15.1 QUANTIZATION OF GAUGE THEORIES This definition satisfies the requirement (15.23)

J d [U'U] =» J V [A'(x) + A(x)] = J V [A(®)] (15.25)

because, at each point X* (or more properly, in each volume a), A(xt) = A: is integrated from -oo to oo, and hence, if the integral exists,

Returning to Eq. (15.20), note that A-1 (A) is gauge invariant because a-1 (a"') = jdU 6 [F (a"'17)] = J d(U'U)S [f (a^)]

Now A(A) can be explicitly evaluated using fields defined in coordinate space, but it is more straightforward if the fields are treated in momentum space. Hence <9MAM(x) —> — iA^A^fc), and all the gauge conditions we have mentioned so far can be written

where

= Lorentz gauge

Coulomb gauge.

Now A(A) will be evaluated for the general case of a non-Abelian gauge transformation because it will be needed later. Working with the infinitesimal transformations given in Eq. (13.31), the gauge transformations are

where the transformations have been written in momentum space, the indices a, b, c are color indices, and /a(,c are the structure constants of the group. Dividing space up into discrete cells, so that A^(ki) = A^, the gauge transformation can be written

where the gauge transformation matrix is

where G = ^AqG0 is an arbitrary Hermitian, traceless matrix in the gauge space. Then consider the more general relation

This quantity is still gauge invariant. Furthermore, working around the new field Ag which satisfies the new gauge condition

we can show that A(A) is independent of G. To demonstrate this, return to Eq. (15.35) and generalize it to include G:

J ia

because Asatisfies the new gauge condition.

Finally, following the discussion leading to Eq. (15.17), we may insert the constraint imposed by the gauge condition in the form

where Af(a) is the (infinite) constant f P(G) trG2 = r

00 ia ia

This constant normalizes the integrals over each Gf and approaches infinity as the number of cells n, into which the spatial integrals are divided, approaches infinity. However, since it is a constant, it may be discarded leading to the following generating function for gauge theories:

2-ttq

Next, we simplify this by letting AM —► Ajj7 '. Then, everything in the integral is explicitly gauge invariant except the 6[F(AU) - G], and we get

G1 eiA(Ul)

Z0\j) = J dU Jv[a^g]a(a^) 6[F(A)~ J dl) J V[AfiG}A{A)8\F(A) -

integral over gauge integral over degrees of degrees of freedom freedom fixed by the gauge iA

Hence the propagator is

There are two common choices for the (arbitrary) gauge parameter a:

Note that a = 0 corresponds to a limiting case where G = 0. The Feynman gauge is the one used in Chapters 10 and 11 and is used for many QED calculations.

(iv) The choice a = oo, which would eliminate the gauge fixing term, gives a singular propagator. A finite a is necessary and hence a gauge fixing term is necessary.

(v) In QED where current conservation takes on the simple form /cMjM = 0 (for on-shell particles) the second term in the propagator vanishes and all results are independent of a explicitly. Gauge invariance (independence of a) and current conservation are again the same constraint.

We now turn to a discussion of the quantization of QCD.

The final application of path integrals will be to the determination of the Feynman rules for QCD. We already have the basic ones, given in Fig. 13.1. Our focus here will be on gauge fixing and the appearance of ghosts in loop diagrams.

We have already done the bulk of the work necessary to obtain the Feynman rules for QCD. The candidate generating function is just the one given in Eq. (15.47). The principal differences are that:

• fiiv^" now contains the covariant derivatives, which include AM terms. When expanded out, they generate a kinetic energy term of the same form as in QED, plus three-gluon (/I3) and four-gluon (A4) interaction terms. These generate the 3g and 4g couplings given in Eq. (13.57).

• A(A) is now dependent on A. This means that it can no longer be discarded., as we did in QED. This Faddeev-Popov term generates new interactions which we want to discuss now.

