Scattering and Decay of Particles

All human things are subject to decay, and when fate summons, monarchs must obey. John Dryden (1631-1700)

9 Scattering and Decay of Particles

So far we have discussed only free particles. If we want to detect the presence and detailed properties of any of them, it is necessary to perform scattering experiments or to observe their decay products. In this chapter, we shall develop an appropriate quantum mechanical description of such processes.

9.1 Quantum-Mechanical Description

We begin by developing the appropriate quantum-mechanical tools for describing the scattering process.

9.1.1 Schr¨odinger Picture Consider a quantum mechanical system whose Schr¨odinger equation

Hˆ ψ (t) = i∂ ψ (t) (9.1) | S i t| S i cannot be solved analytically (here we use natural units withh ¯=1). The standard method of deriving information on such a system comes from perturbation theory: The Hamilton operator is separated into a time-independent part Hˆ0, whose Schr¨odinger equation

Hˆ ψ (t) = i∂ ψ (t) (9.2) 0| S i t| S i is solvable, plus a remainder Vˆ , to be called interaction. The operator Hˆ0 is also called the unperturbed Hamiltonian, and the interaction Vˆ is often called the perturbation. For the sake of a simple notation we have omitted the hats on top of all operators without danger of confusion. The time-dependent state ψS(t) carries a subscript S to indicate the fact that we are dealing with the Schr¨odinger| i picture. Observables are operators OˆS (p, x, t) whose matrix elements between states yield transition amplitudes. The interaction will be assumed to be of ﬁnite range. This excludes, for the moment, the most important interaction potential, the Coulomb

660 9.1 Quantum-Mechanical Description 661 potential. As an example, we may assume the unperturbed Hamiltonian to describe a particle in a Rosen-Morse potential well: const x Vˆ x = V (x)= . (9.3) h | | i cosh2 x | | If the constant is sufficiently negative, the system has one or more discrete bound states. The important Coulomb potential does not fall directly into the class of potentials under consideration. It must first be modified by multiplying it with a very small exponential screening factor. The consequences of an infinite range that would be present in the absence of screening will have to be discussed separately. The interaction Vˆ may in general depend explicitly on time. The results will later be applied to quantum field theory. Then Hˆ0 will be a sum of the second-quantized Hamilton operator of all particles involved, and Vˆ will be some as yet unspecified short-range interaction between them. If the particles are all massive, all forces are of short range. Moreover, the vacuum state is a discrete state which is well separated from all other states by an energy gap. The lowest excited state contains a single particle with the smallest mass at rest. This mass is the energy gap.

9.1.2 Heisenberg Picture The other, equivalent, description of the system is due to Heisenberg. It is based on time-independent states which are equal to the Schr¨odinger states ψ (t) at a | S i certain ﬁxed time t = t0:

ψ ψ (0) = Uˆ(t, t )−1 ψ (t) . (9.4) | H i≡| S i 0 | S i

For simplicity, we choose t0 = 0. When using these states, the time dependence of the system is carried by time-dependent Heisenberg operators

−1 OˆH (t)= Uˆ(t, 0) OˆSUˆ(t, 0). (9.5)

Here Uˆ(t, 0) is the unitary time evolution operator introduced in Section 1.6 which governs the motion of the Schr¨odinger state. It has the explicit form [recall (1.269)]

t −i dt′ Hˆ (t′) Uˆ(t, 0) = Tˆ e 0 . (9.6) R If Hˆ has no explicit time dependence, which we shall assume from now on, then Uˆ(t, 0) has the explicit form Uˆ(t, 0) = e−iHt/ˆ ¯h and satisﬁes the unitarity relations

Uˆ −1(t, 0) = Uˆ †(t, 0) = Uˆ(0, t). (9.7)

Then the relation (9.5) becomes

iHt/ˆ ¯h −iHt/ˆ ¯h OˆH (t)= e OˆSe . (9.8) 662 9 Scattering and Decay of Particles

9.1.3 Interaction Picture As far as perturbation theory is concerned it is useful to introduce yet a third description called the Dirac or interaction picture. Its states are related to the previous ones by

ˆ ˆ ψ (t) = eiH0t ψ (t) = eiH0tUˆ(t, 0) ψ . (9.9) | I i | S i | H i In the absence of interactions, they become time independent and coincide with the Heisenberg states (9.4). The interactions, however, drive ψI (t) away from ψH (t). When using such states, the observables in the interaction picture are

iHˆ0t −iHˆ0t iHˆ0t −iHtˆ iHtˆ −iHˆ0t OˆI (t)= e OˆSe = e e OˆH e e . (9.10)

These operators coincide, of course, with those of the Heisenberg picture in the absence of interactions. By definition, the unperturbed Schrödinger equation Hˆ ψ(t) 0| i can be solved explicitly implying that the time dependence of the operators OÎ (t), d Oˆ (t) = [Hˆ , Oˆ (t)], (9.11) dt I 0 I is completely known. If the Schrödinger operator OˆS is explicitly time-dependent, this becomes d ˙ Oˆ (t) = [Hˆ , Oˆ (t)] + [Oˆ (t)] , (9.12) dt I 0 I S I where

˙ ˆ ˙ ˆ [Oˆ (t)] eiH0tOˆ (t)e−iH0t. (9.13) S I ≡ S 9.1.4 Neumann-Liouville Expansion A state vector in the interaction picture moves according to the following equation of motion:

ˆ ˆ Vˆ (t) eiH0tVˆ e−iH0t (9.15) I ≡ S is the interaction picture of the potential VˆS. It is useful to introduce a unitary time evolution operator also for the interaction picture. It determines the evolution of the states by

ψ (t) = Uˆ (t, t ) ψ (t ) . (9.16) | I i I 0 | I 0 i 9.1 Quantum-Mechanical Description 663

Obviously, UÎ (t, t0) satisfies the same composition law as the previously introduced time translation operator Uˆ(t, t0) in the Schrödinger picture:

′ ′ UÎ (t, t0)= UÎ (t, t )UÎ (t , t0). (9.17) Using the differential equation (9.14) for the states we see that the operator UÎ (t, t0) satisfies the equation of motion

i∂tUÎ (t, t0)= VÎ (t)UÎ (t, t0). (9.18) The equation of motion (9.18) can be integrated, resulting in the Neumann- Liouville expansion [recall (1.200), (1.201)]:

t ( i)2 t UÎ (t, t0) = 1 i dt1VÎ (t1)+ − Tˆ dt1dt2VÎ (t1)VÎ (t2)+ ... − Zt0 2! Zt0 t ′ ′ Tˆ exp i dt VÎ (t ) . (9.19) ≡ − Zt0

The expansion holds for t > t0 and respects the initial condition UÎ (t0, t0) = 1. Note that the differential equation (9.18) for UÎ (t, t0) implies that UÎ (t, t0) satisfies an integral equation:

t ′ ′ ′ UÎ (t, t0)=1 i dt VÎ(t )UÎ (t , t0). (9.20) − Zt0 ′ Indeed, we may solve this equation by iteration, starting with UÎ (t , t0) = 1 which yields the lowest approximation:

t UˆI (t, t0) = 1 i dt1VˆI(t1). (9.21) − Zt0 Reinserting this into (9.2), we obtain the second approximation

t ( i)2 t UÎ (t, t0) = 1 i dt1VÎ (t1)+ − Tˆ dt1dt2VÎ (t1)VÎ(t2). (9.22) − Zt0 2! Zt0 Continuing this iteration we recover the full Neumann-Liouville expansion (9.19). For t < t0, we may use the relation (9.7) according to which Uˆ(t, t0) is simply the −1 inverse Uˆ (t0, t), and an expansion is again applicable. The time-ordering operator T makes sure that the operators VÎ(ti) in the expansion appear in chronological order, with all later VÎ (ti) standing to the left of earlier ones. If Vˆ and therefore Hˆ has no explicit time dependence, the Schrödinger state is governed by the simple exponential time evolution operator:

ˆ ψ (t) = Uˆ(t, t ) ψ (t ) = e−iH(t−t0) ψ (t ) . (9.23) | S i 0 | S 0 i | S 0 i Combining this with (9.9), we see that

iHˆ0t −iHˆ (t−t0) −iHˆ0t0 UˆI (t, t0)= e e e . (9.24) 664 9 Scattering and Decay of Particles

If the potential Vˆ depends explicitly on time, this has to be replaced by [recall (9.6)]

t ˆ −i dt′ Hˆ (t′) ˆ iH0t t −iH0t0 UˆI (t, t0) = e Tˆ e 0 e R iHˆ0t −iHˆ0t0 = e Uˆ(t, t0)e . (9.25)

For a time-independent Hˆ , this relation shows that UˆI (t, t0) satisﬁes the same unitarity relations (9.7) as Uˆ(t, t0): ˆ −1 ˆ † ˆ UI (t, 0) = UI (t, 0) = UI (0, t). (9.26) The Heisenberg representation (9.5) of an arbitrary operator is obtained from the interaction representation (9.10) via

ˆ ˆ ˆ ˆ −1 −iH0t ˆ iH0t ˆ OHˆ (t) = U (t, 0)e OI(t)e U(t, 0) ˆ −1 ˆ ˆ = UI (t, 0)OI(t)UI (t, 0). (9.27)

We have observed above that the operator in the interaction picture OˆI(t) has a particularly simple time dependence. Its movement is determined by the unperturbed equation of motion (9.11). Thus (9.27) establishes a relation between the complicated time dependence of the Heisenberg operator OˆH (t) in the presence of interaction and the simple time dependence of OˆI(t) which reduces to the Heisenberg operator in the absence of an interaction.

9.1.5 Møller Operators Consider now a scattering process in which two initial particles approach each other from a large distance outside the range of their interactions. This implies that at a very early time t , their states obey the unperturbed Schrödinger equation, → −∞ i.e., that their wave function ψI (t) is time-independent. The same thing will be true a very long time after the scattering has taken place. Let us study the behavior of the operator UÎ (t, t0) in the two limits of large positive and negative time arguments. In order to make all expressions well defined mathematically it is convenient to introduce a simple modification of the potential by multiplying it with a switching factor e−η|t|:

Vˆ e−η|t|Vˆ Vˆ η(t), Vˆ (t) e−η|t|Vˆ (t) Vˆ η(t), (9.28) → ≡ I → I ≡ I with an inﬁnitesimal parameter η. As far as physical observations are concerned, such a factor must have little relevance since η can be chosen so that Vˆ remains unchanged over any ﬁnite interval of time. However, an immediate consequence of the prescription (9.28) is that for t , the state vector in the interaction → ±∞ picture becomes time-independent, due to (9.14): Let us denote the limiting states by ψ in , i.e., | out i

ψI (t) ψ in . (9.29) | it→∓∞ → | out i 9.1 Quantum-Mechanical Description 665

Using the time evolution operator, the limit (9.29) may be written as

UˆI (t, 0) ψI(0) ψ in , (9.30) | it→∓∞ → | out i i.e., the operator Uˆ becomes a constant in the limit t . Inverting this relation I → ∓∞ we may write

(±) ψI (0) = UˆI (0, ) ψ in Ωˆ ψ in . (9.31) | i ∓∞ | out i ≡ | out i The operators

Ωˆ (±) Uˆ (0, ) (9.32) ≡ I ∓∞ were ﬁrst studied extensively by Møller1 and are named after him. Their most important property is the following:

(±) (±) (±)† (±)† Hˆ Ωˆ = Ωˆ Hˆ0, Ωˆ Hˆ = Hˆ0Ωˆ . (9.33)

To derive this, we consider, for simplicity, only the most common situation that Vˆ has no explicit time dependence except for the very slow switching factor (9.28). Then we multiply the explicit representation (9.24) of the time-evolution operator by a factor e−itHˆ and ﬁnd

−itHˆ −itHˆ itaHˆ −itaHˆ0 −i(t−ta)Hˆ −i(ta−t)Hˆ0 −itHˆ0 −itHˆ0 e Uˆ(0, ta)= e e e = e e e = Uˆ(0, ta t)e . − (9.34) In the limit t , this yields a → ±∞ ˆ ˆ e−itH Uˆ(0, )= Uˆ(0, )e−itH0 . (9.35) ∓∞ ∓∞ The time derivative of this at t = 0 yields precisely the ﬁrst equation in (9.33). The second follows by Hermitian conjugation. Let ψ in be a steady-state solution of the time-independent unperturbed | out i Schr¨odinger equation:

Hˆ0 ψ in = E ψ in . (9.36) | out i | out i

Then due to (9.33) [or directly (9.35)], the interacting state ψI (0) solves the full Schr¨odinger equation with the same energy: | i

Hˆ ψ (0) = E ψ (0) . (9.37) | I i | I i In the laboratory, one does not really observe steady states but wave packets, which can be obtained from superpositions of steady-state solutions with diﬀerent momenta. The packets of free particles approach each other and enter into the range

1C. Møller, Kgl. Danske Vidensk. Selsk. Mat.Fys. Medd 23, No. 1, (1945). 666 9 Scattering and Decay of Particles of the interaction potential. Also there, the total energy remains the same, due to energy conservation. The amplitude for the scattering process is found as follows. Let ain be a complete set of incoming eigenstates of the unperturbed Schr¨odinger equation| i

Hˆ a = E a , (9.38) 0| ini a| ini where a denotes the collection of all quantum numbers of these states. By applying the interaction operator Uˆ (t, ) to the states a , these are transformed into I −∞ | ini the time-dependent aI (t) . After a long time, these develop into time-independent states a . The energies| iE remain, of course, unchanged. | outi a Let us analyze the outgoing states

S = b Uˆ ( , ) a . (9.41) ba h in| I ∞ −∞ | ini The same name is also often used sloppily for the scattering operator

Sˆ Uˆ ( , ) (9.42) ≡ I ∞ −∞ itself, which can be written as a product of the two Møller operators (9.32):

S = Ωˆ (−)†Ωˆ (+). (9.43)

Since UˆI (t, t0) is a unitary operator, the S-matrix is also unitary and satisﬁes

SS† = S†S =1. (9.44)

The Møller operators, however, are not unitary. As we shall see in the next section, (±)† (±) the operators Ωˆ are inverse to Ωˆ if Hˆ0 has no bound states. Under multiplication of the states from the left-hand side one has

Ωˆ (±)†Ωˆ (±) =1. (9.45)

In contrast, under multiplication from the right-hand side, the product yields a projection operator onto the subspace of continuous states of the Hamiltonian Hˆ :

(±) (±)† Ωˆ Ωˆ = Pˆcontinuum. (9.46)

Note that only in an inﬁnite-dimensional Hilbert space can it happen that the left- inverse is not equal to the right-inverse. 9.1 Quantum-Mechanical Description 667

9.1.6 Lippmann-Schwinger Equation The solution (9.31) of Eq. (9.37) can usually not be given analytically but only in form of a perturbation series. This is found by inserting the Neumann-Liouville expansion (9.19) into Eq. (9.31):

0 0 t ′ ′ 2 ′ ′′ ′ ′′ ψI (0) = ψin i dt VÎ (t ) ψin +( i) dt dt VÎ (t )VÎ (t ) ψin + .... | i | i− Z−∞ | i − Z−∞ Z−∞ | i (9.47)

With the help of (9.15), and the damping factor as in Eq. (9.28), we see that the second term is equal to

0 ηt iHˆ0t −iHˆ0t i dte e Vˆ e ψin . (9.48) − Z−∞ | i

By assumption, ψin is an eigenstate of Hˆ0 with energy E, and Vˆ is independent of time. Thus we can| performi the integration and obtain

In the second denominator we have replaced 2iη by iη since η is infinitesimally small and its actual size is irrelevant. The infinite sum is recognized as an iterative solution of the so-called Lippmann-Schwinger equation2 1 ψI (0) = ψin + Vˆ ψI (0) . (9.51) | i | i E Hˆ + iη | i − 0 The fact that this state solves the Schrödinger equation (9.37) is easily verified by multiplying both sides by E Hˆ , to obtain − 0 (E Hˆ ) ψ (0) =(E Hˆ ) ψ + Vˆ ψ (0) . (9.52) − 0 | I i − 0 | ini | I i The first term on the right-hand side vanishes, due to (9.36). The remaining equation coincides with Eq. (9.37). Comparing (9.51) with (9.31), we identify the Møller operator as

(+) 1 Ωˆ = UˆI (0, )=1+ V.ˆ (9.53) −∞ E Hˆ + iη − 0 2For a discussion see M. Gell-Mann and M.L. Goldberger, Phys. Rev. 91, 398 (1953). 668 9 Scattering and Decay of Particles

Making use of this relation, one often writes the scattering state ψI (0) in (9.51) as ψ(+) and the equation itself as | i | i

(−) 1 (−) ψ = ψout + Vˆ ψ , (9.55) | i | i E Hˆ iη | i − 0 − where the Møller operator is

(−) 1 Ωˆ = UˆI (0, )=1+ V.ˆ (9.56) ∞ E Hˆ iη − 0 − The only diﬀerence with respect to (9.53) is the sign of the iη-term. The operators

ih¯ ih¯ Gˆ0(E) , Gˆ(E) (9.57) ≡ E Hˆ + iη ≡ E Hˆ + iη − 0 − are the resolvents of the free and interacting Hamiltonians, respectively [recall (11.8)]. The second is related to the ﬁrst by the equation

i Gˆ(E)= Gˆ (E) Gˆ (E)Vˆ Gˆ(E). (9.58) 0 − h¯ 0

This can easily be veriﬁed by multiplying (9.58) with [Gˆ(E)]−1 =(E Hˆ )/ih¯ from − the right:

i −1 i i 1= Gˆ0(E) Gˆ0(E)Vˆ Gˆ(E) [Gˆ(E)] = Gˆ0(E)(E Hˆ ) Gˆ0(E)V.ˆ (9.59) − h¯ −h¯ − − h¯

