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Two-loop renormalization scale dependence of the Isgur-Wise function
PhysicsLetters B 305 ( 1993 ) 157-162 North-Holland
PHYSICS LETTERS B
Two-loop renormalization scale dependence of the Isgur-Wise function E. Bagan 1 and P.
Gosdzinsky
Grup de Fisica Tebrica, Departament de Fisica and Institut de Fisica d'Altes Energies, UniversitatAutbnoma de Barcelona, E-08193 Bellaterra (Barcelona), Spain
Received 22 February 1993
The two-loopanomalousdimension of a current consistingof two heavy quarks with velocitiesv and v' (relevant to the renorrealization scale dependenceof the Isgur-Wisefunction) is computedwithin the frameworkof the Heavy Quark EffectiveTheory. Our result agreeswith Korchemskyand Radyushkin'scalculationof the cusp anomalousdimensionof Wilson lines, thus resolving previous discrepanciesin the literature.
1. Introduction
Hadrons made up of one heavy quark (such as the b quark) and light degrees of freedom have the following remarkable property: at long distances their effective QCD interactions exhibit a very large (approximate) symmetry which is not apparent in the original lagrangian. This is due to the fact that one can consider the heavy quark to be a static source of color. Thus, in the hadron the "light cloud" interactions with a color field independent of the mass, mQ, and the spin of the heavy quark itself, provided rnQ is much larger than AQCD. These spin and flavor symmetries can be combined into an SU(2NQ) symmetry, NQ being the number of heavy quark species, sometimes referred to as Isgur-Wise (IW) symmetry. Georgi's effective lagrangian [ 1 ], written in terms of velocity-dependent heavy quark (effective) fields, h~, implements the IW symmetry. It allows for an easy calculation of the asymptotic rnQ dependence of matrix elements of the currents governing the leptonic and semileptonic decays of hadrons with a heavy quark. All this provides a framework that has become known as Heavy Quark Effective Theory (HQET). In semileptonic decays of such hadrons, the original heavy quark Q carrying a 4-velocity v can be "kicked" by an appropriate weak current and become a heavy quark Q' with 4-velocity v'. It is well known that in the limit m Q ~ , the form factors parametrizing such decays are not independent but proportional to a universal function of y = v . v ' , the so called IW function, ~iw(Y) [2]. It is therefore very important for phenomenology to determine the scaling properties of ~iw (Y). This question has been investigated to two loop accuracy by several authors [ 3,4 ] following different approaches. Unfortunately their results are in conflict. The approach of ref. [ 3 ] consists in computing the anomalous dimension, 7HH(Y), of the current J r = ~v' Fhv, where F i s some gamma matrix, within the framework of the HQET. The calculation of ref. [ 4 ] goes along a completely different line. It is based upon the observation that the (classical) trajectories of the two infinitely heavy quarks, which are straight lines labeled by their 4-velocities, v, v', can be replaced by Wilson lines joining at a cusp. From the UVdivergence resulting from the sudden change of direction at the cusp (the infinite acceleration gives rise to an infinite bremsstrahlung) one can easily obtain the cusp anomalous dimension, 7cusp(Y), which is expected to be equal to 7HH(Y)" l Alexandervon Humboldt fellow. 0370-2693/93/$ 06.00 © 1993 ElsevierScience Publishers B.V. All rights reserved.
