Sub-leading logarithmic mass-dependence in heavy-meson form-factors

Volume 257, number 3,4

PHYSICS LETTERS B

28 March 1991

Sub-leading logarithmic mass-dependence in heavy-meson form-factors ¢r Xiangdong Ji and M.J. Musolf Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Received 27 December 1990

We calculate the two-loop anomalous dimension and the first-order coefficient function of weak-interaction currents in the heavy-quark effective field theory. Together with the two-loop QCD fl-function, these quantities form the complete sub-leading logarithmic corrections to the mass-factorization of weak form factors.

The asymptotic freedom of q u a n t u m chromodynamics ( Q C D ) enables one to study physics associated with large m o m e n t u m scales perturbatively. This approach has proven highly successful in the study of deep-inelastic lepton-nucleon scattering, where the large scale is the f o u r - m o m e n t u m transfer squared, Q2. Using the operator-product-expansion, one can factorize matrix elements of the relevant non-local operators into matrix elements of effective local operators times the so-called coefficient functions, C(Q2). Matrix elements o f local operators are much more easily calculated than are those of non-local operators, while the C ( Q 2 ) can be calculated using perturbation theory. Moreover, the coefficient functions contain the dominant effects o f short-distance Q C D physics, typically appearing in the form of large logarithms. With the aid of the renormalization group, one can sum these logarithms to all orders in perturbation theory, and generate a new expansion in the Q C D coupling at the scale o f the physical four-mom e n t u m transfer. A similar approach has recently been taken in studying the physics of heavy mesons composed of quarks with mass mQ >> AQCD, the Q C D confinement scale [ 1 - 3 ] . The simplest case first considered by ¢~ This work is supported in part by funds provided by the US Department of Energy (DOE) under contract # DE-AC0276ER03069.

Voloshin and Shifman [ 1 ] and Politzer and Wise [ 3 ] ( V S P W ) is the matrix element governing the leptonic decay of a pseudo-scalar meson ( M ) consisting of a heavy quark (Q) and a light anti-quark (Cl=

fi, d, s): fpP,, = <0IA~Q I M ( P ) > .

( 1)

Here, A qQ = ~]7~YsQ is the axial current, P~ is the fourm o m e n t u m of the heavy meson, and fp is the decay form factor. VSPW argued that the heavy-quark-mass (mQ) dependence of the form factor is non-analytic in the mQ--,oe limit because the QCD-radiative processes create quarks with virtuality of order of rn~. At one-loop order, for example, Q C D corrections to matrix elements of the form ( 1 ) generate a large logarithm ln(mQ//~) (/~~AQcD) involving two widelyseparated m o m e n t u m scales. VSPW show that leading logarithms of the type c~s(/~)nlnn(mQ//~) can be factored out o f the matrix element into a perturbatively calculable coefficient, C(/t), and summed to all orders in perturbation theory:

=C(/~) + .... (2) The remaining factor < 0IA ~Q 1i~I ( P ) ) defines the matrix element of an effective, heavy-quark current which is well-behaved in the r n Q - ~ limit. Additional terms on the right-hand side ofeq. (2) are sup-

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PHYSICS LETTERS B

pressed relative to the leading term by inverse powers of the heavy quark mass. This "mass-factorization" is physically interesting for two reasons. First, the effective matrix elements (0]A~Q I ]~l ( P ) ) exhibit Isgur-Wise spin and flavor symmetries [4], which imply simple relations between different heavy meson form-factors. Second, because the symmetry-breaking effects associated with finite heavy-quark masses can be calculated perturbatively, heavy-quark decays provide another testing-ground for perturbative QCD. A convenient framework for systematically factorizing out large-scale physics is provided by effective field theory [5]. Precursors to such a theory for heavy-quark systems were introduced by Eichten and Feinberg, who developed a heavy-quark-mass expansion scheme for heavy quarkonium [ 6 ]. Politzer and Wise used an effective theory approach to calculate the anomalous dimension for the weak currents in coordinate space [3]. Eichten and Hill generalized their calculation to m o m e n t u m space [ 7 ]. Recently, Georgi formalized these ideas by constructing a Lorentz covariant effective field theory [ 5 ]. In term of this theory, we write the heavy-quark expansion for any operator 0 containing a heavy-quark field: O ( m Q ) ~ O(mQ =oo, f l ) C ( m Q / f l , + i(=l),p~

O~(fl) )

Oi#( mQ =oo, ]1) 1171Q Ci,a(mQ/,tt , OL([2) ) .

