## Quantum Field Theory

### Contents

Action Principle
Path Integral
Field Equation
Second Quantization
Noether's Theorem and Charge Conservation
Green's Function and Renormalization
Perturbation Theory and S Matrix
Feynman Diagram
Quantum Electrodynamics (QED)
From Coulomb Scattering to Deep Inelastic Scattering
Compton Scattering
Lamb Shift
Renormalizable Theories
The Divergence
Yang-Mills Theory
Weak Interaction
The Standard Model
The SO(10) Group and Unification
Lattice Theory
Unitarity Method
Quantum Vacuum
Spontaneous Symmetry Breaking
Gyromagnetic Ratio and Anomalous Magnetic Moment
Axion
Preons
A footnote - Least Action Principle and Path Integral

### Action Principle

Formulation of theoretical physics usually starts with the "Action Principle", which was originally used to derive the equation of motion for a particle in classical mechanics. The "action" for such case is:

 ---------- (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
between the end points to yield a value for S[q(t)] is responsible for its dependence on the entire path of q(t) from one end point to the other. It is equal to the area under the curve of L with respect to the L = 0 (horizontal) axis. (see further explanation in footnote)

The "Action Principle" states that by minimizing the action S, i.e, by taking S = 0 with a small variation of path - q, we can derive the Euler-Lagrange equation:

 S / q(t)  = ---------- (1b)
from which the trajectory of the particle can be computed.

For a particle with mass m moving in a potential V(q) :

L = (m/2)(dq/dt)2 - V(q), ---------- (1c)

the Euler-Lagrange equation has the explict form with the variable of the actual path labeled as x :

m d2x/dt2 = -dV/dx = F, ---------- (1d)

which is the Newtonian equation of motion, where the negative gradient of the potential is the force F.

The potential for the Hooke's law is V(x) = (k/2)x2, where k is the spring constant, the equation of motion then becomes:

d2x/dt2 + (k/m) x = 0, ---------- (1e)

which is the Newton's law for the motion of harmonic oscillator. The general solution is:

x = A ei(k/m)1/2t + B e-i(k/m)1/2t ---------- (1f).

Note that there would be no oscillation when m or k = 0.

The form of Lagrangian can be constrained by imposing various conditions. The most obvious one is to demand that it should be independent of the orientation of the coordinate system. Under this condition the Lagrangian has to be a scalar in three dimensional space (for classical mechanics) or four dimensional space-time (for relativistic mechanics). Other conditions can also be specified in constructing the Lagrangian. The Noether's Theorem states that each of this condition corresponding to a conservation law:

Time independence Energy conservation
Space independence Momentum conservation
Rotational independence Angular momentum conservation

In other word, the conservation laws are the consequence of Lagrangian symmetries under the transformations of time, space, and rotational angle respectively. Symmetry in physical systems carries a different meaning than simple geometrical invariance. Instead of checking whether an experimental arrangement looks identical when rotated (geometrical invariance), we want to know if the laws of physics are invariant, i.e., if objects behave in the same way when the system is rotated. For example, it can be shown that if the Lagrangian is time independence then
 - L = constant ---------- (1g)
if L is given by Eq.(1c), then it becomes:

(m/2)(dx/dt)2 + V(x) = constant ---------- (1h)

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.
Although the laws of nature may seem to be simple and symmetrical, the real world isn't. It is messy and complicated. The reason is related to the fact that we do not observe the laws of nature, but instead its outcome or solution such as Eq.(1f), which is the solution of Eq.(1e). For example, the law in Eq.(1e) dictates that it is symmetrical (invariant) under time reversal:
t' -t; but its solution in Eq.(1f) is not so unless A = B (see more in the section on "Spontaneous Symmetry Breaking").

### Path Integral

In quantum theory all the paths in Figure 01c are possible, not just the one corresponding to S = 0. However, not all paths are equally likely; we must find an appropriate weighting for the paths. According to Feynman, each path is weighted by the factor eiS/. The classical path is still the one that makes S stationary under small changes of path. In its vicinity paths have a strong tendency to add up constructively, while far from the classical path (the ones that go high up on the sides of the trough) the phase factors will tend to produce cancellations. The amount the particle can stray from the classical path depends on the magnitude of the corresponding action relative to . The steeper the slope of the trough, the closer it is approaching the
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:

This sum can be evaluated by breaking up every path into an infinite number of intermediate points: q1, q2, ..., qn, ...qN, and then takes the sum of each point, e.g., qn in all the paths from - to +. For the free field case where L = (m/2)(dq/dt)2, the functional integrations can be performed analytically. The final result is:

= |m/h(tb - ta)|1/2 exp{[im(qb - qa)2] / [2(tb - ta)]} ---------- (1i)

If we vary one of the end points, e.g. qb = x, it can be shown that the transition amplitude in Eq.(1i) satisfies the time-dependent Schrodinger's equation in one dimension for a non-relativistic free particle:
 ---------- (1j)
The link between Eqs.(1i) and (1j) illustrates the equivalence of the path integral and canonical quantization in quantum theory. The wave function in the Schrodinger's equation can now be equated to the transition amplitude: (x,t) = , where q(ta) = x(ta) can be conveniently chosen at the origin of a coordinate system and with ta= 0. (see further explanation in footnote)

### Field Equation

Quantum field theory can be developed by adopting the path integral formulism as mentioned earlier, or by combining the field equation with canonical quantization as shown in the next sub-topic. Anyway, the concept of Action principle can be generalized to derive the field equation with some modifications for the action. A similar procedure for minimizing the action can be used to derive the Euler-Lagrange equation, which is now called the field equation. Thus, the action of the field is now expressed in the form:

where is the Lagrangian density, is the th component of the field, denotes the space-time component,
,
and the integration is over a four-volume R. Similar to the case of particle Lagrangan in Eq.(1a) S has to be a scalar if is an invariant function of its arguments. The form of is further constrained by demanding that it is invariant under certain transformations such as space-time translation, Lorentz transformation, the gauge transformation, or the conformal transformation, etc.... This is to ensure that the equation of motion (field equation) is unchanged (symmetrical) under these operations. The corresponding conservation laws for these symmetries are summarized in Table 01 below.

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

Note: The gauge symmetries in Table 01 are global (meaning everywhere), local gauge symmetries (at specific point) leads to the interactions with gauge bosons. See the sub-topic on Noether's Theorem for more details about charge conservation.

The "Action Principle" for field theories states that if we perform an arbitrary variation of the field, + , subject to the boundary condition = 0 for t = t1 and t = t2, then the solution of the variational problem S = 0 yields a set of the Euler-Lagrange equations from which the equations of motion, or field equations can be derived:
 ---------- (1k)
For the neutral scalar meson field

which yields the Klein-Gordon Equation according to Eq.(1k):
 ---------- (1l)
where m represents the mass of the one component scalar field , which is a function of x, y, z, and t (collectively represented by x in the equation),

are the d'Alembertian operator in 4-dimensional space-time and the Laplacian operator in 3-dimensional space respectively. The repeated dummy index in the equations is understood to be summed over the 4 space-time coordinates.

The field can be expressed in a series expansion in terms of the harmonic functions and the coefficients ck's, where k is a four dimensional vector related to the momentum and energy of the particle:
 ---------- (2)
which is just a Fourier Series where the coefficients are to be determined by the field:
 ---------- (3)
where , and the time x0 is set to zero after the time derivation has been performed.