157
Volume 305, number 1,2
PHYSICS LETTERS B
6 May 1993
As anticipated, the two expressions are found to disagree. Furthermore, as noted in ref. [ 4 ], the asymptotic behavior ( y ~ oo ) of the result in ref. [ 3 ] does not satisfy linearity in log y. This violates the non-abelian exponentiation theorem of Wilson line averages [ 5 ]. Therefore, since ref. [4] does not use the standard perturbative approach based entirely on the H Q E T lagrangian, both a consistent two-loop calculation Yrm(Y) in the H Q E T and an explicit check of the identity 7nu (Y) = Y¢,sp(Y) are still missing. The aim of this letter is twofold. First, to provide the missing calculation of 7Hn (Y) and hence have a completely independent check of (its relation with) the cusp anomalous dimension computed in ref. [ 4 ]. Second, to illustrate with an example the convenience and simplicity of the method for evaluating loop integrals introduced in ref. [6 ]. To be more precise, we shall compute the renormalization constant, ZnH (Y), of the current Jr=[i~,Fhv as a series in powers o f t = ( y - 1 ) / ( y + 1 ). In ref. [6], this "r-expansion" proved very well suited for computing the three-current vacuum correlator relevant to the H Q E T sum rule estimation of ~lw(Y). The results of ref. [ 6 ] have recently been confirmed in ref. [ 7 ]. For many phenomenological applications it is sufficient to truncate the r-expansion at order r 2. Thus in the present letter we shall only retain the first three terms of the series. The letter is organized as follows: In section 2 we briefly outline our method for the kind of two-loop integrals that will show up in the evaluation of ZHn(y). In section 3 we present the actual calculation. Finally, in section 4 we state our conclusions.
2. The r-expansion The relevant two-loop 1PI diagrams are shown in fig. 1 (the crossed diagrams of a, c and d have been omitted in the figure). Since the overall UV divergence is independent of the external momenta, we can set q. v= q'. v' = ~o in order to simplify the calculation. After some algebra, the diagrams of fig. 1 can be expressed in terms of integrals of the form
~oDIOD~_I a {
1
'~"{
l b
1
l
~"{
~c(
1
I
~'{
~nl
l
1
~n'
V
where D = 4 - 2e is the space-time dimension. Next, we expand these integrals in powers of r as explained in ref. [ 6 ]. We sketch the important steps: ( I ) Introduce two new variables, v+ = ( v + v ' ) / 2 and v_ = ( v - v ' ) / 2 , and expand the v and v' dependent terms in eq. (1) in powers of v_ .l and v .k. (2) Scale the loop variables according to
v
158
v' a
b
c
d
e
f
Fig. 1. 1PI two-loop diagrams contributing to the renorrnalization of the current Jr.
Volume 305, number 1,2
PHYSICSLETTERSB
6 May 1993
(2) and introduce the notation ~-- v + / ~ + , such that ~2= 1. (3) Get rid of the ~. ( l - k ) + co denominators. This can be easily achieved by multiplying the integrand by
1 = 1_ ( [f:. ( l - k ) +co] + 03.k+ co) - (~.l+co)}, co
(3)
and/or making an appropriate shift in the variables k and/or l where necessary. After some algebra, the resulting expression can be written in terms of integrals of the form /~, ~i v " v,,.... v_v .....
d ld kl(.~...l.pk.,...k~)
(4)
(12),(k2) b[ ( / _ k ) 2 ]C(~.l+co).(~.k+co),.,
where the indices in parentheses are to be symmetrized. (4) Evaluate the integral in (4). It must be of the form ~ k d k ( ~ ) U , ...... where (~)u,...~ are symmetric tensors built up from gu, and G. Fortunately, only the simplest cases where p + q ~ 4 are required if one wishes to truncate the r-expansion at O (rE). In order to obtain the coefficients dk, we contract all the indices of the integral in (4) with the appropriate number of metric tensors gU~ and velocities ~". This will provide us with a set of equations for ~¢kthat can be solved in terms of integrals of the form
dDldDk f (12)'(kE)b[ ( l - k ) 2] c(ffl+co)'(~.k+co) m"
(5)
One can easily evaluate these integrals following the procedure described in detail in ref. [ 8 ]. Steps ( 1 ) - ( 4 ) lead quite naturally to an expansion in powers of v 2 /v 2 = - r .
3. Calculation of 7nil(Y) Let jR be the renormalized effective current. One has
J~ = Zr( y ) ~Fhv, = ZnH ( Y ) h-° Fh °, , ZHH ( y ) = Z g l Zr( y ) .
