% Decay constant

\newcommand{\fb}{F_{\rm B}}
%\newcommand{\Ospin}{{\cal O}_{\rm spin}}

% action, lagrangian

\def\Sstat{{S_{\rm h}}}
\def\Sastat{{S_{\bar{\rm h}}}}
\def\Shqet{S_{\rm HQET}}
\def\Srel{S_{\rm rel}}
\def\Lstat{{\cal L}_{\rm stat}}
\def\qcdpar{{\cal C}_{\nf}}
\def\qcdparred{{\cal C}_{\nf-1}}
\def\hqetpar{{\cal C}_{\rm HQET}}
\newcommand{\lnu}[1]{{\cal L}^{(#1)}}
\newcommand{\lcoeff}[2]{\omega^{(#1)}_#2}
\newcommand{\acoeff}[2]{\alpha^{(#1)}_#2}
\newcommand{\minv}{1/m_\mrm{b}}


% Static approximation fields

%\newcommand{\heavy}{\psi_{\rm h}}
\newcommand{\heavy}{\psi_{\rm \phantom{\bar{h}}\hspace{-0.12cm}h}}
\newcommand{\heavyb}{\overline{\psi}_{\rm h}}
\newcommand{\aheavy}{\psi_{\bar{\rm h}}}
\newcommand{\aheavyb}{\overline{\psi}_{\bar{\rm h}}}
\newcommand{\light}{{\psi}_{\rm l}}
\newcommand{\lightb}{\overline{\psi}_{\rm l}}
\newcommand{\rhol}{\rho_{\rm l}}
\newcommand{\rholb}{\overline{\rho}_{\rm l}}
\newcommand{\rholprime}{\rho_{\rm l}\kern1pt'}
\newcommand{\rholbprime}{\overline{\rho}_{\rm l}\kern1pt'}
\newcommand{\rhoh}{\rho_{\rm h}}
\newcommand{\rhohbprime}{\overline{\rho}_{\rm h}\kern1pt'}
\newcommand{\zetal}{{\zeta_{\rm l}}}
\newcommand{\zetalb}{\overline{\zeta}_{\rm l}}
\newcommand{\zetalprime}{\zeta_{\rm l}\kern1pt'}
\newcommand{\zetaiprime}{\zeta_{i}\kern1pt'}
\newcommand{\zetajprime}{\zeta_{j}\kern1pt'}
\newcommand{\zetalbprime}{\overline{\zeta}_{\rm l}\kern1pt'}
\newcommand{\zetaibprime}{\overline{\zeta}_{i}\kern1pt'}
\newcommand{\zetajbprime}{\overline{\zeta}_{j}\kern1pt'}
\newcommand{\zetah}{{\zeta_{\rm h}}}
\newcommand{\zetahb}{\overline{\zeta}_{\rm h}}
\newcommand{\zetahprime}{\zeta_{\rm h}\kern1pt'}
\newcommand{\zetahbprime}{\overline{\zeta}_{\rm h}\kern1pt'}
\newcommand{\Astat}{A^{\rm stat}_0}
\newcommand{\Ahqet}{A^{\rm HQET}_0}
\newcommand{\Vstat}{V^{\rm stat}_0}
\newcommand{\Vhqet}{V^{\rm HQET}_0}
\newcommand{\Akstat}{A^{\rm stat}_k}
\newcommand{\Vlstat}{V^{\rm stat}_l}
\newcommand{\Akhqet}{A^{\rm HQET}_k}
\newcommand{\Vkstat}{V^{\rm stat}_k}
\newcommand{\Vkhqet}{V^{\rm HQET}_k}
\newcommand{\Phqet}{P^{\rm HQET}}
\newcommand{\Scalarhqet}{S^{\rm HQET}}
\newcommand{\Argi}{(A^{\rm RGI})_0}
\newcommand{\Aren}{(A^{\rm stat}_{\rm R})_0}
\newcommand{\Vren}{(V^{\rm stat}_{\rm R})_0}
\newcommand{\ArenSF}{(A^{\rm stat}_{\rm SF})_0}
\newcommand{\ArenMSbar}{(A^{\rm stat}_{\overline{\rm MS}})_0}
\newcommand{\ArenLat}{(A^{\rm stat}_{\rm lat})_0}
\newcommand{\ArenCA}{(A_{\rm CA})_0}
\newcommand{\Astatimpr}{(A^{\rm stat}_{\rm I})_0}
\newcommand{\Vstatimpr}{(V^{\rm stat}_{\rm I})_0}

\newcommand{\zetalone}{\zeta_1}
\newcommand{\zetalbtwo}{\overline{\zeta}_2}
\newcommand{\zetalbprimeone}{\overline{\zeta}_1\kern1pt'}
\newcommand{\zetalprimetwo}{\zeta_2\kern1pt'}

\newcommand{\zh}{Z_{\rm h}}

\newcommand{\chiAstatSFMSbar}{\chi_{\rm A}^{\rm stat}}
\newcommand{\chiAstat}{(\chi_{\rm A}^{\rm stat})}
\newcommand{\chiAstatMSbarLat}{(\chi_{\rm A}^{\rm stat})_{\msbar,\rm lat}}
\newcommand{\chiAstatMSbarLatOne}{(\chi_{\rm A}^{\rm stat})_{\msbar,\rm lat}^{(1)}}
\newcommand{\chiAstatSFMSbarOne}{(\chi_{\rm A}^{\rm stat})^{(1)}}

\newcommand{\SigmaAstat}{\Sigma_{\rm A}^{\rm stat}}
\newcommand{\sigmaAstat}{\sigma_{\rm A}^{\rm stat}}
\newcommand{\sigmaAzerostat}{\sigma_{\rm A,0}^{\rm stat}}
\newcommand{\sigmaAonestat}{\sigma_{\rm A,1}^{\rm stat}}
\newcommand{\sigmaAtwostat}{\sigma_{\rm A,2}^{\rm stat}}
\newcommand{\sigmaAnstat}{\sigma_{{\rm A},n}^{\rm stat}}
\newcommand{\sigmamag}{\sigma_{\rm spin}}
\newcommand{\Sigmamag}{\Sigma_{\rm spin}}


\newcommand{\gamSF}{\gamma^{\rm SF}}
\newcommand{\gamMSbar}{\gamma^{\overline{\rm MS}}}

