High Energy Physics (HEP) System¶

Overview¶

The High Energy Physics (HEP) system provides a complete System of Quantities and System of Units, along with physical constants, tailored for particle physics, nuclear physics, and high-energy experiments. It was designed with input from members of the high energy physics community to address real-world requirements of large-scale HEP software projects.

The mp-units HEP system implements:

System of Quantities: A dimensional system with base quantities (length, time, energy, electric charge, temperature, amount of substance, luminous intensity, angle) that differs from the International System of Quantities (ISQ)
System of Units: Base units millimetre (mm), nanosecond (ns), megaelectronvolt (MeV), elementary charge (eplus), kelvin (K), mole (mol), candela (cd), radian (rad), steradian (sr)
Physical constants: From CODATA 2014, 2018, and 2022 recommendations, allowing users to choose the appropriate version for their application
Interoperability: Designed to work alongside existing HEP frameworks (CLHEP, Gaudi, Geant4, ROOT) during migration

Collaboration with the HEP community

The HEP system design was developed through consultation with members of the HEP community. The goal is to provide a modern, type-safe units library that can eventually replace hand-maintained unit constants across multiple HEP frameworks while maintaining compatibility during migration.

System of Quantities¶

The HEP System of Quantities defines a dimensional system fundamentally different from the International System of Quantities (ISQ). While both systems describe physical relationships between quantities, HEP makes different choices for base quantities that are more natural for particle physics. The two systems are mutually incompatible by design — there is no implicit conversion between them.

Base Quantities and Dimensions¶

The HEP system uses eight base quantities:

Base Quantity	Dimension Symbol	Same as ISQ?
*length*	L	✓ (yes)
*time* (duration)	T	✓ (yes)
*energy*	E	✗ (ISQ uses mass M)
*electric charge*	Q	✗ (ISQ uses electric current I)
*thermodynamic temperature*	Θ	✓ (yes)
*amount of substance*	N	✓ (yes)
*luminous intensity*	I	✓ (yes)
*angle*	α	✗ (no)

Key Differences from ISQ¶

Two base quantities differ from ISQ, making the two systems dimensionally incompatible — the compiler rejects any attempt to mix quantities from both:

quantity<isq::length[si::metre]>  L_isq = 1. * si::metre;
quantity<hep::length[hep::metre]> L_hep = 1. * hep::metre;

// auto bad = L_isq + L_hep;  // Compile error — different dimensional systems ✓

Energy instead of Mass:

In particle physics, energy (E) is more fundamental than mass (M) due to mass-energy equivalence (\(E = mc²\)). When physicists say "electron mass = 0.511 MeV", they're using natural units shorthand where c=1, but they're really referring to rest energy. In the HEP system, energy is the base quantity while mass is derived.

using namespace mp_units::hep::unit_symbols;

// Access mass constants (dimension: M = E/L²T⁻²)
quantity electron_m = m_e;  // ≈ 0.511 MeV/c²
quantity proton_m = m_p;    // ≈ 938.3 MeV/c²

// Calculate rest energies from mass (dimension: E)
quantity electron_rest_energy = m_e * c2;  // E = mc² ≈ 0.511 MeV
quantity proton_rest_energy = m_p * c2;    // E = mc² ≈ 938.3 MeV

// In natural units where c=1, physicists often say "electron mass = 0.511 MeV"
// but in HEP system with explicit units, we distinguish mass from rest energy

Electric Charge instead of Electric Current:

Electric charge (Q) is treated as a base quantity rather than electric current. This reflects the discrete nature of charge in particle physics (multiples of elementary charge) and makes charge conservation explicit.

// In HEP: electric charge is a base quantity
quantity proton_charge = 1. * eplus;
quantity electron_charge = -1. * eplus;
quantity up_quark_charge = 2. / 3. * eplus;

// Current is derived: I = Q/T
quantity current = proton_charge / (1. * ns);

Derived Quantities¶

All other quantities are derived from these base quantities through dimensional analysis:

Force: F = E/L (energy per length)
Power: P = E/T (energy per time)
Electric Current: I = Q/T (charge per time)
Electric Potential: V = E/Q (energy per charge)
Magnetic Field: B = E·T/(Q·L²) (dimension: ETQ⁻¹L⁻²)
Mass: M = E·T²/L² (derived from \(E = Mc²\), where \(c = L/T\))

This dimensional structure ensures dimensional consistency while using quantities natural to particle physics calculations.