(3)

For example, 0 may be the weak current e.g., ~ to indicate that the matrix elements of the operators on different sides of the equation are taken between different wave functions. Let us illustrate the meaning of this expansion by considering matrix elements of the vector current J~, taken between meson states. The left-hand side (LHS) is calculated with meson wave functions in the full QCD theory. The mass dependence in the current is implicit because it appears only in its matrix elements. Because the anomalous dimension of the vector current is zero, the matrix element between physical states is independent of the renormalization scale, p. On the right-hand side (RHS), the notation mQ-----~ indicates that the heavy-quark field appearing in the operators is an effective field, hQ; consequently, the meson states to be used in the RHS

J~,=(IF,Q. Here, we use

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28 March 1991

are states in the effective theory. The matrix elements of the operators on the RHS are independent of mQ, since we have factored logarithmic dependence into the Cip and inverse powers of the heavyquark mass explicitly. The index i sums over operators with different canonical dimensions and p over different operators of the same dimension. Although the effective operators and coefficient functions are constructed to maintain the identity of eq. (3) at the level of its matrix element, the renormalization properties of the operators in effective theory differ from those in the full theory. The effective current operator Ju(mQ = ~ ), for instance, has a non-zero anomalous dimension and therefore depends on the renormalization point. Because the entire matrix element is p-independent, the/~-dependence of the COt) must exactly cancel that of the effective operators. Consequently, the coefficient functions satisfy the renormalization group equation,

( # ~--~fl-- ~i) C,( mQ/ ]l, ol(ll ) ) = 0 ,

(4)

where 7i is the anomalous dimension of the corresponding operators. To leading logarithmic order, QCD corrections to the Isgur-Wise symmetry relations are multiplicative factors consisting of ratios of large logarithms. Although one expects these logarithms to represent the dominant perturbative correction, higher-order "sub-leading" logarithmic contributions need not be negligible. A part of the sub-leading logarithmic correction to C(mq/#, as(/t)) was studied in ref. [2]. The result, however, does not have a well-defined meaning because it depends on the ultraviolet (UV) subtraction scheme. Only the sum of all subleading logarithmic corrections is independent of any particular subtraction scheme. The full set of spin-dependent effects of this order is well-defined and has been studied in ref. [8]. The goal of this paper is to complete the study of refs. [ 1,2,8] on the sub-leading logarithmic corrections to the effective current's coefficient function. The l/mQ power corrections in eq. (3), which are numerically comparable to the QCD correction we are interested in, have been recently calculated by Falk, Grinstein and Luke [9-11 ]. We begin with the general solution of the renormalization group equation (4);

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PHYSICS LETTERS B

C(mQ/N, O~(N)) = C(1, o~(mQ))

g(I;'lQ) dg/,

(5)

g(i~) wherefl(g) is the normal QCD fl-function. The heavyquark mass mQ appeared in eq. (5) sets the scale below which the effective theory is used. However, quark masses are renormalization-point-dependent in the (MS) scheme. In fact, to the first order in perturbation theory, we have [ 12 ] r~(N) = rh/(ln

(6)

f l / A Q c D ) -}'m/fl° ,

where rh is the invariant mass and 7m= --4. Therefore, we must define mQ in eq. (5). In leading logarithmic order, we take mQ=~h because any other choice brings in difference in C(#) only at higherorder in coupling. To subleading logarithmic order, we take mQ as the solution of fft(mQ)=mQ, which guarantees that as scale N
g2

( g2 "~2

7= 7o 1-~5~2 + 7 1 \ 1 6 9 2 ] + ....

[ fl=-g

g2

(g2)"

/~o1 - ~ ~ +fl, ~

(7)

] + ....

(8)

the two-loop anomalous dimension ~1, the two-loop beta function fll and the first-order coefficient function Cl at # = mQ. The former two are to be calculated in the effective field theory. The first order coefficient function is determined by requiring that matrix elements of the effective current operator equal those of the operator in the full theory at N = mQ. In the effective field theory the mass of the heavy quark is infinite. It cannot be easily created or annihilated unless there is an external source. Hence, we neglect closed heavy-quark loops to leading-order in 1/mQ. A state containing a heavy quark is characterized by its velocity v~, which never changes in radiative processes. Virtual heavy-quark lines in loops remain distinctively quark or antiquark in character depending upon the external line to which they connect. The momentum of the quark is generically written as mQv1,+k ~ (mQ--~ov in the end), where the offshell, residual momentum, k,, is independent of mQ. The Feynman rules for the heavy quark propagator, i/v. k, and the heavy-quark/gluon vertex, -igvut a, can be easily derived for the effective theory as was done in ref. [ 8 ]. Before giving our two-loop results, we summarize the one-loop renormalization of the effective theory in the dimensional regularization and modified minimal-subtraction (MS) scheme. The heavy quark wave function renormalization constant was calculated by Politzer and Wise [ 3 ]

g2

C(1, ol~)= 1 +G ~

28 March 1991

g2

+-..,

(9)

