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In HQET, hybrid logarithms become ultraviolet logarithmic divergences governed by the renormalization group, with corresponding anomalous dimensions. 2) possesses the SU (2) heavy-quark spin symmetry [9]. If there are nh heavy-quark ﬁelds with the same velocity, it has the ¯ D ¯∗ SU (2nh ) spin–ﬂavour symmetry. For example, if we consider the B → D, transitions with v = v, to the leading order in 1/mc,b , then the Lagrangian has SU (4) spin–ﬂavour symmetry, which relates all form factors to the B → B form factor at zero momentum transfer, which is equal to 1.

1 Renormalization of QCD 37 The term with c1 can be absorbed by a rescaling of µ; therefore, we shall assume c1 = 0. 5) with a new β (αs ): β0 = β0 , β1 = β1 , β2 = β2 + c2 , . . The ﬁrst two β-function coeﬃcients β0,1 are the same in all mass-independent (or minimal) schemes; the higher coeﬃcients β2,3,... ). In this book, we shall always use the MS scheme. The information contained in Zα is equivalent to that in β(αs ). 8) where Z1 starts from the order αs , Z2 starts from α2s , and so on. Then dZα1 = −β(αs ) , d log αs dZα2 = β 2 (αs ) , d log αs dZα3 = −β 3 (αs ) , .

19) One can obtain the anomalous dimension γ(αs ) from Z1 , the coeﬃcient of 1/ε in Z, and vice versa. The coeﬃcients Z2 , Z3 , . . contain no new information; at each order in αs , they can be reconstructed from lower-loop results. 2 Gluon Propagator The bare (unrenormalized) gluon propagator −iDµν (p) has the structure (Fig. 21) is the free gluon propagator, and the gluon self-energy (polarization operator) iΠµν (p) is the sum of one-particle-irreducible gluon self-energy diagrams (which cannot be separated into two disconnected parts by cutting a single gluon line).