Quantum Field Theory

Contents

Action Principle

---------- (1a)

where the Lagrangian L in unit of energy is a function of the position q and velocity =dq/dt of the particle, and the integration on time t is over the trajectories from "a" to "b" as shown in Figure 01a. Thus S has the unit of erg-sec. Note that S is not just a simple function of t - rather it is a function of the entire set of points q(t). It is a function of the function q(t), or a "functional" of q(t). In other words, to say S is a functional of the function q(t) means that S is a number whose value depends on the form of the function q(t), where t is just a parameter used to specify the form of q(t). The action S is sometimes written as S[q(t)] to emphasize that it depends on the form of q(t). Figure 01a shows three different forms of

Figure 01a Functional [view large image]

q(t) (blue, green, and red), the corresponding Lagrangian L, which is determined by q(t), and the action S[q(t)], which would have three different numbers equal to the areas under L indicated by either the blue, green or the red curve. The integration of the Lagrangian L

S / q(t)  =

---------- (1b)

- L = constant ---------- (1g)

which is the conservation of total energy (kinetic energy + potential energy) for a particle moving in a potential V(x). The momentum is not conserved in this case. However the dependence on x is eliminated in the free field case where V(x) = constant. It follows from Eq.(1h) that dx/dt = constant. This is the Newton's first law, which states that the object would not experience acceleration if there is no external force acting on it. Thus both the momentum m(dx/dt) and energy (m/2)(dx/dt)2 are conserved; or in term of symmetry, the system is now independent of time and space. Figure 01b shows schematically the similarity and difference between geometrical and theoretical physics symmetries. By symmetry, they both mean something is unchanged after some sort of rearrangement.

Figure 01b Symmetry [view large image]

However, in the case of geometrical symmetry it is the configuration that remains the same after the operation. Whereas in theoretical physics the invariance is about the form of the equations before and after the transformation.

Path Integral

iS/

classical theory. The classical trajectory is recovered as 0. In general, some of the paths may indicate movement at greater than light speed, or in violation of energy-momentum conservation law. Those paths would be related to the virtual particles; they need only to obey the uncertainty principle. Their contribution to the transition amplitude is usually very small. Figure 01c shows the different paths in 3D perspective dimensions, each one of which contributes a value to the action S (defined by Eq.(1a)) as indicated by an arrow or the axis

Figure 01c Action Principle [view large image]

labelled [q(t)]. The quantum mechanical (probability) amplitude for going from q(ta) to q(tb), where "a" and "b" denoting the end points, is known as the Feynman Path Integral:

---------- (1j)

Field Equation

Symmetry	Group	Conservation Law
Space-time translation	Poincare (x x + a)	Energy-momentum
Space-time rotation	Lorentz (x = x, = -)	Angular momentum, velocity of light
Interal rotation of complex field	U(1) in electromagnetic interaction	Electric charge
Internal rotation of lepton or quark field	SU(2) in weak interaction	3rd component of weak isospin
Internal rotation of quark field	SU(3) in strong interaction	Color charge

Table 01 Conservation Laws

---------- (1k)

---------- (1l)

---------- (2)

---------- (3)

Second Quantization

---------- (4)

---------- (5)

---------- (6)

Noether's Theorm and Charge Conservation

	The Noether's theorem in field theory takes the form of the conservation of current: where the four-current J is defined as: , where is the index for the space-time, indicates the component of the field, and represents a small variation in the th parameter denoted as .
Figure 01d Gauss Theorem [view large image]	From this conserved current, we can also establish a conserved charge by integrating the equation over a volume in the three dimensional space. Applying the Gauss theorem (see Figure 01d), the term with spatial derivative can be converted into a surface integral, which

or,  in a more familiar form:

,

Green's Function and Renormalization

---------- (7)

---------- (8)

---------- (9a)

---------- (9b)

Perturbation Theory and S Matrix

---------- (10)

---------- (11)

	states. It has to satisfy the unitary condition, i.e., f S*fiSfi = 1, which guarantees that probability is conserved in the process. Such relationship indicates that the matrix (Sfi) has an inverse, which in turn implies that it is possible to return to the initial state from the final state at least in principle although the probability is almost zero in practice so that the
Figure 01e S-Matrix [view large image]	second law of thermodynamics is "almost" never violated. This property is also related to the conservation of information, which caused so much trouble for Stephen Hawking.

---------- (12) ---------- (13)

---------- (14)

---------- (15)

Feynman Diagram

---------- (16)

	Figure 01e corresponds to the scattering of two nucleons by exchanging a pion. The internal line represents the probability amplitude for a virtual particle to travel from one place to another (x y) in a given time with greater than light speed, or to travel with off mass-shell 4-momentum k, which could have arbitrary value in violation of energy-
Figure 01e Nucleon-Nucleon Scattering [view large image]	momentum conservation but allowed by the uncertainty principle. Mathematically, it is expressed by the Green's function:

Table 02 Feynman Rules [view large image]	Table 03 Momentum Representations [view large image]

As an example to demonstrate the power of Feynman diagram, the nucleon-nucleon scattering in Figure 01e will be used to evaluate the corresponding S matrix from the Feynman diagram (in momentum space, see Table 03):


1. Collect the 4 factors from external nucleon lines (for both incoming and outgoing).
2. Multiply 2 coupling constants and 2 delta functions from the vertex x and y respectively.
3. Write down a propagator for the internal line.
4. Integrate over internal momenta. In this case of tree diagram, the delta functions enforce the rule for energy-momentum conservation such that the internal momenta k = p₁ - p₃ = p₄ - p₂. Thus, the integration can be carried out trivially. Such is not the case for a loop diagram (see for example the self-energy diagram in Figure 03e).
5. The integration of the k-space yields: -ig₀²/[(p₁ - p₃)² - m₀²]
6. The final result is: S⁽²⁾ = (m/V)²/(_p1p2p3p4)^1/2 {(-ig₀²)/[(p₁ - p₃)² - m₀²]} ,
   where m and m₀ is the mass of the nucleon and pion respectively.
7. The scattering cross section is proportional to the squared S matrix: |S⁽²⁾|².

Quantum Electrodynamics (QED)

---------- (24)

---------- (32)

---------- (36)

---------- (37)

The photon-electron processes are described by substituting Eq.(38b) to H_I in the S matrix expansion Eq.(15):


• The first order graphs -
  
  where the electron field has been decomposed to:
  
  and the symbol X may stand for an interaction with an external classical potential, the emission of a photon, or the absorption of a photon. The realization of a particular process depends entirely on the initial and final states.



• The second order graphs (tree diagrams) -
  
  
• Two-photon annihilation - Pair of an electron and positron into two gamma rays.
• Compton scattering - It occurs when an electron and a photon collide and scatter elastically.
• Moller scattering - It is the scattering of two relativistic electrons.
• Bremsstrahlung - It is the process by which radiation is emitted from an electron as it moves past a nucleus.



• Second order graphs (loop diagrams) -
  
  
• Vacuum polarization - It produces a correction to the coupling constant (electric charge) and contributes to the Lamb shift, which is a small splitting of hydrogen atom energy levels caused by the interaction with the virtual pair (of electron and positron).
• Vertex correction - A correction to the electron vertex function, which contributes to the anomalous electron magnetic moment.
• Self-energy - The interaction of a charged particle with its own field and it usually gives rise to infinite self-energy and infinite mass.
• Mass renormalization - The observed mass is generated by combining the bare mass and the (calculated) divergent mass.

From Coulomb Scattering to Deep Inelastic Scattering

		This formula is related to the probability of an incident particle with kinetic energy Ek scattered into an solid angle d in the direction . The experimental apparatus to verify the Rutheford scattering is shown in Figure 02b. Figure 02c is the measured angular distribution of the scattered
Figure 02a Impact Parameter	Figure 02b Rutherford Scattering [view large image]	particles with incident energy of 15 Mev.

