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research32 min studyLesson 728 in Physics

Collinear splitting amplitudes

Separate helicity-level splitting factors from color- and spin-summed kernels and state the time-like collinear and infrared assumptions.

Before you begin
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The question for today

What you’ll understand

  • Evaluate the same-helicity gluon splitting factor.
  • Relate its near-collinear bracket singularity to an invariant mass.
  • Distinguish amplitude factorization, splitting kernels and regulated emission probabilities.

Two nearly parallel massless momenta generate a different universal limit from a single soft momentum. Their small combined invariant mass allows a parent line to approach its mass shell, and the amplitude factorizes into a splitting factor and a lower-point amplitude. Helicity labels and an azimuthal complex phase remain visible at amplitude level. Averaging or summing those data produces familiar evolution kernels only after the relevant color, spin, normalization and phase-space conventions are supplied.

A small pair mass defines the collinear limit

Choose a time-like collinear pair with a common energy sign in the analytic incoming description. In the strict limit their momenta approach z times a null parent and one minus z times that parent, with z between zero and one. The real near-collinear process has a small positive pair invariant. Exactly at the endpoint the parent is null; away from it, the lower-point on-shell replacement is part of a limiting construction rather than an exact identity for a massive pair.

For real future-directed representatives the angle and reverse-square brackets vanish together and their product is the small invariant. Each bracket also carries an azimuthal phase that invariants alone cannot retain. For complex kinematics one bracket may vanish without the other, explaining the connection to three-point factorization branches. Space-like initial-state splittings and mixed energy signs require their own crossing prescription; the simple real momentum fraction interval should not be indiscriminately reused there.

pa→zP,pb→(1−z)P,P2=0,0<z<1,sab=⟨ab⟩[ba]→0p_a\to zP,\qquad p_b\to(1-z)P,\qquad P^2=0,\quad 0<z<1,\qquad s_{ab}=\langle ab\rangle[ba]\to0
These are limiting momenta for the specified time-like collinear configuration, in natural units and the module's metric. The parent in the reduced amplitude is on shell. z is dimensionless and sab has energy-squared units; an exact finite-angle sum has small nonzero squared mass.↔ Scroll wide equations horizontally.

A helicity splitting factor retains a complex phase

For adjacent positive-helicity gluons with a positive-helicity parent in the reduced tree amplitude, the leading splitting factor is the reciprocal angle bracket divided by the square root of z times one minus z. Coupling powers are stripped as before. Other daughter and parent helicity combinations use different numerators or the conjugate bracket, and the general factorization includes a sum over the parent helicity.

The factor has inverse-energy units, appropriate for increasing multiplicity by one. In the real time-like slice its squared magnitude is inversely proportional to both the pair invariant and z times one minus z. This singular structure is a limiting factor in an amplitude, not a normalized probability distribution. The soft endpoints z near zero or one overlap another singular region and require careful treatment when integrating or avoiding double counting.

An(…,a+,b+,…)∼1z(1−z)⟨ab⟩An−1(…,P+,…),∣Split∣2=1z(1−z)sabA_n(\ldots,a^+,b^+,\ldots)\sim\frac{1}{\sqrt{z(1-z)}\langle ab\rangle}A_{n-1}(\ldots,P^+,\ldots),\qquad |\mathrm{Split}|^2=\frac{1}{z(1-z)s_{ab}}
The first expression is the stated leading tree helicity channel with coupling stripped. The magnitude relation uses a common future-directed real representative and the bracket convention [ba]=⟨ab⟩*. It does not include color factors, spin sums, flux or phase-space normalization.↔ Scroll wide equations horizontally.

Kernels combine channels and require prescriptions

Unpolarized splitting kernels combine appropriate helicities and color factors. For gluon splitting, the real-emission kernel contains both soft endpoint singularities and a regular term proportional to z times one minus z. Its symmetry under interchanging the daughters is a useful diagnostic. One selected same-helicity factor cannot be substituted for the complete spin-summed kernel, since the other helicity channels contribute different z dependences.

