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many exclusive @xmath0-decay amplitudes in the framework of scet contain @xmath0-meson distribution amplitudes  @xcite . the amplitude of the decay @xmath1 at large photon energies is given , up to power corrections , by a convolution of a hard part ( perturbatively calculable ) and the @xmath0-meson distribution amplitude  @xcite . amplitudes of some decays , e.g. , @xmath2 at large pion energies , contain both factorizable and non - factorizable contributions  @xcite . factorizable parts of decay amplitudes contain light - cone distribution amplitudes of the initial @xmath0-meson and final hadron(s ) . they describe large - distance ( soft ) structure of these hadrons , and can not be calculated in perturbation theory . the theory of hadronic distribution amplitudes in qcd is reviewed in  @xcite . quark  antiquark distribution amplitudes of @xmath0-meson in hqet were introduced and investigated in  @xcite . they are defined as fourier transforms of matrix elements of some gauge - invariant bilocal operators between @xmath0-meson and vacuum . renormalization of these operators was calculated in one - loop approximation . however , an unusual term @xmath3 in the one - loop renormalization constants was erroneously omitted in  @xcite . the correct evolution equation for the leading - twist distribution amplitude was derived in  @xcite at one loop . the evolution kernel contains , in addition to terms obtained earlier  @xcite , an unusual term @xmath4 . the method of solution of the evolution equation is also discussed in  @xcite in detail . quark  antiquark  gluon distribution amplitudes of @xmath0-meson and their relations to quark  antiquark ones ( based on equations of motion ) are discussed in  @xcite . sum rules for the quark antiquark distribution amplitudes were obtained in  @xcite . a simple model of these distribution amplitudes at a low normalization scale ( of order of hadronic scale ) was proposed . radiative corrections to the perturbative term and the quark - condensate term were later calculated  @xcite .    in these lectures , we first briefly discuss what is hqet ( sect . [ s : hqet ] ) . a much more detailed presentation can be found in the textbooks  @xcite . after a short discussion of @xmath5 ( sect  [ s : fb ] ) , quark antiquark distribution amplitudes are introduced ( sect . [ s : qq ] ) . quark  antiquark gluon distribution amplitudes and their relations to two - particle ones are discussed in sect . [ s : qqg ] . [ s : evol ] is the central ( and longest ) one . here renormalization of light - cone bilocal quark operators in hqet is considered in one - loop approximation . a detailed derivation of the evolution kernel is presented , based on the methods of  @xcite . finally , sum rules for the distribution amplitudes are briefly discussed in sect . [ s : sr ] . let s consider a heavy antiquark with mass @xmath6 and momentum @xmath7 in qcd . here @xmath8 is a fixed 4-velocity ( @xmath9 ) , and the residual momentum @xmath10 , as well as momenta of all light quarks and gluons , are supposed to be small ( compared to @xmath6 ) . the propagator of this heavy antiquark can be written as @xmath11 the leading term here is the hqet propagator . this can be graphically presented as @xmath12{$mv+k$ } } \end{picture } } = \raisebox{-5.5mm}{\begin{picture}(22,8 ) \put(11,5){\makebox(0,0){\includegraphics{fr2.eps } } } \put(11,0){\makebox(0,0)[b]{$mv+k$ } } \end{picture } } + \mathcal{o}\left(\frac{1}{m}\right)\ , . \label{graphprop}\ ] ] in a vertex sandwiched between such propagators , we may substitute @xmath13 projector can also be inserted near external legs , so that all qcd vertices can be replaced by the hqet ones , @xmath14{$\mu$ } } \put(8,9){\makebox(0,0)[t]{$a$ } } \end{picture } } = i g_0 t^a ( -v^\mu)\ , , \label{vertt}\ ] ] up to @xmath15 corrections . these feynman rules can be obtained from the lagrangian @xmath16 the antiquark field is our main field here , it satisfies @xmath17 @xmath18 is the conjugate field for @xmath19 . here the covariant derivatives are @xmath20 qcd tree diagrams are reproduced by hqet up to @xmath21 corrections ( they can also be reproduced , if we add the appropriate @xmath22 terms into the lagrangian ) .    the heavy quark chromomagnetic moment is @xmath23 by dimensionality . at the leading order in @xmath22 , the heavy - quark spin does not interact with the gluon field . therefore , it may be rotated at will , without changing the physics ( heavy - quark spin symmetry ) . it may even be switched off ( superflavour symmetry ) . we shall work with the spinless heavy antiquark , @xmath24 during most of these lectures , because this greatly simplifies reasoning and calculations . here again @xmath25 is the main ( scalar ) field , and @xmath26 is its conjugate . so , tree qcd diagrams , expanded in @xmath27 to some order , are reproduced by the corresponding hqet diagrams . but what about loops ? here things are not so simple  @xcite . let s consider , for example , the heavy  light two - point diagram ( fig . [ f : thresh ] ) with @xmath28 , where the residual momentum @xmath10 is small . by choosing @xmath8 along @xmath29 we can always ensure @xmath30 . let s consider the integral @xmath31 ^ 2 ( -l^2)}\ , , \label{thresholdint}\ ] ] which has neither ultraviolet ( uv ) nor infrared ( ir ) divergences . there are two regions of the loop momentum @xmath32 in this integral :    * hard region @xmath33 . expanding the integrand in @xmath34 , @xmath32 , we have @xmath35 ^ 2 ( -l^2 ) } + 4 \frac{(m + l \cdot v ) \omega}{\left[m^2-(mv+l)^2\right]^3 ( -l^2 ) } + \cdots \label{hardintegrand}\ ] ] * soft region @xmath36 . expanding the integrand in @xmath10 , @xmath37 , we have @xmath38 ^ 2 ( -l^2 ) } + 2 \frac{(k+l)^2}{\left[-2 m ( k+l ) \cdot v\right]^3 ( -l^2 ) } + \cdots \label{softintegrand}\ ] ]    ( 40,11.5 ) ( 20,4)(0,0 ) ( 20,-4)(0,0)[b]@xmath39 ( 20,11.5)(0,0)[t]@xmath32 ( 7.5,-4)(0,0)[b]@xmath29 ( 32.5,-4)(0,0)[b]@xmath29    the contribution of the hard region is @xmath40 + \cdots \right]\,,\\ & \frac{\mu^{2\varepsilon } i_h}{\gamma(1+\varepsilon ) } = - \frac{1}{2\varepsilon } + \log\frac{m}{\mu } + \left ( \frac{1}{\varepsilon } - 2 \log\frac{m}{\mu } - 1 \right ) \frac{\omega}{m } + \cdots \end{split } \label{hardcontr}\ ] ] where the on - shell massive two - point integrals ( fig .  [ f : os1 ] ) @xmath41 are @xmath42 this contribution is ir divergent . ( 32,11.5 ) ( 16,4)(0,0 ) ( 16,-4)(0,0)[b]@xmath43 ( 16,11.5)(0,0)[t]@xmath32 ( 3.5,-4)(0,0)[b]@xmath44 ( 28.5,-4)(0,0)[b]@xmath44    the contribution of the soft region is @xmath45 + \cdots \right]\,,\\ & \frac{\mu^{2\varepsilon } i_s}{\gamma(1+\varepsilon ) } = \frac{1}{2\varepsilon } - \log\frac{-2\omega}{\mu } - \left ( \frac{1}{\varepsilon } - 2 \log\frac{-2\omega}{\mu } - \frac{1}{2 } \right ) \frac{\omega}{m } + \cdots \end{split } \label{softcontr}\ ] ] where the hqet two - point integrals ( fig . [ f : hqet1 ] , @xmath46 ) @xmath47 are @xmath48 this contribution is uv divergent . ( 32,11.5 ) ( 16,4)(0,0 ) ( 16,-4)(0,0)[b]@xmath49 ( 16,11.5)(0,0)[t]@xmath32 ( 3.5,-4)(0,0)[b]@xmath10 ( 28.5,-4)(0,0)[b]@xmath10    the complete result is finite : @xmath50 this expansion can be easily continued if desired . what about higher loops ? let s consider , for example , the two - loop two - point diagram with a small external residual momentum . there are several regions of the loop momenta in this diagram ( fig . [ f : regions ] ) . in each of them , some momenta are hard ( @xmath51 ) , some are soft ( @xmath52 ) . as we have already seen , a heavy - quark line with a soft residual momentum becomes an hqet line . soft massless lines are shown by dashed lines in the figure . ( 152,18 ) ( 16,9)(0,0 ) ( 56,9)(0,0 ) ( 96,9)(0,0 ) ( 136,9)(0,0 )    for example , in the second diagram in fig . [ f : regions ] , the left loop is hard . it contains a single scale @xmath6 . all external momenta of this loop are soft , including those of the lines belonging to the other loop . we can expand the integrand in these small momenta ; after taking the loop integral , we obtain a polynomial in these momenta . now from the point of view of the soft loop ( large distances ) this hard loop is just a local vertex . this loop contains a single scale @xmath53 , and we can calculate it , obtaining a non - analytical function of the external residual energy @xmath53 .    in a general multiloop diagram , hard lines must always form loops , so that momentum conservation can hold after neglecting all soft momenta . there can be several disconnected hard parts ; each of them must contain at least one heavy line ( a subdiagram consisting of only light lines and having soft external momenta has no reason to be hard ) . from the point of view of the soft part , hard parts are just local vertices . there may appear a soft subdiagram connected to the rest of the diagram only at such a vertex . such a subdiagram is scaleless and hence vanish . for example , we could consider one more region in fig . [ f : regions ] : when the only soft line is the middle light line , all the rest are hard . but this soft line forms a loop containing one local vertex ( the integrated hard loop ) , and hence it vanishes .    in the usual hqet formalism , lagrangian contains local operators multiplied by matching coefficients . qcd operators are also expanded in hqet operators with matching coefficients . these matching coefficients are the only quantities in the theory which depend on the hard scale @xmath6 . diagrammatically , they come from hard loops in qcd diagrams . local operators produce vertices polynomial in their external momenta . they appear in hqet diagrams , which contain only the soft scale @xmath53 . these hqet diagrams are soft parts of qcd diagrams . during these lectures , we shall mostly live in a world with a heavy antiquark having @xmath54 . physics in the real world is the same , up to @xmath22 corrections . we shall work in the @xmath8 rest frame . the ground - state @xmath55-wave @xmath56 meson has the quantum numbers @xmath57 . there are 2 @xmath58-wave excited mesons with @xmath59 and @xmath60 . the heavy light quark current @xmath61 has no definite parity ; the currents with parity @xmath62 are @xmath63 they have the quantum numbers of @xmath55-wave @xmath64 mesons and @xmath58-wave @xmath65 mesons . the ground - state meson @xmath66 has a dirac wave function @xmath67 which satisfies @xmath68 and is normalized by @xmath69 . the matrix element of @xmath70 from @xmath66 to vacuum is @xmath71 where the one - meson states are normalized by the non - relativistic condition @xmath72    the correlator of the heavy light currents ( fig . [ f : cor ] ) , @xmath73 has the structure @xmath74 therefore , the correlators of the currents with definite parity are @xmath75 the spectral density of , say , the correlator with @xmath76 is @xmath77 where @xmath78 is the residual energy of the ground - state meson , and the dots mean contribution of excited states . ( 56,24 ) ( 28,14)(0,0 ) ( 1,0)(0,0)[b]@xmath79 ( 55,0)(0,0)[b]@xmath80    now we shall return to the real world with @xmath59 heavy antiquark for a while , but still with @xmath81 . the @xmath55-wave ground - state meson turns into a degenerate doublet with @xmath82 , @xmath83 . the two @xmath58-wave mesons turn into two degenerate doublets : with @xmath54 , @xmath84 and with @xmath85 , @xmath86 . all heavy  light currents @xmath87 reduce , due to  ( [ lowercompon ] ) , to 4 ones with @xmath88 the first 2 currents , with @xmath89 anticommuting with @xmath90 , have the quantum numbers of the ground - state @xmath91 , @xmath83 mesons ; the second 2 currents , with @xmath89 commuting with @xmath90 , have the quantum numbers of the @xmath58-wave @xmath92 , @xmath84 mesons .    