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Sandbox Physics
Scattering amplitudes

L3 · Chapter 4

Why four-gluon scattering reduces to one line

Sandbox Physics ·

The same four-gluon amplitude comes from exchange and contact diagrams or from one line of spinor products. Many terms in the long calculation compensate for redundancy in polarization variables. Change the reference direction and individual diagrams move; their complete sum stays fixed.

Fix a reproducible collision

Study massless Yang–Mills theory at tree level, cyclic order (1,2,3,4)(1,2,3,4) and outgoing helicities (−−++)(--++). Set incoming beam energy E=1E=1 as the unit. Two physical beams approach along the longitudinal axis. The all-outgoing momenta below put the final state in the longitudinal plane.

Direct substitution verifies zero squared mass for every external leg and conservation of total four-momentum. For this ordering the backward endpoint has a massless exchange pole; the forward amplitude stays finite. The page uses a finite angle window away from the endpoints.

k1=(−1,0,0,−1),k2=(−1,0,0,1),k3=(1,sin⁡ϑ,0,cos⁡ϑ),k4=(1,−sin⁡ϑ,0,−cos⁡ϑ)k_1=(-1,0,0,-1),\quad k_2=(-1,0,0,1),\quad k_3=(1,\sin\vartheta,0,\cos\vartheta),\quad k_4=(1,-\sin\vartheta,0,-\cos\vartheta)
Energies and spatial momenta are measured in beam-energy units. Negative-energy vectors label physical incoming particles.
s12=4,s23=−2(1+cos⁡ϑ),∑iki=0s_{12}=4,\qquad s_{23}=-2(1+\cos\vartheta),\qquad\sum_i k_i=0
The exchange channels in this color order involve adjacent pairs. Relabeling legs changes the channel labels.

Polarization needs a reference direction

An external polarization εi\varepsilon_i describes its spin state and is transverse to its momentum. A spinor expression introduces another null momentum qiq_i to specify the transverse frame. This reference is a choice of description, not an additional incoming particle.

The sandwich notation below contracts the two spinors with a fixed set of Dirac matrices to form a four-vector. We use its transversality and the fact that changing reference adds only a component along the external momentum.

Changing reference adds a term proportional to kik_i. If replacing an external polarization by its own momentum makes the full amplitude vanish, that added term has no effect. This is the on-shell tree-level Ward check used here. All required diagrams must participate.

ε+μ(k;q)=⟨q∣γμ∣k]2⟨qk⟩,k⋅ε+=0\varepsilon_+^\mu(k;q)=\frac{\langle q|\gamma^\mu|k]}{\sqrt2\langle qk\rangle},\qquad k\cdot\varepsilon_+=0
Avoid a reference collinear with the external leg, where this expression has a zero denominator. Negative helicity uses the corresponding opposite-chirality expression.
εi↦εi+βki⟹ΔA4=βA4(…,εi=ki,…)=0\varepsilon_i\mapsto\varepsilon_i+\beta k_i\quad\Longrightarrow\quad\Delta A_4=\beta A_4(\ldots,\varepsilon_i=k_i,\ldots)=0
This relation uses on-shell external legs, momentum conservation and a complete tree amplitude. Gauge-dependent terms cancel between diagrams in the complete sum.

Source: Lance J. Dixon · R. Britto

Compare both routes in the same conventions

The direct route contracts two three-gluon vertices and a propagator in each of the planar s12s_{12} and s23s_{23} exchange channels, then adds the four-gluon contact term. The contact term has no internal propagator but is essential for the gauge cancellation.

The second route evaluates Parke–Taylor at the same helicities. It contains no polarization reference. Both routes strip the coupling and color trace and retain the same overall phase. Comparing only squared moduli could hide a sign or phase mistake.

Worked example: scatter at a right angle. Then s12=4s_{12}=4 and s23=−2s_{23}=-2. Use ∣⟨ij⟩∣2=∣sij∣|\langle ij\rangle|^2=|s_{ij}| to square the compact expression. Its numerator is 444^4 and denominator 4⋅2⋅4⋅24\cdot2\cdot4\cdot2, giving ∣A4∣2=4|A_4|^2=4. With the same spinor phases, the diagram sum must agree as a complex number as well.

