A Byte of PhysicsLogo

Fundamental particles

In current experiments, quarks, leptons, gauge bosons, and the Higgs boson show no resolved internal structure. The Standard Model treats them as elementary fields. Protons, neutrons, atomic nuclei, and atoms are composite.

“Fundamental” is evidence-dependent: it means no smaller constituents are required or observed within tested scales, not that future experiments are logically forbidden from finding deeper structure.

Fermions and bosons

Particles are classified by spin:

The spin–statistics connection is a result of relativistic quantum field theory, not a statement that particles are little classical spheres physically rotating.

Quarks

There are six quark flavors arranged in three generations:

GenerationUp-type quarkDown-type quark
1updown
2charmstrange
3topbottom

Up-type quarks have electric charge \(+\tfrac23e\); down-type quarks have \(-\tfrac13e\). Every quark also carries color charge and participates in the strong interaction. Color has three basis labels conventionally called red, green, and blue; these names do not refer to visible color.

Free isolated quarks have not been observed. Confinement binds them into color-neutral hadrons. A baryon contains three valence quarks, while a meson contains a valence quark and antiquark, together with gluons and transient quark–antiquark contributions in the full quantum state.

The proton's valence content is \(uud\), giving

\[Q_p=\frac23e+\frac23e-\frac13e=+e\]

The neutron's \(udd\) content gives net charge zero. These sums check electric charge but do not calculate hadron mass or internal dynamics.

from fractions import Fraction

quark_charge = {"u": Fraction(2, 3), "d": Fraction(-1, 3)}
proton_charge = sum(quark_charge[q] for q in ["u", "u", "d"])
neutron_charge = sum(quark_charge[q] for q in ["u", "d", "d"])
print("Proton and neutron charge in units of e:", proton_charge, neutron_charge)
assert proton_charge == 1 and neutron_charge == 0

Most proton and neutron mass comes from strong-interaction energy in the bound state, not simply from adding the Higgs-generated rest masses of three valence quarks.

Leptons

The six leptons also form three generations:

GenerationCharged leptonAssociated neutrino
1electronelectron neutrino
2muonmuon neutrino
3tautau neutrino

Charged leptons have charge \(-e\) and participate in electromagnetic and weak interactions. Neutrinos have zero electric charge and participate in the weak interaction and gravity. None of the leptons carries color charge.

Neutrinos have small nonzero masses and flavor states are mixtures of mass states. A neutrino produced with one flavor can later be detected as another flavor. This oscillation is evidence that the minimal original massless-neutrino version of the Standard Model is incomplete, though the Standard Model framework can be extended to include neutrino mass.

Antiparticles

Each fermion has an antiparticle with the same mass and spin but opposite additive charges. The electron's antiparticle is the positron, with charge \(+e\). Antiquarks carry opposite electric and color charges to their corresponding quarks.

Particle–antiparticle annihilation converts their energy and momentum into allowed final states; it does not make energy disappear. Pair production is the reverse possibility when sufficient energy and all conservation laws can be satisfied.

Some neutral bosons are their own antiparticles. Electric neutrality alone does not guarantee this: a neutron is neutral but its antineutron is distinct because other quantum numbers differ.

Interaction carriers

The Standard Model gauge bosons are:

InteractionGauge bosonKey coupling
Electromagneticphotonelectrically charged particles
Weak\(W^+,W^-,Z^0\)quarks and leptons
Strongeight gluonscolor-charged fields

The photon and gluons are massless in the Standard Model. The \(W\) and \(Z\) bosons are massive. Gluons themselves carry color charge, so they interact with one another. Photons do not carry electric charge.

Calling a boson a “force carrier” is shorthand for interactions in quantum field theory. It should not be interpreted as classical pellets thrown back and forth. Virtual particles in perturbation calculations are internal mathematical terms, not directly detected particles obeying ordinary on-shell relations.

Gravity is not included in the Standard Model and no graviton has been experimentally established.

The Higgs field and Higgs boson

The Higgs field has a nonzero vacuum value. Electroweak symmetry breaking gives mass to the \(W\) and \(Z\) bosons, while fermion masses arise through Yukawa couplings. The Higgs boson is a spin-zero excitation of this field.

It is inaccurate to say that the Higgs mechanism supplies all visible mass. Most mass of ordinary matter resides in protons and neutrons and arises mainly from quantum chromodynamics binding energy.

Conservation bookkeeping

Every proposed particle reaction must conserve total energy, vector momentum, angular momentum, and electric charge. Other useful quantum numbers include baryon number, lepton-family numbers, color, and flavor.