### Second Quantization

Quantization of the field is accomplished by demanding the coefficients ck's to satisfy the following commutation rules:
 ---------- (4)
If a number operator Nk = ck*ck is defined such that it operates on the state vector |nk to generate:
Nk|nk = nk|nk
where nk is the number of particles in the k state; it can be shown that
ck*|nk = (nk+1)1/2|nk+1
ck|nk = nk1/2|nk-1
Thus ck* increases the number of particles in the k state by 1, while ck reduces the number of particles in the k state by 1. They are called creation and annihilation operator respectively. The complete set of eigenvectors is given by:
 ---------- (5)
for all values of kl and nl. They form an abstract space called the Fock space with all the eigenvectors orthogonal (perpendicular) to each others and the norm (length) equal to 1.

In particular, the vacuum state is:
 ---------- (6)
which corresponds to no particle in any state - the vacuum.

See more about second quantization in "Quantization and Field Equations".

### 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
would vanish if the three-current diminishes sufficiently on the surface. Under this condition, we obtain:
,
which implies the charge = constant. Note that the conservation of charge would not be valid if there is source or sink within the volume.

The Noether's theorem can be illustrated in more details by using the two-component Klein-Gordon Equation as an example. The explicit form of the Lagrangian density in this case is:

,
where . These two independent scalar fields can be varied by the internal phase transformation:
 or,  in a more familiar form: ,
where is the parameter corresponding to ( is omitted for single parameter). This U(1) symmetry generates the four-current:. It follows that the conserved charge corresponding to this current is:

where N and N' represent the sums of number operators (in momentum space). In this form, N can be interpreted as the number of particles carrying charge -e (with various momentum), while N' is the number of anti-particles with charge +e. The sum of these numbers is a constant within a volume containing no source or sink.

### Green's Function and Renormalization

The mathematics becomes more complicated when there is interaction with the field. The simplest case is to include the source of the field in the free field equation. An additional term is inserted to the right of Eq.(1l) :
 ---------- (7)
where the label "o" designates quantities associated with the "bare field".

Invoking the Green's function technique, the solution of this equation is given by:

 ---------- (8)
where the first term is the free field solution and the Green's function G(x - y) inside the integral is the solution of the equation with a point source at y in the form:
 ---------- (9a)
where (x - y) is the delta function. Similarly, the Green's function for the ferminon is defined by the equation:
 ---------- (9b)
where m0 is the mass of the fermion, and are the gamma matrices associated with the fermion. In Dirac representation, the matrices can be expressed explicitly as:

where g11 = g22 = g33 = 1, g44 = -1, g = 0 for ; the 5 matrix converts a vector to pseudo-vector (axial vector) with parity +1.

The chiral representation is in the form:
.
It splits the Dirac equation into 2 self-contained pieces (for the left-handed and right-handed leptons respectively) more suitable for formulating the Standard Model.

It can be shown that the "bare field" can be expressed in terms of ck's similar to the case of the free field, but these coefficients are now modified by an additional term related to the structure of the source. As a result the norm (length) of the eigenvectors are no longer equal to 1. To recover this definition, they have to be "renormalized" by the renormalization constant Z, which has the values ; it is equal to 1 for free field, 0 for a point source, and depends on the structure of the source in general. The renormalized field, mass, and energy are and EnR = Z-1Eno respectively. The physical mass mR is the experimentally observed mass, mo is an unspecified parameter (called "bare mass") which together with Z-1 determine a value in agreement with experiment. Since Z-1 is infinite for a point source, mo has to be slightly more or slightly less infinite to yield a finite value for mR. This technique of replacing the ignorance in detailed structure of the source by measurement is the more general definition of renormalization, although it is now more often referred to as the method to cancel the infinities in quantum field theory. For example, it involves a many-body Schrodinger equation to compute the viscosity in the Navier-Stokes equations. Since it is impossible to obtain the solution for such a complicated system, its value is determined by experiment instead. A renormalizable theory is one in which the details of a deeper scale are not needed to describe the physics at the present scale, save for a few experimentally measurable parameters (see more in the section about "Renormalizable Theories").

### Perturbation Theory and S Matrix

It is not possible to obtain an analytical solution for the field equation with the field itself in the interaction term. A perturbation theory was developed to obtain approximate solutions step by step. For example, the interaction between a charged fermion and the photon is in the form , where A is the electromagnetic vector potential and is the 4 components spinor separated into 2 parts (in Dirac representation, and together form a 4-components field for either the fermion or anti-fermion; while in chiral representation they stand for the left-handed and right-handed fermions called Weyl spinors), which now appears on both sides in Eq.(8) (with (y) replaced by the interaction term, replacing , and G defined by Eq.(9b)). An iteration procedure yields a sum of integrals as shown in the formula below:
 ---------- (10)
In this form the unknown field on the left hand side is now expressed in terms of all known quantities on the right hand side. The free field solution is denoted by 0, and the integration is over all the space-time x', x'', x''', ... Note that each of the following term is multiplied by the power of e, from e1, to e2, ... Since e2=1/137 for the electromagnetic interaction, computation on a few terms would be sufficient to obtain a result with acceptable accuracy.

Another formulism is the S-matrix expansion. It is the transition amplitude expressed in a series as the result of the iteration procedure on the transition operator, which transforms the system from an initial state (at negative infinity time) to a final state (at positive infinity time) as shown in the formula below.
 ---------- (11)
where HI involves the interaction fields (integrating over all space and multiplied by the coupling strength) and
t1 > t2 > ... > tn-1. In this picture the fields obey the free field equations, the interaction enters via HI. It can be shown that the green's function and S-matrix formulations are equivalent.

It was thought that since we cannot measure the fields directly, so we should not talk about it, while we do measure S-matrix elements, so this is what we should be mindful about. Essentially, by considering the conservation of probability,
analyticity in the energy variables, and various invariences and symmetries, the S matrix can be expressed in a form that is in agreement with some experiments, e.g., the pion-nucleon scattering amplitude. Such method is called dispersion relations by its similarity with the formula for index of refraction in classical optics. It is now realized that analyzing the S-matrix alone is not sufficient, information on the quantum fields is also necessary. See also Unitarity Method.

The S-matrix elements of the expansion are taken between the initial state (i) at t = - and the final state (f) state at t = +, e.g., Sfi = f|S|i. Usually, one initial state can produce one or more final states as shown in Figure 01e, where three different initial states are taken as examples - namely, the electron positron scattering, the Compton scattering and the deep inelastic scattering. Each of this process produces many final states, but only a few have been shown just for illustration purpose. The Sfi is a complex number in general. It is called probability amplitude and is related to the probability of going from the i to f
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.

Note: For example in the electron positron scattering process, there are three possible finally states as illustrated in Figure 01e. The sum of the probabilities for each one of these has to be: S*11S11 + S*21S21 + S*31S31 = 1 to insure that the final state is in one of the three possibilities.