Z2 = 1 + ~

2CFN~,

(12)

where N e = 2 / e + l n 47r-y and CF= (N 2 - 1 )/2N for an SU (N) color group. The quark-gluon vertex renormalization is easily computed

and substituting them into ( 5 ), we obtain C(mQ/N, as(fl) ) = Co(g)(Ogs (mQ))m/2flo

g2 Z, = 1+ ]6~7~2(2CF--CA)N,, where c~(rno) is accurate up to the second order, ol(mQ)=

47r ( f l , lnln(rn~/A2)) floln(m~)/A2) 1- flgln(rn~)/A2 ) . (ll)

We call the corrections to the leading order in (9) sub-leasing logarithmic because they represent a sum to all orders in perturbation theory of terms of the form o~7 ln~- lmQ/a. Eqs. ( 7 ) - (10) imply that in order to determine C(N) to first order in the physical coupling, we need

(13)

where CA=N. Although Z~ and Z2 differ from the corresponding renormalization constants in the full theory, the gluon's wavefunction renormalization Z3 is the same, up to the number of dynamical fermions. Similarly, the one-loop fl-function retains the same structure in the effective theory

I1 --~TvNf= 4 flo =-g-CA 11 -- 2Nr,

(14)

where Tv= ½and Nf is the number of quark flavors with mass less than mo. The reason for this is simple: the color gauge-symmetry holds in the renormalized 411

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Volume 257, n u m b e r 3,4

effective theory and thus the Slavnov-Taylor identities are still valid. According to these identities, the heavy-quark fl-function is the same as the pure gluonic /?-function which, or course, is identical in the full and effective theory. Therefore, we will use the two-loop /?-function calculated in ref [ 13 ]. 34r2A _ 4 ( ~ C A + C F ) T v N r = I O 2 _ ~ N f . fll ~- ~-~-

(o)

(b)

(c)

(d)

(el

(f)

(cj)

(h)

(i)

(15)

The two-loop anomalous dimension of weak currents has three parts, 71 = 7fo, + 7¢ff+ 7vertex,

( 16 )

where 7fu. is the contribution from two-loop wavefunction renormalization of the heavy quark in the full theory, yen-the same quantity in the effective theory, and 7vmex the anomalous dimension of the twoloop irreducible vertex. The first part can be taken from ref. [13] 7ru,, = Cv( L~CA - 2 T v N f - ~Cv ) •

( 17 )

The second part can be obtained by calculating the same two-loop diagrams with the Feynman rules in the effective theory, which are shown in fig. 1 (we have omitted the diagram with three-gluon vertex which has no contribution to the anomalous dimension). The result we obtain is 38

Yell----CF( - 3 - C a q - ~ T F N f ) •

( 18 )

Notice that there is no C~ term due to a cancellation between the one-loop quark self-energy-insertion diagram (a) and the quark-gluon vertex-insertion diagram (b) in fig. 1. All the two-loop irreducible vertex diagrams are shown in fig. 2. To simplify the calculation, we set the light-quark mass and its external momentum zero. The infrared (IR) singularity is regulated by keeping the heavy quark off shell. All the integrals can be per-

Co)

(b]

(c)

Fig. 1. T w o q o o p corrections to the heavy-quark propagator in the effective field theory.

412

Fig. 2. F e y n m a n diagrams for the second-order anomalous dim e n s i o n of weak current operators. The double lines represent the effective heavy-quark propagator.

formed analytically with Feynman parametrizations. The summed contribution of these diagrams is yvmex = C u2( 4 - 3 ~ 8z 2 ) + C F C A ( - - 4 q - 27~2) .

(19)

Adding all three contributions, we obtain the two-loop anomalous dimension ~1 ~___CF CA ( -- ~ -'}-21'~2 ) "}- /,~' ' 2F,E~5 -- "511,8-2"+~CFTFNr ) 254 56__2.20 ~,r ----- - - ~ - - - ~ : t -r y i Y r .

