Tests des Standardmodells der Teilchenphysik

Spezialfach Kern-Teilchen-Astrophysik

Tests of the Standard Model of ParticlesWinter Semester 2017

Lecture 3

Dr. Zinonas Zinonoszinonos@mpp.mpg.de

Max-Planck-Institut fur Physik, TUM

November 2, 2017

1 / 37

Overview

Particle Interactions and Useful Variables

Phase Space

Decay Rates

Cross Section

2 / 37

Particle Interactions

Particle interactions are described in terms of Feynman diagrams

(a) Scattering (b) Annihilation (c) Interaction

1. Anti-particle arrows point in negative time direction

2. Energy, momentum, angular momentum, etc conserved at all interaction vertices

3. Intermediate particles are ”virtual: E2 6= |p|2 +m2

3 / 37

Mandelstam VariablesIn particle interactions there are three particularly handy Lorentz invariant quantities: s, t & u

Consider any kind of a 2 particles→ 2 particles process: 1 + 2→ 3 + 4

(a)s-channel

(b) t-channel (c) u-channel

The 4-momenta pµ1 , pµ2 , pµ3 , and pµ4 of the 2 incoming and 2 outgoing particles satisfy 8constraints: the on-shell conditions for each particle

p21 = m2

1, p22 = m2

2, p23 = m2

3, p24 = m2

4, (1)

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and the net 4-momentum conservation

pµ1 + pµ2 = pµ3 + pµ4 . (2)

Altogether, this gives us 4× 4− 8 = 8 independent momentum variables, and the number ofindependent Lorentz-invariant combinations of these variables is only 8− 6 = 2.However, for practical purposes it’s is often convenient to use 3 Lorentz-invariant variables witha fixed sum,

s = (p1 + p2)2 = (p3 + p4)2, (3)

t = (p1 − p3)2 = (p4 − p2)2, (4)

u = (p1 − p4)2 = (p3 − p2)2. (5)

s + t + u = m21 + m2

2 + m23 + m2

4 . (6)

Indeed,

s + t + u = (p1 + p2)2 + (p1 − p3)2 + (p1 − p4)2 (7)

= 3p21 + p2

2 + p23 + p2

4 + 2(p1p2) − 2(p1p3) − 2(p1p4) (8)

= p21 + p2

2 + p23 + p2

4 + 2p1 × (p1 + p2 − p3 − p4 = 0) (9)

= p21 + p2

2 + p23 + p2

4 (10)

= m21 + m2

2 + m23 + m2

4 . (11)5 / 37

The s, t, and u are called Mandelstam variables after Stanley Mandelstam who introducedthem in 1958.

Variable s: s = (p1 + p2)2 = (E1 + E2)2 − (p1 + p2)2

⇒The center-of-mass energy is a scalar product of two four-vectors 7−→ Lorentz invariant⇒Since this is a Lorentz-invariant quantity, it can evaluated in any inertial frame. Mostconvenient, the center-of-mass frame (CM):

p∗1 = (E∗1 , p∗), p∗2 = (E∗2 , −p∗) (12)

⇒ s = (E∗1 + E∗2 )2 (13)

In the center-of-mass frame where p1 + p2 = p1 + p4 = 0,√s is the total energy of the

colliding particles,√s = E1 + E2 = E3 + E4.

Also, for an elastic collision in the CM frame, t parametrizes the scattering angle according tot = −(p3 − p1)2 = −p2 × (1− cos θ).

Hence, the Lorentz-invariant definitions (3) translate the CM-frame energy and the CM-framescattering angle to any other frame of reference.