A general result for the A(A) term was given in Eq. (15.38). In the generalized Lorentz gauges it is

This determinant will make the volume integration over V[A] dependent on A and will influence the path densities in a non-uniform manner. Any such influence is an effective interaction which must be taken into account in all calculations. This can be done by writing the factor A as an exponential. Since A multiplies the rest of the path integral, its phase must be added to the other terms in the action, and these new terms will describe new interactions which can be calculated in the usual fashion. Fortunately, the technique for doing this has already been given in Eq. (14.145).

To convert the determinant into an exponential, we introduce two new anti-commuting, colored, scalar fields, denoted by c and c. Since such particles violate the connection between spin and statistics, they are referred to as ghosts, and we expect them to appear only in loops and not to exist outside the region of interactions. Using the identity relating a determinant to an integral over Grassmann variables, Eq. (14.145), we write

cc\ exp

= JvMe'J**™*™-^ f d4pd4p'facbc^p')ip'"Al(p'-p)cb(p)

Fourier transforming the exponent to position space gives

and hence

From the form of this expression, we deduce that ghost fields behave like massless scalar fields, with a propagator of the conventional form

Furthermore, ghosts interact with the gluon fields with a 4>3 type interaction. Since the interaction with c is different than with c, the ghost lines should be oriented, and we have the following addition to Feynman Rule 1:

• the operator | |

gfabc (p + k |
\300LX |

at each gluon ghost vertex, where the | |

outgoing ghost (dotted in the diagram) |
/ P< c |

has momentum p + k and color a, |
Fig. 15.1 |

the incoming gluon has polarization | |

color b, and momentum k, and the in | |

coming ghost has momentum p and | |

color c. |

The sign of this term follows from the observation that the term in the Feynman rules is —iHint — ¿Ant- The momentum of the c field (outgoing) is elp x so that dn ip'n' giving

-i ip'P gfabc = p'^gfabc and precisely the above result when p' = p + k is substituted. The dot in the diagram of Fig. 15.1 tells which line has the momentum attached to it.

In constructing closed loops, c must pair with c, and since the momentum is associated with c, we have the rule:

• a ghost line cannot be dotted at both ends. Finally, since ghosts anticommute:

The full set of Feynman rules for QCD is given in Appendix B [MP 78]. Any QCD diagram can be calculated from these rules, and they will be used in the next section and in Chapter 17.

As an illustration of the use of the Feynman rules for ghost loops, the second order ghost contribution to the gluon self-energy, given by the Feynman diagram shown in Fig. 15.2, is

This diagram, and others which contribute to the gluon self-energy in second order, will be evaluated in Chapter 17, where it will be shown that the effective QCD coupling constant g approaches zero at high energies, a property referred to as asymptotic freedom.

The appearance of ghosts in QCD may appear a bit mysterious. In the next section it is shown that ghosts are necessary in order to preserve unitarity.

Even though ghosts never appear in external states, ghost loops play an important role in QCD. Since they are a consequence of the quantization of a field with a gauge symmetry, it is expected that they will be needed in order to maintain gauge invariance, but it is perhaps less obvious that they are also needed to maint .in unitarity. In this section we will look at a simple example which illustrates now ghosts help to maintain both gauge invariance and unitarity. [See also the discussions in Cheng and Li (1984) and Aitchison and Hey (1982).]

First, consider the annihilation of a quark-antiquark pair into two gluons. To second order in g2 there are three diagrams which describe this process, as shown in Fig. 15.3. Omitting the polarization vectors of the final state gluons, these three diagrams are

^Aab = -92\hKv(p2,s2)^ —Î—1 7"«(Pl,Sl) Ml3ab = -92\K^bV(p2,S2)^ 1 , 7"«(Pl>«l)

Note that the sum of these three diagrams is symmetric under interchange of the two final state gluons.

