Particles and Forces: the Standard Model

Beginner

Start with what you may already know: an atom has a tiny nucleus made of protons, which carry positive charge, and neutrons, which carry none, with negative electrons around it. For a long time these three looked like the final building blocks. They are not all final: each proton and each neutron is a little bag of three smaller particles called quarks. Two kinds are enough for ordinary matter, “up” and “down”. A proton is up-up-down; a neutron is up-down-down.

Check yourself: if one up quark in a proton turned into a down quark, what would the proton become? Answer: a neutron. This really happens in some radioactive nuclei.

What holds quarks together? Gluons, the carriers of the strong force. In the animation they are the springs between the jiggling quarks. Like rubber bands, they pull harder the farther they are stretched, so a quark can never be pulled out alone: the energy you spend creates new quarks instead. Each quark also has a “colour” (red, green or blue; just a name, not a real colour), and a hopping gluon carries colour from one quark to another, so all three colours are always present.

Electrons belong to a different family, the leptons. Two electrons are both negative, so they repel, more strongly the closer they get. On the right of the animation, two electrons slow down, trade a photon, a particle of light, at their closest point, and fly apart: this trade is how physicists picture the electric push.

Nature has just four basic forces: gravity, electromagnetism, the strong force and the weak force, which drives some radioactive decays. And every particle has an antimatter twin with the opposite charge; when the two meet, both vanish in a flash of light.

Intermediate

Units for the very small

Particle physicists measure energy in electronvolts: 1 eV is the energy an electron gains across one volt, $1.602\times 10^{-19}\mathrm{J}$, and 1 MeV is a million of them. Because $E=m{c}^2$, a mass can be quoted as a rest energy over $c^2$: the electron is 0.511 MeV/c², the proton 938 MeV/c².

Lengths inside nuclei are measured in femtometres, $1\mathrm{fm}=10^{-15}\mathrm{m}$, and the proton in the animation is about 1.7 fm across. One constant links energy and length, $\hslash c\approx 197.3\mathrm{MeV}\cdot \mathrm{fm}$, with $\hslash =h/2\pi$: dividing it by an energy gives a distance.

Twelve particles of matter

The Standard Model sorts matter into twelve fermions, six quarks and six leptons, in three generations. The first (up and down quarks, electron and its neutrino) contains everything atoms are made of. The second (charm, strange, muon) and the third (top, bottom, tau), each with a neutrino, repeat the same charges with heavier particles.

Up-type quarks (u, c, t) carry $+\frac{2}{3}e$ and down-type quarks (d, s, b) carry $-\frac{1}{3}e$, where $e$ is the elementary charge; the electron, muon and tau carry $-e$ and the neutrinos nothing. The two nucleons then come out right:

• $p=\mathrm{uud}:Q=\frac{2}{3}+\frac{2}{3}-\frac{1}{3}=+1$
• $n=\mathrm{udd}:Q=\frac{2}{3}-\frac{1}{3}-\frac{1}{3}=0$

Quarks carry a second kind of charge, colour, with three values named red, green and blue; the names are just labels. Only colourless combinations exist on their own: three quarks of different colours make a baryon such as the proton, and a quark with an antiquark of the matching anticolour makes a meson such as the pion ${\pi }^{+}=\mathrm{u}\bar{\mathrm{d}}$, of charge $\frac{2}{3}+\frac{1}{3}=+1$. Together they are the hadrons. Leptons have no colour and feel no strong force.

Forces as exchanged particles

In this picture a force is the exchange of a boson, a messenger particle. The photon $\gamma$ carries electromagnetism between electric charges, eight gluons carry the strong force between colour charges, and the $W^{+}$, $W^{-}$ and $Z^0$ carry the weak force, in which the W turns one kind of quark or lepton into another and so drives β decay. The Higgs boson, found in 2012, is a ripple of the field that gives particles their masses, not a force carrier. Gravity is not part of the model.