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All Lorentz-invariant combinations of the four momenta pµ1 , pµ2 , pµ3 , and pµ4 can be expressed interms of the Mandelstam variables. For example, the Lorentz products kµpµ of any twomomenta are

2(p1p2) = (p1 + p2)2 − p21 − p2

2 = s − m21 − m2

2 , (14)

2(p3p4) = (p3 + p4)2 − p23 − p2

4 = s − m23 − m2

4 , (15)

2(p1p3) = p21 + p2

3 − (p1 − p3)2 = m21 + m2

3 − t, (16)

2(p2p4) = p22 + p2

4 − (p2 − p4)2 = m22 + m2

4 − t, (17)

2(p1p4) = p21 + p2

4 − (p1 − p4)2 = m21 + m2

4 − u, (18)

2(p2p3) = p22 + p2

3 − (p2 − p3)2 = m22 + m2

3 − u. (19)

In particular, for an elastic scattering of 2 identical-mass particles

s + t + u = 4m2, (20)

2(p1p2) = 2(p3p4) = s − 2m2, (21)

2(p1p3) = 2(p2p4) = 2m2 − t, (22)

2(p1p4) = 2(p2p3) = 2m2 − u . (23)

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For future reference, let us write similar formulæ for the e−e+ → µ−µ+ pair-production in theultra-relativistic energy limit,

s + t + u = 2m2µ + 2m2

e ≈ 2m2µ, (24)

2(p1p2) = s − 2m2e ≈ s, (25)

2(p3p4) = s − 2m2µ, (26)

2(p1p3) = 2(p2p4) = m2µ + m2

e − t ≈ m2µ − t, (27)

2(p1p4) = 2(p2p3) = m2µ + m2

e − u ≈ m2µ − u. (28)

(29)

and for the e−e+ → γγ annihilation process p− + p+ → k1 + k2,

s + t + u = 2m2e , (30)

2(p−p+) = s− 2m2e , (31)

2(k1k2) = s, (32)

2(p−k1) = 2(p+k2) = m2e − t, (33)

2(p−k2) = 2(p+k1) = m2e − u . (34)

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Fermi’s Golden RuleThe physical quantities, or experimental observables, weneed to calculate in particle physics concerningtransition between states are:

I decay rates

I cross sectionsTransition rates are calculated with Fermi’s Golden Rule :

Γi→f = 2π|Tfi|2ρ(Ef ) (35)

where

I Γi→j : number of transitions per unit time from initial state |i〉 to final state |f〉I Tfi: transition matrix element

Tfi = 〈f |H|i〉+∑i 6=j

〈f |H|j〉 〈j|H|i〉Ei − Ej

+ . . . , H = perturbing Hamiltonian (36)

I ρ(Ef ): density of final states

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Rates depend on

I matrix element or amplitude M which contains all the dynamical information of theprocess

I density of states which encapsulates the process kinematics

For a proper calculation of decay and scattering processes we need

I relativistic calculations of particle decay rates and cross sections of scattering processes

σ =|M|2

flux× (phase space) (37)

where the phase space factor is just kinematics: it depends on the particle masses,energies, and momenta of the participants −→ reflects the fact that a given process ismore likely to occur the more ”room to develop” there is in the final state

I relativistic treatment of spin-half particles (Dirac Equation)

I relativistic evaluation of the interaction amplitude (Matrix Element)

I need a relativistic versionof the Golden Rule, derived from quantum filed theory (S-Matrix)– out of scope of this lecture

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Particle Decay RatesConsider the two-body decay: i → 1 + 2

Aim to calculate the decay rate in first order perturbation theoryusing plane-wave descriptions of the particles – Bornapproximation

ψ = Ne−ip·x = Ne−i(Et−p·x) (38)

In order to calculate decay rates we need in a Lorentz invariant form:1. The wave-function normalization N2. The transition matrix element Tfi from perturbation theory3. An expression for the density of states ρ(Ef )

Wave-function normalization:I Using a non-relativistic formulation so farI Non-relativistic form: plane wave normalized to one particle in a cube of dimension α

ρ =

∫d3x ψ∗ψ = N2α3 = 1 ⇒ N2 =

1

α3=

1

V(39)

11 / 37

Phase Space (non-relativistic)

