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Sandbox Physics
Scattering amplitudes

L3 · Chapter 5

Why a soft gluon forgets most of the collision

Sandbox Physics ·

A sufficiently low-energy gluon mainly sees its two adjacent hard legs. Nearly parallel final-state legs similarly factor into a splitting and a lower-point amplitude. These limits predict complex processes from simpler pieces, provided momentum conservation accompanies the comparison.

Recognize soft emission in adjacent denominators

Use five-gluon cyclic order (1,2,3,5,4)(1,2,3,5,4), with negative helicities on legs one and two and an added positive-helicity leg 55. Comparing the four- and five-point Parke–Taylor denominators shows that only the link from 33 to 44 has been split. This produces the ratio below.

Scale the new momentum uniformly as k5=λkˉ5k_5=\lambda\bar k_5 at fixed direction. Both spinors scale as λ\sqrt\lambda. Each bracket containing the soft leg supplies a square root, so the soft factor grows inversely with its energy. At leading order this long-wavelength emission retains only the adjacent hard directions.

A5(1−,2−,3+,5+,4+)=i⟨12⟩4⟨12⟩⟨23⟩⟨35⟩⟨54⟩⟨41⟩A_5(1^-,2^-,3^+,5^+,4^+)=i\frac{\langle12\rangle^4}{\langle12\rangle\langle23\rangle\langle35\rangle\langle54\rangle\langle41\rangle}
Compare with the four-point expression: one adjacent bracket becomes two brackets containing the new leg, with the numerator unchanged.
S(3,5+,4)=⟨34⟩⟨35⟩⟨54⟩,A5⟶S(3,5+,4)A4\mathcal S(3,5^+,4)=\frac{\langle34\rangle}{\langle35\rangle\langle54\rangle},\qquad A_5\longrightarrow\mathcal S(3,5^+,4)A_4
The inverse-energy units supply the dimensional difference between four- and five-point tree amplitudes.
λ5,λ~5∝λ⟹S∝λ−1\lambda_5,\widetilde\lambda_5\propto\sqrt\lambda\quad\Longrightarrow\quad\mathcal S\propto\lambda^{-1}
This is a uniform soft limit. A complex analytic limit that scales only one spinor has different power counting.

Source: Lance J. Dixon · R. Britto

Preserve momentum while shortening one leg

Shrinking only one momentum with all others fixed breaks conservation. A resulting discrepancy mixes a limiting approximation with an invalid scattering configuration. Instead choose a three-body final-state path that conserves momentum at every step.

Take total center-of-mass energy two. Hard leg 33 points along unit vector n3\mathbf n_3. Leg 55 has energy λ\lambda and angle α\alpha to it. Energy conservation fixes the recoil energy. Requiring that recoil to remain massless then determines the hard-leg energy.

Worked example: choose λ=0.2\lambda=0.2 and α=π/2\alpha=\pi/2. The result is E3=8/9E_3=8/9 and E4=41/45E_4=41/45. Orthogonal selected momenta give recoil magnitude (8/9)2+(1/5)2=41/45\sqrt{(8/9)^2+(1/5)^2}=41/45, matching its energy. Energies sum to two and spatial momenta cancel by construction.

E3=2(1−λ)2−λ(1−cos⁡α),k5=(λ,λn5),k4=(2−E3−λ,−E3n3−λn5)E_3=\frac{2(1-\lambda)}{2-\lambda(1-\cos\alpha)},\quad k_5=(\lambda,\lambda\mathbf n_5),\quad k_4=(2-E_3-\lambda,-E_3\mathbf n_3-\lambda\mathbf n_5)
A soft scan lowers the energy parameter at fixed angle. A collinear scan lowers the angle at finite energy parameter.
(2−E3−λ)2=E32+λ2+2E3λcos⁡α(2-E_3-\lambda)^2=E_3^2+\lambda^2+2E_3\lambda\cos\alpha
Solving this mass-shell condition gives the energy above and ensures that the recoil energy equals its spatial momentum magnitude.

