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QCD basis for factorization in decays of heavy mesons
Volume 255, number 4
PHYSICS LETTERS B
21 February 1991
QCD basis for factorization in decays of heavy mesons M i c h a e l J. D u g a n l a n d B e n j a m i n G r i n s t e i n 2 Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA Received 8 November 1990; revised manuscript received 13 November 1990
The amplitudes for nonleptonic decays of heavy mesons are notoriously difficult to calculate. Factorization has been used in an attempt to transcribe results from model calculations of semileptonic decay rates of heavy mesons. We propose that factorization holds perturbatively in QCD, provided one looks in the appropriate kinematic regime. This is studied with the use of a novel expansion in which factorization holds in leading order, which includes all orders in the loop expansion. Our results are contrasted with large-N expansion arguments.
1. Introduction The rate for purely h a d r o n i c decays o f heavy mesons is very difficult to compute. While exclusive resonant semileptonic decay rates o f heavy mesons and baryons can by systematically a p p r o x i m a t e d from first principles [ 1-12 ] at one k i n e m a t i c point, no such statement can be m a d e in the purely hadronic case. In the semileptonic case one can use h a d r o n i c models to c o m p u t e these exclusive modes [ 13 ]. In an a t t e m p t to extend these model calculations to the purely h a d r o n i c case [ 14], it has been suggested that the hadronic a m p l i t u d e involved "factorizes". M o r e concretely, for the decay B ~ I ) n one assumes < D x l (9~ IB> ~ ( D I J U l B > ( ~ l J u IO> •
(1.1)
HerejU stands for the V - A current, and the local o p e r a t o r (9, = b-yu( 1 - 7 ~ ) c ayu( 1 - y s ) d
( 1.2 )
is a term in the effective h a m i l t o n i a n for weak interactions at low energies, ~ r r = GF/v/~ Vcb V*d (9~+ .... The a s s u m p t i o n ( I. 1 ) allows one to use the c o m p u t e d B o I 3 matrix elements encountered in the study o f the semileptonic decays. In this p a p e r we introduce a systematic Q C D based expansion for which factorization holds in the leading order. F u r t h e r m o r e , we will argue that the corrections to ( 1.1 ) are under control. We will be able to categorize the hadronic decays for which factorization should hold, and those for which it should not. O f course, factorization m a y hold even for the latter, but for reasons that we cannot explain. At first sight, factorization is a ridiculous idea. First, it does not hold at one loop in p e r t u r b a t i o n theory. That is, the p e r t u r b a t i v e a m p l i t u d e for b--, cod does not factorize. A n d second, the perturbative renormalization p o i n t d e p e n d e n c e o f the c u r r e n t - c u r r e n t o p e r a t o r in ( 1.1 ) is different from that o f the nonlocal p r o d u c t o f currents. F o r one thing, the former mixes u n d e r scaling with a product o f currents that transform as octets under colorSU(3): = b-TayU( 1 - y s ) c
I~TaTu(1 -
ys)d.
(1.3)
E-mail address: [email protected],@huhepl.hepnet or @huhepl.harvard.edu. 2 E-mail address: [email protected], @huhepl.hepnet or @huhepl.harvard.edu. 0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
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Our methods address these two reservations successfully. Factorization holds in the large-Nlimit o f Q C D [ 15 ]. This is hardly an approximation scheme that one would trust in phenomenological applications. For one thing, the corrections are order ½ (times a combinatorial factor measuring the larger number of graphs encountered at subleading order). Even if factorization held empirically only to 30%, it would be hard to muster enough confidence to say that this was expected from the large-N approximation. Another approach to factorization is given by Bjorken [ 16 ]. This is based on ideas of color transparency, first applied to high energy hadronic scattering [ 17 ]. While this approach is intuitive, it is not strictly derived from QCD. It should be clear that our approach is different. We prove a certain property of Green functions in QCD to all orders in the loop expansion. Moreover, our method allows us to calculate logarithmic corrections to factorization (see section 4 below). The main idea presented here is that factorization holds to all orders in perturbation theory in a very specific kinematic limit. Consider the four quark Green function for the current-current operator in ( 1.1 ). We focus on the case where the two light quarks are highly collinear and all quarks are close to on-shell. We then consider the limit in which the two heavy quark masses become arbitrarily large, in a fixed ratio. We then expand the Green function in inverse powers of the heavy masses and the large energy transferred to the light quark pair. We claim the leading term exhibits factorization. Our choice of momenta is expected to be the dominant contribution to the physical hadronic decay. Our result holds also for Green functions with additional light degrees of freedom, provided these are also collinear and carry nonnegligible energy. In order to prove this statement, we will replace the propagator of the energetic collinear light quarks by one that retains only the leading term in an expansion in inverse powers of the large energy. The resulting "large energy effective theory" is much simpler than the original one. If the momentum transferred to the light quarks is p, then the coupling to gluons is only through the combination p.A. The proof of perturbative factorization of Green functions will then follow trivially if one works in light-cone gauge, p.A = 0. We can prove that Green functions factorize in the kinematic region described above. In order to relate this result to physical amplitudes we must make additional assumptions. Namely, that the physical amplitude is dominated by collinear quarks and gluons with nonvanishing momentum fraction. While we feel this is a safe assumption, a categoric statement requires understanding of nonperturbative effects that are beyond the scope of our effort here. Throughout this work, whenever we relate our results for Green functions to physical amplitudes, these assumptions are implicitly understood (although we will remind the reader, from time to time, that this is the case). The paper is organized as follows. In section 2 we will introduce the effective theory. In section 3 we apply it to prove factorization of Green functions. We also discuss there the effects of the spectator quark and why the octet-octet operator is suppressed. In section 4 we apply the machinery just developed to compute the leading logarithmic corrections to factorization, and summed them up by standard renormalization group techniques. We close in section 5 by making several new predictions that contrast this method to large-N techniques, and summarizing its limitations.