(6)
Here, h-°, h°,, are the bare (heavy quark) effective fields and Zh is their wave function renormalization constant [8 ]. The renormalization constant Z r ( y ) is chosen to cancel the UV-divergent part of the 1PI diagrams. The anomalous dimension 7HH(Y)is given by
(7 )
YHH(Y) =/.t ~ log ZHH(Y) ,
where # is the renormalization scale of the MS scheme in which our calculation has been performed. It is convenient to introduce analogous definitions of ~h and ~r(y). From eq. (6) it follows that 7HH(Y) = yr(y) _~h.
(8)
The two-loop anomalous dimension of the heavy-quark field has already been computed in ref. [ 8 ] to be 2
~h=2(a--3)CF-~+[(½a2+4a--L~)CA+-~TRNf]CF(-~)
,
(9)
where Nf is the number of light flavors, TR = ½, Cv = (N 2 _ 1 ) / 2Nc, CA= Nc (Nc is the number of colors). In the MS scheme, one has 159
Volume 305, number 1,2 o~
Zr(y)=l+ ~
PHYSICS LETTERSB
[SZl(y)+aSZ~]+ ~
[½
6 May 1993
5Z'2(y)+eSZ2(y)]+O(ol3),
(10)
where the dependence on t is shown explicitly. 8Zl (y), fiZ¢ are a (gauge parameter) independent. The oneloop functions 8Z, (y) and 8Z'~ are well known (see for instance ref. [9 ] ) and given by
_r~ {2yI°g(Y+x~-I)x/~Z~I 8Z,(y)--,~Fk
+ 1 ,)
8Z~[=-Cv.
(11)
In eq. (10) only 5Z2(y) is a genuine two-loop quantity, whereas 8Z'2(y) can be obtained from the one-loop IPI diagram shown in fig. 2 through the identity (12)
~)Zt2(y) = [ 8 Z 1( y ) .-~-a 8Z•] [ 8 Z 1(y) -~-a 8 Z 7 +rio] --~-a ~ 8 Z ~ ,
where flo (78) is the first coefficient of the expansion of the fl-function (of the gauge parameter anomalous dimension, ya) in powers of a/4rc They are given by
flo=4TRNf- ~CA,
7~=½( ~--a)CA--4TRNf .
(13)
Eq. ( 12 ) will be a nontrivial check of our result. From eq. (10) one gets
[SZt(y)+aSZ,]-4 ~
7r(y)=-2
8Z2(y).
(14)
Next, we have computed the relevant contribution of the 1PI diagrams in fig. 1 and fig. 2. For the sake of simplicity, we have carried out the calculations in the Feynman gauge, a = 1 (note that Yn~ (Y) is gauge invariant). From the final result one can read offSZ, (y), 8Z~, 8Z2 (y) and 8Z'2(y). Recalling that r = ( y - 1 )/(y+ I ), one gets
~)Z,(y)+SZ~=CF(2+ 3rt-fsr 8 - , 6 2) + O ( r 3 ) , 8 376 80 TR ] + r 2 [ ( ~ - - ~ 32 _~NfTR+r[(~__~r2)CA ,6~:~,~ -- ~'~Nf I~A--~-~NfTR]}+O(F3),
8Z2(y)_.~CF{?CA
(15)
whereas 8Z~ (y) is seen to satisfy eq. (12). One can easily check that 8Z, (y) + 8Z7 agrees with the expansion of the exact result in ( 11 ). From eq. (15) one finally obtains 7HH(y) =
_
~CF(~r+8r2+~{r[(~727t2)CA__~.~Nf ] 2 0 TR +r2[ (Z364
__ ~5 ~ 2 ) CA
-SNfTR]}+O(r3)).
(16)
For convenience of those who wish to reproduce our calculation, we also give the UV-divergent part of the separate two-loop diagrams in fig. 1, 9~k), k=a .... , f
~ ) -- -A2C~[2+]r+ ~3r '6 2+ t
( ~8r - + - ~ 344 , "2 ) ] ,
~b)=A2CF{(2TRNf_~CA)(I_F -~r 8 + ~,6r 2) +¢[--CA +r(12~-TRNf--12-~CA)+r2( ~TRNf--698 r~c~)]}, -
+ T3~(2)r] ,
~ d ) = J 2 C F ( 2 C F --CA)[1 +4r+8r2+~'4 -- 172 2 x) ] , t~r~-~-~3r
v
160
v'
(17)
Fig. 2. IPI one-loopdiagram contributing to the renormalization of the current Jr.
