Two-Loop Correction to the Leptonic Decay of Quarkonium

Home , Quarkonium

CERN-TH/97-353

hep-ph/9712302

December 1997

Two-lo op Correction to the Leptonic Decay

of Quarkonium

M. Beneke and A. Signer

Theory Division, CERN, CH-1211 Geneva 23

and

V.A. Smirnov

Nuclear Physics Institute, Moscow State University,

119889 Moscow, Russia

Abstract

Applying asymptotic expansions at threshold, we compute the two-lo op QCD cor-

rection to the short-distance co ecient that governs the leptonic decay ! l l

of a S -wave quarkonium state and discuss its impact on the relation b etween the

quarkonium non-relativistic wave function at the origin and the quarkonium decay

constant in full QCD.

PACS Nos.: 13.20.Gd, 12.38.Bx

Quarkonium decays played an imp ortant role in establishing Quantum Chromo dynamics

(QCD) as a weakly interacting theory at short distances. Calculations of heavy quarko-

nium decays usually pro ceed under the assumption that the heavy quark-antiquark

b ound state is non-relativistic and that the decay amplitude factorizes into the b ound

state wave function at the origin and a short-distance quark-antiquark annihilation am-

plitude. One can then explain the small width of the S -wave spin-triplet charmonium

state J= , b ecause it can decay only through electro-magnetic annihilation or annihila-

tion into at least three gluons [1]. Today's understanding of quarkonium b ound states

has re ned this picture and allows us to calculate relativistic corrections systematically

at the exp ense of intro ducing further non-p erturbative parameters that characterize the

b ound state. Such calculations can be done most transparently in the framework of

a non-relativistic e ective eld theory (NRQCD) [2,3] that implements the factoriza-

tion of contributions to the (partial) decay widths from di erent length scales. Besides

p otentially large relativistic corrections, the size of radiative corrections to the quark-

antiquark annihilation amplitudes has always b een a matter of concern. The one-lo op

radiative corrections to the decays ! l l (where l = e; ) [4], ! light hadrons,

! + light hadrons [5] are large and question the practicability of factorization for

charmonium and, p erhaps, even b ottomonium. [Here and in the following we use as

a lab el for any S -wave spin-triplet state, i.e. J= or for charmonium and (nS ) for

b ottomonium.]

In this Letter we address this question and rep ort on the calculation of two-lo op

short-distance corrections to the leptonic decay ! l l . This is the rst (and also

`easiest') two-lo op matching calculation for quarkonium decays and it can also b e used to

connect the decay constant of the meson de ned in QCD with the non-relativistic wave

function at the origin, which app ears in NRQCD and p otential mo dels. In turn, this

wave function is an imp ortant input parameter for the prediction of other quarkonium

decays and also quarkonium pro duction cross sections.

We recall that decays leptonically through interaction with the electro-magnetic

current. The partial decay rate, neglecting the tiny lepton masses, is exactly given by

2 2 2

4e f

Q em

( ! l l )= ; (1)

where M is the mass of , the ne structure constant and e the electric charge

em Q

of the heavy quark in units of the electron charge. The decay constant f is de ned

through the following matrix element of the electro-magnetic current:

h (p)jQ Qj0i =(i)f M (p): (2)

[ (p)isthe p olarization vector and p the momentum.] The leptonic decay rates are

known exp erimentally [6]. For J= we nd f = (405 15) MeV.

The decay constant parametrizes the strong interaction e ects and contains long-

and short-distance contributions. For quarkonium the short-distance scale is 1=M and

the long-distance (b ound state) scales are 1=(M v ) and 1=(M v ), where v is the (small)

characteristic velo city of the 's quark constituents. The short-distance contributions

can b e isolated, and calculated in p erturbation theory,by matching the vector currentin

QCD onto a series of op erators in NRQCD. Up to corrections of order v , the matching

relation is given by

" !

i y

; h (p)jQ Qj0i = (p) C h j j0i()

s 0 i

; C

s 1

y 2 4

+ h j D j0i()+ O(v ) ; (3)

where (not to b e confused with the meson) and denote non-relativistic two-spinors,

D the spatial covariant derivative and m the heavy quark mass. [More details on

notation and NRQCD can b e found in Ref. [3]. Note, however, that we use a relativistic

normalization of states also for the matrix elements in NRQCD.] The matrix elements on

the right-hand side are de ned in the rest frame and (p) is the matrix that p erforms

the Lorentz b o ost into this frame. The matching co ecients C and C are expressed

0 1

as series in the strong coupling and account for the short-distance QCD corrections.

