because of the bound state Eq. (12.47). [Alternatively, Eq. (12.56) is another way to obtain the bound state Eq. (12.47).]

Next, look at the single poles. This is more complicated. There are terms from the single poles and terms from the expansion of the coefficient of the double poles near p2 = mg, the residue of the double poles.

First look at the terms involving 7Z. These do not contribute because

The expansion of the coefficient of the double pole terms near P2 — Mß will generate terms proportional to dT/dP2 and dT/dP2. By an argument similar to

reducible irreducible

Fig. 12.17 Examples of reducible and irreducible diagrams. Reducible diagrams can be separated into two parts by a line which "cuts" only the two heavy particles.

Note that essential features of the equation are that the integration is over all four components of the internal momentum (and hence it is sometimes referred to as a "four-dimensional" equation in the literature) and that both of the particles are off-shell. Any equation of the general form (12.40) with the choices (12.62) is properly referred to as a Bethe-Salpeter equation.

The choice of the kernel V defines the approximation in which the BS equation is being employed. In principle, V can include any Feynman diagram which is two-particle irreducible [recall the discussion surrounding Eq. (12.42)]. Examples of reducible and irreducible diagrams for the BS equation are shown in Fig. 12.17. If the kernel is the sum of all two-particle irreducible diagrams, then the conventional view is that the solution of the BS equation should give the exact result for the scattering amplitude. In this case the equation can be viewed as producing and summing all diagrams which have a two-particle cut (are two-particle reducible) by combining diagrams which have no such cut. However, because the infinite sum of two-particle irreducible graphs is probably as difficult to calculate as the amplitude M. itself, and since the kernel V exists order-by-order in perturbation theory, the kernel is usually approximated by the first few terms of its perturbation expansion. In theories where boson exchange is believed to describe the important physics, such as photon exchange in atomic physics, gluon exchange in perturbative quantum chromodynamics (QCD), and meson (in particular pion) exchange in low energy nuclear physics, V is often approximated by the lowest order one-boson exchange diagram. In this approximation, the solution to the BS equation can be regarded as the exact sum of the ladder diagrams. For the 4>i example we have been discussing, this gives

A1A2

Note that this kernel is independent of P2, and hence the bound state normalization condition for the BS wave function assumes the simpler form (12.61).

If it is desired to sum the ladder and crossed ladder diagrams, then the kernel V must include all irreducible crossed ladder diagrams. These diagrams to sixth order are shown in Fig. 12.18. Since the number of irreducible crossed ladder

Once again, the physical content of this equation depends on the approximation one makes for the kernel V (p, p'; P). In the OBE approximation, the kernel is

A1A2

This is the same as (12.63), but with the important difference that p2 = p'2 = m2. In the large m2 limit, the kernel (12.65) reduces to a form in coordinate space which is an instantaneous, local potential. Specifically, using the definition (12.49) for the bound state wave function, which for this equation is

the bound state equation becomes mi-(P-p)

In the m2 -> 0 limit, and taking P = (W, 0) with W = m2 + E, this equation reduces to

(m2 +p2 - E2) ip (p, P) = -2mi J (p-k)i> (fc, P) , (12.68) where the effective potential is

geff

H2+q2

with geff defined as in Eq. (12.7). Equation (12.68) is a Klein-Gordon equation for a particle of mass mi and energy E in an instantaneous scalar potential. In coordinate space it is simply

with tir

which is precisely Eq. (4.7) with U(r) = 2miVs(r) and ip(x) = il)(r)e~iEt. We see that the spectator equation has the property that it reduces to a one-body equation in the limit when the mass of the on-shell particle approaches infinity. We will refer to this property as a one-body limit [Gr 82].

the discussion in Sec. 15.2, we know that the meson pole contributions from ladders and crossed ladders cancel as m2 -> oc, and since these contributions dominate all of the higher order kernels shown in Fig. 12.20, they will all approach zero as m2 —> oo, leaving only the OBE term, which therefore gives the exact result for the sum of all ladders and crossed ladders in this limit.

In a similar fashion, it may be shown that the spectator equation for a Dirac particle of mass mi and a scalar particle of mass m2 exchanging a scalar meson of mass p. reduces, in the m2 —> oo limit, to a Dirac equation. This is left as an exercise (see Prob. 15.1).

In the m2 —► oo limit, the relativistic wave function (12.49) for a bound state in the spectator formalism is

where E0 is the bound state energy of particle 1. If the vertex function T is a constant, the coordinate space form of this wave function is

which is the familiar asymptotic S-state wave function. Hence the propagator factor in (12.72) gives the asymptotic part of the wave function, while the vertex function T contains all of the dynamical information contained in the intermediate and short-range part of the wave function.

Next, note that the normalization condition (12.61) for the wave function (12.72) becomes [don't forget the minus sign associated with the integral in (12.64)]

= J d3p2EoiP2(p,P) , where we have chosen M = (27r)~3/2. In coordinate space this becomes

where, as before, ip(x) = ip(r) e~lEot. Note that we have recovered the precise form of the Klein-Gordon normalization, Eq. (4.14).