Now let us take the nucleon-pion system as an example of S-matrix application:
 ---------- (12)---------- (13)

where Eq.(12) is the free field equation for the pion, and Eq.(13) is the free field equation for the nucleon (the Dirac equation). Expressing in terms of the field itself, it can be shown that the quantization rules in Eq.(4) become:
 ---------- (14)
where {a,b} = ab + ba is the anticommunition expression, and the quantities on the right-hand side are the Green's functions for the pion and nucleon respectively (see Eqs.(9a) and (9b)). Since the interaction HI = go the nth order term in the S-matrix expansion Eq.(11) has the explicit form:
 ---------- (15)
where the integration runs from - to + over all space-time, the symbol N is the normal-order operator, which shifts all the creation operators to the left (to avoid infinite vacuum energy), while T is the time-order operator, which re-arranges the fields so that the one associated with later time is on the left (to make sure a particle is created before its annihilation). The T product can be expanded into the N products and pairings, e.g.,

If the coupling constant g0 is a small number less than 1, the successive higher terms would be getting even smaller as the proportional constant is in the form of (g0)n. Therefore the perturbation series can be terminated up to certain term depending on the requirement of accuracy.

### Feynman Diagram

The first order term is:
 ---------- (16)
The mathematical entities inside the integral can be represented graphically by the following conventions:

which translates S(1) into a graph called the Feynman diagram:

It represents the process of the annihilation of a pair of nucleon and anti-nucleon and the creation of a pion.

In the next higher order term the normal operator will generate Green's function such as or SF(x - y) from the pairings, e.g.,

The pairing is referred to as propagator or internal line and graphically represented by a line running from a vertex at x to another vertex at y as shown in the diagram below:

which represents the annihilation of a pair of nucleon and anti-nucleon through a virtual nucleon.

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:

where is a small positive real constant - a mathematical device taking advantage of the technique of contour integral; + 0 will be taken after the integration. The subscript F refers to the Feynman prescription for integrating the Green's functions. They are in a form such that positive energy solutions are carried forward in time, while negative energy solutions are carried backward in time. The latter solution can be interpreted as the anti-particle with positive energy moving forward in time.

Another graph such as the one below:

this loop diagram represents the process of virtual pair creation and annihilation - the vacuum fluctuation (time runs horizontally in this graph). It is this kind of graphs, which give rise to divergent results.

The Feynman rules are summarized in the tables below:

#### Table 03 Momentum Representations [view large image]

where in Table 02, the + and - superscripts refer to the positive frequency (eikx) and negative frequency (e-ikx) terms as shown in the Fourier decompositions below (also see Eq.(2)) . In the tables, N represents the nucleon and represents the anti-nucleon. In evaluating the S matrix, it is sometimes advantageous to go over to momentum space via the Fourier expansions:

Then the Feynman rules (see Table 03) can be expressed in terms of the energy p and k = (mo2 + k2)1/2, and 4-momentum p and k, for the fermion and boson respectively, while v(p,s) and u(p,s) are Dirac spinors with spin s representing the nucleon and anti-nucleon respectively . The appearance of m0 in the formulas for both fermion and boson is rather confusing, it just indicates the rest mass for whichever particle in the process.

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.
Evaluation of the S matrix with the Feynman rules in Table 02 yields the same result. For example, the external lines together with the contribution from the propagator would combine into a factor of exp[i(k+p3-p1)x], which gives a delta function with the same argument upon integrating over the x-space, and similarly for the y-space integration. Then the k-space integration can be performed exactly as outlined above. The rest is to collect all the factors associated with each of the steps. In fact, this is the procedure to derive the Feynman rules in momentum space.

It is worthwhile to repeat once again that Feynman diagrams can be divided into two types, "trees" and "loops", on the basis of their topology. Tree diagrams only have branches. They describe process such as scattering, which yields finite result and reproduces the classical value. Loop diagrams, as their name suggests, have closed loops in them such as the one for vacuum fluctuation. The loop diagrams involve "off mass-shell" virtual particles and is usually divergent (becomes infinity). Such virtual particles can appear and disappear violating the rules of energy and momentum conservation as long as the uncertainty principle is satisfied. They are said to be "off mass-shell", because they do not satisfy the relationship E2 = p2c2 + m2c4.

### Quantum Electrodynamics (QED)

Quantum electrodynamics, or QED, is a quantum theory of the interactions of charged particles with the electromagnetic field. It describes mathematically not only all interactions of light with matter but also those of charged particles with one another. QED is a relativistic theory in that Albert Einstein's theory of special relativity is built into each of its equations. That is, the equations are invariant under a transformation of space-time. The QED theory was refined and fully developed in the late 1940s by Richard P. Feynman, Julian S. Schwinger, and Shin'ichiro Tomonaga, independently of one another. Because the behavior of atoms and molecules is primarily electromagnetic in nature, all of atomic physics can be considered a test laboratory for the theory. Agreement of very high accuracy makes QED one of the most successful physical theories so far devised.

The formulism for QED is very similar to the nucleon-pion system in Eqs.(12) and (13). While Eq.(13) for fermion is readily applicable (with appropriate value for ko, which is proportional to the mass of the fermion); Eq.(12) is replaced by the Maxwell equations:

where E, B are the electric and magnetic field respectively, j is the current density, is the charge density, and c is the velocity of light.

By virtue of the anti-symmetric property of Eq.(21), the continuity equation for the charge-current density is automatically satisfied, i.e., where = /x. The energy-momentum tensor of the electromagnetic field can be expressed as:
, where .
In curved space-time Eq.(23b) remains unchanged, while Eq.(23a) becomes , where g is the determinant of the metric tensor ( g = g11g22g33g44 in isotropic space-time). In general, the field component and the source are multiplied by a factor of (-g)½ with the presence of gravity. The energy-momentum tensor is in a more complicated form: . See "temporal waves" from early universe as dark energy.

Another form of the Maxwell equations is in terms of the four-vector potentials A defined by:
 ---------- (24)
This definition is used to simplify computations. It has a redundancy in the degree of freedom for the four-vector potentials A when Eq.(23a) is expressed in this form. By imposing the Lorenz condition to the arbitrariness,

By the way, Eq.(23b) is automatically satisfied by the relationship between the anti-symmetric tensor fields and the four-vector potentials in Eq.(24).

The 3-vector potential A can always be decomposed into a transverse component and a longitudinal component (parallel to the direction of motion) as shown in Eq.(29) such that the transversality condition Eq.(30) (which dictates that the transverse components are perpendicular to the direction of motion) and the irrotational condition of the longitudinal component Eq.(31) are satisfied:

It can be shown that and A0 together give rise to the instantaneous static Coulomb interactions between charged particles with the total Hamiltonian in the form (the Hamiltonian H is related to the Lagrangian L by the formula H = p - L):
 ---------- (32)
where and A0 have been completely eliminated in favor of the instantaneous Coulomb interaction in the last term. It is instantaneous in the sense that its value at t is determined by the charge distribution at the same instant of time. This formalism was derived by E. Fermi in 1930. Since the separation into different terms is not relativistically covariant, nor is the transversality condition itself, the whole formalism appears noncovariant; each time we perform a Lorentz transformation, we must simultaneously make a gauge transformation to obtain a new set of A and A0. It had been shown that it is possible to develop manifestly covariant calculational techniques starting with this Hamiltonian. It is also possible to construct a formalism, which preserves relativistic covariance at every stage.

