Reparametrization Invariance and Partial Re-Summations of the Heavy Quark Expansion

Reparametrization invariance and the expansion of currents in the heavy quark effective theory

The coefficients appearing at leading and subleading order in the $1/m$ expansion of bilinear heavy quark currents are related to each other by imposing reparametrization invariance on both the effective current operators and the short-distance coefficient functions in the heavy quark effective theory. When combined with present knowledge about the leading order coefficients, the results allow to calculate all coefficients appearing at order $1/m$ to next-to-leading order in renormalization-group improved perturbation theory. They also provide a meaningful definition of the velocity transfer v…

Reparametrization Invariance Constraints on Inclusive Decay Spectra and Masses

Reparametrization invariance relations are determined for the inclusive decay spectra of heavy hadrons. They connect the leading order result to the lambda_1 term, and the lambda_2 and rho_2 terms, and imply that the lambda_1 and rho_2 terms in the total rate occur in the combinations 1+lambda_1/(2m_Q^2) and lambda_2-rho_2/m_Q. The relations are satisfied by the known results for the hadronic decay tensors for B -> X e nu, B -> X gamma and B -> X e+ e- decays. An interesting field-theory result, the connection between currents in full and effective theories and the change in currents under fie…

中微子振幅分解，$S$ 矩阵重相位不变性与重参数化对称性

Neutrino amplitude decomposition, $S$ matrix rephasing invariance, and reparametrization symmetry

本文通过重参数化对称性变换映射中微子振荡S矩阵的不同重相位振幅，揭示了该对称性的量子力学本质，并证明其意味着Denton微扰理论中不存在纯1-3交换对称性，进而引发对微扰级数收敛性的研究。

Reparametrization invariance of B decay amplitudes and implications for new physics searches in B decays

When studying B decays within the Standard Model, it is customary to use the unitarity of the CKM matrix in order to write the decay amplitudes in terms of only two of the three weak phases which appear in the various diagrams. Occasionally, it is mentioned that those two weak phases can be used in order to describe any decay amplitude, even beyond the Standard Model. Here we point out that, when describing a generic decay amplitude, the two weak phases can be chosen completely at will, and we study the behavior of the decay amplitudes under changes in the two weak phases chosen as a basis. Of…

Reparametrization Invariance, the controversial extraction of $α$ from $B\toππ$ and New Physics

The extraction of the weak phase $α$ from $B\toππ$ decays has been controversial from a statistical point of view, as the frequentist vs. bayesian confrontation shows. We analyse several relevant questions which have not deserved full attention and pervade the extraction of $α$. Reparametrization Invariance proves appropriate to understand those issues. We show that some Standard Model inspired parametrizations can be senseless or inadequate if they go beyond the minimal Gronau and London assumptions: the single weak phase $α$ just in the $ΔI=3/2$ amplitudes, the isospin relations and experime…

Comment on Reparametrization Invariance of Quark-Lepton Complementarity

We study the complementarity between quark and lepton mixing angles (QLC), the sum of an angle in quark mixing and the corresponding angle in lepton mixing is $π/4$. Experimentally in the standard PDG parametrization, two such relations exist approximately. These QLC relations are accidental which only manifest themselves in the PDG parametrization. We propose reparametrization invariant expressions for the complementarity relations in terms of the magnitude of the elements in the quark and lepton mixing matrices. In the exact QLC limit, it is found that $|V_{us}/V_{ud}| + |V_{e2}/V_{e1}| + |V…

Properties of derivative expansion approximations to the renormalization group

Approximation only by derivative (or more generally momentum) expansions, combined with reparametrization invariance, turns the continuous renormalization group for quantum field theory into a set of partial differential equations which at fixed points become non-linear eigenvalue equations for the anomalous scaling dimension $η$. We review how these equations provide a powerful and robust means of discovering and approximating non-perturbative continuum limits. Gauge fields are briefly discussed. Particular emphasis is placed on the rôle of reparametrization invariance, and the convergence of…

Renormalization Group Scaling of the 1/m^2 HQET Lagrangian

The operator mixing matrix for the dimension six operators in the heavy quark effective theory Lagrangian is computed at one loop. The results are shown to be consistent with constraints from the equations of motion, and from reparametrization invariance.

Perturbative expansion of the QCD Adler function improved by renormalization-group summation and analytic continuation in the Borel plane

We examine the large-order behaviour of a recently proposed renormalization-group-improved expansion of the Adler function in perturbative QCD, which sums in an analytically closed form the leading logarithms accessible from renormalization-group invariance. The expansion is first written as aneffective series in powers of the one-loop coupling, and its leading singularities in the Borel plane are shown to be identical to those of the standard "contour-improved" expansion. Applying the technique of conformal mappings for the analytic continuation in the Borel plane, we define a class of improv…

Light-Quark, Heavy-Quark Systems: An Update

We review many of the recently developed applications of Heavy Quark Effective Theory techniques. After a brief update on Luke's theorem, we describe striking relations between heavy baryon form factors, and how to use them to estimate the accuracy of the extraction of $|V_{cb}|$. We discuss factorization and compare with experiment. An elementary presentation, with sample applications, of reparametrization invariance comes next. The final and most extensive chapter in this review deals with phenomenological lagrangians that incorporate heavy-quark spin-flavor as well as light quark chiral sym…

Two-Loop Anomalous Dimension of the Chromo-Magnetic Moment of a Heavy Quark

We find that the anomalous dimension (in the $\bar{MS}$ scheme) of the chromo-magnetic operator $g_s\bar h_vσ_{μν} G^{μν} h_v$ in the heavy-quark effective theory is γ_{mag} = (C_Aα_s / 2π) [1 + ( 17/18 C_A - 13/18 T_F n_f)α_s/π+ O(α_s^2)] generalizing the one-loop expression known previously. To derive the two-loop result, we use the reparametrization invariance and the virial theorem. After performing infrared subtractions, all two-loop integrals are of propagator type and are evaluated by a recurrence relation for tensor integrals.

Renormalizing Heavy Quark Effective Theory at O(1/m_Q^3)

We present a calculation of the renormalized HQET Lagrangian at order O(1/m_Q^3) in the one particle sector. The anomalous dimensions of local operators and time ordered products of dimension 7 contributing at this order are calculated in the one loop approximation. We show that a careful treatment of the time ordered products is necessary to arrive at a gauge independent renormalized lagrangian. Our result sets the stage for an investigation of reparametrization invariance at O(1/m_Q^3).

$V_{cb}$ determination from inclusive $b \to c$ decays: an alternative method

The determination of $V_{cb}$ relies on the Heavy-Quark Expansion and the extraction of the non-perturbative matrix elements from inclusive $b\to c$ decays. The proliferation of these matrix elements complicates their extraction at $1/m_b^4$ and higher, thereby limiting the $V_{cb}$ extraction. Reparametrization invariance links different operators in the Heavy-Quark expansion thus reducing the number of independent operators at $1/m_b^4$ to eight for the total rate. We show that this reduction also holds for spectral moments as long as they are defined by reparametrization invariant weight-fu…

On the role of power expansions in quantum field theory

Methods of summation of power series relevant to applications in quantum theory are reviewed, with particular attention to expansions in powers of the coupling constant and in inverse powers of an energy variable. Alternatives to the Borel summation method are considered and their relevance to different physical situations is discussed. Emphasis is placed on quantum chromodynamics. Applications of the renormalon language to perturbation expansions (resummation of bubble chains) in various QCD processes are reported and the importance of observing the full renormalization-group invariance in pr…