2. Large energy effective theory The high mass limit of the Green functions can be obtained through the use of the heavy quark effective theory ( H Q E T ) . In the case of hadronic decays the momentum p of the light quarks also scales with the large masses. Therefore, the H Q E T does not make explicit the leading large mass dependence. This is easily remedied by constructing a large energy effective theory (LEET) for the light quarks. The momenta of the b and c quarks are, respectively, Pb = mbV+ kb and Pc = mcv' + kc. We take the momenta of the light quarks to be xp + k and ( 1 - x ) p - k , 0 < x < 1. Thus, the energy transferred to the light quark system is E = v.p = (m 2 _ m2 )/2mb. Note that E scales with the heavy masses, while p 2 must remain fixed, p2 ~ 0. Also, 584
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the "residual" momentum k does not scale with the heavy masses. It will be useful to write p = E n , where n is a fixed null vector, with v. n = 1. The LEET is constructed by replacing the light quark propagator through x/~+~
0 1
(2.1)
(xp+k) z ~ 2 n.k"
This propagator is to be used only on the light quarks that propagate out of the current-current operator. Notice that the propagator is independent of E. One can prove, using the methods of ref. [ 5 ], that the replacement in (2.1) will give the leading term of Green functions in an expansion in inverse powers of E, up to logarithmic corrections that can be determined by renormalization group arguments. There is a concise way of writing the Feynman rules of the LEET. The propagator is i n.k '
(2.2)
the quark-gluon-quark vertex is -igsTan ~ ,
(2.3)
and one associates a factor of ½# with each quark. The LEET can be expressed as a lagrangian for the light quarks which carry the large momentum, ~QgLEET ~-- i~utn"Dq/•
(2.4)
In addition, the LEET quark field ~ satisfies the condition #q/=0.
(2.5)
It is apparent that there is an independent SU(2 ) internal symmetry for each of the two light LEET quarks. One can covariantize the theory in the manner of ref. [ 6 ] by summing over null vectors n (and labeling the spinors ~u,). We see that the LEET is very similar to the HQET. The main distinction is that the vector n in the former is null, while v in the latter is time-like and normalized to unity.
3. Factorization
To prove that factorization holds, one simply needs to consider the light-cone gauge n.A = 0. It is seen that in this gauge the LEET quarks decouple and factorization trivially holds separately for ~9~and (92. This proves that factorization holds for Green functions in the kinematic region of interest. To prove that it holds for the amplitude in eq. ( 1.1 ) we will draw from our assumption on the behaviour of the hadronic wave function, namely, that it is dominated by collinear quarks and gluons with nonvanishing momentum fraction. One can neglect the effects of the octet-octet operator in this leading order since, once factorized, the octet current has no overlap with the physical hadronic state X. Thus, the matrix element of ~ are suppressed by powers of E. For example, graphs with a gluon emitted from either heavy quark and recoiling with the LEET quarks will carry an additional factor of 1 / E if the gluon carries nonnegligible energy. Similarly, it is straightforward to argue that the spectator light quark ends up in the D meson. Every graph where it goes into the state X is power suppressed. One simply needs to follow the large energy flow through the graph and make every light degree of freedom that carries large energy a LEET degree of freedom. The choice of light-cone gauge might seem artificial. How then, one may ask, do the quarks hadronize into a state X? The situation is similar to the choice of temporal gauge v.A = 0 in the HQET. In the latter case all 585
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anomalous dimensions can be reproduced in temporal gauge and sense can be made of the hadronic state being forced to be a heavy quark surrounded by light "stuff" [ 18 ]. This argument successfully addresses the two reservations drawn from perturbative reasoning described in the introduction. The point is that they do not apply in the kinematic regime we are concerned with here. Of course, we were able to prove this in a particular gauge, and the result may not be obvious in other gauges. In particular, Green functions may not display factorization, but the violations to factorization must be gauge artifacts that disappear in going to a physical amplitude. At this point one could naively say that factorization holds as given in eq. ( 1.1 ) up to terms suppressed by powers of E. This is not quite right, as there is additional anomalous dependence on E in the leading term. We now turn our attention to the computation of this correction.