the correlators ( fig . [ f : cor ] ) are @xmath93 or @xmath94 where @xmath76 for @xmath89 anticommuting with @xmath90 and @xmath95 for commuting . for @xmath96 and @xmath97 , the correlators are @xmath98 and @xmath99 . their spectral densities are @xmath100 and @xmath101 , where @xmath102 and dots mean contribution of higher states . therefore , @xmath103    the usual definition of the decay constants is @xmath104 where the relativistic normalization @xmath105 of single - meson states is used ( this normalization becomes meaningless in the limit @xmath106 , and therefore can not be used in hqet ) . comparing  ( [ fb ] ) for @xmath0 or @xmath107 at rest with  ( [ fb ] ) , we arrive at @xmath108 up to @xmath22 corrections . let s consider the correlator  ( [ cor5 ] ) with @xmath76 . equating the ground - state contributions to the spectral densities of the left - hand side and the right - hand one , we obtain @xmath109 therefore , @xmath110 where the matrix @xmath111 is defined up to a phase factor . we can re - write this result for the relativistic normalization of the meson state , in covariant notation : @xmath112 where @xmath113 ( of course , one can re - define the phases of @xmath114 and hence of @xmath115 ) . after safely returning to the ideal world with a @xmath92 heavy antiquark , we want to invent an operator which probes more details of the structure of @xmath0-meson than the local current  ( [ j ] ) . to this end , we consider a bilocal gauge - invariant operator @xmath116 q(z)\ , , \label{ot0}\ ] ] where @xmath117 = p \exp \left [ - i \int_x^y a_\mu(z ) d z^\mu \right]\ , , \label{exy}\ ] ] and @xmath118 its matrix element from the ground - state meson to vacuum has 2 dirac structures : @xmath119 u\ , . \label{phit}\ ] ] because @xmath120 .    in what follows , we shall often use light - front components of vectors . let s introduce ( in the @xmath8 rest frame ) two light - like vectors @xmath121 light - front components of any vector @xmath122 are defined as @xmath123 we have @xmath124 in particular , @xmath125 we shall also use light - front components of @xmath126 : @xmath127    the definition  ( [ phit ] ) can be re - written as @xmath128 u\ , . \label{ot}\ ] ] if we introduce the operators @xmath129 then @xmath130 the @xmath0-meson distribution amplitudes are the fourier transforms of these functions : @xmath131 they are normalized by @xmath132 the function @xmath133 is the leading - twist distribution amplitude , and @xmath134  the subleading - twist one ( though there is no good definition of twist in hqet ) . we can formally introduce the operators @xmath135 then @xmath136 the distribution amplitudes describe the distribution in the light - front component @xmath137 of the light - quark momentum in @xmath0-meson .    the expansion of the operators  ( [ opmt ] ) in @xmath138 reads @xmath139 or for matrix elements @xmath140 we can also reconstruct @xmath141 from @xmath142 : @xmath143 or for matrix elements @xmath144 ( you can easily check this by substituting  ( [ on ] ) into  ( [ intdn ] ) ; integration in @xmath145 yields @xmath146 ) . the first moments @xmath147 can be found from the equations of motion . the equation of motion for the heavy antiquark is @xmath148 and therefore we obtain @xmath149 the vector part has the structure @xmath150 the equation of motion of the light quark is @xmath151 and using @xmath152 we obtain @xmath153 finally , we arrive at @xmath154    let s now consider the second moments . they involve two new ( nonperturbative ) hadronic parameters : @xmath155 where @xmath156\,,\quad \vec{h } = i \vec{d}\times\vec{d}\,,\quad \vec{\alpha } = \gamma^0\vec{\gamma}\,,\quad \vec{\sigma } = - \vec{\gamma } \gamma_5 \gamma^0\,.\ ] ] now we can calculate all matrix elements with 2 derivatives . from  ( [ d0 ] ) and  ( [ dvec ] ) , using the heavy - antiquark equation of motion  ( [ heavyeom ] ) , we immediately find @xmath157 using the definition  ( [ lambdaeh ] ) of @xmath158 , we immediately find @xmath159 the second spatial derivatives have the structure @xmath160 where the second coefficient follows from the definition  ( [ lambdaeh ] ) of @xmath161 . we find @xmath162 using the light - quark equation of motion  ( [ lighteom ] ) : @xmath163 u\ , . \label{dij}\ ] ] finally , we arrive at @xmath164    what about @xmath0-meson distribution amplitudes in the real world with a @xmath65 heavy antiquark ? it has 4 distribution amplitudes , as any pseudoscalar meson : @xmath165 \gamma_5 q(z)|b{>}_r = - i f_b m \tilde{\varphi}_p\,,\\ & { < } 0|{q^*_v}(0 ) [ 0,z ] \gamma^\mu \gamma_5 q(z)|b{>}_r = f_b \left [ i \tilde{\varphi}_{a1 } p^\mu - m \tilde{\varphi}_{a2 } z^\mu \right]\,,\\ & { < } 0|{q^*_v}(0 ) [ 0,z ] \sigma^{\mu\nu } \gamma_5 q(z)|b{>}_r = i f_b \tilde{\varphi}_t ( p^\mu z^\nu - p^\nu z^\mu)\ , . \end{split } \label{qcdps}\ ] ] similarly to  ( [ trf ] ) , we have @xmath166 \gamma q(z)|m{>}_r = f \operatorname{tr}\gamma \left[\tilde{\varphi}_+ + \frac{\tilde{\varphi}_--\tilde{\varphi}_+}{2 t } \rlap / z\right ] \mathcal{m}\ , . \label{trfphi}\ ] ] therefore these 4 qcd distribution amplitudes can be expressed via 2 hqet ones : @xmath167 the qcd distribution amplitudes are usually considered as functions of the longitudinal momentum fraction @xmath168 . it is usually assumed that @xmath169 at @xmath170 , and @xmath171 . therefore , we shall assume that @xmath172 at @xmath173 , and @xmath174 . the qcd distribution amplitudes are normalized as @xmath175 ( to derive the last formula , we used the @xmath138 expansion  ( [ omn ] ) and the first moments  ( [ mom1 ] ) ) . the vector meson @xmath107 is described by 6 distribution amplitudes in qcd . all 10 distribution amplitudes ( 4 for @xmath0 plus 6 for @xmath107 ) are expressed via 2 hqet distribution amplitudes @xmath176 . this is a consequence of the heavy - quark spin symmetry . we do nt present formulae for @xmath107 here ; they can be found in  @xcite . now we shall discuss relations of quark  antiquark distribution amplitudes and quark  antiquark  gluon ones following from the equations of motion . in order to apply them , we need to differentiate with respect to the coordinates of the light quark and the heavy antiquark separately . therefore , we go slightly off the light cone . the generalization of  ( [ phit ] ) to an arbitrary @xmath177 is @xmath178 u\ , , \label{phiz}\ ] ] where @xmath179 , and the fixed - point gauge @xmath180 is used to simplify notation .    