An integrated evolution equation requires virtual contributions, endpoint distribution prescriptions, factorization scales and a subtraction scheme. Polarized or azimuthally sensitive observables may retain spin correlations lost in a scalar kernel. Loop splitting amplitudes also acquire regulator-dependent singularities and corrections. The universal leading tree limit remains a valuable consistency check of a candidate amplitude, while a phenomenological shower or parton evolution calculation must state the additional ingredients that make its probabilities and scale dependence meaningful.

Work it through

Bracket size controls a collinear enhancement

In the stated helicity channel use z=1/4 and a real near-collinear representative with ⟨ab⟩=0.020 GeV and [ba]=0.020 GeV. Find sab, the splitting factor, and its squared magnitude. How does the latter change if both brackets halve at fixed z?

  1. The invariant is sab=0.0202=0.000400 GeV2s_{ab}=0.020^2=0.000400\,\mathrm{GeV^2}. The momentum-fraction product is z(1−z)=3/16, so its square root is √3/4.

  2. The factor is 1/[(3/4)(0.020)]=200/3≈115.471/[(\sqrt3/4)(0.020)]=200/\sqrt3\approx115.47 GeV⁻¹. Squaring gives 40000/3≈13333.3 GeV⁻², agreeing with the inverse-invariant expression.

  3. Halving both brackets reduces sab by four. The amplitude factor doubles and its squared magnitude increases by four. This statement holds for the leading factor at fixed momentum fraction; it does not integrate the collinear region or include a physical detector cut.

The result

The pair invariant is 4.00×10⁻⁴ GeV² and the splitting factor is about 115.47 GeV⁻¹. Its squared magnitude scales inversely with sab and grows fourfold under the stated change.

The living notebook

Explore the model

Interactive lab

Interactive lab · probability

Inspect the momentum-fraction product as an analogy

The Bernoulli variance p(1−p) has the same algebraic form as z(1−z). Reset and set both controls to p=0.25 and n=1; predict mean 0.25 and variance 0.1875. For a fresh comparison, Reset and set both controls to p=0.50 and n=1; predict mean 0.50 and variance 0.25. The reciprocal square roots entering the splitting normalization are 4/√3 and 2.

Inspect the momentum-fraction product as an analogyThese are exact binomial probabilities for independent trials with one fixed success probability. Playback tours the distribution; it does not resample outcomes. Current readings: Mean · np: 5; Variance · np(1−p): 2.5; Total probability: 1. All parameters are adjustable with the labeled controls below.03581000.070.150.220.3Successes kP(X = k)k = 0 · P = 9.77e-4
Exact probabilityHighlighted outcomeMean np
Mean · np
5
Variance · np(1−p)
2.5
Total probability
1
t = 0 s

These are exact binomial probabilities for independent trials with one fixed success probability. Playback tours the distribution; it does not resample outcomes. 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 forThis is an algebraic comparison with exact binomial readouts. A momentum fraction is not a Bernoulli probability, and the lab does not implement collinear parton dynamics or a splitting kernel.

Pause & reason

Check your understanding

What is lost when a helicity-level splitting amplitude is replaced by an unpolarized scalar kernel?

Your turn

Try it yourself

Use the real-emission unpolarized kernel Pgg(z)=2CA[z/(1−z)+(1−z)/z+z(1−z)]P_{gg}(z)=2C_A[z/(1-z)+(1-z)/z+z(1-z)] with CA=3. Evaluate it at z=1/2 and z=1/4, check daughter-exchange symmetry, and explain why these values are not emission probabilities.

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

Give me a hint

At one quarter the three terms are 1/3, 3 and 3/16. A kernel value is not an integral over a normalized outcome space.

Compare with the solution

At one half the bracket is 1+1+1/4=9/4, giving 13.5. At one quarter it is 169/48, giving 169/8=21.125. Exchanging z with 1−z swaps the first two terms and leaves the third unchanged, so the kernel is symmetric. These dimensionless values require coupling, evolution measure and endpoint/virtual prescriptions; their exceeding one is not a probability violation. This full helicity-combined real kernel differs from the single channel in the worked example.

Keep the idea

Take it with you

  • Time-like collinear factorization uses an on-shell limiting parent and a declared momentum-fraction range.
  • Helicity splitting factors contain phase information and have inverse-energy dimensions.
  • Spin-summed kernels and infrared-regulated evolution require additional channels, measures and prescriptions.

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