VOLUME 80, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 13 APRIL 1998

inimal

sics.

Non-Abelian Dipole Radiation and the Heavy Quark Expansion

Andrzej Czarnecki* and Kirill MelnikovInstitut für Theoretische Teilchenphysik, Universität Karlsruhe, D-76128 Karlsruhe, Germany

Nikolai Uraltsev†

Department of Physics, University of Notre Dame du Lac, Notre Dame, Indiana 46556(Received 9 September 1997)

Dipole radiation in QCD is derived to the second order inas. A powerlike evolution of thespin-singlet heavy quark operators is obtained to the same accuracy. We give anO sa2

s d relationbetween the short-distance low-scale running heavy quark mass and the mass in the modified msubtraction scheme. We discuss the properties of an effective QCD couplingasdd

s sEd which governsthe dipole radiation. We argue that this coupling is advantageous for describing heavy quark phy[S0031-9007(98)05823-2]

PACS numbers: 12.39.Hg, 12.38.Bx

nfisa

the

y

e--

ed

e

-ns

, by

ns

etive

forof

Theoretical description of heavy flavor decays benfits from a strong hierarchy between the mass of thdecaying quark and the typical scale of the strong inteactions,mbscd ¿ LQCD. The current level of precisionrequires an accurate treatment of nonperturbative effeeven in beauty decays. A consistent genuinely QCD-basframework for this is provided by the heavy quark expansion (HQE) which combines the Wilson operator producexpansion (OPE) with the nonrelativistic expansion.allows a simultaneous precise description of both pertubative and nonperturbative effects.

An important class of applications of the HQE aresemileptonic weak transitions betweenb andc quarks, inwhich both initial and final quarks are heavy. The most informative predictions can be made for the transition amptudes in the so-called small velocity (SV) limit [1], whenthe velocities of heavy hadrons in initial and final stateare small. The heavy quark in this case plays a role ofslowly moving source of the color Coulomb field whichaffects the light degrees of freedom in the hadrons. Thphysical picture is formalized in the Wilsonian approacby integrating out high-momentum degrees of freedomfull QCD and resorting to an effective low-energy theory

A peculiarity of the effective theory for heavy fla-vor hadrons is its essentially Minkowskian nature. Anprocess in which the velocity of the heavy quarkQchanges involves gluon radiation with energy and mometum in the whole range up to the quark masses. On tother hand, when the energy loss is small comparedmQ, the radiation off the heavy colored particles is almost a classical effect, and therefore a universal proceindependent description is possible. In the small velocilimit it is the familiar dipole radiation, with some pecu-liarities due to the non-Abelian nature of QCD.

Even though the dipole radiation has the most obvioumanifestation in the SV processes, it is relevant imany other situations, from the zero-recoil transitionto inclusive decay widths. The reason is that the OPensures certain universality of the effects of soft physic

0031-9007y98y80(15)y3189(4)$15.00

e-er-

ctsed-t

Itr-

-li-

sa

ishof.

y

n-heto-ss-ty

snsEs

originating at the momentum scale well belowmQ for allgenuinely short-distance observables.

In this Letter we derive the non-Abelian dipole radiatioto O sa2

s d and use it to obtain the power evolution oa number of spin-singlet heavy quark operators to thorder. In particular, a gauge-invariant relation betweenshort-distance low-scale running heavy quark mass andmodified minimal subtraction scheme (MS) massmQsmQdis given, which is important since the two-loop accuracin heavy quark processes is becoming a standard.

The radiation by a heavy charged particle when its vlocity changes is well known from classical electrodynamics. It is obtained from the Liénard-Wiechert retardpotentialsAms$r , td at $r ! ` and has the form

1v

dIsvddv

­a

p

µ1

j $yjln

1 1 j $yj

1 2 j $yj2 2

∂1v

­23

a

p

$y2

v1 O s $y4d , (1)

wherev is frequency,Isvd intensity, anda is the finestructure constant. We will be interested only in thdipole term~ $y2 in QCD, and do not consider multipoleradiation proportional to higher powers of$y.