The matching co ecients and matrix elements in NRQCD individually dep end on the

factorization scale . C is de ned (as in Ref. [7]) such that C =1+O( ) and [4]

1 1 s

m 2C (m )

Q F s Q s

C ; =1 +c (m =) + :::; (4)

0 s 2 Q

where C =(N 1)=(2N ) and N =3 is the number of colours. We now discuss the

F c c

calculation of the two-lo op matching co ecient c .

Since the matching co ecient contains only short-distance e ects, it can b e obtained

by replacing the quarkonium state on b oth sides of Eq. (3) by a free quark-antiquark

pair of on-shell quarks at small relativevelo city. In terms of this on-shell matrix element,

the matching equation can b e rewritten as

1 2

Z = C Z Z + O (v ); (5)

2;QCD QCD 0 2;NRQCD NRQCD

where Z are the on-shell wave function renormalization constants in QCD and NRQCD

and the amputated, bare electro-magnetic annihilation vertices in QCD and NRQCD.

The Feynman diagrams for at two lo ops are shown in Fig. 1. Since the current

QCD

J = need not b e conserved in NRQCD, we allowed for its renormalization, J =

i bare

Z J ,on the right-hand side. We then obtain C by calculating all other quantities in

J ren 0

Eq. (5) in dimensional regularization and using the mo di ed minimal subtraction scheme

( MS scheme) [8].

The matching calculation is considerably simpli ed, if one uses the threshold ex-

pansion of Ref. [9] to compute directly as an expansion in v . The threshold

QCD

expansion is obtained by writing down contributions corresp onding to hard (l m ),

2 2

soft (l m v ), p otential (l m v , l m v ) and ultrasoft (l m v ) regions. The

Q 0 Q i Q Q 2

D D

1 2

D D

3 4

D D { D

5 6 9

Figure 1: Diagrams that contribute to . Symmetric diagrams exist for D . The last

QCD 2;3;5

diagram summarizes vacuum p olarization contributions from massless fermions (D ), gluons

(D ), ghosts (D ) and the massive fermion with mass m (D ).

7 8 Q 9

contributions from soft, p otential and ultrasoft lo op momenta can all b e identi ed with

diagrams in NRQCD that app ear in the calculation of . Hence, they drop out of

NRQCD

the matching relation Eq. (5) and it suces to compute the contribution to the threshold

expansion of the diagrams in Fig. 1, where all lo op momenta are hard. [The threshold

expansion is not only convenient; it also provides an implicit de nition of NRQCD in di-

mensional regularization. This is necessary, b ecause dimensionally regularized NRQCD

is not given by the dimensionally regularized Feynman integrals constructed from the

vertices and propagators of NRQCD. In order to avoid that the cut-o for the e ective

theory be treated as larger than m , dimensionally regularized NRQCD has to be sup-

plemented by a prescription for expanding the Feynman integrands. This prescription is

provided to all orders as part of the diagrammatic threshold expansion metho d [9].]

We brie y describ e the calculation of the hard contributions to . [Details of the

QCD

calculational metho d and a solution to the recurrence algorithm for the two-lo op integrals

will b e given in a long write-up of this Letter.] The spinor structure of the on-shell matrix

element in QCD is conventionally parametrized bytwo form factors, F and F , of which

1 2

only the combination F + F is required here. Since terms of order v are not needed

1 2

to determine C [see Eq. (5)], we may set the relative momentum to zero and compute

the form factors directly at threshold. The form factors have Coulomb singularities at

threshold and diverge as 1=v . However, these singularities app ear only in the soft,

p otential and ultrasoft contributions, and the hard contribution is well-de ned directly 3

1 1

Colour factor nite

2 2 2

35 (3)

9 5 133 5 ln 2 27 81

D C

32 64 24 128 96 12 8

733 3 43 971

D C +

16 32 192 576

2 2 2

21 (3)