The equation (9.58) for Gˆ(E) can be solved iteratively by the geometric series

i i 2 Gˆ(E)= Gˆ0(E) Gˆ0(E)Vˆ Gˆ0(E)+ Gˆ0(E)Vˆ Gˆ0(E)Vˆ Gˆ0(E)+ .... (9.60) − h¯ h¯ The Møller operators (9.53) and (9.56) can be expressed with the help of the free resolvent as i i Ωˆ (+) =1 Gˆ (E)V,ˆ Ωˆ (−) =1 Gˆ (E∗)V.ˆ (9.61) − h¯ 0 − h¯ 0 9.1 Quantum-Mechanical Description 669

9.1.7 Discrete States The Møller operators have interesting nontrivial properties with respect to the discrete spectrum of the unperturbed Hamiltonian Hˆ0 and the perturbed Hamiltonian Hˆ . Let us ﬁrst understand the diﬀerence between the quasi-unitarity relations (9.45) and (9.46). We denote by ϕ the continuum, and by ϕ the bound states of Hˆ , | ai | βi 0 similarly by ψ(±) the continuum and by ψ the bound states of Hˆ which solve the | a i | βi Lippmann-Schwinger equations (9.54) or (9.55), respectively. These states satisfy the completeness relations

(±) (±) ϕa ϕa + ϕβ ϕβ =1, ψa ψa + ψβ ψβ =1. (9.62) a | ih | | ih | a | ih | | ih | X Xβ X Xβ Only the continuum states carry the distinctive label ( ), since only these depend on the sign of the small quantity η in the Lippmann-Schwinger± equations (9.54) and (9.55). The bound states are uniquely determined by the condition of quadratic integrability. The Møller operators Ωˆ (±) transform the continuum states ϕ into scattering | ai states ψ(±) . Thus we can expand | i ˆ (±) ˆ (±) ˆ (±)† ˆ † (±) Ω = UI (0, )= ψa ϕa , Ω = UI (0, )= ϕa ψa . (9.63) ∓∞ a | ih | ∓∞ a | ih | X X From these expansions, we extract the following important properties of the Møller operators:

Ωˆ (±)† ψ(±) = ϕ , Ωˆ (±)† ψ =0. (9.64) | a i | ai | βi The second relation implies that

(±)† Ωˆ Pbd =0. (9.65) We further ﬁnd

The operator on the right-hand side is the projection on the bound states of Hˆ0. In the opposite order, the product yields

Ωˆ (±)Ωˆ (±)† = ψ(±) ϕ ϕ ψ(±) =1 ψ ψ =1 Pˆ , (9.67) | a ih a| bih b | − | βih β| − bd Xa,b Xβ where Pˆbd projects onto the bound states of Hˆ . The different orders of the products differ by the projection onto the bound states of Hˆ . For the S-matrix, on the other hand, this dependence on the order disappears and we find

S†S = [Ωˆ (−)†Ωˆ (+)]†[Ωˆ (−)†Ωˆ (+)]= Ωˆ (+)†Ωˆ (−)Ωˆ (−)†Ωˆ (+) = Ωˆ (+)† 1 Pˆ Ωˆ (+) − bd = 1 P 0 , (9.68) − bd 670 9 Scattering and Decay of Particles and

SS† = Ωˆ (−)†Ωˆ (+)Ωˆ (+)†Ωˆ (−) = Ωˆ (−)† 1 Pˆ Ωˆ (−) = Ωˆ (−)†Ωˆ (−) =1. (9.69) − bd The projection operator Pˆbd has dropped out because of (9.65). By deﬁnition, the S-matrix is calculated only between states in the subspace of the Hilbert space formed by the continuous eigenstates of Hˆ0. In this subspace, the 0 † † projection operator Pbd vanishes and we ﬁnd S S = 1 and SS = 1, in either order, showing that the S-matrix is unitary. From the property (9.33) of the Møller operator we derive immediately that S does not change the energy of the incoming continuum states since [Hˆ0,S] = 0:

(−)† (+) (−)† (+) (−)† (+) Hˆ0S = Hˆ0Ωˆ Ωˆ = Ωˆ Hˆ Ωˆ = Ωˆ Ωˆ Hˆ0 = SHˆ0. (9.70)

For scattering states which all lie in the continuum of Hˆ0, the size of η is irrelevant as long as it is very small and has a fixed sign. In contrast, a discrete eigenstate of Hˆ0 has a singular η-dependence in the limit of small η. In order to understand this we observe that, for any short-range potential, the two scattering particles lie practically all the time outside of each other’s range. Only for a small time interval do they interact. The potential has no effect for large positive and negative times where the wave packets are separated by a large distance. The switching factor is therefore physically irrelevant for sufficiently small η. It cannot influence the physics of the system. For bound states, however, the particles interact with each other all the time with the same strength. In order to understand this consider a specific discrete state, assuming it to be well separated from the lower and higher states by a finite energy gap ∆. Then, for η ∆, the time dependence of Vˆ η(t)= ≪ e−η|t|Vˆ is so slow that the interaction is, over a long time interval, incapable of causing transitions from the discrete state to a neighbor state. Such a slow time dependence is called adiabatic. For an adiabatically time-dependent potential Vˆ η(t), the discrete state ψ(t) is at any time t0 a very good approximation to the solution of the time-independent| i Schrödinger equation that contains the time-independent potential Vˆ (t0). The energy E(t) of the discrete state ψ(t) is at any finite time equal to E. Only for very large times t > 1/η does| thei interaction become so weak that the state ψ(t) becomes an eigenstate| | ± of Hˆ , and the energy E(t) goes | i 0 against the associated eigenvalue E0 of the unperturbed Hamilton operator Hˆ0.

9.1.8 Gell-Mann--Low Formulas The Møller operators relate the wave functions ψ of the interacting system, that satisﬁes the Schr¨odinger equation | i

Formally, this correspondence can be specified in a unique way by multiplying Vˆ by a coupling constant g, and by continuing g smoothly from g = 0, where the system is free, to the value g = 1, where it is fully interacting. The energy E(g) and the associated discrete state ψ(g) are continuous functions of g interpolating between | i E0 and E, and between ψ0 and ψ , respectively. | i ˆ (±|) i ˆ η The two Møller operators Ω of Eq. (9.32) associated with the potential VI carry the nondegenerate discrete state ψ0 into two different solutions of the interacting Schrödinger equation (9.71) as| followsi (first Gell-Mann --Low formula):3

(±) η 1 ψ = lim Uˆ (0, ) ψ0 . (9.73) η→0 I (±) | i ∓∞ | izη To have a ﬁnite limit, the transformed state has been divided by a normalization (±) factor zη deﬁned by the expectation value

z(±) ψ Uˆ η (0, ) ψ . (9.74) η ≡h 0| I ∓∞ | 0i The phase determines the energy shift via the second Gell-Mann --Low formula:

∂ η ∆E = E E0 = lim g ϕ∓. (9.75) − ∓ η→0 ∂g In order to prove the statements (9.73) and (9.75), let us denote the states on the right-hand side of (9.73) without the normalization factors by

η(±) Uˆ η (0, ) ψ . (9.76) | i ≡ I ∓∞ | 0i For small but ﬁxed η, these states satisfy the Schr¨odinger equation

Hˆ + Vˆ η(0) η(±) = E η(±) , (9.77) 0 | i η| i h i where Eη is some η-dependent energy. The proof is based on the observation that in front of the state ψ , the η-dependent Møller operators | 0i Ωˆ (±) Uˆ η(0, ) (9.78) η ≡ I ∓∞ satisfy the obvious commutation rule

Hˆ , Uˆ η (0, ) ψ = Hˆ η(±) E η(±) . (9.79) 0 I ∓∞ | 0i 0| i − 0| i h i We now expand Uˆ η (0, ) according to (9.19) as I ∓∞ 0 −ig dt eηtVˆ (t) ˆ η ˆ −∞ I UI (0, ) = T e −∞ R ∞ n 0 0 n ( ig) η ti = − dt1 dtne i=1 Tˆ VˆI (t1) VˆI(tn) . (9.80) n! −∞ ··· −∞ ··· nX=0 Z Z P 3M. Gell-Mann and F. Low, Phys. Rev. 84 , 350 (1951). 672 9 Scattering and Decay of Particles

The operators VˆI (t) in the interaction picture have the time dependence (9.11), which yields d Vˆ (t)= i Hˆ , Vˆ (t) . (9.81) dt I 0 I h i Using (9.80) and (9.81) we ﬁnd

∞ n 0 0 η ( ig) n ˆ ˆ η i ti H0, UI (0, ) = i − dt1 dtne =1 −∞ − n! −∞ ··· −∞ h i nX=0 Z Z P n ∂ Tˆ VÎ (t1) VÎ (ti) VÎ (tn) . (9.82) × ∂ti ··· ··· Xi=1 Since the integrand is symmetric in the time variables, we may replace the sum n i=1 ∂/∂ti by n-times the time derivative ∂/∂tn. Taking the derivative ∂/∂tn to the ηti leftP of the switching factors e , the integral becomes

∞ n 0 0 0 n ( ig) ∂ η ti i − n dt1 dtn−1 dtn e i=1 Tˆ VÎ(t1) VÎ (tn) − n! −∞ ··· −∞ −∞ × ∂tn ··· nX=0 Z Z Z h P i ∞ n 0 0 n ( ig) η ti I +iη − n dt1 dtne i=1 Tˆ VÎ (t1) Vˆ (tn) . (9.83) n! −∞ ··· −∞ ··· nX=0 Z Z P The second sum is equal to ∂ iηg Uˆ η (0, ) . (9.84) ∂g I −∞ The first sum contains integrals over pure time derivatives which amount to pure surface terms. Moreover, since the integrand vanishes at t = , only the t =0 n −∞ n -term contributes. The time tn = 0 is the latest of all time variables. Thus it can be moved to the left of the time-ordering symbol, and we arrive at the equation ∂ Hˆ , Uˆ η (0, ) = gVˆ (0)Uˆ η (0, )+ iηg Uˆ η (0, ) . (9.85) 0 I −∞ − I I −∞ ∂g I −∞ h i In a similar way we derive ∂ Hˆ , Uˆ η (0, + ) = gVˆ (0)Uˆ η (0, + ) iηg Uˆ η (0, + ) . (9.86) 0 I ∞ − I I ∞ − ∂g I ∞ h i An important property of these formulas is that, in the presence of a level shift, the second term on the right-hand side proportional to η is nonzero in the limit η 0. To see this, take Eq. (9.85), and rewrite it in a slightly different way as →

Hˆ Uˆ η (0, ) = Hˆ + gVˆ (0) Uˆ η (0, ) I ∓∞ 0 I I ∓∞ h i ∂ = Uˆ η (0, ) Hˆ iηg Uˆ η (0, ) . (9.87) I ∓∞ 0 ± ∂g I ∓∞ 9.1 Quantum-Mechanical Description 673

(±) If this equation is applied to the state ψ0 , it shows that the state η of (9.76) satisfies the modified Schrödinger equation| i | i

(±) ∂ (±) Hˆ η = E0 iηg η . (9.88) | i ± ∂g ! | i If we had taken the limit η 0 carelessly, we would have arrived at the equation Hˆ η(±) = E η(±) , and would→ have concluded erroneously that η are solutions | i 0| i | i∓ of the fully interacting Schr¨odinger equation with energy E0. For the continuum states, this conclusion is indeed correct, and we have

Hˆ Uˆ (0, )= Uˆ (0, ) Hˆ (9.89) I ∓∞ I ∓∞ 0 rather than (9.87). This agrees with Eq. (9.33). For the discrete states, the energy is in general shifted by the interaction. Hence the derivative g(∂/∂g) η(±) must diverge like 1/η, so that it gives a ﬁnite contribution in (9.88). The origin| i of the divergence is a diverging normalization of the state η(±) , and the divergence is | i eliminated by making use of the states on the right-hand side of (9.73):

(±) (±) 1 ψη η (±) . (9.90) | i≡| izη These have a definite limit for η 0. For these states, Eq. (9.88) implies → ˆ (±) ∂ (±) (±) H ψη = E0 iηg log zη ψη . (9.91) | i ± ∂g ! | i The level shifts caused by the interaction are given by the formula ∂ ∆E = lim iηg log z(±). (9.92) η→0 ∂g η Note that, in the limit η 0, the two states ψ(±) coincide. This is seen as → | η i follows. The state η(+) evolves after an infinitely long time into Uˆ η ( , ) ψ . | i I ∞ −∞ | 0i There it becomes again an eigenstate of Hˆ0 due to the commutation relation (9.70). (+) Thus it must coincide with ψ up to some overall factor eiϕη (recall that ψ is | 0i | 0i nondegenerate by assumption). Similarly, we may evolve the state ψ0 backwards (−) ˆ η | i in time via η into UI ( , ) ψ0 , i.e., into the initial state rather than ψ0 . | i ∞ η−∞ | i | i Note that η η eiϕ . When forming ψ from η , the pure phase | i− ≡ | i+ | ηi∓ | i± between ψ and Uˆ η ( , ) ψ cancels. Thus the two states ψ must in- | 0i I ∞ −∞ | 0i | ηi∓ deed be identical. The phase ϕη associated with the total time evolution operator Uˆ η ( , ), defined by I ∞ −∞ η eiϕ ψ Uˆ η ( , ) ψ , (9.93) ≡h 0| I ∞ −∞ | 0i η is obviously related to the previously introduced phases ϕ∓ by the relations ϕη = ϕη ϕη =2ϕη = 2ϕη . (9.94) − − + − − + 674 9 Scattering and Decay of Particles

This equation can also be written in terms of matrix elements as

ψ Uˆ η ( , ) ψ = ψ Uˆ η ( , 0) ψ ψ Uˆ η (0, ) ψ , (9.95) h 0| I ∞ −∞ | 0i h 0| I ∞ | 0ih 0| I −∞ | 0i which may come somewhat as a surprise. In general, there is certainly an identity containing a sum over a complete set of orthonormal eigenstates in the middle: ˆ η ˆ η ˆ η ψ0 UI ( , ) ψ0 = ψ0 UI ( , 0) ψn ψn UI (0, ) ψ0 . (9.96) h | ∞ −∞ | i n h | ∞ | ih | −∞ | i X It is the adiabatic slowness of the switching-on process of the interaction which does not permit a transition to an excited intermediate state, making (9.95) valid. The level shift can also be expressed in terms of the η-dependent S-matrix:

Sη = Uˆ η ( , )= Uˆ η ( , 0) Uˆ η (0, ) . (9.97) I ∞ −∞ I ∞ I −∞ Indeed, by multiplying Eq. (9.88) for η(+) by ψ(+) from the left, we find | i h η | ψ(±) Hˆ E η(±) = ∆E ψ(±) η(±) = iηg ψ(±) ∂ η(±) , (9.98) h η | − 0| i h η | i ± h η | g| i such that we obtain, with the abbreviation ∂ ∂/∂g: g ≡ (±) (±) ψ ∂g η ∆E = iηg h η | | i (9.99) (±) (±) ± ψη η h | i ψ Uˆ η ( , 0) ∂ Uˆ η (0, ) ψ = iηg h 0| I ±∞ g I ∓∞ | 0i. (9.100) ± ψ Uˆ η ( , 0) Uˆ η (0, ) ψ h 0| I ±∞ I ∓∞ | 0i Similarly, taking the equation for η(±) and multiplying it by ψ(±) from the right, h | | η i we find ˆ η ˆ η ψ0 ∂gUI ( , 0) UI (0, ) ψ0 ∆E = iηg h | ±∞ ∓∞ | i. (9.101) ± ψh Uˆ η ( , 0) Uîη (0, ) ψ h 0| I ±∞ I ∓∞ | 0i Adding the two results with the upper sign gives η ∂ ∆E = i g log ψ Sη ( , ) ψ . (9.102) 2 ∂g h 0| ∞ −∞ | 0i

Note that the matrix element ψ Sη ( , ) ψ is a pure phase eiϕη . This h 0| ∞ −∞ | 0i is seen as follows. The state η(+) develops after an inﬁnitely long time into Uˆ η ( , ) ψ . There it becomes| i again an eigenstate of Hˆ due to the com- I ∞ −∞ | 0i 0 mutation relation (9.70). Thus it must coincide with ψ0 up to some overall factor iϕη | i e (recall that ψ0 is nondegenerate by assumption). Hence we arrive at the level shift formula | i η ∂ ∆E = g ϕη. (9.103) −2 ∂g 9.2 Scattering by External Potential 675

One sometimes ﬁnds this result stated in the opposite direction:

g ′ η 1 dg ′ ϕ = 2 ′ ∆E(g ). (9.104) − η Z0 g To lowest order in g, where ∆E g, the formula becomes ∝ g ϕη 2 ∆E + O g2 . (9.105) ≈ − η This formula may be used for a lowest-order estimate of the change of a particle mass due to interactions. Actually, in applications, the exponential switching factor is somewhat difficult to handle. A much more direct way of cutting off the interaction uses a step function along the time axis, i.e., it assumes the total interaction time T to be finite. The connection between T and η is found from the correspondence between integrals

∞ 2 T/2 dte−η|t| = dt = T. (9.106) Z−∞ η → Z−T/2 With this, we may rewrite (9.104) as

g ′ η dg ′ ϕ = T ′ ∆E(g ). (9.107) − Z0 g 9.2 Scattering by External Potential

After this development, it is straightforward to derive scattering amplitudes for a particle in an external potential V (x) of a ﬁnite range.