In QED the same relation holds, with1yv dIydv giv-ing the probability of radiating soft photon(s) with energy v. Moreover, there are no higher-order correctioin aem providedv ø m. The effects of the photon in-teraction with vacuum fluctuations are also suppressedpowers ofvyme, and the soft radiation in QED is gov-erned byas0d.

Because of gluon self-interaction the dipole radiatioin non-Abelian theory is different. Our main interest liein the domain wherev is large compared toLQCD (thenonperturbative multipole expansion of the color-singlsystems was discussed in [2]). In this regime perturbatcalculations can be performed.

Below we introduce some standard notations usedheavy quarks. Let us consider a process of scatteringa color-singlet “weak” currentJ with momentumq on a

© 1998 The American Physical Society 3189

VOLUME 80, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 13 APRIL 1998

o

e

pees

-

e

-

-

t

,

-

era-

n

e-is

heavy quarkQ in the SV kinematics. For simplicity, theinitial quark is assumed to rest. The initialQ and finalstateQ quarks can have arbitrary masses; however, bomasses must be large, so that the nonrelativistic expanscan be applied. The SV limit$y ­ $qymQ , j $yj ø 1, iskept by adjusting $q appropriately. For simplicity weconsider the case of equal masses,mQ ­ mQ , althoughnothing depends on this assumption. The currentJ musthave a nonvanishing tree level nonrelativistic limit (e.gJ

y0 ­ Qg0Q or J

yS ­ QQ); otherwise it can be arbitrary.

Inclusive processes of the scattering of heavy quarare described by an appropriate structure functioW sq0, $qd which is a sum of all transition probabilitiesinduced by J into final states with momentum$q andenergymQ 1 q0. The optical theorem relates it to thediscontinuity of the forward transition amplitudeT sq0, $qdat physical values ofq0,

Tsq0, $qd ­i

2mQ

Zd4x e2iqxkQjTJsxdJys0djQl,

W sq0, $qd ­ 2 Im T sq0, $q d . (2)

In the heavy quark limit the spin degrees of freedom bcome irrelevant, and we assume averaging over spin sta(also color, for perturbative calculations). The structur

functions have a thresholdqmin0 ­

q$q2 1 m2

Q 2 mQ ømQ $y 2y2 corresponding to elastic transitions. The variable v ­ q0 2 qmin

0 measures the hadron excitation energy in the final state.

The nonrelativistic expansion and the perturbative treament are both justified ifLQCD ø v ø mQ, and this hi-erarchy is assumed in what follows. The nonperturbativaspects will be addressed later. In perturbative calculatiowe can use quark states instead of actual heavy hadroas was tacitly assumed above. The structure functionWtakes the following form in the heavy quark limit:

W sv, $yd ­ N dsvd 12 $y 2

3dsvd

v1 O s $y 4d . (3)

At $y ­ 0 only the elastic peak is present. The dipolradiation is described bydsvd, and N is a (velocity-dependent) factor depending on the current.

Motivated by the dipole radiation in QED, we definethe dipole couplinga

sdds by projecting Eq. (3) on its

second term,

CFa

sdds svdpv

­ lim$y!0

limmQ!`

32 $y 2

Wsv, $ydRv

0 Wsv0, $yd dv0. (4)

Here v is assumed to be positive. The denominatoin the last ratio eliminates the overall normalization othe current J. In this form the heavy quark limitmQ ! ` yields a finite result. The normalization integraincludes the elastic peak, which makes it infrared safearbitrary $y. On the other hand, the exact upper limidoes not matter in the SV kinematics, since it affec

3190

thion

.,

ksn

e-tese

--

t-

ensns,

e

rf

lattts

the ratio only by O s $y2d. In this way, the inelasticSV structure function determines the effective QCDrunning coupling driving the dipole radiation of gluons.(Brodsky suggested calling it “radiation charge.” A moreprecise term could also be a “dipole radiation charge,” tdistinguish from soft radiation in the ultrarelativistic case.)It is a dimensionless function of the ratioLQCDyv.

The OPE and factorization of the infrared effects ensurthat asdd

s svd is universal: it does not depend on thechoice of the weak currentJ or on the ratio of thequark masses. Also, there is no dependence on the tyof the initial heavy hadron as soon as the onset of thquark-hadron duality is passed. This property alwayholds perturbatively. The couplingasdd

s obeys the usualrenormalization group (RG) equation with the standardfirst two terms in theb function. Because of its physicaldefinition, the evolution ofasdd

s is properly defined whenheavy flavor thresholds are passed.