Volume 305, number 1,2
PHYSICS LETTERS B
6 May 1993
@~e) =AEC~[ 2 + ~ r + ~s6r2 - ~(4+ ~Qr+ 36~56rZ)] ,
~ ~ =AzCF( C A - 2CF)~( - 2 - ~ r - 46~sSr2) ,
(17 cont'd)
where 32 = ( - 2~0) -4' (o~/4z~e)2/'. The one-loop amplitude of fig. 2 reads - A l C F [ 2 + ~ r t a 3 r S16 2 + ~ ( 4 r . F ~ r 2 ) ] - -
_ A I C F . ~ot{ 4 T R N f _ 9 C A +r(~-TRNf---~CA)+r 32 sS 2 (~TRNf--z-~-CA) 64 176 lO .+E[..g_CA__~TRNf + r(-~TRNf---cCA) s 34 8 122 +r 2 (gTRNf---iTCA) ]} ,
(18)
where A~= ( - 2 o ~ ) - : ' ( a / 4 n e ) F . The O ( a 2) contributions appear after the bare coupling constant and gauge parameter, ao, have been written in terms of (the renormalized) a and a ( a = 1 ).
4. Conclusions and discussion We have computed the anomalous dimension, 7ian (Y), of the current Jr = [[~,Fh~ up to O ( a 2, r 2 ) within the HQET using the techniques introduced in ref. [6]. Note that 7an(V= V')=0, as it should because of current conservation. In the HQET, the anomalous dimension of Jr is independent of F and no mixing of operators takes place. Our result agrees (disagrees) with the expansion of the result of ref. [4] (ref. [3] ). This provides both, a confirmation of Korchemsky and Radyushkin's exact result as well as an explicit example that the HQET and the Wilson line approaches are equivalent. We would like to conclude by discussion very briefly the scaling properties of the IW function. The (renormalization ) scale dependence of ~w (Y) is given implicitly by
(°)2
dlOg~lw(Y) _~HH(Y)= ~--~o a RH(Y)+ ~ /z d/t
?nH(y)+ ... ,
(19)
where y~H (y) and 7~H (y) can be simply read off ( 16). To two loop accuracy, the solution of ( 19) reads [ 10,1 1 ] 8//o
\Tfin(y)
flo
'
(20)
where ~iw (Y) is the renormalization group invariant (scale independent) IW function [6] and flo, fl~ are the first two coefficients of the fl-function, normalized as in ( 13 ).
Acknowledgement One of us (P.G.) acknowledges gratefully a grant from the Generalitat de Catalunya.
References [ 1 ] H. Georgi, Phys. Lett. B 240 (1990) 447. [2] N. Isgur and M.B. Wise, Phys. Lett. B 232 (1989) 113; B 237 (1990) 527. [3] X. Ji, Phys. Lett. B 264 (1991) 193. [4] G.P. Korchemsky and A.V. Radyushkin, Phys. Lett. B 279 (1992) 359.
161
Volume 305, number 1,2
PHYSICS LETTERS B
[ 5 ] J.G.M. Gatheral, Phys. Lett. B 133 ( 1983 ) 90; J. Frenkel and J.C. Taylor, Nucl. Phys. B 246 (1984) 231. [6] E. Bagan, P. Ball and P. Gosdzinsky, Phys. Lett. B 301 (1993) 249. [ 7] M. Neubert, preprint SLAC-PUB-5992 (November 1992). [8] D.J. Broadhurst and A.G. Grozin, Phys. Lett. B 267 ( 1991 ) 105. [9] A.F. Falk, H. Georgi, B. Grinstein and M. Wise, Nucl. Phys. B 343 (1990) 1. [ 10] P. Ball, Ph.D. thesis, Heidelberg preprint HD-THEP-92-9 [in German]. [ 11 ] E. Bagan, P. Ball, V.M. Braun and H.G. Dosch, Heidelberg preprint HD-THEP-91-36.