System of Units (Base Units)¶

The HEP System of Units provides concrete measurement scales for the base quantities. Unlike SI (which uses metre, second, kilogram, ampere), the HEP system uses units natural to particle physics:

Quantity	Unit	Symbol	Why?
*Length*	millimetre	mm	Detector geometries are typically millimetre-scale; avoids frequent powers of 10³
*Time*	nanosecond	ns	Particle lifetimes and detector timing naturally in nanoseconds
*Energy*	megaelectronvolt	MeV	Particle energies range from MeV to TeV; MeV provides good numerical range
*Electric Charge*	elementary charge	eplus	Charge is quantized in multiples of e; makes charge quantum numbers explicit
*Temperature*	kelvin	K	Same as SI (no HEP-specific convention)
*Amount*	mole	mol	Same as SI (for material composition)
*Luminosity*	candela	cd	Same as SI (for detector calibration)
*Angle*	radian	rad	Same as SI (for angular measurements)

Specialized Quantities of the Same Kind¶

Beyond the basic dimensional safety, the HEP system provides specialized quantities to distinguish physically distinct concepts that share the same dimension. This addresses real problems in particle physics code where mixing conceptually different values is physically meaningless but dimensionally valid.

The HEP system provides specialized quantities for:

Energy: total_energy, kinetic_energy, rest_mass_energy, binding_energy, excitation_energy, ionization_energy, threshold_energy, Q_value, and more
Length: path_length, displacement, decay_length, radiation_length, interaction_length, impact_parameter, wavelength, range, and more
Time: proper_time, coordinate_time, mean_lifetime, half_life, time_of_flight
Mass: rest_mass, invariant_mass, effective_mass, reduced_mass
Momentum: momentum, transverse_momentum
Angle: scattering_angle, opening_angle, azimuthal_angle, polar_angle
Dimensionless (special): lorentz_factor (γ = E/E₀), relativistic_beta (β = v/c)
Interaction: cross_section (σ), number_density (n), decay_constant (λ)

For complete hierarchy trees and all available specialized quantities, see the HEP System Reference.

Distinct Quantity Subkinds¶

Some HEP quantities share the same dimension, unit, and even the same parent quantity, yet represent fundamentally incompatible physical concepts that should never be added to or compared with each other. The HEP system uses the is_kind specifier to enforce this at compile time. See Creating Distinct Quantity Kinds with is_kind for the general mechanism.

Relativistic Time: `proper_time` and `coordinate_time`¶

Both are sub-quantities of duration, but they live in different reference frames and are related by the Lorentz factor, not by addition:

\[\tau = \frac{t}{\gamma}\]

Adding a proper time to a coordinate time is a classic special-relativity mistake — one is Lorentz-invariant, the other is frame-dependent:

quantity tau = 10. * hep::ns * proper_time;  // particle rest frame
quantity t   = 25. * hep::ns * coordinate_time;  // lab frame

// auto wrong = tau + t;  // Compile error — different kinds ✓

To convert between them, the Lorentz factor must be applied explicitly:

quantity gamma = 2.5 * lorentz_factor;
quantity t_lab = coordinate_time(tau * gamma);  // explicit physics conversion

Material Characteristic Lengths: `radiation_length` and `interaction_length`¶

These are properties of a material, not distances travelled by a particle. They appear as denominators in physics expressions (number of radiation lengths traversed = path / X₀), never as addends.

radiation_length (X₀): mean distance over which an electron loses 1/e of its energy through Bremsstrahlung — an electromagnetic material property.
interaction_length (λ_I): mean free path before a hadronic nuclear interaction — a nuclear material property.
nuclear_interaction_length: a specialisation of interaction_length for nuclear (as opposed to hadronic-elastic) interactions.

All three are distinct from each other and from geometric lengths:

quantity X0      = 88.97 * hep::mm * radiation_length;      // lead X₀
quantity lambda_I = 194.4 * hep::mm * interaction_length;   // lead λ_I
quantity path    = 50. * hep::mm * path_length;

// auto wrong1 = X0 + lambda_I;  // Compile error — different kinds ✓
// auto wrong2 = X0 + path;      // Compile error — different kinds ✓

// Correct usage: use as a denominator
quantity n_X0 = path / X0;                             // dimensionless: number of X₀ traversed

Dimensionless Relativistic Quantities¶

lorentz_factor (γ) and relativistic_beta (β) are dimensionless quantities with specific physical meaning that must not be mixed with generic dimensionless values or with each other:

lorentz_factor (γ = E/E₀): characterises relativistic time dilation and length contraction; ranges from 1 (at rest) to ∞.
relativistic_beta (β = v/c): velocity expressed as a fraction of the speed of light; ranges from 0 to 1. Related to γ by \(\beta = \sqrt{1 - 1/\gamma^2}\).
phase: quantum mechanical phase angle — a cyclic, dimensionless quantity incompatible with both γ and β and with the angular angle kind.