\newcommand{\latstep}{\Sigma_{\rm A}^{\rm stat}}
\newcommand{\contstep}{\sigma_{\rm A}^{\rm stat}}

% Static approximation correlation functions

\newcommand{\fastat}{f_{\rm A}^{\rm stat}}
\newcommand{\fastatimpr}{f_{\rm A}^{\rm stat, I}}
\newcommand{\fdeltaastat}{f_{\delta\rm A}^{\rm stat}}
\newcommand{\fonestat}{f_{1}^{\rm stat}}
\newcommand{\foneav}{f_{1}^{\rm av}}
\newcommand{\fonehh}{f_{1}^{\rm hh}}
\newcommand{\fastatren}{(f_{\rm A}^{\rm stat})_{\rm R}}
\newcommand{\fonestatren}{(f_{1}^{\rm stat})_{\rm R}}
\newcommand{\Xren}{X_{\rm R}}
\newcommand{\XLat}{X_{\rm lat}}
\newcommand{\Xmatch}{X_{\rm match}}
\newcommand{\Xnew}{\Xi}
\def\caa{C_{\rm AA}}
\def\cdaa{C_{\rm \delta AA}}
\def\cada{C_{\rm A\delta A}}
\def\caahqet{C_{\rm AA}^{\rm HQET}}
\def\daa{D_{\rm AA}}

% Light-light correlation functions

\newcommand{\fone}{f_1}
\newcommand{\kone}{k_1}

% Masses

\newcommand{\mcrit}{m_{\rm c}}
\newcommand{\mcritone}{m_{\rm c}^{(1)}}
\newcommand{\kc}{\kappa_{\rm c}}
\newcommand{\mMSbar}{\overline{m}_{2,\overline{\rm MS}}}

% Matrices appearing in the correlation functions

\newcommand{\Hlight}{H_{\rm l}}
\newcommand{\Hheavy}{H_{\rm h}}
\newcommand{\Klight}{K_{\rm l}}
\newcommand{\Kheavy}{K_{\rm h}}

% Static approximation renormalizations

\newcommand{\zastat}{Z_{\rm A}^{\rm stat}}
\newcommand{\zastatSF}{Z_{\rm A, SF}^{\rm stat}}
\newcommand{\zastatRGI}{Z_{\rm A, RGI}^{\rm stat}}
\newcommand{\zastatrgi}{Z_{\rm A, RGI}^{\rm stat}}
\newcommand{\zastatMSbar}{Z_{\rm A, \overline{\rm MS}}^{\rm stat}}
\newcommand{\zastatLat}{Z_{\rm A, lat}^{\rm stat}}
\newcommand{\BAstat}{B_{\rm A}^{\rm stat}}
\newcommand{\zvstat}{Z_{\rm V}^{\rm stat}}
\newcommand{\zvkstat}{Z_{\rm\bf V}^{\rm stat}}
\newcommand{\zakstat}{Z_{\rm\bf A}^{\rm stat}}
\newcommand{\ltworgi}{\lambda_2^{\rm RGI}}
%\newcommand{\zmag}{Z_{\rm mag}}
\newcommand{\zmag}{Z_{\rm spin}}
\newcommand{\vspin}{V_{\rm spin}}
\newcommand{\vspinsf}{V_{\rm spin}^{\rm SF}}
\newcommand{\vspinrgi}{V_{\rm spin}^{\rm RGI}}
\newcommand{\vthree}{V_{3}}
\newcommand{\vthreesf}{V_{3}^{\rm SF}}
\newcommand{\vthreergi}{V_{3}^{\rm RGI}}
\newcommand{\vfour}{V_{4}}
\newcommand{\vfoursf}{V_{4}^{\rm SF}}
\newcommand{\vfourrgi}{V_{4}^{\rm RGI}}
%\newcommand{\zmagsf}{Z_{\rm mag}^{\rm SF}}
%\newcommand{\zmagrgi}{Z_{\rm mag}^{\rm RGI}}
%\newcommand{\zmagsfp}{Z_{\rm mag}^{\rm SF'}}
%\newcommand{\zmagrgip}{Z_{\rm mag}^{\rm RGI'}}
\newcommand{\zmagsf}{Z_{\rm spin}^{\rm SF}}
\newcommand{\zmagrgi}{Z_{\rm spin}^{\rm RGI}}
\newcommand{\zmagsfp}{Z_{\rm spin}^{\rm SF'}}
\newcommand{\zmagrgip}{Z_{\rm spin}^{\rm RGI'}}
\newcommand{\sk}{S_{k}}
\newcommand{\pol}{{\cal P}}

\def\dmstat{\delta m}
\def\dmstathat{\widehat{\delta m}}

% Light quark renormalizations

\newcommand{\zmMSbar}{Z_{\rm m, \overline{\rm MS}}}

% Static approximation improvement coefficients

\def\castat{c_{\rm A}^{\rm stat}}
\def\cstat{c_{\rm h}}
\def\dastat{d_{\rm A}^{\rm stat}}
\def\bastat{b_{\rm A}^{\rm stat}}
\def\bstat{b_{\rm h}}
\def\cvstat{c_{\rm V}^{\rm stat}}
\def\bvstat{b_{\rm V}^{\rm stat}}

% Martin Kurth's stuff

\newcommand{\ola}{\overleftarrow}
\newcommand{\vectq}{{\it \bf q}}
\newcommand{\vectp}{{\it \bf p}}
\newcommand{\vectn}{{\it \bf 0}}

\newcommand{\cttilde}{\widetilde{c}_{\rm t}}

\newcommand{\gbarLat}{\bar{g}_{\rm lat}}
\newcommand{\gLat}{g_{\rm lat}}

\newcommand{\fonehhtree}{f_1^{\rm hh,(0)}}
\newcommand{\fonehhoneloop}{f_1^{\rm hh,(1)}}
\newcommand{\zastattree}{Z_{\rm A}^{\rm stat,(0)}}
\newcommand{\zastatoneloop}{Z_{\rm A}^{\rm stat,(1)}}
\newcommand{\SigmaAstatoneloop}{\Sigma_{\rm A}^{\rm stat,(1)}}
\newcommand{\sigmaAstatoneloop}{\sigma_{\rm A}^{\rm stat,(1)}}
\newcommand{\sigmaAstattwoloop}{\sigma_{\rm A}^{\rm stat,(2)}}
\newcommand{\sigmaAstatnloop}{\sigma_{\rm A}^{{\rm stat,}(n)}}
\newcommand{\deltaAstat}{\delta}
\newcommand{\deltaAstatoneloop}{\delta^{(1)}}