The spectator equation has been used as the foundation for the calculation of higher order QED corrections in simple atomic systems [EK 91] and for the relativistic treatment of nucleon-nucleon scattering [GV 92].

12.7 EQUIVALENCE OF TWO-BODY EQUATIONS 399

The two different relativistic equations we have discussed so far correspond to two different ways of calculating the scattering matrix, and it is usually assumed that the exact answer could be obtained from either equation if the kernel included all irreducible diagrams. However, since the kernel is always approximated by a few irreducible diagrams, an approximate calculation of M using one equation will differ from an approximate calculation using another, and it is important to know how to compare the two approximations. Alternatively, by carefully choosing the kernels, it is possible to obtain the same solution for M from two different equations. In this sense different equations are equivalent. We will discuss this now.

We assume that the same solution for M. has emerged from two different equations, and ask how their kernels must be related by this fact. Specifically, assume that m = vi + J vigim = Vi + ¡mglvl f m = v2+ / v2g2m = v2 + mg2v2

Discretizing the integrals, so that V g and m become matrices, these equations imply m = ( 1 - ViGi)-1 Vi = V2 (1 - g2v2)~l (12.77)

As an illustration of the content of this equation, suppose that equation 1 is the BS equation and Vi is the OBE approximation. Then Eq. (12.78) tells us that the kernel V2 of the spectator equation which exactly sums the ladder diagrams (since Vi does this) is given by the solution of the equation

Iterating this equation and then setting p = p generate the infinite series of diagrams shown in Fig. 12.21. The difference of the propagators g i — g2 is equivalent diagrammatically to the open circle on the heavy particle line, as shown in Fig. 12.20. We conclude that the use of the spectator equation to sum the ladder diagrams is extremely inefficient. The kernel for this operation is an infinite series of terms, and evaluating it by solving (12.79) is just as difficult as solving the original BS equation in ladder approximation.

in the upper half plane, and hence can never contribute to the imaginary part of M. The proof will be carried out for physical values of W only. These are real values of W > mi 4- m2.

To prove these statements, we will prove that the poles are ordered in the following sequence along the real k0 axis:

where the ordering of the pairs (7,6), (5,1), and (2,3) is indeterminate. Using W = Ei(p) + E2(p), these inequalities become

8 < (7,6) (7,6) <5: (7,6) <1: 5 <(2, 3): 1 < (2,3) (2,3) <4:

-Ei < E2(p) - uj -Ei{p)-u< -Ei E2 (p) - u) < E2 Ei (p) - Ei < w E2< w + E2(p) w < Ei(p) + Ei ,

where inequalities involving p' are similar to those involving p and need not be considered explicitly. The only inequality which requires any demonstration is

and a similar one for E2. To prove this we square both sides giving the requirement

2m? + p2 + k2 - 2\Jmf + p2 jm\ + k* < n2 +p2 + k2 - 2p ■ k .

The minimum value of J1 occurs when p ■ k = pk, and rearranging terms and squaring again give the requirement

(2m? + 2pk - n2) < 4 (m? + m2 (p2 + k2) + p2k2) .

Expanding out these terms shows that this equality is always satisfied, even if p, is very small, because p2 + k2 > 2pk.

Hence the exact result for the imaginary part of A4box can be obtained from the pinching of poles 1 and 5, which from Eq. (12.4) gives

/crfc

12.10 DISPERSION RELATIONS AND ANOMALOUS THRESHOLDS 405

The imaginary part of this propagator therefore restricts the intermediate particles to their mass shell, and in this way generates the correct two-body unitary cuts shown in Fig. 12.24. It is also a three-dimensional equation. Carrying out the integration over s gives, in the rest frame.

Gbbs

showing that the relative energy is no longer an independent variable.

The two-body BBS equation was designed to preserve two-body unitarity, but, in fact, this is a feature it shares in common with the other two equations we have discussed previously. To compare the BBS propagator with the spectator propagator we must first remove the factor of n/E2 contained in the spectator integral operator (12.64) and factor the denominator,

£?i [Ei+E2-W - ie] [Ei +E2 + W) ' In the same form, the spectator propagator is r s(p u \-_¿(£2 ~ kg)_

Note that these two propagators are identical along the unitarity cut (when W — Ei + E2) and differ only in how they describe the physics away from the unitarity cut.

One of the significant features of the BBS equation is that it treats the two particles symmetrically [this is most easily seen from the original form (12.94)], and hence it is easy to use the BBS equation for the description identical particles, r'urthermore, the only singularities of the BBS propagator are those associated with 'he unitarity cut. The spectator equation shares neither of these features. It hai additional singularities and can only be used to describe identical particles if it is explicitly symmetrized (or anti-symmetrized) by including channels in which either particle 1 or particle 2 is on-shell [GV 92],

The construction of relativistic two-body equations and the comparison between different methods are active areas of current research, and we will leave the subject at this point.