		The agreement between theoretical prediction and experimental data overruled J. J. Thomson's "Plum Pudding Model", in which electrons are suspended in a pudding-like positively charged substance that contains most of atom's mass. The success of Rutherford's theory gives him the distinction of having discovered the atomic nucleus. However, there is a problem. When the incident energy is over about 27.5 Mev, the experimental data start to deviate from the theoretical curve. Figure 02d
Figure 02c Cross Section vs Angles [view large image]	Figure 02d Cross Section vs Energy	shows the deviation at a fixed angle of 60o.

		region: , q2, and 0 x = q2/(2M) 1 (elastic scattering corresponds to x = 1), W1F1(x), W2F2(x). The F1(x) and F2(x) are finite and depend only on x. This relation is called Bjorken scaling. The usefulness of the parton model is that we can compare the scaling behavior of F1 and F2 against the various predictions for spin-0 and spin-1/2 partons, e.g., F1(x) = 0, and 2xF1(x) = F2(x) respectively for each case. Experimentally, the spin-1/2 model is reasonably satisfied (see Figure 02f). This result suggests that the partons
Figure 02e Deep Inelastic Scattering [view large image]	Figure 02f Form Factor [view large image]	are, in fact, just the quarks. Thus we have progressed from the discovery of atomic nucleus to unravelling the structure of the nucleon in about 100 years.

Compton Scattering

The quantum theory of light postulates that photons behave like particles except for the absence of any rest mass. If this is true, then it should be possible for us to treat collisions between photons and, say, electrons in the same manner as billiard-ball collisions are treated in classical mechanics. Figure 02g shows how such a collision might be represented, with an X-ray photon striking an electron at rest and being scattered away from its original path while the electron receives an impulse and begins to move. Using just

Figure 02g Compton Scattering [view large image]

kinematics the change in the wavelength of the X-ray can be expressed in the form: ' - = o (1 - cos) ---------- (38f)

	The Compton scattering belongs to the second order graphs in the Feynman diagrams. Since there is no loop diagram involved, it is still susceptible to classical treatment. The cross section has been computed by J. J. Thomson in the 1900's: d()/d = (ro2/2) (1 + cos2) ---------- (38g) where ro = e2/mc2 = 2.82x10-13 cm stands for the classical radius of the electron.
Figure 02h Scattering Cross Section[view large image]	The experimental scattering cross section at different incident energy are shown in Figure 02h. At the low energy range of about l Kev, the cross section in Eq.(38g) shows good agreement with minima at = 90o and 270o, and maxima at = 0o and 180o. Asymmetric patterns appear at higher energy with the maximum shifted to the forward direction.

Lamb Shift

The energy levels calculated from the Dirac equation with the electron in a Coulomb potential is given by: where Z is the number of positive charges, m is the mass of the electron, and is the fine structure constant. The formula predicts that the energy levels of the hydrogen atom should depend only on the principal quantum number n and the total angular momentum j = L + S, so that the energy levels of the 2S1/2 (n=2, j=1/2) and 2P1/2 (n=2, j=1/2) states should be degenerate (the same). In 1947, Lamb and Retherford demonstrated that this is not the case. The 2P1/2 and 2S1/2 energy levels are separated by more than 1000 MHz (see Figure 03a).

Figure 03a Lamb Shift [view large image]

• All the terms in V^(eff) come from the radiative corrections.
• The first and second terms arise from the vertex and self-energy corrections. Since _min represents the photon mass, which is equal to zero, it introduces a divergence into the formulation. However, it is eventually cancelled by the infrared correction and disappears completely in the level shift calculation. These two terms contribute the majority of the Lamb shift with a correction of 1011.41 MHz.
• The third term is related to the vacuum polarization. It actually brings down the Lamb shift by -27.13 MHz.
• The last two terms come from the effect of anomalous magnetic moment interaction or spin-orbit interaction. These two terms together create a Lamb shift of 67.82 MHz.
• Those terms with the ⁽³⁾(x) is applicable only to the ²S_1/2 state, which has a high probability to be near the center of the Coulomb potential. On the other hand, the term with the quantum number L vanishes for the ²S_1/2 state since L = 0 in this case.

		more accordingly. By summing the contributions from : fine structure, hyperfine splittings, radiative corrections, recoil and proton size, ..., the total computed 2S1/2F=1 - 2P3/2F=2 energy difference (the dominate transition) is : E = 209.9779(49) - 5.2262(rp)2 + 0.0347(rp)3 where rp is the root-mean-square charge radius of the proton, numbers in parenthesis indicate uncertainty, and the energy is expressed in unit of mev = 10-3ev. The first term
Figure 03b Muonic Lamb Shift [view large image]	Figure 03c Laser Resonance [view large image]	in the formula is dominated by vacuum polarization, which causes the 2S states to be more tightly bound than the 2P states (Figure 03b,c).

	The traditional assumption for proton mass distribution is about 75% concentrated in a central core (size rp) with the other 25% lying outside in the halo up to 1.4 fm. An August 2010 calculation suggests that the discrepancy
Figure 03d Latest Proton Size [view large image]	between the original proton size and the latest (2010) experimental data can be reconciled if the halo band extends 4.7 times as far as previously thought (Figure 03d).

Renormalizable Theories

1. Pauli-Villars regularization - A fictitious particle of mass M is introduced to modify the propagator such that it behaves now as 1/k⁴, which is usually enough to render all graphs finite in the off mass-shell integral. Then the limit of M² is taken at the end of calculation to decouple the unphysical particle from the theory.
2. Dimensional regularization - The action is generalized to arbitrary dimension d, where there are regions in complex d space in which the Feynman integrals are all finite. The Feynman graphs pick up poles in d space to absorb the divergences of the theory into the physical parameters as d is reduced to four.
3. Lattice regularization - This is the most widely used in QCD for non-perturbative calculations. The lattice spacing serves as the cutoff for the space-time integrals.

Graph	Diagram	F	B	D	D'	D'+1	Parameter(s)
	Vacuum	2	1	0	0	1	(Ignored)
	Photon Vacuum Polarization	2	0	2	0	1	Z3-1
	Electron Self-energy	1	1	1	1	2	m and Z2-1
	Electron-Photon Vertex	2	1	0	0	1	Z1-1
	Photon-Photon Scattering	4	0	0	0 >	0 >	Nil (Convergent)
	Electron-Electron Scattering	2	2	-2	-2	-1	Nil (Convergent)

Table 04 Degree of Divergence

In Table 04:


• The time axis in the Feynman diagrams runs horizontally from left to right.
• The processes are related to quantum electrodynamics (QED).
• The number of Ferimon and boson internal lines (propagator) F and B can be counted from each diagram.
• The degree of divergence D is computed from Eq.(39a) with n = 0 for gauge theories. D is often referred as "primitive" divergence. For example, the electron-electron scattering diagram has no primitive divergence since D = -2. However, there is a subdiagram in its upper corner that looks like the vertex diagram, which is know to have D = 0 producing logarithmic divergence. In other words, divergences in subdiagrams are not counted as primitive divergences for the whole diagram.
• A gauge theory has a high degree of symmetry, which allows some primitive divergences to be cancelled and reduces the degree of divergence to D'. This is the case for the photon self-energy and the photon-photon scattering diagrams, which are actually primitively convergent.
• It is known that the divergences of a diagram are summarized in D' + 1 divergent parameters. The vacuum diagram and its divergence is usually ignored, on the grounds that it is not related to any external particles and cannot be measured experimentally. If we assume that the bare mass m₀, and the bare charge e₀ are adjustable infinite quantities, then they can be combined with the divergent parameters to re-formulate the theory into one that does not involve infinities. The mass m_R, and charge e_R in this new form will be "dressed" and agree with experimental measurements. The bare and renormalized quantities are related by the following formulas:
  
  m_R = m₀ - m
  e_R = [(Z₂Z₃^1/2)/Z₁]e₀ = (Z₃^1/2)e₀
  
  It also requires that the fields should be transformed like:
  
  _R = Z₂^1/2₀
  A_R = Z₃^1/2A₀
• The vacuum diagram in which virtual e^-e⁺ pairs and virtual photons are created and annihilated together, give terms that modify the vacuum energy. Energy shifts in perturbation theory are to be expected, but since there is no unperturbed vacuum with which to compare, such shifts are not measurable. The cosmological constant of general relativity gives a measure of the vacuum energy density that is very small. Thus, the vacuum energy density, whatever its origin, is taken to be zero.