4. The logarithmic corrections We will calculate the anomalous dependence on heavy masses and E that is missing in eq. ( 1.1 ) in the leading logarithmic approximation. This is not expected to give an accurate result, since the separation of scales in the real world is not very large. Nevertheless, it is the first term in a consistent expansion. Moreover, only the subleading logs are expected to compete with the leading logs. The strong coupling constant runs substantially between mb and me, and the sub-sub-leading logs are expected to really be suppressed relative to the previous ones. Even if in the real world all three of the scales rob, rnc and E = (m 2 _ m 2 )~2rob are close, it is instructive to consider two possible limiting regions of parameter space: (i) mb ~ mc >>E, and (ii) m b ~ E >> mc. In both cases we start the construction of the HQLEET by "integrating in" the heavy b quark. In case (i) we then proceed to integrate in the heavy c quark and finally the LEET quarks, while in case (ii) we first integrate in the LEET quarks and then the heavy c quark. The procedure is standard. We need the anomalous dimensions for the operators (9~in (1.2) and (1.3). We have done this using an MS scheme in dimensional regularization (in Feynman gauge). For the theories with four quarks, one heavy and three quarks and two heavies and two quarks these are, respectively
g2 Y= ~5~2 (_04/3
-62),
g2 [ 2 Y= ~5~2\ _ 2 / 3
-
(4.1)
33)
and
g2 ( ( 8 / 3 ) [ 1 - v ' . v r ( v ' . v ) ] 7= ~ 2
0 2 5 / 6 + ( 1/3)v' .vr( v' .v)
0
)
(4.2)
"
Here
1
r(x) = f l U 2 5
ln(x+~)
.
(4.3)
For the theories with two LEET quarks there is no mixing, as is seen readily in light-cone gauge n.A = 0. We are then interested in the anomalous dimension of the singlet-singlet operator ~0~only. for the theories with one heavy, one light and two LEET quarks, and with two heavies and two LEET quarks, these are, respectively, g2 r = ~--~z2 and
gZ 8 7= ~--~z~ [ 1 - v ' . v r ( v ' . v ) ]
.
(4.4)
We have done the computation both in light-cone and in Feynman gauges. The results agree, and they correspond to the anomalous dimension of the heavy-light and heavy-heavy current, respectively. 586
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The matrix elements of the octet-octet operator are 1/E suppressed. It seems inconsistent to compute their anomalous dimensions without regard to 1/E corrections to the formalism of section 2. Indeed, if one attempts the calculation without these further corrections, one discovers unexpected infrared divergences [involving, e.g., In (E2/p 2) ]. Resolution of this issue must await the development of the LEET at order 1/E. It is straightforward to integrate the renormalization group equation to obtain the leading-log effective hamiltonian. Using eqs. ( 4 . 1 ) - ( 4 . 4 ) in case (i) gives
[3 (I5~1 ~ f f l B ) ~
(as(Mw)~- - 1 2 / 2(ots(mb) ~ -- 12/25 3, \ Ors(rob) ]
\ot,s(mc)]
, -3/25 ] 2(oq(Mw).~ 6/23(Ots(mb)~ + 3 \ Ors(rob) ] \a'~(mc)} J
aL GF \ a"(/2) ] ~ V~bV*"d(f)IJUlB)(nlJ~'IO) ' it
x(Ot:'(m¢)~
(4.5)
while in case (ii)
L31-1(OLs(MW)~-12/23(Olls(mb)~12/25 2(as(Mw)~ 6/23(OLs(mb)~ ,' --3/25]
(Dnl~fflB)~
\as(mb).]
\ ~--~-~}
+ 3\as(rob) ]
GF Xf ~ot's(E) ) "~'~'fot"(rn~)'¢ t ~ ) t"-~ VcbV:d(E)lf'lB>
.