15 5 715 319 ln 2

D C C

3 F A

32 64 16 384 576 8 16

2 2 2

31 (3)

3 39 251 3 ln 2

D C (C 2C ) 0

4 F A F

16 16 32 1152 8 16

2 2

3 (3)

761 1157 ln 2 9 19

+ + D C (C 2C )

5 F A F

32 64 384 1152 6 4

1 5 355 5

D C T n

6 F F f

8 48 288 48

19 53 6787 95

D C C +

7 F A

128 768 4608 768

1 1 361 5

+ D C C

8 F A

128 768 4608 768

1 13 145 5

D C T +

9 F F

4 48 96 72

2 2

21 99 637 733

Sum C + (3)

32 64 12 384 576

2 2 2

11 49 4811 209 ln 2

C C + 4 (3)

F A

32 192 8 1152 576 3

1 5 355 5

C T n

F F f

8 48 288 48

1 13 145 5

C T +

F F

4 48 96 72

2 2 2 2

Table 1: Co ecientof( = ) (e m =(4 )) for the hard contribution to the diagrams

2 2

of Fig. 1 evaluated at threshold q = 4m . D include a factor of 2 to account for the

2;3;5

corresp onding symmetric diagrams. For SU (3) the colour factors are C = 4=3, C = 3,

F A

T =1=2. The numb er of light (massless) quark avours is denoted by n .

at threshold in dimensional regularization. The lo op integrals simplify considerably, once

the relative momentum is set to zero, since they then dep end only on a single scale. We

then pro ject on the form factor F + F and reduce all integrals to integrals without

1 2

numerators. These integrals can be further reduced to `simple' integrals and two non-

trivial two-lo op integrals by means of recurrence relations derived from integration by

parts in the lo op momenta [10]. The solution to the recurrence relations is the dicult

part of the calculation. The remaining non-trivial two-lo op integrals can be calculated

explicitly using standard Feynman parameters. The results obtained for the diagrams

of Fig. 1 are listed in Table 1.

After summing all the diagrams, multiplying by the two-lo op QCD on-shell wave

function renormalization constant [11], and p erforming (one-lo op) coupling and mass

renormalization the result still contains p oles in = (4 d)=2 (where d is the space-

time dimension). Since the wave function renormalization constant Z in NRQCD

2;NRQCD

equals 1 up to higher-order terms in v , not needed here, these p oles are attributed to 4

an anomalous dimension of J , which rst arises at the two-lo op order. As a consequence

the matrix element h j j0i()is factorization scale-dep endent. We de ne it in the

MS scheme and obtain the anomalous dimension for the NRQCD vector current J :

dlnZ

J s

= = C (2C +3C ) +O( ): (6)

J F F A

dln 6

The scale dep endence is comp ensated by the scale dep endence of the two-lo op matching

co ecient c (m =) in Eq. (4). Separating the di erent colour group factors, the nal

2 Q

result for c (m =)inthe MS scheme is:

2 Q

c (m =) = C c + C C c + C T n c + C T c ; (7)

2 Q 2;A F A 2;N A F F f 2;L F F 2;H

! " #

79 1 23 (3)

c = ln +ln2 + ; (8)

2;A

6 36 8 2

! " #

89 1 5 151 13 (3)

+ c = ln ln 2 ; (9)

2;N A

4 144 6 72 4

; (10) c =

2;L

2 22

c = + : (11)

2;H

9 9

Here (3) = 1:202 ::: and we have taken all fermions with masses less than m as

massless, which is a go o d approximation even for m = m , the b ottom quark mass, in

Q b

which case we neglect m , the charm quark mass. The co ecient c prop ortional to n ,

c 2;L f

the numb er of light fermions, has b een recently obtained by Braaten and Chen [7]. Note

also that the C -term of the form factors F close to threshold has b een calculated by

1;2

Hoang [12], using the absorptive parts of the form factors obtained in Ref. [13]. Because

the result in Ref. [12] contains hard and soft (p otential, ultrasoft) contributions, it is not

p ossible to extract the matching co ecient c from Ref. [12]. [The structures in c

2;A 2;A

that cannot arise from the small lo op momentum regions agree with Hoang's result.]