A carrier with mass can only travel a limited distance, so it gives a short-ranged force:
$R\approx \frac{\hslash }{mc}=\frac{\hslash c}{m{c}^2},\hslash c\approx 197.3\mathrm{MeV}\cdot \mathrm{fm}$
The photon is massless, so the electric force reaches arbitrarily far and keeps the inverse-square form of Coulomb's law:
$F=\frac{k{e}^2}{r^2}\text{(photon exchange)}$

The W has 80.4 GeV/c², so $R\approx 197.3/80400\mathrm{fm}\approx 0.0025\mathrm{fm}$. Gluons are massless too, but colour stays locked inside hadrons; the force between neighbouring nucleons is a colourless leftover, carried mainly by pions of 140 MeV/c², so $R\approx 197.3/140\mathrm{fm}\approx 1.4\mathrm{fm}$.

Worked example: the electrons in the animation

On the right of the animation two electrons approach head-on, stop for an instant 100 nm apart and fly back out. At closest approach, with $k=8.99\times 10^9\mathrm{N}\cdot {\mathrm{m}}^2/{\mathrm{C}}^2$,
$F=\frac{k{e}^2}{r^2}=\frac{8.99\times 10^9\cdot (1.60\times 10^{-19}{)}^2}{(1.00\times 10^{-7}{)}^2}\approx 2.3\times 10^{-14}\mathrm{N}$
that is 23 femtonewtons, the peak of 23.1 fN on the $F$ readout when the $r$ readout shows 100 nm.

At the start the electrons are 343 nm apart, 3.43 times farther, so the force is $3.43^2\approx 11.8$ times weaker, $23.1/11.8\approx 1.96\mathrm{fN}$ (the readout shows 2.0 fN). The red $F$ arrows are drawn to scale, so they grow about twelvefold on the way in.

At closest approach both electrons are at rest, so the whole energy of the pair is electric potential energy, $k{e}^2/r=2.3\times 10^{-21}\mathrm{J}$ or 14.4 meV. At the start 4.2 meV of it is still potential and 10.2 meV is kinetic: each electron has 5.1 meV, a speed of about 42 km/s. The real encounter lasts about 8 ps, so the 9 s loop slows it down by a factor of $10^{12}$.

Worked example: annihilation in a PET scanner

A positron emission tomography (PET) scan uses a tracer, such as sugar labelled with fluorine-18, whose nucleus turns a proton into a neutron, $\mathrm{p}\to \mathrm{n}+e^{+}+{\nu }_e$, conserving charge ($+1=0+1+0$) and lepton number ($0=0-1+1$). The positron stops within a millimetre and meets an electron; both are almost at rest, so their total momentum is almost zero.

A single photon cannot carry away the energy, because a photon always has momentum $E/c$. Two photons flying apart back to back can, each with the rest energy of one particle:
$e^{+}+e^{-}\to \gamma +\gamma ,E_{\gamma }=m_e{c}^2=511\mathrm{keV}\text{(at rest)}$
Here $m_e{c}^2=9.109\times 10^{-31}\cdot (2.998\times 10^8{)}^2=8.187\times 10^{-14}\mathrm{J}$, which is 511 keV, and the wavelength is $\lambda =hc/E=1239.8\mathrm{keV}\cdot \mathrm{pm}/511\mathrm{keV}\approx 2.43\mathrm{pm}$, some 200 000 times shorter than green light.

A ring of detectors records both photons within a few nanoseconds; the annihilation lies on the line joining the hits. If one photon arrives 200 ps earlier, the point is $c\Delta t/2=3.00\times 10^8\cdot 2\times 10^{-10}/2=3\mathrm{cm}$ from the middle of the line, towards that detector.

Reading the space-time diagram

Above the electrons the intermediate layer draws a space-time diagram: position runs across, time runs upward, and a dashed line marks the present. Each electron traces a purple worldline, slanted while far apart, vertical at $t=4.5\mathrm{s}$ when both stand still 100 nm apart, then bending away. The steeper a worldline, the slower the particle.

The gold wiggle at that instant is the exchanged photon $\gamma$, joining two vertices (the black dots). The smooth worldlines are a classical picture; the quantum description sums interaction diagrams, each vertex being one emission or absorption.