9.2.1 The T - Matrix Imagine a scattering center with a region of nonzero potential concentrated around the origin. A wave packet of a given average momentum p and energy E approaches this region along the x-axis from negative inﬁnity. As long as it is far from the scattering center, it follows the free-particle Schr¨odinger equation

Hˆ ψ(t) = i∂ ψ(t) , (9.108) 0| i t| i 2 with the unperturbed Hamiltonian Hˆ0 = p /2M. As the particle approaches the potential, the state is driven by the full Hamiltonian Hˆ = Hˆ0 +Vˆ . In the interaction representation we study the time evolution of a state which diagonalizes the unperturbed Hamiltonian Hˆ0. An artficially modified Schrödinger equation shows best how the incoming states change due to an interaction Vˆ exp ( η t ). The modifica- I − | | tion by the artificial switching factor exp ( η t ) enables us to remove the scattering potential long before the scattering process− happens.| | This frees us from the necessity of assuming the incoming wave to be localized in a packet long before the scattering 676 9 Scattering and Decay of Particles process. Instead, we can take it to be a pure plane wave, a momentum eigenstate p solving the time-independent Schrödinger equation | i

Hˆ p = Ep p , (9.109) 0| i | i 2 with an energy Ep = p /2M. The incoming wave function is 1 x p = eipx/¯h. (9.110) h | i √V The particle has the same probability everywhere in space, also in the scattering region. The amplitude for the incoming state p to be scattered into another | i eigenstate with momentum p′ is given by the matrix elements | i p′ Sˆ p = p′ Uˆ ( , ) p . (9.111) h | | i h | I ∞ −∞ | i Decomposing Uˆ ( , ) into the product of Møller operators, I ∞ −∞ Uˆ ( , )= Uˆ ( , 0) Uˆ (0, )= Ωˆ (−)†Ωˆ (+), (9.112) I ∞ −∞ I ∞ I −∞ we can also write

p′ Sˆ p = p′ (−) p(+) , (9.113) h | | i h | i where p(±) are the states | i p(±) Ωˆ (±) p . (9.114) | i ≡ | i The time evolution does not change the energy under the assumption of an inﬁnites- imally small η, such that these states satisfy the full Schr¨odinger equation with the 2 same energy E = Ep = p /2M:

(±) (±) Hˆ p = Ep p . (9.115) | i | i It is useful to take advantage of this property of the scattering process by rewriting the matrix element (9.111) in another way, using the explicit operator (9.24):

The first term in brackets is nonzero only if the momenta are equal, p′ = p, in which case the energies are also equal, Ep′ = Ep. The right-hand side is unaffected by the limit t in the prefactor. 2 → −∞ We shall imagine the system to be enclosed in a box of a large volume V (not to be confused with the potential Vˆ ). Then the momenta are discrete and the first term is simply δp′,p. It describes the amplitude for the direct beam to appear behind the scattering region, containing all unscattered particles. The second term describes scattering. Here the ′ limit t2 in the prefactor is nontrivial. If E = E, it oscillates so fast that it can be set→ equal ∞ to zero, by the Riemann-Lebesgue6 lemma (see Ref. [13] on p. 387). For E′ = E, it is identically equal to unity so that

′ ei(E −E)t2 0, E′ = E, lim ′ = ′ 6 (9.118) t2→∞ E E + iη ( i/η, E = E. − − Such a property deﬁnes a δ-function in the energy. Only the normalization needs adjustment, which is done as follows:

′ i(E −E)t2 i e ′ lim ′ = δ(E E). (9.119) t2→∞ 2π E E + iη − − Indeed, if we integrate the left-hand side over a smooth function f(E′) and set E′ E + ξ/t , then the E′-integral can be rewritten as ≡ 2 i ∞ dξ eiξ f (E + ξ/t2) . (9.120) −2π −∞ t ξ/t iη Z 2 2 −

In the limit of large t2, the function f(E) can be taken out of the integral, and the contour of integration can then be closed in the upper half-plane giving e−ηt2 . Since η is an inﬁnitesimal quantity, this is the same result as the one that would be obtained from the right-hand side of (9.119). The δ-function (9.28) ensures the conservation of energy in the scattering process. Another way to verify formula (9.119) is to rewrite the left-hand side as

1 t2 ′ dtei(E −E−iη)t, (9.121) 2π Z−∞ which obviously tends to δ(E′ E) in the limit t . − 2 →∞ Let us also express the Lippmann-Schwinger equation in terms of wave functions. Multiplying (9.54) by the local states x from the left, and identifying the interacting states ψ(+) with the states p(+) hin| Eq. (9.114) that arises from an incoming | i | i particle wave of momentum p, the left-hand side is the full Schr¨odinger wave function

ψ(+)(x) x p(+) , (9.122) ≡h | i 678 9 Scattering and Decay of Particles which is an eigenstate of Eq. (9.37):

Hˆ p(+) = E p(+) . (9.123) | i | i The ﬁrst term on the right-hand side is the plane wave (9.110). The second term in (9.54),

sc 3 ′ ′ ′ ′ (+) x p = d x x Gˆ0(E) x Vˆ (x ) x p , (9.126) h | i Z h | | i h | i ′ where x Gˆ0(E) x are the matrix elements of the free-particle resolvent deﬁned in Eq. (9.57):h | | i

′ 1 ′ x Gˆ0(E) x = x x . (9.127) h | | i h |E Hˆ + iη | i − 0 Thus the result of the full Lippmann-Schwinger equation is the wave function

x p(+) = x p + x p sc. (9.128) h | i h | i h | i It solves the interacting Schr¨odinger equation (9.123) and shows explicitly the free incoming wave plus the scattered wave. Asymptotically, the scattered wave is usually an outgoing asymptotically spherical wave emerging from the scattering center. We are now prepared to introduce the so-called T -matrix as the quantity

p′ Tˆ(E) p p′ Vˆ p(+) = p′ Vˆ Ωˆ (+) p , (9.129) h | | i≡h | | i h | | i so that the S-matrix decomposes into a direct-beam contribution plus an energy- conserving scattering contribution as follows:

p′ Sˆ p = p′ p 2πiδ(E′ E) p′ Tˆ(E) p . (9.130) h | | i h | i − − h | | i In terms of Tˆ, the Lippmann-Schwinger equation (9.54) reads

Equation (9.61) for the Møller operator implies that the T -matrix obeys the implicit equation 1 Tˆ(E)= Vˆ + Vˆ Tˆ(E). (9.132) E Hˆ iη − 0 − Its matrix elements satisfy an integral equation for the T -matrix. Note that Tˆ(E) can be expressed in terms of the Møller operator (9.53) as

Tˆ = Vˆ Ωˆ (+). (9.133)

If we multiply the state vectors (9.131) by ket vectors x from the left, we obtain the position representation of the Lippmann-Schwinger equation:h | 1 1 x p(+) = x p + x Vˆ p(+) = x p + x Tˆ(E) p . (9.134) h | i h | i h |E Hˆ +iη | i h | i h |E Hˆ + iη | i − 0 − 0 The ﬁrst term x p is once more the incoming wave, and the second term is the T -matrix representationh | i of the scattered wave (9.125): 1 1 x p sc x Vˆ p(+) = x Tˆ p . (9.135) h | i ≡h |E Hˆ + iη | i h |E Hˆ + iη | i − 0 − 0 9.2.2 Asymptotic Behavior Let us calculate the asymptotic form of the scattered wave (9.135) in the form (9.126), where it is expressed in terms of the free-particle resolvent in Eq. (9.57):

sc i 3 ′ ′ ′ ′ (+) x p = d x x Gˆ0(E) x Vˆ (x ) x p . (9.136) h | i −h¯ Z h | | i h | i The matrix elements of the resolvent have the Fourier decomposition

d3pV ih¯ x ˆ x′ x p p x′ G0(E) = ih¯ 3 h | | i Z (2πh¯) h | iE Ep + iη h | i − ′ d3p eip(x−x )/¯h = ih¯ . (9.137) (2πh¯)3 E p2/2M + iη Z − Introducing p 2M(E + iη), the momentum integral yields E ≡ q ′ ′ d3p eip(x−x )/¯h 2Mi 1 eipE|x−x |/¯h x Gˆ (E) x′ = 2Mih¯ = . (9.138) h | 0 | i − (2πh¯)3 p2 p2 − h¯ 4π x x′ Z − E | − | For each x′, this is a wave function whose time dependence is given by an exponential e−iEt/¯h. It represents an outgoing spherical wave emerging from x′. Had we chosen the opposite sign of iη in the denominator of (9.137), the exponential would have ′ been e−ipE |x−x |/¯h which, together with the time dependence e−iEt/¯h, would have 680 9 Scattering and Decay of Particles represented an incoming spherical wave. With the choice (9.138), the scattered wave (9.136) is a superposition of outgoing spherical waves emerging from all points x′ in the potential V (x′). As such it is an adaptation of the ancient Huygens principle of light waves to Schr¨odinger’s material waves. The outgoing spherical waves observed at some point x come from all points x′ in the scattering potential V (x′). They appear as a common outgoing spherical wave whose shape is found by going in (9.138) to the asymptotic regime of large x. As long as x′ x = r, we can expand | |≪| | x x′ r xˆ x′ + ..., (9.139) | − | ≡ − · so that (9.136) simpliﬁes to

ip r/¯h r→∞ E ′ sc e 2M 3 ′ −ipExˆ·x /¯h ′ ′ (+) x p 2 d x e V (x ) x p . (9.140) h | i −−−→− 4πr h¯ Z h | i

Since the scattering is elastic the energy of the incoming particles Ep is equal to the energy of the outgoing particles Ep′ . Those arriving at the point x emerge with a momentum p′ whose size is equal to that of p and whose direction is xˆ. Thus we deﬁne the outgoing momentum in the spherical wave as

′ p = pE xˆ, (9.141) and rewrite the exponent in (9.140) as

′ ′ ′ e−ipExˆ·x /¯h e−ip ·x /¯h. (9.142) → With the scattered wave (9.140), the total wave function (9.134) may be written as a sum of an incoming plane wave and an outgoing spherical wave

ipr/¯h (+) 1 ipx/¯h e x p e + fp′p . (9.143) h | i −−−→r→∞ √V r !

The prefactor of the spherical wave fp′p is called the scattering amplitude. Explicitly, it is given by the integral4:

1 2M 3 ′ −ip′·x′/¯h ′ ′ (+) ′ √ fp p = V 2 d x e V (x ) x p . (9.144) − 4π h¯ Z h | i From the asymptotic behavior (9.143), where the wave function is decomposed into an incoming plane wave and an outgoing spherical wave, we deduce immediately that the square of fp′p determines the scattering cross section. The argument goes as follows: The incoming wave has the particle current density v j = (i/M)ψ∗∇ψ = . (9.145) − V 4For the phase convention see, for example, the textbook by L.D. Landau and E.M. Lifshitz, Quantum Mechanics, Pergamon Press, London, 1965. 9.2 Scattering by External Potential 681

ipr The outgoing spherical wave has an amplitude fp′pe /r√V , and describes radial particle ﬂow with current density

′ 2 xˆ v j = fp′p . (9.146) | | r3 V The ratio between the two j′ / j is identified with dσ/r2dΩ, and defines the differential cross section: | | | |

dσ 2 = fp′p . (9.147) dΩ | | If the potential is weak, the full wave function x p(+) on the right-hand side can be approximated by the incoming plane wave x hp |, andi one obtains the scattering amplitude in the Born approximation h | i

1 2M ′ ′ ′ 3 ′ −i(p −p)·x /¯h ′ fp p 2 d x e V (x ). (9.148) ≈ −4π h¯ Z The momentum diﬀerence q p′ p is equal to the momentum transfer of the scattering process. Thus the≡ scattering− amplitude in the Born approximation is, up to a factor 2M/h¯2, equal to the Fourier components of the potential at the momentum transfer− q:

′ ′ V˜ (q) d3x′ e−i(p −p)·x /¯hV (x′). (9.149) ≡ Z For a potential V (x)= gδ(x), where V˜ (q)= g, the scattering amplitude in the Born approximation is M fp′p = g. (9.150) −2πh¯2 The Born formula (9.148) will be generalized to D dimensions further down in Eq. (9.249).

9.2.3 Partial Waves The scattering amplitudes are often analyzed separately for each angular momentum of the scattered particles. For equal incoming and outgoing energies Ep′ = Ep, which implies that also the momenta are equal p′ = p = p, the T -matrix p′ Tˆ p in Eq. (9.129) has a partial-wave expansion | | | | h | | i (2π)3 ∞ l p′ Tˆ p = T (p) Y (pˆ′)Y (pˆ). (9.151) h | | i V l lm lm Xl=0 mX=−l

Here Ylm(pˆ) are the spherical harmonics deﬁned in (4F.1), which are now associated with the momentum direction pˆ:

1/2 m 2l +1 (l m)! m imϕ Ylm(pˆ)= Ylm(θ,ϕ) ( 1) − Pl (cos θ)e . (9.152) ≡ − " 4π (l + m)! # 682 9 Scattering and Decay of Particles

They satisfy the orthogonality relation (4F.3), which we shall write in the form

2 ∗ ′ ′ d pˆ Yl′m′ (pˆ)Ylm(pˆ)= δl lδm m, (9.153) Z where d2pˆ denotes the integral over the surface of a unit sphere:

R 1 2π d2pˆ d cos θ dϕ. (9.154) Z ≡ Z−1 Z0 It is useful to view the spherical harmonics as matrix elements of eigenstates l m of angular momentum [recall (4.846)] with localized states pˆ on the unit sphere:| i h | pˆ l m Y (pˆ). (9.155) h | i ≡ lm The latter satisfy the orthogonality relation:

pˆ′ pˆ = δ(2)(pˆ′ pˆ), (9.156) h | i − where δ(2)(pˆ′ pˆ) is the δ-function on a unit sphere: − δ(2)(pˆ′ pˆ) δ(cos θ′ cos θ)δ(ϕ′ ϕ). (9.157) − ≡ − − With the help of the addition theorem for the spherical harmonics

2l +1 l P (cos ∆ϑ)= Y (pˆ′)Y ∗ (pˆ′), cos ∆ϑ pˆ′ pˆ, (9.158) 4π l lm lm ≡ · mX=−l the partial-wave expansion (9.151) becomes

(2π)3 ∞ 2l +1 p′ Tˆ p = T (p) P (pˆ′ pˆ), (9.159) h | | i V l 4π l · Xl=0 where P (cos θ) P 0(cos θ) are the Legendre polynomials (4F.7). l ≡ l A similar expansion of the S-matrix (9.130) requires decomposing the scalar ′ product p p = δp′,p into partial waves. In a large volume V , this is replaced by (2π)3δ(3)(hp′ | ip)/V , which is decomposed into a directional and a radial part as − 3 3 (2π) (3) ′ (2) ′ (2π) 1 ′ δp′ p = δ (p p)= δ (pˆ pˆ) δ(p p). (9.160) , V − − V p2 −

The directional δ-function on the right-hand side has a partial-wave expansion which is, in fact, the completeness relation (4E.1) for the states l m , | i ∞ l l m l m =1, (9.161) | ih | Xl=0 mX=−l 9.2 Scattering by External Potential 683 after being sandwiched between the localized states pˆ on the unit sphere. Using the matrix elements (9.155) and the orthogonality relation| i (9.156), we ﬁnd from (9.161):

∞ l δ(2)(pˆ′ pˆ)= Y (pˆ′)Y ∗ (pˆ). (9.162) − lm lm Xl=0 mX=−l The radial part of (9.160) can be rewritten as

3 3 3 (2π) 1 ′ (2π) 1 dE ′ (2π) 1 ′ 2 δ(p p)= 2 δ(E E)= δ(E E), (9.163) V p − V p dp − V ρE −

2 ′ ′2 where E = Ep = p /2M, E = Ep′ = p /2M, p p , with ≡| | p2 ρ = Mp (9.164) E ≡ dE/dp being proportional to the density of energy levels on the E-axis in a large volume V . The quantity ρ V/(2π)3 is the density of states. With the help of ρ we can E × E rewrite the sum over all momentum states as follows: d3pV V 2p = 3 = d ˆ 3 dE ρE. (9.165) p (2π) (2π) X Z Z Z Thus we ﬁnd the expansion

3 ∞ l ′ (2π) ′ 1 ′ p p = δp′,p = Ylm(pˆ )Ylm(pˆ) δ(E E). (9.166) h | i V ρE − Xl=0 mX=−l In a partial-wave expansion for the S-matrix (9.130), the overall δ-function for the energy conservation is conventionally factored out and one writes:

3 ∞ l ′ (2π) 1 ′ ′ p Sˆ p = δ(E E) Sl(p) Ylm(pˆ )Ylm(pˆ). (9.167) h | | i V ρE − Xl=0 mX=−l Recalling (9.130), we ﬁnd the relation between the partial-wave scattering amplitudes Sl(p) and those of the T -matrix:

S (p)=1 2πiρ T (p). (9.168) l − E l The potential scattering under consideration is purely elastic. For such processes, the unitarity property (9.44) of the S-matrix implies

d3p′′ V p′ ˆ† p′′ p′′ ˆ p p′ p 3 S S = . (9.169) Z (2π) h | | ih | | i h | i 684 9 Scattering and Decay of Particles

