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Discrete Symmetries and CP ViolationFrom Experiment to Theory$

Marco Sozzi

Print publication date: 2007

Print ISBN-13: 9780199296668

Published to Oxford Scholarship Online: January 2008

DOI: 10.1093/acprof:oso/9780199296668.001.0001

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(p.495) APPENDIX C SCATTERINGIN QUANTUM THEORY

(p.495) APPENDIX C SCATTERINGIN QUANTUM THEORY

Source:
Discrete Symmetries and CP Violation
Publisher:
Oxford University Press

You don't understand anything until you learn it more than one way.

M. Minsky

Here is a brief review of the description of scattering in quantum field theory. The purpose is to collect some useful formulæ and derive some results in detail to help avoid the confusion which can sometimes arise due to the alternative descriptions adopted in the literature to deal with this vast subject; the reader is directed to standard textbooks such as Newton (1982) or Weinberg (1995) for details.

C.1 Scattering

Because of the impossibility of following the details of a microscopic interaction, all measurements in particle physics are performed by scattering experiments, so that the quantities of interest are the probability amplitudes for a given transition. In order to deal with interacting particles the Hamiltonian can be split in the form H = H 0 + H int, where H 0 describes the free motion and H int describes all the interactions among the particles, assumed to vanish in the remote past and future, when they are sufficiently far apart.

It is useful to work with states that have simple transformations under the Lorentz group: this is the case for the (rather uninteresting) single-particle states and for states consisting of several non-interacting particles, which are direct products of single-particle states. One therefore introduces the so-called ‘in’ and ‘out’ states, which are eigenstates of the full Hamiltonian

(C.1)
H | α ( in,out ) > = E α | α ( in,out ) >
(where α stands for the full specification of number and type of particles, momenta, spins, etc.). For these states the particle content is specified at time t = –∞ and t = +∞respectively: e.g. the state |α (in)〉 contains the set of particles specified by the quantum numbers α at t = –∞, and not necessarily at other times. Such (p.496) specification in the far past and future corresponds to free states at such time |α〉 = |i, f〉, since H int → 0 for t → ±∞:

(C.2)
H 0 | i > = E i | i > H 0 | f > = E f | f >
For a scattering experiment the particle content at those asymptotic times can be conveniently specified plane waves (actually wave packets of finite extension in time and space)164. The connection between the free states and the ‘in’ and ‘out’ states which is appropriate for wave packets (superpositions of energy eigenstates) is given by the conditions

e i H t | α ( in ) > t e i H 0 t | α > e i H t | α ( out ) > t + e i H 0 t | α >
so that

(C.3)
| α ( in ) > = lim t e i H t e i H 0 t | α > | α ( out ) > = lim t + e i H t e i H 0 t | α >

The amplitudes for finding in the remote future the system in the (free) final state |f〉, when it was prepared to be in the (free) initial state |i〉 in the remote past are the set of numbers of interest in scattering theory

(C.4)
S f i < f ( out ) | i ( in ) >
which define the S (scattering) matrix. In other words, the ‘in’ and ‘out’ states form two (complete) sets of basis states spanning the Hilbert space, and the coefficients S. describe the change of basis among them, or how a given ‘in’ state is expanded in terms of ‘out’ states:

(C.5)
| i ( in ) > = f S f i | f ( out ) >
In absence of interactions the ‘in’ and ‘out’ states coincide, and Sfi = δfi.

The S matrix is unitary, since165

γ S γ β * S γ α = γ < γ ( out ) | β ( in ) > * < γ ( out ) | α ( in ) > = γ < β ( in ) | γ ( out ) > < γ ( out ) | α ( in ) > = < β ( in ) | α ( in ) > = δ β α

(p.497) The analogous relation Σγ SγαS.*γβ = δβα (not equivalent to the previous one for infinite matrices) also holds. The unitarity of the S matrix is the expression of probability conservation: any given state |i (in)〉 whose content is specified in the remote past will evolve into some of the states |f (out)〉 whose content is specified in the remote future with unity probability:

(C.6)
f | S f i | 2 = f S f i * S f i = 1
and analogously any given state |f (out)〉 should be obtainable from some |i (in)〉 state. This also requires

(C.7)
f | S f i | 2 = f | S i f | 2
Ascattering operator S is also introduced, which is defined to have matrix elements between free states corresponding to the elements of the S matrix166:

(C.8)
< f | S | i > S f i

The unitarity of the S matrix implies that of the S operator:

δ β α = γ < γ | S | β > * < γ | S | α > = < β | S | γ > * < γ | S | α > = < β | S S | α >
due to the completeness of the free states, so that

(C.9)
S S = S S = 1
We recall that in the quantum-mechanical interaction picture of time evolution (see e.g. Sakurai (1985)), only the evolution of the state vectors due toH 0 is factored out, so that their time evolution only depends on H int, while the operators O evolve according to H 0:

i t | ψ ( t ) > ( I ) = H int ( I ) | ψ ( t ) > ( I ) d O ( I ) ( t ) d t = O ( I ) ( t ) t + i [ H 0 ( I ) , O ( I ) ( t ) ]

(p.498) (the superscript (I) denotes quantities in the interaction picture). The relations with Schrödinger and Heisenberg pictures are

| ψ ( t ) > ( I ) = e i H 0 t | ψ ( t ) > ( S ) = e i H 0 t e i H t | ψ > ( H ) O ( I ) ( t ) = e i H 0 t O ( S ) e i H 0 t = e i H 0 t e i H t O ( H ) ( t ) e i H t e i H 0 t
The operatorU(I)(t, t 0) describing time evolution from time t 0 to t in the interaction picture satisfies the equation

(C.10)
i t U ( I ) ( t , t 0 ) = H int ( I ) ( t ) U ( I ) ( t , t 0 )
with the condition U(I)(t 0, t 0) = 1, and its action on free states is

| i ( in ) > ( I ) = U ( I ) ( 0 , ) | i > ( I ) | i ( out ) > ( I ) = U ( I ) ( 0 , ) | f > ( I )
i.e. the time evolution operator U(I) transforms the free |i, f〉 (eigenstates of H 0) into |i (in)〉, | f (out)〉 (eigenstates of H) so that

S f i = lim t 2 lim t 1 ( I ) < f | U ( I ) ( t 2 , 0 ) U ( I ) ( 0 , t 1 ) | i > ( I )
and a formal expression for the S operator is obtained

(C.11)
S U (I) ( ∞, −∞)
which is not of much practical help, since knowing U(I) and the time evolution of the states amounts to having already solved the scattering problem.

The explicit form of the time evolution operator in the interaction picture is

(C.12)
U ( I ) ( t 2 , t 1 ) = e i H 0 t 2 e i H ( t 2 t 1 ) e i H 0 t 1
which is consistent with eqns (C.3).

Another expression for the scattering operator is obtained (omitting the superscripts in the following) from the Lippmann–Schwinger integral equation for the ‘in’ states (Sakurai, 1985)

(C.13)
| i ( in ) > = | i > + lim ε 0 1 E i H 0 + i ε H int | i ( in ) >
from which, using eqn (C.12) and (C.2)

(C.14)
S f i = lim t 2 < f | e i H 0 t 2 e i H t 2 | i ( in ) > = lim t 2 e i ( E f E i ) t 2 < f | i ( in ) > = δ f i + lim t 2 e i ( E f E i ) t 2 E f E a + i ε < f | H int | i ( in ) > = δ f i i ( 2 π ) δ ( E f E i ) T f i

(p.499) thus defining the transition matrix, with elements

(C.15)
T f i < f | H int | i ( in ) > = < f ( out ) | H int | i >
and a corresponding operator T , in analogy with (C.8), such that

(C.16)
< f | T | i > T f i
In a shorthand notation

(C.17)
S = 1 i ( 2 π ) δ ( E ) T
In a relativistic theory, energy and momentum are treated on the same footing and one writes instead167

S f i = δ f i ( 2 π ) 4 i δ 4 ( p f p i ) f i
whereMfi is a Lorentz-invariant amplitude.