The proton obeys a similar rule for colour. Four gluons hop during each loop, and whenever none is flying you can count one red, one green and one blue quark.

Common mistakes

• The proton's mass is not the sum of its quark masses: they add up to about 9 MeV/c², the proton has 938. About 99 % is the energy of the gluon field and of the quarks' motion, via $E=m{c}^2$.
• The weak force is short-ranged because its carriers are heavy ($R\approx 0.0025\mathrm{fm}$ for the W), not because it is the weakest: gravity is far weaker, yet reaches arbitrarily far.
• A positron and an electron at rest cannot annihilate into one photon: a photon of 1022 keV would carry momentum $E/c$, but the total is zero. Two back-to-back photons of 511 keV balance both.
• Neutral does not mean being your own antiparticle: the antineutron has baryon number $-1$ and annihilates with a neutron, whereas the photon and the ${\pi }^0$ are their own antiparticles.

Where you meet it

PET scanners use antimatter every day, and cosmic-ray muons, heavy electrons, cross every square centimetre at sea level about once a minute. The Sun shines because the weak force turns protons into neutrons in its core (see Fission, Fusion and E = mc²), and carbon dating counts β decays of carbon-14 (see Radioactivity and Half-Life).

Colliders steer particles around rings with magnets, using the circular motion of Magnetic Force on Moving Charges:
$R=\frac{p}{eB}\text{(circular accelerator)}$
A momentum of 1 GeV/c corresponds to $BR=3.336\mathrm{T}\cdot \mathrm{m}$, so protons of 7000 GeV/c in the 8.33 T bending magnets of the Large Hadron Collider move on a circle of radius $R=7000\cdot 3.336/8.33\mathrm{m}\approx 2.80\mathrm{km}$.

Advanced

The range of a force and the mass of its carrier

Legend

$\varphi$	Field of the carrier around the source; the vertical scale is arbitrary.
$r$	Distance from the source, drawn in units of $R$.	$\mathrm{m}$
$R$	Range of the force, $R=\hslash /(mc)$; at $r=R$ the solid curve is at $1/e$ of the dashed one.	$\mathrm{m}$
$m$	Mass of the carrier: zero for the dashed curve, positive for the solid one.	$\mathrm{kg}$

Why does the electric force reach across the universe while the nuclear force stops dead after about a femtometre? This section shows that the answer is the mass of the particle that carries the force, and then turns the measured range of the nuclear force into a prediction of a particle's mass.

Figure 1 shows the field $\varphi$ that a static source sets up at distance $r$, for two kinds of carrier. The dashed curve ($m=0$) is for a massless carrier. It is Coulomb's $1/r$, which contains no length scale and so never cuts off. The solid curve ($m>0$) is for a carrier of mass $m$. It is the dashed curve multiplied by an extra factor that is close to 1 only very near the source and shrinks with distance, so the solid curve sinks further and further below the dashed one. Look at the two dots above the end of the bracket $R$: there the solid curve is down to $1/e\approx 0.37$ of the dashed one. At $2R$ the ratio is 0.14, at $3R$ it is 0.05, and beyond a few $R$ the field is gone for practical purposes. The length $R$ is the range of the force; the rest of the section shows what sets it.

The contrast in nature is stark. The nuclear force between protons and neutrons fades within about 1.4 fm, while the electric force reaches arbitrarily far. In 1935 Hideki Yukawa turned that contrast into a prediction. Picture a nucleon at rest that emits a carrier of mass $m$. Creating the carrier breaks energy conservation by $m{c}^2$. The energy–time uncertainty relation tolerates such a debt only for a time of about $\hslash /(m{c}^2)$, where $\hslash$ is the reduced Planck constant; then the carrier must be absorbed again. Even at nearly the speed of light it covers at most $c$ times that time, so the range is
$R\approx \frac{\hslash }{mc}=\frac{\hslash c}{m{c}^2},\hslash c\approx 197.3\mathrm{MeV}\cdot \mathrm{fm}$
The heavier the carrier, the shorter the loan and the shorter the range. A massless carrier can be borrowed indefinitely, so its range is infinite.