% Rainer Sommer's stuff

\newcommand{\fhh}{f^{\rm hh}}
\newcommand{\zastathh}{Z_{\rm A, SF,hh}^{\rm stat}}
\newcommand{\gamhh}{\gamma^{\rm SF,hh}}
\newcommand{\Fhatstat}{ {\Phi}^{\rm stat} }
\newcommand{\FhatstatRGI}{\Phi^{\rm stat}_{\rm RGI}}

% Jochen Heitger's stuff

\newcommand{\mumatch}{\mu_{\rm m}}

\newcommand{\Fm}{{\rm fm}}
\newcommand{\psib}{\psi_{\rm b}}
\newcommand{\bpsid}{\overline{\psi}_{\rm d}}
\newcommand{\mbMSbar}{m_{{\rm b},\MS}}
\newcommand{\mhbare}{m_{\rm bare}}

\newcommand{\mB}{m_\mrm{B}}
\newcommand{\mBstar}{m_\mrm{B^*}}
\newcommand{\mbav}{m_{\rm B}^\mrm{av}}
\newcommand{\mbsplitt}{\Delta m_{\rm B}}
\newcommand{\Mb}{M_{\rm b}}
\newcommand{\Fbbare}{F_{\rm B}^{\rm bare}}
\newcommand{\Fbsstat}{F_{{\rm B}_{\rm s}}^{\rm stat}}
\newcommand{\Fbstat}{F_{{\rm B}}^{\rm stat}}
\newcommand{\Aistat}{(A^{\rm stat}_{\rm I})_{0}}
\newcommand{\Arstat}{(A_{\rm R}^{\rm stat})_{0}}

\newcommand{\oldSF}{{\rm old}}
\newcommand{\newSF}{{\rm new}}
\newcommand{\gamSFold}{\gamma^{{\rm SF},\oldSF}}
\newcommand{\zastatold}{Z_{{\rm A},\oldSF}^{{\rm stat}}}
\newcommand{\SigmaAstatold}{\Sigma_{{\rm A},\oldSF}^{{\rm stat}}}

\newcommand{\Phibare}{\Phi_{\rm bare}}
\newcommand{\Phihqet}{\Phi^{\rm HQET}}
\newcommand{\Phiqcd}{\Phi^{\rm QCD}}
\newcommand{\Phistat}{\Phi^{\rm stat}}
\newcommand{\PhistatR}{\Phi^{\rm stat}_{\rmR}}
\newcommand{\Yr}{Y_{\rm PS}}
\newcommand{\XRGI}{X_{\rm RGI}}
\newcommand{\XRGIspin}{X_{\rm RGI}^{\rm spin}}

\newcommand{\PhiRGI}{\Phi_{\rm RGI}}
\newcommand{\Phirgi}{\Phi_{\rm RGI}}
\newcommand{\Phiinter}{\Phi_{\rm inter}}
\newcommand{\PhiSF}{\Phi_{\rm SF}}
\newcommand{\Phisf}{\Phi_{\rm SF}}
\newcommand{\PhiMSbar}{\Phi_{\overline{\rm MS}}}
\newcommand{\Phimatch}{\Phi_{\rm match}}
\newcommand{\FhatRGI}{\hat{F}_{\rm RGI}}
\newcommand{\FhatMSbar}{\hat{F}_{\overline{\rm MS}}}
\newcommand{\FhatSF}{\hat{F}_{\rm SF}}
\newcommand{\ZPhi}{Z_{\rm RGI}}
\newcommand{\Ztil}{\widetilde{Z}}
\newcommand{\Cps}{C_{\rm PS}}
\newcommand{\Cpshat}{\widehat{C}_{\rm PS}}
\newcommand{\Cmagmatch}{C^{\rm match}_{\rm mag}}
\newcommand{\Cmag}{C_{\rm mag}}
\newcommand{\Cspin}{C_{\rm spin}}

\newcommand{\Nn}{N_{n}}
\newcommand{\Neff}{N_\mrm{eff}}

\newcommand{\zb}{z_{\rm b}}

\newcommand{\omegaspin}{\omega_\mrm{spin}}
\newcommand{\omegakin}{\omega_\mrm{kin}}
\newcommand{\gamdifstat}{\Xi^{\rm stat}}
\newcommand{\gamdifone}{\Xi^{\rm kin}}
\newcommand{\gamdif}{\Xi}

\newcommand{\ampl}{{\cal A}}
\newcommand{\amplqcd}{{\cal A}_\mrm{qcd}}
\newcommand{\amplhqet}{{\cal A}_\mrm{hqet}^\mrm{RGI}}
\newcommand{\amplmsbar}{{\cal A}_\mrm{hqet}^\msbar}
\newcommand{\Ospin}{{\cal O}_\mrm{spin}}
\newcommand{\Ospinrgi}{{\cal O}_\mrm{spin}^{\rm RGI}}
\newcommand{\OspinSF}{{\cal O}_\mrm{spin}^{\rm SF}}
\newcommand{\OspinMS}{{\cal O}_\mrm{spin}^{\msbar}}
\newcommand{\Ospinlat}{{\cal O}_\mrm{spin}^{\rm lat}}
\newcommand{\Okin}{{\cal O}_\mrm{kin}}
\newcommand{\zahqet}{Z_{\rm A}^{\rm HQET}}
\newcommand{\zvhqet}{Z_{\rm V}^{\rm HQET}}
\newcommand{\cahqet}{c_{\rm A}^{\rm HQET}}
\newcommand{\cvhqet}{c_{\rm V}^{\rm HQET}}
\newcommand{\zakhqet}{Z_{\rm\bf A}^{\rm HQET}}
\newcommand{\zvkhqet}{Z_{\rm\bf V}^{\rm HQET}}
\newcommand{\cakhqet}{c_{\rm\bf A}^{\rm HQET}}
\newcommand{\cvkhqet}{c_{\rm\bf V}^{\rm HQET}}
\newcommand{\cavhqet}{c_{\rm av}^{\rm HQET}}
\newcommand{\cdiffhqet}{c_{\rm diff}^{\rm HQET}}