\ a's(E) ]
J (4.6)
Here ot,, a', and a ~' refer to the strong coupling constant appropriate for a theory with five, four and three flavors of quarks, and the anomalous dimensions appropriate for a heavy-light and a heavy-heavy current are [7] ai=-6
and
aL=~7[v"vr(v"v)-- 1] .
(4.7)
Notice that the explicit dependence on the scale/2 on the right-hand side of (4.5), (4.6) cancels against the Isgur-Wise function (i.e., the B--,I3 matrix element). As stated earlier, the real world b and c quarks have masses somewhat in between cases (i) and (ii), and is probably closer to the latter than to the former. To get an idea of the size of the correction factor in eq. (4.6), we compute it at/2=m¢. This choice leaves out the last factor (i.e., [ a " (mc)/a~ (/2)]aL), which in any case belongs with the Isgur-Wise function. Using mb = 5.0 GeV, me= 1.8 GeV and A~4~D=230 MeV this factor is 1.15, as opposed to a factor of 1.03 if the running below mb is neglected.
5. Discussion
Our method makes a number of predictions which distinguish it from other approaches to factorization. First, our method suggests that the amplitude for processes that involve the decay of a heavy meson into two light mesons are suppressed relative to that for a heavy into a heavy plus a light meson. Indeed, the need to kick the spectator quark to high energy makes the leading contribution already suppressed by 1/E. We have not proved that factorization holds in these processes. In our method factorization only holds at zeroth order in 1/E. Our method also suggests that factorization holds for processes that involve a heavy meson and multiple light mesons in the final state, provided the light mesons are all highly collinear and energetic. In this limit the amplitude for vacuum to many pions is computable, say, using chiral lagrangian techniques (provided the relative m o m e n t u m is smaller than the mass of the p), or could be extracted from experiment, say, from x decays. This is to be contrasted with predictions from the large-N limit, where factorization holds in the whole kinematic range. We expect deviations from factorization as the final state pions become acollinear. These issues can be studied experimentally. Finally, we point out that our methods apply straightforwardly to the nonleptonic decays of heavy baryons. 587
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For example, one could study the process Ab-~Acn. The new hadronic matrix element [ 11,12] could be extracted from the corresponding semileptonic decay Ab-}Acev. Much work remains to be done. The validity of the LEET should be established. It seems interesting to explore the consequences of the enlarged symmetry group of the LEET. It is most important to understand the nature of the subleading 1/E corrections. Our implicit assumption that the wavefunction for the final state hadron is not singularly peaked at x = 0 or l must be put on a firm basis. And the numerical correction to factorization, as in eqs. (4.5), (4.6), needs to be computed in sub-leading-log order for it to be reliable.
Acknowledgement We are grateful to Adam Falk, Howard Georgi and Michael Luke for their healthful skepticism. B.G. is grateful to the Milton Fund of the Harvard Medical School and the Alfred P. Sloan Foundation for partial support. This work was supported in part by NSF grant PHY-87-14654.
References [ 1 ] M.B. Voloshin and M.A. Shifman, Yad. Fiz. 45 (1987) 463 [Sov. J. Nucl. Phys. 45 (1987) 292]. [ 2 ] H.D. Politzer and M.B. Wise, Phys. Lett. B 206 ( 1988 ) 681; B 208 ( 1988 ) 504. [ 3 ] N. Isgur and M.B. Wise, Phys. Lett. B 232 (1989) 113; B 237 (1990) 527. [4] E. Eichten and B. Hill, Phys. Lett. B 234 (1990) 511. [5] B. Grinstein, Nucl. Phys. B 339 (1990) 253. [6] H. Georgi, Phys. Lett. B 240 (1990) 447. [7] A.F. Falk, H. Georgi, B. Grinstein and M.B. Wise, Nucl. Phys. B 343 (1990) 1. [ 8 ] E. Eichten and F.L. Feinberg, Phys. Rev. Lett. 43 ( 1979 ) 1205; Phys. Rev. D 23 ( 1981 ) 2724; E. Eichten, in: Field theory on the lattice, Proc. Intern. Symp. (Seillac, France, 1987), eds. A. Billoire et al., Nucl. Phys. B (Proc. Suppl.) 4 (1988) 170. [9] G.P. Lepage and B.A. Thacker, in: Field theory on the lattice, Proc. Intern. Symp. (Seillac, France, 1987), eds. A. Billoire et al., Nucl. Phys. B (Proc. Suppl.) 