The size of the two-lo op correction to C [Eq. (4)] given by Eqs. (7){(11) is enormous.

We de ne the (scale-dep endent) non-relativistic decay constant as

y NR

h j j0i()= (i)f ()M ; (12)

in analogy with Eq. (2). The non-relativistic decay constant is related to the wave

NR 2 2

function at the origin by M (f ) =12j (0)j . Using Eq. (3), we obtain

8 (m )

s Q s

1 f = + ::: f (m ): (13) (44:55 0:41n )

Q f

With (m ) 0:35 and (m ) 0:21 the second-order correction exceeds the rst-

s c s b

order correction even for the b ottomonium states. For charmonium, the second-order 5

term is almost twice as large as the already sizeable rst-order correction. [Note that

the BLM estimate of the two-lo op correction [7] is far o the exact two-lo op result.]

Perturbative matching at the scale = m do es not seem to work. Can the factorization

of short- and long-distance e ects still b e useful?

A novel, and p erhaps unexp ected, asp ect at the two-lo op level is the factorization

scale dep endence of the non-relativistic decay constant and, hence, the quarkonium wave

function at the origin. The scale dep endence is large, esp ecially due to the non-ab elian

term in Eq. (6). This scale dep endence of the wave function indicates the limitation of

the non-relativistic p otential mo del approach already at leading order in v , since the

wave functions obtained from solving the Schrodinger equation are scale-indep endent.

Despite this shortcoming it may b e argued that the wave function/non-relativistic decay

constant obtained from p otential mo dels corresp onds { if to anything { to the wave

function evaluated at a scale typical for the b ound state and not m . This p oint of

view is also evident if the wave function is computed non-p erturbatively using lattice

NRQCD, in which case the ultraviolet cut-o /factorization scale is also much smaller

than m . We consider = 1 GeV as an adequate b ound state scale for b ottomonium

and charmonium, as the applicability of p erturbation theory prevents us from taking yet

smaller scales. We then nd, for b ottomonium (m = 5 GeV, n = 4):

b f

0 1

8 (m ) (m )

s b s b

NR NR

@ A

f = 1 + ::: f (1 GeV) 0:81 f (1 GeV): 1:74

(nS )

(nS ) (nS )

(14)

The numerical factor 0.81 is stable against variations of the scale of the coupling at

xed factorization scale 1 GeV. At this factorization scale the second-order correction is

numerically insigni cant. Although we do not know whether the three-lo op correction

would also be small at the low factorization scale, we tend to consider Eq. (14) as an

accurate prediction. This prediction could be tested, if f (1 GeV) were accurately

(nS )

known, for example from NRQCD lattice simulations. The large scale dep endence raises

the theoretical question (the answer to which we p ostp one to the long write-up) as to

2 2

whether it is necessary to resum the logarithms ln(m = ) to all orders.

The situation is less favourable for charmonium states. Since m 1:5 GeV, the

size of the second-order correction is altered little for scales > 1 GeV. We have not

succeeded to nd a trustworthy interpretation of Eq. (13) and conclude that the fac-

torization of short-distance and long-distance e ects may not be useful in practice for

charmonium. Since the leptonic decay is the simplest conceivable decay, this puts into

question the p ossibility to obtain universal relations b etween various charmonium decays

and pro duction pro cesses through the use of NRQCD [3,14]. This p essimistic conclu-

sion may be biased by our use of the MS factorization scheme. It is conceivable that

other factorization schemes or relations between physical observables, from which the

wave function at the origin is eliminated, exhibit b etter convergence prop erties of their

p erturbative series. A de nitive conclusion on this issue can be obtained only once a

second quarkonium decay is computed to two-lo op order. 6

While this pap er was b eing written, Czarnecki and Melnikov [15] also considered the

two-lo op matching of the electro-magnetic current, also using the asymptotic expansion

metho d of [9]. After correction of a trivial normalization error (A. Czarnecki and K.

Melnikov, private communication), their result agrees with ours.

Acknow ledgements. We thank G. Buchalla for useful discussions and A. Hoang for cor-

resp ondence. The work of V.S. has b een supp orted by the Russian Foundation for Basic

Research, pro ject 96{01{00726, and by INTAS, grant 93{0744.

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