\newcommand{\fakin}{f_{\rm A}^{\rm kin}}
\newcommand{\faspin}{f_{\rm A}^{\rm spin}}
\newcommand{\kvstat}{k_{\rm V}^{\rm stat}}
\newcommand{\kvkin}{k_{\rm V}^{\rm kin}}
\newcommand{\kvspin}{k_{\rm V}^{\rm spin}}
\newcommand{\kdeltavstat}{k_{\delta \rm V}^{\rm stat}}
\newcommand{\fonekin}{\fone^{\rm kin}}
\newcommand{\fonespin}{\fone^{\rm spin}}
\newcommand{\konekin}{\kone^{\rm kin}}
\newcommand{\konespin}{\kone^{\rm spin}}

\def\zzetah{Z_{\zeta_ {\mrm{h}}}}

\newcommand{\ratastat}{R^{\rm stat}_{\rm A}}
\newcommand{\ratakin}{R^{\rm kin}_{\rm A}}
\newcommand{\rataspin}{R^{\rm spin}_{\rm A}}
\newcommand{\ratdastat}{R_{\delta \rm A}}
\newcommand{\ratonep}{R^{\rm P}_{1}}
\newcommand{\ratonev}{R^{\rm V}_{1}}
\newcommand{\ratone}{R_1}
\newcommand{\rata}{R_{\rm A}}
\newcommand{\ratv}{R_{\rm V}}
\newcommand{\ratonestat}{R^{\rm stat}_{1}}
\newcommand{\ratonekin}{R^{\rm kin}_{1}}
\newcommand{\ratonespin}{R^{\rm spin}_{1}}
% Definitions and abbreviations


% Roman letters in math formulae

\newcommand{\mrm}[1]{\mathrm{#1}}

\def\rmd{{\rm d}}
\def\rmD{{\rm D}}
\def\rme{{\rm e}}
\def\rmO{{\rm O}}
\def\Or{{\rm O}}
\def\rmR{{\rm R}}

% Real and integer numbers

\def\rz{\blackboardrrm}
\def\gz{\blackboardzrm}
\def\Im{{\rm Im}\,}
\def\Re{{\rm Re}\,}

% Expectation values

\newcommand{\MV}[1]{\langle#1\rangle}
\newcommand{\mvl}[1]{\left\langle#1\right\rangle}
\newcommand{\ketbra}[3]{\langle#1|#2|#3\rangle}

% Special relations and symbols

\def\defeq{\mathrel{\mathop=^{\rm def}}}
\def\proof{\noindent{\sl Proof:}\kern0.6em}
\def\endproof{\hskip0.6em plus0.1em minus0.1em
\setbox0=\null\ht0=5.4pt\dp0=1pt\wd0=5.3pt
\vbox{\hrule height0.8pt
\hbox{\vrule width0.8pt\box0\vrule width0.8pt}
\hrule height0.8pt}}
\def\frac#1#2{\hbox{$#1\over#2$}}
\def\dual{\mathstrut^*\kern-0.1em}
% in amsmath already \def\mod{\;\hbox{\rm mod}\;}
\def\ring{\mathaccent"7017}
\def\lvec#1{\setbox0=\hbox{$#1$}
    \setbox1=\hbox{$\scriptstyle\leftarrow$}
    #1\kern-\wd0\smash{
    \raise\ht0\hbox{$\raise1pt\hbox{$\scriptstyle\leftarrow$}$}}
    \kern-\wd1\kern\wd0}
\def\rvec#1{\setbox0=\hbox{$#1$}
    \setbox1=\hbox{$\scriptstyle\rightarrow$}
    #1\kern-\wd0\smash{
    \raise\ht0\hbox{$\raise1pt\hbox{$\scriptstyle\rightarrow$}$}}
    \kern-\wd1\kern\wd0}
\newcommand{\grtsim}{\raisebox{-.6ex}{$\stackrel{\textstyle{>}}{\sim}$}}
\newcommand{\simas}[1]{\raisebox{-.1ex}{
            $\stackrel{\small{#1}}{\sim}$}}
\newcommand{\toas}[1]{\raisebox{-.1ex}{
            $\stackrel{\small{#1}}{\longrightarrow}$}}
\newcommand{\ggas}[1]{\raisebox{-.1ex}{
             $\stackrel{\small{#1}}{\gg}$}}
\newcommand{\permille}{\textperthousand}
% Lattice derivatives

\def\nab#1{{\nabla_{#1}}}
\def\nabstar#1{\nabla\kern-0.5pt\smash{\raise 4.5pt\hbox{$\ast$}}
               \kern-4.5pt_{#1}}
\def\drv#1{{\partial_{#1}}}
\def\drvstar#1{\partial\kern-0.5pt\smash{\raise 4.5pt\hbox{$\ast$}}
               \kern-5.0pt_{#1}}
\newcommand{\drvsym}[1]{\widetilde{\drv{#1}}}
\newcommand{\dzero}{\widetilde{\drv{0}}}

% Lattice momenta

\def\momp#1#2{
    \setbox0=\hbox{${#1}$}\setbox1=\hbox{${#1}_{#2}$}
    {#1}_{#2}\kern-\wd1\kern\wd0
    \smash{\raise4.5pt\hbox{$\scriptscriptstyle +$}}}
\def\momm#1#2{
    \setbox0=\hbox{${#1}$}\setbox1=\hbox{${#1}_{#2}$}
    {#1}_{#2}\kern-\wd1\kern\wd0
    \smash{\raise4.5pt\hbox{$\scriptscriptstyle -$}}}
\def\mompm#1#2{
    \setbox0=\hbox{${#1}$}\setbox1=\hbox{${#1}_{#2}$}
    {#1}_{#2}\kern-\wd1\kern\wd0
    \smash{\raise4.5pt\hbox{$\scriptscriptstyle \pm$}}}
\def\smomp#1#2{
    \setbox0=\hbox{${#1}$}\setbox1=\hbox{${#1}_{#2}$}
    {#1}_{#2}\kern-\wd1\kern\wd0
    \smash{\raise3pt\hbox{$\scriptscriptstyle +$}}}
\def\smomm#1#2{
    \setbox0=\hbox{${#1}$}\setbox1=\hbox{${#1}_{#2}$}
    {#1}_{#2}\kern-\wd1\kern\wd0
    \smash{\raise3pt\hbox{$\scriptscriptstyle -$}}}
\def\smompm#1#2{
    \setbox0=\hbox{${#1}$}\setbox1=\hbox{${#1}_{#2}$}
    {#1}_{#2}\kern-\wd1\kern\wd0
    \smash{\raise3pt\hbox{$\scriptscriptstyle \pm$}}}

