§6.2Nucleon Structure

§6.1 looked at quarks from the outside: make a pair and watch it turn into jets. This section looks from the inside — fire an electron into a proton and see what it bounces off.

It is Rutherford’s experiment, run sixty years later on a target a hundred thousand times smaller, and it produced the same kind of surprise: something hard and point-like inside something that had looked smooth.

The apparatus

Two invariants, from two measured numbers

import numpy as np
mp, E = 0.9382720882, 17.0            # proton mass, and the SLAC beam energy

print("  theta     E'    Q^2 GeV^2     nu     x       W GeV    regime")
for th, Ep in [(6, 15.5), (6, 12.0), (10, 12.0), (10, 6.0), (18, 6.0), (26, 4.0), (34, 2.5)]:
    t = np.radians(th); nu = E - Ep
    Q2 = 4 * E * Ep * np.sin(t / 2)**2      # Eq. (6.11)
    x = Q2 / (2 * mp * nu)                  # Eq. (6.16)
    W2 = mp**2 + 2 * mp * nu - Q2           # Eq. (6.12)
    W = np.sqrt(W2) if W2 > 0 else float('nan')
    reg = ("UNPHYSICAL x>1" if x > 1 else "quasi-elastic" if W < 1.3
           else "resonance region" if W < 2.0 else "deep inelastic")
    print(f"  {th:5.0f}  {Ep:6.1f}  {Q2:9.2f}  {nu:7.2f}  {x:6.3f}  {W:7.3f}   {reg}")
print()
print("x = 1 is the ELASTIC limit: there Q^2 = 2 m_p nu exactly and W = m_p.")
print("x > 1 is forbidden -- nothing inside a proton is heavier than the proton,")
print("so no constituent can absorb more momentum than the whole of it.")

  theta     E'    Q^2 GeV^2     nu     x       W GeV    regime
    6    15.5       2.89     1.50   1.026    0.899   UNPHYSICAL x>1
    6    12.0       2.24     5.00   0.238    2.833   deep inelastic
   10    12.0       6.20     5.00   0.661    2.016   deep inelastic
   10     6.0       3.10    11.00   0.150    4.292   deep inelastic
   18     6.0       9.98    11.00   0.484    3.397   deep inelastic
   26     4.0      13.76    13.00   0.564    3.393   deep inelastic
   34     2.5      14.53    14.50   0.534    3.682   deep inelastic

x = 1 is the ELASTIC limit: there Q^2 = 2 m_p nu exactly and W = m_p.
x > 1 is forbidden -- nothing inside a proton is heavier than the proton,
so no constituent can absorb more momentum than the whole of it.

Scaling: the surprise

The cross-section from a target with structure is described by two structure functions $W_1(Q^2,\nu)$ and $W_2(Q^2,\nu)$ — one sensitive to the current distribution, one to the charge distribution. In the SLAC kinematics $W_1$ is negligible, so Eq. (6.15) reduces to

Everything hangs on how $W_2$ behaves. And what SLAC found is that in the deep inelastic region it barely depends on $Q^2$ at all.

Feynman’s reading: partons and x

Feynman’s move was to look at the proton in a frame where it moves very fast. Then the transverse momenta of its constituents are negligible, time dilation freezes their interactions with each other, and the electron scatters off one free constituent — the impulse approximation. He called them partons parton Feynman's name for the hard constituents seen in DIS, before they were identified with quarks and gluons. defined in §6.2 — open in glossary .

Let that parton carry a fraction $x$ — Bjorken x Bjorken x x = Q²/2m_pν, the fraction of the nucleon's momentum carried by the struck parton in the infinite-momentum frame. defined in §6.2 — open in glossary — of the proton’s four-momentum. If it is massless and stays massless after being struck,

which is Eq. (6.16). And now the payoff: if the parton is point-like, the structure function cannot depend on $Q^2$ — only on $x$. Define the dimensionless version, Eq. (6.17):

and Bjorken’s scaling law Bjorken scaling the observation that F₂ depends on x alone, not on Q² — the signature of point-like constituents, exactly as a Q²-independent form factor was for the nucleus. defined in §6.2 — open in glossary says $F_2$ is a function of $x$ alone. The data agree, and that is why quarks stopped being bookkeeping and became objects.