in order to apply the light - quark equation of motion  ( [ lighteom ] ) , we apply @xmath181 to this definition . the differentiation of the right - hand side is straightforward . in the left - hand side we write @xmath182 the first two terms yield zero . in the fixed - point gauge @xmath183 where @xmath184 . the matrix element of the operator containing @xmath185 contains 4 structures : @xmath186 i g_{\nu\mu}(uz ) z^\nu [ uz , z ] q(z)|m { > } = { } \\ & - f \left [ ( v_\mu \rlap / z - t \gamma_\mu ) ( \tilde{\psi}_a - \tilde{\psi}_v ) + i \sigma_{\mu\nu } z^\nu \tilde{\psi}_v - z_\mu \tilde{\psi}_x + \frac{z_\mu}{t } \rlap / z \tilde{\psi}_y \right ] u\ , , \end{split } \label{psi}\ ] ] and is parametrized by 4 quark  antiquark gluon distribution amplitudes ( note that this matrix elements vanishes when multiplied by @xmath187 ) .    equating the coefficients of @xmath67 and @xmath188 , we obtain two consequences of the light - quark equation of motion : @xmath189    similarly , to use the heavy - antiquark equation of motion we apply @xmath190 the first two terms in the last operator yield zero . now we move the heavy antiquark ; the center of the fixed - point gauge must be constant , and we have to use the gauge @xmath191 : @xmath192 we arrive at two consequences of the heavy - antiquark equation of motion : @xmath193    the functions @xmath194 are some new ( non - leading ) quark  antiquark distribution amplitudes . we are not interested in them here . there are 2 equations involving only our familiar distribution amplitudes @xmath195 , namely , ( [ eom1 ] ) and a combination of  ( [ eom2 ] ) and  ( [ eom3 ] ) : @xmath196    the quark  antiquark  gluon distribution amplitudes in the momentum space are defined as @xmath197 performing fourier transform of the equations  ( [ eom5 ] ) , we obtain @xmath198 where @xmath199\,,\\ & j(\omega ) = - 2 \frac{d}{d\omega } \int_0^\omega d\rho \int_{\omega-\rho}^\infty \frac{d\xi}{\xi } \left[\psi_a(\rho,\xi ) + \psi_x(\rho,\xi)\right ] - 4 \int_0^\omega d\rho \int_{\omega-\rho}^\infty \frac{d\xi}{\xi } \frac{\partial}{\partial\xi } \psi_v(\rho,\xi)\,.\end{aligned}\ ] ] if we insist on having @xmath200 , then the condition @xmath201 = 0 \label{cond0}\ ] ] must be satisfied . it should follow from vanishing of @xmath202 at @xmath173 ( behaviour of the three - particle distribution amplitudes at the boundaries is discussed in  @xcite ) . if we new @xmath203 , we could solve the equations  ( [ eomm ] ) for @xmath176 . this is not very realistic , because we do nt know them . the solution is a sum of two terms : @xmath204 where @xmath205 is the solution of  ( [ eomm ] ) in the case if all quark  antiquark  gluon distribution amplitudes vanish ( it is called the wandzura  wilczek part of the solution ) , and @xmath206 is induced by the gluonic terms . the wandzura wilczek part is @xmath207 ( fig . [ f : ww ] ) ; it satisfies the normalization conditions  ( [ norm ] ) . note that @xmath208 is the maximum value of @xmath137 of the light quark if we assume that @xmath0-meson ( having the residual energy @xmath78 ) consists of the on - shell heavy antiquark ( always having zero residual energy ) and the on - shell light quark . the gluon - induced part is given by some explicit integrals of @xmath203 ; we do nt present these long expressions here , they can be found in  @xcite . ( 42,45 ) ( 21,24)(0,0 ) ( solid line ) and @xmath134 ( dashed line),title="fig : " ] ( 31,0)(0,0)[b]@xmath208 ( 41,0)(0,0)[b]@xmath53    the moments  ( [ omn ] ) also consist of two contributions . the wandzura wilczek parts are @xmath209 the gluon - induced part does not contribute to the zeroth moments ( normalization ) and the first ones ; its contribution to the second moments  ( [ mom2 ] ) is expressed via the normalizations of the quark gluon distribution amplitudes : @xmath210 the contributions to the higher moments are expressed via the moments of @xmath203 ; the explicit formulae can be found in  @xcite . until now , we neglected renormalization of the considered operators . in this section we shall discuss renormalization of the leading - twist @xmath0-meson distribution amplitude . let s summarize what we know about the bare operators of interest . there are 3 families of such operators , or 3 `` representations '' ( @xmath138 , @xmath53 , @xmath211 ) : @xmath212 q(z)\,,\\ & o_+(\omega ) = { q^*_v}(0 ) \gamma_+ \delta(i d_+ - \omega ) q(0)\,,\\ & o_+^{(n ) } = { q^*_v}(0 ) \gamma_+ ( i d_+)^n q(0)\ , . \end{split } \label{bare}\ ] ] they can be converted into each other : @xmath213 as we shall see , not all of these relations survive renormalization . we shall calculate matrix elements of these bare operators , therefore , we need the feynman rules for them . if we retain @xmath214 in all brackets @xmath215 in @xmath216 , we obtain the quark antiquark vertex with @xmath217 shown in fig . [ f : oqq ] . when transformed to the @xmath53-representation  ( [ barerel ] ) , this gives @xmath218 , as expected from the form  ( [ bare ] ) of @xmath219 . ( 100,23 ) ( 16,10.5)(0,0 ) ( 16,23)(0,0)[t]@xmath211 ( 7,10)(0,0)@xmath29 ( 32,10)(0,0)[l]@xmath220 ( 66,10.5)(0,0 ) ( 66,23)(0,0)[t]@xmath53 ( 57,10)(0,0)@xmath29 ( 82,10)(0,0)[l]@xmath221    now