The Divergence

Figure 03e is the Feynman diagram in momentum space for the electron self-energy (the insert portrays the unperturbed propagator). Using the Feynman rules, the S matrix can be written down for the transition of an electron from state (p, s) to state (p', s') in the form: where the self-energy correction (p) is a 4X4 matrix given by:                               -[A + (ip + m)B + ], and m is, at this point, finite but arbitrary.

Figure 03e Electron Self-energy [view large image]

Counting powers of the off mass-shell 4-momentum |k| = (k2 - E2)1/2 in the numerator and denominator of the integral suggests that it diverges linearly like d|k|. However, due to cancellations in the integrand, it diverges only logarithmically like d|k|/|k| log().

---------- (39b)

---------- (39c)

This divergent correction to be absorbed by the electric charge is denoted by 1/Z1. The other divergent correction is labeled as Z3 = 1/(1+C), where C is another infinite number from vacuum polarization (not shown here, it modifies the photon line AR = Z31/2A0). In the S-matrix expansion, each term has a number of the interaction Hamiltonian HI -e00 0 = -eRR R (integration over 3-D space is omitted here) if the renormalized charge further amalgamates the factor Z2, and Z31/2, i.e., eR = [(Z2Z31/2)/Z1]e0 = (Z31/2)e0 since the "Ward's Identities" has shown that Z1 = Z2. Thus in terms of the renormalized quantities, all the divergent terms disappear from the QED formulation, which yield amazing predication with an accuracy up to one part in trillions. As one part of Eq.(39c) gives rise to Z2, which is divergent at |k| , the other part diverges logarithmically at low value of |k| as |k| 0. This is an example of what is known

Figure 03f Vertex Correction [view large image]

as the "infrared divergence". A possible way to dispose of this difficulty is to temporarily assign a very small but finite mass to the photon; this can be accomplished by modifying the photon propagator in Eq.(39c) according to the following prescription:

...

which increases as Q increases (or, equivalently, as the probing distance becomes shorter, see Figure 03g). Similar vacuum polarization effects occur in QCD with similar Feynman graph but for gluon exchange between quarks and antiquarks. There is an additional contribution from the gluon self-coupling as shown in Figure 03g. The effective coupling constant for strong interaction for large Q is now in the form: where nf = 6 is the effective number of quark flavours for large Q, and -1 is called the

Figure 03g Effective Coupling Constant [view large image]

confinement length since s becomes infinite when Q = . The phenomenon of asymptotic freedom in QCD is associated with the fact that s 0 as Q .

Yang-Mills Theory

		attempt to formulate a gauge theory for the weak interaction. It is a generalization of the U(1) gauge theory for QED with non-abelian group (for the internal space). It fails to account for the mass of the gauge bosons in the interaction. However, it provides a good framework for QCD in strong interaction as its mediating bosons are massless. Since non-abelian gauge bosons can be emitted and absorbed from the gauge bosons themselves (see Figure 03i), they create anti-shielding for the quark anti-quark pairs in the virtual particle cloud (the Yukawa cloud). The color charge becomes weaker at smaller probing distance, and leads to
Figure 03h Yang-Mills [view large image]	Figure 03i Gluon Vertex [large image]	asymptotic freedom. Conversely, at larger distances, the color charge increases, so that the quarks tends to bind more tightly together giving rise to quark confinement, which is the flip side of asymptotic freedom.

The Yang-Mills theory appears to be the intermediate stage between QED and the Standard Model. A comparison shows the progress made by the Yang-Mills theory since QED and the problems that prevent its development to a full-fledged theory of weak interaction.


• Comparison of the Yang-Mills formula with Eq.(13) and Eq.(41b) reveals that they differ by a factor of "-i" in the derivative term. This is the result of adopting different convention for using either covariant or contravariant component (lower and upper indexing) and for using different definition on the time coordinate x₀. Such different conventions can be translated into one another by noting that _k = -i ^k where k = 1, 2, 3 and x₀ = it in the Yang-Mills theory; while x₀ = t in some other notations.
• The field equation for the fermion in Eq.(39f) appears to be similar in form to the one for QED (see Eq.(13)) with the interaction Hamiltonian in Eq.(38b), except that the abelian gauge boson field A is now replaced by the non-abelian gauge boson fields ^ab^a.
• The field equation for the gauge boson in Eqs.(39e) and the definition of f in Eq.(39h) is similar to the electromagnetic field equation in Eqs.(23) and (24), except an additional non-linear term, indicating that the b's interact among themselves (with the same coupling constant, also see Figure 03i) and is responsible for the anti-shielding effect.
• While the f in Eq.(39d) can be identified to W in Eqs.(41a) and (42a) for the Standard Model. The Yang-Mills theory fails to account for the chiral nature of the fermion fields in weak interaction. Parity violation in weak interaction has not been discovered until 1957 - four years later than the introduction of the Yang-Mills theory.
• The mass for the vector bosons in weak interaction is absent in the Yang-Mills theory. Only the fermion mass is hand written into the formulation. The Standard Model takes care of both via the interaction with the Higgs field in Eq.(41c). The problem with the gauge-field mass is recalled fondly in an anecdote by C. N. Yang (1922-).
• The Lagrangian density Eq.(39d) in the Yang-Mills theory is virtually identical to the QCD's in Eq.(43), except that the number of fermions (quarks) is six altogether. Such data were not available to Yang and Mills in 1953.
• For convenience, most of the formulas in quantum field theory are expressed in natural units with = 1, c =1, then becomes dimensionless. Otherwise, the mass term in the formulism would be m/c; while the interaction term would appear as (/c)b.
• The Yang-Mills theory is applicable to the early universe when the temperature was higher than 250 Gev (see table on "A History of Cosmic Expansion"). The Higgs field, which endows mass to most of the elementary particles, was still in its symmetrical state, and was unable to interact with them. Thus, all particles are massless and the Yang-Mills theory is perfectly suitable to describe them.

Weak Interaction

Of the 4 different kinds of forces in the universe, gravity is attractive; electromagnetic force can be attractive or repulsive; the strong force provides a powerful attraction at short distance; the weak force works in very short distance, its action is to change one form of particle into another such as the beta decay of a free neutron in Figure 03j (strong force in stable nucleus inhibits the neutrons from beta decay). Note that the simple picture in Figure 03j already embraced the conservation of mass-energy, charge, and lepton number (nl).

Figure 03j Weak Interaction in Picture [view large image]

The strength of these forces varies according to the separation between particles, but in term of their coupling constants the ratio between the strong, electric, weak, and gravity is about 1 : 10-2 : 10-13 : 10-38. The understanding of the weak force took many twists and turns.

• Discovery of neutrino - Beta decay was first observed in 1900. It caused a great deal of anxiety among physicists at

  that time because while both the alpha and gamma rays are emitted with discrete spectra following the law of mass-energy conservation, beta rays appeared to exhibit a continuous spectrum. The problem persisted until 1930 when Pauli suggested that an invisible entity called neutrino shares the decay energy with the electron - the two particles together restore the conservation of mass-energy to its rightful place. The neutrino was dully discovered in 1953. It interacts only through the feeble weak interaction and gravity making its detection very difficult. Later on it is realized that

  Figure 03k Lifetime of Decays [view large image]

  the elusive entity has to be anti-neutrino to converse lepton number (Figure 03j). It is now understood that all fermions feel the weak force which is characterized by long lifetime as shown in Figure 03k.

