--- title: Natural Units and Scales module: Units and Kinematics moduleNumber: 2 lessonNumber: 1 order: 201 summary: > Setting $\hbar = c = 1$ collapses mass, momentum, and energy into a single unit, the GeV, and turns lengths and times into inverse energies through the conversion $\hbar c = 197.3$ MeV·fm. This lesson fixes the natural-unit conventions used for the rest of the course, converts cross sections between barns and GeV$^{-2}$, and shows how to restore factors of $\hbar$ and $c$ by dimensional analysis. topics: [Units and Kinematics] draft: false sources: - book: Griffiths ref: "Appendix — Units and Natural Units; Ch. 3 opening" - book: Thomson ref: "§1.3 Natural units" - book: PDG ref: "Physical constants; CODATA values (pdg.lbl.gov)" --- Every calculation in this course carries factors of $\hbar$ 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 $\hbar$, 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, $$ \hbar = 1.0546 \times 10^{-34}\ \text{J·s}, \qquad c = 2.9979 \times 10^{8}\ \text{m/s}. $$ Natural units are defined by the two conditions $$ \hbar = 1, \qquad c = 1. $$ 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 $E = mc^2$ and $E^2 = (pc)^2 + (mc^2)^2$. Setting $\hbar = 1$ then ties that common energy dimension to inverse time and inverse length, since $\hbar$ 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 $\hbar = 1$ enforces. $$ % caption: The SI-to-natural dimensional dictionary. Each mechanical quantity % carries a single power of energy once c and hbar are set to one; the exponent d % is the mass dimension. Areas (cross sections) sit at d = -2, times and lengths % at d = -1. \begin{tikzpicture}[font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \draw[black, thick] (-4.6,-2.7) rectangle (4.6,2.3); \draw[black] (0,-2.7) -- (0,2.3); \draw[black] (-4.6,1.5) -- (4.6,1.5); \node[font=\small, black!70] at (-2.3,1.9) {quantity}; \node[font=\small, black!70] at (1.9,1.9) {mass dimension}; \foreach \yy/\q/\dm in { 1.0/{energy, mass, momentum}/{d = +1}, 0.3/{inverse length, inverse time}/{d = +1}, {-0.4}/{length, time}/{d = -1}, {-1.1}/{area, cross section}/{d = -2}, {-1.8}/{action, angular momentum}/{d = 0}} { \node[anchor=west, black!75] at (-4.4,\yy) {\q}; \node[black!75] at (1.9,\yy) {\dm}; } \end{tikzpicture} $$ 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: | Scale | Energy | Physics | |---|---|---| | eV | $1\ \text{eV}$ | atomic binding, optical photons | | keV | $10^{3}\ \text{eV}$ | X-rays, inner-shell electrons | | MeV | $10^{6}\ \text{eV}$ | nuclear binding, electron mass $0.511$ MeV | | GeV | $10^{9}\ \text{eV}$ | proton mass $0.938$ GeV, hadron scale | | TeV | $10^{12}\ \text{eV}$ | LHC collision energy, $\sqrt s = 13.6$ TeV | 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 $0.938\ \text{GeV}/c$. $$ % caption: A logarithmic ruler of energy scales from the atomic eV to the Planck % energy, spanning 28 decades. The subatomic physics of this course occupies the % MeV-to-TeV window; the Planck scale at 1.2 times 10 to the 19 GeV marks where % gravity becomes strong. \begin{tikzpicture}[font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \draw[black!70, thick, -{Stealth[length=2.4mm]}] (0,0) -- (12.2,0); \node[black, right, font=\scriptsize] at (12.2,0) {log energy}; \foreach \x/\lab/\en in { 0.4/eV/atom, 2.4/keV/{X-ray}, 4.4/MeV/nucleus, 6.4/GeV/proton, 8.4/TeV/LHC, 11.6/{10^{19} GeV}/Planck} { \draw[black] (\x,-0.14) -- (\x,0.14); \node[above, font=\scriptsize] at (\x,0.18) {$\lab$}; \node[black, below, font=\scriptsize] at (\x,-0.18) {\en}; } \draw[acc!70, very thick] (4.4,-0.5) -- (8.4,-0.5); \node[acc, below, font=\scriptsize] at (6.4,-0.62) {this course}; \end{tikzpicture} $$ ## Converting to SI Two conversion factors carry the SI content, one for length and one for time. Both follow from restoring $\hbar$ and $c$ where $\hbar = c = 1$ hid them. The length conversion uses the combination $$ \hbar c = 197.327\ \text{MeV·fm} = 0.197327\ \text{GeV·fm}, $$ with $1\ \text{fm} = 10^{-15}$ m. Because $\hbar c$ has dimension energy times length and equals unity in natural units, a length $L$ in fermis corresponds to an inverse energy through $$ L\ [\text{fm}] = \frac{L}{0.1973}\ \text{GeV}^{-1}, \qquad\text{so}\qquad 1\ \text{GeV}^{-1} = 0.1973\ \text{fm}. $$ A quantity naturally sized in inverse GeV is therefore a fraction of a fermi, matching the hadronic length scale. The time conversion uses $\hbar$ alone, $$ \hbar = 6.582 \times 10^{-25}\ \text{GeV·s}, \qquad\text{so}\qquad 