Inserting the partial-wave decomposition (9.167) and using (9.165) as well as the orthogonality relation (9.153), we derive from (9.169) the unitarity property of the partial waves of elastic scattering

∗ Sl (p)Sl(p)=1, (9.170) implying that Sl(p) is a pure phase factor:

2iδl(p) Sl(p)= e . (9.171)

For the amplitudes Tl(p) in (9.168), this implies

1 iδl(p) Tl(p)= e sin δl(p). (9.172) −πρE

The quantities δl(p) are the phase shifts observed in the scattered waves of angular momentum l with respect to the incoming waves. In order to show this we use the expansion formula

∞ iz cos ϑ π l e = i Jl+1/2(z)(2l + 1)Pl(cos ϑ), (9.173) r2z l=0 X where the Bessel functions Jl+1/2(z) can be expressed in terms of spherical Bessel functions jl(z) as 2z Jl+1/2(z)= jl(z). (9.174) s π The expansion (9.173) may be viewed as a special case of the Dirac bracket expansion

∞ l ∞ 1 ipx 2l +1 x p = e = ψl(pr) Ylm(xˆ)Ylm(pˆ)= ψl(pr)Pl(xˆ pˆ), (9.175) h | i √V 4π · Xl=0 mX=−l Xl=0 where r x , and ≡| | 1 l ψl(pr)= i 4πjl(pr) (9.176) √V are partial-wave amplitudes of the plane wave. Far from the scattering region, the spherical Bessel functions have the asymptotic behavior

sin(z πl/2) j (z) − , (9.177) l → pr implying for the partial waves

1 l sin(pr πl/2) ψl(pr)= i 4π − . (9.178) √V pr 9.2 Scattering by External Potential 685

We now turn to the partial-wave expansion of the scattered wave (9.135) in the Lippmann-Schwinger equation. Inserting a complete set of free momentum eigenstates into the right-hand side, we obtain d3p′ V 1 x p sc = x p′ p′ Tˆ p . (9.179) h | i (2π)3 h | iE E′ + iη h | | i Z − Inserting the partial-wave decompositions (9.159) and (9.175), the right-hand side becomes d3p′V 1 x p′ p′ Tˆ p (2π)3 h | iE E′ + iη h | | i Z − ∞ l ∞ ′ 1 ′ l ′ ′2 jl(p r)Tl(p) = Ylm(xˆ )Ylm(pˆ)i 4π dp p . (9.180) √V 0 E E′ + iη Xl=0 mX=−l Z − Adding this to the incoming wave function, we ﬁnd the total partial waves:

∞ ′ tot 1 l ′ ′2 jl(p r)Tl(p) ψl (pr)= 4πi jl(pr)+ dp p . (9.181) √V " 0 E E′ + iη # Z − For large r, we use the asymptotic form (9.177) to write the second term in the brackets as ′ ′ ∞ T (p′) ei(p r−πl/2) e−i(p r−πl/2) dp′ p′2 l − . (9.182) 0 E E′ + iη 2ip′r Z − Being at large r, the integral is oscillating very fast as a function of p′. It receives a sizable contribution only from a neighborhood of the pole term in the energy.5 There we can approximate dE E E′ (p p′), (9.183) − ≈ dp − and approximate p′ p in all terms, except for the fast oscillating exponential. The ensuing integral ≈

′ ′ 1 T (p′) ei(p r−πl/2) e−i(p r−πl/2) p2 dp′ l − (9.184) dE/dp p p′ + iη 2ip′r Z − can be extended to the entire p′-axis with a negligible error for large r. The part containing the second exponential is evaluated by closing the contour of integration in the lower half-plane. There the integrand is regular so that the contour can be contracted to a point yielding zero. In the part with the ﬁrst exponential we close the contour in the upper half-plane and deform it to pick up the pole at p′ = p + iη. Then Cauchy’s theorem yields

′ ′ ′ 1 T (p′) ei(p r−πl/2) e−i(p r−π/2l) 2πp2 eip r p2 dp′ l − = T (p)e−iπl/2 , (9.185) dE/dp p p′ + iη 2ipr −dE/dp l 2p′r Z − 5This is a consequence of the Riemann-Lebesgue lemma cited in Ref. [13] on p. 387. 686 9 Scattering and Decay of Particles so that the total wave function (9.181) becomes

r→∞ ipr tot 1 l e ψl (pr) 4π i jl(pr) 2πρE Tl(p) −−−→ √V " − 2pr # l r→∞ 1 4πi ipr l −ipr ipr e ( 1) e 2πiρETl(p) e . (9.186) −−−→ √V 2ipr − − − h i Using (9.168), this is equal to

r→∞ l tot 1 4πi ipr l −ipr ψ (pr) Sl(p)e ( 1) e . (9.187) l −−−→ √V 2ipr − − h i Inserting the unitary form (9.171) for elastic scattering, this becomes

r→∞ 1 4π sin(pr πl/2+ δl(p)) ψtot(pr) ileiδl(p) − . (9.188) l −−−→ √V p r

This shows that δl(p) is indeed the phase shift of the scattered waves with respect to the incoming partial waves (9.178). At small momenta, the l = 0(s-wave) phase shift dominates the scattering process since the particles do not have enough energy to overcome the centrifugal barrier. For short-range potentials, the s-wave phase shift diverges for p 0, and the leading two orders in p may be parametrized as follows →

1 1 2 p cot δ0 = + reﬀ p + .... (9.189) −as 2

The parameter as is the s-wave scattering length, and reﬀ is called the eﬀective range of the scattering process. The direct small-k expansion of δ0 is

3 2 as 3 δ0(p)= asp asreﬀ p + .... (9.190) − − − 3 !

For small p, only the s-wave scattering length is relevant. There the s-wave scattering amplitude behaves like [recall (9.172)]

1 2 2 3 as 2 2 T0(p) δ0(p)+iδ0(p) δ0 (p)+ ... 1 iasp+(asreﬀ as)k +... . ≈−πMp − 3 ≈ πM − − h (9.191)i Inserting this into (9.159), we see that in the limit p 0: → a a 1 p′ Tˆ p (2π)3 s =2π s , (9.192) h | | i→ 4π2V M M V which by (9.262) is also equal to gR/M. 9.2 Scattering by External Potential 687

For a nonzero angular momentum l, the eﬀective-range formula (9.189) leads to the small-p behavior 2l+1 1 1 2 p cot δl = + reﬀ l p + .... (9.193) −al 2 Although (9.189) and (9.193) are small-p approximations, the range of validity may extend to quite large momenta if the potential is of short range. It may include a resonance in the scattering amplitude, in which case one may use the approximation

4π f (p) . (9.194) l ≡ p cot δ (p) ip l − In general, one may parametrize the S-matrix element in the vicinity of a resonance by the unitary ansatz

E ER iΓR/2 S (p)= e2iδl(p) − − , (9.195) l ≡ E E + iΓ /2 − R R where ER is the energy of the resonance and ΓR its decay rate. At every resonance or bound state, δl(p) passes through π/2 plus an integer multiple of π. This is guaranteed by Levinson’s theorem [4], according to which the number of bound states in a potential is equal to the difference [δ (0) δ ( )]/π. l − l ∞ We may also write 4π Γ /2 f (p) − R , (9.196) l ≡ p E E + iΓ /2 − R R and −2ip aeff l(p) Sl(p)= e , (9.197) with 1 Γ (E E ) a a + arctan R − R . (9.198) eff l ≡ l 2p (E E )2 +Γ2 /4 − R R 9.2.4 Off-Shell T -Matrix In scattering experiments, only matrix elements are observables in which the energy 2 ′2 E is equal to the energies Ep p /2M = Ep′ p /2M of incoming and outgoing particles. For mathematical≡ purposes, however,≡ one also defines a T -matrix for unequal energies E = Ep = Ep′ , as the matrix elements of the operator 6 6 1 Tˆ(E)= Vˆ + Vˆ Tˆ(E). (9.199) E Hˆ + iη − 0 This is called the off-shell T -matrix. It can be expressed in terms of the resolvent Gˆ0(E) in (9.57) as

Tˆ(E)= Vˆ + Vˆ Gˆ0(E)Tˆ(E). (9.200) 688 9 Scattering and Decay of Particles

This is an implicit equation for Tˆ(E) that can be solved iteratively by the geometric series Tˆ(E)= Vˆ + Vˆ Gˆ0(E)Vˆ + Vˆ Gˆ0(E)Vˆ Gˆ0(E)Vˆ + .... (9.201) Comparison with (9.60) shows that we may also write, instead of (9.200),

Tˆ(E)= Vˆ + Vˆ Gˆ(E)V.ˆ (9.202)

There are two further equivalent equations:

Gˆ0(E)Tˆ(E)= Gˆ(E)V,ˆ (9.203) and Gˆ (E)Tˆ(E)Gˆ (E)= Gˆ(E) Gˆ (E). (9.204) 0 0 − 0 The latter can also be written as

Tˆ(E)=(E Hˆ )Gˆ(E)(E Hˆ ) (E Hˆ ). (9.205) − 0 − 0 − − 0 This equation gives a useful relation between the oﬀ-shell matrix elements of the T -matrix and those of the resolvents:

′ 2 ′ ′ p Tˆ(E) p = (E Ep) [ p Gˆ(E) p p Gˆ (E) p ] h | | i − h | | i−h | 0 | i 2 ′ ′ = (E Ep) p Gˆ(E) p (E Ep) p p . (9.206) − h | | i − − h | i Let us express the right-hand side in terms of the spectral representation of the resolvents. We take the completeness relation for the states (9.109):

p p =1, (9.207) p | ih | X and multiply this by the operator 1/(E Hˆ0). Using the eigenvalue equation (9.109), we obtain the spectral representation of− the free resolvent: p p Gˆ(E)= | ih | . (9.208) p E Ep + iη X − The interacting resolvent Gˆ(E) has a similar representation. Let n be the complete set of eigenstates of the full Hamiltonian [consisting of the set of| i scattering states ψ(+) plus the set of bound states ψ on the right-hand side of Eq. (9.62)]: | a i | βi Hˆ n = E n . (9.209) | i n| i Then the completeness relation n n = 1 (9.210) n | ih | X leads to the spectral representation n n Gˆ(E)= | ih | . (9.211) n E En X − 9.2 Scattering by External Potential 689

Rewriting E Ep =(E E )+(E Ep′ ), then Eq. (9.206) becomes − − n n − ′ ′ p n n p ′ ′ p Tˆ(E) p = (En Ep′ )(E Ep)h | ih | i +(E Ep) p n n p p p . h | | i n − − E En +iη − " n h | ih | i−h | i# X − X (9.212)

The second term vanishes due to the completeness relation (9.210), and can be dropped. This is true only if the potential has no hard core forbidding completely the presence of a particle inside some core volume Vcore. In this case, the wave functions x n vanish identically inside V , and the wave functions are complete h | i core only outside Vcore. Then the second term yields a correction to the ﬁrst term

′ D i(p′−p)x p Tˆcorr(E) p = (E Ep) d x e , (9.213) h | | i − − ZVcore ′ ′ which is zero on shell. Using the identity (E Ep′ ) p n = p Hˆ Hˆ n = n − h | i h | − 0| i p′ Vˆ n , the ﬁrst term in (9.212) takes the form h | | i ′ ˆ ′ p V n n p p Tˆ(E) p = (E Ep)h | | ih | i . (9.214) h | | i n − E En + iη X −

Replacing further E Ep by (E E )+(E Ep), this becomes − − n n − p′ Vˆ n n Vˆ p p′ Tˆ(E) p = p′ Vˆ n n p + h | | ih | | i . (9.215) h | | i n h | | ih | i n E En + iη X X − Let us now focus our attention upon a potential without bound states. Then we can use, instead of the discrete states n in the completeness relation (9.210), the continuous set of solutions p(+) of the| Lippmann-Schwingeri equation (9.131) with | i E = Ep. Using Eq. (9.133), we ﬁnd

p′ Vˆ p′′(+) p′′(+) Vˆ p p′ Tˆ(E) p = p′ Vˆ p′′(+) p′′(+) p + h | | ih | | i . (9.216) h | | i ′′ h | | ih | i ′′ E Ep′′ + iη Xp Xp − Replacing the states in the ﬁrst term p′′(+) by the result of the Lippmann-Schwinger equation (9.131), we obtain h |

p′ Tˆ(E) p = p′ Vˆ p(+) h | | i h | | i p′ Vˆ p′′(+) p′′(+) Vˆ p p′ Vˆ p′′(+) p′′(+) Vˆ p + h | | ih | | i + h | | ih | | i . (9.217) ′′ − Ep Ep′′ + iη E Ep′′ + iη  Xp − −   Now we observe that the matrix elements p′ Vˆ p(+) are the half-on-shell matrix h | | i elements of the T -matrix, whose energy E is equal to the energy Ep of the incoming state p : | i ′ (+) ′ p Vˆ p = p Tˆ(Ep) p . (9.218) h | | i h | | i 690 9 Scattering and Decay of Particles

The left-hand momentum is arbitrary. Hence we can rewrite Eq. (9.217) as

′ ′ p Tˆ(E) p = p Tˆ(Ep) p h | | i h | | i ′ ′′ ′′ ∗ 1 1 + p Tˆ(Ep) p p Tˆ(Ep) p + .(9.219) ′′ h | | ih | | i −Ep Ep′′ + iη E Ep′′ + iη ! Xp − − After solving the Lippmann-Schwinger equation (9.131) to find p′ Tˆ(E) p , this h | | i equation supplies us with the full off-shell T -matrix. For a hard-core potential, we must add the correction term (9.213). The off-shell T -matrix has a partial-wave expansion

(2π)3 ∞ l p′ Tˆ(E) p = T (p′,p; E) Y (pˆ′)Y (pˆ). (9.220) h | | i V l lm lm Xl=0 mX=−l If we want to calculate the oﬀ-shell T -matrix, we usually must do this separately for each angular momentum. This has been done for various potentials. The result for a hard-core potential of radius a will be given in Eq. (9.287).

9.2.5 Cross Section The square of the matrix elements of the T -matrix is experimentally observable as differential cross section. This quantity is defined by the probability rate dP˙ (t) of scattering an incoming particle into a solid angle dΩ=ˆ d cos θdϕ around the direction pˆ = (sin θ cos ϕ, sin θ sin ϕ, cos θ). Forming the ratio of this with the incoming particle current density j, i.e., with the rate of incoming particles per area, we obtain the differential cross section

dσ dP˙ (t) 1 = . (9.221) dΩ dΩ j The total rate of scattered particles is given by the integral over this in all space directions dσ σ = dΩ . (9.222) Z dΩ The two quantities, which have the dimension of an area, are determined by the matrix elements of the T -matrix, as we shall now demonstrate. The time-dependent version of the incoming wave functions (9.175) is

1 −i(E0t−px) x p, t = ψp(x, t)= e . (9.223) h | i √V The associated incoming current density is 1 ↔ 1 p 1 j = ψ† (x, t)∇ψ (x, t)= = v, (9.224) 2Mi V M V 9.2 Scattering by External Potential 691

′ the factor 1/V reﬂecting the normalization p p = δp′,p of one particle per total volume. In order to ﬁnd the rate of outgoingh particles,| i we form the time derivative

dP (t) d d = p′ Uˆ (t, t ) p 2 = p′ Uˆ (t, t ) p p′ Uˆ (t, t ) p ∗ dt dt|h | I 0 | i| dth | I 0 | ih | I 0 | i = 2 Im p′ Vˆ (t)Uˆ (t, t ) p p′ Uˆ (t, t ) p ∗ h | I I 0 | ih | I 0 | i h i = 2 Im p′ Vˆ (t)Uˆ (t, 0)Uˆ (0, t p p′ Uˆ (t, 0)Uˆ (0, t ) p ∗ ,(9.225) h | I I I 0i| ih | I I 0 | i h i which yields, in the limit t , the expression 0 → −∞ dP (t) = 2 Im p′ Vˆ (t)Uˆ (t, 0) p(+) p′ Uˆ (t, 0) p(+) ∗ . (9.226) dt h | I I | ih | I | i h i

Inserting the explicit time dependence of VˆI(t) from Eq. (9.15), and of UˆI (t, 0) from Eq. (9.24), and using the fact that Hˆ acting on p and Hˆ on p(+) produce the 0 | i | i same energy eigenvalues, this becomes

dP (t) ˆ ˆ ˆ ˆ ˆ ˆ = 2 Im p′ eiH0tVˆ e−iH0t eiH0te−iHt p(+) p′ eiH0te−iHt p(+) ∗ dt h | | ih | | i h i = i p′ V p(+) p′ p(+) ∗ p′ V p(+) ∗ p′ p(+) . (9.227) − h | | ih | i −h | | i h | i h i We now use the Lippmann-Schwinger equation (9.131) to rewrite this as

dP (t) =2 p′ p Im p Tˆ p +2πδ(E E′) p′ Tˆ(E) p 2. (9.229) dt h | i h | | i − |h | | i| The first term gives the rate of the transmitted particles in the direct beam. The second term gives the rate of scattered particles. If we collect the scattered particles 3 3 of all final momenta in a sum p′ = d pV/(2π) , we find the total probability rate P R dP d3p′V ′ p′ ˆ p 2 = 3 2πδ(E E) T (E) . (9.230) dt Z (2π) − |h | | i| There exists a simple mnemonic rule for deriving this formula. The probability for a particle to go from the initial to the final state is given by the absolute square of the scattering amplitude (9.130):

′ 2 ′ ′ ′ 2 Pp′p = p Sˆ p = p p 2πiδ(E E) p Tˆ(E) p . (9.231) |h | | i| |h | i − − h | | i| 692 9 Scattering and Decay of Particles