% Units

\def\MeV{{\rm MeV}}
\def\GeV{{\rm GeV}}
\def\TeV{{\rm TeV}}
\def\fm{{\rm fm}}

\def\fpi{F_{\pi}}

% Constants

\def\euler{\gamma_{\rm E}}
\newcommand{\hlf}{{\textstyle{\frac{1}{2}}}}

% Fields

\def\psibar{\overline{\psi}}
\def\psiclass{\psi_{\rm cl}}
\def\psibarclass{\psibar_{\rm cl}}
\def\psitilde{\widetilde{\psi}}
\def\rhoprime{\rho\kern1pt'}
\def\rhobar{\bar{\rho}}
\def\rhobarprime{\rhobar\kern1pt'}
\def\rhobartilde{\kern2pt\tilde{\kern-2pt\rhobar}}
\def\rhobartildeprime{\kern2pt\tilde{\kern-2pt\rhobar}\kern1pt'}
\def\etabar{\bar{\eta}}
\def\chibar{\overline{\chi}}
\def\phibar{\overline{\phi}}
\def\zetabar{\overline{\zeta}}
\def\zzetaprime{\zeta\kern1pt'}
\def\zetabarprime{\zetabar\kern1pt'}
\def\zetar{\zeta_{\raise-1pt\hbox{\rm R}}}
\def\zetabarr{\zetabar_{\raise-1pt\hbox{\rm R}}}
\def\phieff{\phi_{\rm eff}}
\def\phiimpr{\phi_{\kern0.5pt{\rm I}}}
\def\phir{\phi_{{\rm R}}}
\def\psibars{\psibar_s}
\def\psid{\psi_d}
\def\ar{A_{{\rm R}}}
\def\vr{V_{{\rm R}}}
\def\pr{P_{{\rm R}}}
\def\sr{S_{{\rm R}}}
\def\aimpr{A_{{\rm I}}}
\def\pimpr{P_{{\rm I}}}
\def\vimpr{V_{{\rm I}}}

% Dirac matrices

\def\dirac#1{\gamma_{#1}}
\def\diracstar#1#2{
    \setbox0=\hbox{$\gamma$}\setbox1=\hbox{$\gamma_{#1}$}
    \gamma_{#1}\kern-\wd1\kern\wd0
    \smash{\raise4.5pt\hbox{$\scriptstyle#2$}}}

% chiral variations

\def\da{\delta_{{\rm A}}}
\def\dv{\delta_{{\rm V}}}
\def\delsig{\delta_{{\sigma}}}

% Improvement coefficients

\def\csw{c_{\rm sw}}
\def\ca{c_{\rm A}}
\def\cv{c_{\rm V}}
\def\cs{c_{\rm s}}
\def\ct{c_{\rm t}}
\def\cttil{\widetilde{c}_{\rm t}}

\def\ba{b_{\rm A}}
\def\bv{b_{\rm V}}
\def\bp{b_{\rm P}}
\def\bs{b_{\rm S}}
\def\bm{b_{\rm m}}
\def\bg{b_{\rm g}}
\def\bzeta{b_{\zeta}}

% Correlation functions
 
\def\fa{f_{\rm A}}
\def\fv{f_{\rm V}}
\def\kv{k_{\rm V}}
\def\fp{f_{\rm P}}
\def\fx{f_{X}}
\def\faimpr{f_{\rm A}^{\rm I}}
\def\kvimpr{k_{\rm V}^{\rm I}}
\def\fvr{f_{\rm V}^{\rm R}}
\def\fda{f_{\delta{\rm A}}}

\def\etax{\eta_{X}}
\def\etaa{\eta_{\rm A}}
\def\etap{\eta_{\rm P}}

% Gauge group, parameters

\def\nf{N_{\rm f}}
\def\Nf{N_{\rm f}}

\def\SUtwo{{\rm SU(2)}}
\def\SUthree{{\rm SU(3)}}
\def\SUn{{\rm SU}(N)}
\def\tr{\,\hbox{tr}\,}
\def\Ad{{\rm Ad}\,}
\def\CF{C_{\rm F}}
\def\cf{C_{\rm F}}

% Lagrangian

\newcommand{\lag}[1]{{\mathcal{L}}_{\rm {#1}}}

% Action

\def\Sg{S_{\rm G}}
\def\Sf{S_{\rm F}}
\def\Seff{S_{\rm eff}}
\def\Simpr{S_{\rm impr}}
\def\Simprf{S_{\rm F\!,impr}}
\def\Zf{{\cal Z}_{\rm F}}
\def\op#1{{\mathcal O}_{\rm #1}}
\def\opi#1#2{{\cal L}_{#1}^{(#2)}}
\def\opa#1#2{{\mathcal A}_{#1}^{(#2)}}
\def\opprime#1{\setbox0=\hbox{${\cal O}$}\setbox1=\hbox{${\cal O}_{\rm #1}$}
    {\cal O}_{\rm #1}\kern-\wd1\kern\wd0
    \smash{\raise4.5pt\hbox{\kern1pt$\scriptstyle\prime$}}\kern1pt}
\def\ophat#1{\widehat{\cal O}_{\rm #1}}
\def\opbar#1{\overline{\mathcal O}_{\rm #1}}
\def\ophatprime#1{\setbox0=\hbox{$\widehat{\cal O}$}
    \setbox1=\hbox{$\widehat{\cal O}_{\rm #1}$}
    \widehat{\cal O}_{\rm #1}\kern-\wd1\kern\wd0
    \smash{\raise4.5pt\hbox{\kern1pt$\scriptstyle\prime$}}\kern1pt}
\def\bop#1{{\cal L}_{\rm #1}}
\def\bopprime#1{\setbox0=\hbox{${\cal O}$}\setbox1=\hbox{${\cal O}_{\rm #1}$}
    {\cal L}_{\rm #1}\kern-\wd1\kern\wd0
    \smash{\raise4.5pt\hbox{\kern1pt$\scriptstyle\prime$}}\kern1pt}
\def\blag#1{{\cal B}_{#1}}
\def\blagprime#1{\setbox0=\hbox{${\cal B}$}\setbox1=\hbox{${\cal B}_{#1}$}
    {\cal B}_{#1}\kern-\wd1\kern\wd0
    \smash{\raise5.2pt\hbox{\kern1pt$\scriptstyle\prime$}}\kern1pt}

