Particle Physics Formula Sheet

Particle Physics Formula Sheet Special Relativity Spacetime Coordinates: x µ = (c t, x) x µ = (c t, x) x = (x, x, x ) = (x, y, z) 4-Momentum: p µ = (E/c, p) p µ = (E/c, p) p = (p, p, p ) = (p x, p y, p z ) Derivatives: µ = ( x µ = c ) t, µ = ( c ) t, = ( x, x, ) ( x = x, y, ) z Metric: η µν = η µν = η µ ν = δ µ ν = Raising and Lowering Indices: a µ = η µν a ν a µ = η µν a ν Einstein Summation Convention: a µ b µ = a b + a b + a b + a b = a b a b Lorentz Transformations: a µ = Λ µ ν a ν η µν = Λ µ αλ ν β η αβ a µ = Λ µ ν a ν η µν = Λ µ α Λ ν β η αβ Properties of Lorentz Transformations:. η µν = Λ µ αλ ν β η αβ. det Λ µ ν =. (Λ ) µ α (Λ ) α ν = (Λ ) µ ν 4. Λ Boost Along the z-axis: γ β γ Λ µ ν = β γ γ with β = v c γ = x µ = Λ µ ν x ν = ( γ (c t β z), x, y, γ (z β c t) ) β p µ = Λ µ ν p ν = ( γ (E/c β p z ), p x, p y, γ (p z β E/c) )

Rotation About the y-axis: Λ µ cos θ sin θ ν = sin θ cos θ x µ = Λ µ ν x ν = ( c t, x cos θ z sin θ, y, z cos θ + x sin θ ) p µ = Λ µ ν p ν = ( E/c, p x cos θ p z sin θ, p y, p z cos θ + p x sin θ ) Relativistic Kinematics Conservation of 4-Momentum in Any Process: i=initial p µ i = j=final p µ j Massive Particles: Massless Particles: p µ p µ = E c p p = m c p µ p µ = p µ = p (, ε ) with ε ε = Important Matrices Matrices are written out explicitly, and 4 4 matrices are written out in terms of blocks. The identity matrix is = Pauli Matrices: σ = σ i = i σ = σ i σ j = δ ij + i ɛ ijk σ k [ σ i, σ j ] = i ɛ ijk σ k { σ i, σ j } = δ ij Dirac Matrices: γ = γ i σ i = σ i γ 5 = i γ γ γ γ = { γ µ, γ ν } = η µν /a = a µ γ µ = a γ a γ a γ a γ

Relativistic Wave Equations Klein-Gordon Equation (Spin- Particle): ( µ µ + m ) φ = µ µ = c t Dirac Equation: (i γ µ µ m) ψ = Maxwell s Equations (Massless Spin- Particle): µ F µν = with F µν = µ A ν ν A µ µ F µν = µ F νλ + ν F λµ + λ F µν = { E = B E t = { B = E + B t = Solutions of the Dirac Equation Some terms in these equations include implicit factors of the or 4 4 identity matrices. Plane-wave solutions: ψ(x) = { a e i pµ xµ u(p) (fermions) a e i pµ xµ v(p) (anti-fermions) Dirac Spinors: Conjugate Spinors: Conjugate Matrices: (/p m c) u (s) (p) = (/p + m c) v (s) (p) = ū (s) (p) (/p m c) = v (s) (p) (/p + m c) = ū = u γ v = v γ ψ = ψ γ Ā = γ A γ for any 4 4 matrix A. Complete Set of Plane-wave Solutions: u () (p) = N c p z E+m c c (p x+i p y) E+m c c (p x i p y) E+m c v () c p z (p) = N E+m c u () (p) = N v () (p) = N c (p x i p y) E+m c c p z E+m c c p z E+m c c (p x+i p y) E+m c E + m c with N = c

Fermi s Golden Rule Decays in the Rest Frame of the Decaying Particle: Γ = S p 8π m c M Scattering in the CM Frame: d σ = d Ω CM c S M p f 8π (E + E ) p i with p i = p = p and p f = p = p 4 Elastic Scattering in the Lab Frame (m = m, m 4 = m ): d σ = d Ω Lab p S M 8π m p (E + m c ) p p E cos θ Particle Data Mass in MeV/c, liftime in seconds, charge in units of the proton charge. Leptons (Spin /) Generation Flavor Charge Mass Lifetime First e -.5999 ν e Second µ - 5.659.97 6 ν µ Third τ - 776.88.9 ν τ 4

Quarks (Spin /) Generation Flavor Charge Mass Other Quantum Numbers First d -/ 7 u / Second s -/ Strangeness = - c / Charm = + Third b -/ 4 Bottomness = - t / 74 Topness = + (Note: Quark masses are approximate.) Baryons (Spin /) Baryon Quark Content Charge Mass Lifetime p uud 98.7 n udd 99.565 885.7 Λ uds 5.68.6 Σ + uus 89.7 8. Σ uds 9.64 7.4 Σ dds - 97.45.48 Ξ uss 4.8.9 Ξ dss -..64 Λ + c udc 86.5. Baryons (Spin /) Baryon Quark Content Charge Mass Lifetime uuu, uud, udd, ddd,,, - 5.6 4 Σ uus, uds, dds,, - 85.8 Ξ uss, dss, - 5 6.9 Ω sss - 67 8. Vector Mesons (Spin ) Meson Quark Content Charge Mass Lifetime ρ u d, (u ū d d )/, d ū,,, - 775.5 4 4 K u s, d s, d d, s ū,,, - 894 ω (u ū + d d )/ 78.6 8 ψ c c 97 7 D c d, c ū, u c, d c,,, - 8 Υ b b 946 5

Pseudoscalar Mesons (Spin ) Meson Quark Content Charge Mass Lifetime π ± u d, d ū ± 9.57.6 8 π (u ū d d )/ 4.977 8.4 7 K ± u s, s ū ± 49.68.4 8 K, K d s, s d 497.65 KS KL : 8.95 : 5. 8 η (u ū + d d s s )/ 6 547.5 5. 9 η (u ū + d d + s s)/ 957.78. D ± c d, d c ± 869..4 D, D c ū, u c 864.5 4. D s ± c s, s c ± 968. 5. B ± u b, b ū ± 579..6 B, B d b, b d 579.4.5 Baryons, Mesons, and the Eightfold Way Light Baryons (l = ) n p udd dud udu duu Σ Σ Λ Σ + dsd sdd usd sud+dsu sdu uds dus+usd sud dsu+sdu usu suu Ξ Ξ dss sds uss sus The Baryon Octet χ : Antisymmetric in and dud ddu uud udu udd ddu uud duu dds dsd dus dsu+uds usd uus usu dds sdd uds sdu+dus sud uus suu sud sdu+dus dsu uds+usd usd dsu+uds sdu dus+sud sds ssd sus ssu dss ssd uss ssu χ : Antisymmetric in and χ : Antisymmetric in and 6

+ ++ ddd ddu+dud+udd uud+udu+duu uuu Σ Σ Σ + dds+dsd+sdd uds+usd+dus+dsu+sud+sdu 6 uus+usu+suu Ξ Ξ dss+sds+ssd uss+sus+ssu Ω The Baryon Decuplet sss χ S : Completely Symmetric uds usd+dsu dus+sud sdu 6 χ A : Completely Antisymmetric Light Mesons (l = ) K K + K K + π π π + ρ ρ ρ + η η φ ω K K K K Pseudoscalar Meson Nonet (Octet plus Singlet η ) Vector Meson Nonet (Octet plus Singlet ω) 7

QED Feynman Rules. Make sure that fermion are labelled with an arrow that indicates whether they are particles or anti-particles.. Associate a 4-momentum p, p,..., p n with each external line, and draw an arrow next to each line pointing forward in time. Associate a 4-momentum q, q,... q m with each internal line, and draw an arrow next to each line (the direction is arbitrary for internal line).. External lines contribute the following factors Incoming Outgoing Electron = u Electron = ū Positron = v Positron = v Photon = ɛ µ Photon = ɛ µ 4. Each vertex contributes a factor i g γ µ, where the dimensionless coupling g is given by g = e 4π/ c. 5. Internal lines (propagators) contribute the following factors } = i (γµ q µ + m c) = i η µν q m c q 6. For each vertex, include a factor of the form (π) 4 δ (4) (±k ± k ± k ) () where the k i are the 4-momenta entering the vertex. The sign of each term is plus for a momentum arrow pointing into the vertex and minus if the momentum arrow points out of the vertex. 7. For each internal line with momentum q include a factor and integrate. d 4 q (π) 4 () 8. After performing integrals and using the various delta functions, the result will still contain an overall factor Replace this with a factor of i to obtain the amplitude M. (π) 4 δ (4) (p + p +... p n ). () 9. If there are multiple diagrams, include appropriate antisymmetrization factors. Include an extra minus sign between diagrams that differ only by exchange of two incoming (or outgoing) fermions or antifermions, or by exchange of an incoming fermion and outgoing antifermion (or vice versa). 8

Casimir s Trick For a process with initial spin states s, final spin states s [ū (s) (p A ) Γ u (s ) (p B )] [ū (s) (p A ) Γ u (s ) (p B )] = Tr [ Γ (/p A + m A c) Γ (/p B + m B c) ] s,s [ v (s) (p A ) Γ v (s ) (p B )] [ v (s) (p A ) Γ v (s ) (p B )] = Tr [ Γ (/p A m A c) Γ (/p B m B c) ] s,s [ū (s) (p A ) Γ v (s ) (p B )] [ū (s) (p A ) Γ v (s ) (p B )] = Tr [ Γ (/p A + m A c) Γ (/p B m B c) ] s,s [ v (s) (p A ) Γ u (s ) (p B )] [ v (s) (p A ) Γ u (s ) (p B )] = Tr [ Γ (/p A m A c) Γ (/p B + m B c) ] s,s where u (s) (p) and v (s) (p) are Dirac spinors, their conjugates are ū (s) (p) and v (s) (p), and Γ and Γ are any complex 4 4 matrices. Factors of m c appearing in the traces on the right-hand side of these relations have implicit factors of the 4 4 identity matrix. Trace Identities If A and B are matrices and α is a complex number Tr [ A + B ] = Tr [ A ] + Tr [ B ] Tr [ α A ] = α Tr [ A ] Cyclic property of the Trace Tr [ A A... A n A n ] = Tr [ A n A A... A n ] =... = Tr [ A... A n A n A ] Gamma Matrix Identities If a µ and b µ are 4-vectors, and /a = γ µ a µ, then {γ µ, γ ν } = η µν implies /a /b + /b /a = a b Other contractions of gamma matrices follow from the anti-commutation relations γ µ γ µ = 4 γ µ γ ν γ µ = γ ν The trace of an odd number of gamma matrices is always zero. Some other traces are Tr [ I 4 4 ] = 4 Tr [ γ µ γ ν ] = 4 η µν ( Tr [ γ µ γ ν γ λ γ σ ] = 4 η µν η λσ η µλ η νσ + η µσ η νλ). Factors of the 4 4 identity matrix are implicit in many of these identities. 9