In practice one needs the relation betweenasdds and the

standardaMSs . We calculated it toO sa2

s d using methodsof Ref. [3],

asdds smdp

­aMS

s smdp

1

∑µ53

2 ln 2

∂b0

2

2 CA

µp2

62

1312

∂∏ µas

p

∂2

,

(5)whereCA ­ Nc for the SUsNcd gauge group andb0 ­113 CA 2

23 nf . The term,CA in Eq. (5) represents the gen-

uine (non-BLM) second-order effect; it has a purely nonAbelian origin. The BLM (Brodsky-Lepage-MacKenzie)terms [4] which account for the leading effect ofas run-ning are readily computed to any order in this case (se[5]). The termb

20a3

s and the expression for the case ofmassive flavors are given in [6]. Even in the Abelian theory with light flavors,asdd

s smd deviates from theV -schemecouplingasV d

s smd [4]; their commensurate scales are somewhat different. This is a purely kinematic effect;asdd

s smdincorporates integration over the invariant mass of the lighfermion pair in the interval from0 to m, while in asV d

s thefermion loop enters at a fixed spacelike momentum.

A similar effective coupling for emission of softgluons in the ultrarelativistic casej $yj ! 1 has also beencalculated to two loops [7]; its exact definition, howeveris less direct. It differs fromasdd

s in the second-orderterms, although the difference is not significant.

Construction of the HQE requires a precise definition of the basic objects of the effective theory, suchas the heavy quark mass, kinetic operator, and all othcomposite operators. For consistency, an explicit sepration scalem between the low- and high-momentumdomains is introduced. All operators then depend om. A normalization-point–independent pole mass of theheavy quark, although appearing in purely perturbativcalculations at a given order, cannot be completely defined when nonperturbative effects are addressed. Th

VOLUME 80, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 13 APRIL 1998

,

on

-

y

applies also to the parameterL measuring the differ-ence between the heavy hadron massMHQ

andmQ in theheavy quark limit. Instead, in the OPE one must use tshort-distance massmQsmd andLsmd ­ limmQ!` MHQ 2

mQsmd with LQCD ø m ø mQ [8]. The low-scale short-distance mass can be defined in different ways. Howevusing theMS mass ifm ø mQ is unnatural and should beavoided [9].

A physically appropriate gauge-invariant scheme sugested in [5,9,10] is based on the SV sum rules relatithe moments of the SV structure functions to the locheavy quark operators [11]. The normalization point is itroduced as an upper cutoff in the integral over excitatioenergy. For example, for a heavy hadronHQ one defines

Lsmd ; limmQ!`

fMHQ2 mQsmdg

­ lim$y!0

limmQ!`

2$y 2

Rm

0 vW sv, $yd dvRm

0 W sv, $yd dv,

m2p smd ;

kHQjQsi $Dd2QjHQlm

kHQ jQQjHQlm

­ lim$y!0

limmQ!`

3$y 2

Rm0 v2W sv, $yd dvRm

0 W sv, $yd dv, (6)

etc. These operator relations suggest that the perturbaevolution is described by one and the same functioasdd

s smd of Eq. (4),

dLsmddm

­43

CFasdd

s smdp

,dm2

p smddm2

­ CFasdd

s smdp

.

(7)

The samea2s running of m2

p was obtained in [12] fromthe zero-recoil transitions.

A similar m dependence holds for the renormalizeslope of the Isgur-Wise (IW) function when it is defined i

he

er,

g-ngaln-

n

tiven,

dn

the same physical way [9] based on the Bjorken sum rule

mdf%2smd 2

14 g

dm­

43

2asdds smd3p

. (8)

Because of short-distance effects, the observable transitiform factors do not have a literal heavy quark limit at$y fi

0 but vanish, and require factoring out the perturbativesuppression due to gluon radiation, to yield the effectivem-dependent IW functionjsy2; md. In analogy with Eqs. (6),it is convenient to definejsy2; md for smally as

jsy2; md ­ limmQ!`

Fsydf 1

2p

Rm

0 W sv, $yd dvg1y2,

% 2smd ­ 22djsy2; md

dy2

Éy­0

,

whereF is the form factor, say, of the vector currentQg0QandW is the corresponding structure function. This is thescheme suggested in [9] for which Eq. (8) holds.