  
  
• Fermi's Theory - In 1934 Fermi expressed the beta decay mathematically by considering the interaction of the 4 fermions at a single point as shown in Figure 03la and in the formula below:

  	M = GF(pun)(eu) where GF is the weak coupling constant, the u's are the Dirac spinors for the proton, neutron, electron, and anti-neutrino; the factor contain the essence of the weak interaction effects, in case it is equated to the gamma matrices, it would prescribe the
  Figure 03l Fermi's Theory [view large image]	current-current interaction (see Table 05 below); M has been labeled as (probability) amplitude, matrix element (of the interaction), (phenomenological) Lagrangian, or (interaction) Hamiltonian similar to the one for QED as shown in Eq.(38b).

  This equation describes the event occurs at a single point. It could be a good approximation for a very short range force, if it doesn't go up with increasing energy and if it is renormalizable. The problem with high energy has been resolved by inserting into the equation a propagator term 1/(M_W² - q²) corresponding to mediation with a massive boson, e.g., the W^- (q is the magnitude of 4-momentum for the boson, see Figure 03lb). The problem with non-renormalization remained for a long time until the successful formulation of the Standard Model in 1971.



• Parity Violation - Parity is the operation P that reverses the coordinate (x,y,z) to (-x,-y,-z) of a system (e.g., a particle or collection of particles). It is equivalent to a mirror reflection followed by a rotation through 180^o. The parity can be +1 or -1 for a system depending on whether the corresponding field remains the same or flips a sign, i.e., in simple

  mathematical expression: Pu(x,t) = u(-x,t). The overall parity is given by the multiplication of its parts, e.g. (+1)x(-1)x(+1) = (-1) etc. It has been long held that parity is conserved before and after all particle interactions. The problem started in the 1950's when it was observed that the and mesons seem to be identical in every respect but their decay products appeared to be very different. While the ends up with three mesons the decays into two. Since the has parity of -1, that means the two end products have different parity, one of them must violate parity. Further study by T. D. Lee and C. N. Yang strongly suggested that is the case.

  Figure 03m Parity Violation [view large image]

  Almost immediately C. S. Wu devised and carried a test, which confirmed the idea conclusively (Figure 03m). The result also implied that only the electrons with left-handed spin participate in the weak interaction. The three Chinese physicists shared

  the 1957 Nobel Prize for their efforts in identifying this peculiar behaviour in weak interaction.
  
  
• V - A interaction - It is obvious that the Fermi's Theory is not able to accommodate the experimental fact of parity violation. Theorists were looking for a way to construct a realistic formula from the various types of fermion current as shown in Table 05, which lists the scalar, vector, and tensor under coordinate transformation (where is the Dirac spinor simimlar to the u here). A pseudo or axial is added in front for those involve the ⁵:
  .

  		After a long search for the magic formula, it turns out that parity violation can be expressed in combination of vector and axial vector (V - A)
  Table 05 Types of Fermion Current [view large image]	Figure 03n Beta Decay, Modern Version	interaction, e.g., for the electron anti-neutrino portion:

  [_e(1 - ⁵)u]/2 = _eLu_L, where the subscript L indicates that only the left-handed elements are involved. There is no right-handed neutrino, and the right-handed electron only participates in the electromagnetic interaction. A similar expression is derived for the d quark (in neutron) and the u quark (in proton) as shown in Figure 03n. The updated version of M for beta decay is now in the form :
  
  where g_w is the weak coupling constant and V_ud is the u-d element in the CKM matrix. It is the relative probability that the down quark decays into up quark in the weak interaction.




Gauge Theory - The weak interaction in gauge theory (as mentioned in the previous section) belongs to the SU(2) group, which is represented by 2x2 matrices. Thus the Dirac spinor has to be split into one part for the right-handed lepton (electron only) and the other for the left-handed neurtino and electron. This 2-elements representation is known as Weyl spinor. The mediating bosons appear naturally as the result of local gauge invariance. The concept of gauge invariance can best be visualized with an example, which portrays a boat floating on the water surface with the depth of water at 3 meters (Figure 03o). The gauge (the water level marker) at the left registers a reading of 6 meter on the surface and 3 meter on the bottom, while the measurements from the other one at the right are 9 and 6 meter correspondingly. The differences in the readings are the result of moving the right marker further down, but the depth of water remains the same at 3 meters - the picture has not been changed by moving the gauge around. In theorietical physics it is the governing equation that

Figure 03o Gauge Invariance [view large image]

is unchanged (invariant) under the change of some parameters, which turn out to be the phase angles in an abstract space. Local gauge invariance produces mediating bosons, while global gauge invariance is related to conservation law.

The following mathematical derivation shows the appearance of gauge boson from the requirement of local gauge invariance. The Dirac equation for the free electron has the form :
i - m = 0
The gauge transformation e^ig(x), where (x) is the gauge parameter and a function of x, and g is the coupling constant. The transformation change the original Dirac equation to :
i - g() - m = 0
The invariance is clearly spoiled by the extra term in the middle. The equation can be made invariant only if a boson interaction term -gA is inserted into the original Dirac equation since A A - () for the Lorentz gauge according to classical electromagnetism. In other word, the partial derivative should be replaced by the covariant derivative D = + igA. Thus the gauge invariance form of the Dirac equation is :
iD - m = 0
The mass term would be removed in the Standard Model because the mass for all kinds of particles is generated via the Higgs mechanism. The Dirac spinor would also be replaced by the two-elements Weyl spinor representing the left-handed neutrino and electron in the Standard Model.

The Standard Model

Meaning of the symbols in the Lagrangina density :
• The symbols and are indices for the space-time components running from 1 to 4. Whenever an index appears in both the subscript and superscript, it signifies a summation over these components.
• W^a is related to the three gauge (vector) bosons with the index "a" running from 1 to 3.
• F is the anti-symmetric tensor for the electromagnetic field as shown in Eq.(24), where the vector potential A is now denoted by B.
• f^abc is the antisymmetric tensor such that f¹²³ = +1, f²¹³ = -1, f¹¹³ = 0, ... etc.
• Since there is no right-handed neutrino in nature, the fermionic field consists of a left-handed isodoublet and a right-handed isosinglet as shown below -
  

  ---------- (42f)

  This curious feature, that the electron is split into two parts is a consequence of the fact that the weak interactions violate parity and are mediated by the V-A interactions, where V stands for vector and A stands for axial vector (also known as pseudovector, which changes sign under a parity transformation). The axial vector interaction is hidden in the last term of Eq.(42d). The asymmetric forms for the fermion fields in Eq.(42f) is a way to portrait the chiral nature of the objects in weak interaction - the left-handed version is different from the right-handed one. Note that in Eq.(42c) the right-handed field R does not participate in the weak interaction involving the vector bosons W^a.
• are the Pauli matrices as shown in Eq.(10) in the appendix on "Groups" with i running from 1 to 3. The four 4X4 metrices are constructed from the Pauli matrices and the identity matrix.
• g and g' are the coupling constants associated with the W and B boson respectively. G_e is the Yukawa coupling constant, which defines the strength of the interaction between the Higgs field and the lepton fields.
• The first three terms in Eq.(41c) are responsible for generating the mass of the gauge bosons, while the last term takes care of the fermion mass.
• is the Higgs doublet with four real components, three of which will be absorbed by the W, and Z⁰ to become massive. The diagram below shows a simplified Higgs field in the shape of a Mexican hat:
  
