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Quantum Field Theory


The Standard Model

The Standard Model, based on the gauge group SU(3) X SU(2) X U(1), is one of the great successes of the gauge theory. It can describe all known fundamental forces except gravity. However, the Standard Model is certainly not the final theory of particle interactions. It was created by crudely splicing the electroweak theory and the theory of quantum chromodynamics (QCD). It involves 18 unknown parameters, cannot explain the origin of the quark masses or the various coupling constants. The theory is rather unwieldy and inelegant. Nevertheless, not only is it renormalizable, it can explain a vast number of results from all areas of particle physics. In fact, there is no piece of experimental data that violates the Standard Model. The following is a crude attempt to provide a glance of the subject matter by introducing the Lagrangian density for the Standard Model. The field equations are derived by minimizing the action, which is related to the Lagrangian density. Thus instead of writing down the field equations explicitly such as in Eqs.(17) - (20) or Eqs.(13) and (38a), the dynamics of the electro-weak interaction can be expressed in term of the Lagrangian density:

This is known as the Weinberg-Salam Model. The lepton Lagrangian density in Eq.(40) consists of three parts. 1 is the gauge bosons part; 2 is the fermionic part; and 3 is the scalar Higgs sector, which generates mass for the gauge bosons and the fermions. Experimentally, the predictions of the Weinber-Salam model have been tested to about one part in 103 or 104. It has been one of the outstanding successes of the field theory, gradually rivaling the predictive power of QED.

The above formulism can be carried over to the electro-weak interactions between quarks with the massless neutrino replaced by the up quark u (which has mass), and the electron replaced by the down quark d. In order to get the correct quantum numbers, such as the charge, the covariant derivatives are different from the lepton as shown below:

where the coupling constants g and g' are also different from the case of leptons.

The theory for the strong interaction is called quantum chromodynamics (QCD), which has the Lagrangian density:
---------- (43)
Since the gauge bosons (the gluons) are massless, the Lagrangian density appears to be in a much simpler form than the electro-weak interactions in Eq.(40).

The Lagrangian for the Standard Model then consists of three parts:
---------- (43a)
where WS stands for the Weinberg-Salam model, lept. and qurk. stand for the leptons and quarks that are inserted into the WS model with the correct SU(2)XU(1) assignments. It is assumed that both the leptons and quarks couple to the same Higgs field in the usual way. From this form of the Standard Model, several important conclusions can be drawn. First, the gluons from QCD only interact with the quarks, not the leptons. Thus, symmetries like parity are conserved for the strong interactions. Second, the chiral symmetry, which is respected by the QCD action in the limit of vanishing quark masses, is violated by the weak interactions. Third, quarks interact with the leptons via the exchange of W and Z vector mesons.

The Standard Model can be written either in mathematical form as summarized in Eq.(43a), or in pictorial form, using the Feynman diagrams as shown in Figure 04. It is convenient to rewrite Eq.(43a) into a shorthand notation in describing the Feynman diagrams:
---------- (44)
where G stands for the gluon field, W for the vector mesons, F for the photon, H for the Higgs, and fi for the fermions.

It was mentioned earlier that there is a certain amount of mixing between the d quark and s quark from different generations. Although the Standard Model does not explain the origin of this mixing. The Cabbibbo angle, however, allows us to parametrize our ignorance. It is found that the most general form of mixing can be expressed by the mixing matrix V:

D' = V D,
where

Flavor Mixing cij = cosij, sij = sinij, and the phase angle tunrs some matrix elements into complex numbers, thereby violates CP invariance (CP invariance demands that V* = V).

Experimentally, the mixing angles ij are either smaller than or comparable to the Cabibbo angle 1 ~ 15o. Thus, the quark mixing is relatively unimportant. A similar mixing matrix exists in neutrino mixing between the flavor states (e, , ) and the mass (mixed) states (1, 2, 3). Neutrino mixing is large in comparison to the quark mixing as shown in
Figure 04h. It leads to the detection of neutrino mass.

Figure 04h Flavor Mixing
[view large image]


The mixing angle must have the same value for every electroweak process. It is observed to have the same value everywhere, to an accuracy of about one percent. Other successful predictions include the existence of the W and Z bosons, the gluon, the charm and the top quarks. Z boson decays have been confimred by LEP in 20 million of such events.
However, the Standard Model contains 26 free parameters:

3 coupling constants + 2 Higgs parameters + 2 x [3 generations x (2 fermions masses) + 4 CKM parameters] + 1 instanton§

For massless neutrinos and no leptonic mixing angles, there are still 19 free parameters. With so much arbitrariness, the Standard Model should be considered only as the first approximation to the true theory of subatomic particles, i.e., it is an effective theory to be explained by more fundamental theory.

    Following is a list of subjects that the Standard Model fails to explain. Either it is not in the formulation or it is just plugged into the theory without explanation of its origin.

  1. The cosmological constant or vacuum energy.
  2. Dark energy.
  3. The inflaton in the first fraction of a second of the Big Bang.
  4. Matter-antimatter asymmetry.
  5. Cold dark matter.
  6. The form of the Higgs field.
  7. Hierarchy problem - huge Higgs boson mass implies huge mass for all elementary particles.
  8. Gravity.
  9. Masses of the quarks and leptons.
  10. Three generations of elementary particles.
Half of the list above from 1 to 5 is related to cosmology and astronomy. It emphasizes that understanding of the largest and the smallest phenomena must come together. Supersymmetry can address items 1, 3 - 7, while superstring theory may be able to explain items 8, 9, 10. Thus item 2 about dark energy remains to be the most enigmatic subject in physics and astronomy.

Pear-shaped Nucleus In spite of these shortcomings as mentioned above, the Standard Model has been proven to be remarkably resilient under various verifications including the latest measurement for the mass of the Higgs particle. The most recent attempt (in 2013) to break SM is to measure the shape of some nuclei, which would become pear-shaped (Figure 05a) in the presence

Figure 05a Pear-shaped Nucleus [view large image]

of permanent electric dipole moment (EDM). Since EDM would violate the T symmetry (and thus also introduce CP violation) in SM, the detection of specific radiation patterns (from the pear-shaped nuclei) will indirectly indicate the necessity of new physics. It is
found that radon (Z=86) shows only modest enhancement of the octupole patterns (mostly from vibrational deformation), whereas radium (Z=88) yields strong enhancement (as intrinsic deformation). It is expected that thorium (Z=90) and uranium (Z=92) may exhibit even stronger patterns (to be confirmed by experiments with the next generation accelerators).

§ Instantons is the solution of the Euclidean version of Yang-Mills equations. The purpose is to probe the nonperturbative realm of gauge theories. It is called instanton because it creates an almost instantaneous blip (peak) in the Lagrangian. They are not particles and have no direct physical interpretation. Rather, it reveals that the vacuum of Yang-Mills theory actually consists of an infinite number of degenerate vacua, so the true vacuum must be a superposition of all of them.

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