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research33 min studyLesson 622 in Physics

Read QCD vertices from one gauge-covariant action

Expand the non-Abelian field strength, track vertex powers, and connect gauge-required interference to controlled short-distance predictions.

Before you begin
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The question for today

What you’ll understand

  • Derive three- and four-gluon interactions from the field strength.
  • Track amplitude powers separately from probability powers.
  • Identify the gauge and kinematic assumptions behind a diagram calculation.

QCD differs from electrodynamics because its gauge generators do not commute. The gluon field strength therefore contains a term quadratic in the gauge potential. Squaring that field strength produces gluon self-interactions with coefficients fixed by the same coupling that appears in the quark vertex. These are related pieces of one gauge theory, rather than independent forces that can be tuned arbitrarily. We use natural units, a Minkowski metric with positive time component, and an explicit sign convention for the covariant derivative.

Choose a derivative and keep its signs

Take the covariant derivative to be the ordinary derivative plus i times the coupling times the matrix-valued gauge potential. The commutator of two derivatives equals i times the coupling times the field strength. Since the matrix commutator of generators contains i times a structure constant, the component field strength has a minus sign on its quadratic potential term in this convention. Reversing the covariant-derivative convention changes several signs together, without changing measurable predictions.

The quark kinetic term then contains a quark-antiquark-gluon interaction linear in the gauge potential and proportional to g s. Its generator connects the incoming and outgoing color components while its Dirac matrix couples the vector field to the quark current. Flavor is conserved by that QCD vertex. A gluon can split into an allowed quark-antiquark pair, but it does not turn an up quark into a down quark as a charged weak interaction can.

Dμ=∂μ+igsTaAμa,[Dμ,Dν]=igsTaFμνa,Fμνa=∂μAνa−∂νAμa−gsfabcAμbAνcD_\mu=\partial_\mu+i g_sT^aA^a_\mu,\quad [D_\mu,D_\nu]=i g_sT^aF^a_{\mu\nu},\qquad F^a_{\mu\nu}=\partial_\mu A^a_\nu-\partial_\nu A^a_\mu-g_sf^{abc}A^b_\mu A^c_\nu
The dimensionless coupling gs multiplies Hermitian generators with [Tᵃ,Tᵇ]=ifᵃᵇᶜTᶜ. In four spacetime dimensions A has mass dimension1, F dimension2, and D dimension1. These signs use D=∂+igsA.↔ Scroll wide equations horizontally.

Expand the gauge term by powers of the field

Write the field strength as a derivative term plus a quadratic term. The gauge kinetic action contains the square of the derivative term, their cross term, and the square of the quadratic term. These respectively give the free gauge propagation, a three-gluon interaction proportional to one coupling, and a four-gluon interaction proportional to its square. Derivatives in the cubic term supply momentum factors. The quartic contact graph can have the same overall coupling order as two cubic vertices joined by a propagator.

That counting explains why gluon scattering at its lowest nonzero order needs several diagrams. A sum of exchange and contact contributions is required by gauge consistency. An individual graph depends on gauge and need not satisfy the physical polarization test on its own. In covariant quantization, gauge fixing and ghost fields organize the unphysical components, particularly in loops. Ghosts are not extra observed outgoing hadrons; their role is to maintain the consistency of internal gauge calculations.

LQCD=∑fqˉf(iγμDμ−mf)qf−14FμνaFaμν,Vqqˉg∼gs,Vggg∼gs,Vgggg∼gs2\mathcal L_{\mathrm{QCD}}=\sum_f\bar q_f(i\gamma^\mu D_\mu-m_f)q_f-\frac14F^a_{\mu\nu}F^{a\mu\nu},\qquad V_{q\bar qg}\sim g_s,\quad V_{ggg}\sim g_s,\quad V_{gggg}\sim g_s^2
The Lagrangian has mass dimension4. qf has dimension3/2 and mf dimension1. The vertex symbols state coupling order only, excluding signs, color tensors, Lorentz tensors and momentum factors.↔ Scroll wide equations horizontally.

Power counting is useful before integrating

A tree-level two-to-two QCD amplitude commonly starts at g s squared. Its squared amplitude therefore starts at g s to the fourth power, or alpha s squared after absorbing factors of four pi into the coefficient. An extra resolved gluon introduces another coupling in the amplitude and another power of alpha s in its leading rate. Interference between a one-loop amplitude and a tree amplitude also contributes at the next order, so real and virtual corrections belong in the same accuracy statement.