Note that the long range instantaneous Coulomb interaction does not imply a force travelling faster than the speed of light. Although the intereaction is instantaneous, it can be considered as the interaction between two overlapping Coulomb tails (clouds of virtual photons) of the two charged particles, so there is no need for the interaction to travel from one point to another in zero time.

See a slightly different treatment for derivation of the free field equations in "Quantization and Field Equations".

QED concerns mainly with the transverse components, which account for the electromagnetic radiation of accelerating charged particles. The transverse electromagnetic fields provide a simple and physically transparent description of a variety of processes in which real photons are emitted, absorbed, or scattered. The three basic equations for the free-field case are:

where A satisfies the transversality condition in Eq.(30).

Eq.(33b) is in a form very similar to the Klein-Gordon Equation Eq.(1l) or Eq.(12) except that the mass term vanishes (because the photon has no rest mass) and the field is a vector (instead of scalar) with two transverse components (polarization) perpendicular to each other. Thus A can be expressed in Fourier series similar to Eq.(2):
The quantization rules for the electromagnetic field is very similar to that in Eq.(4):
 ---------- (36)
where the ak's are related to the ck's by:
 ---------- (37)
Construction of the eigenvectors follows exactly the same way as in Eqs.(5) and (6) with an additional index for polarization.

Interaction between photon and fermion, e.g., electron can be introduced by demanding local gauge invariance for the formulism. With this constraint on the quantum field theory, the ordinary derivative in Eq.(13) becomes the covariant derivative:

and the interaction takes the form:

where e is the coupling constant. (See appendix on "Abelion/non-Abelion Groups and U(1), SU(2), SU(3)" for a discussion about the concept of gauge or phase transformation.)

The photon-electron processes are described by substituting Eq.(38b) to HI 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.
In summary QED rests on the idea that charged particles (e.g., electrons and positrons) interact by emitting and absorbing photons, the particles of light that transmit electromagnetic forces. These photons are virtual; that is, they cannot be seen or detected in any way because their existence violates the conservation of energy and momentum. Interaction also occurs by the exchange of virtual electron/positron. The exchange of virtual photon is manifested as the "force" in the electromagnetic interaction, because the interacting particles change their speed and direction of travel as they release or absorb the energy of a photon. Photons also can be emitted in a free state, in which case they may be observed. The interaction of real particles occurs in a series of graphs of increasing complexity. In the first order graph, no virtual photon or virtual electron/positron is involved; in the second-order graph, there are either one virtual photon or one virtual electron/positron; and so forth. The graphs correspond to all the possible ways in which the particles can interact by the exchange of virtual photons and virtual electrons/positrons, and each of them can be represented graphically by means of the Feynman diagrams. Besides furnishing an intuitive picture of the process being considered, this type of diagram prescribes precisely how to calculate the observable quantity involved.

### From Coulomb Scattering to Deep Inelastic Scattering

Coulomb scattering is the scattering of charged particles off one another as a result of the electrostatic force between them. Such process belongs to the first order graphs in the Feynman diagrams. Since there is no virtual particles involved, classical mechanics can be used to show that for a given kinetic energy Ek of the incident particle, the scattering angle is related inversely to the impact parameter b as depicted in Figure 02a. It also shows that for a given the incident particle can probe closer to the target (smaller b) with higher Ek. Since there is no way to measure b experimentally, the cross section = b2 is defined such that the incident particle is initially directed anywhere within such area. The Rutherford scattering formula is derived by identifying the alpha particles to be the incident beam:

d(,Ek)/d = (Ze2/4Ek)2[1/sin4(/2)] ---------- (38c)

where Z is the number of charges in the "nucleus", which contains all the positive charges and most of the mass in an atom.
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 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 02d Cross Section vs Energy

shows the deviation at a fixed angle of 60o.

The relativistic correction has been obtained by N. F. Mott within the framework of QED but without recourse to perturbation theory. The improved formula has the form:

d(,E)/d = [(Ze2E)/(2|p|2c2)]2 {[(1 - 2sin2(/2)]/sin4(/2)} ---------- (38d)

where p is the momentum of the incident particle, E = (mo2c4+p2c2)1/2, and =|p|c/E.

When the incident energy increases to over 1 Gev, the collision becomes inelastic. Under the impact of such high energy, the target nucleon is likely to disintegrate. High energy scatterings are used to probe the structure of nucleons. According to the parton model, at high enough energy the probe would see only the point-like parton inside the structure instead of seeing the whole nucleon as a coherent object (see Figure 02e). In general, the inelastic scattering cross section takes on the form:

d2/dq2d = (42/q4) (E'/EM) [(M/)W2(q2,)cos2(/2) + 2W1(q2,)sin2(/2)] ---------- (38e)

where in the limit when the mass of the incident particle is negligible, q2 = (k - k')24EE'sin2(/2), = (pq)/ME - E'. They are the 4-momentum transfer squared, and the energy transfer respectively representing such transfers from the electron to the "off mass-shell" photon with q2 0 (see Figure 02e for the definition of the various notations).

In Eq.(38e) the cross section is expressed in term of the simple Rutherford scattering formula. The W1 and W2 in the rest of the equation are dimensionless form factors. It is used to correct the point particle description. In general the precise structure of the nucleon is unknown, but the form of W1,2 is severely restricted by Lorentz invariance and electromagnetic current conservation. This is an extremely general and important tool in the absence of a complete theory. In the deep inelastic
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 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)
where o = (h/mc) = 2.43x10-10 cm is the Compton wavelength, which is actually the de Broglie wavelength for the electron moving at a speed of c/.
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.
The relativistic correction has been derived within the framework of QED. It is known as the Klein-Nishina formula:

d()/d = (ro2/2) ('/)2 (/' + '/ + sin2) ---------- (38h)

where , and ' is the frequency before and after the scattering as related in Eq.(38f). Already in the late 1920's, the angular distribution of -ray scattering had been measured to show that the observed deviation from the Thomson formula is precisely what one expects from Eq.(38h). The agreement achieved then is one of the earliest quantitative triumphs of the Dirac theory, which is an attempt to unify the theories of quantum mechanics and special relativity. It is the linearized version of the Schrodinger equation in consistence with special relativity, i.e., it is Lorentz invariant as shown in Eq.(13). The Dirac equation describes particles with spin-1/2. Note also that the deep inelastic scattering is just a special kind of Compton scattering with virtual photon hitting a necluon (such as the proton) instead of real photon colliding an electron as shown in Figure 02e.

### 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]

This small shift in energy levels is caused by various high-order radiative corrections as mentioned earlier. The regular Coulomb potential is now supplemented by an effective potential V(eff) (up to the 2nd order):
• 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 2S1/2 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 2S1/2 state since L = 0 in this case.
By taking the average of V(eff) with the corresponding hydrogen wave functions, it yields the Lamb shift in the form

The total shift amounts to 1052.10 MHz. When other smaller contributions such as the 4th order radiative corrections and the mass corrections are included into the calculation, the final theoretical value is 1057.864 0.014 MHz. Comparing with the experimental value of 1057.862 0.020 MHz, it provides an excellent indicator of the basic correctness of QED. Actually, it is instrumental in restoring confidence to QED from the dubious methodology of renormalization.