What is actually inside

The distributions split into valence and sea valence and sea quarks the three quarks that fix the spectroscopy, versus the qq̄ pairs continually created and annihilated in the colour field; the sea dominates below x ≈ 0.3. defined in §6.2 — open in glossary quarks, and the measurements — electrons on hydrogen and deuterium, then neutrinos and antineutrinos, which see different quarks because the weak charged current turns $d \to u$ but not $u \to d$ — determine the distributions parton distribution function (PDF) f(x), the probability density for a parton of flavour f to carry momentum fraction x; measured, not calculated, and the reason a hadron collider's luminosity is a distribution rather than a number. defined in §6.2 — open in glossary separately.

import numpy as np
xs = np.linspace(1e-5, 1 - 1e-9, 400000)
shape = lambda a, b: xs**a * (1 - xs)**b

# The valence normalisations are NOT free: a proton has exactly two u and one d.
Au = 2 / np.trapezoid(shape(-0.5, 3.0), xs)
Ad = 1 / np.trapezoid(shape(-0.5, 4.0), xs)
uv, dv = Au * shape(-0.5, 3.0), Ad * shape(-0.5, 4.0)
print(f"counting rules: int u_v dx = {np.trapezoid(uv, xs):.3f} (must be 2), "
      f"int d_v dx = {np.trapezoid(dv, xs):.3f} (must be 1)")

Muv, Mdv = np.trapezoid(xs*uv, xs), np.trapezoid(xs*dv, xs)
print(f"valence u carries {Muv:.3f} of the momentum, valence d {Mdv:.3f}"
      f"  ->  {Muv+Mdv:.3f} together")

# The sea normalisation is then fixed by the MEASURED total, not chosen.
sea = 0.50 - (Muv + Mdv)
print(f"the measurement says quarks + antiquarks carry 0.50, so the sea supplies {sea:.3f}")
print()
print(f"  everything the electron and the neutrino can see: {Muv+Mdv+sea:.2f}")
print(f"  everything they cannot:                          {1-0.50:.2f}")
print()
print("Half the proton's momentum belongs to partons with neither an electric")
print("nor a weak charge.  That is the gluon, found by subtraction.")

counting rules: int u_v dx = 2.000 (must be 2), int d_v dx = 1.000 (must be 1)
valence u carries 0.224 of the momentum, valence d 0.092  ->  0.315 together
the measurement says quarks + antiquarks carry 0.50, so the sea supplies 0.185

everything the electron and the neutrino can see: 0.50
everything they cannot:                          0.50

Half the proton's momentum belongs to partons with neither an electric
nor a weak charge.  That is the gluon, found by subtraction.

Scaling, broken — and why that is good news

At HERA (30 GeV electrons on 800 GeV protons, so $\sqrt s \approx 310$ GeV) the range opened to $2.7 < Q^2 < 30\,000$ GeV². Over most of $x$, $F_2$ is flat and scaling holds. But at small $x$ it rises with $Q^2$ — the scaling law fails scaling violation the slow rise of F₂ with Q² at small x, predicted by DGLAP: better resolution splits one parton into two of smaller x. The size of the violation measures α_s. defined in §6.2 — open in glossary , and it fails exactly as predicted.

The mechanism is Fig. 6.14 and it is almost embarrassingly simple. A quark with fraction $x$ emits a gluon and drops to $x'$. Look with poor resolution and you see one object at $x$; look with better resolution and you see two, at $x'$ and $x-x'$. Better resolution finds more partons, each softer. So the small-$x$ population grows with $Q^2$ and the large-$x$ population shrinks.

Where this goes next

• §6.3 finally writes the theory these two sections have been demanding: SU(3), and gluons that carry the charge they mediate.
• §6.5 justifies the impulse approximation that Feynman had to assume — asymptotic freedom is what makes a struck parton behave as free.
• §9.14 is where PDFs stop being a result and become a tool: every LHC cross-section is a convolution of two of them.
• §1.8 is the elastic form factor whose steep fall is the control curve in Fig. 6.10.