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Particle physics models

Connect relativistic kinematics, particle reactions, decays, and counting experiments through definitions, derivations, and twenty worked examples.

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Matter pathway: atom → solid → liquid → gas → plasma. Quantum mechanics and quantum field theory provide foundations across the pathway; they are not additional phases. This is a connected modeling route, not a universal heating curve. Actual phases depend on pressure, composition, and kinetics.

1. Four-momentum and invariant mass

Definitions & inputs. pμ=(E/c,p); m is rest mass; s is squared centre-of-mass energy when c=1.

  1. Lorentz invariance fixes the mass shell.

    pμpμ=m2c2⇒E2=p2c2+m2c4p^\mu p_\mu=m^2c^2\Rightarrow E^2=p^2c^2+m^2c^4

  2. Express energy and momentum in a chosen inertial frame.

    E=γmc2,p=γmv,γ=(1−v2/c2)−1/2E=\gamma mc^2,\quad p=\gamma mv,\quad\gamma=(1-v^2/c^2)^{-1/2}

  3. Add four-momenta before taking the invariant; masses do not generally add.

    M2c4=(∑iEi)2−c2∣∑ipi∣2M^2c^4=(\sum_iE_i)^2-c^2|\sum_i\mathbf p_i|^2

  4. Expand the incoming invariant for collider or fixed-target kinematics.

    c=1:s=ma2+mb2+2(EaEb−pa⋅pb)c=1:\quad s=m_a^2+m_b^2+2(E_aE_b-\mathbf p_a\cdot\mathbf p_b)

Interpretation. Kinematic accessibility does not imply a reaction has an appreciable cross section.

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2. Two-body decay and lifetimes

Definitions & inputs. M is parent rest mass, m1,m2 daughter masses, Γ energy width, τ proper mean lifetime.

  1. Use energy and momentum conservation in the parent rest frame.

    M=E1+E2,p1=−p2M=E_1+E_2,\quad\mathbf p_1=-\mathbf p_2

  2. Subtract daughter mass-shell equations and solve.

    E1=(M2+m12−m22)/(2M)E_1=(M^2+m_1^2-m_2^2)/(2M)

  3. Integrate a constant decay probability per proper time.

    dN/dt=−N/τ⇒N(t)=N0e−t/τdN/dt=-N/\tau\Rightarrow N(t)=N_0e^{-t/\tau}

  4. Energy width sets proper lifetime; time dilation sets mean flight distance.

    τ=ℏ/Γ,ℓlab=βγcτ\tau=\hbar/\Gamma,\quad\ell_{\rm lab}=\beta\gamma c\tau

Interpretation. A mean decay length is not a deterministic stopping distance.

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3. Cross sections and experimental counts

Definitions & inputs. σ is interaction cross section, luminosity ℒ is incident flux overlap per area per time, ε selection efficiency, B branching fraction.

  1. Rate is luminosity times effective interaction area.

    R=Lσ,Lint=∫LdtR=\mathcal L\sigma,\quad\mathcal L_{\rm int}=\int\mathcal Ldt

  2. Include the desired decay channel and detection/selection losses.

    Nselected=LintσBϵN_{\rm selected}=\mathcal L_{\rm int}\sigma B\epsilon

  3. Independent event counts follow a Poisson model.

    P(N∣μ)=e−μμN/N!,σN=μP(N|\mu)=e^{-\mu}\mu^N/N!,\quad\sigma_N=\sqrt\mu

  4. Invert a background-subtracted counting model; propagate background and calibration uncertainties separately.

    σ≃(Nobs−Nbg)/(LintBϵ)\sigma\simeq(N_{\rm obs}-N_{\rm bg})/(\mathcal L_{\rm int}B\epsilon)

Interpretation. A counting estimate is not automatically a discovery significance.

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4. Symmetries, interactions, and effective models

Definitions & inputs. ψ denotes fermion fields, Aμ gauge fields; g a coupling; Oi effective operators with scaling dimension di; Λ an energy cutoff.

  1. Gauge interactions enter through covariant derivatives.

    Dμ=∂μ+igAμ,L⊃ψˉiγμDμψD_\mu=\partial_\mu+igA_\mu,\quad\mathcal L\supset\bar\psi i\gamma^\mu D_\mu\psi

  2. Higher-dimension interactions are suppressed at low energy under a controlled effective expansion.

    LEFT=Llow+∑iCiΛdi−4Oi\mathcal L_{\rm EFT}=\mathcal L_{\rm low}+\sum_i\frac{C_i}{\Lambda^{d_i-4}}\mathcal O_i

  3. Check scale separation before truncating an effective theory.

    E/Λ≪1E/\Lambda\ll1

Interpretation. The Standard Model combines strong, weak, and electromagnetic interactions; gravity requires a separate framework here.

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Graphical worked example

Exponential decay of a constant-width unstable population in its rest-frame lifetime convention. X axis: Elapsed time / mean lifetime (dimensionless). Y axis: Surviving fraction (dimensionless).
Exponential decay of a constant-width unstable population in its rest-frame lifetime convention. Related worked calculation · Download SVG · Plot data

Twenty worked examples

Open a problem to see its defined inputs, assumptions, equation, numerical substitution, result, and interpretation. Values are illustrative analytical exercises.

Example 01. Photon wavelength

Definitions & inputs. Photon energy E=1 MeV; hc=1.239841984 MeV pm.

  1. Choose the governing model and isolate the requested quantity.

    λ=hc/E\lambda=hc/E

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    λ=1.239841984/1\lambda=1.239841984/1

  3. Evaluate the expression; the result uses the units shown.

    Result=1.239842 pm\mathrm{Result}=1.239842\ {\rm pm}

Interpretation. Higher photon energy corresponds to shorter wavelength.

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Example 02. Lorentz factor at 0.8c

Definitions & inputs. β=v/c=0.8.

  1. Choose the governing model and isolate the requested quantity.

    γ=(1−β2)−1/2\gamma=(1-\beta^2)^{-1/2}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    γ=(1−0.82)−1/2\gamma=(1-0.8^2)^{-1/2}

  3. Evaluate the expression; the result uses the units shown.

    Result=1.666667 \mathrm{Result}=1.666667\

Interpretation. This factor multiplies rest energy and dilates mean lifetime.

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Example 03. Electron kinetic energy

Definitions & inputs. Electron mc²=0.511 MeV and γ=3.

  1. Choose the governing model and isolate the requested quantity.

    K=(γ−1)mc2K=(\gamma-1)mc^2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    K=(3−1)(0.511)K=(3-1)(0.511)

  3. Evaluate the expression; the result uses the units shown.

    Result=1.022 MeV\mathrm{Result}=1.022\ {\rm MeV}

Interpretation. Total energy includes rest energy; kinetic energy does not.

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Example 04. Momentum from total energy

Definitions & inputs. Particle E=5 GeV, mc²=3 GeV.

  1. Choose the governing model and isolate the requested quantity.

    pc=E2−m2c4pc=\sqrt{E^2-m^2c^4}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    pc=25−9pc=\sqrt{25-9}

  3. Evaluate the expression; the result uses the units shown.

    Result=4 GeV\mathrm{Result}=4\ {\rm GeV}

Interpretation. The momentum itself is 4 GeV/c.

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Example 05. Head-on photon invariant mass

Definitions & inputs. E1=2 GeV, E2=8 GeV, angle θ=π.

  1. Choose the governing model and isolate the requested quantity.

    Mc2=2E1E2(1−cos⁡θ)Mc^2=\sqrt{2E_1E_2(1-\cos\theta)}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Mc2=2(2)(8)(2)Mc^2=\sqrt{2(2)(8)(2)}

  3. Evaluate the expression; the result uses the units shown.

    Result=8 GeV\mathrm{Result}=8\ {\rm GeV}

Interpretation. Opposite photon momenta can produce a nonzero system invariant mass.

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Example 06. Symmetric collider energy

Definitions & inputs. Two equal-mass beams have E=10 GeV each and opposite momenta.