The same result follows exactly when relativity and quantum mechanics are combined. A free quantum of mass $m$ obeys $E^2=p^2{c}^2+m^2{c}^4$. In quantum mechanics energy and momentum act on a wave as the operators $i\hslash \partial /\partial t$ and $-i\hslash \nabla$. Putting them in place of $E$ and $p$ and dividing by ${\hslash }^2{c}^2$ turns the energy relation into a wave equation for the field $\varphi$:
$\frac{1}{c^2}\frac{{\partial }^2\varphi }{\partial t^2}-{\nabla }^2\varphi +\frac{m^2{c}^2}{{\hslash }^2}\varphi =0$
The first two terms make up the ordinary wave equation, which on its own describes waves moving at $c$. The third is the mass term, the only place where $m$ enters.

Around a static point source nothing changes in time, so the first term drops out and the equation becomes ${\nabla }^2\varphi =\varphi /R^2$, with $R=\hslash /(mc)$. To solve it away from the source, write $\varphi =u/r$: for a spherically symmetric field this turns the equation into $d^{2}u/d{r}^2=u/R^2$, whose solution that dies away at large $r$ is $u=e^{-r/R}$. Hence
$\varphi (r)\propto \frac{e^{-r/R}}{r},R=\frac{\hslash }{mc}$
These are exactly the two curves of Figure 1. With $m=0$ the range is infinite, the exponential equals 1 and $\varphi \propto 1/r$: the dashed Coulomb curve. With a mass, $e^{-r/R}$ is the extra factor of the solid curve, and it kills the field beyond a few $R$.

Now run the argument backwards. A range of 1.4 fm calls for $m{c}^2=\hslash c/R=197.3\mathrm{MeV}\cdot \mathrm{fm}/1.4\mathrm{fm}\approx 140\mathrm{MeV}$, a carrier of about 140 MeV/c². The pion, found in 1947, has just that mass. Dozens more strongly interacting particles, hadrons, followed, and quarks later explained their regular families. The table in the animation's expert layer shows the order that emerged: three generations of quarks and leptons with identical charges, differing only in mass, and the force carriers in a column of their own.

Symmetry and the origin of forces

Why these particular carriers, and why is the photon massless? The modern answer is symmetry: the forces are not added to the theory by hand but are demanded by symmetries of its laws.

The link between symmetry and conservation is older. A symmetry is a change that leaves the laws unchanged. If the laws are the same at every place, momentum is conserved; if they are the same at all times, energy is conserved; if they are the same in every direction, angular momentum is conserved. Emmy Noether proved in 1918 that this is general: every continuous symmetry of the laws brings its own conserved quantity.

Quantum mechanics adds a symmetry with no classical counterpart. A particle is described by a complex amplitude $\psi$, and only the probability $|\psi {|}^2$ is ever measured. Multiplying every amplitude by the same phase factor $e^{i\delta }$ therefore changes nothing observable, since $|e^{i\delta }\psi {|}^2=|\psi {|}^2$. Any angle $\delta$ will do, so this is a continuous symmetry, and the conserved quantity that Noether's theorem attaches to it is electric charge.

Now make the demand stronger and let the phase $\delta (x,t)$ differ from point to point and from moment to moment. The probabilities are still unchanged, but the wave equation is not. It contains derivatives of $\psi$, and the derivative of $e^{i\delta }\psi$ picks up an extra term in the derivatives of $\delta$. The laws can keep this local symmetry only if a new field is added whose own change cancels those extra terms exactly. That field is the electromagnetic potential, and its quantum is the photon. A mass term for this field would itself break the symmetry, so the photon must be massless, which by the previous section is why the electric force has an infinite range.

The same recipe, applied to richer symmetries, produces the other carriers. Colour symmetry, called $\mathrm{SU}(3)$ because it mixes the three colours, needs $3^2-1=8$ fields: the eight gluons. The electroweak symmetry $\mathrm{SU}(2)\times \mathrm{U}(1)$ needs $3+1=4$ fields, which appear as $W^{+}$, $W^{-}$, $Z^0$ and the photon.