The discoverer of Lamb Shift - Willis Eugene Lamb, Jr. passed away on May 15, 2008. Lamb was born in Los Angeles on 12 July 1913. He studied at the University of California, Berkeley, where he obtained his first degree (in chemistry) in 1934 and his PhD (in physics, under the supervision of Robert Oppenheimer) in 1938. He had taken up posts at various institutions, and became professor of physics and optical sciences at the University of Arizona since 1974. His effort on discovering and measuring the Lamb shift earned him the Nobel Prize of Physics in 1955.

More than sixty years after the celebrated QED triumph, precision optical spectroscopy of H atoms and the corresponding calculation have improved tremendously and reach a point where the proton size is the limiting factor when comparing experiment with theory. A much better way to determine the proton radius is possible by measuring the Lamb shift in muonic hydrogen (atom formed by proton and negative muon). Since the Bohr radius is proportional to 1/m and the muon is about 200 times heavier than the electron, the wave function of its S state overlapped more with the proton charge cloud and shifted
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 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).
Muonic hydrogen atoms are produced in a cavity filled with H2 gas and bombarded by a beam of low energy muons. Most of them are in highly excited states with n ~ 14 initially. About 99% quickly de-excite to the 1S ground state (blue in Figure 03b,a), but ~ 1% populate the longer living 2S state with a lifetime of about 1 microsecond (red in Figure 03b,a). Tuneable laser pulses with a frequency of around 5 THz are used to illuminate the cavity for inducing transitions from the 2S to the 2P state within the narrow time window (green in Figure 03b,b,c). By recording the 2 kev K X-ray events (red in Figure 03b,b), a resonance curve is obtained with a peak at 49.88188(76) THz (Figure 03c) corresponding to E = 206.2949(32) mev or rp = 0.84184(36)(56) fm (1 fm = 10-13cm) with numbers in parentheses indicating experimental and computational uncertainties respectively. This new value of rp is significantly different from the previous measurements from H atom spectroscopy (CODATA), and from electron-proton scattering experiment (Figure 03c). Much more works are needed before we can find out the cause of the discrepancy.

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

As mentioned earlier, it is very difficult to compute the viscosity from the Schrodinger equation, although the former must somehow follow from the latter. However, in relativistic theory of elementary particles, the form of the equations is highly restrictive by the requirement of relativistic invariance. For that reason it is actually not difficult to carry out the renormalization procedure in a relativistic theory, at least in the perturbation framework.

It is also mentioned that divergences (infinities) appear with certain loop diagrams in QED. The origin of these infinities can be traced back to the singular nature of the inverse-square law 1 / r2 as r 0. But there is no direct evidence that this law is still valid below a distance of some 10-16 cm. If QED is renormalizable, then the detail at the deeper scale is not important. The inverse-square law can be replace by a constant 1 / r02 for r r0 ~ 10-16 cm. It turns out that QED is indeed renormalizable. The schemes to make the force law less singular is called regularization. There are three popular schemes :
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.
In layman's language the renormalization procedure goes like this: one theoretical calculation produces a quantity Z which involves an infinity. Since ebare is a factitious entity corresponding to no interaction, it can be assumed to be in such form as ebare = eobs/Z. In other word, eobs = Zxebare so that ebare exactly cancel out the infinity in Z leaving behind eobs, which is to be determined by experiment. BTW, such procedure works for subtraction as well as multiplication, e.g., mobs = mbare - mZ.

Mathematically, the divergences come from the integration of the 4-momentum p (or sometimes denoted as k) for the virtual particles (both fermion and boson) to infinity. The integration "4-volume" element d4p contributes a p4 term, which can be "diluted" by a p-1 term from each fermion propagator (internal line), a p-2 term from each boson propagator, and in addition the coupling constant in each vertex will contribute a (-n) power of p if its unit has n-dimension of p (in order to keep the total dimension of the graph to be the same). The origin of the other negative power p terms can be traced ultimately to the number of spacetime derivatives (within individual term) in either the Lagrangian or the field equation. While the Dirac equation involves only first order derivatives, the field equations for most boson are written down in second order derivatives. Thus, if we denote the Divergence as D, then the degree of divergence can be written as :

D = 4 - F - 2B - nV ---------- (39a)

where F and B are respectively the number of fermion and boson propagators (internal lines), V is the number of vertices in the Feynman diagram, which is divergent if D 0. The coupling constants g for any gauge theory is dimensionless, so n = 0. Depending on whether n = 0, n > 0, or n < 0, the corresponding theories are renormalizable, super-renormalizable, or non-renormalizable.

According to the formula for D, only a small variety of graphs in a renormalizable theory are divergent. It is summerized in Table 04 below:

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 mR, and charge eR in this new form will be "dressed" and agree with experimental measurements. The bare and renormalized quantities are related by the following formulas:

mR = m0 - m
eR = [(Z2Z31/2)/Z1]e0 = (Z31/2)e0

It also requires that the fields should be transformed like:

R = Z21/20
AR = Z31/2A0
• 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.
For a super-renormalizable theory, n > 0, so there are even fewer divergent diagrams than the renormalizable theories. Moreover, if V is large enough, Eq.(39) indicates that all diagrams would be primitively convergent.

A non-renormalizable theory has n < 0, so D' grows indefinitely with V. Since D' + 1 divergent parameters are required in each case, a non-renormalizable theory contains an infinite number of divergent parameters. Even if we had an infinite number of bare parameters to adjust, there would be an infinite number of parameters in the theory. It would take a prohibited long time to measure all of them. The gravitational constant G = 6.7087x10-39 c (Gev / c2)-2 in quantum gravity has a negative energy dimension of n = -2 and hence it is non-renormalizable. Presumably some new physics will happen at the scale of 1019 Gev to turn quantum gravity into a renormalizable or a finite theory.

### The Divergence

This section will examine the origin and the forms of the divergence. The formulation will adopt the Feynman rules in momentum space as shown in Table 03 with a few minor changes (such as 5 ) to converse the system from nucleon-pion to electron-photon. Since the divergence arises from the virtual photon with off mass-shell momentum k in the internal line, the main focus will be directed to the integration of k in the S matrix.

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().
It can be shown that A =
where the self-energy becomes infinite as the cut-off (in the upper limit of the k space integration) is made to go to infinity. This is referred to as ultraviolet divergence since it is caused by the infinite value of |k|. The physical mass of the electron is now the difference between the bare mass m0 and , which represents the contribution from the virtual electron-positron pairs in the vicinity of the electron. It is noticed that even when the cutoff is taken to be the Planck energy of 1019 Gev, the correction to the bare mass is /m ~ 0.178. Anyway, the two terms m0 and together render a finite physical mass ,i.e., mobs = mR = m = m0 - . The exact cutoff point is not important as long as mR agrees with observation.

Since the modification occurs in the propagator (from x2 to x1), it is necessary to consider its alteration as shown in Figure 03e from the unperturbed state with bare mass m0 (representing the state of no electromagnetic interaction). By using the identity 1/(X+Y) = 1/X - (1/X)Y(1/X) + with (1/X) = and Y = -i[A + (ip + m)B], it can be shown that the correction to the electron propagator (in momentum space) is given by:

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

where the term has been moved into the denominator, Z2 = 1/(1 - B), and B is another infinity in the (p) expansion about the point -ip = m. The factor Z2 is absorbed into the electron line R = Z21/20, where the subscript 0 denotes bare state and the subscript R the renormalized state. Only the A and B terms are divergent in the Taylor series for (p), and R becomes finite as the result of one infinity from 0 is divided by another infinity from (1-B).