  1. Choose the governing model and isolate the requested quantity.

    s=2E\sqrt s=2E

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    s=2(10)\sqrt s=2(10)

  3. Evaluate the expression; the result uses the units shown.

    Result=20 GeV\mathrm{Result}=20\ {\rm GeV}

Interpretation. Total momentum vanishes in the centre-of-mass frame.

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Example 07. Fixed-target centre-of-mass energy

Definitions & inputs. Projectile total E=10 GeV; projectile and stationary target mass energies m=1 GeV.

  1. Choose the governing model and isolate the requested quantity.

    s=2m2+2mE\sqrt s=\sqrt{2m^2+2mE}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    s=2+20\sqrt s=\sqrt{2+20}

  3. Evaluate the expression; the result uses the units shown.

    Result=4.690416 GeV\mathrm{Result}=4.690416\ {\rm GeV}

Interpretation. A fixed-target experiment has less centre-of-mass energy than equal head-on beam energies.

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Example 08. Two-photon decay

Definitions & inputs. Parent rest energy Mc²=0.135 GeV.

  1. Choose the governing model and isolate the requested quantity.

    Eγ=Mc2/2E_\gamma=Mc^2/2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Eγ=0.135/2E_\gamma=0.135/2

  3. Evaluate the expression; the result uses the units shown.

    Result=0.0675 GeV\mathrm{Result}=0.0675\ {\rm GeV}

Interpretation. Both photons have equal and opposite momenta in this frame.

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Example 09. Massive two-body daughter energy

Definitions & inputs. M=5 GeV, m1=1 GeV, m2=2 GeV in c=1 units.

  1. Choose the governing model and isolate the requested quantity.

    E1=(M2+m12−m22)/(2M)E_1=(M^2+m_1^2-m_2^2)/(2M)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    E1=(25+1−4)/10E_1=(25+1-4)/10

  3. Evaluate the expression; the result uses the units shown.

    Result=2.2 GeV\mathrm{Result}=2.2\ {\rm GeV}

Interpretation. The other daughter carries 2.8 GeV, conserving energy.

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Example 10. Lifetime from a decay width

Definitions & inputs. Γ=1 keV=10⁻⁶ GeV; ℏ=6.582119569×10⁻²⁵ GeV s.

  1. Choose the governing model and isolate the requested quantity.

    τ=ℏ/Γ\tau=\hbar/\Gamma

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    τ=6.582119569×10−25/10−6\tau=6.582119569\times10^{-25}/10^{-6}

  3. Evaluate the expression; the result uses the units shown.

    Result=6.58212×10−19 s\mathrm{Result}=6.58212\times10^{-19}\ {\rm s}

Interpretation. A broader width means a shorter proper lifetime.

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Example 11. Survival after two lifetimes

Definitions & inputs. Elapsed proper time t=2τ.

  1. Choose the governing model and isolate the requested quantity.

    N/N0=e−t/τN/N_0=e^{-t/\tau}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    N/N0=e−2N/N_0=e^{-2}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.1353353 \mathrm{Result}=0.1353353\

Interpretation. About 13.5% survive.

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Example 12. Mean flight length

Definitions & inputs. βγ=10, proper τ=2.2 µs, c=299792458 m/s.

  1. Choose the governing model and isolate the requested quantity.

    ℓ=βγcτ\ell=\beta\gamma c\tau

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    ℓ=10(299792458)(2.2×10−6)\ell=10(299792458)(2.2\times10^{-6})

  3. Evaluate the expression; the result uses the units shown.

    Result=6595.434 m\mathrm{Result}=6595.434\ {\rm m}

Interpretation. Individual decay locations follow a distribution.

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Example 13. Produced collision events

Definitions & inputs. Integrated luminosity 100 fb⁻¹; cross section 2 fb.

  1. Choose the governing model and isolate the requested quantity.

    N=LintσN=\mathcal L_{\rm int}\sigma

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    N=100(2)N=100(2)

  3. Evaluate the expression; the result uses the units shown.

    Result=200 \mathrm{Result}=200\

Interpretation. The inverse-area units cancel.