Hidden symmetry: the Higgs field

Legend

$V$	Potential energy density of the Higgs field.	$J/{\mathrm{m}}^3$
$\varphi$	Value of the Higgs field (one component of it), given as an energy in GeV, as is usual in particle physics.	$\mathrm{J}$
$v$	Value of the field at the two lowest points, $\varphi =\pm v$; $v\approx 246\mathrm{GeV}$.	$\mathrm{J}$

The previous section leaves a puzzle. The same symmetry that keeps the photon massless forbids mass terms for the W and Z as well, yet they are heavy, about 80 and 91 GeV/c². The way out is that the laws keep the symmetry while the vacuum, the state of lowest energy, does not. Figure 2 shows how that can happen.

Figure 2 is a cross-section of the potential energy density $V$ of the Higgs field, plotted against the field's value $\varphi$ (here $\varphi$ is the Higgs field, not the field of Figure 1). The curve is symmetric: it treats $+\varphi$ and $-\varphi$ alike. Yet its symmetric point, $\varphi =0$, is a hilltop, marked by the pale ball. Any small disturbance sends the field rolling down, along the dashed path, until it comes to rest in one of the two troughs, at $\varphi =+v$ (the dark ball) or at $\varphi =-v$. Now turn the curve about the vertical axis. It becomes a Mexican hat, whose trough is a whole circle of equally low points. The field must settle at one of them, so the state it picks is less symmetric than the laws, which allow every one. The distance of the trough from zero is the value $v\approx 246\mathrm{GeV}$, and the field holds this value throughout space.

Particles moving through this everywhere-present background interact with it, and that interaction is what we measure as their mass. The W gets $m_W{c}^2=gv/2\approx 80\mathrm{GeV}$, where $g\approx 0.65$ is the weak coupling. Each fermion gets $m_f{c}^2=y_{f}v/\sqrt{2}$, set by its own coupling $y_f$ to the field: the top quark has $y_t\approx 1$, the electron only $y_e\approx 3\times 10^{-6}$, which is why their masses differ by a factor of some 300 000. The photon belongs to the one combination of electroweak fields that the vacuum leaves unbroken, so it stays massless.

The field itself can also ripple about its resting value $v$, like a ball wobbling in the trough. The quantum of that ripple is the Higgs boson, found in 2012 at 125 GeV/c².

Mirror symmetry and CP

Not every symmetry of everyday physics holds for the weak force. Mirror reflection fails, and its failure, found in 1957, was a surprise.

In that year cobalt-60 nuclei were cooled to a very low temperature and lined up in a magnetic field, so that their spins all pointed the same way. They emitted β electrons mostly in the direction opposite to the nuclear spin. Now imagine the experiment seen in a mirror perpendicular to the spin axis. The nuclei rotate in planes parallel to the mirror, so their sense of rotation, and hence their spin, is unchanged; but the electrons travelling along the axis are reversed. The mirror image would therefore show electrons leaving mostly along the spin, which never happens. The weak force can tell a process from its mirror image.

Combining the mirror with the exchange of matter and antimatter, an operation called CP, almost restores the symmetry, but not quite: neutral K and B mesons break it, kaons by about two parts in a thousand and B mesons by tens of per cent in some decays.

Some conservation laws, by contrast, have never been seen to fail. Electric charge, baryon number $B$ ($+\frac{1}{3}$ for each quark, so $+1$ for a proton) and lepton number $L$ ($+1$ for each lepton, $-1$ for each antilepton) hold in every reaction observed. The decay $\mathrm{p}\to e^{+}+{\pi }^0$, for example, would conserve charge but change $B$ from 1 to 0 and $L$ from 0 to $-1$; searches show that in that channel protons outlive $10^{34}$ years, far longer than the age of the universe.