4 (1988) 199; W.E. Caswell and G.P. Lepage, Phys. Lett. B 167 (1986) 437. [ 10] A. Falk and B. Grinstein, Phys. Lett. B 247 (1990) 406. [ 11 ] N. Isgur and M.B. Wise, CALT-68-1626 (1990), to appear in Nucl. Phys. B.; H. Georgi, HUTP-90/A046 (1990), to appear in Nucl. Phys. B.; T. Mannel, W. Roberts and Z. Ryzak, preprint HUTP-90/A047, DESY-90-101. [ 12] M. Luke, preprint HUTP-90/A052.; H. Georgi, B. Grinstein and M.B. Wise, preprint HUTP-90/A052 and CALT-68-1664. [ 13 ] B. Grinstein, M.B. Wise and N. Isgur, Phys. Rev. Lett. 56 ( 1986 ) 298; T. Altomari and L. Wolfenstein, Phys. Rev. Lett. 58 (1987) 1583; N. Isgur, D. Scora, B. Grinstein and M.B. Wise, Phys. Rev. D 39 (1989) 799; M. Wirbel, B. Stech and M. Bauer, Z. Phys. C 29 (1985) 637. [ 14] M. Bauer, B. Stech and M. Wirbel, Z. Phys. C 34 (1987) 103. [ 15 ] G. 't Hooft, Nucl. Phys. B 75 (1974) 461; M. Fukugita, T. Inami, N. Sakai and S. Yazaki, Phys. Lett. B 72 (1977) 237. [ 16] J.D. Bjorken, Nucl. Phys. B (Proc. Suppl.) 11 (1989) 325. [ 17] S. Brodsky and A. Mueller, Phys. Lett. B 206 (1988) 685, and references therein. [ 18 ] M. Dugan, B. Grinstein and L. Randall, in preparation.
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QCD basis for factorization in decays of heavy mesons M i c h a e l J. D u g a n l a n d B e n j a m i n G r i n s t e i n 2 Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA Received 8 November 1990; revised manuscript received 13 November 1990
The amplitudes for nonleptonic decays of heavy mesons are notoriously difficult to calculate. Factorization has been used in an attempt to transcribe results from model calculations of semileptonic decay rates of heavy mesons. We propose that factorization holds perturbatively in QCD, provided one looks in the appropriate kinematic regime. This is studied with the use of a novel expansion in which factorization holds in leading order, which includes all orders in the loop expansion. Our results are contrasted with large-N expansion arguments.
1. Introduction The rate for purely h a d r o n i c decays o f heavy mesons is very difficult to compute. While exclusive resonant semileptonic decay rates o f heavy mesons and baryons can by systematically a p p r o x i m a t e d from first principles [ 1-12 ] at one k i n e m a t i c point, no such statement can be m a d e in the purely hadronic case. In the semileptonic case one can use h a d r o n i c models to c o m p u t e these exclusive modes [ 13 ]. In an a t t e m p t to extend these model calculations to the purely h a d r o n i c case [ 14], it has been suggested that the hadronic a m p l i t u d e involved "factorizes". M o r e concretely, for the decay B ~ I ) n one assumes < D x l (9~ IB> ~ ( D I J U l B > ( ~ l J u IO> •
(1.1)
HerejU stands for the V - A current, and the local o p e r a t o r (9, = b-yu( 1 - 7 ~ ) c ayu( 1 - y s ) d
( 1.2 )
is a term in the effective h a m i l t o n i a n for weak interactions at low energies, ~ r r = GF/v/~ Vcb V*d (9~+ .... The a s s u m p t i o n ( I. 1 ) allows one to use the c o m p u t e d B o I 3 matrix elements encountered in the study o f the semileptonic decays. In this p a p e r we introduce a systematic Q C D based expansion for which factorization holds in the leading order. F u r t h e r m o r e , we will argue that the corrections to ( 1.1 ) are under control. We will be able to categorize the hadronic decays for which factorization should hold, and those for which it should not. O f course, factorization m a y hold even for the latter, but for reasons that we cannot explain. At first sight, factorization is a ridiculous idea. First, it does not hold at one loop in p e r t u r b a t i o n theory. That is, the p e r t u r b a t i v e a m p l i t u d e for b--, cod does not factorize. A n d second, the perturbative renormalization p o i n t d e p e n d e n c e o f the c u r r e n t - c u r r e n t o p e r a t o r in ( 1.1 ) is different from that o f the nonlocal p r o d u c t o f currents. F o r one thing, the former mixes u n d e r scaling with a product o f currents that transform as octets under colorSU(3): = b-TayU( 1 - y s ) c
I~TaTu(1 -
ys)d.
