Consequences of symmetries in renormalizing collinear effective theory

We consider effects of symmetries on renormalization properties of the collinear effective theory. We investigate which types of operators are possible in the effective theory satisfying gauge invariance, reparameterization invariance and residual energy invariance. Each symmetry puts a constraint on the possible structure of the theory, and there can appear only specific combinations of operators in the effective Lagrangian satisfying all the symmetry requirements. And the final effective Lagrangian is not renormalized to all orders in αs as long as no other nonlocal operators are induced at higher order. We explicitly prove this at one loop by renormalizing one-gluon vertices and discuss their features. Strong interaction processes which involve energetic, massless particles can be described by the collinear effective theory [1–4]. It has been applied to sum Sudakov logarithms [1], to prove factorization [5,6] and study power corrections [7]. Symmetry properties such as reparameterization invariance, residual energy invariance, and gauge invariance have been investigated [8,9]. The collinear effective theory offers a systematic way to organize physical quantities in powers of a small parameter λ ∼ p⊥/E, in which a massless energetic quark moves with energy E, and transverse momentum p⊥. The momentum P μ of an energetic particle can be decomposed as P μ = n · p 2 n + p ⊥ + k, (1) where p = 1 2 (n · p)n + p ⊥ is the label momentum of order λ and λ, respectively. The momentum k is the residual momentum of order λ, which represents small fluctuation due to the strong interaction. 1 E-mail address: [email protected] Preprint submitted to Elsevier Preprint 1 February 2008 In the collinear effective theory, we classify fields into three classes according to their momenta as collinear, soft and ultrasoft (usoft) fields. Their typical momenta scale as E(λ, 1, λ), E(λ, λ, λ), and E(λ, λ, λ), respectively. In processes involving collinear quarks, the relevant fields are collinear quarks ξn, collinear gluons A μ n and usoft gluons A μ u. The effective Lagrangian can be derived from the full QCD in terms of the collinear quark spinor ξn which satisfies / n/ n 4 ξn = ξn, / nξn = 0, (2) where n = 0, n = 0 and n · n = 2. The effective Lagrangian is written as L = ξn [ n · iD + i/ D⊥ 1 n · iD i/ D⊥ / n 2 ξn, (3) where D = D c +D μ u is a covariant derivative under collinear and usoft gauge transformations. The covariant derivatives Dc and Du are defined as iD c = P μ − gAμn, iD μ u = i∂ μ − gAμu. (4) Here P is the operator which extracts label momenta from collinear fields. For example, if we apply P to a collinear spinor ξn with label momentum p , we get P ξn = n · p 2 n + p ⊥ )