The transition matrix elements give the differential transition rate (probability per unit time, normalized to the number of particles) as

d Γ ( i f ) = ( 2 π ) 4 δ 4 ( p f p i ) j = 1 , N i ( 1 2 E j ) V ( 1 N i ) | f i | 2 k = 1 , N f [ d p k ( 2 π ) 3 2 E k ]
where Ni and Nf are the number of particles in the initial and final states, V the normalization volume and the last factor is the Lorentz-Invariant Phase Space volume element for the final state particles, denoted as d(LIPS). For the decay of a particle of mass M (Ni = 1) the above expression gives the relativistic generalization of Fermi's golden rule

(C.18)
Γ ( i f ) = ( 2 π ) 4 2 M δ 4 ( p f p i ) | f i | 2 d ( LIPS )
The cross section (transition rate normalized to relative flux) for the collision of two particles of masses m 1,m 2 (Ni = 2) is

(C.19)
σ ( i f ) = ( 2 π ) 4 δ 4 ( p f p i ) 1 4 ( p 1 p 2 ) 2 m 1 m 2 | f i | 2 d ( LIPS )
The unitarity of the S matrix (C.9) gives the following relation for the transition matrix:

(C.20)
T i f * T f i = 2 π i δ ( E n E i ) T n f * T n i

(p.500) (valid for Ei = Ef), sometimes written formally as

(C.21)
T T = 2 π i δ ( E ) T T
The introduction of the T operator defined in eqn (C.16) trades the complication of the ‘in’ and ‘out’ states for that of having to deal with an operator which does not share the simple properties of the Hamiltonian, such as Hermiticity. Indeed the description of scattering can be done entirely in terms of free states, the ‘in’ and ‘out’ states being required only in the connection of the transition matrix elements with the Hamiltonian.

C.2 Approximations

If the interaction is weak, i.e. the matrix elements of H int are much smaller than those of H 0, then the ‘in’ states are not too different from free states, so that instead of the exact equation (C.15) a good approximation is

(C.22)
T f i < f | H int | i >
This is the Born approximation: in this case, by comparing with eqn (C.16) the operator T is seen to coincide with the interaction Hamiltonian Hint and is therefore Hermitian; this can also be seen from the unitarity relation (C.20) in the limit in which the right-hand side, of second order in T , can be neglected. In this approximation therefore the following relation holds for the transition amplitudes

(C.23)
T f i T i f * ( Born approximation )
Higher-order approximations are obtained considering that eqn (C.10) for the time evolution operator in the interaction picture can be written as an integral equation

(C.24)
U ( t , t 0 ) = 1 i t 0 t d t H int ( t ) U ( t , t 0 )
which when iterated gives

(C.25)
U ( t , t 0 ) = n = 0 ( i ) n t 0 t d t 1 t 0 t 1 d t 2 t 0 t n 1 d t n H int ( t 1 ) H int ( t 2 ) H int ( t n ) = n = 0 ( i ) n n t 0 t d t 1 t 0 t 1 d t 2 t 0 t d t n θ ( t , t 1 ) θ ( t 1 t 2 ) θ ( t n 1 t n ) H int ( t 1 ) H int ( t 2 ) H int ( t n )

(p.501) where the step function orders the interaction terms in time (note that H int (t) and H int (t′) need not commute if tt′). The above expression is often written as

(C.26)
U ( t , t 0 ) = n = 0 ( l ) n n ! t o t d t 1 t o t d t 2 t o t d t n T [ H int ( t 2 ) H int ( t n ) ]
by introducing the time-ordering operator168 T

T [ O ( t 1 ) O ( t n ) ] p θ ( t α 1 , t α 2 ) θ ( t α n 1 t α n ) O ( t 1 ) O ( t n )
where the sum is over all permutations of indexes αi = 1, 2, …, n; T reorders a product of time-labelled operators according to their times, so that the latest one occurs first:

T [ O ( t 1 ) O ( t n ) ] = O ( t i ) O ( t j ) O ( t k ) ( t i > t j > t k )
allowing one to write formally

(C.27)
U ( t , t 0 ) = Texp ( i t 0 t d t H int ( t ) )
The usefulness of the above series expansion is when Hint is small enough so that the first few terms suffice. While this is often not the case, (e.g. when dealing with weak transitions among particles which also interact through strong or electromagnetic interactions) the formalism can be adapted to deal with the case in which the interaction contains two terms, one strong and one weak:

(C.28)
H int = H S + H W
The above discussion can be repeated, introducing new states |α S(in,out)〉 defined as what the ‘in’ and ‘out’ states would be if HW were absent, and obtaining

(C.29)
T f i = < f S ( out ) | H W | i ( i n ) > + < f S ( out ) | H S | i S ( i n ) >
For a transition which cannot be driven by HS due to some selection rule (which makes the existence of HW relevant) the second term in eqn (C.29) vanishes and

(C.30)
T f i = < f S ( out ) | H W | i ( i n ) >
This is analogous to eqn (C.15), but now the state appearing on the left is not a free state but rather one in which the particles do interact according to HS , the strong but ‘uninteresting’ part of the interaction. This leads to a picture in which, although (p.502) the complete interaction H int is not weak, the first-order approximation in which the effect of HW on the states can be neglected in eqn (C.30) might still be valid:

(C.31)
T f i < f S ( out ) | H W | i S ( i n ) >
This is the distorted Born approximation, valid at first order in HW but to all orders in HS; it is the same as the previous case except that eigenstates of H 0 + HS are used instead of free states (eigenstates of H 0). The difference is due to the presence of interactions induced by HS in the initial and final states, among the particles participating in the transition induced by HW; note however that for single particle states (e.g. the initial state for a decay process) there are no interactions and |i S(in)〉 = |i〉.

Introducing operators TS , TW for the strong and weak parts of the interaction:

(C.32)
T = T S + T W
the scattering operator can be written as

(C.33)
S = S S i ( 2 π ) δ ( E ) T W

(C.34)
Where

(C.35)
S S = 1 l ( 2 π ) δ ( E ) T S
The unitarity condition (C.9) can now be written as

S S S S + i ( T W S S S S T W ) + ( 2 π ) δ ( E ) T W T W = 1
which at zeroth order in TW gives S S S S = 1 and at first order

(C.36)
T W = S S T W S S
instead of the simple Hermiticity of TW in the Born approximation.

In the case of strongly interacting particles the first-order approximation in the weak interaction which induces a transition is not simply equivalent to taking the (Hermitian) weak Hamiltonian as the transition matrix.

Notes:

(164) The above is also valid in the Heisenberg picture of motion, in which the states do not depend explicitly on time but contain the whole ‘history’ of time evolution. The ‘in’ states are those containing incoming plane waves and outgoing spherical waves, while the ‘out’ states contain outgoing plane waves and incoming spherical waves.

(165) This proof requires the completeness of the ‘in’ and ‘out’ states: in the case at hand in which all interactions vanish at times t → ±∞these can be the scattering states, specified at those times as containing free particles, since no bound states can exist; in the general case in which bound states are possible they should also be included in the complete set.

(166) Note that this is not the operator Ŝ which is sometimes defined to map an ‘in’ scattering state with a given plane-wave particle content at t = –∞into the ‘out’ state with the same particle content at t = +∞: |α (in)〉 = ‘Ŝ|α (out)〉; such an operator is guaranteed to exist (and be unitary) byWigner's theorem (Chapter 1) if the set of scattering states is complete, i.e. in absence of bound states.

(167) Care should be taken as several alternative definitions of the transition matrix can be found in the literature, differing by 2π factors, the inclusion of delta functions or energy factors (see e.g. Donoghue et al. (1992) for a discussion); the one in eqn (C.15) is not Lorentz-invariant and keeps a close connection to the non-relativistic theory. Of course the final expressions for physical quantities such as cross sections or decay rates are the same in every notation.

(168) Not to be confused with the time reversal operator T!