Subatomic Physics: Particle Physics Lecture 3 Measuring Decays, Scatterings and Collisions

Particle lifetime and width Particle decay modes Particle decay kinematics Scattering cross sections Collision centre of mass energy

Lots of examples used today. Learn the concepts, not the details about the particular decays or scatterings!

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Particle Lifetime

Typical • Particle lifetime, !, is the time taken for the Force sample to reduce to 1/e of original sample. Lifetime • Different forces have different typical lifetimes. Strong 10"20 - 10"23 s • Also define total decay width, ! " ℏ/!. Electromag 10"20 - 10"16 s dN N Γ = = N Weak 10"13 - 103 s dt − τ − ￿

N(t) = N0 exp( t/τ) = N0 exp( Γt/￿) ⇒ − − • In its own rest frame particle travels v! = #c! before decaying. • In the lab, time is dilated by $. • If ! is large enough, energetic particles travel a measurable distance L = $#c! in lab. example of a recent c!= measurement of a this experimental data was used to determine !12 particle lifetime !("b) = (1.30 ± 0.15) # 10 s 2 This is an example. Learn the concepts, not Decay Modes the details about this particular decay. Particles can have more than one possible final state or decay mode. e.g. The KS meson decays 99.9% of the time in one of two ways: + 0 0 K π π−, K π π S → S → • Each decay mode has its own matrix element, M. Fermi’s Golden Rule gives us the partial decay width for each decay mode:

+ + 2 0 0 0 0 2 Γ(KS π π−) (KS π π−) Γ(K π π ) (K π π ) → ∝ |M → | S → ∝ |M S → | • The total decay width is equal to the sum of the decay widths for all the allowed decays.

0 0 + Γ(K ) = Γ(K π π ) + Γ(K π π−) S S → S → • The branching ratio, BR, is the fraction of time a particle decays to a particular final state: + 0 0 + Γ(KS π π−) 0 0 Γ(KS π π ) BR(KS π π−) = → BR(KS π π ) = → → Γ(KS) → Γ(KS)

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Particle Decay Kinematics

• Most particles decay. e.g. KS meson can decay as: + K π π− S → • Reconstruct the mass of a particle from the momenta of the decay products:

p = p initial final + ￿p(KS) = ￿p(π ) + p(π−)

Invariant mass from + " squaring each side ... KS!! ! decay. Not perfect 2 due to limited accuracy of 2 + (M(KS)) = p(π ) + p(π−) measurements 2 2 ￿ + ￿ + = p(π ) + p(π−) + 2p(π ) p(π−) · ￿ + ￿ 2 ￿ ￿ 2 + + = m(π ) + m(π−) + 2 E(π )E(π−) 2 p￿ (π ) p￿ (π−) − · • M(KS) is reconstructed￿ invariant￿ ￿ mass￿ of KS 4 Decay Kinematics II

pb pd Decay of an unstable A b d particle at rest: mb md MA A ! b d p = (MA, 0) p = (Eb, p￿b) p = (Ed, p￿d) A b d Before After • Four-momentum conservation: p = p + p p = p p p￿b = p￿d A b d ⇒ b A − d ⇒ − 2 2 2 (p ) = (p ) + (p ) 2p p = M 2 + m2 2M E = m2 b A d − A · d A d − A d b 2 2 2 MA + md mb Ed = − For moving particles, apply ⇒ 2MA appropriate Lorentz boost.

+ + • Example: π µ ν µ work in rest frame of pion. m" % 0 2 →2 mπ + mµ 2 2 E = = 109.8 MeV p￿ν = p￿µ = Eµ mµ = 29.8 MeV/c µ 2m | | | | − π ￿ 5

Cross Section see JH D&R §2

• We have a beam of particles incident on a target (or another beam). a a b b (E, p￿a) (Ea, p￿a) (Eb, p￿b)

• Flux of incident beam, f : number of particles per unit area per unit time. • Beam illuminates N particles in target. • We measure the scattering rate, dw/d#, number of particles scattered in given direction, per unit time per unit solid angle, d#. dw dσ = fN d&/d# is differential dΩ dΩ cross section

• Integrate over the solid angle, rate of scattering: w = fNσ • Define luminosity, L = f N • Scattering rate w = σ L

6 Collision Centre ofn Mass Energy, √ s

• For a collision define Lorentz-invariant quantity, s: square of sum of four- momentum of incident particles: s = (p + p ) (p + p ) a b · a b = (p )2 + (p )2 + 2 p p a b a · b = m2 + m2 + 2(E E p￿ p￿ cos θ) a b a b − | a|| b| • !s=ECM is the energy in centre of momentum frame, energy available to crate new particles!