(1.3)
E-mail address: [email protected],@huhepl.hepnet or @huhepl.harvard.edu. 2 E-mail address: [email protected], @huhepl.hepnet or @huhepl.harvard.edu. 0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
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Our methods address these two reservations successfully. Factorization holds in the large-Nlimit o f Q C D [ 15 ]. This is hardly an approximation scheme that one would trust in phenomenological applications. For one thing, the corrections are order ½ (times a combinatorial factor measuring the larger number of graphs encountered at subleading order). Even if factorization held empirically only to 30%, it would be hard to muster enough confidence to say that this was expected from the large-N approximation. Another approach to factorization is given by Bjorken [ 16 ]. This is based on ideas of color transparency, first applied to high energy hadronic scattering [ 17 ]. While this approach is intuitive, it is not strictly derived from QCD. It should be clear that our approach is different. We prove a certain property of Green functions in QCD to all orders in the loop expansion. Moreover, our method allows us to calculate logarithmic corrections to factorization (see section 4 below). The main idea presented here is that factorization holds to all orders in perturbation theory in a very specific kinematic limit. Consider the four quark Green function for the current-current operator in ( 1.1 ). We focus on the case where the two light quarks are highly collinear and all quarks are close to on-shell. We then consider the limit in which the two heavy quark masses become arbitrarily large, in a fixed ratio. We then expand the Green function in inverse powers of the heavy masses and the large energy transferred to the light quark pair. We claim the leading term exhibits factorization. Our choice of momenta is expected to be the dominant contribution to the physical hadronic decay. Our result holds also for Green functions with additional light degrees of freedom, provided these are also collinear and carry nonnegligible energy. In order to prove this statement, we will replace the propagator of the energetic collinear light quarks by one that retains only the leading term in an expansion in inverse powers of the large energy. The resulting "large energy effective theory" is much simpler than the original one. If the momentum transferred to the light quarks is p, then the coupling to gluons is only through the combination p.A. The proof of perturbative factorization of Green functions will then follow trivially if one works in light-cone gauge, p.A = 0. We can prove that Green functions factorize in the kinematic region described above. In order to relate this result to physical amplitudes we must make additional assumptions. Namely, that the physical amplitude is dominated by collinear quarks and gluons with nonvanishing momentum fraction. While we feel this is a safe assumption, a categoric statement requires understanding of nonperturbative effects that are beyond the scope of our effort here. Throughout this work, whenever we relate our results for Green functions to physical amplitudes, these assumptions are implicitly understood (although we will remind the reader, from time to time, that this is the case). The paper is organized as follows. In section 2 we will introduce the effective theory. In section 3 we apply it to prove factorization of Green functions. We also discuss there the effects of the spectator quark and why the octet-octet operator is suppressed. In section 4 we apply the machinery just developed to compute the leading logarithmic corrections to factorization, and summed them up by standard renormalization group techniques. We close in section 5 by making several new predictions that contrast this method to large-N techniques, and summarizing its limitations.
2. Large energy effective theory The high mass limit of the Green functions can be obtained through the use of the heavy quark effective theory ( H Q E T ) . In the case of hadronic decays the momentum p of the light quarks also scales with the large masses. Therefore, the H Q E T does not make explicit the leading large mass dependence. This is easily remedied by constructing a large energy effective theory (LEET) for the light quarks. The momenta of the b and c quarks are, respectively, Pb = mbV+ kb and Pc = mcv' + kc. We take the momenta of the light quarks to be xp + k and ( 1 - x ) p - k , 0 < x < 1. Thus, the energy transferred to the light quark system is E = v.p = (m 2 _ m2 )/2mb. Note that E scales with the heavy masses, while p 2 must remain fixed, p2 ~ 0. Also, 584
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the "residual" momentum k does not scale with the heavy masses. It will be useful to write p = E n , where n is a fixed null vector, with v. n = 1. The LEET is constructed by replacing the light quark propagator through x/~+~
0 1
(2.1)
(xp+k) z ~ 2 n.k"
This propagator is to be used only on the light quarks that propagate out of the current-current operator. Notice that the propagator is independent of E. One can prove, using the methods of ref. [ 5 ], that the replacement in (2.1) will give the leading term of Green functions in an expansion in inverse powers of E, up to logarithmic corrections that can be determined by renormalization group arguments. There is a concise way of writing the Feynman rules of the LEET. The propagator is i n.k '
(2.2)
the quark-gluon-quark vertex is -igsTan ~ ,
(2.3)
and one associates a factor of ½# with each quark. The LEET can be expressed as a lagrangian for the light quarks which carry the large momentum, ~QgLEET ~-- i~utn"Dq/•
(2.4)
In addition, the LEET quark field ~ satisfies the condition #q/=0.