  
  This form of field exhibits a drastic effect called "spontaneous symmetry breaking". When the origin of the field is shifted to v = (-m²/2)^1/2 (the scalar field at minimum energy) by the transformation ' = - v, the fields W^a and B recombine and reemerge as the physical photon field A, a neutral massive vector particle Z, and a charged doubled of massive vector particles W. In terms of the Weinberg angle (mixing angle) tan = g'/g,
  
  Thus, the transformation from (B, W³) to (A, Z) can be considered as a rotation of the mixing angle.
  The masses of the gauge bosons are obtained from the formula:
  
  where we have used the known constants for G_F = 1 / v²2^1/2 = 1.14x10^-5 Gev^-2, e = g sin_W = (4)^1/2 = 0.3028, and from the measurement of sin²_W = 0.226 0.004 to evalutate the masses for the gauge bosons. The theoretical values are in good agreement with M_W = 80.4 Gev, and M_Z = 91.19 Gev determined by experiments.
• The mass of the electron is generated by the last term in Eq.(41c). Together with shift of the Higgs field v, it emerges as m_e = G_ev. While the neutrino remains massless. This formulation is necessary because the usual mass term in the Dirac equation is not gauge invariant when the left-handed component L and the right-handed component R of the fermion wave functions transform differently under SU(2). Thus it is assumed that the fermions start out with no mass, they acquire the mass by interacting with the Higgs field. Each fermion mass is therefore an unknown parameter (determined experientially now) in the Standard model. The attempt to find a theory which can predict these masses, remains an important but unsolved problem.

Meaning of the symbols in the QCD Lagrangina density :
• F^a is similar to the SU(2) gauge field in Eq.(42b). This is essentially the Yang-Mills field developed back in 1953 by generalizing the gauge invariance used in QED. It represents the eight massless gluons carrying the SU(3) "colour" force with the index "a" running from 1 to 8.
• The index "i" is taken over the flavors, which labels the up, down, strange, charm, top, and bottom quarks. The flavor index is not gauged; it represents a global symmetry. However, the quarks also carry the local colour SU(3) indices red, green, and blue (which is suppressed here). In other words, quarks come in six flavors and three colours, but only the colour index participates in the local gauge symmetry. It is the colour force, which binds the quarks together.
• The mass of the quarks m_i is derived from the interaction with the Higgs field in a formula similar to Eq.(41c).
• = .

---------- (43)

---------- (44)

Explanation of the terms in Eq.(44), and Feynman diagrams in Figure 04:

The square terms G2, ... are the energy-momentum tensor of the boson fields, they also include terms that describe interaction among themselves except for the photon, which don't have this kind of interaction (see Figure 04a, b, and g). The represents the covariant derivative. These terms signify interaction with the gauge bosons. The fjHfk terms indicate interaction of the fermions with the Higgs. The cjk's are the coupling constants, which determine the masses of all the quarks and leptons.

Figure 04 Standard Model [view large image]

The actual numerical values of these coupling constants are not given by the theory and must be measured from experiments.

• In Figure 04c, the vector mesons interacts with the left-handed electrons and right-handed anti-neutrino (not shown) only, while the right-handed electron interacts only with the photon.
• In Figure 04d, the last diagram actually involves a quark from the second generation, the strange quark, as well as a quark from the first generation, the up quark. This is an example of quark mixing, in which quarks of different generations get jumbled up. The amount of mixing is controlled by a parameter known as the Cabibbo angle ₁ such that the mixed state
  d' = d cos₁ + s sin₁.
• Interactions between quark and qluon can be expressed similar to Figure 04d.
• The Higgs boson interacts with just about all elementary particles (Figure 04e, and f) except the neutrino and gluon. These are the interactions that give the vector mesons and fermions their masses after spontaneous symmetry breaking. The neutrinos and gluons remain massless in the Standard Model. The annihilation of the vector mesons in Figure 04e indicates the short lifetime (and thus the short interaction range) of these particles.
• Figure 04g shows the Higgs interacts with itself. These self-interactions generate the Mexican hat potential so crucial for spontaneous symmetry breaking.
• Only the diagrams for the first generation are given in Figure 04; there are similar diagrams for the other fermion families.
• Figure 04 also leaves out diagrams that can be obtained by exchanging all particles with their antiparticles.

where

cij = cosij, sij = sinij, and the 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.

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.

	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

The SO(10) Group and Unification

Gauge-Group / Handedness	SU(3)	SU(2)	U(1)
Left-handed	ur2/3, uw2/3, ub2/3, dr-1/3, dw-1/3, db-1/3	0, e-1
Right-handed	ur2/3, uw2/3, ub2/3, dr-1/3, dw-1/3, db-1/3		e-1	0

Table 06 Fermions in Standard Model

An alternate arrangement unifies all these fermions into the 10 dimensional SO(10) group with the introduction of two color charges for weak interaction: The 5 color charges include the red (R), white (W), blue (B) in strong interaction, and the green (G) and purple (P) in weak interaction. Each one is represented by a separate 2-dimensional plane (with '+' and '-' components) to fit into the 10 dimensions in SO(10). Figure 05b lists all the fermions of the standard model in the first column. The left-handed species are not labeled, while the right-handed ones are represented by the anti-particles of the left-handed fermions and denoted with the superscript "c" since these two kinds are similar except for the opposite charges. The electric charge is now generalized to the hypercharge Y = - ( R + W + B ) / 6 + ( G + P ) / 4. The rules for assigning color charge to the fermions are such that complementary color charges, e.g., (+ +) or (- - -), ..., cancel each other. The particle carries the color charge corresponding to the left over '+' or '-' sign in the strong color charges (R, W, B) and to the single '+' sign in the weak color charges (G, P). Consequently, all the right-handed fermions do not participate in weak interaction (the cause of parity violation), and all the leptons do not take part in strong interaction. The hypercharge Y is computed by substituting +1 or -1 into the formula according to the '+' or '-' entry in the table respectively. Its value is actually the average of the sub-grouping. For example, the left-handed neutrino and electron carries 0 and -1 electric charge respectively, thus the average is -1/2.

Figure 05b SO(10) Group [view large image]

The right-handed neutrino N is the odd-man/woman out in the standard model. It is there to provide a mechanism for the mass of the neutrinos. It does not involve in any of the interactions - that's why it has not been observed (smelling like dark matter ?).

The SO(10) group breaks into many schemes. One of them is the SU(3)SU(2)U(1) corresponding to the charges of (R, W, B), (G, P), and Y respectively. It is this particular grouping that fit all the elementary fermions snugly together without much room to maneuver. The breaking down to various subgroups occurred long time ago soon after the Big Bang.

Charge	Gauge Field(s)	Gauge Particle(s)
r	grr', grw', grb', gluon fields	Gluons from red color charge
w	gww', gwr', gwb', gluon fields	Gluons from white color charge
b	gbb', gbw', gbr', gluon fields	Gluons from blue color charge
g	ggg', ggp', vector meson fields	W vector mesons
p	gpp', gpg', vector meson fields	Z0 vector meson
e	A, electromagnetic field	Photon

Table 07 Gauge Bosons

	They interact by gluon exchange such that the blue quark emits a blue-antired gluon thereby transforming itself into a red quark, while the red quark absorbs this gluon to become a blue quark. In other word, the gluon manages the creation and annihilation of the red-antired color charge pair, while fetches and donates the blue color charge from one quark to another
Figure 05c Strong Interaction	resulting in the exchange of color charge on the quarks during the course of the interaction.

energy, which enables penetration through the shielding by the virtual particles, the coupling strength of the various kinds of charges would merge into one unifying point. As shown by the diagram on the left of Figure 05d, it almost works, but not quite (the experimental errors is indicated by the width of the lines). The expectation comes true by introducing supersymmetry (SUSY), it even brings the outcast gravity to close proximity of the merging point (see diagram on the right of Figure 05d). This additional symmetry

Figure 05d Unification by SUSY [view large image]

requires a new partner for all the particles. They have the same charges (of all kinds) as their known partner, but different masses (heavier) and spins (integer 1/2 integer). Supersymmetry in effect doubles the number of particles in the standard model.