% Renormalization schemes

\def\alphaSF{\alpha_{\rm SF}}
\def\alphaTP{\alpha_{\rm TP}}
\def\alphaqq{\alpha_{\rm q\bar q}}
\def\alphaMSbar{\alpha_{\msbar}}
\def\bMS{b^{\msbar}}
\def\dMS{d^{\msbar}}
\def\dSF{d^{\rm SF}}

\def\gms{g_{\ms}}
\def\gbar{\bar{g}}
\def\gbsq{\bar{g}^2}
\def\gbarMS{\gbar_{\ms}}
\def\gbarMSbar{\gbar_{\msbar}}
\def\gbarSF{\gbar_{\rm SF}}
\def\gbarTP{\gbar_{\rm TP}}
\def\gbarqq{\gbar_{\rm qq}}
\def\gqq{\gbarqq}
\def\gr{g_{{{\rm R}}}}
\def\glat{g_{\lat}}
\def\gSF{g_{{{\rm SF}}}}
\def\gparisi{g_{{{\rm P}}}}

\def\mbare{m_0}
\def\mq{m_{\rm q}}
\def\mqi{m_{\mrm{q},i}}
\def\mqj{m_{\mrm{q},j}}
\def\mqtilde{\widetilde{m}_{\rm q}}
\def\mr{m_{{{\rm R}}}}
\def\mc{m_{\rm c}}
\def\mlat{m_{\lat}}
\def\mSF{m_{{\rm SF}}}
\def\m{m}
\def\mprime{m'}
\def\dm{\Delta m}
\def\r{r}
\def\s{s}
\def\rprime{r'}
\def\sprime{s'}
\def\M{M}
\def\Mprime{M'}
\def\dM{\Delta M}
\def\mbar{\kern1pt\overline{\kern-1pt m\kern-1pt}\kern1pt}
\def\avm{m_{\rm av}}

\def\za{Z_{\rm A}}
\def\zv{Z_{\rm V}}
\def\zp{Z_{\rm P}}
\def\zg{Z_{\rm g}}
\def\zm{Z_{\rm m}}
\def\zM{Z_{\rm M}}
\def\zgm{Z_{\rm g,m}}
\def\zphi{Z_{\phi}}
\def\zmSF{Z_{\rm m}^{\hbox{\rm SF}}}
\def\zmlat{Z_{\rm m}^{\rm lat}}
\def\zzeta{Z_{\zeta}}
\def\sigmap{\sigma_{\rm P}}
\def\sigmam{\sigma_{\rm m}}
\def\sigmamtilde{\tilde{\sigma}_{\rm m}}
\def\sigmamstat{\sigma_{\rm m}^{\rm stat}}
\def\sigmaspin{\sigma^\mrm{spin}}
\def\sigmakin{\sigma^\mrm{kin}}
\def\Sigmakin{\Sigma^\mrm{kin}}
\def\Sigmam{\Sigma_{\rm m}}

\def\xg{X_{\rm g}}
\def\xm{X_{\rm m}}

\def\gtilde{\tilde{g}_0}
\def\mtilde{\widetilde{m}_0}
\def\hop{\kappa}
\def\hopc{\kappa_{\rm c}}

\def\ms{{\rm MS}}
\def\msbar{{\rm \overline{MS\kern-0.05em}\kern0.05em}}
\def\lat{{\rm lat}}

% Blackbord bold types
% [02/01/91 202131045  MEMBER NAME  MACROS   (TEXT13)   M  TEX]
 
\def\blackboardrrm{\mathchoice
{\rm I\kern-0.21 em{R}}{\rm I\kern-0.21 em{R}}
{\rm I\kern-0.19 em{R}}{\rm I\kern-0.19 em{R}}}
 
\def\blackboardzrm{\mathchoice
{\rm Z\kern-0.32 em{Z}}{\rm Z\kern-0.32 em{Z}}
{\rm Z\kern-0.28 em{Z}}{\rm Z\kern-0.28 em{Z}}}
 
\def\blackboardh{\mathchoice
{\rm I\kern-0.14 em{H}}{\rm I\kern-0.14 em{H}}
{\rm I\kern-0.11 em{H}}{\rm I\kern-0.11 em{H}}}
 
\def\blackboardp{\mathchoice
{\rm I\kern-0.14 em{P}}{\rm I\kern-0.14 em{P}}
{\rm I\kern-0.11 em{P}}{\rm I\kern-0.11 em{P}}}
 
\def\blackboardt{\mathchoice
{\rm T\kern-0.52 em{T}}{\rm T\kern-0.52 em{T}}
{\rm T\kern-0.40 em{T}}{\rm T\kern-0.40 em{T}}}
  
\def\su{{\rm su}(N)}
\def\SU{{\rm SU}(N)}
\def\sutwo{{\rm su}(2)}
\def\suthree{{\rm su}(3)}

% Schroedinger functional, effective action, action
 
\def\schrodinger{{\cal Z}}
\def\effaction{\Gamma}
\def\chernsimons{S_{\rm CS}}
 
% Background field, quantum field
 
\def\bvalue{C}
\def\bfield{B}
\def\bfieldtensor{G}
\def\bfunc{b}
\def\bfieldparm{\eta}
\def\qfield{q}
\def\bvaluelat{W}
\def\bfieldlat{V}
\def\bfunclat{v}
 
% Fluctuation operator, Faddeev-Popov operator
 
\def\deltazero{\Delta_0}
\def\deltazerohat{\hat{\Delta}_0}
\def\deltaone{\Delta_1}
\def\deltaonehat{\hat{\Delta}_1}
\def\deltaoneprime{\Delta\kern-1.0pt
    \smash{\raise 4.5pt\hbox{$\scriptstyle\prime$}}
    \kern-1.5pt_{1}}
\def\FP{{\rm FP}}
\def\deltaSone{\Delta^{{\rm S}^1}}
 