Wave-function vanishing at box boundaries 7−→ quantizedparticle momenta (px, py, pz) = 2π

α (nx, ny, nz)

Volume of single state in momentum space:(

2πα

)3= (2π)3

V

Normalization to one particle/unit volume will give number

of states in element d3p = dpxdpydpz =(

2πα

)3= (2π)3

V :

dn = 4πp2dp1

(2π)3/V× 1

V(40)

⇒The number of states dn with magnitude of momentum inthe range p −→ p+ dp, is equal to the momentum spacevolume of the spherical shell at momentum p with thicknessdp divided by the average volume occupied by a single state,(2π)3/V

(a) Wavefunction of a particle confined to a box of

side α satisfies the periodic boundary conditions such

that there are an integer number of wavelengths in each

direction: e.g. ψ(x + α, y, z) = ψ(x, y, z)

(b) Allowed states in the momentum space

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and hencedn

dp=

4πp2

(2π)3with d3p = 4πp2dp (41)

The density of states in Fermi’s golden rule then can be obtained from

ρ(Ef ) =

∣∣∣∣ dndE∣∣∣∣Ef

=

∣∣∣∣dndp dpdE∣∣∣∣Ef

with p = βE (42)

The density of states corresponds to the number of momentum states accessible to a particulardecay and increases with the momentum of the final-state particle.

Hence, all other things being equal, decays to lighter particles, which will be pro- duced withlarger momentum, are favoured over decays to heavier particles.

Integrating over an elemental shell in momentum-space gives

ρ(Ef ) =4πp2

(2π)3× β (43)

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Dirac δ Function

In the relativistic formulation of particle decay rates and process cross sections the ”Diracδ-function” is commonly used.

The Dirac delta can be loosely thought of as a function on the real line which is zeroeverywhere except at the origin, where it is infinite,

δ(x) =

+∞, x = 0

0, x 6= 0(44)

and which is also constrained to satisfy the identity∫ ∞−∞

δ(x) dx = 1. (45)

⇒an ”infinitely narrow spike of unit area”

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∫ +∞

−∞dxδ(x− x0) = 1 (46)∫ +∞

−∞dxδ(x− x0)f(x) = f(x0) (47)

Any function with the above criteria can represent δ(x), e.g. an infinitesimally narrow Gaussianfunction

δ(x) = limσ→0

1√2π σ

e−x2

2σ2 (48)

In relativistic quantum mechanics, the Dirac δ-functions are extremely useful for integrals overphase-space, e.g. particle decay i→ 1 + 2 + . . .+ f∫

dE . . . δ(Ei − E1 − E2 − . . .− Ef );

∫d3p . . . δ(pi − p1 − p2 − . . .− pf ) (49)

express energy and momentum conservation.

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In integrating the phase space, an expression for the δ-function of a function is typicallyrequired δ (f(x))

From the definition of the δ-function∫ y2

y1

dy δ(y) =

1 y1 < y = 0 < y2

0 otherwise(50)

Expressing in terms of y = f(x) where f(x0) = 0 and then change variables∫ x2

x1

dxdf(x)

dxδ(f(x)) =

1 x1 < x = x0 < x2

0 otherwise(51)

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The δ-function is only non-zero at x = x0∣∣∣∣df(x)

dx

∣∣∣∣x0

∫ x2

x1

dx δ(f(x)) =

1 x1 < x = x0 < x2

0 otherwise(52)

The right-hand side can be rearranged and expressed as a δ-function∫ x2

x1

dx δ(f(x)) =1

|df/dx|x0

∫ x2

x1

dx δ(x− x0) (53)

Finally,

δ (f(x)) =

∣∣∣∣df(x)

dx

∣∣∣∣−1

x0

δ(x− x0) (54)

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Golden Rule Revisited

The classical versionΓif = 2π|Tfi|2ρ(Ef ) (55)

can be rewritten by changing the expression for density of states using a δ-function

ρ(Ef ) =

∣∣∣∣ dndE∣∣∣∣Ef

=

∫dE

dn

dEδ(E − Ei) since Ef = Ei (56)

⇒Energy conservation is now taken into account explicitly by δ-function when integrating overall final state energies

Therefore, for quantum mechanical transition rates, the Golden Rule takes the form

Γfi = 2π

∫dn |Tfi|2δ(Ei − E) (57)

where the integral is over all allowed final states of any energy.