(20)

Numerically, for No= 3, the anomalous dimension is about - 4 2 . The remainingO(c~s) term in eq. (10) - the coefficient function c~ - is determined by requiring that the effective theory reproduce the results of the full theory at # = mQ. In the evaluation of Feynman diagrams in the effective theory, we must use the same IR and UV regulators and the same renormalization scheme (MS) as in the full theory in order for C(#) to be renormalization-independent. One-loop matrix elements of weak currents between free-quark states are shown in fig. 3. They have the general form

Volume 257, number 3,4

PHYSICS LETTERS B

•l =Cl +

28 March 1991 ~'l

Yofll

2/~o

2l~

=_4+4# (a)

+

(b)

Fig. 3. One-loop matrix elements in the full (a) and effective (b) theories.

-381-28~2+30Nf 9(33-2Nf)

= Uq(ZFu + s p i n d e p e n d e n t t e r m s ) UQ,

(21)

where Fu=7 u (7~7s) for vector (axial vector) currents and where Z = 1 at tree-level. The spin-dependent terms in the full theory have the form [ 8 ]

(26)

F o r c h a r m e d mesons, the n u m b e r o f light quarks is three and we get At = - 4 . 7 5 _ + 1.33 ~.

(qlJ~,(0) IQ)

(27)

Finally, let us briefly m e n t i o n an application o f our result to the pseudo-scalar D-meson decay form-factor defined in eq. ( 1 ). According to the expansion in eq. ( 3 ) , we have f p ( m c ) = ( mc)-1/2OLs( mc) ;'°/2~

g2

_+ ~

+ 1 2 ( 1 5 3 - 19Nr) (33_2N02

CF~,

(22) × (1 - 6.08" °q~-nc ) ) M o +... ,

where the plus ( m i n u s ) sign corresponds to a vector (axial vector) current. No such spin-dependent terms a p p e a r at one-loop o r d e r in the effective theory matrix elements, so one must a d d them explicitly in order to satisfy the m a t c h i n g condition. Since these terms are UV-finite, their c o n t r i b u t i o n is unambiguous. The r e n o r m a l i z e d coefficient Z, however, depends on the r e n o r m a l i z a t i o n scheme, In the full theory we have Zfu,,-l=~Cv

[

-l+61n

(mo)] ~

,

(23)

which is r e n o r m a l i z a t i o n - p o i n t independent. In the effective theory, one has Ze~--I=~CF

2+61n

,

(24)

which is quark-mass independent. Matching at p = mQ we obtain the coefficient function c, = C v ( - 3 _ + ~ ) .

(25)

Only when c~ is a d d e d to the terms involving ~ and fl~ in eq. (10) do we obtain a renormalization schemei n d e p e n d e n t result for the sub-leading logarithmic correction to the leading-order factorization. The q u a n t i t y o f interest is

(28)

where Mo is the product o f Co(p) in eq. (10) and the matrix element o f the effective current and is indep e n d e n t o f mo The ellipses denote 1/mQ power-suppressed terms considered in refs. [ 9 - 1 2 ] . I f we take as ( m c ) = 0.3, the second term in the large parentheses is - 0 . 1 4 . Therefore, the subleading logarithmic correction to the leading logarithmicfp is about 14%.

[ 1] M.B. Voloshin and M.A. Shifman, Sov. J. Nucl. Phys. 45 (1987) 292. [2] M.B. Voloshin and M.A. Shifman, Sov. J. Nucl. Phys. 47 (1988) 511. [ 3] H.D. Politzer and M.B. Wise, Phys. Lett. B 206 (1988 ) 681; B208 (1988) 504. [4] N. Isgur and M.B. Wise, Phys. Lett. B 232 (1989) 113; B 237 (1990) 527. [ 5 ] H. Georgi, Phys. Lett. B 240 (1990) 447; Weak interactions and modern particle theory (Benjamin/Cummings, Menlo Park, CA, 1984). [6] E. Eichten and F.L. Feinberg, Phys. Rev. Lett. 43 (1975) 1205; Phys. Rev. D 23 (1975) 2724. [7] E. Eichten and B. Hill, Phys. Lett. B 234 (1990) 511. [8] A.F. Falk, H. Georgi and B. Grinstein, Nucl. Phys. B 343 (1990) 1. [9] A.F. Falk and B. Grinstein, Phys. Lett. B 247 (1990) 406. [ 10] A.F. Falk, B. Grinstein and M.E. Luke, preprint HUTP-90/ A044, to be published. [ I 1] M.E. Luke, preprint HUTP-90/A051, to be published. [12] S. Narison, Phys. Lett. B 197 (1987) 405. [ 13] W.E. Caswell, Phys. Rev. Lett. 33 (1973) 224; D.T.R. Jones, Nucl. Phys. B 75 (1974) 730. 413