Lattice Theory

1. There are three color charges in QCD (red, green and blue), whereas QED comes with only two (the positive and negative charges). 2. The force carrier (the photon) in QED does not interact with each other, but the gluons in QCD interact among themselves. This is the reason why the photon fields spread out in QED, while the gluon fields confine to a flux tube in QCD (see Figure 05e). 3. The QCD coupling constant at large distance (~ 10-14cm) approaches to 1 making it impractical to use perturbative method. The lattice theory resorts to the use of path integral by generating thousands of paths weighted according to their likelihood under the particular rule that governs the physical evolution of the system.

Figure 05e Lattice Theory [view large image]

On the other hand, there is a price to pay for putting QCD on lattice:


1. Since the metric is Euclidean, it means that calculations with the theory are limited to the static properties of QCD such as confinement and perhaps the low-energy spectrum of states. It has difficulty calculating scattering amplitudes, which are defined in Minkowski space.
2. Lattice theory explicitly breaks continuous rotational and translational invariance, since space-time is discretized. All that is left is symmetry under discrete rotations of the lattice.
3. It is constrained by the available computational power. Many important effects (e.g., virtual pair creation etc.) are not included in the calculation back in the 20^th century.

	1. The three dimensional space is discretized into a finite, periodic gird as shown in Figure 05f. 2. The fermions are located at the nodes (vertices) of the grid with link (edge) "a" in between. 3. Normal time is converted to Euclidean time, i.e., t it. This 4th dimension will be added to the grid in Figure 05f. It follows that the transition amplitude ~ all pathseiS (where S is the
Figure 05f 3-D Lattice [view large image]	action/) now changes from oscillatory to dumping form: all pathse-S.

	4. In a Feynman diagram, it would be the gauge field that goes between the fermions. However if we want to retain gauge invariant in lattice theory, the link should be identified to: U(Cxy, An) = P exp(igaCxyAnaa),
Figure 05g The Link [view large image]	where Cxy is a path between x and y (Figure 05g), P denotes path ordering, An represents the gauge field at point n, g is the coupling constant, and a are the SU(3) generators in the form of 3X3 matrices.

	5. One way to construct gauge-invariant quantities out of the links is the Wilson loop (Figure 05h), which goes around along the links in a closed path, and takes the trace (Tr) of the product from each link (trace of a matrix A: TrA = ann). It can be shown that in the continuum limit a 0, the Wilson loop reduces to a form proportional to the
Figure 05h Wilson Loop [view large image]	Yang-Mills Lagrangian. 6. The Wilson action for SU(N) group can be expressed as: SW(Uij) = P[Tr(UP + UP) - 2N]

< O(U) > ~ dUijO(Uij)exp[-SW(Uij)], where denotes multiplicaltion of objects with indices i, j. In particular, the average of the Wilson loop is: < W(C) > = < Tr(U1, U2, ... Un)C > ~ exp(-TA) at the strong coupling limit, where A is the area encircled by C, and T is called the string tension. This result implies a static potential V(r) ~ Tr, where r is the separation between the quarks. Since V(r) increases with the value of r, thus it validates the concept of confinement.

Figure 05i Gluon Fields in Grid [view large image]

8. Finally, the lattice theory has to be verified that it goes over to QCD at the continuum limit as the link a 0 and the Euclidean time it is reverted back to the normal time t.

For the simplest possible group with only two elements, if the lattice is 8x8x8x8 in size, then the sum contains 2212 ~ 101228 terms, which is clearly prohibitive even by running all these different paths with a super computer. The Monte Carlo technique (as inspired by winning the game), however, evades this problem by the statistical sampling method wherein it does some random sampling first in order to discover the algorithms that would yield the best fit. The first result from lattice calculation was published by K. G. Wilson in 1974, who set up the Wilson loop to perform the computation in the strong-coupling limit for quark confinement as shown in Figure 05j. The latest effort has derived a 'u' mass of 1.9 Mev and a 'd' mass of 4.4 Mev. Thus the uud quarks in a proton weigh 8.2 Mev, which is only about 1% of the proton's

Figure 05j Confinement [view large image]

measured mass (938 Mev). All the rest of the proton mass comes from the energy that binds the quarks together. It reveals that not only the atoms are mostly empty space; now we know the nuclei inside atoms are also mere puffballs with almost no solid substance.

		QCDOC (QCD On Chip), is a series of computers designed specifically for lattice QCD calculations and dedicated more or less exclusively to that task. The glass-fronted cabinets in Figure 05k, at Brookhaven National Laboratory, hold two QCDOC machines, each with 12288 processors; a third QCDOC is at the University of Edinburgh. A typical lattice has 32 nodes along each of the three spatial dimensions and 128 nodes along the time dimension. That's about
Figure 05k QCDOC [view large image]	Figure 05l Lattice	4 million nodes altogether, and 16 million links between nodes. Some lattice QCD simulations are run on "commodity clusters" - machines assembled out of hundreds or thousands of off-the-shelf computers. Figure 05l is an artist's imagination about the Lattice Theory - blending rows of computers with the nucleons and its quarks within.

(see Figure 05m). The results (filled circles) are in remarkable agreement with the experimental values (horizontal lines). Each symbol (, K and so on) refers to a different type of hadron. The widths of the bands indicate the experimental decay widths, the invert of which are related to the finite lifetime of the particles. The vertical error bars denote the theoretical error estimates. Three of the hadrons (, K and ) have no error bars because they are used to fix the theory's parameters. Since there are (3x3x4) + (8x6) = 84 numbers at each node (quark fields have three flavours, three colours, and four components accounting for spin and antiparticles; gluon fields have eight internal

Figure 05m Masses of Hadrons [view large image]

directions in its symmetry group, and for each direction there are six fields: three electric and three magnetic) no computer can handle the astronomical number crunching in the lattice calculation, certain approximations are adopted to make the task manageable:

1. Lattice Size - Because of asymptotic freedom, the short-wavelength fluctuations of the fields can be replaced by Gaussian random (free) fields. Thus there is a minimum lattice size, which helps to cut down the number of links. The effects of the missing fluctuations can be computed analytically and added back in.
2. Lattice Volume - It is possible to control finite-volume errors by varying the simulated volume and making theoretically informed extrapolations.
3. Small u and d quark masses - Solutions become hard to obtain with the very small physical values of these masses. They are handled by sophisticated, theoretically informed extrapolation from simulations using larger mass values.
4. Finite computer power - Even after acceptable levels of discretization and restriction to finite volume, the space that should be surveyed in the lattice calculation is far too large for even the most powerful modern computer banks to handle. So in place of a complete survey, it is replaced by a statistical sample. This introduces errors that can be estimated by the standard techniques of statistics.

Unitarity Method

In case there are a lot of Feynman graphs to be evaluated in a process, the Unitarity Method can be used to shorten the computing within a manageable time. It is akin to Fluid Dynamics where the motion of individual molecules is collectively aggravated into macroscopic elements governed by a set of equations at another level (known as the Navier-Stokes Equations). It utilizes the fact that the infinities in the loop diagrams are often canceled out leaving a finite result at the end of the calculation. Figure 05n illustrates schematically the idea of one S-matrix corresponding to many Feynman diagrams. It has been shown that together with some conditions imposed on the S matrix, the unitarity method can produce rather accurate result.