A process such as beta-minus decay is

\[n\rightarrow p+e^-+\bar\nu_e\]

Electric charge balances:

\[0=(+1)+(-1)+0\]

At the quark level, one down quark changes to an up quark through the charged weak interaction. Energy and momentum decide whether the process is allowed for a particular bound system.

Some apparent conservation rules are approximate. Individual quark flavor can change in weak interactions. Neutrino mixing means separate flavor lepton numbers need qualification. Baryon and total lepton number are observed to be highly constrained at ordinary energies but have a different theoretical status from exact gauge charge conservation.

particles = {
    "neutron": {"charge": 0, "baryon": 1},
    "proton": {"charge": 1, "baryon": 1},
    "electron": {"charge": -1, "baryon": 0},
    "antineutrino": {"charge": 0, "baryon": 0},
}
for quantity in ["charge", "baryon"]:
    before = particles["neutron"][quantity]
    after = sum(particles[p][quantity] for p in ["proton", "electron", "antineutrino"])
    assert before == after
print("Additive bookkeeping checks passed.")

This check only works for a genuinely additive conserved quantity. It cannot decide angular-momentum coupling, color singlets, or reaction probability by itself.

Rest mass and energy units

Particle masses are commonly quoted as energy divided by \(c^2\), such as \(\mathrm{MeV}/c^2\). Momentum is often reported in \(\mathrm{MeV}/c\). If natural units set \(c=1\), mass, momentum, and energy may all appear in electronvolt-based units. The unit convention must be explicit when data enters ordinary SI calculations.

Unstable particles are resonances with finite lifetimes and widths. Their quoted masses and widths follow conventions and experimental fits, so use a versioned Particle Data Group source rather than copying an unqualified table.

How particles are identified

Detectors rarely photograph a fundamental particle directly. They record ionization, light, deposited energy, timing, and curved charged-particle tracks. Physicists reconstruct candidates and infer masses through energy–momentum relations. For a system,

\[M^2c^4=E_{\mathrm{total}}^2-c^2|\mathbf p_{\mathrm{total}}|^2\]

A peak above modeled background can indicate a resonance, but detector calibration, selection efficiency, alternative processes, and statistical significance must be evaluated. A particle chart summarizes the result of this evidence; it is not itself the experiment.

Reliable particle records

For every entry, state:

The Standard Model is extraordinarily successful but does not explain gravity, dark matter, dark energy, the observed matter–antimatter imbalance, or all facts about neutrino mass. “Fundamental particle” therefore names the best-tested current description, not a finished theory of everything.

A worked Python notebook

Combine up and down valence quarks, holding their total count at three. The question for this calculation is precise: how does valence electric charge change when we change up-quark count in a three-quark state, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

Only integer quark counts are physical; this bookkeeping does not determine allowed bound states or masses. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from 0 to 3 count. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for up-quark count in a three-quark state, expressed in count. The output is valence electric charge, expressed in elementary-charge units. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = q - 1\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 2 count, the expression gives 1 elementary-charge units. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("q-1", locals={"q": q})
reference_input = 2.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "count")
print("Predicted valence electric charge:", reference_output, "elementary-charge units")

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives -1 elementary-charge units. The midpoint, at 1.5 count, gives 0.5 elementary-charge units. The upper endpoint gives 2 elementary-charge units. Read these together with the assumptions: they describe different choices of up-quark count in a three-quark state within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([0.0, 1.5, 3.0])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} count -> {output_value:.6g} elementary-charge units")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows up-quark count in a three-quark state; the vertical axis shows valence electric charge. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(0.0, 3.0, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Up-quark count in a three-quark state (count)")
ax.set_ylabel("Valence electric charge (elementary-charge units)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does valence electric charge change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 2 count is 1 in those ratio units. Locally, the output increases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = 1\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 2.015
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "elementary-charge units")
print("Full expression:", full_prediction, "elementary-charge units")
print(
    "Approximation difference:",
    full_prediction - linear_prediction,
    "elementary-charge units",
)

Here the changed input is 2.015 count. The tangent estimate is 1.015 elementary-charge units, while the full expression gives 1.015 elementary-charge units. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 3.0
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of up-quark count in a three-quark state inside the stated interval, predict whether valence electric charge will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: Only integer quark counts are physical; this bookkeeping does not determine allowed bound states or masses. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of elementary-charge units: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(3.0) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 3.0, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources

Share to: