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Constructing QCD One-loop Amplitudes

This work provides an overview of constructing one-loop amplitudes in Quantum Chromodynamics (QCD) using unitarity techniques. The focus is on the decomposition of one-loop amplitudes into cut-constructible components, dealing with missing rational pieces, and utilizing effective recursion relations. Key approaches involve quad and triple cut techniques to determine rational coefficients of triangle and bubble integrals, enabling the automation of amplitude calculations for a wide range of processes relevant to collider physics. Additionally, comparisons with existing literature are made, establishing the framework’s robustness and applicability.

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Constructing QCD One-loop Amplitudes

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  1. Darren Forde (SLAC & UCLA) Constructing QCD One-loop Amplitudes arXiv:0704.1835 (To appear this evening)

  2. Overview

  3. The unitarity bootstrap Focus on these terms • Cut-constructible from gluing together trees in D=4, • i.e. unitarity techniques in D=4. •  missingrational pieces in QCD. [Bern, Dixon, Dunbar, Kosower] • Rational from one-loop on-shell recurrence relation. [Berger, Bern, Dixon, DF, Kosower] • Alternatively work in D=4-2ε, [Bern, Morgan], [Anastasiou, Britto, Feng, Kunszt, Mastrolia] • Gives both terms but requires trees in D=4-2ε. Unitarity bootstrap technique

  4. One-loop integral basis • A one-loop amplitude decomposes into • Quadruple cuts freeze the integral  boxes [Britto, Cachazo, Feng] Rational terms l l1 l3 l2

  5. Two-particle and triple cuts • What about bubble and triangle terms? • Triple cut  Scalar triangle coefficients? • Two-particle cut  Scalar bubble coefficients? • Disentangle these coefficients. Additional coefficients Isolates a single triangle

  6. Disentangeling coefficients • Approaches, • Unitarity technique, [Bern, Dixon, Dunbar, Kosower] • MHV one-loop cut-constructible by joining MHV vertices at two points, [Bedford, Brandhuber, Spence, Traviglini], [Quigley, Rozali] • Integration of spinors, [Britto,Cachazo,Feng] + [Mastrolia] + [Anastasiou, Kunst], • Solving for coefficients, [Ossola, Papadopoulos, Pittau] • Recursion relations, [Bern, Bjerrum-Bohr, Dunbar, Ita] • Large numbers of processes required for the LHC, • Automatable and efficient techniques desirable. • Can we do better?

  7. Triangle coefficients • Coefficients, cij, of the triangle integral, C0(Ki,Kj), given by Triple cut of the triangle C0(Ki,Kj) Single free integral parameter in l K3 A3 A2 A1 K1 K2 Series expansion in t at infinity Masslessly Projected momentum

  8. six photons 6 λ‘s top and bottom [Nagy, Soper], [Binoth, Heinrich, Gehrmann, Mastrolia], [Ossola, Papadopoulos, Pittau] • 3-mass triangle of A6(-+-+-+)  the triple cut integrand • The complete coefficient. Extra propagator  Box terms No propagator  Triangle Propagator ↔ pole in t, 2 solutions to γ divide by 2 The scalar triangle coefficient

  9. Vanishing integrals • In general higher powers of t appear in [Inf A1A2A3](t). • Integrals over t vanish for chosen parameterisation, e.g.(Similar argument to [Ossola, Papadopoulos, Pittau]) • In general whole coefficient given by

  10. Another Triangle Coefficient • 3-mass triangle coefficient of in the 14:23:56 channel. [Bern, Dixon, Kosower] 2 λ‘s top and bottom Independent of t Series expand in t around infinity

  11. What about bubbles? • The bubble coefficient bj of the scalar bubble integral B0(Kj) Two-particle cut of the bubble B0(Ki) Two free integral parameter in l A2 max y≤4 K1 A1

  12. Non-vanishing Integrals • Similar to triangle coefficients, but depends upon t. • Two free parameters implies Box and triangle coeff’s One extra Pole in y, looks like a triangle Two-particle cut contrib y fixed at pole Contains bubbles

  13. Triple-cut contributions • Example: Extract bubble of three-mass linear triangle, • Cut l2 and (l-K1)2propagators, gives integrand • Complete coefficient. Single pole Series expand y and then t around ∞, No “triangle” terms as set

  14. Triple cut contributions cont. • Multiple poles Can’t choose χso that all integrals in t vanish. • Sum over all triangles containing the bubble, • Renormalisable theories, max of t3. • Integrals over t known, Cij a constant, e.g. C11=1/2 • Gives equivalent, χindependent result

  15. other Applications • Comparisons against the literature • Two minus all gluon bubble coefficients for up to 7 legs. [Bern, Dixon, Dunbar, Kosower], [Bedford, Brandhuber, Spence, Travigini] • N=1 SUSY gluonic three-mass triangles for A6(+-+-+-), A6(+-++--). [Britto, Cachazo, Feng] • Various bubble and triangle coefficients for processes of the type . [Bern, Dixon, Kosower] • Bubble and three-mass triangle coefficients for six photon A6(+-+-+-) amplitude. [Nagy, Soper], [Binoth, Heinrich, Gehrmann, Mastrolia], [Ossola, Papadopoulos, Pittau]

  16. Conclusion

  17. A<n R<n Rn On-shell recursion relations Two reference legs “shifted”, • Recursion using on-shell amplitudes with fewer legs, [Britto, Cachazo, Feng] + [Witten] • Final result independent of the of choice shift. • Complete amplitude at tree level. • At one loop need the cut pieces [Berger, Bern, Dixon, DF, Kosower] • Combining both involves overlap terms. Intermediate momentum leg is on-shell.

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