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Scattering Cross Section, the S-Matrix

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Scattering Cross Section, the S-Matrix PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Dr. Anosh Joseph, IISER Mohali LECTURE 06 Wednesday, January 15, 2020 Topics: Scattering Cross Section, The S-Matrix. We have found a formula for computing the n-point correlation function hΩjT [φ(x1)φ(x2) ··· φ(xn)] jΩi: Let us now focus on computing the quantities that can actually be measured: cross sections and decay rates. These quantities can be connected to a more primitive quantity: the S-matrix. Scattering Cross Section In order to study the behavior of elementary particles we need to perform scattering experiments. We can collide two beams of particles with well-defined momenta and then observe what comes out. The likelihood of any particular final state can be expressed in terms of the cross section, a quantity that is intrinsic to the colliding particles. Let us consider a target at rest. It is made out of particles of type A, with number density ρA (particles per unit volume). Now aim at this target a bunch of particles of type B, with number density ρB and velocity v. Let us take lA and lB as the lengths of the bunches of particles. Then we expect the total number of scattering events to be proportional to ρA; ρB; lA; lB, and the cross- sectional area A common to the two bunches. The cross-section, denoted by σ, is the total number of events divided by all these quantities Number of scattering events σ ≡ : (1) (ρAlA)(ρBlB) A We note that the above definition is symmetric between the A’s and B’s. Thus we could have taken the B’s to be at rest. We could also work in any other reference frame. The cross-section has units of area. It is the effective area of a chunk taken out of one beam, by each particle in the other beam, that subsequently becomes the final state we are interested in. PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 We note that in real life situations, ρA and ρB are not constant in the beams. The particle density is generally larger at the center of the beam than at the edges. We will assume that both the range of the interaction between the particles and the width of the individual particle wave packets are small compared to the beam diameter. We can then consider ρA and ρB to be constant in the following discussions. To compute the event rate in an actual accelerator we must integrate over the beam area. That is, Z 2 Number of events = σ (lAlB) d x ρA(x)ρB(x): (2) If the densities are constant, or if we use the above formula to compute an effective area A of the beams, then we have simply N N Number of events = σ A B ; (3) A where NA and NB are the total numbers of A and B particles. Cross sections for many different processes may be relevant to a single scattering experiment. For example, in e+e− collisions, one can measure the cross sections for the production of µ+µ−, τ +τ −, µ+µ−γ, µ+µ−γγ, etc., processes involving hadron production, and on top of all these the scattering of e+e−. We wish to measure not only what the final-state particles are, but also the momenta with which they come out. In this case, the definition of σ Number of scattering events σ ≡ : (4) (ρAlA)(ρBlB) A still works, but if we specify the exact momenta desired, σ will be infinitesimal. The solution is to define the differential cross section, dσ 3 3 : (d p1 ··· d pn) It is simply the quantity that, when integrated over any small 3 3 d p1 ··· d pn; gives the cross section for scattering into that region of final-state momentum space. The various final state momenta are not all independent. Four components will always be constrained by 4-momentum conservation (the delta function). In the simplest case, where there are only two final-state particles, this leaves only two unconstrained momentum components. They are usually taken to be the angles θ and φ of the momentum of one of the particles. Integrating dσ 3 3 (d p1 ··· d pn) over the four constrained momentum components then leaves us with the usual differential cross 2 / 5 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 section dσ : dΩ A somewhat simpler measurable quantity is the decay rate Γ of an unstable particle A (assumed to be at rest) into a specified final state (of two or more particles). It is defined as Number of decays per unit volume Γ ≡ (5) Number of A particles present The lifetime τ of a particle is then the reciprocal of