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For the special case of a two-body decay , we only needto consider one particle in dn

dn =d3p

(2π)3(58)

since momentum conservation fixes the other

Γfi =

∫d3p1

(2π)3|Tfi|2(2π)δ(Ei − E1 − E2) (59)

However, the explicit momentum conservation is typically included when integrating over themomenta of both outgoing particles by introducing yet another δ-function

Γfi =

∫d3p1

(2π)3

d3p2

(2π)3︸ ︷︷ ︸density of states

|Tfi|2 (2π)4 δ(Ei − E1 − E2)︸ ︷︷ ︸energy cons.

δ(3)(pi − p1 − p2)︸ ︷︷ ︸momentum cons.

(60)

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Lorentz Invariant Phase Space

In non-relativistic Quantum Mechanicswave-functions are normalized to oneparticle/unit volume∫

d3x ψ∗ψ = 1 (61)

On the other hand, when consideringrelativistic effects, the original normalizationvolume contracts by a factor of 1/γ

γ = E/m = 1/√

1− β2 (62)

Therefore, particle density increases by γ =⇒ a relativistically invariant wave-functionnormalization must be proportional to E particles per volume , such that the increase inenergy accounts for the effect of Lorentz contraction.

20 / 37

Wave-functions with the appropriate Lorentz-invariant normalization, denoted as ψ′, arenormalized to 2E particles per unit volume∫

d3x ψ′∗ψ′ = 2E (63)

Hence, ψ′ =√

2E ψ is normalized to 2E per unit volume.

For a general process, a+ b+ · · · → 1 + 2 + · · · , the Lorentz-invariant matrix element, usingwave-functions with a Lorentz-invariant normalization, is defined and related to the transitionmatrix element of Fermis golden rule by

Mij = 〈ψ′1ψ′2 . . .|H ′|ψ′aψ′b . . .〉 =√

2E1 2E2 . . . 2Ea 2Eb . . . Tfi (64)

where the product on the energies includes all initial- and final-state particles.

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For the two-body decay i→ 1 + 2, the process amplitude is

Mij = 〈ψ′1ψ′2|H ′|ψ′i〉 (65)

=√

2E1 2E2 2Ei 〈ψ1ψ2|H|ψi〉 =√

2E1 2E2 2Ei Tfi (66)

Using the above relation between the transition matrix element and the Lorentz invariantmatrix element, Eq. (60) can be written as

Γfi =1

2Ei

∫d3p1

(2π)32E1

d3p2

(2π)32E2(2π)4δ(Ei − E1 − E2)δ(3)(pi − p1 − p2) |Mfi|2 (67)

I Mfi uses relativisticlly normalized wave-fucntions −→ Lorentz invariant

I d3p(2π)32E represents the Lorentz Invariant Phase Space (LIPS) for each final state particle –

the 2E factor arises from the wave-function normalization

I the integral over the phase space is now frame independent , i.e. Lorentz invariant

I Γfi is inversely proportional to the energy of the decaying particle, Ei (expected from timedilation E = γm)

I δ-functions ensure energy-momentum conservation

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Two-Body Decay Rate

Let’s examine further the case of a two-body decay process.Because the integral over the phase space is Lorentzinvariant, we can evaluate it in any frame – the CM frameturns to be the most convenient one!