Neither a vertex diagram nor this counting fixes a numerical cross section. Flux, phase space, spin and color averages, cuts, masses, and infrared treatment are also required. Perturbation theory is useful when its hard scales are large enough and logarithms are controlled. The existence of gluon self-interactions helps explain antiscreening but does not by itself constitute an analytic proof of confinement. Low-energy hadron properties require nonperturbative input or methods; simply inserting a large coupling into a tree graph does not supply them.

M2→2=gs2A0+gs4A1+⋯ ,∣M∣2=gs4∣A0∣2+2gs6Re⁡(A0∗A1)+⋯ ,αs=gs24π\mathcal M_{2\to2}=g_s^2\mathcal A_0+g_s^4\mathcal A_1+\cdots,\qquad |\mathcal M|^2=g_s^4|\mathcal A_0|^2+2g_s^6\operatorname{Re}(\mathcal A_0^*\mathcal A_1)+\cdots,\quad \alpha_s=\frac{g_s^2}{4\pi}
The coefficients include kinematics, color, and loop-normalization factors. This schematic expansion assumes the selected process has a nonzero tree amplitude of order gs²; special processes can start at a different order.↔ Scroll wide equations horizontally.

Work it through

Compare two amplitudes at a changed coupling

In an illustrative fixed-kinematics tree model let a two-to-two amplitude be A g² and an extra-emission amplitude be B g³, with A and B held fixed. Change g from0.5 to1. Find both rate ratios and the change in their rate ratio.

  1. The coupling doubles. The two-to-two amplitude grows by2²=4, so its squared magnitude grows by16. Fixed flux and phase-space factors are being held constant by the stated model.

  2. The extra-emission amplitude grows by2³=8 and its rate by64. Its additional coupling is one power at amplitude level and two at rate level.

  3. The extra-emission rate divided by the two-to-two rate thus grows by64/16=4. This is algebraic power counting, not a precision QCD prediction at the large chosen coupling.

The result

The two rates grow by16 and64 respectively, and their ratio grows by4 under the fixed-coefficient model.

The living notebook

Explore the model

Interactive lab

Interactive lab · scattering

Inspect the tree-level fourth power

For separate runs Reset and set BOTH coupling0.5, angle60°; then Reset and set coupling1, angle60°. Predict weights0.078125 and1.25, a factor16. The displayed graph and weight are schematic; they have no gauge cancellations, color sums, real-emission phase space, or physical cross-section units.

Inspect the tree-level fourth powerA schematic term diagram with dimensionless weight g⁴(1 + cos²θ). Lines denote terms, not particle paths; this is not a process cross-section or loop calculation. Current readings: Coupling factor g⁴: 0.24; Angular factor 1 + cos²θ: 1.329; Illustrative weight: 0.319. All parameters are adjustable with the labeled controls below.Incoming termsOutgoing termsθSchematic interaction term · no particle trajectories
Incoming termsOutgoing termsAngle θ
Coupling factor g⁴
0.24
Angular factor 1 + cos²θ
1.329
Illustrative weight
0.319
t = 0 s

A schematic term diagram with dimensionless weight g⁴(1 + cos²θ). Lines denote terms, not particle paths; this is not a process cross-section or loop calculation. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forSquaring an amplitude changes a second power of coupling into a fourth power in a rate.

Pause & reason

Check your understanding

Why does a four-gluon contact diagram contribute at the same coupling order as two three-gluon vertices joined by a propagator?

Your turn

Try it yourself

In a fixed-coefficient tree model, reduce alpha s from0.20 to0.10. A two-to-two rate begins at alpha s squared and a resolved extra-parton rate begins at alpha s cubed. What fractions of their original rates remain?

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Count powers of alpha s directly rather than treating alpha s as g.

Compare with the solution

The coupling alpha s is halved. The first rate becomes(1/2)²=1/4 of its original value and the extra-parton rate becomes(1/2)³=1/8. Their ratio is halved. Since alpha s=g²/(4pi), the corresponding change of g is a factor1/√2, not1/2. Running scales, coefficients and cuts were deliberately fixed in this illustrative comparison.

Keep the idea

Take it with you

  • The gluon self-couplings follow from the non-Abelian field strength.
  • Gauge consistency relates contact and exchange diagrams.
  • Coupling order alone does not determine a physical cross section.

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