the sum of its decay rates into all possible final states. The particle’s half-life is τ · ln 2: In non-relativistic quantum mechanics, an unstable atomic state shows up in scattering exper- iments as a resonance. Near the resonance energy E0, the scattering amplitude is given by the Breit-Wigner formula 1 f(E) / i : (6) E − E0 + 2 Γ The cross section therefore has a peak of the form 1 σ / : (7) 2 1 2 (E − E0) + 4 Γ The width of the resonance peak is equal to the decay rate of the unstable state. The Breit-Wigner formula Eq. (6) also applies in relativistic quantum mechanics. It gives the scattering amplitude for processes in which initial particles combine to form an unstable particle, which then decays. The unstable particle, viewed as an excited state of the vacuum, is a direct analogue of the unstable non-relativistic atomic state. If we call the 4-momentum of the unstable particle p and its mass m, we can make a relativis- tically invariant generalization of Eq. (6) 1 1 2 2 ≈ 0 (8) p − m + imΓ 2Ep [p − Ep + i(m=Ep)Γ=2] The decay rate of the unstable particle in a general frame is m Γ; Ep in accord with relativistic time dilation. Although the two expressions in Eq. (8) are equal in the vicinity of the resonance, the left-hand side, which is manifestly Lorentz invariant is much more convenient. 3 / 5 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 The S-Matrix We would like to compute cross section in quantum field theory. Let us invoke the concept of wavepackets to model this. We represent the initial states using wavepackets. Then evolve this initial state for a very long time with the time-evolution operator exp (−iHt) of the interacting field theory, and overlap the resulting final state with wavepackets representing some desired set of final-state particles. This gives the probability amplitude for producing that final state, which is simply related to the cross section. A wavepacket representing some desired state jφi can be expressed as Z d3k 1 p jφi = 3 φ(k)jki; (9) (2π) 2Ek where φ(k) is the Fourier transform of the spatial wavefunction, and jki is a one-particle state of momentum k in the interacting theory. In the free theory we would have p y jki = 2Ekakj0i: We have Z d3k hφjφi = 1; if jφ(k)j2 = 1: (10) (2π3) The probability we wish to compute is then 2 P = j hφ1φ2 ··· j φAφBi j ; (11) | {z } | {z } future past where jφAφBi is a state of two wavepackets constructed in the far past, and hφ1φ2 · · · j is a state of several wavepackets (one for each final-state particle) constructed in the near future. The wavepackets are localized in space, so each can be constructed independently of the others. States constructed this way are called in and out states. Note that we are using Heisenberg picture: States are time-independent. But the name we give a state depends on the eigenvalues or expectation values of time-dependent operators. Thus states with different names constructed at different times have a nontrivial overlap, which depends on the time dependence of the operators. We can write the initial state as Z 3 3 −ib·kB d kA d kB φA(kA)φB(kB)e jφAφBiin = 3 3 p jkAkBiin: (12) (2π) (2π) (2EA)(2EB) In the above b represents the impact factor. The factor e−ib·kB accounts for spatial translation. 4 / 5 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 In the convention we are using the reference momentum-space wavefunctions are collinear. That is b = 0. We could expand hφ1φ2 · · · j in terms of similarly defined out states of definite momentum formed in the asymptotic future. 0 1 Z 3 Y d pf φf (pf ) outhφ1φ2 · · · j = @ A outhp1p2 · · · j: (13) (2π)3 p2E f f The probability of scattering in a real experiment can now be related to outhp1p2 · · · jkAkBiin: (14) We note that the conventions for defining the two sets of states are related by time translation outhp1p2 · · · jkAkBiin = lim hp1p2 ··· j kAkBi T !1 | {z } | {z } T −T −iH(2T ) = lim hp1p2 · · · je jkAkBi: (15) T !1 In the last line, states are defined at any common reference frame. Thus the in and out states are related by the limit of a sequence of unitary operators. This limiting unitary operator is called the S-matrix. That is, outhp1p2 · · · jkAkBiin ≡ hp1p2 · · · jSjkAkBi: (16) The S-matrix has the following structure. If the particles in question do not interact at all, S is simply the identity operator. Even if the theory contains interactions, the particles have some probability of simply missing one another.
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