In CM frame, Ei = mi and pppi = 0 and hence

Γfi =1

8π2mi

∫d3p1

2E1

d3p2

2E2δ(mi − E1 − E2)δ(3)(p1 + p2) |Mfi|2 (68)

Integrating over p2 using the momentum-space δ-function

Γfi =1

8π2mi

∫d3p1

4E1E2δ(mi − E1 − E2) |Mfi|2 (69)

The δ-function imposes p2 = −p1 and thus E22 = m2

2 + p21 (here for convenience p1 = |p1|)

In spherical polar coordinates: d3p = p2dp dΩ with dΩ = d cos θ dφ and thus...

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Γfi =1

32π2mi

∫dp1 dΩ δ

(mi −

√m2

1 + p21 −

√m2

2 + p21

)p2

1

E1E2|Mfi|2 (70)

At first sight this integral looks quite tricky, however, the Dirac delta-function will simplify thewhole picture. The expression above has the functional form

Γfi =1

32π2mi

∫dΩ dp1 δ (f(p1)) g(p1) |Mfi|2 (71)

where

g(p1) =p2

1

E1E2and f(p1) = mi −

√m2

1 + p21 −

√m2

2 + p21 (72)

I δ (f(p1)) will impose energy conservation and is only non-zero for p1 = p, p is the solutionof f(p) = 0

I f(p1) determines the CM momenta of the two decay particles, i.e. f(p1) = 0 for p1 = p

Eq. (70) can be then integrated using the δ-function property of Eq. (54)∫dp1 g(p1) δ (f(p1)) =

1

|df/dp1|p

∫dp1g(p1)δ(p1 − p) =

g(p)

|df/dp1|p(73)

24 / 37

It remains to evaluate df/dp1:

df(p1)

dp1= − 2p1

2(m21 + p2

1)− 2p1

2(m22 + p2

1)= − p1

E1− p1

E2= −p1

E1 + E2

E1E2(74)

leading Eq. (73) top21

E1E2/∣∣∣−p1

E1+E2

E1E2

∣∣∣p1=p

=∣∣∣ p1E1+E2

∣∣∣p1=p

and thus

Γfi =1

32π2mi

∫dΩ

∣∣∣∣ p1

E1 + E2

∣∣∣∣p1=p

|Mfi|2 (75)

By imposing energy conservation mi = E1 + E2, f(p1) = 0 (Eq. (72)) we obtain the generalexpression for any two-body decay ,

1

τ≡ Γfi =

|p|32π2m2

i

∫dΩ |Mfi|2 (76)

The momentum of either of the final-state particles in the center-of-mass frame p can beobtained from energy conservation, or equivalently f(p1) = 0 in Eq (72):

|p| = 1

2mi

√(m2

i − (m1 +m2)2) (m2i − (m1 −m2)2) (77)

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Cross Section DefinitionIn scattering processes, the interaction rate per targetparticle Rb is proportional to the incident particle flux Φaand can be written as

Rb = σ × Φa (78)

where σ is the most useful observable in scattering processes, the ”cross section”

⇒Can be thought as the effective cross-sectional area of the target particles (expression of theunderlying quantum mechanical probability) for the interaction to occur

σ =number of interactions per unit time per target

incident flux of particles(79)

Differential cross section :

dσ

dΩ=

number of interactions per unit time per target into solid angle dΩ

incident flux of particles(80)

with

dΩ = d cos θ dφ and σ =

∫dΩ

dσ

dΩ(81)

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Consider a single incoming particle of type α with velocity vα traversing a region of area Acontaining nb particles of type b per unit volume

In time δt a particle of type α traverses regioncontaining nb (vα + vb)Aδt.

The interaction probability P obtained from the effective cross-sectional area occupied bynb(vα + vb)Aδt particles of type b is given by

P =nb(vα + vb)Aδt σ

A= nb v δt σ with v = vα + vb (82)

and thus the rate per particle of type α is

Rα = nb v σ (83)

Considering volume V for the entire reaction rate

R = (nb v σ)× (na V ) = (nb V )(na v) σ = Nb Φa σ (84)

and as anticipated

Rate = Number of targets× Flux× cross section (85)

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Cross Section CalculationConsider the scattering process 1 + 2→ 3 + 4.

