lepton sector of the Standard Model.

proportional to the electron mass. The Feynman rules for all Higgs couplings are summarized in Fig. 15.7.

Finally, we expand the gauge field part of the Lagrangian. This generates many couplings of the photon, W± and Z and will ultimately confirm our charge assignments. First, consider the terms involving the square of d^Alu - d^A1^ and the B^y term. Since the transformation connecting A^ and Z^ to A^ and B^ is orthogonal, and because tr(r+r_) = 1, we have immediately eld

KE terms where FaU is the usual EM field term, and

jfl V

Note the W^W^ term is twice as large, as required of a complex vector field with two real components (recall Prob. 7.3).

Next, we calculate the four-gauge coupling terms. To reduce these, use

63ijA^Ai = i [WlW„ - WlW„] (tujA^Ai + i^ijAiAl) = -iWl [sin 9WAU + cos 6)WZU] - (n ~ n)

(eujA^Al - ienjA1 Ai) = iW^ [sin 9WA„ + cos9WZV] - (n v2

Hence the four-gauge couplings are

Afield

= - W [WlWvW^W* - WlW^WvW] 49 - [sin2 ewAvAv + sin 26wAl/Zv + cos2 6WZVZV

+ g2WlW„ [sin2 9WA^AV + i sin29W {A»Z» + A»ZV + cos2ewztizv] .

Note that all of these couplings conserve charge and that all involve the charged bosons. Finally, the three-gauge couplings can also be found with the help of (15.118). We have

Afield

+ ig sin 9W Av [WW^ - W^W^} + ig cos ewzv [ww^ - wrtwßV]

There are many terms, and it is helpful to separate out those which correspond to the expected electromagnetic couplings of the Wß field. Recalling that e = g sin 6, the terms proportional to e and e2 can be readily isolated from (15.120) and

(15.119). These are also the terms proportional to A2 in (15.119) and A in

(15.120). These terms give:

¿field

If these terms are combined with the kinetic terms for W, we have

¿field

This is precisely the result expected from minimal substitution; if dß —> dß+ieAß, we would obtain (15.122) from the free W Lagrangian in (15.116). Furthermore, the sign identifies W as a field with plus charge e. The electromagnetic couplings of the W+ boson are given in Fig. 15.8, and the self-couplings of the W's and the Z are shown in Fig. 15.9.

At this time, we collect together all of the transformed terms into a Lagrangian density with four terms:

"scalar

These terms are combinations of terms from different parts of the original Lagrangian, collected together for convenience. The first term includes the original lepton part, (15.107), plus the electron mass term which came from the original interaction term. It is

The field part contains the free field terms together with the electromagnetic interactions and the boson mass term:

- I [Wl„ - ie (AnWt - A„Wl)] [W^ + ie (AW - A^W")} ■

• the operator | ||

ie |
(p + p')ß g»a |
Hn |

- (p' + q)„ 9vl\ - (p - q)x g»»} |
puJj^ | |

at each vertex where a photon with |
Fig. 15.8a | |

momentum q and polarization fi is ab | ||

sorbed by a positively charged spin one | ||

boson with incoming momentum p and | ||

polarization v and outgoing momentum | ||

p' and polarization A. |

• the operator | |

-ie2 [2gilvgap | |

—9/117 9 up — 9p.p9ua\ |
v |

o Srr P | |

at each vertex where two photons with | |

polarization p, and i> are emitted from |
Fig. 15.8b |

or absorbed by two weak charged | |

bosons with polarizations a and p. |

Fig. 15.8 Quantum Electrodynamics of a spin one boson. The scalar part includes the Higgs kinetic energy piece and all Higgs interactions:

and finally the "interaction" part includes only the weak interactions of gauge bosons:

These Lagrangians account for the large number of interactions summarized in Figs. 15.6-15.9 and in Appendix B.

The particle content of the transformed Lagrangian was summarized in Table 15.1. There are five independent parameters. After the U gauge transformation, they are e, the absolute value of the electron charge, 6W> the Weinberg angle, me, the electron mass, Mw, the mass of the charged boson, and rn H, the Higgs mass. These are related to the original parameters through

Gev g sin 0vv

gv 2

The parameter gw is only a shorthand for the combination (15.108).

We now look at the high energy behavior of the electroweak interactions.

'Ve close this chapter with a short calculation which illustrates how the Standard Model solves a longstanding problem associated with the weak interactions. This calculation will also illustrate the new Feynman rules given in Figs. 15.6 and 15.9.

As discussed in Sec. 9.10, a massive vector particle (such as the W± or the Z) has three polarization states, owing to the fact that it can always be brought to rest and polarized in any of the three independent directions in space (recall that this is not true for a massless particle). The four-vectors e1M which describe these three states can be conveniently defined in the rest frame by the requirements

where p0 = (m, 0,0,0) is the four-momentum of the particle at rest. These requirements are clearly satisfied by any three-space unit vector with a zero time component, and there are precisely three independent such vectors. The polarization states of a moving particle can then be obtained by boosting the rest frame polarization vectors. This will preserve the requirements (15.129), which can then be used to construct the polarization vectors in an arbitrary frame. For example, if a vector particle has momentum = (E, 0,0,p), then three independent

This good behavior at high energy was obtained only through a delicate cancellation of diverging contributions from different terms. While this cancellation was obtained easily in this example, it presents a serious problem when these Feynman rules are used to calculate loop diagrams. Now the divergences due to the longitudinal polarization will show up in the bad high energy behavior of the spin one propagator; the p^/M2 term in the propagator (9.138) gives strong ultraviolet divergences, which must be canceled by diverging contributions from other terms. This is an unfortunate feature of the Feynman rules in the U gauge (which are the ones given in this chapter), but there are gauges (the gauges) in which the boson propagators do not diverge as p —► oo and in which all terms are individually well behaved at high momentum. The disadvantage of these R^ gauges is that they produce (several) ghosts and many ghost interactions which greatly increase the number of interactions and diagrams which must be calculated. However, it turns out that the advantage of the improved convergence for higher order loop corrections outweighs the disadvantage of having to use ghosts, and the gauges are preferred when loop calculations are to be carried out. For a discussion of the Feynman rules for the R$ gauges, see Cheng and Li (1984).

PROBLEMS

15.1 Prove the relation (15.68). Show that it is true for k\ = z and k2 = —z and then generalize the result.

15.2 Compute the electron-neutrino elastic scattering amplitude to second order in the weak coupling constant g2.

Relativistic Quantum Mechanics and Field Theory

FRANZ GROSS Copyright© 2004 WILEY-VCH Verlag GmbH

CHAPTER 16

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