let s retain a single @xmath222 in @xmath215 . after some combinatorics , this gives @xmath223 ( i \partial_+)^{n - m}\ , . \label{vqqg0}\ ] ] this gives the quark gluon vertex of @xmath216 with @xmath224 shown in fig . [ f : oqqg ] . transforming it to the @xmath53-representation  ( [ barerel ] ) gives two @xmath225-functions ( fig . [ f : oqqg ] ) . of course , the integral in @xmath226 of this vertex vanishes , because @xmath227 does not interact with gluons . ( 100,25 ) ( 16,12.5)(0,0 ) ( 16,25)(0,0)[t]@xmath211 ( 16,0)(0,0)[b]@xmath228 ( 7,12)(0,0)@xmath29 ( 21,7)(0,0)@xmath10 ( 32,12)(0,0)[l]@xmath229     +    ( 100,25 ) ( 16,12.5)(0,0 ) ( 16,25)(0,0)[t]@xmath53 ( 16,0)(0,0)[b]@xmath228 ( 7,12)(0,0)@xmath29 ( 21,7)(0,0)@xmath10 ( 32,12)(0,0)[l]@xmath230    the bare operators @xmath219 can be expressed via the renormalized operators @xmath231 ( @xmath232 is the @xmath233 renormalization scale ) , or vice versa : @xmath234 the renormalization `` matrices '' @xmath235 and @xmath236 are inverse to each other : @xmath237    the renormalized operators @xmath231 obey the renormalization - group equation @xmath238 where the anomalous dimension `` matrix '' is @xmath239 this equation can be derived in two ways : either we differentiate  ( [ ren1 ] ) in @xmath240 and obtain 0 ( because the bare operator is @xmath232-independent ) , or we differentiate  ( [ ren2 ] ) .    at one loop @xmath241 where @xmath242 from  ( [ anomdim ] ) we obtain @xmath243    the matrix element of @xmath219 between a state with the light quark with momentum @xmath29 and the heavy antiquark with momentum @xmath244 ( which are off - shell ) and vacuum is @xmath245\\ { } = { } & { < } 0|o_+(\omega;\mu)|q(p),{q^*_v}(p ' ) { > } + a_s \int z_+^{(1)}(\omega,\omega';\mu ) \delta(p_+-\omega ' ) \gamma_+ d\omega'\ , , \end{split } \label{m}\ ] ] where @xmath246 are the contributions of the one - loop diagrams shown in fig . [ f : m ] , the matrix element of the renormalized operator is finite at @xmath247 , and @xmath248 contains only negative powers of @xmath249 ( in the term with @xmath250 , we may substitute @xmath251 instead of @xmath252 ) . this allows us to find the renormalization `` matrix '' with one - loop accuracy . ( 106,42 ) ( 16,31.5)(0,0 ) between a quark antiquark state and vacuum , title="fig : " ] ( 16,10.5)(0,0 ) between a quark antiquark state and vacuum , title="fig : " ] ( 53,10.5)(0,0 ) between a quark antiquark state and vacuum , title="fig : " ] ( 90,10.5)(0,0 ) between a quark antiquark state and vacuum , title="fig : " ]    let s calculate the first diagram ( fig .  [ f:1 ] ) for @xmath253 it is @xmath254 \left[-k^2-i0\right]}\ , . \label{m1a}\ ] ] the numerator can be simplified as @xmath255    ( 36,25 ) ( 18,13.5)(0,0 ) ( 18,25)(0,0)[t]@xmath53 ( 2,7)(0,0)@xmath29 ( 10,15)(0,0)@xmath10 ( 24,3)(0,0)@xmath256    this diagram can be written in the form @xmath257\ , , \label{m1b}\ ] ] where the term with @xmath258 comes from the second @xmath225-function in  ( [ m1a ] ) . this function is @xmath259 \left[-k^2-i0\right]}\,.\ ] ] now we use @xmath260 parametrization @xmath261 for both denominators , and also write the @xmath225-function as @xmath262 \frac{d\nu}{2\pi}\ , . \label{delta}\ ] ] we obtain @xmath263\ , . \end{split}\ ] ] shifting the integration momentum as @xmath264 to form a full square , we obtain @xmath265\\ & \quad{}\times \int \frac{d\nu}{2\pi } \exp\left[2 i \nu \left ( \frac{\alpha_1}{\alpha_1+\alpha_2 } \omega ' - \omega \right ) \right]\ , \int d^d k ' \exp\left[(\alpha_1+\alpha_2)(k^{\prime2}+i0)\right ] \end{split}\ ] ] the integral in @xmath266 is calculated using the wick rotation @xmath267 : @xmath268 the integral in @xmath269 gives a @xmath225-function . we obtain an integral in two @xmath260-parameters : @xmath270\ , \delta\left(\frac{\alpha_1}{\alpha_1+\alpha_2}\omega'-\omega\right)\ , . \end{split}\ ] ] it is always possible to calculate one integral in a `` radial '' variable @xmath271 in the space of @xmath260-parameters via @xmath89-function . in this case , the most convenient choice of such a variable is @xmath272 . therefore , we insert @xmath273 under the integral sign , and substitute @xmath274 : @xmath275 the final result is @xmath276    functions @xmath277 which appear in the evolution kernel should be understood as distributions : they are always integrated with smooth test functions @xmath278 . the distribution @xmath279_+$ ] is defined by @xmath280_+ \varphi(\omega ' ) d\omega ' = \int f(\omega,\omega ' ) \left(\varphi(\omega')-\varphi(\omega)\right ) d\omega'\ , . \label{plus}\ ] ] therefore , formally we can write @xmath281_+ + \delta(\omega-\omega ' ) \int f(\omega,\omega '' ) d\omega''\ , . \label{plus2}\ ] ] the result  ( [ m1b ] ) , ( [ f1 ] ) of the calculation of the diagram in fig .  [ f:1 ] can be written via a @xmath282-distribution as @xmath283_+ + \delta(\omega-\omega ' ) \left ( \int f_1(\omega,\omega '' ) d\omega '' - \int f_1(\omega'',\omega ) d\omega '' \right ) \biggr]\ , . \label{m1c}\ ] ] the coefficient of @xmath284 here can be calculated by substitution @xmath285 : @xmath286 substituting @xmath287 in the first integral , we have @xmath288 therefore , the final result for @xmath289 ( fig .  [ f:1 ] ) is @xmath290\ , . \label{m1}\ ] ]    now we shall calculate the second diagram ( fig .  [ f:2 ] ) for @xmath291 it is @xmath292 \left[-(p'-k)\cdot v - i0\right]}\ , . \label{m2a}\ ] ]    ( 36,25 ) ( 18,13.5)(0,0 ) ( 18,25)(0,0)[t]@xmath53 ( 36,10)(0,0)@xmath244 ( 33,16)(0,0)@xmath293 ( 15,3)(0,0)@xmath10    this diagram can be written in the form @xmath294\ , , \label{m2b}\ ] ] where the term with @xmath295 comes from the first @xmath225-function in  ( [ m2a ] ) . this function is @xmath296 \left[-(p'-k)\cdot v - i0\right]}\,.\ ] ] now we use @xmath260-parametrization  ( [ alpha ] ) for both denominators ( it is convenient to multiply the linear denominator by 2 before this ) and  ( [ delta ] ) for the @xmath225-function , and obtain @xmath297\,.