Natural Units and Scales

Every calculation in this course carries factors of $\hslash$ and $c$ that never change the physics and always clutter the algebra. Particle physics removes them by choosing units in which both equal one. What remains is a single dimension — energy, measured in electron-volts and their multiples — and a small set of conversion factors that translate back to the SI quantities an experiment reports. This lesson fixes those conventions.

The natural system

The relevant constants are the reduced Planck constant $\hslash$, which sets the scale of quantum action, and the speed of light $c$, which relates space to time and mass to energy. In SI units,

Natural units are defined by the two conditions

These are not approximations. They are a choice of unit for action and for speed, just as choosing to measure angles in radians fixes the unit of arc length. Once made, the choice couples the mechanical dimensions. Setting $c=1$ makes time and length the same dimension, since a length is a time multiplied by a speed; it also makes mass, momentum, and energy the same dimension, through $E=m{c}^2$ and $E^2=(pc{)}^2+(m{c}^2{)}^2$. Setting $\hslash =1$ then ties that common energy dimension to inverse time and inverse length, since $\hslash$ has dimensions of energy times time.

The net result is that every mechanical quantity has dimension equal to a power of energy. Writing $[X]=E^d$ for the mass dimension $d$ of a quantity $X$:

• mass, energy, momentum: $d=+1$. A proton has mass $0.938$ GeV, and one writes its energy, momentum, and mass in the same unit.
• length, time: $d=-1$. A distance is an inverse energy.
• cross section (an area): $d=-2$.
• action, angular momentum: $d=0$, dimensionless — this is what $\hslash =1$ enforces.

Because all quantities share one dimension, an equation's consistency reduces to matching a single exponent. A term of dimension $E^{+1}$ can only equal another term of dimension $E^{+1}$; a candidate formula with mismatched powers of energy is wrong before any number is inserted. This is the practical payoff of the system and the check applied throughout the course.

The GeV and its neighbors

The unit of energy is the electron-volt, the kinetic energy an electron gains across a one-volt potential difference, $1\text{eV}=1.602\times 10^{-19}$ J. The scales of subatomic physics span many decades of it:

The GeV is the working unit of the field: proton and neutron masses, the $W$ and $Z$ masses ($80.4$ and $91.2$ GeV), and the Higgs mass ($125$ GeV) all sit within three decades of it. Masses are quoted as energies. The proton's mass is $0.938$ GeV, understood as $0.938\text{GeV}/c^2$ in SI, but the $c^2$ is suppressed. Likewise a momentum of $0.938$ GeV means $0.938\text{GeV}/c$.

Converting to SI

Two conversion factors carry the SI content, one for length and one for time. Both follow from restoring $\hslash$ and $c$ where $\hslash =c=1$ hid them. The length conversion uses the combination

with $1\text{fm}=10^{-15}$ m. Because $\hslash c$ has dimension energy times length and equals unity in natural units, a length $L$ in fermis corresponds to an inverse energy through

A quantity naturally sized in inverse GeV is therefore a fraction of a fermi, matching the hadronic length scale. The time conversion uses $\hslash$ alone,

A decay width $\Gamma$ measured in GeV converts to a lifetime $\tau =\hslash /\Gamma$ in seconds through this factor; the connection is developed in the lesson on decay rates.

Cross sections and areas

A cross section has dimensions of area, natural dimension $E^{-2}$. Experiments quote it in barns:

a unit chosen so that a typical nuclear cross section is of order one barn. The submultiples millibarn, microbarn, nanobarn, picobarn, and femtobarn descend by factors of a thousand and match progressively rarer processes; LHC Higgs production is measured in picobarns and femtobarns.

To convert a cross section between barns and GeV${}^{-2}$, square the length conversion. From $1{\text{fm}}^2=10\text{mb}$ and $(\hslash c{)}^2=(0.1973\text{GeV⋅fm}{)}^2=0.03894{\text{GeV}}^2{\text{⋅fm}}^2$,

This one factor turns any theoretical cross section, which emerges from a Feynman calculation in GeV${}^{-2}$, into the millibarns or picobarns a detector counts.

Restoring $\hslash$ and $c$

A result computed in natural units is converted to SI by inserting the unique combination of $\hslash$ and $c$ that fixes its dimensions. The procedure is mechanical:

1. Determine the SI dimensions the answer must have (length, time, area, energy).
2. Multiply the natural-unit result, expressed in powers of energy, by powers of $\hslash$ and $c$ chosen to produce those dimensions. Since $[\hslash ]=\text{energy⋅time}$ and $[c]=\text{length/time}$, the two exponents are fixed by requiring the time and length dimensions to come out right.

The rule for which powers to insert has a compact form. If a quantity has natural dimension $E^n$ and one wants it in SI, write it as $(\text{number})\times {\text{GeV}}^n$ and multiply by $(\hslash c{)}^{-n}$ when $n$ is a length-like power, or by the appropriate mix of $\hslash$ and $c$ set by matching both the length and the time dimension separately. In practice the two factors $\hslash c=0.1973$ GeV·fm and $\hslash =6.582\times 10^{-25}$ GeV·s cover every case that arises: lengths and areas use $\hslash c$, lifetimes use $\hslash$, and energies need nothing.

The conventions fixed here — energies and masses in GeV, lengths and times as inverse GeV, cross sections in GeV${}^{-2}$ converted to barns by $0.3894{\text{GeV}}^2\text{⋅mb}$ — are assumed without comment in every later calculation. The next lesson builds the relativistic kinematics on top of them, starting from the energy-momentum four-vector.¹