quantity gamma  = 10. * lorentz_factor;
quantity beta   = 0.995 * relativistic_beta;
quantity phi    = 1.57 * phase;

// auto wrong1 = gamma + beta;   // Compile error — different kinds ✓
// auto wrong2 = gamma + phi;    // Compile error — different kinds ✓
// auto wrong3 = gamma + 1.;     // Compile error — not a plain dimensionless ✓

// Physics: recover beta from gamma
quantity beta_from_gamma = relativistic_beta(sqrt(1. - 1. / pow<2>(gamma / lorentz_factor)));

Physical Constants with CODATA Versions¶

One of the major challenges in HEP software is that physical constants change over time as measurements improve. Different frameworks may use different CODATA recommendations, leading to inconsistencies.

The Problem¶

As noted by John Chapman (currently one of the ATLAS Offline Software coordinators):

HEP community feedback

"HEP Software stacks typically rely on multiple projects. Each of which define their physical constants based on the CODATA recommendations. (Note these do change over time!)

Currently the CODATA version is fixed for a particular release of a project, which can lead to (small) inconsistencies, for example if an older version of one project needs to be used, but another project needs to be updated to pick up new functionality.

I would like to find a way to avoid this. If they all depended on a single external library which we could easily configure to use a particular set of recommendations, then that would make things easier."

The Solution: CODATA Namespaces¶

The mp-units HEP system provides constants organized by CODATA release, allowing explicit version selection:

#include <mp-units/systems/hep.h>

using namespace mp_units;
using namespace mp_units::hep::unit_symbols;

// Default: CODATA 2018 (inline namespace - matches CLHEP, Gaudi, Geant4)
quantity kinetic_energy = 10. * MeV;
quantity beta_2018 = kinetic_energy / (hep::electron_mass * c2);  // Uses CODATA 2018

// Interface with ROOT analysis using CODATA 2022
quantity analyze_with_root()
{
  // All constants now use CODATA 2022 values to match ROOT
  quantity rest_energy = hep::codata2022::electron_mass * c2;  // 0.51099895069 MeV
  quantity momentum = sqrt(pow<2>(kinetic_energy + rest_energy) - pow<2>(rest_energy));
  return momentum;
}

// Reproduce legacy results with CODATA 2014
quantity validate_legacy_analysis(quantity<MeV> E_gamma)
{
  // Use older constants for exact reproducibility
  quantity threshold = 2. * hep::codata2014::electron_mass * c2;  // Pair production
  return (E_gamma > threshold) ? E_gamma - threshold : 0. * MeV;
}

Key Differences Between CODATA Releases¶

Constant	CODATA 2014	CODATA 2018	CODATA 2022
Boltzmann constant (MeV/K)	8.6173303×10⁻¹¹	8.617333262×10⁻¹¹ (exact)	8.617333262×10⁻¹¹ (exact)
Electron mass (MeV/c²)	0.5109989461	0.51099895000	0.51099895069
Proton mass (MeV/c²)	938.2720813	938.27208816	938.27208943
Neutron mass (MeV/c²)	939.5654133	939.56542052	939.56542194
Fine structure constant	7.2973525664×10⁻³	7.2973525693×10⁻³	7.2973525643×10⁻³
Classical electron radius (m)	2.8179403227×10⁻¹⁵	2.8179403262×10⁻¹⁵	2.8179403205×10⁻¹⁵

SI 2019 Redefinition

Since the 2019 SI redefinition, several fundamental constants (Planck constant, Boltzmann constant, Avogadro constant, elementary charge) are exact by definition. This is why the Boltzmann constant doesn't change between CODATA 2018 and 2022 - it's now fixed.

Available Constants¶

Constants that are exact (identical across all CODATA releases):

avogadro_constant (Nₐ)
speed_of_light_in_vacuum (c)
planck_constant (h)
elementary_charge (e)
permeability_of_vacuum (μ₀)

Constants that vary by CODATA release (organized in codata2014, codata2018, codata2022 namespaces):

boltzmann_constant (k_B) - exact since 2018
electron_mass (mₑ)
proton_mass (m_p)
neutron_mass (m_n)
atomic_mass_unit (u)
fine_structure_constant (α)
classical_electron_radius (rₑ)
electron_compton_wavelength (λ_C)
bohr_radius (a₀)
bohr_magneton (μ_B)
nuclear_magneton (μ_N)

API Reference¶

For complete API documentation including all available units, constants, and examples:

HEP System Reference - Complete reference for all dimensions, quantities, units, and prefixes