• Fixed Target Collision, b is at rest. Ea >> ma, mb a 2 2 b s = ma + mb + 2Eamb 2Eamb ≈ (E, p￿a) ECM = 2Eamb ￿ • Collider Experiment, with E = Ea = Eb >> ma, mb, ' = ( a b 2 E = 2E s = 4E CM (Ea, p￿a) (Eb, p￿b)

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Scattering Width

14 40. Plots of cross sections and related quantities • Lifetimes can be very short, e.g. lifetime of $++ baryon (uuu) is 5#10$24 s. • Heisenburg Uncertainty Principle: "max (+p scattering ∆E∆t ￿ !!!!!!!!!!!!!!!!!! 10 2 ≈ "max/e Very short lifetime gives small *t ! • π+ p ! total !"#$$%$&'()#*%+,-.

• *E % ℏ/*t is significant ! 10

+ measurable width π p elastic

Plab GeV/c

-1 2 10 1 10 10 Mass of short lived particles (e.g. 1.2 2 3 4 5 6 7 8 9 10 20 30 40 • #p ++ "s GeV $ ) is not fixed. d # )2.2s, collision3 energy4 5 6 (GeV)7 8 9 10 20 30 40 50 60 10 2 + ++ + ( p!$ !( p scattering! Mass has most-likely value, but π∓ d total

• !!!!!!!!!!!!!!!!!! From analysing the data in the above can take on a range of values. plot it was found: ! ++ 2 π− p total Another example. Learn •m($ ) ~ 1232 MeV/c ++ concepts, not details! •!($ ) ~ 118 MeV

!"#$$%$&'()#*%+,-. 10 8

π− p elastic

Plab GeV/c

-1 2 10 1 10 10

Figure 40.13: Total and elastic cross sections for π±p and π±d (total only) collisions as a function of laboratory beam momentum and total center-of-mass energy. Corresponding computer-readable data files may be found at http://pdg.lbl.gov/current/xsect/. (Courtesy of the COMPAS Group, IHEP, Protvino, August 2005) Collision Examples (in Natural Units)

• To make hadrons, a 45.1 GeV electron beam was fired into a Beryllium target. • Electrons collide with protons and neutrons in Beryllium. 2 • The previous LEP collider at CERN collided electrons s = p(e−) + p(p) and positrons head-on with E(e!) = E(e+) = 45.1 GeV. 2 ￿ 2 2 ￿ + = me + mp s = p(e ) + p(e−) +2(EeEp p￿e p￿p cos θ) 2 2 − | || | = ￿2m + 2(E ￿p + p￿ cos θ) e e e− 2(Eemp) 2 − | || | ≈ 2(E + p￿e+ p￿e ) ≈ | || − | ECM = 2Eemp 4E2 ≈ = ￿√2 45.1 1 E = 2E = 91.2 GeV × × CM = 9.5 GeV • Cross section to create muons pairs was + ! + ! • In fixed target electron energy is $(e e %µ µ )=1.9 nb at ECM = 91.2 GeV wasted providing momentum to • Total integrated luminosity &L dt = 162 pb!1 the CM system rather than to + # + # make new particles. • Nevts(e e !µ µ ) = 162,000 # 1.9 = 307,800

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Summary of Lecture 3 Fermi’s Golden Rule Particle Scattering • Rates of decays and scatterings yield information about the forces and •Cross section, &, probability for particles involved through the matrix decay to happen. Measured in b = "28 2 element, M. 10 m . • M can be related to the quantum •Luminosity, L is a property of beam description of the Standard Model. (s) 2π •Integrated luminosity, !Ldt. T = 2ρ ￿ |M| •Number of events: N = & !Ldt Particle Decay •Two types of scattering experiment: •Lifetime, !, time taken for sample collider and fixed target. to decrease by 1/e. Natural Units: set ℏ=c=1 •Partial width of decay mode, • Measure energies, momenta and mass in GeV !(A"x) # |M(A+x)|2 Relativistic Kinematics •Tot al widt h is sum of all possible p = (E, px, py, pz) = (E, p￿ ) decay widths, != /! 2 ℏ p = E2 p￿ 2 = m2 •Branching ratio, proportion of − decays to given final state, Centre￿ of￿ Mass energy s = (p + p )2 BR (A"x) = !(A"x)/! a b ECM = √s 10