162
6 May 1993
PHYSICS LETTERS B
Two-loop renormalization scale dependence of the Isgur-Wise function E. Bagan 1 and P.
Gosdzinsky
Grup de Fisica Tebrica, Departament de Fisica and Institut de Fisica d'Altes Energies, UniversitatAutbnoma de Barcelona, E-08193 Bellaterra (Barcelona), Spain
Received 22 February 1993
The two-loopanomalousdimension of a current consistingof two heavy quarks with velocitiesv and v' (relevant to the renorrealization scale dependenceof the Isgur-Wisefunction) is computedwithin the frameworkof the Heavy Quark EffectiveTheory. Our result agreeswith Korchemskyand Radyushkin'scalculationof the cusp anomalousdimensionof Wilson lines, thus resolving previous discrepanciesin the literature.
1. Introduction
Hadrons made up of one heavy quark (such as the b quark) and light degrees of freedom have the following remarkable property: at long distances their effective QCD interactions exhibit a very large (approximate) symmetry which is not apparent in the original lagrangian. This is due to the fact that one can consider the heavy quark to be a static source of color. Thus, in the hadron the "light cloud" interactions with a color field independent of the mass, mQ, and the spin of the heavy quark itself, provided rnQ is much larger than AQCD. These spin and flavor symmetries can be combined into an SU(2NQ) symmetry, NQ being the number of heavy quark species, sometimes referred to as Isgur-Wise (IW) symmetry. Georgi's effective lagrangian [ 1 ], written in terms of velocity-dependent heavy quark (effective) fields, h~, implements the IW symmetry. It allows for an easy calculation of the asymptotic rnQ dependence of matrix elements of the currents governing the leptonic and semileptonic decays of hadrons with a heavy quark. All this provides a framework that has become known as Heavy Quark Effective Theory (HQET). In semileptonic decays of such hadrons, the original heavy quark Q carrying a 4-velocity v can be "kicked" by an appropriate weak current and become a heavy quark Q' with 4-velocity v'. It is well known that in the limit m Q ~ , the form factors parametrizing such decays are not independent but proportional to a universal function of y = v . v ' , the so called IW function, ~iw(Y) [2]. It is therefore very important for phenomenology to determine the scaling properties of ~iw (Y). This question has been investigated to two loop accuracy by several authors [ 3,4 ] following different approaches. Unfortunately their results are in conflict. The approach of ref. [ 3 ] consists in computing the anomalous dimension, 7HH(Y), of the current J r = ~v' Fhv, where F i s some gamma matrix, within the framework of the HQET. The calculation of ref. [ 4 ] goes along a completely different line. It is based upon the observation that the (classical) trajectories of the two infinitely heavy quarks, which are straight lines labeled by their 4-velocities, v, v', can be replaced by Wilson lines joining at a cusp. From the UVdivergence resulting from the sudden change of direction at the cusp (the infinite acceleration gives rise to an infinite bremsstrahlung) one can easily obtain the cusp anomalous dimension, 7cusp(Y), which is expected to be equal to 7HH(Y)" l Alexandervon Humboldt fellow. 0370-2693/93/$ 06.00 © 1993 ElsevierScience Publishers B.V. All rights reserved.