Excluding the direct beam, this is equal to

′ 2 ′ 2 Pp′p = [2πδ(E E)] p Tˆ(E) p . (9.232) − h | | i| Imagining the world to exist only for a ﬁnite time ∆t, the δ-function of the energy at zero argument is really ﬁnite:

∆t/2 dt iEt ∆t δ(E) E=0 = e = . (9.233) | −∆t/2 2π 2π Z E=0

Thus we can write (9.232) as

′ ′ 2 Pp′p = ∆t 2πδ(E E) p Tˆ(E) p . (9.234) − |h | | i| Integrating (9.234) over the final phase-space volume, we find the probability rate for going to all final states: dP P d3p′V ′ p′ ˆ p 2 = = 3 2πδ(E E) T (E) , (9.235) dt ∆t Z (2π) − |h | | i| leading directly to Eq. (9.230). This formula is written in natural units with c = h¯ = 1. In proper physical units, the right-hand side carries a factor 1/h¯. With this factor, Eq. (9.235) is known as Fermi’s golden rule. The alert reader will have noted that, in going from (9.232) to (9.234), we have done an operation that is forbidden in mathematics: we have calculated the square of a distribution with the heuristic formula

δ(0)δ(∆E). (9.236)

In the theory of distributions, such operations are illegal. Distributions form a linear space, implying that they can only be combined linearly with each other. In the present context, there are two ways of going properly from (9.232) to (9.234). One is to use wave packets for the incoming and outgoing particles. This was done in the textbook [1]. Another way proceeds via a more careful evaluation of the δ-functions on a large but ﬁnite interval ∆t where one uses, instead of δ(E′ E), − the would-be δ-function ′ ∆t/2 ′ sin ∆t(E E)/2 δ(E′ E) dt ei(E −E)t = | − |. (9.237) − ≈ −∆t/2 E′ E /2 Z | − | Squaring this expression yields sin2[∆t(E′ E)/2] [δ(E′ E)]2 − . (9.238) − ≈ (E′ E)2/4 − To get the rate we must divide this by the time interval ∆t, so we obtain a factor 1 sin2[∆t(E′ E)/2] [δ(∆E)]2 − . (9.239) ∆t ≈ ∆t(E′ E)2/4 − 9.2 Scattering by External Potential 693

This function has a sharp peak around E′ E = 0. The area under it is 2π. Hence we obtain in the limit ∆t : − →∞ 1 sin2[∆t(E′ E)/2] [δ(∆E)]2 − 2πδ(E′ E), (9.240) ∆t ≈ ∆t(E′ E)2/4 −−−→∆t→∞ − − and thus once more the previous heuristic result (9.234). There exists also a more satisfactory solution that is free of the restriction of distributions to linear combinations. This is possible by defining consistently products of distributions in a unique way. We simply require that path integrals, which are completely equivalent to quantum mechanics, must share with the Schrödinger equation the property of being independent of the coordinates in which they are formulated [2]. The momentum integral can be split into an integral over the final energy and the final solid angle dΩ. For a nonrelativistic particle in D dimensions, this goes as follows D ∞ ∞ d p 1 D−1 1 D = D dΩ dpp = D dΩ dE ρE Z (2π) (2π) Z Z0 (2π) Z Z0 ∞ M D−2 = D dΩ dE p . (9.241) (2π) Z Z0 The energy integral removes the δ-function in (9.229) and, by comparison with (9.221), we can identify the differential cross section as

D−2 dσ MVp 2 1 = 2π Tp′p , (9.242) dΩ (2π)D | | j where we have set ′ p Tˆ(E) p Tp′p, (9.243) h | | i ≡ for brevity. Inserting j = p/MV from (9.224), this becomes

2 2 D−3 dσ M V p 2 = Tp′p . (9.244) dΩ (2π)D−1 | | If the scattered particle moves relativistically, we have to replace M in (9.241) with E = √p2 + M 2 inside the momentum integral, so that

D ∞ ∞ d p 1 D−1 1 rel D = D dΩ dpp = D dΩ dE ρE Z (2π) (2π) Z Z0 (2π) Z Z0 ∞ 1 D−2 = 3 dΩ dEE p . (9.245) (2π) Z ZM In the relativistic case, the initial current is not proportional to p/M, which is the nonrelativistic velocity of the incoming particle beam, but to the relativistic velocity vrel = p/E, so that the incoming current is 1 p j = . (9.246) V E 694 9 Scattering and Decay of Particles

Hence the cross section becomes, in the relativistic case,

2 2 D−3 dσ E V p 2 = Tp′p . (9.247) dΩ (2π)D−1 | |

Let us compare this with the result in Eq. (9.147), where we introduced a scattering amplitude fp′p whose absolute square is equal to the diﬀerential cross section:

dσ 2 = fp′p . (9.248) dΩ | | Thus we can identify [compare (9.148)]

MV fp′p Tp′p, (9.249) ≡ −(2π)(D−1)/2 where we have chosen the sign to agree with the convention in Landau and Lifshitz’ textbook. Using the partial-wave expansion (9.151) and (9.159), we obtain for fp′p, in the three-dimensional nonrelativistic case, the expansion

∞ l ∞ ′ ∗ 2l +1 ′ fp′p = f (p) Y (pˆ )Y (pˆ)= f (p) P (pˆ pˆ), (9.250) l lm lm l 4π l · Xl=0 mX=−l Xl=0 where from (9.172)

4π2 2π 4π f (p)= ρ T (p)= [S (p) 1] = eiδl(p) sin δ (p). (9.251) l − p E l ip l − p l

This guarantees that the radial particle current which leaves the scattering zone 2 diﬀers, on the unit sphere, by a factor fp′p from the incoming current, leading | | directly to the diﬀerential cross section (9.248).6

9.2.6 Partial Wave Decomposition of Total Cross Section The contribution of the various partial waves to the total cross section is obtained by forming the absolute square of (9.250). After this, one integrates over all directions of the outgoing beam and uses the orthogonality relation (9.153) and the addition theorem (23.12) to express the total cross section as a sum over partial waves:

∞ ′ ′ 2 ′ ∗ ′ ′ ∗ ′ ∗ σ = dpˆ fp p = fl(p)fl (p) dpˆ Ylm(pˆ )Ylm(pˆ)Yl′m′ (pˆ )Yl′m′ (pˆ) | | ′ ′ Z Xl=0 Xlm lXm Z ∞ ∞ 2l +1 = f (p) 2 Y (pˆ)Y ∗ (pˆ)= f (p) 2 P (1). (9.252) | l | lm lm | l | 4π l Xl=0 Xlm Xl=0 6In D dimensions, the partial-wave expansion (9.251) is given by (9B.41). 9.2 Scattering by External Potential 695

Since Pl(1) = 1, this is equal to

∞ ∞ 2l +1 4π ∞ σ = σ = f (p) 2 = (2l + 1) sin2 δ (p). (9.253) l | l | 4π p2 l Xl=0 Xl=0 Xl=0 As mentioned at the end of Subsection 9.2.3, there can be a resonance at small p if the eﬀective range is large enough and a is positive. In the s-wave, we can approximate f0(p) in (9.251) by the expression (9.194):

4π f (p) . (9.254) 0 ≡ k cot δ (p) ik 0 − Inserting the approximation (9.189), this becomes

4π f (p) , (9.255) 0 1 1 ≈ + r k2 ik −a 2 eff − corresponding to the general resonance formula (9.197) for the S-matrix. In nuclear scattering, this is called a Feshbach resonance. It leads to a contribution to the cross section in Eq. (9.253): 4πa2 σ . (9.256) 0 ≡ (1 ar k2/2)2 + a2k2 − eff The different values of the parameter a for various depths of a typical potential are illustrated in Fig. 9.1.

Figure 9.1 Behavior of wave function for diﬀerent positions of a bound state near the continuum in a typical potential between atoms.

9.2.7 Dirac δ -Function Potential As an example, consider a local interaction that may be approximated by a δ- function

V (x)= g δ(3)(x). (9.257) 696 9 Scattering and Decay of Particles

Between plane waves, this has the matrix elements

g ′ g p′ V p = d3xe−ip xδ(3)(x)eipx = , (9.258) h | | i V Z V so that the Born approximation to the scattering amplitude is M fp′p = g . (9.259) −2πh¯2 3 2 Since g has the dimension energy times length , orh ¯ /M times length, fp′p has the dimension length, making the cross section an area seen by the incoming beam. For the δ-function potential, it is instructive to calculate also the next correction to the Born approximation. According to Eq. (9.132), it is given by d3p′′V 1 T ′ = V ′ + V ′ ′′ V ′′ + ... p p p p (2π)3 p p E E(p′′)+ iη p p Z − 1 d3p′′ 1 = g + g2 + ... . (9.260) V " (2π)3 E E(p′′)+ iη # Z − Thus, for zero incoming momentum and energy E, the coupling constant g is eﬀec- tively replaced by d3p 1 g = g g2 + ..., (9.261) R − (2π)3 E(p) iη Z − and the T -matrix has the isotropic value 1 Tp′p = gR. (9.262) |p|=0,E=0 V

The integral on the right-hand side diverges. A finite result can be obtained by assuming the initial g to be infinitesimally small such as to compensate for the divergence. This is the first place where one is confronted with a renormalization problem that is typical for all quantum field theories with a local interaction. A systematic treatment of this problem will be presented in Chapters 10 and 11. Here this problem occurs for the first time, in the context of quantum mechanics. This is due to the fact that the Schrödinger equation for the δ-function potential does not have a proper solution. Nevertheless, it is possible to bypass the problem by renormalization and to solve the Lippmann-Schwinger equation (9.54) after all, yielding the full scattering amplitude (9.260). The zero-energy expression (9.261) is extended by repetition to a geometric series

3 3 2 2 d p 1 3 d p 1 gR = g g + g + .... (9.263) − (2π)3 E(p) iη " (2π)3 E(p) iη # Z − Z − The series for the inverse of this can be summed up to 1 1 d3p 1 = + . (9.264) g g (2π)3 E(p) iη R Z − 9.2 Scattering by External Potential 697

Comparing (9.262) with (9.192), we see that the renormalized coupling determines the s-wave scattering length as being M a = g . (9.265) s 2πh¯2 R Together with (9.262) this agrees with (9.192). The identification of the renormalized coupling constant gR with the scattering length may be attributed to the use of a so-called pseudopotential for which the Schrödinger equation can be properly solved. Instead of the potential gδ(3)(x) which in a Schrödinger equation picks out the value of a wave function at the origin when (3) forming the product, one uses the condition V (x)ψ(x) = gδ (x)∂rrψ(x). This 2 ensures that the s-wave scattering length is as = MgR/2πh¯ as for a hard sphere of 7 radius as. Note that (9.265) holds only for a particle of mass M in an external potential gδ(3)(x). A two-body system involves the relative Schrödinger equation which contains the reduced mass Mred = 1/(1/M1 +1/M2) rather than M. Hence in a many-body system, gR is related to the scattering length by

2 gR = (4πh¯ /M)as. (9.266)

A remark is useful in connection with the divergence of Eq. (9.267). It can be made finite by cutting off the momentum integral at some large value Λ to find

1 1 4π Λ 1 4π = +2M 3 dp = +2M 3 Λ. (9.267) gR g (2π) Z0 g (2π)

Since Λ is very large, one might be tempted to conclude that gR must be positive for any g. However, as we shall see in Chapters 7 and 8, this conclusion is not allowed in renormalizable quantum ﬁeld theories. According to the rules of renormalization, any divergence may be cancelled by a counterterm, and the sign of it is arbitrary. Only the renormalized quantity gR is physical, and it must be ﬁxed by experiment.

9.2.8 Spherical Square-Well Potential For a square-well potential

V , r

2 d 2 d 2 l(l + 1) + + K ψl(r)=0, (9.269) (dr2 r dr " − r2 #)

7See K. Huang, Statistical Mechanics, Wiley, N.Y., 1963 (p.233). For higher partial waves see Z. Idziaszek and T. Calarco, (quant-ph/0507186). 698 9 Scattering and Decay of Particles where K 2M(E + V )/h.¯ (9.270) ≡ 0 There is only one solution which isq regular at the origin:

ψl(r)= Ajl(Kr), (9.271) with an arbitrary normalization factor A. For positive energy E, the solution in the outer region r a receives an admix- ≥ ture of the associated spherical Bessel function nl(kr):

ψl(r)= B [cos δl(k)jl(kr) + sin δl(k)nl(kr)] , (9.272) where k √2ME/h,¯ (9.273) ≡ and B, δl(k) must be determined from the boundary condition at r = a. The associated Bessel functions nl(z) have the asymptotic behavior orthogonal to (9.177): cos(z πl/2) n (z) − , (9.274) l → − pr The continuity of the wave function ﬁxes the ratio B/A. The continuity of the logarithmic derivative is independent of this ratio and ﬁxes the phase shifts by the equation j′(Kr ) cos δ (k)j′(kr ) sin δ (k)n′(kr ) K l 0 = k l l 0 − l l 0 . (9.275) j (Kr ) cos δ (k)j (kr ) sin δ (k)n (kr ) l 0 l l 0 − l l 0 For an s-wave, this reduces to

K cot Kr0 = k cot(kr0 + δ0). (9.276)

Inserting (9.270) and (9.273), the resulting binding energy are plotted in Fig. 9.2. For small incoming energy, this equation becomes

K cot K r k cot(kr + δ ), (9.277) 0 0 0 ≈ 0 0 where K √2MV /h¯, which is solved by 0 ≡ 0 tan K0r0 δ0(k)= r0 1 k + .... (9.278) − − K0r0 This determines the phase shift to be

tan K0r0 as = r0 1 . (9.279) − − K0r0 The next term in the small-k expansion of (9.278) is

3 k 3 2 3 3as r0 3asr0 2as 2 , (9.280) 6 − − − K0 ! 9.2 Scattering by External Potential 699

Figure 9.2 Behavior of binding energy (in units of V ) and scattering length in an | 0| attractive square well potential of depth V0 and radius R0 for identical bosons of reduced | | 2 2 2 mass Mred. The abcissa shows the potential depth in units of π ¯h /8MredR0. implying an eﬀective radius deﬁned in (9.190):

3 r0 r0 1 reﬀ = 2 + 2 . (9.281) 2 − 6as 2asK0 At negative energies E, there are bound states which decrease for large r. The only solution of (9.269) with this property is the spherical Bessel function of the third kind (Hankel function of the ﬁrst kind):8

ψ (k)= Bh(2)(kr) B [j (kr) in (kr)] , (9.282) l l ≡ l − l which behaves like e−(κr−lπ/2)/kr for E < 0 where k = i√ 2ME iκ. The energies of the bound states are found from the continuity of the− logarithm≡ ic derivative at r = r0: h(2)′(iκr ) j′(Kr ) iκ l 0 = K l 0 . (9.283) (2) j (Kr ) hl (iκr0) l 0 For the s-wave, this becomes K cot Kr = κ. (9.284) 0 − For a repulsive potential, Eq. (9.278) can be continued analytically to

tanh K0 r0 δ0(k)= r0 1 | | k + .... (9.285) − − K0 r0 ! | | 8M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965, Subsection 10.1.1. 700 9 Scattering and Decay of Particles

In the hard-core limit V , this yields an s-wave scattering length 0 →∞ a = r , (9.286) s − 0 and a vanishing eﬀective range. For the hard-core potential, equation (9.205) has been solved [3]. The lth partial wave amplitude is

2 2 ′2 ′ l 2πh¯ a kE k ′ ′ Tl(k ,k; E) = ( 1) − ka jl(k a)jl+1(ka)+(k k ) − − (2πM)3 ( k2 k′2 ↔ − h(2) (ka) k a l+1 j (k′)j (k) , (9.287) − E (2) l l  hl (ka)  where k √2ME/h¯.  E ≡ 9.3 Two-Particle Scattering

Usually, the ﬁnal state has two or more particles leaving the interaction zone. The interaction energy is an integral over some local interaction density

3 Vˆ = d x ˆint(x), (9.288) Z H integrated over all space at a ﬁxed time x0 = t = 0. In this case, the matrix elements of the potential factorizes as

′ 3 ′ ipxˆ −ipxˆ ′ p Vˆ p = d x p e înt(0)e p = V δp′ p p înt(0) p , (9.289) h | | i Z h | H | i h |H | i where p and p′ are the sum over all initial and final momenta, respectively. Thus the matrix elements of Vˆ always carry a δp′ p-factor that ensures total momentum conservation. The same thing is true for the T -matrix Tp′p. This factor is customarily removed by defining the so-called t-matrix

Tp′p = V δp′ p tp′p. (9.290)

The Born approximation is now

′ tp′p p ˆ (0) p . (9.291) ≈h |Hint | i Replacing the Kronecker δ for momentum conservation by its continuum limit,

3 (2π) (3) ′ δp′ p δ (p p), → V − we can write (9.290) as

3 (3) ′ Tp′p = (2π) δ (p p) tp′p, (9.292) − 9.3 Two-Particle Scattering 701 i.e., the matrix elements (9.130) of the S-matrix become

′ ′ 4 (4) ′ p Sˆ p = p p i(2π) δ (p p) tp′p. (9.293) h | | i h | i − − In a world of a finite volume V existing for a finite time T , the probability of going from an initial state p to a final state p′ is given by the absolute square of (9.293). By analogy with (9.233), we have

d4p TV δ(4)(p) = eipx = , (9.294) p=0 (2π)4 (2π)4 Z p=0

so that we obtain the transition rate [compare (9.234)]. Thus we can write (9.232) as 4 (4) ′ 2 P = TV (2π) δ (p p) tp′p . (9.295) − | | This has to be summed over all ﬁnal particle states, and we obtain the rate dP = V (2π)4δ(4)(p′ p) t 2. (9.296) dt − | fi| ﬁnalX states

We have changed the notation for the t-matrix tp′p and written it as t fi to emphasize the validity for a more general initial state i and a larger variety of possible ﬁnal states f . | i | i If the particles 1′, 2′,...,n′ leave the interaction zone, the sum over all ﬁnal states in a volume V consists of an integral over the phase space

3 ′ 3 ′ 3 ′ n d p1 d p2 d pn = V 3 3 3 , (9.297) p′ ,p′ ,... (2π) (2π) ··· (2π) 1X2 Z Z Z and (9.296) becomes

3 ′ 3 ′ 3 ′ dP n+1 d p1 d p2 d pn 4 (4) ′ 2 = V 3 3 3 (2π) δ (p p) t fi , (9.298) dt Z (2π) Z (2π) ··· Z (2π) − | | ′ ′ ′ ′ where p = p1 + p2 and p = p1 + p2 + ... + pn. By dividing out the current density of the incoming particles j, we find from this the cross section dP 1 σ = . (9.299) dt j By omitting some of the final-state integrations, we may extract from this any desired differential cross section.