Momentum transfer and the coupling constant

Legend

$F$	Electric force between the two electrons; it is repulsive.	$\mathrm{N}$
$t$	Time in the animation, slowed down about $10^{12}$ times.	$\mathrm{s}$
$t_c$	Moment of closest approach, in the middle of the run; the curve is symmetric about it.	$\mathrm{s}$
$F_{\max }$	Largest force, reached at closest approach: 23.1 fN.	$\mathrm{N}$
$\Delta t$	Width of the rectangle of height $F_{\max }$ with the same area as the curve, $\Delta t=2p/F_{\max }$: how long the push lasts.	$\mathrm{s}$
$p$	Momentum of one electron at the start of the run; the shaded area is $2p$.	$\mathrm{kg}\cdot m/s$

How does exchanging a particle push two others apart? One concrete collision, between the two electrons of the animation, shows how much momentum passes between them and how long the push lasts. The same exchange picture then explains why electromagnetic calculations can be made so precise.

Figure 3 plots the force $F$ between the two electrons against time $t$; it is the red graph in the animation's expert layer. Three features are worth reading off. First, the curve is symmetric about $t_c$, the moment of closest approach, when the electrons are 100 nm apart, and it peaks there at $F_{\max }=23.1\mathrm{fN}$. Second, at the start and the end of the run the electrons are 343 nm apart and still push each other with 1.96 fN, so the curve does not reach zero at its edges. Third, the shaded area under the curve, labelled $2p$, is the impulse: the total momentum the force hands over.

Why is that area $2p$? Each electron begins with momentum $p$ directed towards the other. The repulsion slows it to rest at $t_c$ and then drives it back out, so it ends with momentum $p$ directed away. Its momentum has changed by $2p$, and by Newton's second law, $F=dp/dt$, the change in momentum is the area under the force–time graph:
$\int Fdt=\Delta p=2p$
The dashed rectangle turns this area into a duration. It has the same area as the curve and the height $F_{\max }$, so its width is $\Delta t=2p/F_{\max }$. A constant push of $F_{\max }$ lasting $\Delta t$ would hand over the same momentum, so $\Delta t$ measures how long the collision lasts: it is the width of the bump.

Now the numbers. At closest approach both electrons are at rest. Moving out from 100 nm to 343 nm, their electric potential energy falls by 10.2 meV, and the two equal electrons share it equally, so at 343 nm each has kinetic energy $K=5.1\mathrm{meV}$. With the electron mass $m_e$, its momentum is $p=\sqrt{2m_{e}K}=3.9\times 10^{-26}\mathrm{kg}\cdot m/s$, and the area is $2p=7.7\times 10^{-26}\mathrm{N}\cdot \mathrm{s}$. Dividing by the peak force gives $\Delta t\approx 3.3\mathrm{ps}$, the real width of the bump. The animation slows time by a factor of about $1.1\times 10^{12}$ (it shows 8.2 ps in 9 s), which stretches the bump to 3.7 s on screen.

The $2p$ is carried across by a photon, but an unusual one. In the centre-of-mass frame each electron ends with the energy it began with, so the photon delivers momentum but no energy. A free photon always has $E=pc$, so a photon with momentum and no energy cannot exist on its own: it is virtual, allowed only for the brief time between its emission and its absorption. Treating the electrons as following classical paths is still fair, because their reduced wavelength $\hslash /p\approx 2.7\mathrm{nm}$ at the start is far below the 100 nm closest approach.

The full quantum description is a sum of terms, each drawn as a diagram in which particle lines meet at points called vertices. One exchanged photon gives the lowest-order term. Each further photon adds two vertices, and each vertex brings a factor of the electron's charge $e$. Two extra factors of $e$ make $e^2$, which in dimensionless form, with the vacuum permittivity ${\epsilon }_0$, is the fine-structure constant
$\alpha =\frac{e^2}{4\pi {\epsilon }_0\hslash c}\approx \frac{1}{137}$
Each extra photon therefore suppresses its term by roughly $1/137$, and so a few terms give the electron's magnetic moment to one part in $10^{12}$.

The other forces have couplings of very different sizes: the strong coupling near 1 GeV is of order 1, so this expansion fails there; the weak coupling at nuclear energies is about $10^{-5}$; and gravity between two protons is a mere $6\times 10^{-39}$.