The vertex correction in Figure 03f modifies the first order interaction of the electron with the external potential as shown in the small insert. According to the Feynman rules, the propagator of vertex correction is involved in modification of the charge:
 ---------- (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:
where min is the "mass" of the photon. Under such scheme, virtual photons of very low energies (min) do not get emitted or absorbed. The net result is that, for small values of q = p' - p, that part of Eq.(39c) can be written as follows:
 ...
where the second term containing the generalized Pauli matrices can be identified as the second order correction to the electron magnetic moment. It can be shown that the infrared divergence is cancelled by the similar divergence arisen from the bremsstrahlung process to all orders in perturbation theory. Thus, QED is free of divergence seemingly out of a mathematical miracles.

It can be shown that for large momentum transfer Q = |p' - p|, the effective coupling constant in QED including vacuum polarization is modified to:

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

The Yang-Mills (Figure 03h) theory was originally devised to discribe the strong interaction between nucleons (neutrons and protons, now recognize as not elementary) mediating by gauge bosons - the pion in this case. It was also involved in the early
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 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 mathematical formulation starts with the local gauge transformation of the fermion field:

' = exp(i(x))

where (x) is a function of x denoting the phase angles in the internal space, and represents three 2x2 non-commuting matrices (generators) for the SU(2) group; and eight 3x3 non-commuting matrices for the SU(3) group. In other word, this kind of transformation is non-abelian.

The problem with the construction is that the derivatives of the fermion field are not covariant under this transformation. It can be shown that the formulation is invariant only if the derivative is replaced by the covariant derivative:

D = - iaba

where the space-time component is labeled by the index , the index "a" is for the phase angles, is the coupling constant, and b = aba represents the new fields (the gauge bosons) introduced to render an invariant theory under the local gauge transformation. Then the Lagrangian density takes the form:

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 aba.
• 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.
Its historical development includes the discovery of neutrino, the introduction of Fermi's theory, the violation of parity, the selection of V - A interaction, its adoption to the gauge theory and finally incorporation into the Standard Model (see next section). All these subjects will be presented in some details below.

### 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 Standard Model 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.
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.
• Wa 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.
• fabc 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 Wa.
• 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. Ge 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 Wa 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 GF = 1 / v221/2 = 1.14x10-5 Gev-2, e = g sinW = (4)1/2 = 0.3028, and from the measurement of sin2W = 0.226 0.004 to evalutate the masses for the gauge bosons. The theoretical values are in good agreement with MW = 80.4 Gev, and MZ = 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 me = Gev. 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.
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:

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).

Meaning of the symbols in the QCD Lagrangina density :
• Fa 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 mi is derived from the interaction with the Higgs field in a formula similar to Eq.(41c).
• = .
The Lagrangian for the Standard Model then consists of three parts:
 ---------- (43)
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.(43), or in pictorial form, using the Feynman diagrams as shown in Figure 04. It is convenient to rewrite Eq.(43) 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.

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.
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
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.
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.

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.

### The SO(10) Group and Unification

The 16 fermions in the standard model are summarized in Table 06, where the superscript denotes electric charge, and the subscript denotes color charge. All the quarks in the SU(3) gauge group participate in strong interaction, all left-handed fermions undertake weak interaction, and those fermions carrying electric charge are eligible for electromagnetic interaction.

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.
• In addition, the SO(10) has a 16-dimensional spinor representation, which fits exactly into the 16 fermions in Figure 05b.

• To complete the list of all particles in the standard model, table 07 below displays all the force mediating bosons, where the single quote (') denotes anti-charge similar to the + charge of the positron in the electromagnetic interaction:

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

There is actually only 8 independent bosons for SU(3) and 3 independent bosons for SU(2) since the trace (trace of a matrix A: TrA = ann) of their matrix representations is equal to zero - a condition to reduce the number of independent bosons by one. This is a requirement for the unitarity of the gauge group.

Since the force mediating gluon itself carries color charges, the corresponding Feynman diagram is interpreted in slightly different way than those in QED. Figure 05c shows two quarks with blue and red colors moving toward each other initially.
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.

Since all the color charges are SO(10) group members that can be turned into each others, it is expected that at high probing
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

The lattice theory uses brute force calculation by computer to obtain low-energy features of the hadron world in QCD. There are many problems associated with theoretical evaluation in this particular area of quantum field 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 20th century.
Implementation of lattice theory (with some special features for QCD):

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]
7. For any gauge-invariant operator O, its average < O(U) > is proportional to:
< 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.

Figure 05i shows a typical pattern of activity in the gluon fields from a quark within the grid. In Figure 05j, all the gluon fields are concentrated in a "flux tube" as an anti-quark is added nearby to cancel the fluctuations shown in Figure 05i.

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 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.

A paper published in November 2008 unveiled the masses of many light hadrons calculated with the lattice methodology
(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.
It seems that such huge effort to calculate the masses of the hadrons is a waste of time and money since they are known experimentally already. Actually, it provides another proof for the correctness of QCD, and the developed technique can be used to compute other interesting quantities that are very difficult to measure experimentally.

### 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.
One of the great successes of this approach in the 1960s was the use of dispersion relations to calculate constraints on the pion-nucleon cross sections. These relations were used extensively in strong interaction physics where a renormalizable field theory eluded physicists for many decades. With the successful development of field theory, this method fell out of favor for many years until the 21st century when it is used to obtain the expected cross sections (from the Standard Model) produced by LHC. The calculated cross section (as a function of colliding energy) is compared with the experimental data, any excess would indicate a new process or new particle is involved. The researchers then re-opened the case against the infinities
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

In classical physics, empty space is called the vacuum. The classical vacuum is utterly featureless. However, in quantum theory, the vacuum is a much more complex entity. The uncertainty principle allows virtual particles (each corresponding to a quantum field) continually materialize out of the vacuum, propagate for a short time and then vanish. These zero-point vibrations mean that there is a zero-point energy associated with any quantum field. Since there are an infinite number of harmonic oscillators per unit volume, the total zero-point energy density is, in fact, infinite. The process of renormalization is usually implemented to yield a zero energy density for the standard quantum vacuum, which is defined as no excitation of field quanta, i.e., no real particles are present. In other word, the quantum vacuum is at a state of minimum energy - the ground state.

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.
As the collision energies increase, new resonance spikes occur at 3.096 Gev and 3.687 Gev. It was discovered subsequently that they are the products of a new type of quark called "charm", which together with the "strange" quark form the second generation of quark. With this discovery the number of mesons is expanded into the 16-plet of spin-0 mesons generated by SU(4) flavour symmetry. Similarly, the spikes at 10 Gev and 10.40 Gev in Figure 06b lead to the discovery of the bottom quark for the third generation. The SU(4) flavour symmetry has to be enlarged to SU(5) so that the basic multiplet of spin-0
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:

V = - 4s / (3r) + ar ,

with which the Schrodinger equation in non-relativistic quantum mechanics yields a most satisfactory match to the observed meson mass level (see Figure 06c). This formula combines a Coulomb potential at short ranges with an attractive potential rising linearly at longer distance, giving rise to the ever-increasing forces of quark confinement. Thus, the masses of the c mesons provide direct support for the QCD picture of inter-quark forces containing both asymptotic freedom at short ranges and confining forces at longer distance.

