[Paper Review] Massive particles and unitarity cuts

Ruth Britto, Edoardo Mirabella|arXiv (Cornell University)|Feb 11, 2012

Particle physics theoretical and experimental studies29 references3 citations

TL;DR

This paper extends the spinor integration formalism to single cuts of one-loop amplitudes, enabling analytical computation of tadpole and on-shell bubble coefficients in the presence of massive particles. It introduces an off-shell continuation method that allows finite evaluation of unitarity cuts in the channel of a single massive fermion, achieving exact cancellation between loop diagrams and counterterms in the on-shell scheme.

ABSTRACT

We present an extension of the spinor integration formalism of one loop amplitudes from the double-cut to the single-cut case. This technique can be applied for the computation of the tadpole coefficients. Moreover we describe an off-shell continuation of one loop amplitudes that allows a finite evaluation of the unitarity cuts in the channel of a single massive external fermion.

Motivation & Objective

• To develop a systematic method for computing tadpole and on-shell bubble coefficients in one-loop amplitudes with massive particles.
• To address the challenge of divergent single-cut integrals in massive theories using an off-shell continuation technique.
• To enable finite evaluation of unitarity cuts in the channel of a single massive external fermion.
• To ensure exact cancellation between loop diagrams and counterterms in the on-shell renormalization scheme.

Proposed method

• Extends the spinor integration formalism from double cuts to single cuts using a parametrization of loop momentum in terms of spinor variables and complex coordinates $ z, \bar{z}, t $.
• Applies the generalized Cauchy formula to evaluate contour integrals over $ z $ and $ \bar{z} $, focusing on the leading $ \Lambda \to \infty $ behavior of primitives to isolate tadpole contributions.
• Uses the OPP decomposition to separate physical and spurious terms, treating coefficients as unknowns and solving via single-cut equations.
• Introduces an off-shell continuation of one-loop amplitudes to regulate divergences and enable finite evaluation of unitarity cuts in the massive fermion channel.
• Performs an expansion in $ \xi $ around $ \xi = 0 $ to extract the $ \mathcal{O}(\xi^0) $ terms and verify cancellation between loop and counterterm diagrams.
• Applies the method to the external leg correction diagram and counterterm, showing exact cancellation in the on-shell scheme.

Experimental results

Research questions

• RQ1 How can the spinor integration formalism be extended to single cuts to compute tadpole coefficients in massive theories?
• RQ2 What is the behavior of single-cut integrals in the presence of massive particles, and how can divergences be handled systematically?
• RQ3 Can unitarity cuts in the channel of a single massive external fermion be evaluated in a finite and unambiguous way?
• RQ4 How do loop diagrams and counterterms cancel in the on-shell scheme when massive particles are involved?
• RQ5 What role does the off-shell continuation play in enabling finite evaluation of unitarity cuts for massive external fermions?

Key findings

• The single-cut integral diverges in general, but the leading $ \Lambda \to \infty $ behavior of the primitive functions allows isolation of tadpole contributions via $ \Lambda^2 $ dependence.
• Tadpole primitives are purely rational, while higher-point integrands are logarithmically suppressed, enabling algorithmic selection of tadpole terms by power counting.
• The method successfully computes the tadpole coefficient $ a(0) $ for the integrand $ I = \frac{2\ell \cdot R}{D_0 D_1} $, assuming non-degenerate masses and non-vanishing Gram determinant.
• The external leg correction amplitude $ \mathcal{M}_A $ is found to contain a term proportional to $ A_0(m) $, explicitly given by $ \frac{g^2 C_F}{16\pi^2} \left( \frac{2}{\xi\gamma} - \frac{1}{m^2} \right) \bar{u}_k \mathcal{A}_{cc_{\rm ext}} A_0(m) $.
• The counterterm $ \mathcal{M}^{\rm ct} $, expanded in $ \xi $, contains a $ \frac{2}{\xi\gamma} A_0(m) $ term that exactly cancels the $ \frac{2}{\xi\gamma} $ divergence in $ \mathcal{M}_A $, ensuring finiteness.
• After summing $ \mathcal{M}_A + \mathcal{M}^{\rm ct} $, all $ \xi $-dependent and $ \bar{k} $-dependent terms cancel, confirming exact cancellation in the on-shell scheme.