Colour, confinement and asymptotic freedom

Legend

$V$	Potential energy of the quark–antiquark pair (Cornell potential).	$\mathrm{J}$
$r$	Separation of the quark and the antiquark.	$\mathrm{m}$
${\alpha }_s$	Strong coupling; sets the depth of the Coulomb-like term.
$\sigma$	String tension: the slope of the straight term $\sigma r$.	$J/m$
$r_1$	Separation at which the two terms are equal in size and cancel, so $V=0$; about 0.3 fm.	$\mathrm{m}$
$r_2$	A separation of 1 fm, for comparison with $r_1$.	$\mathrm{m}$
$\Delta V$	Rise of $V$ from $r_1$ to $r_2$: about 0.8 GeV.	$\mathrm{J}$

No quark has ever been seen on its own. The reason lies in the force between a quark and an antiquark: it resembles the electric force at short range, but at long range it stops weakening altogether.

Figure 4 plots the potential energy $V$ of a quark and an antiquark against their separation $r$. The solid curve is the Cornell potential, a simple formula that fits the masses of heavy quark–antiquark pairs well:
$V(r)=-\frac{4}{3}\frac{{\alpha }_s\hslash c}{r}+\sigma r$
Its two terms are drawn dashed. The first, labelled ${\alpha }_s$ (dashed blue), is Coulomb-like: it varies as $-1/r$, its depth is set by the strong coupling ${\alpha }_s$, and it dominates at short range, pulling $V$ steeply down. The second, labelled $\sigma$ (dashed gold), is a straight line whose slope $\sigma$ is called the string tension; it dominates at long range and keeps rising without limit. Setting the sizes of the two terms equal, $\frac{4}{3}{\alpha }_s\hslash c/r_1=\sigma r_1$, gives the separation where they balance:
$r_1=\sqrt{\frac{4{\alpha }_s\hslash c}{3\sigma }}\approx 0.3\mathrm{fm}$
for ${\alpha }_s=0.3$ and $\sigma \approx 0.9GeV/fm$. Because the terms have opposite signs, they cancel there, and the solid curve crosses $V=0$ at $r_1$.

Why should the energy grow in proportion to the separation? Gluons, unlike photons, carry the charge of their own force, colour, so they interact with one another. Electric field lines between two charges spread out through all of space. The colour field lines between a quark and an antiquark instead pull one another together into a narrow tube, a string, with a fixed energy per unit length. Stretching the tube by a length $\Delta r$ costs the energy $\sigma \Delta r$, and that is the straight term.

At long range, then, the attractive force, of size $dV/dr$, settles to the constant $\sigma$. In everyday units, $\sigma =0.9\times 10^9\cdot 1.602\times 10^{-19}\mathrm{J}/10^{-15}\mathrm{m}\approx 1.4\times 10^5\mathrm{N}$, the weight of 15 tonnes, pulling on a single quark.

Pulling the pair apart therefore costs ever more energy. In Figure 4, stretching it from $r_1$ to $r_2=1\mathrm{fm}$ raises $V$ by $\Delta V$, about 0.8 GeV, the vertical bar. That is already enough to create a new light quark–antiquark pair. So the string does not set a quark free: it snaps, and the new quark and antiquark at the broken ends join the old ones to make two colourless hadrons. A quark knocked hard out of a proton shows up in a detector not as a free quark but as a narrow jet of hadrons.

The strong coupling also changes with distance, and gluon self-interaction is again the cause: it makes ${\alpha }_s$ shrink at short range. Higher collision energies probe shorter distances, and ${\alpha }_s$ is about 0.12 at the 91 GeV of the Z mass, 0.3 at 2 GeV, and of order 1 near 1 GeV. Two behaviours follow. Quarks struck hard, probed at short range, behave almost as free particles: this is asymptotic freedom. Slow quarks, at long range, stay bound: this is confinement. The springs in the animation are a harmonic caricature of the string: their pull grows with the stretch, while the real string's pull stays constant, yet both keep the quarks inside.