1\ \text{GeV}^{-1} = 6.582 \times 10^{-25}\ \text{s}. $$ A decay width $\Gamma$ measured in GeV converts to a lifetime $\tau = \hbar/\Gamma$ in seconds through this factor; the connection is developed in the lesson on decay rates. $$ % caption: The hbar-c bridge as a labelled conversion. A length in fermis enters % on the left, is divided by 0.1973 GeV per fermi, and leaves as an inverse energy % in inverse GeV. Reading the bridge backward converts an inverse energy to a % length. \begin{tikzpicture}[font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \node[draw=black, thick, minimum width=2.2cm, minimum height=1.0cm, align=center] (len) at (-4.0,0) {length\\in fm}; \node[draw=acc, very thick, fill=acc!10, minimum width=3.0cm, minimum height=1.0cm, align=center] (mid) at (0,0) {divide by\\0.1973 GeV fm}; \node[draw=black, thick, minimum width=2.4cm, minimum height=1.0cm, align=center] (inv) at (4.2,0) {inverse energy\\in inverse GeV}; \draw[-{Stealth[length=2.4mm]}, black, thick] (len) -- (mid); \draw[-{Stealth[length=2.4mm]}, black, thick] (mid) -- (inv); \draw[-{Stealth[length=2.4mm]}, black, thick] (inv) to[bend right=32] node[below, black, font=\scriptsize]{multiply back} (len); \end{tikzpicture} $$ ## Cross sections and areas A cross section has dimensions of area, natural dimension $E^{-2}$. Experiments quote it in barns: $$ 1\ \text{barn} = 10^{-28}\ \text{m}^2 = 100\ \text{fm}^2, $$ 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 $(\hbar c)^2 = (0.1973\ \text{GeV·fm})^2 = 0.03894\ \text{GeV}^2\text{·fm}^2$, $$ (\hbar c)^2 = 0.3894\ \text{GeV}^2\text{·mb}, \qquad\text{so}\qquad 1\ \text{GeV}^{-2} = 0.3894\ \text{mb} = 389.4\ \mu\text{b}. $$ 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. $$ % caption: Areas across the subatomic range on a logarithmic scale. One barn is the % nuclear geometric area; a proton-proton total cross section is tens of millibarns; % rare electroweak processes fall to picobarns and femtobarns, each step down a % factor of one thousand from the last. \begin{tikzpicture}[font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \draw[black!70, thick] (0,0) -- (0,-6.0); \foreach \yy/\unit/\proc in { -0.3/barn/{nuclear area}, -1.4/millibarn/{proton-proton total}, -2.5/microbarn/{hadron production}, -3.6/nanobarn/{heavy-quark pairs}, -4.7/picobarn/{Higgs production}, -5.8/femtobarn/{rare decays}} { \fill (0,\yy) circle (2.0pt); \node[anchor=west, font=\scriptsize] at (0.25,\yy) {\unit}; \node[black, anchor=west, font=\scriptsize] at (2.55,\yy) {\proc}; } \node[black, rotate=90, anchor=south, font=\scriptsize] at (-0.5,-3.0) {smaller area, rarer process}; \end{tikzpicture} $$ ## Restoring $\hbar$ and $c$ A result computed in natural units is converted to SI by inserting the unique combination of $\hbar$ 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 $\hbar$ and $c$ chosen to produce those dimensions. Since $[\hbar] = \text{energy·time}$ and $[c] = \text{length/time}$, the two exponents are fixed by requiring the time and length dimensions to come out right. > **Worked example.** The Compton wavelength of a particle of mass $m$ is > $\lambda_C = 1/m$ in natural units, a length equal to an inverse energy. To > restore SI, a length must appear, so multiply by $\hbar c$, which carries one > power of length: > > $$ > \lambda_C = \frac{\hbar c}{m c^2} = \frac{\hbar}{mc}. > $$ > > For the electron, $m c^2 = 0.511$ MeV, so > $\lambda_C = 197.3\ \text{MeV·fm} / 0.511\ \text{MeV} = 386\ \text{fm} = > 3.86 \times 10^{-13}$ m. The dimensional check and the number agree with the > textbook value. 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 $(\hbar c)^{-n}$ when $n$ is a length-like power, or by the appropriate mix of $\hbar$ and $c$ set by matching both the length and the time dimension separately. In practice the two factors $\hbar c = 0.1973$ GeV·fm and $\hbar = 6.582 \times 10^{-25}$ GeV·s cover every case that arises: lengths and areas use $\hbar c$, lifetimes use $\hbar$, 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.[^units] [^units]: Griffiths, _Introduction to Elementary Particles_, Appendix on units, and the opening of Ch. 3, set the natural-unit conventions; Thomson, _Modern Particle Physics_, §1.3, gives the same system with the $\hbar c$ and cross-section conversions. Constant values follow CODATA via the Particle Data Group, _Review of Particle Physics_, [pdg.lbl.gov](https://pdg.lbl.gov): $\hbar c = 197.3269804$ MeV·fm and $\hbar = 6.582119569 \times 10^{-25}$ GeV·s.