% Seeley-DeWitt expansion, zeta functions
 
\def\seeleycoeff#1#2{\alpha_{#1}({#2})}
\def\zetafunc#1#2{\zeta(#1|#2)}
\def\zetaprime#1{\zeta'(0|#1)}
 
% Hamilton operator, Hilbert space, transfer matrix
 
\def\projector{\blackboardp}
\def\ham{\mathbb{H}}
\def\trans{\mathbb{T}}
\def\opX{{\,\mathbb{X}\,}}
\def\opA{{\,\mathbb{A}_0}}
\def\Tr{{\rm Tr}}
\def\diag{{\rm diag}}
 
\def\initial{|\rm i_{0}\rangle} 
\def\final{\langle \rm f_{0}|} 
\def\initialt{\langle \rm i_{0}|} 
\def\finalt{| \rm f_{0}\rangle} 
\def\initialpi{|\rm i_{\pi}\rangle} 
\def\finalpi{\langle \rm f_{\pi}|} 
\def\initialpi{|\rm i_{\pi}\rangle} 
\def\initialpit{\langle \rm i_{\pi}|} 
\def\pf{{\cal Z}}
\def\vecp{{\bf p}}
\def\vecs{{\bf s}}
\def\vecsigma{{\bf \sigma}}
\def\vecD{{\bf D}}
\def\vecB{{\bf B}}

% Plaquette field, covariant derivatives
 
\def\del#1{{\Delta_{#1}}}
\def\delstar#1{\Delta\kern-1.0pt\smash{\raise 4.5pt\hbox{$\ast$}}
               \kern-4.0pt_{#1}}
\def\nab#1{{\nabla_{#1}}}
\def\nabstar#1{\nabla\kern-0.5pt\smash{\raise 4.5pt\hbox{$\ast$}}
               \kern-4.5pt_{#1}}
\def\lnab#1{{\overleftarrow{\nabla}_{#1}}}
\def\lnabstar#1{\overleftarrow{\nabla}\kern-0.5pt\smash
             {\raise 4.5pt\hbox{$\ast$}}\kern-4.5pt_{#1}}
\def\cdev#1{D\kern-0.2pt\smash{\raise 4.2pt
            \hbox{$\scriptstyle\phantom{\ast}$}}
            \kern-4.8pt_{#1}}
\def\cdevstar#1{D\kern-0.2pt\smash{\raise 4.2pt
                \hbox{$\scriptstyle\ast$}}
                \kern-4.8pt_{#1}}
\def\plaq{P}
\def\dplaq{d^*\!P}
 
% Abelian background field
 
\def\angle{\phi^{\vphantom{\prime}}}
\def\angleprime{\phi'}
\def\anglevector{\Phi^{\vphantom{\prime}}}
\def\anglevectorprime{\Phi'}
\def\temptrans{\Theta}
\def\setS{{\cal S}}
\def\perm{\sigma}
 
% Group of gauge transformations, field space etc.
 
\def\gaugegroup{{\cal G}}
\def\largegaugegroup{\hat{\gaugegroup}}
\def\cartangroup{{\rm C}_N}
\def\gaugealgebra{{\cal L}}
\def\fluctuationspace{{\cal H}}
\def\ga#1{\gaugealgebra_{#1}}
\def\fs#1{\fluctuationspace_{#1}}
\def\gap#1{\ga{#1}({\bf p})}
\def\fsp#1{\fs{#1}({\bf p})}
\def\gaugefunction{F}

% Calculation of determinants
 
\def\amat{{\cal A}}
\def\bmat{{\cal B}}
\def\cmat{{\cal C}}
\def\solmat{M}
\def\phat{\hat{p}}
\def\cgh{\cos{\gamma\over2}}
\def\funcspace{{\cal F}}
\def\dim{d_{\funcspace}}
 
\newcommand{\Dsl}{D \kern-.65em/}

\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bi}{\begin{itemize}}
\newcommand{\ei}{\end{itemize}}
\newcommand{\bes}{\begin{eqnarray}}
\newcommand{\ees}{\end{eqnarray}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\newcommand{\bean}{\begin{eqnarray*}}
\newcommand{\eean}{\end{eqnarray*}}

\newcommand{\eq}[1]{eq.~(\ref{#1})}
\newcommand{\Eq}[1]{Eq.~(\ref{#1})}
\newcommand{\eqs}[1]{eqs.~(\ref{#1})}
\newcommand{\Eqs}[1]{Eqs.~(\ref{#1})}
\newcommand{\fig}[1]{Fig.~\ref{#1}}
\newcommand{\Fig}[1]{Figure~\ref{#1}}
\newcommand{\sect}[1]{Sect.~\ref{#1}}
\newcommand{\Sect}[1]{Section~\ref{#1}}
\newcommand{\app}[1]{App.~\ref{#1}}
\newcommand{\App}[1]{Appendix~\ref{#1}}
\newcommand{\APP}[1]{\ref{#1}}
\newcommand{\tab}[1]{Table~\ref{#1}}
\newcommand{\Tab}[1]{Table~\ref{#1}}
\newcommand{\Ref}[1]{Ref.~\cite{#1}}
\newcommand{\Refs}[1]{Refs.~\cite{#1}}
\newcommand{\myref}[1]{\cite{#1}}
\newcommand{\subsubsubsection}[1]{\vspace{2mm} \noindent {\it #1}\vspace{2mm}}

\newcommand{\MSbar}{\overline{\rm MS}}
\newcommand{\MS}{\MSbar}
\newcommand{\alphaMS}{\alpha_{\MS}}
\newcommand{\Oa}{{\rm O}(a)}
\newcommand{\Oasq}{{\rm O}(a^2)}
\newcommand{\Lmax}{L_{\rm max}}
\newcommand{\lmax}{L_{\rm max}}

% One-loop coefficients etc.