Worked problem: how heavy could the photon be?

The first section showed that only a massless carrier gives an infinite range. How sure are we that the photon is truly massless? The answer has two parts: what a small photon mass would do to the animation, and what measurements allow.

Suppose the photon had a small mass $m_{\gamma }$. By the first section, the electrons' potential energy would gain the factor $e^{-r/R}$ and become $k{e}^2{e}^{-r/R}/r$, with Coulomb's constant $k=1/(4\pi {\epsilon }_0)$ and $R=\hslash /(m_{\gamma }c)$. The force is minus the derivative of this energy with respect to $r$. Differentiating $1/r$ gives the familiar $1/r^2$, and differentiating $e^{-r/R}$ adds a second term, $r/R$ times the first:
$F=\frac{k{e}^2}{r^2}{e}^{-r/R}\left(1+\frac{r}{R}\right)$
What mass would make the cut-off visible in the animation, with $R$ equal to the 100 nm closest approach?
$m_{\gamma }{c}^2=\frac{\hslash c}{R}=\frac{197.3\mathrm{MeV}\cdot \mathrm{fm}}{10^8\mathrm{fm}}\approx 2\mathrm{eV}$
At the same separations the force would be multiplied by $e^{-1}(1+1)=2/e=0.74$ at 100 nm, where $r/R=1$, and by $e^{-3.43}(1+3.43)=4.43e^{-3.43}=0.14$ at 343 nm, where $r/R=3.43$. The readouts would show about 17.0 fN and 0.28 fN instead of 23.1 fN and 1.96 fN, and their ratio would grow from 11.8 to about 60.

Measurements of the magnetic field carried by the solar wind put the photon's mass below about $10^{-18}\mathrm{eV}/c^2$, so
$R>\frac{197.3\times 10^6\mathrm{eV}\cdot \mathrm{fm}}{10^{-18}\mathrm{eV}}\approx 2\times 10^{26}\mathrm{fm}=2\times 10^{11}\mathrm{m}$
more than the distance from the Earth to the Sun. The inverse-square law therefore holds at least out to the Earth's orbit.

A heavy carrier is the opposite case. For the $Z^0$, of 91.2 GeV/c², $R=197.3/91200\mathrm{fm}=2.2\times 10^{-18}\mathrm{m}$. At the electrons' 100 nm its force is suppressed by the factor $e^{-4.6\times 10^{10}}$, and it becomes as strong as the electromagnetic force only in collisions near 100 GeV.

Limits of the Standard Model

The model's reach is impressive. Collisions of a few TeV probe distances of about $\hslash c/E\approx 10^{-19}\mathrm{m}$, and at that resolution no quark or lepton has shown any size. Yet the model needs about two dozen measured inputs, masses, couplings and mixing angles that it cannot predict, and it leaves out at least four things.

• Neutrino masses: oscillations, in which neutrinos change type as they travel, show that neutrinos have mass, below about 0.5 eV/c², under a millionth of the electron's; the model has none.
• Dark matter: the motions of galaxies and the cosmic microwave background demand about five times more matter than atoms contain.
• The matter–antimatter imbalance of the universe, far larger than the model's CP violation can produce.
• Gravity: its coupling between two particles of energy $E$, $G{E}^2/(\hslash c^5)$, reaches order 1 only near the Planck energy, $1.2\times 10^{19}\mathrm{GeV}$, some $10^{15}$ times beyond collider energies.

Where this leads

Coulomb's Law is single-photon exchange, and Electromagnetic Waves are made of photons. The range formula and annihilation rest on $E^2=p^2{c}^2+m^2{c}^4$ from Special Relativity, the phases that generate forces are the amplitudes of Wave–Particle Duality and Schrödinger's Equation and Tunnelling, and the link between symmetry and conservation is already at work in Angular Momentum and Its Conservation. β decay in Radioactivity and Half-Life is W exchange, and the mass behind Fission, Fusion and E = mc² is mostly the energy of the gluon field and of the quarks moving in it. Gravity as Curved Spacetime describes the one force the model leaves out.