Starting from Fermi’s Golden Rule:

Γfi =

∫d3p3

(2π)3

d3p4

(2π)3δ (E1 + E2 − E3 − E4)× (86)

δ(3) (p1 + p1 − p3 − p4) |Tfi|2

where Tfi stands for the transition matrix for a normalization 1/unit volume,we can write

rate

volume= (flux of particles 1)× (number density of particles 2)× σ (87)

= n1(v1 + v2)× n2 × σ (88)

and thus for 1 target particle per unit volume : rate = (v1 + v2)× σ ⇒ σ =Γfiv1+v2

giving

σ =1

v1 + v2︸ ︷︷ ︸not L.I.

∫d3p3

(2π)3

d3p4

(2π)3︸ ︷︷ ︸not L.I.

(2π)4δ (E1 + E2 − E3 − E4) δ(3) (p1 + p1 − p3 − p4) |Tfi|2︸ ︷︷ ︸not L.I.

(89)28 / 37

In order to obtain a Lorentz invariant form of the cross-section we must:

I use wave-functions normalized to 2E particles per unit volume: ψ′ =√

2E ψ

I define a Lorentz invariant matrix element Mfi = (2E1 2E2 2E3 2E4)1/2Tfi

Therefore,

σ =1

2E1 2E2 (v1 + v2)

∫d3p3

2E3(2π)3

d3p4

2E4(2π)3(2π)4δ (E1 + E2 − E3 − E4)× (90)

δ(3) (p1 + p1 − p3 − p4) |Mfi|2

and the integral is now termed in a Lorentz invariant form.

The quantity F = 2E12E2(v1 + v2) can be written in terms of a four-vector scalar product(Lorentz invariant flux):

F = 2E1 2E2(v1 + v2) = 4E1 E2

(|p1|E1

+|p2|E2

)(91)

= 4 (|p1|E2 + |p2|E1) (92)

29 / 37

To prove that this is Lorentz invariant we first consider

p1 · p2 = pµ1p2µ = E1E2 − |p1||p2| cos θ = E1E2 + |p1||p2| for θ = π (93)

leading to

F 2

16− (pµ1p2µ)

2= (|p1|E2 + |p2|E1)

2 − (E1E2 + |p1||p2|)2 (94)

= |p1|2(E22 − |p2|2) + E2

1(|p2|2 − E22) = |p1|2m2

2 − E21m

22 (95)

= −m21m

22 (96)

F = 4

√(p1 · p2)

2 −m21m

22 (97)

Consequently, the cross section is a Lorentz invariant quantity. Special cases:

CM Frame p1 = −p2 = p

F = 4E1E2(v1 + v2)

= 4E1E2(|p|/E1 + |p|/E2)

= 4|p|(E1 + E2) = 4|p|√s

Target at rest p2 = 0

F = 4E1E2(v1 + v2)

= 4E1m2v1 = 4E1m2|p1|/E1

= 4m2|p1|30 / 37

Two-to-two Particle Scattering

Let’s specialize our cross section formulation on the 2→ 2 scattering case in the CM frame:p1 + p2 = 0 and

√s = E1 + E2

Eq. (98) will become

σ =1

4|p∗i |√s (2π)2

∫d3p3

2E3

d3p4

2E4δ(√s− E3 − E4

)δ(3) (p3 + p4) |Mfi|2 (98)

This integral is exactly the same that appeared in the particle decay calculation in Eq. (68) butwith m1 replaced by

√s in Eq. (76), thus

σ =1

4|p∗i |√s (2π)2

|p∗f |4√s

∫dΩ∗ |Mfi|2 (99)

σ =1

64π2s

|p∗f ||p∗i |

∫dΩ∗ |Mfi|2 (100)

31 / 37

In the case of elastic scattering : |p∗f | = |p∗i | and thus

σelastic =1

64π2s

∫dΩ∗ |Mfi|2 (101)

To calculate the total cross section , which is Lorentz invariant, the formula of Eq. (100) issufficient.