12.10 DISPERSION RELATIONS AND ANOMALOUS THRESHOLDS

We saw in Sec. 11.7 that the vacuum polarization diagram 11.10 satisfied a dispersion relation n(<?2) = n(o)

This relation follows from the observation that n(g2) is a real analytic function of its argument [i.e., II* (ç2) = n(ç2*)], with a cut which lies along the real q2 > 4m2

As this argument shows, scattering amplitudes and vertex functions generally satisfy dispersion relations. Even though they are initially defined only along the real axis, they can usually be analytically continued to the complex plane, and since they are real over some interval of the real axis, the analytic continuation must satisfy the real analytic property

everywhere. When dispersion relations are combined with generalized unitarity, which specifies the imaginary part as the product of the initial and final state scattering amplitudes (or vertex functions), as discussed in Sec. 12.8 above, they can be an important tool for understanding the properties of matrix elements.

It is our purpose in this section to show how these ideas can be used to gain i .sight into the structure of bound states. In so doing, we point out that care must oe exercised in applying dispersion theory to amplitudes involving bound states, because of the existence of anomalous thresholds.

As an example, consider the spectator vertex function for a bound state in the (p3 theory, introduced in Eq. (12.66). As we have previously indicated, this vertex function can depend on only two independent four-vectors (the third is fixed by four-momentum conservation) and must be a scalar function of these two four-vectors. Only three scalar variables can be formed from the two four-vectors (they are p2, P2, and p ■ P), and if we take the heavy particle to be on-shell (along with the bound state), there is only one scalar variable remaining, which we will choose to be the mass (squared) of the light off-mass-shell particle. In the notation of Fig. 12.26, this variable is u = (P + p)2, and the vertex function will be regarded as a complex function of this variable,

To use dispersion theory to study this amplitude, we must know the location of its singularities in the complex u plane, and these are found (typically) by examining one of the simplest Feynman diagrams which contributes to the vertex. Such a diagram is shown in Fig. 12.26.

This diagram gives the following amplitude:

(ml - k2 - u) {m\ - (P - k)2 - it) (M2 - (P + P - kf) '

where p and P are the four-momenta of the initial heavy particle and bound state, respectively, and the masses are the same as the ones we introduced earlier in the chapter. Note that we have "turned the heavy particle around," so that the amplitude we are considering is actually the virtual process m2 + Mb m i

where m2 is the antiparticle associated with particle 2, instead of the original process

This new amplitude is an analytic continuation of the original amplitude and is more convenient to use with the dispersion relation.

The utility of the dispersion relation is evident if we use it to evaluate (12.102). As suggested by Fig. 12.26, we would expect the unitarity cut to begin at u — (mi + n)2, the threshold for the production of a real intermediate state consisting of two particles of mass mi and /¿, and hence the dispersion integral should have the form n i(m, du' Imf(u')

where Im f(u') is evaluated from the generalized unitarity relation. Taking the general form of the unitarity relation from Eq. (12.83), we have

where p2 is the two-body phase space factor and k is the four-momentum of the intermediate on-shell particle 1. A quick estimate of the function for negative values of u can be obtained by approximating f(u) by a single pole at mass uq which would give f(u. (12.107)

Since the cut in (12.105) begins at u = (mi + ¡x)2, we expect uo > (mi + fi)2. Now, evaluating u in the frame in which the bound state is at rest and assuming the state is weakly bound so that Mb — mi + m2 and eB = m\ + m2 — Mb is

12.10 DISPERSION RELATIONS AND ANOMALOUS THRESHOLDS 409

where p2 is the square of the three-momentum of either of the bound particles in the rest system of the bound state. Inserting this into (12.107) gives f(u) = r(p,p)

where 62 = 2m\tB - 2m\(mi + m2 - Mb) and all terms of order p4 have been discarded. If this estimate is to agree with the Hulthdn model we discussed in Sec. 12.1, then we require u0 - m\ + 62 Si (p + S)2

However, if u0 — ("H + p)2, this would imply that mi ~ 6, which is clearly not satisfied for a loosely bound state where S « mi. Our estimate does not workl The resolution of this problem lies in the fact that the dispersion integral has an anomalous threshold which lies far below the normal threshold at (mi + p.)2. It can be shown (Prob. 12.3) that if M% > m2 + m2+ which is certainly the case for the loosely bound system we are considering here, then the exact location of the threshold is not at u0 = (mi + p)2 but is instead at

where A (a, b, c) = 2 a2b2 + 2a2c2 + 2 b2c2 -a4 -b4- c4. In the limit m2 oc, the above expression becomes uq =m2 + p? + 2p6 ,

which agrees precisely with the requirement (12.110). The dispersion estimate, including the anomalous threshold, is in beautiful agreement with the estimate derived from the wave function.

To understand the origin of the anomalous threshold [Cu 61], it is helpful to regard the dispersion integral (12.105) as a function of the complex variable Mjg and to analytically continue this function from small values of M\ (where there are only normal thresholds) to large values (where the anomalous thresholds will appear). To carry out the analytic continuation it is convenient to write the imaginary part, Eq. (12.106), in the following form:

WB = Merit

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