Split one leg into two almost parallel legs

In the collinear limit write k3≃zPk_3\simeq zP and k5≃(1−z)Pk_5\simeq(1-z)P. The positive-helicity pair has a vanishing angle bracket. Insert λ3≃zλP\lambda_3\simeq\sqrt z\lambda_P and λ5≃1−zλP\lambda_5\simeq\sqrt{1-z}\lambda_P in the cyclic denominator to read off the universal splitting factor.

Along our real path use z=E3/(E3+λ)z=E_3/(E_3+\lambda). Map the comparison four-point state to a unit-energy parent along k3+k5\mathbf k_3+\mathbf k_5 and an opposite unit-energy recoil. This explicitly conserves momentum. At finite angle the mapping contributes subleading differences that vanish in the collinear limit.

A5⟶Split⁡−(3+,5+)A4(…,P+,4),Split⁡−(3+,5+)=1z(1−z)⟨35⟩A_5\longrightarrow\operatorname{Split}_{-}(3^+,5^+)A_4(\ldots,P^+,4),\qquad\operatorname{Split}_{-}(3^+,5^+)=\frac{1}{\sqrt{z(1-z)}\langle35\rangle}
The splitting subscript uses the standard opposite-parent-helicity notation. The parent here has positive helicity.
s35=2E3λ(1−cos⁡α)≃E3λα2s_{35}=2E_3\lambda(1-\cos\alpha)\simeq E_3\lambda\alpha^2
The real collinear amplitude grows inversely with the square root of the invariant. This real-path square-root singularity differs from a simple pole in a complex analytic channel.

Source: Lance J. Dixon

A residue identifies an on-shell intermediate state

Near an internal channel P2=0P^2=0, a general tree amplitude has a pole whose coefficient is a product of two on-shell subamplitudes, summed over intermediate helicities. Multiply by P2P^2 and take the limit to extract this residue. The next chapter uses complex momentum to reconstruct the amplitude from such local information.

Exactly real, massless three-point kinematics forces all momenta to be parallel. Generic three-point factorization residues therefore use complex momenta. Real collinear splitting is closely related but employs approximate kinematics along a specified real path.

lim⁡P2→0P2An=∑hAL(…,Ph) i AR((−P)−h,…)\lim_{P^2\to0}P^2 A_n=\sum_h A_L(\ldots,P^h)\,i\,A_R((-P)^{-h},\ldots)
The propagator phase belongs to our tree-amplitude convention. Color ordering permits only channels assembled from adjacent legs.

Source: Lance J. Dixon

Trace an approach instead of testing one tiny number

For the soft scan compare A5/(SA4mapped)A_5/(\mathcal S A_4^{\rm mapped}); for the collinear scan compare A5/(Split⁡A4mapped)A_5/(\operatorname{Split} A_4^{\rm mapped}). Evaluate the four-point amplitude at the conserving mapped configuration and keep spinor phases continuous. A ratio approaching one along a path gives more information than one favorable point.

Record mass-shell and momentum residuals, absolute errors and relative errors as the scan parameter decreases. Eventually floating-point errors can grow while physical subleading terms shrink. Separating those effects identifies a useful asymptotic window.

Try the reasoning in an experiment

Approach low energy and small angle separately

  1. Fix the emission direction and lower its energy tenfold. Predict the amplitude, soft factor and normalized ratio before comparing.
  2. Keep energy finite and reduce the splitting angle. Compare the collinear approximation and check mass-shell and momentum residuals.
  3. Plot limiting errors on logarithmic axes. Identify where the approximation improves. Assess the precision reached in this range and describe what a smaller-parameter offline scan should record.

Exercises and explanations

  1. Lower the soft energy tenfold at fixed direction. How do the leading amplitude and its squared modulus change? Do these directly give event counts?

    Read the explanation

    The amplitude grows tenfold and its squared modulus grows a hundredfold. Counts also involve soft phase space, virtual corrections and an infrared-safe observable definition. A single-emission singularity is insufficient to predict a measurable total rate.

  2. Delete the added leg from the cyclic five-point denominator. Why might the resulting four-point algebraic expression fail to describe a physical four-point process at finite soft energy?

    Read the explanation

    The remaining four original momenta generally do not conserve momentum. An algebraic cancellation can be exact, but interpreting the shortened expression as a physical amplitude requires a conserving map and the appropriate limit.

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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