157
Volume 305, number 1,2
PHYSICS LETTERS B
6 May 1993
As anticipated, the two expressions are found to disagree. Furthermore, as noted in ref. [ 4 ], the asymptotic behavior ( y ~ oo ) of the result in ref. [ 3 ] does not satisfy linearity in log y. This violates the non-abelian exponentiation theorem of Wilson line averages [ 5 ]. Therefore, since ref. [4] does not use the standard perturbative approach based entirely on the H Q E T lagrangian, both a consistent two-loop calculation Yrm(Y) in the H Q E T and an explicit check of the identity 7nu (Y) = Y¢,sp(Y) are still missing. The aim of this letter is twofold. First, to provide the missing calculation of 7Hn (Y) and hence have a completely independent check of (its relation with) the cusp anomalous dimension computed in ref. [ 4 ]. Second, to illustrate with an example the convenience and simplicity of the method for evaluating loop integrals introduced in ref. [6 ]. To be more precise, we shall compute the renormalization constant, ZnH (Y), of the current Jr=[i~,Fhv as a series in powers o f t = ( y - 1 ) / ( y + 1 ). In ref. [6], this "r-expansion" proved very well suited for computing the three-current vacuum correlator relevant to the H Q E T sum rule estimation of ~lw(Y). The results of ref. [ 6 ] have recently been confirmed in ref. [ 7 ]. For many phenomenological applications it is sufficient to truncate the r-expansion at order r 2. Thus in the present letter we shall only retain the first three terms of the series. The letter is organized as follows: In section 2 we briefly outline our method for the kind of two-loop integrals that will show up in the evaluation of ZHn(y). In section 3 we present the actual calculation. Finally, in section 4 we state our conclusions.
2. The r-expansion The relevant two-loop 1PI diagrams are shown in fig. 1 (the crossed diagrams of a, c and d have been omitted in the figure). Since the overall UV divergence is independent of the external momenta, we can set q. v= q'. v' = ~o in order to simplify the calculation. After some algebra, the diagrams of fig. 1 can be expressed in terms of integrals of the form
~oDIOD~_I a {
1
'~"{
l b
1
l
~"{
~c(
1
I
~'{
~nl
l
1
~n'
V
where D = 4 - 2e is the space-time dimension. Next, we expand these integrals in powers of r as explained in ref. [ 6 ]. We sketch the important steps: ( I ) Introduce two new variables, v+ = ( v + v ' ) / 2 and v_ = ( v - v ' ) / 2 , and expand the v and v' dependent terms in eq. (1) in powers of v_ .l and v .k. (2) Scale the loop variables according to
v
158
v' a
b
c
d
e
f
Fig. 1. 1PI two-loop diagrams contributing to the renorrnalization of the current Jr.
Volume 305, number 1,2
PHYSICSLETTERSB
6 May 1993
(2) and introduce the notation ~-- v + / ~ + , such that ~2= 1. (3) Get rid of the ~. ( l - k ) + co denominators. This can be easily achieved by multiplying the integrand by
1 = 1_ ( [f:. ( l - k ) +co] + 03.k+ co) - (~.l+co)}, co
(3)
and/or making an appropriate shift in the variables k and/or l where necessary. After some algebra, the resulting expression can be written in terms of integrals of the form /~, ~i v " v,,.... v_v .....
d ld kl(.~...l.pk.,...k~)
(4)
(12),(k2) b[ ( / _ k ) 2 ]C(~.l+co).(~.k+co),.,
where the indices in parentheses are to be symmetrized. (4) Evaluate the integral in (4). It must be of the form ~ k d k ( ~ ) U , ...... where (~)u,...~ are symmetric tensors built up from gu, and G. Fortunately, only the simplest cases where p + q ~ 4 are required if one wishes to truncate the r-expansion at O (rE). In order to obtain the coefficients dk, we contract all the indices of the integral in (4) with the appropriate number of metric tensors gU~ and velocities ~". This will provide us with a set of equations for ~¢kthat can be solved in terms of integrals of the form
dDldDk f (12)'(kE)b[ ( l - k ) 2] c(ffl+co)'(~.k+co) m"
(5)
One can easily evaluate these integrals following the procedure described in detail in ref. [ 8 ]. Steps ( 1 ) - ( 4 ) lead quite naturally to an expansion in powers of v 2 /v 2 = - r .
3. Calculation of 7nil(Y) Let jR be the renormalized effective current. One has
J~ = Zr( y ) ~Fhv, = ZnH ( Y ) h-° Fh °, , ZHH ( y ) = Z g l Zr( y ) .