(2.5)
It is apparent that there is an independent SU(2 ) internal symmetry for each of the two light LEET quarks. One can covariantize the theory in the manner of ref. [ 6 ] by summing over null vectors n (and labeling the spinors ~u,). We see that the LEET is very similar to the HQET. The main distinction is that the vector n in the former is null, while v in the latter is time-like and normalized to unity.
3. Factorization
To prove that factorization holds, one simply needs to consider the light-cone gauge n.A = 0. It is seen that in this gauge the LEET quarks decouple and factorization trivially holds separately for ~9~and (92. This proves that factorization holds for Green functions in the kinematic region of interest. To prove that it holds for the amplitude in eq. ( 1.1 ) we will draw from our assumption on the behaviour of the hadronic wave function, namely, that it is dominated by collinear quarks and gluons with nonvanishing momentum fraction. One can neglect the effects of the octet-octet operator in this leading order since, once factorized, the octet current has no overlap with the physical hadronic state X. Thus, the matrix element of ~ are suppressed by powers of E. For example, graphs with a gluon emitted from either heavy quark and recoiling with the LEET quarks will carry an additional factor of 1 / E if the gluon carries nonnegligible energy. Similarly, it is straightforward to argue that the spectator light quark ends up in the D meson. Every graph where it goes into the state X is power suppressed. One simply needs to follow the large energy flow through the graph and make every light degree of freedom that carries large energy a LEET degree of freedom. The choice of light-cone gauge might seem artificial. How then, one may ask, do the quarks hadronize into a state X? The situation is similar to the choice of temporal gauge v.A = 0 in the HQET. In the latter case all 585
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anomalous dimensions can be reproduced in temporal gauge and sense can be made of the hadronic state being forced to be a heavy quark surrounded by light "stuff" [ 18 ]. This argument successfully addresses the two reservations drawn from perturbative reasoning described in the introduction. The point is that they do not apply in the kinematic regime we are concerned with here. Of course, we were able to prove this in a particular gauge, and the result may not be obvious in other gauges. In particular, Green functions may not display factorization, but the violations to factorization must be gauge artifacts that disappear in going to a physical amplitude. At this point one could naively say that factorization holds as given in eq. ( 1.1 ) up to terms suppressed by powers of E. This is not quite right, as there is additional anomalous dependence on E in the leading term. We now turn our attention to the computation of this correction.
4. The logarithmic corrections We will calculate the anomalous dependence on heavy masses and E that is missing in eq. ( 1.1 ) in the leading logarithmic approximation. This is not expected to give an accurate result, since the separation of scales in the real world is not very large. Nevertheless, it is the first term in a consistent expansion. Moreover, only the subleading logs are expected to compete with the leading logs. The strong coupling constant runs substantially between mb and me, and the sub-sub-leading logs are expected to really be suppressed relative to the previous ones. Even if in the real world all three of the scales rob, rnc and E = (m 2 _ m 2 )~2rob are close, it is instructive to consider two possible limiting regions of parameter space: (i) mb ~ mc >>E, and (ii) m b ~ E >> mc. In both cases we start the construction of the HQLEET by "integrating in" the heavy b quark. In case (i) we then proceed to integrate in the heavy c quark and finally the LEET quarks, while in case (ii) we first integrate in the LEET quarks and then the heavy c quark. The procedure is standard. We need the anomalous dimensions for the operators (9~in (1.2) and (1.3). We have done this using an MS scheme in dimensional regularization (in Feynman gauge). For the theories with four quarks, one heavy and three quarks and two heavies and two quarks these are, respectively
g2 Y= ~5~2 (_04/3
-62),
g2 [ 2 Y= ~5~2\ _ 2 / 3
-
(4.1)
33)
and
g2 ( ( 8 / 3 ) [ 1 - v ' . v r ( v ' . v ) ] 7= ~ 2
0 2 5 / 6 + ( 1/3)v' .vr( v' .v)
0
)
(4.2)
"
Here
1
r(x) = f l U 2 5
ln(x+~)
.
(4.3)
For the theories with two LEET quarks there is no mixing, as is seen readily in light-cone gauge n.A = 0. We are then interested in the anomalous dimension of the singlet-singlet operator ~0~only. for the theories with one heavy, one light and two LEET quarks, and with two heavies and two LEET quarks, these are, respectively, g2 r = ~--~z2 and
gZ 8 7= ~--~z~ [ 1 - v ' . v r ( v ' . v ) ]
.