Figure 05n Unitarity Method [view large image]

The conditions include :

1. Unitarity or conservation of probability.
2. Analyticity in the various energy variables. An analytic function is infinitely differentiable, and can be expressed by Taylor series in some neighborhood of every point.
3. Lorentz invariance.
4. Crossing symmetry, which means the S-matrix of two different processes can be converted to each others, e.g., the Compton effect and pair annihilation, etc.

	produced by supergravity in the 1960s. The argument was rather indirect because it was not feasible to evaluate 1020 terms in the three virtual-graviton loops. The unitarity method has now taken up the challenge and showed that those infinite quantities are indeed finite. The next tests would be the four and five loops computations. Along the way, the
Figure 05o Graviton and Gluon [view large image]	technique has also revealed that a graviton can be considered as two copies of gluon (Figure 05o).

Quantum Vacuum

Another kind of vacuum structure is prescribed by the Dirac theory of spin-1/2 particles. It admits the existence of both positive- and negative-energy particles: E = (m02c4 + p2c2)1/2. The concept of negative-energy entities is wholly alien to our knowledge of the universe. All things of physical significance are associated with varying amounts of positive energy. To get around the problem, Dirac proposed an energy spectrum containing all electrons in the universe (see Figure 06a). In addition to the normal positive-energy spectrum, it also contains the negative-energy variety, which spans the spectrum from -m0c2 down to negative infinity. All the negative-energy levels are filled, thus the positive-energy particle is inhibited from transition into these lower energy states. Thus, there is no observable effect in the real world. Only when there is enough energy available, e.g., E 2m0c2, a real particle and anti-particle pair with positive-energy can be created from this unseen sea of negative-energy particles.

Figure 06a Negative Energy

The particle is the electron originally resided in the negative energy region, while the anti-particle (positron) can be interpreted as the hole in the vacated energy level acquiring a mass m0. The law of charge conservation demands that this anti-particle carries a positive charge.

Many interesting things occur when the electron meets the hole (see Figure 06a). Over the past three decades, the use of e⁺e^- collisions to probe the vacuum have yielded a great deal of information about the nature of the strong, weak, and electromagnetic interactions, and have played a major role in establishing the Standard Model of elementary particles. There are many special features of the e⁺e^- experiments:


• The e⁺e^- pair often annihilate into a vacuum state of pure energy (in the form of electromagnetic radiation or other kind of bosons). All the quantum numbers of the initial particles disappear by cancellation. The energy is then free to sample the hidden, negative-energy content of the vacuum without the inhibiting effects of conservation laws. Such energy can be in the form of photon in diagram (a) of Figure 06a boosting any particle pair from the negative-energy sea.
• When an electron and positron are collided head-on, with equal and opposite momentum, the center of mass of the reaction is absolutely stationary. In such circumstance, the angular distribution of created particles can be measured directly and significant asymmetries detected much more easily.
• Since the e⁺e^- pair can annihilate to a vacuum state, it means that the reactions are very clean. There is no debris surviving from the initial state to mask interesting new effects or to confuse the detection.
• The range of phenomena open to study depends upon the total energy of the collision. But while the e⁺e^- beams are kept in orbit they emit their energy in the form of synchrotron radiation. So, to achieve very high energies, it is necessary to build very large rings and to input a lot of radio frequency power to replenish the lost energy. Therefore, the search for new particles has led to the construction of increasingly more powerful and bigger accelerators.

The basic e⁺e^- reactions:


• Electromagnetic processes -
  • The elastic scattering as shown in diagram (b), Figure 06a.
  • Creation of ^+- as shown in diagram (c), Figure 06a. The single photon must be virtual (off-shell) as it does not obey energy-momentum conservation.
  • Production of two real photons as shown in diagram (d), Figure 06a.
  Accurate measurement of these electromagnetic effects is used to test the validity of QED at very high energies. It confirms that the leptons are indeed fundamental point-like particles with size < 10^-16 cm.
• Annihilation into Z⁰ boson - The e⁺e^- can annihilate into Z⁰ boson involving weak interaction (diagram (f), Figure 06a). The virtual Z⁰ boson is free to explore the "weak" content of the vacuum. The electroweak theory predicts that slightly more ⁺s should emerge in one hemisphere than in the other (with the reverse asymmetry for the ^-s). Such parity violation was duly confirmed in 1981.
• Hadron (composite particles containing quarks) productions - One of the most significant quantities in particle physics is the ratio R of the cross section for e⁺e^- hadrons to that for e⁺e^{- +-}, as a function of the colliding energy:

  ---------- (45)

  It compares a well-understood reaction (muon-pair production) with the class of less well-known reactions (hadron production). The ratio R is relatively straight-forward to observe experimentally. Only two charged "prongs" are seen to emerge in muon-pair production whereas, almost invariably, more emerge from a hadronic final state (digram (e), Figure 06a). So the ratio can be obtained from the numbers of these different events. Surprisingly, the ratio R is constant over large energy ranges, indicating that the complicated hadronic state is produced in much the same way as the simple muon pair. In this case, the virtual photon is probing the negative-energy sea of hadrons in the vacuum instead of electrons or muons.

Closer examination of diagram (e), Figure 06a reveals that the intermediate product is a pair of quarks as shown in the insert within Figure 06b. The quark confinement process creates the observed hadrons in the final state. As the quarks are observed to be point-like (in deep inelastic scattering) and spin 1/2, the intermediate process e+e- q is very similar to the process e+e- +-, the only difference being that the charges on the quarks are only some fractions of that on the muons. This explains the constancy of the

Figure 06b Reaction Ratio [view large image]

ratio R mentioned earlier and displayed in Figure 06b. As for the pronounced spikes which punctuate the curve in Figure 06b. These shapes are formed at certain energies of the e+e- collision, when the q pair have just the correct mass to appear as a single-meson resonance. They are the SU(3) flavour symmetry mesons with mass ~ 1 Gev.

mesons is now expanded to a 25-plet. It is further realized that the c mesons form a spectrum from the different values for the spin and the orbital angular momentum of the constituent quarks as shown in Figure 06c. In the same notation for the atomic spectra, S, P and D in the diagram refer respectively to orbital angular momentum 0, , and 2. The resemblance to the atomic spectrum is understandable because of asymptotic freedom as the c and bound themselves together loosely. The force between the quarks can be formulated

Figure 06c Meson Spectrum [view large image]

as a potential acting in the vicinity of a colour charge. Thus instead of the Coulomb potential in the case of the atom, the quarks interact via the potential:

As the beam energy is increased, the quark and antiquark are produced with very large momenta, moving in opposite directions known as two-jet event. The fragmentation into hadrons then takes place, preferentially along the direction of the motion the quark and antiquark, resulting in jets of hadrons which become more and more collimated as energy is increased (see left diagram of Figure 06d). The measurement of the angular distribution

Figure 06d Two- and Three-Jet Event [view large image]

of jet axes confirms that the spin of quarks is indeed 1/2. At even higher energy the quark or antiquark radiating a gluon, which forms a separate jet of its own. This three-jet event is shown in the right diagram of Fgiure 06d.

Spontaneous Symmetry Breaking

The existence of asymmetric solutions to a symmetric theory is common to many branches of physics. The reason lies in the fact that the symmetric state is not the state of minimum energy, i.e., the ground state, and that in the process of evolving towards the ground state, the intrinsic symmetry of the system has been broken. Figure 06e shows that the initial position of the marble on top of the bump is symmetric but not in a state of minimum

Figure 06e Symmetry Breaking [view large image]

energy. A small perturbation will cause the rotational symmetry to be broken and the system to assume the ground state configuration. When the symmetry of a physical system is broken in this way, it is often referred to as "spontaneous symmetry breaking".

	This idea can be applied to account for the mass of gauge boson in the electroweak interaction in the Standard model. Let us start by examine how such unstable symmetry can arise mathematically. Considering the Lagrangian for a scalar field with the potential V: L = (1/2) - V() ---------- (46) where V() = (1/2)m22 + (/4!)4, and 4! = 4x3x2x1.
Figure 06f Scalar Field [view large image]	If m2 > 0, the system has real mass, the potential exhibits a minimum at the origin, where = 0. This system is associated with a unique vacuum (see Figure 06f).