(6)
Here, h-°, h°,, are the bare (heavy quark) effective fields and Zh is their wave function renormalization constant [8 ]. The renormalization constant Z r ( y ) is chosen to cancel the UV-divergent part of the 1PI diagrams. The anomalous dimension 7HH(Y)is given by
(7 )
YHH(Y) =/.t ~ log ZHH(Y) ,
where # is the renormalization scale of the MS scheme in which our calculation has been performed. It is convenient to introduce analogous definitions of ~h and ~r(y). From eq. (6) it follows that 7HH(Y) = yr(y) _~h.
(8)
The two-loop anomalous dimension of the heavy-quark field has already been computed in ref. [ 8 ] to be 2
~h=2(a--3)CF-~+[(½a2+4a--L~)CA+-~TRNf]CF(-~)
,
(9)
where Nf is the number of light flavors, TR = ½, Cv = (N 2 _ 1 ) / 2Nc, CA= Nc (Nc is the number of colors). In the MS scheme, one has 159
Volume 305, number 1,2 o~
Zr(y)=l+ ~
PHYSICS LETTERSB
[SZl(y)+aSZ~]+ ~
[½
6 May 1993
5Z'2(y)+eSZ2(y)]+O(ol3),
(10)
where the dependence on t is shown explicitly. 8Zl (y), fiZ¢ are a (gauge parameter) independent. The oneloop functions 8Z, (y) and 8Z'~ are well known (see for instance ref. [9 ] ) and given by
_r~ {2yI°g(Y+x~-I)x/~Z~I 8Z,(y)--,~Fk
+ 1 ,)
8Z~[=-Cv.
(11)
In eq. (10) only 5Z2(y) is a genuine two-loop quantity, whereas 8Z'2(y) can be obtained from the one-loop IPI diagram shown in fig. 2 through the identity (12)
~)Zt2(y) = [ 8 Z 1( y ) .-~-a 8Z•] [ 8 Z 1(y) -~-a 8 Z 7 +rio] --~-a ~ 8 Z ~ ,
where flo (78) is the first coefficient of the expansion of the fl-function (of the gauge parameter anomalous dimension, ya) in powers of a/4rc They are given by
flo=4TRNf- ~CA,
7~=½( ~--a)CA--4TRNf .
(13)
Eq. ( 12 ) will be a nontrivial check of our result. From eq. (10) one gets
[SZt(y)+aSZ,]-4 ~
7r(y)=-2
8Z2(y).
(14)
Next, we have computed the relevant contribution of the 1PI diagrams in fig. 1 and fig. 2. For the sake of simplicity, we have carried out the calculations in the Feynman gauge, a = 1 (note that Yn~ (Y) is gauge invariant). From the final result one can read offSZ, (y), 8Z~, 8Z2 (y) and 8Z'2(y). Recalling that r = ( y - 1 )/(y+ I ), one gets
~)Z,(y)+SZ~=CF(2+ 3rt-fsr 8 - , 6 2) + O ( r 3 ) , 8 376 80 TR ] + r 2 [ ( ~ - - ~ 32 _~NfTR+r[(~__~r2)CA ,6~:~,~ -- ~'~Nf I~A--~-~NfTR]}+O(F3),
8Z2(y)_.~CF{?CA
(15)
whereas 8Z~ (y) is seen to satisfy eq. (12). One can easily check that 8Z, (y) + 8Z7 agrees with the expansion of the exact result in ( 11 ). From eq. (15) one finally obtains 7HH(y) =
_
~CF(~r+8r2+~{r[(~727t2)CA__~.~Nf ] 2 0 TR +r2[ (Z364
__ ~5 ~ 2 ) CA
-SNfTR]}+O(r3)).
(16)
For convenience of those who wish to reproduce our calculation, we also give the UV-divergent part of the separate two-loop diagrams in fig. 1, 9~k), k=a .... , f
~ ) -- -A2C~[2+]r+ ~3r '6 2+ t
( ~8r - + - ~ 344 , "2 ) ] ,
~b)=A2CF{(2TRNf_~CA)(I_F -~r 8 + ~,6r 2) +¢[--CA +r(12~-TRNf--12-~CA)+r2( ~TRNf--698 r~c~)]}, -
+ T3~(2)r] ,
~ d ) = J 2 C F ( 2 C F --CA)[1 +4r+8r2+~'4 -- 172 2 x) ] , t~r~-~-~3r
v
160
v'
(17)
Fig. 2. IPI one-loopdiagram contributing to the renormalization of the current Jr.