A4=A12+A23+Acontact=i⟨12⟩4⟨12⟩⟨23⟩⟨34⟩⟨41⟩A_4=A_{12}+A_{23}+A_{\rm contact}=i\frac{\langle12\rangle^4}{\langle12\rangle\langle23\rangle\langle34\rangle\langle41\rangle}
The first equality describes the diagram calculation. The final expression supplies the independent compact route.
Jab=(εa ⁣⋅ ⁣εb)(ka−kb)+2(εa ⁣⋅ ⁣kb)εb−2(εb ⁣⋅ ⁣ka)εaJ_{ab}=(\varepsilon_a\!\cdot\!\varepsilon_b)(k_a-k_b)+2(\varepsilon_a\!\cdot\!k_b)\varepsilon_b-2(\varepsilon_b\!\cdot\!k_a)\varepsilon_a
For transverse on-shell external legs, a three-gluon vertex contraction reduces to this four-vector current. Every dot product uses the same metric.
A4=i2[J12 ⁣⋅ ⁣J34s12+J23 ⁣⋅ ⁣J41s23+2(ε1 ⁣⋅ ⁣ε3)(ε2 ⁣⋅ ⁣ε4)−(ε1 ⁣⋅ ⁣ε2)(ε3 ⁣⋅ ⁣ε4)−(ε1 ⁣⋅ ⁣ε4)(ε2 ⁣⋅ ⁣ε3)]A_4=\frac i2\left[\frac{J_{12}\!\cdot\!J_{34}}{s_{12}}+\frac{J_{23}\!\cdot\!J_{41}}{s_{23}}+2(\varepsilon_1\!\cdot\!\varepsilon_3)(\varepsilon_2\!\cdot\!\varepsilon_4)-(\varepsilon_1\!\cdot\!\varepsilon_2)(\varepsilon_3\!\cdot\!\varepsilon_4)-(\varepsilon_1\!\cdot\!\varepsilon_4)(\varepsilon_2\!\cdot\!\varepsilon_3)\right]
The first two terms are exchange diagrams. The remaining three form the contact diagram, giving an explicitly reproducible tensor calculation.
∣A4∣2=s122s232=4(1+cos⁡ϑ)2|A_4|^2=\frac{s_{12}^2}{s_{23}^2}=\frac{4}{(1+\cos\vartheta)^2}
This fixed-order, fixed-helicity, coupling-stripped result still needs color and spin treatment and phase space to predict a cross section.

Source: Lance J. Dixon

Make redundancy large and watch cancellation deteriorate

A reference close to an external direction produces a small denominator in the polarization representation. Individual diagrams can become huge although their sum remains ordinary. Finite-precision subtraction loses significant digits. The representation is becoming ill conditioned; the interaction has not suddenly grown stronger.

Define a cancellation estimate κsum\kappa_{\rm sum} by dividing the sum of individual magnitudes by the magnitude of their sum. It measures sensitivity of this sum to relative perturbations of its terms. Near an amplitude zero, relative error itself becomes unreliable and must be accompanied by absolute error.

Useful checks compare the two complex amplitudes, replace each leg’s polarization in turn for a Ward test, and record mass-shell and momentum residuals. A stable sum of reference-dependent diagrams is a direct numerical demonstration of gauge redundancy.

κsum=∣A12∣+∣A23∣+∣Acontact∣∣A4∣\kappa_{\rm sum}=\frac{|A_{12}|+|A_{23}|+|A_{\rm contact}|}{|A_4|}
This estimates cancellation for the chosen decomposition. A full input-condition analysis would also vary all independent kinematic inputs.

Try the reasoning in an experiment

Drag a reference direction and follow the cancellations

  1. At the default collision angle, record the three diagram contributions, their complex sum and the compact answer.
  2. Hold the physical collision fixed and drag only the polarization reference. Predict and compare individual terms, total amplitude, Ward residuals and cancellation estimate.
  3. Explore the full permitted reference circle and compare its largest and smallest cancellation estimates. The page keeps references separated from the planar external legs; the analytic exercise below explores stronger cancellation.

Exercises and explanations

  1. Removing the contact diagram still gives a close answer at one angle. Is that sufficient?

    Read the explanation

    No. Accidental agreement at one point cannot replace a model identity. Change angle and reference and apply the Ward replacement to each leg. A missing contact term generally breaks these relations.

  2. Two terms are approximately 101010^{10} and −1010+1-10^{10}+1, each with relative error 10−1510^{-15}. What absolute error can their sum retain?

    Read the explanation

    Each term can have absolute error of order 10−510^{-5}, which can remain in their sum. The answer is of order one, so its relative error greatly exceeds the relative input error. A better reference or higher precision can improve the calculation.

Sources and reproduced scope

Worked examples and schematic figures are authored for this course. Numerical values are calculated model references. The published comparison recalculates a specified analytic CDD curve; it does not copy experimental data, source artwork or the paper’s numerical optimization points. Source locations and reproduced scope are listed below.

  1. Lance J. Dixon, A brief introduction to modern amplitude methods, arXiv:1310.5353v1 (2013)

    Section 2, equation (5), color decomposition; section 3.1, equations (14), (19), (21)–(27), spinors and all-outgoing conventions; section 3.3, equations (41)–(46), polarizations; section 4.1, equations (55)–(57), soft factors; section 4.3, equations (68)–(70), MHV and collinear factors; sections 4.5 and 5.1, equations (74)–(84), complex three-point seeds and recursion. Our conserving scan and four-point residue example are derived extensions.

  2. R. Britto, F. Cachazo, B. Feng and E. Witten, Direct Proof Of Tree-Level Recursion Relation In Yang-Mills Theory, hep-th/0501052v2; Physical Review Letters 94, 181602 (2005)

    Section 2 introduces null-momentum spinors, constructs the shift in equation (2.1), and derives recursion in equation (2.7); section 3 controls large-parameter behavior; section 4 proves the MHV construction. The course keeps Dixon’s overall amplitude phase and specializes the original shift to legs one and four. Its soft scan uses the MHV function as input and derives the conserving real-momentum path separately.

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