pi·~pf =pipfcosθ, (2.35) as shown inFigure 2.4. In the CM frame, s= (E1+E2)2 = (E3+E4)2 is just the square
1 2
3
4 pi
−pi
pf
−pf
θ
(i)
1 2
3
4 θ3
(ii)
1
2
3 4
(iii)
Figure 2.4
Scattering kinematics in (i) the center of mass, (ii) the lab, and (iii) the Breit frames.
of the total energy. In the physical scattering region,s≥(m1+m2)2 ands≥(m3+m4)2. Usingp2=p1+p2−p1 one finds
E1= s+m21−m22
2√s −−−−−→m
1=m2
√s
2 , (2.36)
so that
pi=q
E12−m21=
s−(m1+m2)21/2
s−(m1−m2)21/2
2√s
−−−−−→m
1=m2
[s−4m21]1/2
2 ,
(2.37)
with similar expressions for the other particles. This is sometimes written as pi= λ1/2(s, m21, m22)
2√s , (2.38)
where
λ(x, y, z)≡x2+y2+z2−2xy−2xz−2yz. (2.39) Thet anduvariables, which describe the momentum transfer between particles 1 and 3 and between 1 and 4, respectively, are given in the CM by
t=m21+m23−2E1E3+ 2pipfcosθ−−−−−→m1=m3
m2=m4
−2p2(1−cosθ) =−4p2sin2θ 2 ≤0 u=m21+m24−2E1E4−2pipfcosθ−−−−−→m1=m4
m2=m3
−2p2(1 + cosθ) =−4p2cos2θ 2 ≤0,
(2.40)
where the last expressions are for elastic scattering (m1 = m3, m2 = m4 or m1 = m4, m2=m3), for whichpi=pf ≡p. Note thatt anduare negative at high energies (e.g., the last expressions in (2.40) hold withp∼√s/2 when the masses can be neglected), but may be positive at low energies if the masses are not all equal.
Fixed target experiments are carried out in thelaboratoryframe, in which 2 is at rest, p1= (E1, ~p1), p2= (m2, ~0 ), (2.41) with
E1=s−m21−m22 2m2
. (2.42)
A sometimes useful relation between the CM and laboratory variables is
pi=|~p1|m2/√s, (2.43)
wherepi is the CM momentum in (2.37). From energy and momentum conservation, E3+E4=E1+m2, |p~4|2=|~p1|2+|~p3|2−2|~p1||~p3|cosθ3, (2.44) so thatE3andE4can be expressed in terms ofsand the laboratory scattering angleθ3for particle 3. The relations between the laboratory variables andtanduare straightforward.
For example,
t=m23+m21−2E1E3+ 2|~p1||~p3|cosθ3, u=m22+m23−2m2E3. (2.45) These formulae become especially simple in the special case m1 =m3 = 0 andm2=m4, e.g.,
p3
p1
= 1
1 +mp1
2(1−cosθ3), t=−2p1p3(1−cosθ3), (2.46) withp1≡ |~p1|=E1 andp3≡ |~p3|=E3.
Yet another frame, especially useful for theoretical purposes, is theBreit or brick wall frame (e.g., Hagedorn, 1964; Renton, 1990), in which the scattered particle 3 simply reverses the direction of 1. Form1=m3, the momenta are
p1= (E1, ~p), p3= (E1,−~p), (2.47) so that
t=−4|~p|2. (2.48)
We will see an example in discussing the simple parton model inSection 5.5.
2.3.5 The Cross Section and Decay Rate Formulae
In this section we sketch the derivation of the relation of the transition amplitude Mf i to the cross section or decay rate. The results apply to any field theory, not just Hermitian scalars.
Two-Body Scattering
First consider the cross section for the 2 → n process i → f, where |ii = |~p1~p2i and
|fi=|~pf1· · ·~pfni, as shown inFigure 2.5. (Particle-type and spin labels are suppressed.) As described inAppendix B, the transition matrix elementUf i and transition (scatter-ing) amplitudeMf i are related by(B.1), so the transition probability is
|Uf i|2=
(2π)4δ4(X
k
pfk−p1−p2)Mf i
2
. (2.49)
pf1
To interpret the square of the delta function, it is convenient to temporarily assume a finite volume V for space and a total transition time T, which can be taken to ∞ at the end.
Then
where we have used one delta function to replace the integrand of the other integral by unity.
A scattering cross section is defined as the transition rate divided by the relative flux.
The differential cross section to scatter intoQ
kd3p~fk is therefore factors correct for the normalization of the covariant states,4 and thenfactors ofV /(2π)3 represent the density of momentum states of the n final particles. Note that the factors of V andT cancel in the final expression in (2.51), and that all the particles are on-shell (E2=~p2+m2). The flux factor|β~1−β~2|is evaluated usingβ~j=~pj/Ej.
It should be intuitively clear that the differential cross section is the same in the lab frame and in collinear frames related to the lab by boosts along the~p1 direction, such as the CM frame, but isnot the same in arbitrary frames. This can be seen by the fact that the factor E1E2|β~1−β~2| in the denominator is equal to the Lorentz invariant quantity [(p1·p2)2−m21m22]1/2 in collinear frames. In fact, the cross section formula is often written
4The inner producth~p|~p0i= (2π)32Epδ3(~p−~p0) = 2Ep
Rd3xe−i(~p−~p0)·~xgoes to 2EpV δ~p ~p0 in a finite volumeV.
in terms of that quantity, though that is only strictly valid in the collinear frames.5 Theδ4 function and the scattering amplitudeMf i in (2.51) are manifestly Lorentz invariant. The final state phase space factors are also invariant, as can be seen in (2.4). The differential cross section can be integrated over the ranges of momenta of interest. The total cross section for the specific process is obtained by integrating over all final momenta,
σ=Y
l
Sl
Z
dσ, (2.52)
where the statistical factorSl≡1/l! must be included for any set oflidentical final particles to avoid multiple counting of the same final state.
Now, consider the example of 2→2 scattering in the CM, as illustrated inFigure 2.4(i).
For a given CM energy√s=ECM, the initial and final momentapi,pf, and the energies Ea, a = 1· · ·4, are fixed by (2.36) and (2.37). It is convenient to introduce spherical coordinates, with the z axis along ~pi, so that ~pf has polar angle θ and azimuthal angle ϕ. For spin-0 particles (or unpolarized initial particles with non-zero spin) the scattering amplitudeMf i is independent ofϕby rotational invariance.
The differential cross section is given by dσ= (2π)4δ4(p3+p4−p1−p2) The implicit~p4 integral can be done using the four-momentum conservation,
δ4(p3+p4−p1−p2)d3~p3
The differential cross section is therefore dσ
5In applications to astrophysics one is usually interested in the thermal average ofσ|β~1−β~2|, so the flux factor cancels (e.g., Kolb and Turner, 1990).
m2→m4. The last expression in (2.57) is for elastic scattering, for whichpi =pf. Closely
wheret is the Mandelstam invariant given by (2.40), anddΩ≡dϕ dcosθis the solid angle element. The second form is useful ifdϕis not integrated over.
In our example of the Hermitian scalar field with VI given by (2.22), the differential cross section at tree level is obtained from (2.30) and (2.57),
dσ where the 1/2 is because the final particles are identical.
One can also use (2.53) to calculate the cross section in the lab frame, using
E1E2|β~1−β~2|=E1m2|β~1|=p1m2. (2.62) The phase space integral can be carried out explicitly in the lab frame, or can be obtained by Lorentz transforming the CM result. In analogy with the derivation of (2.56)
δ4(p3+p4−p1−p2)d3~p3
where θ3 is the laboratory scattering angle of particle 3.dE4/dE3 can be calculated using the second equation in (2.44), yielding
dσ
Consider the 1 → n scattering of a single particle from a static source (i.e., potential scattering), as illustrated inFigure 2.5. This may be an approximation to scattering from a heavy target particle. In particular, suppose Lcontains an interaction term
LI(x) =Lp(x) Φ(0, ~x), (2.66)
where Φ(0, ~x) is the static source andLpinvolves ordinary quantum fields. Then, in analogy with Equation(B.4)from Appendix B, the tree-level transition amplitude fori→f, with
|ii=|p~1iand|fi=|p~f1· · ·~pfni, is is the Fourier transform of Φ, q= (0, ~q), and we have used translation invariance for the matrix element. Then, carrying out thexintegral,
Uf i= 2πδ(Ef−E1)Mf i, (2.69) where
Mf i=hf|iLp(0)|iiΦ(~˜ pf−~p1) (2.70) andpf =P
kpfk. Proceeding as in (2.51), the differential cross section is dσ= |Uf i|2
so that energy but not 3-momentum is conserved, as expected. For the important special case of elastic scattering, i.e.,n= 1 withm2=m1 andp2≡pf1,
|~p1,2|=βE1,2≡p, withβ ≡ |β~1|. (2.72) But,
δ(E2−E1)d3~p2=p2E2dcosθdϕ, (2.73) whereθ andϕare the polar and azimuthal scattering angles. Therefore,
dσ
dcosθdϕ = 1
16π2|Mf i|2 → dσ dcosθ = 1
8π|Mf i|2. (2.74) The last form holds when Mf i depends only on
|~q|2=|~p2−~p1|2= 2p2(1−cosθ) = 4p2sin2θ
2, (2.75)
as in the case of spinless or unpolarized particles and a radially symmetric source. As a simple example, consider
LI =−1
2φ2Φ(r), (2.76)
where Φ(r) is the spherically symmetric Yukawa potential Φ(0, ~x) =κe−µr
r ←→ Φ(~˜ q) = 4πκ
µ2+|~q|2, (2.77)
withr≡ |~x|. Φ can be thought of as arising from the exchange of a heavy scalar of massµ between a nucleus and the scattered particle described byφ. (This could be an approximate model for the contribution to pion-nucleon scattering from the exchange of a heavy scalar resonance, such as theσor thef0(980).) From (2.70) and (2.74) we obtainMf i=−iΦ(~˜ q) 2.5). Similar to (2.49) and (2.51), one has
|Uf i|2=V T(2π)4δ4X
where Γ is the decay rate of particle 1 and τ is its lifetime, and the second expression is specialized to the rest frame of the decaying particle. The total decay rate is obtained by integrating over the final particle phase space,
Γ =Y
l
Sl Z
dΓ, (2.81)
whereSl≡1/l! is a statistical factor forl identical particles, analogous to (2.52).
For a 2-body decay, 1→2 + 3, Equation (2.80) simplifies to dΓ = (2π)4δ4(p3+p2−p1)
p22+m22,3 in the second line. The quantity in square brackets evaluates to p2
where the expression forp2is analogous to (2.37).
As an example, consider three distinct Hermitian scalar fieldsφi, i= 1· · ·3, with masses mi. If m1 > m2+m3 it is possible for particle 1 to decay into 2 + 3 as shown in Figure 2.6, provided there is an interaction term to drive the decay. The simplest appropriate
p2 p3
p1
2 3
1
−iκ
Figure 2.6