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.

The next topic is about the Higgs field, which prescribes yet another kind of structure to the quantum vacuum.

### 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).

When m2 < 0, the vacuum at = 0 is unstable; a particle would prefer to move down the potential to a lower-energy state at the bottom of one of the wells at = v = (-6m2/)1/2. The states at = 0, and = v are referred to as false vacuum, and true vacuum respectively (Fiigure 06f). Eq.(46) indicates that the system is symmetrical under -. If the origin at the false vacuum is shifted to v, i.e., ' = - v, then the Lagrangian in Eq.(46) becomes:

L = (1/2)'' - V'(') ---------- (47)

with V'(') = |m|2'2 + (1/6)v'3 + (/4!)'4. In this way, the scalar particle has acquired a positive mass squared given by 2|m|2, but the original symmetry between and - has been spontaneously broken because the field has been shifted (resulting in the occurrence of the '3 term) and there is a true vacuum at ' = 0. Note that the valley in diagram (b) has a depth of (3/2)(m4/). The zero point of the potential can be shifted up or down without any effect on subsequent calculation.

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]

In this case there is a circle of degenerate minima giving by (Re)2 + (Im)2 = v2, where v = (-4m2/)1/2 (see Figure 06g). Therefore, there are infinitely many possibilities for the stable configuration: any scalar field satisfying the condition for minimum energy will do. The Lagrangian in Eq.(48) is invariant under the global gauge transformation:

= (1 + i2)ei/(2)1/2

which is just a rotation in the 1-2 plane. In analogy to the previous example, the origin of one of the field components, e.g., 1 is shifted to a point at the circle of minimum energy:
= 1 - v
= 2
In terms of , and , the Lagrangian is transformed to:

Once again, the symmetry is spontaneously broken as before. The mass term can be easily located by looking for the one with quadratic fields. It reveals that the field acquires a (positive) mass squared of -2m2. The novel feature in this example is that the field remains massless. Such massless modes, which arise from the degeneracy of the ground state after spontaneous symmetry breaking, are called "Goldstone bosons". The appearance of Goldstone bosons seems to be in contrary to real world experience since no such massless, spin-0 particle exists. However, if the global gauge transformation in the formulation is replaced by local gauge transformation, it can be shown that the Goldstone boson is absorbed by the formerly massless gauge boson, which has now acquired a mass. The corresponding model is called "scalar electrodynamics", but when it it spontaneously broken it is then referred to as the "Higgs mechanism". Before the symmetry breaking the Lagrangian for the interacting scalar field and electromagnetic vector potential A has the form:

If the vector potential is transformed as:

where q is the coupling constant and v is the scalar field at the true vacuum as defined earlier. By a suitable choice of local gauge transformation such as:

' = e-iq, with q = tan-1(/)

then the scalar field components become:

' = H = cos(q) + sin(q)
' = 0

The Lagrangian re-emerges in the form:

which shows that the symmetry is broken (by the odd power terms in H), and the Goldstone mode has been completely removed by the gauge-transformed boson A', which has acquired a mass qv. The remaining scalar field H has also acquired a mass (-2m2)1/2. The total number (four) of degrees of freedom is unaltered. Instead of a massless gauge boson, having two (transverse) modes, plus a complex field composed of two real components, we now have a massive vector field A' having three modes - two transverse and one longitudinal (a requirement for massive spin-1 boson), plus one real scalar field H. This is just a theoretical model to illustrate the effect of spontaneous symmetry breaking. Of course, the photon remains massless in the real world. It requires both spontaneous symmetry breaking (with v0) and the coupling of the gauge field to the scalar field (q0) to acquire a mass. A more complicated version of this model is applicable to the electroweak interaction in the Standard model. It has been shown subsequently by 't Hooft that the spontaneous symmetry breaking formulation remains renormalizable; the ultraviolet divergences encountered are no worse than those occurring in QED.

In the Standard model the scalar field is identified as the Higgs field responsiable for the mass of fermions and gauge bosons. Supersymmetry increases the number of Higgs particles to five with masses ranging from about 100 - 400 Gev. The Higgs fields are supposed to permeated throughout the universe uniformly and isotropically since the Big Bang. The spontaneous symmetry breaking occurred at a temperature corresponding to about 250 Gev in the electroweak ear soon after the end of inflation. Above that temperature (the phase is known as the symmetric phase) all the particles become massless. The period after the transition is called the Higgs phase. In a way, the Higgs fields are similar to the hypothetical ether of the pre-relativity era, but with a crucial difference - that it is formulated as a relativistic invariant theory, which would not prescribe an absolute frame of reference.

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.

### Gyromagnetic Ratio and Anomalous Magnetic Moment

The anomalous magnetic moment for the electron and muon illustrates the progress of our understanding in particle physics from classical mechanics to quantum theory, quantum field theory, the Standard Model, and beyond.

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).

In quantum theory, the interaction between an electron and a magnetic field can be portrayed by the Feynman diagram in Figure 08a, which shows a photon from the magnetic field is absorbed by the electron and thus altered its trajectory. The gyromagnetic ratio derived from the Dirac equation takes the form: e = /L= -g e/(2me), where g = 2 is related to the fact
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 08b Vertex Cor-rection [view large image]

The more accurate calculation including higher loop diagrams up to the 4th order term yields the following expression:
g - 2 = 2 ( /2 - 0.328 (/2)2 + 1.181 (/2)3 - 1.510 (/2)4) = 0.0023193042800. The experimental value is: g - 2 = 0.0023193043768 (in agreement with the calculated value to ten significant figures).

In calculating the effects of the cloud of virtual particles, we need to include not just the effects of virtual photons and virtual electron-positron pairs, but also virtual quarks, virtual Higgs particles, and, in fact, all the particles of the Standard Model. It turns out, though, that because of the larger muon mass, any such heavy particles would affects the muon magnetic moment more than the electron magnetic moment. The muon g - 2 has been calculated with the Standard Model to a precision of 0.6 ppm (parts per million). The calculated value with the combined effect is g - 2 = 0.0023318360. A remarkable fact is that the muon g - 2 factor not only can be predicted to high precision, but also measured to equally high precision. The measurements at Brookhaven National Laboratory in 2001/2004 (Figure 09) yields an average value of g - 2 = 0.0023318416. Thus, the comparison of measurement and theory provides a sensitive test of the Standard Model. If there is physics not included in the
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.

### 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.

A report in March, 2006 claimed the detection of axions in laboratory. However, the results contradict other observations and do not fit with constraints deduced from astrophysics. Since axions interact only weakly with baryonic matter, they would be able to stream out from the central cores of stars into space virtually unobstructed, carrying energy away with them and cooling the stellar cores. This cooling is more effective the heavier the axions are, and if each axion had a mass greater than 0.01 ev this would affect the appearance of stars and the way supernovas exploded. Further experiments such as the one shown in Figure 10 will confirm (or refute) the discovery.