(4.4)
We have done the computation both in light-cone and in Feynman gauges. The results agree, and they correspond to the anomalous dimension of the heavy-light and heavy-heavy current, respectively. 586
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The matrix elements of the octet-octet operator are 1/E suppressed. It seems inconsistent to compute their anomalous dimensions without regard to 1/E corrections to the formalism of section 2. Indeed, if one attempts the calculation without these further corrections, one discovers unexpected infrared divergences [involving, e.g., In (E2/p 2) ]. Resolution of this issue must await the development of the LEET at order 1/E. It is straightforward to integrate the renormalization group equation to obtain the leading-log effective hamiltonian. Using eqs. ( 4 . 1 ) - ( 4 . 4 ) in case (i) gives
[3 (I5~1 ~ f f l B ) ~
(as(Mw)~- - 1 2 / 2(ots(mb) ~ -- 12/25 3, \ Ors(rob) ]
\ot,s(mc)]
, -3/25 ] 2(oq(Mw).~ 6/23(Ots(mb)~ + 3 \ Ors(rob) ] \a'~(mc)} J
aL GF \ a"(/2) ] ~ V~bV*"d(f)IJUlB)(nlJ~'IO) ' it
x(Ot:'(m¢)~
(4.5)
while in case (ii)
L31-1(OLs(MW)~-12/23(Olls(mb)~12/25 2(as(Mw)~ 6/23(OLs(mb)~ ,' --3/25]
(Dnl~fflB)~
\as(mb).]
\ ~--~-~}
+ 3\as(rob) ]
GF Xf ~ot's(E) ) "~'~'fot"(rn~)'¢ t ~ ) t"-~ VcbV:d(E)lf'lB>
.
\ a's(E) ]
J (4.6)
Here ot,, a', and a ~' refer to the strong coupling constant appropriate for a theory with five, four and three flavors of quarks, and the anomalous dimensions appropriate for a heavy-light and a heavy-heavy current are [7] ai=-6
and
aL=~7[v"vr(v"v)-- 1] .
(4.7)
Notice that the explicit dependence on the scale/2 on the right-hand side of (4.5), (4.6) cancels against the Isgur-Wise function (i.e., the B--,I3 matrix element). As stated earlier, the real world b and c quarks have masses somewhat in between cases (i) and (ii), and is probably closer to the latter than to the former. To get an idea of the size of the correction factor in eq. (4.6), we compute it at/2=m¢. This choice leaves out the last factor (i.e., [ a " (mc)/a~ (/2)]aL), which in any case belongs with the Isgur-Wise function. Using mb = 5.0 GeV, me= 1.8 GeV and A~4~D=230 MeV this factor is 1.15, as opposed to a factor of 1.03 if the running below mb is neglected.
5. Discussion
Our method makes a number of predictions which distinguish it from other approaches to factorization. First, our method suggests that the amplitude for processes that involve the decay of a heavy meson into two light mesons are suppressed relative to that for a heavy into a heavy plus a light meson. Indeed, the need to kick the spectator quark to high energy makes the leading contribution already suppressed by 1/E. We have not proved that factorization holds in these processes. In our method factorization only holds at zeroth order in 1/E. Our method also suggests that factorization holds for processes that involve a heavy meson and multiple light mesons in the final state, provided the light mesons are all highly collinear and energetic. In this limit the amplitude for vacuum to many pions is computable, say, using chiral lagrangian techniques (provided the relative m o m e n t u m is smaller than the mass of the p), or could be extracted from experiment, say, from x decays. This is to be contrasted with predictions from the large-N limit, where factorization holds in the whole kinematic range. We expect deviations from factorization as the final state pions become acollinear. These issues can be studied experimentally. Finally, we point out that our methods apply straightforwardly to the nonleptonic decays of heavy baryons. 587
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For example, one could study the process Ab-~Acn. The new hadronic matrix element [ 11,12] could be extracted from the corresponding semileptonic decay Ab-}Acev. Much work remains to be done. The validity of the LEET should be established. It seems interesting to explore the consequences of the enlarged symmetry group of the LEET. It is most important to understand the nature of the subleading 1/E corrections. Our implicit assumption that the wavefunction for the final state hadron is not singularly peaked at x = 0 or l must be put on a firm basis. And the numerical correction to factorization, as in eqs. (4.5), (4.6), needs to be computed in sub-leading-log order for it to be reliable.
Acknowledgement We are grateful to Adam Falk, Howard Georgi and Michael Luke for their healthful skepticism. B.G. is grateful to the Milton Fund of the Harvard Medical School and the Alfred P. Sloan Foundation for partial support. This work was supported in part by NSF grant PHY-87-14654.
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