9.3.1 Center-of-Mass Scattering Cross Section Let us assume the collision to take place in the center-of-mass frame with zero total momentum and p = p = p and a total energy E . Then the δ-function ﬁxes 1 − 2 CM 702 9 Scattering and Decay of Particles

′ ′ ′ p = p1 = p2, and the equality of the total energies before and after the collision implies that−

′ ′ E1 + E2 = E1 + E2 = ECM. (9.300) In order to be more explicit we have to distinguish the case of nonrelativistic and relativistic particles. It is useful to treat the relativistic case ﬁrst and obtain the nonrelativistic result by going to the limit of small velocities. In the relativistic case with initial momenta p = p = p , ﬁnal momenta p′ = p′ = p′ , and CM 1 − 2 CM 1 − 2 energies

2 2 ′ ′ 2 ′ 2 E1,2 = pCM + M1,2, E1,2 = pCM + M1,2 , (9.301) q q the δ-function for the total energy reads, in the center-of-mass frame, with zero total momentum

δ p′ 2 + M ′ 2 + p′ 2 + M ′ 2 E . (9.302) CM 1 CM 2 − CM q q This is zero unless p′ makes the argument zero, which happens at 1 p′ = I (M, M ′ , M ′ ) , (9.303) CM 2M 1 2 where

2 2 2 2 I(M, M1, M2) M (M1 + M2) M (M1 M2) (9.304) ≡ q − q − − and E = M √s. (9.305) CM ≡ The energy ECM is equal to the total mass, since the momentum in the center-of-mass frame is zero. Another notation for M is √s, where s is the so-called Mandelstam variable. The individual energies of the two particles are 1 E = p2 M 2 = M 2 M 2 M 2 , 1,2 CM − 1,2 2M ± 1 − 2 q 1 h i E′ = p′ 2 M ′2 = M 2 M ′2 M ′2 . (9.306) 1,2 CM − 1,2 2M ± 1 − 2 q h i To evaluate the momentum integral over the δ-function (9.302), we use the formula δ (f(p′) f(p′ )) = [f ′(p′)]−1 δ(p′ p′ ) (9.307) − CM − CM to write

δ p′2 + M ′ 2 + p′2 + M ′ 2 E 1 2 − CM q q ′ ′ ′ −1 ′ ′ = [(∂/∂p )(E1 + E2)] δ(p pCM) ′ ′ − ′ ′ E1E2 ′ ′ E1E2 ′ ′ = ′ ′ ′ δ(p pCM)= ′ δ(p pCM). (9.308) pCM (E1 + E2) − pCMECM − 9.3 Two-Particle Scattering 703

Then we obtain from (9.298) the total rate of scattered relativistic particles

3 ′ ′ ′ dP (t) V pCME1E2 2 = 2 dΩ t fi . (9.309) dt (2π) ECM Z | | Since the t-matrix elements depend, in general, on more than just the initial and ′ ﬁnal total momenta p and p , they have been denoted generically by t fi. The initial current density is given by the relative velocity between two incoming particles in the center-of-mass frame: p p 1 p E 1 j = 1 2 = CM CM . (9.310) E1 − E2 V E1E2 V Dividing dP (t)/dt by j as demanded by (9.299) and omitting the integration over the solid angle in (9.309), we obtain the diﬀerential cross section in the center of mass: ′ dσ 1 pCM 1 ′ ′ 4 2 = 2 2 E1E2E1E2V t fi . (9.311) dΩCM (2π) pCM ECM | | It is useful to realize that the combination of incoming currents and energies j × 2V E1E2 is a Lorentz-invariant quantity coinciding with I(M, M1, M2) of Eq. (9.304). This follows immediately by inserting (9.303) into (9.310): 1 j = I(M, M1, M2). (9.312) 2V E1E2 Thus we can write down an explicit formula for the cross section (9.299) in any frame of reference: 3 ′ 3 ′ 3 ′ 2V E1E2 n d p1 d p2 d pn (4) ′ 2 σ = V 3 3 3 δ (p p) t fi . (9.313) I(M, M1, M2) Z (2π) Z (2π) ··· Z (2π) − | | 9.3.2 Laboratory Scattering Cross Section In most experiments, the directly measured quantity is the laboratory cross section where an initial particle of momenta p = p and energy | 1| L E (E ) = p2 + M 2 (9.314) L ≡ 1 Lab L 1 q hits the particle 2 at rest. Then the incoming current is p v j = L = L , (9.315) ELV V rather than (9.310). The cross section of two particles going into two particles in the laboratory is from (9.298) and (9.299):

3 ′ 3 ′ d p1V d p2V 4 (4) ′ 2 V σ = 3 3 (2π) δ (p p) t fi . (9.316) Z (2π) (2π) − | | j 704 9 Scattering and Decay of Particles

The δ-function for the total energy depends on the final laboratory momenta p1L p , p p as follows: ≡ | 1L| 2L ≡| 2L| δ p′ 2 + M ′ 2 + p′ 2 + M ′ 2 E M . (9.317) 1L 1 2L 2 − L − 2 q q ′ ′ In contrast to the center-of-mass frame, the final momenta p1L and p2L are no longer equal, which complicates the further evaluation. We want to express them as functions of the initial energy and the scattering angle θL. For this, some kinematic considerations are necessary. The total mass introduced in (9.305) is given by the Poincaré-invariant quantity

M 2 = s (p + p )2 = M 2 + M 2 +2M E . (9.318) ≡ 1 2 1 2 2 L Solving this we obtain the invariant expressions

E = M 2 M 2 M 2 M 2 /2M (9.319) L − 1 − 1 − 2 2 pL = I (M, M1, M2) /2M2. (9.320) Observe that by inserting the last equation into the frame-independent expression (9.312), we ﬁnd the correct current (9.315) in the laboratory frame. The relation between the laboratory and center-of-mass energies is

2 M2EL + M1 E1CM = , (9.321) 2 2 M1 + M2 +2ELM2 q 2 M2EL + M2 E2CM = . (9.322) 2 2 M1 + M2 +2ELM2 q The relation between the laboratory and the center-of-mass momenta is

M1 pCM = pL . (9.323) 2 2 M1 + M2 +2ELM2 q The relation between the scattering angles in the two frames of reference is more involved. To ﬁnd it, we consider a Lorentz invariant quantity that is formed from the momentum transfer, the so-called Mandelstam variable t:

t (p p′ )2 = M 2 + M ′ 2 + 2(p p′ E E′ ). (9.324) ≡ 1 − 1 1 1 1 1 − 1 1 ′ ′ In either frame, the scattering angles appear in the scalar product p1p1 = p1p1 cos θ. A calculation in the center-of-mass frame leads to the formula

1 2 ′ 2 ′ cos θCM = ′ (t M1 M1 +2E1CME1CM) 2pCMpCM − − 1 = ′ ′ (9.325) I(M, M1, M2)I(M, M1, M2) [M 4 + M 2(2t M 2 M 2 M ′ 2 M ′ 2)+(M 2 M 2)(M ′ 2 M ′ 2)]. × − 1 − 2 − 1 − 2 1 − 2 1 − 2 9.3 Two-Particle Scattering 705

In the laboratory frame, we take advantage of the fact that p2 = (M2, 0) and calculate t using energy-momentum conservation as

t =(p p′ )2 =(p′ p )2 = M ′ 2 + M 2 2M E′ . (9.326) 1 − 1 2 − 2 2 2 − 2 2L It is convenient to introduce another Lorentz-invariant Mandelstam variable u as

u (p p′ )2 =(p′ p )2 = M 2 M ′ 2 2M E′ . (9.327) ≡ 1 − 2 1 − 2 2 − 1 − 2 1L The three Mandelstam variables s,t,u have the sum

2 2 ′ 2 ′ s + t + u = M1 + M2 + M1 + M2. (9.328) This implies that all different kinematical configurations of a two-body scattering process can be represented in an invariant way by a point in an equilateral triangle 2 2 ′ 2 ′ 2 of height M1 + M2 + M1 + M2 , the so-called Mandelstam triangle. The heights over the three sides represent the quantities s,t,u. Similar considerations can be made for the process of a particle decaying into three particles. Then a Poincaré-invariant way of picturing the distribution of the final products in a Mandelstam triangle is known as Dalitz plot. The laboratory energy of all particles can be calculated from (9.319) and

1 2 ′ 2 E2L = (M2 + M2 t), (9.329) M2 − ′ 1 2 ′ 2 E1L = (M2 + M1 u). (9.330) M2 − By evaluating the invariant (9.324) in the laboratory frame, we ﬁnd the scattering angle in this frame:

4 2 2 2 ′ 2 2 2 ′ 2 (M M1 M2 )(M2 + M1 u)+2M2 (t M1 M1 ) cos θL = − − − ′ ′ − − . (9.331) I(M, M1, M2)I(M, M1, M2) The argument of the total energy δ-function (9.317) requires the knowledge of ′ ′ the momenta p1,p2 of the final particles as a function of the initial energy EL and the scattering angle θL. These can now be calculated as follows. From (9.318) we find the invariant M as a function of EL. This is inserted into (9.331) to obtain the invariant t as a function. Together with (9.328), this fixes the invariant u. Inserted ′ into Eqs. (9.330), we find the final energy E1L, and from energy conservation also ′ E2L. These determine the final momenta. It is useful to generalize the definition of the scattering amplitude f that was introduced in (9.249) for the scattering of a nonrelativistic particle on an external potential. The generalization holds also in present case of two relativistic particles scattering on each other in the center-of-mass frame. . As before, we define f so that its absolute square gives the differential cross section: dσ = f 2. (9.332) dΩCM | | 706 9 Scattering and Decay of Particles

Then we see from (9.311) that f should be identiﬁed with

′ 1 pCM 1 4 ′ ′ f = V E1E2E1E2 t fi. (9.333) −2π sp E CM CM q

In the case of elastic scattering on a very heavy particle of mass M2 (choosing the second perticle to be the heavier one), this reduces back to the original deﬁnition. In a more general scattering process in which two particles collide and produce a ﬁnal state with n particles, the production cross section is given by

n 3 V d piV 4 (4) ′ 2 σ fi = (2π) δ (p p) t fi , (9.334) j " (2π) # − | | iY=1 Z

2 with the prefactor V/j = V E1E2/pCMECM, as given by (9.310). Scattering processes in a center-of-mass frame with zero total momentum can be observed directly in colliding beam experiments. There one has two beams impinging almost with opposite momenta (see Fig. 9.3). If Vˆint is the volume of the intersecting

Figure 9.3 Geometry of particle beams in a collider. The angle of intersection α is greatly exaggerated. The intersection deﬁnes the interaction volume Vˆint.

zone, the so-called interaction volume, and n1, n2 are the particle densities in the two beams (particles per unit volume), the so-called luminosity of the colliding beam is obtained from the relative velocity vrel by forming the product

I(M, M1, M2) L = vrelVˆintn1n2 = Vˆintn1n2. (9.335) 2E1E2

The rate of particle scattering is obtained by multiplying this with the diﬀerential cross section dσ/dΩCM:

dN dσ = L . (9.336) dtdΩCM dΩCM

Note that the luminosity increases linearly if the beams are focused upon a smaller 2 area (since n1n2 increases like 1/area while Vˆint increases like area). 9.4 Decay 707

9.4 Decay

The decay rate of an unstable particle into two particles is directly given by Eq. (9.298):

3 ′ 3 ′ dP 2 d p1 d p2 4 (4) ′ 2 = V 3 3 (2π) δ (p p) t fi . (9.337) dt Z (2π) Z (2π) − | | Evaluating this in the center-of-mass frame of the decaying particle yields

3 ′ ′ ′ d V pCME1E2 2 Γ= P (t)= 2 dΩ t fi , (9.338) dt (2π) M Z | | where M is the mass of the initial particle. If the particle has spin s and decays into ′ ′ ﬁnal spins s1,s2, an unpolarized initial state and all possible ﬁnal spin orientations ′ ′ ′ ′ ′ ′ m1 = s1,...,s1 and m2 = s2,...,s2 lead to an isotropic distribution, and we can simplify− (9.338) to −

3 ′ ′ ′ d V pCME1E2 4π 2 Γ= P (t)= 2 t fi . (9.339) dt (2π) M 2s +1 m′ ,m′ | | X1 2

9.5 Optical Theorem

The unitarity of the S-matrix (9.44),

SˆSˆ† = Sˆ†Sˆ =1, (9.340) implies an important theorem for the imaginary part of the diagonal elements of the T -matrix. Writing

S = δ i(2π)4δ(4) (p p ) t , (9.341) fi fi − f − i fi we see that

i t t† = t† (2π)4δ(4) (p p ) t , (9.342) − ii − ii − if f − i fi Xf where the sum runs over all possible many particle states. According to Eq. (9.333), the left-hand side is related to the imaginary part of the forward scattering amplitude fii as follows

n 3 2πECM d piV 4 (4) 2 2 Im fii = 3 (2π) δ (pf p1) t fi . (9.343) V E1E2 − n " (2π) # − | | X iY=1 Z Now, the right-hand side, if divided by the relative current density j = pCMECM/E1E2V , is equal to 1/V times the sum over all cross sections σ fi, i.e., 708 9 Scattering and Decay of Particles it is equal to j/V times the total cross section (j/V )σ [see (9.316)]. For the − − tot forward scattering amplitude fii this implies the simple relation 2π fii = σtot. (9.344) pCM − This relation is referred to as the optical theorem (in reference to a theorem re- lating the total absorption of light to the imaginary part of the forward-scattering amplitude).

9.6 Initial- and Final-State Interactions

Some inelastic scattering and decay processes are caused by a weak interaction Hamiltonian Hˆw, which can be treated in Born approximation. This can happen while initial- and ﬁnal states are still governed by strong interactions with a Hamil- tonian Hˆs. Suppose the total Hamiltonian contains these interactions additively, i.e., Hˆ = Hˆ0 + Vˆs + Vˆw. (9.345)

In speciﬁc applications, Hˆw may be the weak interaction of beta decay, say, and Hˆs a combination of electromagnetic and strong interactions. Alternatively, Hˆw may be an electromagnetic interaction, and Hˆs the strong interaction. For time-independent interactions, the Møller operators are given by (9.32) and read explicitly

ˆ (±) i(Hˆ0+Vˆ w+Vˆs)t −iHˆ0t Ω = UˆI (0, ) = lim e e . (9.346) ∓∞ t→∓∞ The right-hand sides can obviously be decomposed into a product

Uˆ (0, )= Uˆ (0, ) Uˆ (0, ) , (9.347) I ∓∞ Iw ∓∞ I,s ∓∞ where

i(Hˆ0+Vˆs+Vˆw)t −i(Hˆ0+Hˆs)t UˆIw (0, ) = lim e e , (9.348) ∓∞ t→∓∞ i(Hˆ0+Vˆs)t −iHˆ0t UˆIs (0, ) = lim e e . (9.349) ∓∞ t→∓∞ Thus we can write the Møller operators as

(±) (±) (±) Ω = Ωw Ωs , (9.350) with Ω(±) = Uˆ (0, ) , Ω(±) = Uˆ (0, ) . (9.351) w Iw ∓∞ s Iw ∓∞ Using this, we can express the scattering amplitude as follows:

S = f Uˆ ( , 0)Sˆ Uˆ (0, ) i , (9.352) ﬁ h | Is ∞ w Is −∞ | i 9.7 Tests of Time-Reversal Violations 709 where, by analogy with (9.42),

Sˆ Uˆ ( , ) = Ω(−)†Ω(+). (9.353) w ≡ Iw ∞ −∞ w w For a simple description of the eﬀects of the initial- and the ﬁnal-state interactions it is convenient to introduce, in a slight variation of (9.114), the notation

i( ) Uˆ (0, ) i = Ω(+) i , (9.354) | ± i ≡ Is ∓∞ | i w | i where the symbols ( ) refer now to the activity of Hˆ in the time evolution. The ± w scattering matrix elements become

j Sˆ i = j( ) Sˆ i(+) . (9.355) h | | i h − | w| i

Under the assumption of a weak Vw, we replace Sˆw by its Born approximation and write ∞ Sˆw i dtVˆIw(t), (9.356) ≈ − Z−∞ where

ˆ ˆ ˆ ˆ ˆ Vˆ (t) ei(H0+Vs+Vw)tVˆ e−i(H0+Vs)t (9.357) Iw ≡ w is the interaction representation of the weak interaction in the presence of the initial- and ﬁnal-state interactions. The initial and ﬁnal states are eigenstates of the operator Hˆ0 +Vˆs [see (9.15)]. The integration over t in (9.356) can now be done in (9.356), and leads to a δ-function of energy conservation [recall the derivation of (9.130)], so that

j Sˆ i = j i 2πiδ(E′ E) j( ) Vˆ i(+) . (9.358) h | | i h | i − − h − | w| i For a local interaction which arises from an integral over an interaction density as in Eq. (9.288), we find that the initial- and final-state interactions are accounted for by the modified t-matrix

′(−) (+) tp′p p ˆ (0) p . (9.359) ≈h |Hint | i For the amplitude of multiparticle scattering (9.296), the corresponding expression is

t f( ) ˆ (0) i(+) . (9.360) ﬁ ≈h − |Hint | i 9.7 Tests of Time-Reversal Violations

If the Hamiltonian is invariant under time-reversal, the time translation operator U(t, t0) fulﬁlls the equation (7.123). This has the consequence that the scattering operator Sˆ of Eq. (9.42) satisﬁes

†Sˆ = Sˆ†. (9.361) T T 710 9 Scattering and Decay of Particles

To derive observable consequences from this, we ﬁrst use the equations (7.118) and (7.119) to deduce trivially the equality for any matrix element between an initial state i , obtained by applying one or more particle creation operators upon the vacuum,| i and a ﬁnal state f , obtained likewise: | i f Sˆ i = f 1 Sˆ i ∗. (9.362) h | | i h | A | i Invoking now time-reversal invariance by applying (9.361), we obtain f Sˆ i = f †Sˆ† i ∗ = f Sˆ† i ∗ = i Sˆ f . (9.363) h | | i h |T T | i h T | | T i h T | | T i Here iT denotes the time-reversed version of the state i , in which all creation operators| i are replaced by the right-hand sides of (7.113), (7.322| i ), etc. Thus momenta and spin directions are reversed, and there is a phase factor ηT for each particle.