Volume 305, number 1,2
PHYSICS LETTERS B
6 May 1993
@~e) =AEC~[ 2 + ~ r + ~s6r2 - ~(4+ ~Qr+ 36~56rZ)] ,
~ ~ =AzCF( C A - 2CF)~( - 2 - ~ r - 46~sSr2) ,
(17 cont'd)
where 32 = ( - 2~0) -4' (o~/4z~e)2/'. The one-loop amplitude of fig. 2 reads - A l C F [ 2 + ~ r t a 3 r S16 2 + ~ ( 4 r . F ~ r 2 ) ] - -
_ A I C F . ~ot{ 4 T R N f _ 9 C A +r(~-TRNf---~CA)+r 32 sS 2 (~TRNf--z-~-CA) 64 176 lO .+E[..g_CA__~TRNf + r(-~TRNf---cCA) s 34 8 122 +r 2 (gTRNf---iTCA) ]} ,
(18)
where A~= ( - 2 o ~ ) - : ' ( a / 4 n e ) F . The O ( a 2) contributions appear after the bare coupling constant and gauge parameter, ao, have been written in terms of (the renormalized) a and a ( a = 1 ).
4. Conclusions and discussion We have computed the anomalous dimension, 7ian (Y), of the current Jr = [[~,Fh~ up to O ( a 2, r 2 ) within the HQET using the techniques introduced in ref. [6]. Note that 7an(V= V')=0, as it should because of current conservation. In the HQET, the anomalous dimension of Jr is independent of F and no mixing of operators takes place. Our result agrees (disagrees) with the expansion of the result of ref. [4] (ref. [3] ). This provides both, a confirmation of Korchemsky and Radyushkin's exact result as well as an explicit example that the HQET and the Wilson line approaches are equivalent. We would like to conclude by discussion very briefly the scaling properties of the IW function. The (renormalization ) scale dependence of ~w (Y) is given implicitly by
(°)2
dlOg~lw(Y) _~HH(Y)= ~--~o a RH(Y)+ ~ /z d/t
?nH(y)+ ... ,
(19)
where y~H (y) and 7~H (y) can be simply read off ( 16). To two loop accuracy, the solution of ( 19) reads [ 10,1 1 ] 8//o
\Tfin(y)
flo
'
(20)
where ~iw (Y) is the renormalization group invariant (scale independent) IW function [6] and flo, fl~ are the first two coefficients of the fl-function, normalized as in ( 13 ).
Acknowledgement One of us (P.G.) acknowledges gratefully a grant from the Generalitat de Catalunya.
References [ 1 ] H. Georgi, Phys. Lett. B 240 (1990) 447. [2] N. Isgur and M.B. Wise, Phys. Lett. B 232 (1989) 113; B 237 (1990) 527. [3] X. Ji, Phys. Lett. B 264 (1991) 193. [4] G.P. Korchemsky and A.V. Radyushkin, Phys. Lett. B 279 (1992) 359.
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[ 5 ] J.G.M. Gatheral, Phys. Lett. B 133 ( 1983 ) 90; J. Frenkel and J.C. Taylor, Nucl. Phys. B 246 (1984) 231. [6] E. Bagan, P. Ball and P. Gosdzinsky, Phys. Lett. B 301 (1993) 249. [ 7] M. Neubert, preprint SLAC-PUB-5992 (November 1992). [8] D.J. Broadhurst and A.G. Grozin, Phys. Lett. B 267 ( 1991 ) 105. [9] A.F. Falk, H. Georgi, B. Grinstein and M. Wise, Nucl. Phys. B 343 (1990) 1. [ 10] P. Ball, Ph.D. thesis, Heidelberg preprint HD-THEP-92-9 [in German]. [ 11 ] E. Bagan, P. Ball, V.M. Braun and H.G. Dosch, Heidelberg preprint HD-THEP-91-36.
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