The RG equations (7) are proportional toasdds . Hence,

the OPE shows that the first-order running is not renormalized in the Abelian theory without light fermions toall orders in the coupling [5], since in such a theoryasdd

sidentically coincides withas0d.

According to [10], one can calculate perturbatively aphysical (gauge-invariant) low-scale running massmQsmdwith m ø mQ by subtracting, order by order, the infraredpart given by the SV sum rules, from the pole massm

poleQ ,

mQsmd ­ fmpoleQ gpert 2 fLsmdgpert 2

12mQsmd

fm2p smdgpert .

This mass determines the nonrelativistic kinetic energterm in the renormalized heavy quark Hamiltonian. Thetwo-loop relation between thismQsmd and theMS mass is

mQsmd ­ m

Ω1 1

43

assmdp

µ1 2

43

m

m2

m2

2m2

∂

1

µassmd

p

∂2∑K 2

83

1m

m

µ8b0

9X1 1

8p2

92

529

∂1

m2

m2

µb0

3X2 1

p2

32

2318

∂∏æ, (9)

as

re

oningr-

for.

E

rk

where

K ­b0

2

µp2

61

7148

∂1

665144

1p2

18

µ2 ln 2 2

192

∂2

16

z s3d ,

X1 ­ ln2m

m2

83

, X2 ­ ln2m

m2

136

,

and m ­ msmd is the MS mass normalized at the scalem. The O sa2

s d relation between the pole mass and thMS mass [13] was used here. We neglected small ter,m3ym2

Q which can be incorporated in the same way,

emsif

necessary. The next-order BLM correction to Eq. (9) hbeen calculated [6].

The nonperturbative effects in the dipole radiation asuppressed by powers ofLQCDyv. In particular, thestructure functions in Eqs. (2) refer to a heavy hadrand not an isolated heavy quark. The effective couplasdd

s , like any other effective charge [14], acquires powesuppressed nonperturbative contributions; they differ,example, inB and Lb even in the heavy quark limitOne can quantify the nonperturbative effects inasdd

s svd byconstructing its1yv expansion; the methods of the OPare applicable here [6].

The nonperturbative effects inasdds are expressed

through the expectation values of local heavy qua

3191

VOLUME 80, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 13 APRIL 1998

ss,d

d-

y

a-

te,

D

operatorsQOkQ. The first nontrivial operator is thekinetic operatorQsi $Dd2Q, giving, by dimension counting,1yv2 effects. However, it does not contribute toasdd

s .Indeed, we can consider the initial heavy quark movinwith the spacelike momentumj $pj ø v. The structurefunction for such a state is directly related to the onerest, and is modified only in terms, $p 2 $y 4 which can beneglected. Since the kinetic expectation value for sua state is nonzero, the corresponding coefficient functmust vanish.

There is one spin-singlet, so-called Darwin operatOD ­ 2iQDkD0DkQ yielding 1yv3-suppressed terms(for simplicity, we consider the case of aB meson). TheOPE relates its effect [6] to the logarithmic anomaloudimensiongD of OD ,

dDasdds svd

asdds svd

­3pgDsvd8asdd

s svdr

3Dsvdv3

ø 2

µassmdassvd

∂213y2b0 39r3Dsmd

32v3

ø 2

µ0.55 GeV

v

∂3

,

where the sum rule for the Darwin term [9,15,16] waused, and

md

dmr3

Dsmd ­ gDsmdr3Dsmd, gDsmd ø 2

13asdds smd

4p,

r3Dsmd ­

kBjOD jBlm

2MBø

2pas

9f2

BMB

ø 0.1 GeV3 at assmd ø 1 .

We also employed the factorization estimate ofr3D [15];

the anomalous dimensions were calculated in [17]. Wsee that the nonperturbative effects are expected toout quickly at large energy,dnpasdd

s svd , sLQCDyvdk ,k $ 3.