	The formulation can be generalized to complex scalar field with two independent components corresponding to positively and negatively charged fields. In a slightly different notations, the Lagrangian for a complex scalar field has the form similar to Eq.(46): L = ()(*) - V(,*) ---------- (48) where V() = m2* + (/4)(*)2.
Figure 06g Complex Scalar Field [view large image]

i

-iq

As each term in the Lagrangian of the Standard model represents a different process, Figure 06h shows the various Higgs interactions in the form of Feynman diagrams. Diagram (a) represents a fermion emitting or absorbing a Higgs particle. Diagram (b) shows the corresponding process for the gauge bosons. They can also interact simultaneously with two Higgs, as shown in (c), which also represents a gauge boson scattering a Higgs particle. The Higgs also interacts with itself, as shown in diagrams (d) and (e), which are related to the shape of the scalar potential (Figure 06g). Diagram (f) depicts an electron acquiring its mass.

Figure 06h Higgs Field Interaction [view large image]

Gyromagnetic Ratio and Anomalous Magnetic Moment

A classical electron moving around a nucleus in a circular orbit has an orbital angular momentum, L=mevr, and a magnetic dipole moment, = -evr/2, where e, me, v, and r are the electron´s charge, mass, velocity, and radius, respectively. A classical electron of homogeneous mass and charge density rotating about a symmetry axis has an angular momentum, L=(3/5)meR2, and a magnetic dipole moment, = -(3/10)eR2, where R and are the electron´s classical radius and rotating frequency, respectively. The classical gyromagnetic ratio of an orbiting or a spinning electron is defined as the ratio of the

Figure 07 Classical g-ratio [view large image]

magnetic moment to the angular momentum. In both cases one finds cl = /L= -e/(2me). The minus sign indicates that is in the opposite direction to L (see Figure 07).

		that the spin of the electron is equal to /2. If the vertex correction as shown in Figure 08 is taken into account, then g = 2 ( 1 + /2), where = e2/(4c) ~ 1/137.036 is the fine structure constant giving g - 2 = 0.002322814. The extra term arises from the electron self-interaction, in which it emits and reabsorbs a virtual photon, making a loop in the Feynman diagram as shown in Figure 08b. The same process also applies to the muon.
Figure 08a Quantum Description [view large image]	Figure 08b Vertex Cor-rection [view large image]

current theory, and such new physics is of a nature that will affect the muon's spin, then the measurement would differ from the theory. This is what appears to have been observed, although there are several interpretations of the result that must be considered. One of the missing pieces in the theoretical calculation is the exotic particles predicted by the theory of supersymmetry. Although these particles are rare and unstable their mere existence in the vacuum would modify observable quantities such as the muon magnetic moment.

Figure 09 g-2 Experiment [view large image]

Axion

Axion was postulated to explain why CP violation is not observed in strong interaction, although it should be according to the Standard Model. The CP violation in SM arises from a certain non-zero parameter related to the QCD vacuum (see instanton). It was shown in 1997 that the parameter could be driven to zero by a Higgs field. A side effect of the transformation is the introduction of a new particle - the axion. If axions do exist, they each would have a

Figure 10 Axion [view large image]

mass of around 10-5 ev, but there could be so many of them in the Universe, that they contribute a large proportion of the overall mass in the form of dark matter.

Preons

		With a few tweaks of the conventional fermion table, it can be re-arranged into a form shown in Figure 11, which appears to be a shortened version of the periodic table compiled in 1869. This new periodic table has increasing mass along both column (from top to bottom) and row (from left to right with two exceptions) similar to the old periodic table. The orderly arrangement of the old table was eventually explained by the internal structure of the atoms. Thus it is surmised that the same kind of explanation can be applied to the fermions and
Figure 11 A New Periodic Table [view large image]	Figure 12 Preon Model [view large image]	the bosons (not shown in Figure 11). The hypothetical constituent is now dubbed as preon. It is hoped that a new model with the

A Footnote - Least Action Principle and Path Integral

		This footnote is supposed to clarify the hand-waving argument in the main text with the help of simple mathematics and graphic. The first task is to show that there is a minimum for the action S (for a given Lagrangian L as shown in Eq.(49)) as the paths varies between two fit points. The trajectory of the particle is calculated from the equation derived from the "principle of least action" as shown in Eq.(50). ---------- (49)
Figure 11 Paths [view large image]	Figure 12 Actions [view large image]	S / q(t) = ---------- (50)

		Then along came quantum mechanics and the method of path integral to calculate the transition amplitude between two points, e.g., from ta = 0 to tb = T : The sum is supposed to include all kinds of paths imaginable. However, for demonstration purpose the family of paths considered in the previous section is
Figure 13 Path Integral, Real [view large image]	Figure 14 Path Integral, Img [view large image]	sufficient (see Figure 11). The Planck's constant can also be absorbed into the action S so that the computational procedure can be carried over to evaluate the sum.

An alternate way to perform the path integral is to sum over all the paths at different time t and then put them all together as shown in Figure 15, where the time coordinate is divided into N parts of equal interval . Thus T = N, and dt = 0. The transition amplitude <0|X> can be written as : <0|X> = Ae(i/)L(qn - qn-1)dq1dqN-1 ---------- (55) where A is the normalization constant. For a particle in free field L = (m/2)(dq/dt)2, A = (m/ih)N/2 Eq.(55) can be evaluated analytically : <0|X> = Ae(im/2)(qn - qn-1)2dq1dqN-1 = (m/ihT)1/2exp(imX2/2T) ---------- (56)

Figure 15 Path Integral [view large image]

The integration is carried out by applying the Gaussian integral : e-f(qn-qn-1)2-g(qn+1-qn)2dqn = [/(f+g)]1/2{e-[fg/(f+g)](qn-1-qn+1)2} ---------- (57) repeatedly (N-1) times.

		The real part of the transition amplitude <0|X> is plotted in Figure 16 and 17 with T and X holding constant respectively. Figure 16 shows that the wavelength is shorter at large X signifying higher probability for particle with higher velocity to get there. On the other hand in Figure 17 for a given X, a particle with lower velocity (and hence longer wavelength) would take longer to reach. But the chance of getting there becomes progressively smaller. Numerical computation is again performed by a home computer with Basic programming. The electron mass m is assumed in the calculation, thus (m/h)1/2 ~ 0.4 and (m/2) ~ 0.5. For the case of constant T, X is evaluated
Figure 16 X vs Transition Amplitude [view large image]	Figure 17 T vs Transition Amplitude [view large image]	within the range -3 to +3 in step of 0.001 (in unit of cm); while for the case of constant X, T is evaluated from 0.5 to 5 in step of 0.0001 (in unit of sec).

i(kx-t)

Table 08 Comparison of Method 1 and 2

There is a third method to evaluate the path integral. It is the variant of method 1 with a twist (literally). Followings are the steps on how it is done:


1. Replace the summation over all paths with integration of the parameter "a" from - to +. Thus, the transition amplitude can be rewritten as:
   <0|X>₃ = e^iS/da ---------- (61)


2. Transform the end time T to -iT' in Eq.(60) for the action S. This is sometimes referred as Wick rotation, which rotates the time axis by 90^o from the Minkowskian spacetime to the Euclidean spacetime. The technique alters the oscillating exponential to a decaying exponential making the integration much easier to carry out. Actually, Method 2 has performed the Wick rotations already implicitly.


3. Use the Gaussian integral: C e^-kx2+fx+gdx = C(/k)^1/2e^(f2/4k)+g ---------- (62)
   to perform the integration.
4. Reverse the end time transformation from T' to iT after the integration.
5. The resulting transition amplitude is now in closed form:
   
   <0|X>₃ = (1/iT)(12h/imT)^1/2exp(imX²/2T) (8.94/iT^3/2)exp(imX²/2T) ---------- (63)