Collinear effective theory at subleading order and its application to heavy-light currents

We consider a collinear effective theory of highly energetic quarks with energy E, interacting with collinear and soft gluons by integrating out collinear degrees of freedom to subleading order. The collinear effective theory offers a systematic expansion in power series of a small parameter $\lambda \sim p_{\perp }/E,$ where $p_{\perp }$ is the transverse momentum of a collinear particle. We construct the effective Lagrangian to first order in $\lambda$ and discuss its features, including additional symmetries such as collinear gauge invariance and reparametrization invariance. Heavy-light currents can be matched from the full theory onto the operators in the collinear effective theory at one loop and to order $\lambda .$ We obtain heavy-light current operators in the effective theory, calculate their Wilson coefficients at this order, and the renormalization group equations for the Wilson coefficients are solved. As an application, we calculate the form factors for decays of B mesons to light energetic mesons to order $\lambda$ and at leading-logarithmic order in ${\alpha }_s.$