As an effective QCD coupling,asdds has an additional ad-

vantage: it directly measures the strength of the interactiwhereas such popular couplings asasDd

s or asRds determine

the deviationof the corresponding observables, AdlerD-function DsQ2d or Rse1e2 ! hadronsd, from their tree-level value. From this perspective,asdd

s is closer to theso-calledV -scheme couplingasVd

s related to the heavyquark potentialV sRd [4,18,19]. An exact definition ofasV d

sin higher orders is not evident at the moment, howevEmergence of termsak

s ln as with k $ 4 in its relation toaMS

s [18] may indicate thatasV ds is not a genuinely short-

distance quantity.The couplingaMS

s proved to be convenient in multi-loop calculations, but is not always the best expansionrameter [20]. Usingasdd

s may improve the accuracy ofperturbative estimates in heavy quark decays. Sincevelocity of the charm hadrons is typically small inB de-

3192

g

at

chion

or

s

s

edie

on,

er.

pa-

the

cays,asdds determines the gluon bremsstrahlung and thu

sums the lowest-momentum component of the correctionwhere the effects of higher orders in a differently defineas could be large.

To summarize, we derived the non-Abelian dipoleradiation by a nonrelativistic color particle to the seconorder in as and the two-loop relation between the shortdistance low-scale heavy quark massmQsmd suitable for

the OPE andmMSQ smQd. We discussed the properties of

the effective dipole radiation couplingasdds , useful for

heavy quark physics.N. U. is grateful to I. Bigi, S. Brodsky, Yu. Dokshitzer,

A. Mueller, M. Shifman, A. Vainshtein, and M. Voloshinfor discussions. This work was supported in part bthe NSF Grant No. PHY 92-13313 and by BMBF-057KA92P.

*Present address: Physics Department, Brookhaven Ntional Laboratory, Upton, New York 11973.

†Permanent address: Petersburg Nuclear Physics InstituGatchina, 188350 St. Petersburg, Russia.

[1] M. Voloshin and M. Shifman, Yad. Fiz.47, 801 (1988)[Sov. J. Nucl. Phys.47, 511 (1988)].

[2] M. B. Voloshin, Nucl. Phys. B154, 365 (1979);K. Gottfried, Phys. Rev. Lett.40, 598 (1978).

[3] A. Czarnecki and K. Melnikov, Phys. Rev. D56, 7216(1997).

[4] S. J. Brodsky, G. Lepage, and P. Mackenzie, Phys. Rev.28, 228 (1983).

[5] N. G. Uraltsev, Nucl. Phys.B491, 303 (1997).[6] A. Czarnecki, K. Melnikov, and N. Uraltsev (to be

published); see also Report No. TTP97-33.[7] S. Catani and L. Trentadue, Nucl. Phys.B327, 323 (1989).[8] I. Bigi et al., Phys. Rev. D50, 2234 (1994).[9] I. Bigi, M. Shifman, and N. Uraltsev, Annu. Rev. Nucl.

Part. Sci.47, 591 (1997).[10] I. Bigi et al., Phys. Rev. D56, 4017 (1997).[11] I. Bigi et al., Phys. Rev. D52, 196 (1995).[12] A. Czarnecki, K. Melnikov, and N. Uraltsev, Phys. Rev.

D 57, 1769 (1998).[13] N. Grayet al., Z. Phys. C48, 673 (1990).[14] G. Grunberg, Phys. Lett.95B, 70 (1980); Phys. Rev. D

29, 2315 (1984).[15] I. Bigi et al., Int. J. Mod. Phys. A9, 2467 (1994).[16] I. Bigi et al., Phys. Lett. B339, 160 (1994); C.-K. Chow

and D. Pirjol, Phys. Rev. D53, 3998 (1996).[17] M. Voloshin and M. Shifman, Sov. Phys. JETP64, 698

(1986).[18] T. Appelquist, M. Dine, and I. Muzinich, Phys. Lett.69B,

231 (1977); Phys. Rev. D17, 2074 (1978).[19] M. Peter, Phys. Rev. Lett.78, 602 (1997); Nucl. Phys.

B501, 471 (1997).[20] S. J. Brodsky and H. J. Lu, Phys. Rev. D51, 3652 (1995);

hep-ph/9506322; hep-ph/9409462.