Now, the physical scattering amplitudes are obtained from the expressions (15.60) by contracting them with the polarization vectors of the outgoing gluons, which will be denoted by (these vectors also carry a color index, which will be suppressed for simplicity). As we saw in Sees. 2.5 and 9.10, a massive spin one field has three independent polarization states defined by the requirement p-e = 0 , (15.61)

where p is the four-momentum of the particle [recall Eq. (2.44)]. If the particle is massless, p2 = 0, and this condition does not uniquely specify the polarization states. In particular, for a massless particle traveling in the z direction, the helicity states e±" = q=4s(0,l,±i,0) (15.62)

satisfy condition (15.61), but so does any vector of the form e'± = e± + ap , (15.63)

where a is an arbitrary parameter. To uniquely define two transverse states of a massless particle, we must impose an additional condition which determines a. This condition is equivalent to fixing the gauge. In our discussion of the EM field in Part I, we chose the Coulomb gauge, which is equivalent to the requirement that n ■ e = 0, where, for a particle moving in the ¿-direction, n^ = (0,0,0,1). Returning to the problem under consideration, the requirement that the physics be independent of the gauge translates into the requirement that the amplitudes M give the same result for any polarization vector of the form (15.63). Hence the necessary and sufficient condition for a gauge invariant result is that the sum of the three diagrams in Fig. 15.3 satisfy the conditions = 0 and k^M^ = 0.

Because of the symmetry, it is only necessary to look at one of these relations; the other can be obtained by interchanging k\ <-> k-2 and a <-» b.

506 QUANTUM CHROMODYNAMICS AND THE STANDARD MODEL The first two diagrams give

where we used the Dirac equation to simplify the result. Note that this would be zero if the commutator vanished, which shows that, in QED (where diagram C does not exist), the two diagrams A and B are gauge invariant. The contribution from diagram C is

" (l§2 J fabc^cV(p2,S2) [7"K2 - fcjf ft] u(p1>Sl). (15.65)

Note that the first term in the final expression cancels (15.64), so that the total 'esult is simply kv kl»Mab = {92 fabc\K V(p2, S2) ft tifci.Si)

where T is defined by this equation. Note that T is symmetric under interchange of the two gluons and that therefore k2vM= k^T. Now, because of the condition (15.61), the result (15.66) is zero when contracted with the polarization vector e2. Hence we conclude that the physical scattering amplitude is gauge invariant and that the gluon self-interaction diagram, C, is essential to this result. However, if some other vector, ki for example, is contracted into (15.66), the result is not zero, and this has implications for the unitarity relation, which will be examined now.

As we discussed in Sec. 12.8, the unitarity relation tells us that the imaginary part of a scattering amplitude must be equal to the integral over the product of the amplitudes which describe scattering from the initial state to an intermediate state and from the intermediate state to the final state. Symbolically,

where the integral is over all degrees of freedom of the intermediate state, which must be a physical state with both particles on the mass-shell. The fk includes the two-body phase space factors discussed in Sec. 12.8 (the specific form of which are not needed in the subsequent discussion). This unitarity relation is a profound restriction on any theory and is directly related to the conservation of probability.

Let us see how this restriction applies to the elastic scattering of a qq pair through a two-gluon intermediate state, described by the six diagrams shown in Fig. 15.4. Unitarity tells us that the imaginary part of these diagrams must be equal (with suitable factors) to the "square" of the q + q —► 2g annihilation amplitudes which we have just studied, and in the remainder of this section we will show that this is true, but only because of the presence of the ghost loop diagram, F, which is needed to cancel unwanted contributions from the first five diagrams A-E.

In preparation for this demonstration, we note that the metric tensor can be

508 QUANTUM CHROMODYNAMICS AND THE STANDARD MODEL decomposed as follows flV = sjL + 9^

5 {ki^k-xv + kivk2fij , where ei are transverse polarization states with zero time components and fcx and k2 are "unit" vectors in the direction of fci and k2\ i.e., = k%/k0, where k0 is the energy of either gluon in the overall center of mass frame (see Prob. 15.1). This decomposition is very convenient for the present problem.

Using this result, and drawing on the discussion in Sec. 12.8, the imaginary part of the diagrams A-E is

Im -M scattering

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