9.7.1 Strong and Electromagnetic Interactions Strong and electromagnetic interactions are time-reversal invariant. The matrix elements of the scattering operator Sˆ, the scattering amplitudes, satisfy the identity p ,s ; p ,s Sˆ p ,s ; p ,s = η p , s ; p , s Sˆ p , s ; p ,s , h c 3c d 3d| | a 3a b 3bi h− a − 3a − b − 3b| | − c − 3c d 3di (9.364) with some phase factor η of unit norm. When calculating the unpolarized cross section, we have to sum over the ﬁnal states, and average over the initial states. In ′ the center-of-mass frame, the cross section carries a phase-space factor pCM /pCM , the ratio of ﬁnal and initial center-of-mass momenta. It has to be removed to compare the squared amplitudes. Hence we obtain dσ(ab cd) → pcd 2 (2s + 1)(2s + 1) dΩ = CM c d . (9.365) dσ(cd ab) pab (2s + 1)(2s + 1) → CM ! a b dΩ This is called the principle of detailed balance. It is important for the thermal equilibrium of chemical reactions. This equation was used in 1951 to determine the spin of the pion from a reaction in which two protons collide and produce a pion and a deuteron9: p + p ↽⇀ d + π+. (9.366) The principle of detailed balance says that dσ( ) → 3 q2(2s + 1) dΩ = π π . (9.367) dσ( ) 4 p2 ← p dΩ

This relation was satisﬁed with sπ = 0. 9R.E. Marshak, Phys. Rev. 82, 313 (1951); W.B. Cheston, Phys. Rev. 83, 1118 (1951); R. Durbin, H. Loar, and J. Steinberger, Phys. Rev. 83, 646 (1951); D.L. Clark, A. Roberts, and R. Wilson, Phys. Rev. 83, 649 (1951). 9.7 Tests of Time-Reversal Violations 711

9.7.2 Selection Rules in Weak Interactions Consider a one-particle state i prepared by strong and electromagnetic interactions, which decays into a several-particle| i state b via weak interactions. Let ˆ be an observable which changes sign under time reversal:| i O

ˆ = ˆ. (9.368) OT −O Consider the expectation value of ˆ a long time after the decay. It is given by O ˆ = i Uˆ( , ) ˆ Uˆ( , ) i = i Sˆ† ˆSˆ i . (9.369) hOi h | −∞ ∞ O∧ ∞ −∞ | i h | O | i Using the antiunitarity of the time-reversal operator as expressed by Eqs. (7.118) and (7.119), the right-hand side can be rewritten as

i Sˆ† ˆSˆ i = i 1 Sˆ† ˆSˆ i ∗ = i † Sˆ† ˆSˆ i ∗. (9.370) h | O | i h | A O | i h |T T O | i Time-reversal invariance (9.361) brings this to

i †Sˆ ˆ Sˆ† i ∗ = i Sˆ ˆ Sˆ† i ∗. (9.371) h |T OT T| i h T | OT | T i Since ˆ was assumed to be an observable, the expectation is real, and we can drop O the complex conjugation at the end. This leaves us with the relation

ˆ = i Sˆ† ˆSˆ i = i Sˆ ˆ Sˆ† i . (9.372) hOi h | O | i h T | OT | T i If the decay is very slow, the scattering operator can be replaced by its Born approximation ∞ Sˆ SˆB i dtVˆIw(t), (9.373) ≈ ≡ Z−∞ which is a Hermitian operator, up to a factor i (the direct term in (9.341) does not contribute to a decay process). But then (9.372) becomes

ˆ = i Sˆ ˆSˆ i = i Sˆ ˆ Sˆ i . (9.374) hOi h | BO B| i h T | BOT B| T i If the initial particle is spinless, then

i = η i , η =1. (9.375) | T i T | i | T | Together with (9.368), this implies

ˆ =0. (9.376) hOi Thus, in a weak decay process, the expectation value of an observable that changes sign under time-reversal, is zero. Note that this result is independent of the phase ηT of the decaying particle. Without the weakness assumption, the change to the Hermitian conjugates of the scattering operators in (9.372) would have prevented us from any conclusion. 712 9 Scattering and Decay of Particles

Historically, the invariance under time reversal of weak interactions was appar- ently assured by the weak decays

K+ π0 +¯µ+ + ν, K− π0 + µ− +¯ν. (9.377) −−−→ −−−→ The outcoming muon was initially found to have no average polarization orthogonal to the production plane formed by the momenta p and p , i.e., the operator ˆ = µ π O

(pˆ pˆ ) had no measurable expectation value. Since this operator is odd under µ · µ × π time reversal, the vanishing of ˆ seemed to conﬁrm time-reversal invariance.10 hOi However, as it turned out later, the breakdown of time-reversal invariance in weak interactions is much too small to be detected by a polarization experiment. There exists a much more sensitive experiment which shows the violation dramatically: the K0-K¯ 0 meson system. This will be discussed later.

9.7.3 Phase of Weak Amplitudes from Time-Reversal Invariance If a particle decays weakly into two final particles, time-reversal invariance fixes the phase of the decay amplitude. Let us assume that the final-state interaction is purely elastic, i.e., that there are no other decay channels caused by the final-state interactions, then a weak decay can be treated in lowest-order perturbation theory. It has the amplitude t = f( ) Hˆ i(+) , (9.378) ba h − | w| i where the labels ( ) indicate the states defined in Section 9.6, containing the full effect of the initial-± and final-state interactions. Under the assumption of time- reversal invariance of the weak interactions, we find

f( ) Hˆ i(+) = i ( ) Hˆ f (+) = f (+) Hˆ i ( ) ∗. (9.379) h − | w| i h T − | w| T i h T | w| T − i

If we choose the phase factors ηT of initial and ﬁnal particles to be unity [see (4.226)], we obtain f( ) Hˆ i(+) = f (+) Hˆ i ( ) ∗. (9.380) h − | w| i h T | w| T − i The initial state is a stationary single-particle state if the weak interactions are ignored. Hence i(+) i( ) . (9.381) | i≈| − i We now rotate the matrix element on the right-hand side in space by 180 degrees, we reverse, in the center of mass frame, momentum and spin directions, and bring the time-reversed state back to the original form. Thus we obtain

b( ) Hˆ a(+) = b(+) Hˆ a( ) ∗. (9.382) h − | w| i h | w| − i 10This possibility of testing time-reversal invariance of weak interactions was pointed out by J.J. Sakurai, Phys. Rev. 109, 980 (1958). See also M. Gell-Mann and A.H. Rosenfeld, Ann. Rev. Nucl. Sci. 7, 407 (1957), Appendix A. Appendix 9A Green Function in Arbitrary Dimensions 713

The amplitudes with + and interchanged diﬀer by a pure phase factor, that is precisely the scattering amplitude− for the two-particle scattering in their ﬁnal state:

b( ) Hˆ a(+) = e2iδ b(+) Hˆ a ( ) . (9.383) h − | w| i h | w| T − i The pure-phase nature is a consequence of the elasticity assumption of the two- particle scattering process. Together with (9.382), this ﬁxes the phase:

b( ) Hˆ a(+) = const . eiδ. (9.384) h ∓ | w| i × This equation has many applications in phenomenological analyses [5].

Appendix 9A Green Function in Arbitrary Dimensions

In D dimensions, the Green function (9.138) in the scattered wave (9.136) of the Lippmann- Schwinger equation has the following form:

′ dDp eip(x−x )/h¯ x Gˆ (E) x′ = 2Mi¯h . (9A.1) h | 0 | i − (2π¯h)D p2 p2 Z − E The momentum integral has been calculated in Section 1.12 (also 9.1) of Ref. [2], for E > 0 yielding

(1) ′ Mπ kD−2 H (kE x x ) x Gˆ (E) x′ = E D/2−1 | − | , (9A.2) h | 0 | i ¯h (2π)D/2 (k x x′ )D/2−1 E | − | (1) where kE pE/¯h = 2M(E + iη)/¯h, and H (z) is the Hankel function, whose leading large-z behavior is≡ p

(1) 2 i(z−πν/2−π/4) Hν e , (9A.3) ≈ rπz thus reproducing the previous result (9A.1) for D = 3. In D = 1 dimension, the expression (9A.2) reduces to

′ ′ M ikE |x−x | x Gˆ0(E) x = e , (9A.4) h | | i ¯hkE so that the Lippmann-Schwinger equation (9.134) in one dimension reads

∞ 1 M ′ x ψ(+) = eikx + i eikE |x−x | dx V (x′) x′ ψ(+) , (9A.5) h | k i L1/2 ¯h2k h | k i E Z−∞ where kE = 2M(E + iη) is equal to the wave number k of the incoming particles. As a simple application, consider the case of a δ-function potential p ¯h2 V (x)= δ(x), (9A.6) Ml where (9A.5) becomes 1 1 ∞ x ψ(+) = eikx + eik|x| 0 ψ(+) . (9A.7) h | k i L1/2 ikl h | k i Z−∞ Setting x = 0 on the left-hand side determines 1 ikl 0 ψ(+) = , (9A.8) h | k i L1/2 ikl 1 − 714 9 Scattering and Decay of Particles and Eq. (9A.7) yields the fully interacting wave function 1 1 x ψ(+) = eikx + eik|x| . (9A.9) h | k i L1/2 ikl 1 − The corresponding states ψ(−) are | k i 1 1 x ψ(−) = e−ikx + e−ik|x| . (9A.10) h | k i L1/2 ikl 1 − − In the negative-x regime, the second wave runs to the left and its prefactor is the reflection amplitude 1 r = . (9A.11) ikl 1 − At positive x, the two exponentials run both to the right and the common prefactor ikl t = (9A.12) ikl 1 − is the transmission amplitude. The unitarity relation for this process is r 2 + t 2 = 1. | | | | For an attractive δ-function, there is a bound-state pole at kB = 1/ l , corresponding to the energy | | ¯h2k2 ¯h2 g2M ε = B = = . (9A.13) B − 2M −2l2M − 2 It is easy to verify that x ψ(±) satisfies the Schrödinger equation h | k i ¯h2∂2 ¯h2 x + δ(x) E x ψ(±) =0, (9A.14) − 2M Ml − h | k i with the jump property of the solution at the position of the δ-function ¯h2∂ ¯h2 x [ 0 ψ(±) 0 ψ(±) ]= 0 ψ(±) . (9A.15) 2M h +| k i − h −| k i Ml h | k i Let us also calculate the matrix elements of the resolvent operator (9.57), i.e., the Green function i¯h x Gˆ(E) x′ = x x′ . (9A.16) h | | i h |E Hˆ + iη | i − It has the spectral representation ∞ ˆ ′ dk i¯h (±) (±) ′ x G(E) x = x ψk ψk x . (9A.17) h | | i 2π E Ek + iη h | ih | i Z−∞ − Performing the integral over k we find, from the residue at kE :

M ′ 1 ′ x Gˆ(E) x′ = eikE |x−x | + eikE (|x|+|x |) . (9A.18) h | | i ¯hk ik l 1 E E − This is related to the free Green function (9A.4) via Eq. (9.58). The Green function (9A.18) can also be obtained with the help of the Wronski method11 by combining the two independent solutions

f (x)= √L x ψ(+) , f (x)= f ∗(x) (9A.19) 1 h | kE i 2 1 11See Subsection 3.2.1 in the textbook [2]. Appendix 9B Partial Waves in Arbitrary Dimensions 715 of the homogeneous Schr¨odinger equation (Hˆ E)f(x) = 0 in the form − i¯h ′ 2Mi 1 ′ ′ ′ ′ x Gˆ(E) x = x x = [Θ(x x )f1(x)f2(x )+Θ(x x)f1(x )f2(x)], (9A.20) h | | i h |E Hˆ | i ¯h W (x, x) − − − where W (x, x) is the Wronski determinant W (x, x) f ′ (x)f (x) f (x)f ′ (x′). (9A.21) ≡ 1 2 − 1 2 ˆ 2 ′ Indeed, applying the operator E H = E +¯h∂x/2M to (9A.20), we obtain i¯hδ(x x ). − ikE x −ikE x − For the free system where f1(x) = e , f2(x) = e , and W (x, x)=2ikE, this formula reproduces the above result (9A.4):

′ 2Mi 1 M ikE |x−x | x Gˆ0(E) x = f1(x>)f2(x<)= e . (9A.22) h | | i ¯h W (x, x) ¯hkE In the presence of the δ-potential, where the Wronski determinant is calculated from the solutions (9A.19) as 2ik2 l W (x, x)= E , (9A.23) kE l + i the general formula (9A.20) yields

i¯h M ′ 1 ′ x Gˆ(E) x = x x′ = eikE |x−x | eikE (|x|+|x |) . (9A.24) h | | i h |E Hˆ | i ¯hkE − 1 ikEl − − From this, we ﬁnd the bilocal density of states

1 M ′ 1 ′ ′ ˆ ˆ ikE (x−x ) ikE (|x|+|x |) x ρˆ(E) x = x [G(E+iη) G(E iη)] x = 2 e + e +c.c. . h | | i 2π¯hh | − − | i 2π¯h k ikEl 1 E − (9A.25)

∞ 2 ∞ Integrating this over all energies in the continuum, i.e., by forming 0 dE =h ¯ −∞ kEdkE /M, we obtain ∞ R R 1 ′ dE x ρˆ(E) x′ = δ(x x′) Θ( l) e−(|x|+|x |)l. (9A.26) h | | i − − − l Z0 − According to Eq. (9A.17), the same quantity is equal to ∞ ∞ dk ∞ dk dE x ψ(+) δ(E ¯hk2/2M) ψ(+) x′ = x ψ(+) ψ(+) x′ . (9A.27) 2π h | k i − h k | i 2π h | k ih k | i Z0 Z−∞ Z−∞ Hence Eq. (9A.26), together with the contribution of the single bound state wave function x ψβ = (1/√ l)e−|x|l, yields h | i − ∞ dk x ψ(+) ψ(+) x′ + δ( l) x ψ ψ x′ = δ(x x′), (9A.28) 2π h | k ih k | i − h | β ih β| i − Z−∞ thus proving the completeness relation (9.62) of the states x ψ(+) and x ψ = (1/√ l)e−|x|l. h | k i h | β i − Appendix 9B Partial Waves in Arbitrary Dimensions

In D dimensions, the angular decomposition of the scattering amplitudes proceeds as follows. For ′ ′ equal incoming and outgoing energies Ep′ = Ep, where p = p = p, the T -matrix p T p in Eq. (9.129) has an expansion | | | | h | | i

D ∞ ′ (2π) ′ p T p = Tl(p) Ylm(pˆ )Ylm(pˆ), (9B.1) h | | i V m Xl=0 X 716 9 Scattering and Decay of Particles

where Ylm(pˆ) are the hyperspherical harmonics associated with the momentum direction pˆ. They generalize (4F.1), and their precise form is somewhat tedious to write down. Here we only recall that their magnetic quantum numbers form a D 2-dimensional vector of eigenvalues − (m1,m2,...,mD−2) running through the number

(2l + D 2)(l + D 3)! d = − − (9B.2) l l!(D 2)! − of diﬀerent values for each l. The orthogonality (4F.3) becomes now:

2 ∗ ′ ′ d pˆ Yl′m′ (pˆ)Ylm(pˆ)= δl lδm m, (9B.3) Z where dD−1pˆ denotes the integral over the surface of a unit sphere in D dimensions:

R 2πD/2 dD−1pˆ = S . (9B.4) D ≡ Γ(D/2) Z The spherical harmonics are matrix elements of eigenstates l m of angular momentum [recall (4.846)] with localized states pˆ on the unit sphere: | i | i pˆ l m Y m(pˆ). (9B.5) h | i≡ l The latter are orthogonal: pˆ′ pˆ = δ(D−1)(pˆ′ pˆ), (9B.6) h | i − where δ(D−1)(pˆ′ pˆ) is the δ-function on a unit sphere. The addition− theorem (23.12) for the spherical harmonics is replaced by

2l + D 2 1 (D/2−1) ′ ′ ∗ − Cl (pˆ pˆ)= Ylm(pˆ )Ylm(pˆ), (9B.7) D 2 SD · m − X (α) where the functions Cl (cos ϑ) are the hyperspherical Gegenbauer polynomials, the D-dimensional versions of the Legendre polynomials Pl(cos θ) in Eq. (4F.7). The Gegenbauer polynomials are related to Jacobi polynomials, which are deﬁned in terms of hypergeometric functions by12

1 Γ(l +1+ β) P (α,β)(z) F ( l,l +1+ α + β;1+ β;(1+ z)/2). (9B.8) l ≡ l! Γ(1 + β) 2,1 −

The relation is13 Γ(2ν + l)Γ(ν +1/2) C(ν)(z)= P (ν−1/2,ν−1/2)(z). (9B.9) l Γ(2ν)Γ(ν + l +1/2) l

Thus we can write 1 Γ(l +2ν) C(ν)(z)= F ( l,l +2ν;1/2+ ν;(1+ z)/2). (9B.10) l l! Γ(2ν) 2,1 −

For D = 2 and 3, one has14 1 1 lim C(ν)(cos ϑ)= cos lϑ, (9B.11) ν→0 ν l 2l 12M. Abramowitz and I. Stegun, op. cit., Formula 15.4.6. 13Ibid., Formula 15.4.5. 14I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 8.934.4. Appendix 9B Partial Waves in Arbitrary Dimensions 717

(1/2) (0,0) Cl (cos ϑ)= Pl (cos ϑ)= Pl(cos ϑ). (9B.12) The partial-wave expansion (9B.1) becomes, in D dimensions:

∞ ′ ˆ 2l + D 2 1 (D/2−1) ′ p T p = Tl(p) − Cl (pˆ pˆ). (9B.13) h | | i D 2 SD · Xl=0 − ′ A similar expansion for the S-matrix (9.130) requires expanding the scalar product p p = δp′,p into partial waves. In a large volume V , we replace this by (2π)Dδ(D)(p′ p)/V . Theh |δ-functioni is decomposed into a directional and a radial part −

D D (2π) (D) ′ (D−1) ′ (2π) 1 ′ δp′ p = δ (p p)= δ (pˆ pˆ) δ(p p). (9B.14) , V − − V pD−1 −

The directional δ-function on the right-hand side has a partial-wave expansion which is the completeness relation [compare (9.161] for the states lm , | i ∞ l m l m =1, (9B.15) m | ih | Xl=0 X evaluated between the localized states pˆ on the unit sphere. Using the matrix elements (9B.5) and the orthogonality relation (9B.6), we| i ﬁnd from (9B.15):

∞ l (D−1) ′ ′ ∗ δ (pˆ pˆ)= Ylm(pˆ )Ylm(pˆ). (9B.16) − m Xl=0 X The radial part of (9B.14) can be rewritten as

D D D (2π) 1 ′ (2π) 1 dE ′ (2π) 1 ′ D−1 δ(p p)= D−1 δ(E E)= δ(E E), (9B.17) V p − V p dp − V ρE − where pD−1 ρ . (9B.18) E ≡ dE/dp It allows us to rewrite the sum over all momentum states as follows: dDpV V D−1p = D = D d ˆ dE ρE. (9B.19) p (2π) (2π) X Z Z Z Thus we ﬁnd the expansion

D ∞ ′ (2π) ′ 1 ′ p p = δp′,p = Ylm(pˆ )Ylm(pˆ) δ(E E). (9B.20) h | i V m ρE − Xl=0 X In a partial-wave expansion for the S-matrix (9.130), the overall δ-function for the energy conservation is conventionally factored out and one writes:

D ∞ ′ (2π) 1 ′ ′ p Sˆ p = δ(E E) Sl(p) Ylm(pˆ )Ylm(pˆ). (9B.21) h | | i V ρE − m Xl=0 X

The relation between the partial-wave scattering amplitudes Sl(p) and those of the T -matrix is again given by (9.168). The only change in D dimensions appears in ρE which acquires the form (9B.18). 718 9 Scattering and Decay of Particles

In order to show this we use the expansion formula corresponding to (9.173),

∞ ih cos∆ϑ l + D/2 1 1 (D/2−1) e = al(h) − Cl (cos ∆ϑ), (9B.22) D/2 1 SD Xl=0 − where

a (h) (2π)D/2h1−D/2ilJ (h). (9B.23) l ≡ l+D/2−1 (ν) The expansion (9B.22) follows from the completeness of the polynomials Cl (cos ϑ) at ﬁxed ν, using the integration formulas15

π/2 21−ν Γ(2ν + l) dϑ sinν ϑeh cos ϑC(ν)(cos ϑ) = π h−ν I (h), (9B.24) l l!Γ(ν) ν+l Z−π/2 π/2 1−2ν ν (ν) (ν) 2 Γ(2ν + l) dϑ sin ϑ C (cos ϑ)C ′ (cos ϑ) = π δ ′ . (9B.25) l l l!(l + ν)Γ(ν)2 ll Z−π/2 We now use (9B.22) to expand the incoming free-particle wave

∞ l 1 ipx x p = e = ψ (pr) Y m(xˆ)Y m(pˆ) h | i √ l l l V l=0 m ∞ X X l + D/2 1 1 (D/2−1) = ψl(pr) − Cl (xˆ pˆ), (9B.26) D/2 1 SD · Xl=0 − where r x , and ≡ | | 1 ψl(pr)= al(pr). (9B.27) √V Far from the scattering region, the spherical Bessel functions have the asymptotic behavior

2 Jν (z) sin(z νπ/2+ π/4), (9B.28) → rπz − implying for the partial waves [compare (9.178)]

1 l (D−1)/2 sin pr π[l/2 (D 3)/4] ψl(pr)= i 2 (2π) { − − − }. (9B.29) √V (pr)(D−1)/2 We shall now calculate the asymptotic form of the outgoing waves. The second term of the Lippmann-Schwinger equation (9.131) yields the scattered wave, which reads in conﬁguration space

dDp′V 1 x p sc = x p′ p′ T p . (9B.30) h | i (2π)D h | iE E′ + iη h | | i Z − Inserting here the partial-wave decompositions (9B.1) and (9B.26), we ﬁnd

dDp′V 1 x p′ p′ T p (2π)D h | iE E′ + iη h | | i Z ∞ − 1 ∞ a (p′r)T (p) x′ p ′ ′D−1 l l = Ylm(ˆ )Ylm(ˆ) dp p ′ , (9B.31) √V m 0 E E + iη Xl=0 X Z − 15I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formulas 7.321 and 7.313. Appendix 9B Partial Waves in Arbitrary Dimensions 719 so that the total partial waves, including the scattered waves are ∞ ψ (p′r)T (p) ψtot(pr)= ψ (pr)+ dp′ p′D−1 l l . (9B.32) l l E E′ + iη Z0 − In terms of the spherical Bessel functions of Eq. (9.174), this reads 1 1 ψtot(pr) = 2(2π)(D−1)/2 il (9B.33) l √V (pr)(D−3)/2 ∞ T (p′) j (pr)+ p(D−3)/2 dp′ p′(D+1)/2 j (p′r) l . × l+(D−3)/2 l+(D−3)/2 E E′ + iη Z0 − For large r, we use the asymptotic formula (9.177) to rewrite the integral as [compare (9.182)]

′ ′ ∞ T (p′) ei{p r−π[l/2−(D−3)/4]} e−i{p r−π[l/2−(D−3)/4]} dp′ p′(D+1)/2 l − . (9B.34) E E′ + iη 2ip′r Z0 − The integral can now be approximated as in (9.184): 1 T (p′) 1 dp′ p′(D+1)/2 l ei{pr−π[l/2−(D−3)/4]} e−i{pr−π[l/2−(D−3)/4]} . (9B.35) dE/dp p p′ + iη 2ip′r − Z − Together with the prefactor p(D−3)/2 in (9B.33), this becomes

2πpD−1 eipr e−iπl/2eiπ(D−3)/4 . (9B.36) − dE/dp 2pr In this way we ﬁnd for the wave function (9B.33):

r→∞ 1 2(2π)(D−2)/2 ψtot(pr) (9B.37) l −−−→ √V 2i(pr)(D−1)/2 ei{pr−π[l/2−(D−3)/4]} e−i{pr−π[l/2−(D−3)/4]} 2πiρ T (p)eiπ(D−3)/4eipr . × − − E l h i Expressing Tl(p) in terms of Sl(p) using Eq. (9.168), we have

r→∞ (D−1)/2 tot 1 2(2π) i(pr−π(D−3)/4) l −ip(pr−π(D−3)/4) ψ (pr) Sl(p)e ( 1) e . (9B.38) l −−−→ √V 2i(pr)(D−1)/2 − − h i Inserting the unitary form (9.171) for elastic scattering, this becomes

r→∞ 1 2(2π)(D−1)/2 sin pr + δ (p) π[l/2 (D 3)/4] ψtot(pr) ileiδl(p) { l − − − }. (9B.39) l −−−→ √V p(D−1)/2 r(D−1)/2

This shows again that δl(p) is the phase shift of the scattered waves with respect to the incoming partial waves (9B.29). If we insert (9B.33) on the right-hand side of the partial wave expansion (9B.26), the ﬁrst term gives again the incoming plane wave. The Tl(p)-terms add to this a scattered wave, determined by Eq. (9B.33). The result is a total wave function for large r similar to (9.143):

i(pr−π(D−3)/4) (+) 1 ipx e x p e + fp′p , (9B.40) h | i −−−→r→∞ √ r(D−1)/2 V where fp′p has the partial wave expansion generalizing (9.250):

∞ ∞ l + D/2 1 1 (D/2−1) ′ ′ ′ fp p = fl(p) Ylm(pˆ )Ylm(pˆ)= fl(p) − Cl (pˆ pˆ), (9B.41) m D/2 1 SD · Xl=0 X Xl=0 − 720 9 Scattering and Decay of Particles with (2π)(D+1)/2 (2π)(D−1)/2 2(2π)(D−1)/2 f (p)= ρ T (p)= [S (p) 1] = eiδl(p) sin δ (p). (9B.42) l − p(D−1)/2 E l ip(D−1)/2 l − p(D−1)/2 l

Taking the absolute square of this, and integrating over all directions of the ﬁnal momentum p′, we ﬁnd the partial wave decomposition of the total cross section:

∞ ′ 2 ∗ ′ ∗ ′ ∗ σ = dpˆ fp′p = f (p)f (p) Y m(pˆ )Y m(pˆ)Y ′ (pˆ )Y ′ (pˆ) | | l l l l lm lm Z l=0 lm l′m′ ∞ X X X ∞ 2 l+D/2 1 1 (D/2−1) 2 l+D/2 1 1 (l+D 3)! = fl(p) − Cl (1) = fl(p) − − , | | D/2 1 SD | | D/2 1 SD l!(D 2)! Xl=0 − Xl=0 − − (9B.43) where we have used the special value of the Gegenbauer function16

l +2ν 1 C(ν)(1) = − . (9B.44) l l Inserting (9B.42), the cross section decomposes as

∞ D−1 ∞ 4(2π) l+D/2 1 1 (l+D 3)! 2 σ = σl = D−1 − − sin δl(p). (9B.45) p D/2 1 SD l!(D 2)! Xl=0 Xl=0 − − In three dimensions, this reduces to (9.253).

Appendix 9C Spherical Square-Well Potential in D Dimensions

For a square-well potential (9.268) in D dimensions, the radial Schr¨odinger equation (9.269) becomes d2 D 1 d l(l + D 2) + − + k2 − ψ (r)=0, (9C.1) dr2 r dr E − r2 l where K 2M(E + V )/¯h. (9C.2) ≡ 0 There is only one solution which is regularp at the origin: A ψ (r)= J (Kr), (9C.3) l (Kr)D/2−1 D/2−1+l with an arbitrary normalization factor A. For positive energy E, the solution in the outer region r a receives an admixture of the ≥ associated spherical Bessel function nl(kr): B ψ (r)= cos δ (k)J (Kr) + sin δ (k)N (Kr) , (9C.4) l (kr)D/2−1 l D/2−1+l l D/2−1+l where k √2ME/¯h, (9C.5) ≡ 16M. Abramowitz and I. Stegun, op. cit., Formula 22.2.3. Appendix 9C Spherical Square-Well Potential in D Dimensions 721

and B, δl(k) must be determined from the boundary condition at r = a. The associated Bessel functions ND/2−1+l(z) have the asymptotic behavior orthogonal to (9B.28): 2 Nν (z) cos(z νπ/2+ π/4). (9C.6) →−rπz − The continuity of the wave function ﬁxes the ratio B/A. The continuity of the logarithmic derivative is independent of this ratio and ﬁxes the phase shifts by the equation 1−D/2 ∂r0 [(Kr0) JD/2−1+l(Kr0)] 1−D/2 (9C.7) (Kr) JD/2−1+l(Kr0)) 1−D/2 1−D/2 ∂r0 [cos δl(k)(Kr0) J (Kr0) sin δl(k)(Kr0) N (Kr0)] = D/2−1+l − D/2−1+l . cos δ (k)(Kr )1−D/2J (Kr ) sin δ (k)(Kr )1−D/2N (Kr ) l 0 D/2−1+l 0 − l 0 D/2−1+l 0 This determines the phase shifts as follows:

kJ (kr0)J (K0r0) K0J (K0r0)J (kr0) tan δ (k)= D/2+l D/2−1+l − D/2 D/2−1 . (9C.8) l kY (kr )J (K r ) K J (K r )Y (kr ) D/2+l 0 D/2−1+l 0 0 − 0 D/2+l 0 0 D/2−1+l 0 For the s-wave in D = 3 dimensions, this reduces to k cos kr sin K r K cos K r sin kr tan δ (k)= 0 0 0 − 0 0 0 0 , (9C.9) 0 k sin kr sin K r K cos K r cos kr 0 0 0 − 0 0 0 0 which is, of course, the same as (9.276). In D = 2 dimensions, we obtain kJ (kr )J (K r ) K J (K r )J (kr ) tan δ (k)= 1 0 0 0 0 − 0 1 0 0 0 0 , (9C.10) 0 kY (kr )J (K r ) K J (K r )Y (kr ) 1 0 0 0 0 − 0 1 0 0 0 0 which is, for small k, equal to

π K0r0J1(K0r0) δ0 γ , (9C.11) ≈ 2 J0(K0r0)+ K0r0J1(K0r0)log(kr0e /2) where γ 0.577216 is the Euler-Mascheroni number. ≈ The phase shifts for a repulsive potential are obtained by setting K0 = iκ0 i 2M V0 /¯h and by replacing ≡ | | p 2 J (K r ) iν I (κ r ), Y (K r ) iν iI (κ r ) ( 1)ν K (κ r ) . (9C.12) ν 0 0 → ν 0 0 ν 0 0 → ν 0 0 − π − ν 0 0 In the limit of an inﬁnitely high repulsive potential, the modiﬁed Bessel functions Iν (z) diverge z −z like e /√2πz, while Kν (κ0r0) go to zero exponentially fast like π/2ze . Hence

JD/2−1(kr0) p tan δ0(k) = . (9C.13) V0→∞ YD/2−1(kr0) In D = 3 dimensions, the right-hand side becomes tan kr , implying that δ (k) kr . In − 0 0 ≡ − 0 D = 2 dimensions, the right-hand side is J0(kr0)/Y0(kr0) which has the small-k expansion π 1 tan δ0(k)= γ . (9C.14) 2 log(kr0e /2) For D> 2, the small-k expansion of (9C.13) is D−2 (kr0/2) tan δ0(k) = π , (9C.15) V0→∞ − Γ(D/2)Γ(D/2 1) − which reduces correctly to kr for D = 3, but not to (9C.14) for D = 2, since the small z − 0 behavior of Y0(z) (2/π) log z cannot be obtained as ν 0 -limit of the ν > 0 behavior Yν (z) (1/π)Γ(ν)(z/2)ν≈.17 → → − 17M. Abramowitz and I. Stegun, op. cit., Formulas 9.1.8 and 9.1.9. 722 9 Scattering and Decay of Particles

Notes and References

For more details on scattering theory see S. Schweber, Relativistic Quantum Fields, Harper and Row, N.Y., 1961.

The individual citations refer to:

[1] See Chapter 3 in M.L. Goldberger and K.N. Watson, Collision Theory, John Wiley and sons, New York 1964. [2] The unique extension of the linear space of distributions to a semigroup that includes also their products is developed in the textbook H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Finan- cial Markets, World Scientiﬁc, Singapore 2009 (http://klnrt.de/b5). [3] J.M.J. Van Leeuwen and A.S. Reiner, Physica 27, 99 (1961)); S.A. Morgan, M.D. Lee, and K. Burnett, Phys. Rev. A 65, 022706 (2002). [4] N. Levinson, Dansk. Videnskab. Selskab Mat.-fys. Medd. 25, Np. 9 (1949). [5] G. Takeda, Phys. Rev. 101, 1547 (1956).