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Example 14. Selected decay yield

Definitions & inputs. Nproduced=1000; branching B=0.1; efficiency ε=0.6.

  1. Choose the governing model and isolate the requested quantity.

    Nsel=NBϵN_{\rm sel}=N B\epsilon

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Nsel=1000(0.1)(0.6)N_{\rm sel}=1000(0.1)(0.6)

  3. Evaluate the expression; the result uses the units shown.

    Result=60 \mathrm{Result}=60\

Interpretation. Branching and efficiency reduce expected yield separately.

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Example 15. Poisson relative precision

Definitions & inputs. Expected count μ=400.

  1. Choose the governing model and isolate the requested quantity.

    σN/μ=1/μ\sigma_N/\mu=1/\sqrt\mu

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1/4001/\sqrt{400}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.05 \mathrm{Result}=0.05\

Interpretation. The counting-only fractional uncertainty is 5%.

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Example 16. Cross section after subtraction

Definitions & inputs. Nobs=120, Nbg=20, luminosity 10 pb⁻¹, efficiency 0.5, B=1.

  1. Choose the governing model and isolate the requested quantity.

    σ=(Nobs−Nbg)/(Lintϵ)\sigma=(N_{\rm obs}-N_{\rm bg})/(\mathcal L_{\rm int}\epsilon)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    σ=(120−20)/(10⋅0.5)\sigma=(120-20)/(10\cdot0.5)

  3. Evaluate the expression; the result uses the units shown.

    Result=20 pb\mathrm{Result}=20\ {\rm pb}

Interpretation. This point estimate needs uncertainty propagation for inference.

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Example 17. Momentum from track curvature

Definitions & inputs. Charge magnitude |q|=e, transverse field B=2 T, radius r=1 m.

  1. Choose the governing model and isolate the requested quantity.

    pT[GeV/c]=0.299792458B[T]r[m]p_T[\mathrm{GeV}/c]=0.299792458B[\mathrm T]r[\mathrm m]

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    pT=0.299792458(2)(1)p_T=0.299792458(2)(1)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.5995849 GeV/c\mathrm{Result}=0.5995849\ {\rm GeV}/c

Interpretation. The formula measures transverse momentum relative to the field.

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Example 18. Two-flavor oscillation probability

Definitions & inputs. sin²(2θ)=1; phase Δ=1.267Δm²[eV²]L[km]/E[GeV]=π/4.

  1. Choose the governing model and isolate the requested quantity.

    P=sin⁡2(2θ)sin⁡2ΔP=\sin^2(2\theta)\sin^2\Delta

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    P=1⋅sin⁡2(π/4)P=1\cdot\sin^2(\pi/4)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.5 \mathrm{Result}=0.5\

Interpretation. The supplied phase produces 50% conversion in this model.

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Example 19. Momentum-resolution propagation

Definitions & inputs. pT∝Br; fractional B error 1%, radius error 2%, independent.

  1. Choose the governing model and isolate the requested quantity.

    σp/p=(σB/B)2+(σr/r)2\sigma_p/p=\sqrt{(\sigma_B/B)^2+(\sigma_r/r)^2}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    0.012+0.022\sqrt{0.01^2+0.02^2}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.02236068 \mathrm{Result}=0.02236068\

Interpretation. The combined fractional uncertainty is about 2.24%.

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Example 20. Effective-theory suppression

Definitions & inputs. Leading correction scales as (E/Λ)²; E=100 GeV, Λ=1000 GeV, coefficient one.

  1. Choose the governing model and isolate the requested quantity.

    δ∼(E/Λ)2\delta\sim(E/\Lambda)^2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    δ∼(100/1000)2\delta\sim(100/1000)^2

  3. Evaluate the expression; the result uses the units shown.

    Result=0.01 \mathrm{Result}=0.01\

Interpretation. A one-percent power-counting estimate is not a computed cross-section correction.

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Symbols and units

Each derivation and problem defines its own symbols and inputs. Symbols may be reused with different meanings in other subjects. Keep units consistent, retain sufficient precision during calculation, and apply the stated validity limits.