Axion is also proposed to explain the 20 minutes cycles of X-ray flares coming from the center of the Milky Way. In the 1990s, computer simulations of clouds of dark matter made of axions showed that giant bubbles of these particles would burst out from the clouds. These axion bubbles would expand and contract with a period of 20 minutes - matching the period of infrared and X-ray flares from Sagittarius A*. The model relies on a controversial version of gravity, which proposes that gravity starts to repel as the gravitational field gets stronger. Confirmation by observation would prove the existence of axion and would also raise question about Einstein's General Relativity.

### 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 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
preons can address many of the problems with the Standard Model. This kind of view is also consistent with the observation that the structure of the universe is like an onion or a Russian doll with layers within layers (admittedly a very rough analogy).

Many hypothetical models for the preon have emerged over the years. Figure 12 shows the one proposed in 1979. It posits two kinds of preons and their antimatter version, which could comprise both the fermions and bosons (gluons are not shown here as they are a bit more complicated). This scheme is for the first generation of the fermions. The higher generations are considered as the excited states of generation I. Thus far the construction of the model(s) is still at the stage of numerology playing with specific numerical patterns. There is no clue about its size, mass, spin, interaction, ... etc. It is hope that the experiments at LHC may offer some indications on whether preon is relevant to the future development of particle physics.

### 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 12 Actions [view large image]

 S / q(t) = ---------- (50)
The Lagrangian in the sample computation is taken to be the one for harmonic motion in the form :

L = (m/2)(dq/dt)2 - (k/2)q2 ---------- (51a)

This is the difference between the kinetic and potential energies, i.e., L = T - V, not the total energy E = T + V. It is therefore not a constant or conserved quantity of the motion.

The path in the action of Eq.(49) should be considered as a variable just like the independent variable in a function. It does not represent the actual trajectory of the particle except in some special cases. The equation of motion for the real path is derived from Eqs.(50,51a). To avoid confusion, the actual trajectory is labeled as x instead of q, the result in Eq.(51b) below is just the Newtonian equation of motion with the Hooke`s force law :

m[d(dx/dt)/dt] = -kx ---------- (51b)

The solution of this differential equation is the sine or cosine function or the linear combination of both, e.g.,

x = A cos[(k/m)1/2t] ---------- (51c)

which is very different from the path specified in Eq.(52a) below. For demonstration purpose, the family of parabolic curves :

q = bt - (a/2)t2 ---------- (52a)
dq/dt = b - at ---------- (52b)

will be used to generate the paths between the two end points (0,0), (X,T) as illustrated in Figure 11. If the parameter "a" is taken to be the running index for the different curves, then the other parameter "b" is determined by the end point (X,T) :

b = [X + (a/2)T2]/T ---------- (53)

Each curve in Figure 11 is generated by a Basic program in a home computer with incremental step dt = 0.001 sec :

dq = bdt - atdt ---------- (54)

The action S is calculated by integrating L(q,dq/dt) from (0,0), to (1,4/) with m = 1 gm and substituting q, dq/dt from Eqs.(52a,b) (see also Figure 01a). The paths in Figure 11 and the corresponding actions in Figure 12 are computed by varying the parameter "a" from -50 to +50 in step of 10. The path traces out a straight line for a = 0. For particle in free space, i.e., for k = 0, the correspondence of shortest path (a straight line) and least action (when S is at its minimum) is exact. The correspondence is off when interaction is taken into account (for example with k = 1 gm/sec2, a=(kX/2)/[m-(kT2/10)] 0.6 cm/sec2 when S is at its minimum) as if the force field creates a curve space for which the shortest path is not a straight line anymore. Actually, each paths is similar to a numeral devoid of any physical content by itself. They are one step away from the trajectory of a real particle contrary to what some physicists have it interpreted otherwise. Feynman's expression of "sum over paths" in his original work has been altered to "sum over histories" attaching additional meaning as if the paths are real. Furthermore the Lagrangian involved may or may not describe the actual motion of the particle. Additional constraints have to be imposed by demanding invariance under certain operations and it will ultimately be justified by observations.

The link between least action and shortest path has its root back to the 17th century when physicists in that era discovered that light ray and particles in free space invariably travel in a shortest path between two end points. The "action" is a generalization of the idea to include cases with interaction(s). Thus, to trace backward from the Lagrangian in Eq.(51a) with no interaction, i.e., with k = 0, the action S is reduced to :
S = (mv/2)dq = (mv/2) X = constant (shortest path) = constant (actual trajectory) = least action
where the velocity v=dq/dt is a constant in free space. For cases with interaction, the principle of least action is now used to secure the equation of motion instead of shortest path.

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 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.
The result is presented in Figure 13 and 14 for the real and imaginary parts respectively. The blue curve shows the variation of S as a function of "a", while the red curve (not in scale) is the accumulated sum of <0|X> starting from a = -50. The insert displays a much wider range for <0|X> from a = -400 to a = +400. These diagrams demonstrate that the main contribution to <0|X> comes mainly from paths near the classical path (around the minimum of S). The exponential of the action assumes a pattern of rapid oscillation as its value getting larger, the amplitudes tend to cancel out each other leaving a steady value for <0|X> at the asymptotic limit.

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 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).
The relative probability P that a particle arrives at point X is :

P = <0|X>*<0|X> = |<0|X>|2 = (m/hT) ---------- (58)

These are the formulas originally derived by Feynman in his thesis. The Schrodinger Equation for a free particle can be derived by differentiating the end point (X,T) :
-(/2m)<0|x> = i<0|x> ---------- (59)
where X, T are replaced by x, t in conformity with convention, and <0|x> is now interpreted as wave function. The particular wave function in the form of <0|x> as shown in Eq.(56) has the initial position as well as the initial time specified. While the commonly quoted wave function in the form of ei(kx-t) is not connected to any initial conditions but satisfies the same equation as shown in Eq.(59).

Comparison of the two methods (method 1 for sum over paths and method 2 for sum across paths) is hindered by progressive inaccuracy (in method 1, and in my home computer) as the argument getting larger. However, for the case of free field the action S can be evaluated analytically:

S = (m/2)(X2/T + a2T3/12) ---------- (60)

making the numerical computation less time consuming. The results for the 3 end points in Table 08 are obtained by visual inspection of the asymptotic limits (for method 1, with step for "a" equals to 0.1), and m taking to be the mass of the electron.

X T |<0|X>1| |<0|X>2| |<0|X>2|/|<0|X>1| |<0|X>2|/T |<0|X>1|
1 1.0 8.75 0.40 0.045 0.045
1 1.25 6.28 0.36 0.057 0.046
1 1.5 4.80 0.33 0.067 0.045

#### 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 = eiS/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 90o 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/iT3/2)exp(imX2/2T) ---------- (63)
By defining <0|X>'3 = T<0|X>3, a wave equation can be derived from method 3:
-(/2m)<0|x>'3 = i<0|x>'3 ---------- (64)
with the ratio |<0|X>2|/|<0|X>'3| = (m/2h)/31/2 0.045 ---------- (65)
The extra factor of "T" is introduced by assuming a set of parabolic curves for the paths as shown in Eq.(52a).

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Finally, there is a website for anyone who aspires to a career in theoretical physics. The website provides links to teach people on the intricacy of particle theories. It was created by Gerard 't Hooft, who showed in 1971 that the gauge bosons in Standard Model could be made massive while preserving renormalizability. He is the recipient of the 1999 Nobel Prize.