Abstract
Factorization is possible due to the universal behavior of Yang-Mills theories in soft and collinear limits. Here, we take a small step towards a more transparent understanding of these limits by proving a form of perturbative factorization at tree level using on-shell spinor helicity methods. We present a concrete and self-contained expression of factorization in which matrix elements in QCD are related to products of other matrix elements in QCD up to leading order in a power-counting parameter determined by the momenta of certain physical on-shell states. Our approach uses only the scaling of momenta in soft and collinear limits, avoiding any assignment of scaling behavior to unphysical (and gauge-dependent) fields. The proof of factorization exploits many advantages of helicity spinors, such as the freedom to choose different reference vectors for polarizations in different collinear sectors. An advantage of this approach is that once factorization is shown to hold in QCD, the transition to soft-collinear effective theory is effortless.
Highlights
That perturbative calculations in a strongly coupled theory like quantum chromodynamics (QCD) can ever be related to experimental data is due to two remarkable properties: asymptotic freedom and factorization
Since the reference vector can be chosen to be some momentum in the external state of interest, we can write any matrix element in QCD in terms of helicity spinors associated with physical on-shell momentum
The main result of this paper is a proof at tree-level of factorization for matrix elements of operators in QCD
Summary
That perturbative calculations in a strongly coupled theory like quantum chromodynamics (QCD) can ever be related to experimental data is due to two remarkable properties: asymptotic freedom and factorization. Most useful form, factorization states that cross sections in QCD can be calculated up to power corrections in some small scale λ by convolutions of universal (and often nonperturbative) objects, such as parton-distribution functions, and perturbative, but process-dependent matrix elements. In SCET one assumes, often without a completely rigorous proof, that factorization holds and derives formulae for cross sections in terms of gauge-invariant matrix elements of effective-theory fields with interactions different from those of full QCD. Since the reference vector can be chosen to be some momentum in the external state of interest, we can write any matrix element in QCD in terms of helicity spinors associated with physical on-shell momentum This lets us use only the scaling of external momenta to simplify matrix elements at leading power.
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