\ ] ] shifting the integration momentum as @xmath298 to form a full square , we obtain @xmath299\\ & \quad{}\times \int \frac{d\nu}{2\pi } \exp\left[2 i \nu \left ( \frac{\alpha_2}{\alpha_1 } - \omega '' \right ) \right]\ , \int d^d k ' \exp\left[\alpha_1(k^{\prime2}+i0)\right]\ , . \end{split}\ ] ] taking the integrals in @xmath266 ( [ wick ] ) and @xmath269 , we obtain the integral in two @xmath260-parameters : @xmath300\ , \delta\left(\frac{\alpha_2}{\alpha_1}-\omega''\right)\,.\ ] ] now the best choice of the `` radial '' variable is @xmath301 , so we simply substitute @xmath302 : @xmath303 finally , @xmath304    now we rewrite @xmath305 ( [ m2b ] ) , ( [ f2 ] ) via a @xmath282-distribution : @xmath306_+ + \delta(\omega-\omega ' ) \left ( \int_0^\omega f_2(\omega-\omega '' ) d\omega '' - \int_0^\infty f_2(\omega '' ) d\omega '' \right ) \biggr]\ , . \label{m2c}\ ] ] the coefficient of @xmath284 is @xmath307 we introduced @xmath308 only to regularize possible infrared problems ; we only need the uv divergence of this diagram , which does not depend on @xmath308 . therefore , we may assume @xmath309 . the coefficient of @xmath284 is @xmath310 and the final result for @xmath305 ( fig . [ f:2 ] ) is @xmath311\ , . \label{m2}\ ] ]    this one - loop diagram contains a @xmath3 uv divergence ! it is in the coefficient of @xmath284 given by the integral  ( [ coefdelta ] ) . the function @xmath312 has a @xmath313 uv divergence in the integration in transverse momentum ; the longitudinal integral  ( [ coefdelta ] ) is again uv divergent  ( [ coefdelta2 ] ) . the operator at the vertex of fig . [ f:2 ] is ( after integration in @xmath53 ) a light - like wilson line ; the heavy - antiquark propagator is a time - like wilson line . the vertex correction to a time - like  light - like cusp on a wilson line is known to have a @xmath3 uv divergence at one loop  @xcite .    the third diagram ( fig . [ f:3 ] ) is @xmath314 \left[-k^2-i0\right ] \left[-(p'+p - k)\cdot v - i0\right]}\ , . \label{m3a}\ ] ] the numerator can be simplified as @xmath315 we shall now demonstrate that this diagram is uv - finite and hence does not contribute to the renormalization constant . ( 36,25 ) ( 18,13.5)(0,0 ) ( 18,25)(0,0)[t]@xmath53 ( 2,7)(0,0)@xmath29 ( 10,15)(0,0)@xmath10 ( 36,10)(0,0)@xmath244 ( 36,16)(0,0)@xmath316 ( 18,3)(0,0)@xmath317    using @xmath260-parametrization  ( [ alpha ] ) for the denominators and  ( [ delta ] ) for the @xmath225-function , and obtain @xmath318\ , . \end{split}\ ] ] shifting the integration momentum as @xmath319 we obtain @xmath320\ ; \int \frac{d^d k'}{(2\pi)^d } e^{(\alpha_1+\alpha_2)(k^{\prime2}+i0)}\\ { } = { } & 2 c_f \frac{g_0 ^ 2}{(4\pi)^{d/2 } } \omega \gamma_+ \int d\alpha_1\,d\alpha_2\,d\alpha_3\ , \delta\left(\frac{\alpha_1\omega'+\alpha_3}{\alpha_1+\alpha_2}-\omega\right ) e^{-a}\ , , \end{split}\ ] ] where @xmath321 inserting @xmath322 under the integral sign and making substitutions @xmath323 , @xmath324 , we have @xmath325 , and the integral in the `` radial variable '' @xmath271 is easily calculated : @xmath326 where @xmath327 y + ( \omega - \omega ' ) \frac{\omega}{\omega^{\prime2 } } p^2\,.\ ] ] the integral in @xmath328 is finite at @xmath247 ( and easy to calculate ) . now we are ready to find the renormalization constant @xmath235 . re - expressing @xmath329 in the matrix element  ( [ m ] ) , we have @xmath330\ , . \end{split } \label{mren}\ ] ] the quark - field renormalization constants in qcd and hqet th the feynman gauge are @xmath331 finally , the renormalization constant `` matrix '' at one loop is @xmath332\ , . \end{split } \label{z1res}\ ] ]    the evolution kernel  ( [ anomdim ] ) has the structure @xmath333\ , . \delta(\omega-\omega')\ , . \label{gammastr}\ ] ] the logarithm @xmath334 only appears linearly , to all orders of perturbation theory ; the coefficient of this logarithm is the cusp anomalous dimension  @xcite : @xmath335 ( the two - loop term is also known  @xcite ) . the non - logarithmic part is written as @xmath336 , where the anomalous dimension of the local current @xmath70  ( [ j ] ) is @xmath337 ( the two- and three - loop terms are also known  @xcite ) . it determines the evolution of @xmath338 ; the difference @xmath339 appears in the evolution equation for @xmath340 . the non-@xmath225 term is @xmath341 the evolution kernel has dimensionality of @xmath342 . the evolution equation for the distribution amplitude is @xmath343 \varphi_+(\omega;\mu ) + \int \gamma(\omega,\omega';a_s ) \varphi_+(\omega';\mu ) d\omega ' = 0\ , . \label{evol}\ ] ] how to solve it ? powers @xmath344 are eigenfunctions of the integral operator in  ( [ evol ] ) , by dimensionality : @xmath345 where @xmath346\ , , \label{tildegamma}\ ] ] from  ( [ gamma ] ) ( here @xmath347 is the euler constant ) . they are not eigenfunctions of the whole evolution operator , due to the logarithmic term . we can construct solutions with the power depending on @xmath232 : @xmath348 if the function @xmath349 obeys @xmath350 then @xmath351 will cancel in the evolution equation .    dividing the definition  ( [ xi1 ] ) by @xmath352 and integrating , we obtain @xmath353\ , , \label{xi}\ ] ] where @xmath354 . we shall use @xmath355 as an independent variable ( `` time '' ) in the evolution equation instead of @xmath232 . the function @xmath356 satisfies the evolution equation @xmath357 if @xmath358 at the leading order , @xmath359\,,\ ] ] and @xmath360 - \int_0^\xi \frac{\tilde{\gamma_0}(n+\xi)}{\gamma_0 } d\xi - \frac{\gamma_0}{\gamma_0 } \xi\,.\ ] ] here , from  ( [ tildegamma ] ) , @xmath361 finally , at the leading order , @xmath362 - \xi } \frac{\gamma(1-n-\xi)\gamma(1+n)}{\gamma(1+n+\xi)\gamma(1-n ) } \exp\left[-\left(\frac{\gamma_0}{\gamma_0}+2\gamma_e\right)\xi\right]\ , . \label{uxi}\ ] ]    the distribution amplitude at @xmath363 can be expressed via its moments  ( [ intdn2 ] ) : @xmath364 substituting the solutions  ( [ solform ] ) instead of @xmath344 gives us the solution of the evolution equation  ( [ evol ] ) with initial conditions . at the leading order  ( [ uxi ] ) , @xmath365 } \exp\left[-\left(\frac{\gamma_0}{\gamma_0}+2\gamma_e\right)\xi\right ] \left(\frac{\omega}{\lambda_{\overline{\text{ms}}}}\right)^\xi\\ & { } \times \int_{-\infty}^{+\infty } { < } \omega^{-1-in}{>}_+^{(\mu_0)}\,\omega^{in}\ , \frac{\gamma(1-n-\xi)\gamma(1+n)}{\gamma(1+n+\xi)\gamma(1-n)}\ , \frac{dn}{2\pi}\ , . \end{split } \label{sol}\ ] ]    qualitatively , each `` bin '' in the distribution amplitude at @xmath363 near some finite @xmath366 produces , after evolution to a larger @xmath232 , a radiative tail , slowly decreasing ( as @xmath367 ) at @xmath368 , due to the evolution kernel @xmath369 ( which behaves as @xmath367 at large @xmath53 , by dimensionality ) . therefore , the behaviour of the distribution amplitude at large @xmath53 is @xmath367 , up to logarithms . the normalization integral  ( [ norm ] ) of the distribution amplitude logarithmically diverges at large @xmath53 ; its moments  ( [ omn ] ) with @xmath370 are power - divergent . this means that the expression  ( [ on ] ) for the local operators @xmath216 ( @xmath371 ) via @xmath219 is not valid for the renormalized operators . renormalization of @xmath219 removes uv divergences in transverse momenta ; in order to renormalize @xmath216 , we should also remove longitudinal uv divergences ( at large @xmath53 ) .    as an illustration  @xcite , let s suppose that at a low @xmath363 the distribution amplitude @xmath133 is given by the simple model  ( [ model ] ) . namely , this initial condition is taken at the scale @xmath363 where @xmath372 where the quark model is supposed to work , so that the light quark has only momenta of order @xmath78 ( no radiative tail ) . this function is shown by the solid line in fig . ( [ f : evol ] ) , where @xmath53 is measured in units of @xmath373  ( [ omega0 ] ) . using the leading - order solution  ( [ sol ] ) of the evolution equation  ( [ evol ] ) and supposing @xmath374 , we obtain the distribution amplitude at the scale where @xmath375 ( dashed line ) and @xmath376 ( dashed - dotted line ) . we can see that the main part of the distribution amplitude ( at @xmath377 ) becomes lower and the radiative tail becomes more prominent when @xmath232 grows . ( 100,62 ) ( 50,31)(0,0)-meson distribution amplitude @xmath340,title="fig:",width=377 ] in order to estimate the quark  antiquark distribution amplitudes @xmath176 of @xmath0-meson from qcd sum rules , we consider the correlator of the local current @xmath378  ( [ jp ] ) ( having the quantum numbers of the ground - state meson ) and the bilocal operator @xmath379  ( [ opmt ] ) : @xmath380 we are most interested in its fourier transform @xmath381 analytically continuing it from @xmath382 to @xmath383 , we obtain @xmath384 where @xmath385 is the spectral density . the correlator contains the contribution of the ground - state meson ( written explicitly in  ( [ sr3 ] ) ) and of the continuum of excited states . ( 32,17 ) ( 16,9)(0,0 ) ( 1,0)(0,0)[b]@xmath386 ( 21,0)(0,0)[b]@xmath79 ( 31,17)(0,0)[t]@xmath387    for sufficiently small @xmath388 , we can calculate this correlator theoretically . the perturbative contribution [ f : sr ] ) is given in the fixed - point gauge @xmath389 by the light - quark propagator from @xmath386 to @xmath387 : @xmath390 its fourier transform is @xmath391 inverting the laplace transform  ( [ sr3 ] ) , we find the spectral densities ( the integration contour should be to the right of the singularity at @xmath392 ) : @xmath393 we obtain @xmath394 the perturbative spectral density @xmath395 describes the contribution of the on - shell quark  antiquark intermediate state with energy @xmath249 into the correlator  ( [ sr3 ] ) . the on - shell heavy antiquark has zero energy ; the maximum value of @xmath137 of the on - shell light quark with energy @xmath249 is @xmath396 . this explains the @xmath397-functions in  ( [ rho0 ] ) . the quark condensate contribution is also important here . it contains the vacuum average @xmath398[0,z]q(z){>}\,.\ ] ] this non - collinear quark condensate can be expanded in terms of bilocal condensates  @xcite , the leading term in this expansion is the bilocal quark condensate @xmath399q(x ) { > } = { < } \bar{q}q{>}\,f_s(x^2 ) \label{bilocal}\ ] ] with @xmath400 . this leading term produces the same contribution into @xmath401 : @xmath402 the expansion of the bilocal quark condensate at small @xmath80 is @xmath403 where @xmath404 in general , it can be written as @xmath405 where @xmath406 has the meaning of the virtuality distribution function of vacuum quarks . its moments are expressed via vacuum averages of local operators : @xmath407 in terms of this function , the quark - condensate contribution into the fourier - transformed correlators is @xmath408 in other words , the virtuality distribution function appears directly in the sum rules for the @xmath0-meson distribution amplitudes ! this is similar to the case of non - diagonal sum rules for the pion distribution amplitude  @xcite . the local operator expansion  ( [ local ] ) gives @xmath409 of course , a smooth function can always be expanded in derivatives of @xmath225-function , but such an expansion does not tell us much about the shape of this function , unless we sum an infinite number of terms . the bilocal quark condensate  ( [ bilocal ] ) at large @xmath410 behaves as @xmath411 because the wilson line @xmath412 $ ] can be considered an hqet heavy - quark propagator , and this vacuum average is the correlator of two hqet heavy  light currents , having @xmath0-meson as the lowest - energy intermediate state . of course , the conditions  ( [ momfs ] ) , ( [ asympt ] ) do nt determine the shape of the distribution function @xmath406 in a unique way . they are satisfied by the bakulev mikhailov ansatz  @xcite @xmath413 where the parameters @xmath414 , @xmath415 are determined by  ( [ momfs ] ) . now we equate the theoretical result for the correlators to the result obtained from a phenomenological model of the spectral densities . they have the contribution @xmath416 of the ground - state meson and that of the continuum of excited states . as usual , this contribution is modeled by the perturbative spectral densities  ( [ rho0 ] ) starting from a continuum threshold energy @xmath417 : @xmath418 with this model , the equality of the phenomenological expression for the correlator and the theoretical one , @xmath419 becomes @xmath420 where the perturbative spectral density is integrated over the `` duality interval '' of the ground - state meson ( from @xmath79 to the continuum threshold @xmath417 ) . we obtain the sum rules for the @xmath0-meson distribution amplitudes @xmath421 where the functions @xmath422 describe the effect of subtracting continuum from the perturbative contribution when the spectral density is @xmath423 . the perturbative contributions vanish at @xmath424 , because of the properties of the spectral densities  ( [ rho0 ] ) .    setting @xmath425 in the correlator  ( [ sr1 ] ) , or integrating  ( [ sr2 ] ) in @xmath226 , we obtain the well - known sum rule for @xmath426 : @xmath427 the natural energy scale in this sum rule is @xmath428 with @xmath429 , one finds the optimal value of the continuum threshold @xmath430 ; then a wide plato at @xmath431 $ ] exists , and yields @xmath432 . we divide the sum rules  ( [ sr ] ) by  ( [ sr0 ] ) and take @xmath433 ( in the middle of the plato ) . the results are shown in fig . [ f : p ] , [ f : m ] , where @xmath53 is measured in units of @xmath10 . they are automatically normalized  ( [ norm ] ) . of course , the leading - order sum rules can not tell us at what scale these distribution amplitudes are normalized ; this scale must be low , @xmath434 . the perturbative contributions vanish at @xmath424 . the quark - condensate contribution is the same for both @xmath133 and @xmath134 . here we used the ansatz  ( [ ansatz ] ) . this contribution gives a sharp peak at low @xmath53 and @xmath435 . ] . details of its shape are unknown , but it can not be wider because of the restriction on the first moment  ( [ momfs ] ) . this contribution falls off quickly at larger @xmath53 . therefore , the distribution amplitudes at this low @xmath232 are only non - zero at @xmath377 , in accord with the expectations of the quark model . ( 100,62 ) ( 50,31)(0,0 ) ( solid line ) , perturbative contribution ( dashed line ) , and quark - condensate contribution ( dashed - dotted line),title="fig:",width=377 ]    ( 100,62 ) ( 50,31)(0,0 ) ( solid line ) , perturbative contribution ( dashed line ) , and quark - condensate contribution ( dashed - dotted line),title="fig:",width=377 ]    if we neglect the quark - condensate contribution ( though this is not a good idea ) and also the effect of continuum subtraction , then the perturbative contributions  ( [ pert2 ] ) suggest the following simple model of distribution amplitudes : @xmath436 they are normalized  ( [ norm ] ) ; from the first moments  ( [ mom1 ] ) we obtain @xmath437 these functions ( fig . [ f : model ] ) have something in common with the wandzura  wilczek ones ( fig .  [ f : ww ] ) , but in contrast to them they fall off smoothly at large @xmath53 . ( 42,36 ) ( 21,20)(0,0 ) ( solid line ) and @xmath134 ( dashed line),title="fig : " ] ( 9.5,0)(0,0)[b]@xmath373    radiative corrections to the perturbative spectral density and the dimension-3 quark - condensate contribution for @xmath438 were calculated in  @xcite . with these corrections , one can check that the correlator  ( [ sr2 ] ) satisfies the evolution equation  ( [ rg ] ) . the resulting sum rules should be used together with the sum rules for @xmath426 with radiative corrections  @xcite . the perturbative spectral density is @xmath439\\ x<1:&\displaystyle 2 c_f \frac{\alpha_s}{4\pi } \biggl [ 2 \bigl(\log(1-x)+x\bigr ) \log\frac{\omega}{\mu } + 2 \log^2 ( 1-x)\\ & \displaystyle { } + ( 2x-1 ) \log(1-x ) - x \biggr ] \end{array } \right . \end{split } \label{rho1}\ ] ] where @xmath440 . now the spectral density does not vanish at @xmath441 , it is just suppressed by @xmath442 . therefore , the perturbative contribution to the sum rule  ( [ sr ] ) for @xmath133 does not vanish at @xmath424 . it produces a radiative tail @xmath443 with the magnitude of order @xmath442 .    the radiative correction to the dimension-3 quark - condensate contribution can not be used together with a model of bilocal condensate , because radiative corrections to higher - dimensional contributions are not known . it is vital to use some model of the bilocal quark condensate in the sum rules for the distribution amplitude , because the local operator expansion produces contributions @xmath444 , @xmath445 ,  , which do nt tell us much about the shape of the distribution amplitude . therefore , the full result for this correction can not be used in the sum rule for @xmath133 . fortunately , the authors of  @xcite demonstrated that a part of this correction is universal , and exponentialized into the sudakov factor . we may multiply the bilocal quark - condensate contribution by this sudakov factor . i am grateful to m.  neubert for collaboration on  @xcite and to s.v . mikhailov for useful discussions .
@xmath0-meson light - cone distribution amplitudes are discussed in these lectures in the framework of hqet . the evolution equation for the leading - twist distribution amplitude 
 is derived in one - loop approximation . 
 qcd sum rules for distribution amplitudes are discussed .