Note that for calculating the differential cross section in a rest frame other than the CMframe, the angles in the infinitesimal solid angle dΩ∗ = d cos θ∗ dφ∗ refer to the CM frame

dσ =1

64π2s

|p∗f ||p∗i ||Mfi|2 dΩ∗ (102)

We therefore need to seek for a Lorentz invariant expression for dσ. This can be achieved byexpressing dΩ∗ in terms of the Mandelstam variable t

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(a) Scattering in the CM frame (b) t Mandelstam variable

Using the definition of t, a product of four-vectors and thus Lorentz invariant,

t ≡ q2 = (p1 − p3)2 (103)

we can express dΩ∗ in terms of Lorentz invariant dt,

t = (p1 − p3)2 = p21 + p2

3 − 2p1 · p3 = m21 +m2

3 − 2p1 · p3 (104)

and in the CM frame...

33 / 37

p∗µ1 = (E∗1 , 0, 0, |p∗1|) and p∗µ3 = (E∗3 , |p∗3| sin θ∗, 0, |p∗3| cos θ∗) (105)

p1 · p3 = E∗1E∗3 − |p∗1|p∗3| cos θ∗ (106)

t = m21 +m2

3 − E∗1E∗3 + 2|p∗1|p∗3| cos θ∗ (107)

⇒ dt = 2|p∗1|p∗3|d cos(θ∗) (108)

=⇒ dΩ∗ = d cos(θ∗)dφ∗ =dt dφ

2|p∗1|p∗3|(109)

Hence, we can write dσ as

σ =1

64π2 s

|p∗3||p∗1||Mfi|2 dΩ∗ =

1

2× 64π2 s

1

|p∗1|2|Mfi|2 dφ∗dt (110)

Finally, assuming no dependence of the matrix element on the azimuthal angle, we canintegrate over dφ∗ and obtain

dσ

dt=

1

64π s |pi|2|Mfi|2 (111)

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Lorentz Invariant Differential Cross Section

All parts in the expression of dσ/dt are Lorentz invariant and hence it applied to anyreference frame.

It is worth noting that |pi|2 is a constant scalar quantity fixed by energy-momentumconservation

|pi|2 =1

4s

(s− (m1 +m2)2

) (s− (m1 −m2)2

)(112)

One representative example of how to make use of the invariant expression of dσ/dt is theelastic scattering in the laboratory frame in the limits where the mass of the colliding particlecan be safely neglected, E1 m1

In this limit, |pi|2 = (s−m2)2

4s and hence

dσ

dt=

1

16π(s−m22)2|Mfi|2 for m1 ' 0 (113)

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SummaryWe used a Lorentz invariant formulation of Fermi’s Golden Rule to derive decay rates andcross sections in terms of Lorentz invariant Matrix Element M based on wave-functionsnormalized to 2E/unit volume.

Main Results:

I Particle decay rate / inverse life time:

1

τ≡ Γfi =

|p|32π2m2

i

∫dΩ |Mfi|2 (114)

with |p| being a function of masses of the particles involved in the decay

|p| = 1

2mi

√(m2

i − (m1 +m2)2) (m2i − (m1 −m2)2) (115)

I Scattering cross section in CM frame:

σ =1

64π2s

|p∗f ||p∗i |

∫dΩ∗ |Mfi|2 (116)

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Summary Cont’d

I Invariant differential cross section – valid in all frames:

dσ

dt=

1

64π s |pi|2|Mfi|2 (117)

⇒Have introduced the basis for kinematics of particle decays and scattering process

⇒The dynamics and fundamental particle physics are encapsulated in the matrix element

⇒We have all ingredients for calculations that will follow in this lecture!

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