\newcommand{\cbar}{c_{1,1}}
\newcommand{\vbar}{\bar v}
\newcommand{\pbar}{p_{1,1}}
\newcommand{\bbar}{b_{0,1}}
\newcommand{\ctbar}{c_t^{(1,1)}}
\newcommand{\bzero}{b_0(n_f,0)}
\newcommand{\deltabar}{\delta_{1,1}}
\newcommand{\eps}{\epsilon}

\newcommand{\proj}{{\cal P}}
\newcommand{\smear}{{\cal S}}
\newcommand{\Vbar}{\overline{V}}
\newcommand{\Ubar}{\overline{U}}

\newcommand{\tmin}{t_{\rm min}}
\newcommand{\tmax}{t_{\rm max}}
\newcommand{\vecr}{{\it \bf r}}
\newcommand{\vecd}{{\it \bf d}}
\newcommand{\vecu}{{\it \bf u}}
\newcommand{\vecv}{{\it \bf v}}
\newcommand{\vecx}{{\it \bf x}}
\newcommand{\vecy}{{\it \bf y}}
\newcommand{\vecz}{{\it \bf z}}
\newcommand{\rnod}{r_{0}}
\newcommand{\rc}{r_{\rm c}}
\newcommand{\mbarscale}{1.436\,r_{0}}
\newcommand{\rI}{r_{\rm I}}

\newcommand{\sigplatt}{\Sigma_{\rm P}}
\newcommand{\Sigmap}{\Sigma_{\rm P}}
\newcommand{\sigpcont}{\sigma_{\rm P}}
\newcommand{\Fpi}{F_{\pi}}
\newcommand{\Gpi}{G_{\pi}}
\newcommand{\Fp}{F_{\rm PS}}
\newcommand{\Gp}{G_{\rm PS}}
\def\mp{m_{\rm PS}}
\newcommand{\Fk}{F_{\rm K}}
\newcommand{\Fb}{F_{\rm B}}
\newcommand{\Fbs}{F_{\rm B_{\rm s}}}
\newcommand{\Bb}{B_{\rm B}}
\newcommand{\Gk}{G_{\rm K}}
\newcommand{\Bk}{B_{\rm K}}
\newcommand{\mk}{m_{\rm K}}
\newcommand{\mb}{m_{\rm B}}
\newcommand{\md}{m_{\rm D}}
\newcommand{\mbstar}{m_{\rm B^*}}
\newcommand{\mbs}{m_{\rm B_{\rm s}}}
\newcommand{\mbsstar}{m_{\rm B_{\rm s}^*}}
\newcommand{\mkstar}{m_{\rm K^*}}
\newcommand{\mphi}{m_{\Phi}}
\newcommand{\mv}{m_{\rm V}}
\newcommand{\mpi}{m_{\pi}}
\newcommand{\mrho}{m_{\rho}}
\newcommand{\RWI}{R}
\newcommand{\Rwi}{R}
\newcommand{\mav}{m_{\rm av}}
\newcommand{\strange}{{\rm s}}
\newcommand{\up}{{\rm u}}
\newcommand{\down}{{\rm d}}
\newcommand{\Mlight}{\hat{M}}
\newcommand{\mlight}{\hat{m}}
\newcommand{\Mref}{M_{\rm ref}}
\newcommand{\mref}{m_{\rm ref}}
\newcommand{\mstrange}{m_{\strange}}
\newcommand{\Mstrange}{M_{\strange}}
\newcommand{\beauty}{{\rm b}}
\newcommand{\mbeauty}{m_{\beauty}}
\newcommand{\Mbeauty}{M_{\beauty}}
\newcommand{\mpoleb}{m_{\rm Q,\beauty}}
\newcommand{\charm}{{\rm c}}
\newcommand{\mcharm}{m_{\charm}}
\newcommand{\Mcharm}{M_{\charm}}
\newcommand{\mup}{m_{\up}}
\newcommand{\mdown}{m_{\down}}

\newcommand{\calM}{{\cal M}}
\newcommand{\xr}{X_{\rm R}}
\newcommand{\yr}{Y_{\rm R}}
\newcommand{\rrmm} {1.5736}
\newcommand{\meff}{\Gamma}
\newcommand{\meffp}{\Gamma^{\rm P}}
\newcommand{\meffv}{\Gamma^{\rm V}}
\newcommand{\meffav}{\Gamma^{\rm av}}
\newcommand{\meffdiff}{\Delta_\Gamma}
\newcommand{\meffrel}{\Gamma^{\rm rel}}
\newcommand{\meffstat}{\Gamma^{\rm stat}}
\newcommand{\meffone}{\Gamma_1}
\newcommand{\meffonestat}{\Gamma^{\rm stat}_1}
\newcommand{\meffonekin}{\Gamma^{\rm kin}_1}
\newcommand{\meffkin}{\Gamma^{\rm kin}}
\newcommand{\meffdakin}{\Gamma^{\rm kin}_{\delta A}}
\newcommand{\meffspin}{\Gamma^{\rm spin}}
\newcommand{\meffdastat}{\Gamma_{\delta A}}
\newcommand{\meffhqet}{\Gamma^{\rm HQET}}
\newcommand{\Estat}{E_{\rm stat}}
\newcommand{\Ekin}{E_{\rm kin}}
\newcommand{\Espin}{E_{\rm spin}}

\newcommand{\hypone}{{\rm HYP1}}
\newcommand{\hyptwo}{{\rm HYP2}}


\newcommand{\Gr}{G_{\rm R}}
\newcommand{\Grrel}{G_{\rm R}^{\rm rel}}
\newcommand{\VI}{V_{\rm I}}
\newcommand{\VIsub}{write explicitly}
\newcommand{\rIv}{d_{{\rm I}}}
%\newcommand{\note}[1]{{\bf NOTE: #1}}
\newcommand{\todo}[1]{{

    \bf To Be Done: #1

    }}
\newcommand{\lecture}[2]{{
    \vspace{3mm}

     \vfill\eject \noindent \large \bf Lecture #1. \\[1ex] #2
    \vspace{3mm}
    \setcounter{section}{0}
    %\renewcommand\thesection      {#1. \arabic{section}}

    }}
\newcommand{\blank}{\multicolumn{2}{c}{\mbox{}}}


% Abbreviations for text
 
\newcommand{\SF}{{Schr\"odinger functional }}
\newcommand{\ps}{pseudoscalar}
\newcommand{\stat}{{static effective field theory }}

\def\ALPHA{\hbox{\epsfxsize=1.2 true cm
                           \epsfbox{plots/alpha_color_60.eps}}
                   }
\def\SALPHA{\hbox{\epsfxsize=0.8 true cm
                           \epsfbox{plots/alpha_color_60.eps}}
                   }



































































