Table Of ContentANL-HEP-PR-08-43
Order-α corrections to the quarkonium electromagnetic current
s
at all orders in the heavy-quark velocity
Geoffrey T. Bodwin,1 Hee Sok Chung,2 Jungil Lee,1,2 and Chaehyun Yu2
1High Energy Physics Division, Argonne National Laboratory,
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9700 South Cass Avenue, Argonne, Illinois 60439, USA
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2Department of Physics, Korea University, Seoul 136-701, Korea
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a
(Dated: January 14, 2009)
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1 Abstract
] Wecomputeinorderα thenonrelativisticQCD(NRQCD)short-distancecoefficientsthatmatch
h s
p
quark-antiquarkoperatorsofallordersintheheavy-quarkvelocityvtotheelectromagneticcurrent.
-
p
e Weemployanewmethodtocomputetheone-loopNRQCDcontributiontothematchingcondition.
h
[
Thenew methodusesfull-QCDexpressions as astarting pointto obtain theNRQCD contribution,
4
v thus greatly streamlining the calculation. Our results show that, under a mild constraint on
4
3 the NRQCD operator matrix elements, the NRQCD velocity expansion for the quark-antiquark-
6
2 operator contributions to the electromagnetic current converges. The velocity expansion converges
.
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0 rapidly for approximate J/ψ operator matrix elements.
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:
v PACS numbers: 12.38.Bx,12.39.St, 13.20.Gd, 14.40.Gx
i
X
r
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1
I. INTRODUCTION
The electromagnetic decays of quarkonia through a single virtual photon have played
an important rˆole in the experimental and theoretical development of quarkonium physics.
On the experimental side, the decays of charge-conjugation-odd quarkonium states to a
lepton pair provide unique signals for the detection of those states. On the theoretical
side, the decays of 3S quarkonium states to a lepton pair allow one to determine one of
1
the fundamental parameters of the heavy-quark–antiquark (QQ¯) bound state, namely, the
squareofthewave functionattheorigin. (See, forexample, Ref.[1].) Thesquareofthewave
function at the origin enters into many calculations of quarkonium decay and production
rates.
The expression for the 3S quarkonium decay rate into a lepton pair at leading order in
1
the QCD coupling α and at leading order in v, the Q or Q¯ velocity in the quarkonium
s
rest frame, has been known since the first discovery of quarkonium and is based on the
Van Royen-Weisskopf formula [2] of quantum electrodynamics. The order-α corrections to
s
this formula at leading order in v were calculated in Refs. [3, 4]. The order-α0 relativistic
s
corrections at relative orders v2 and v4 were calculated in Refs. [5, 6], respectively. Order-
α2 corrections to the decay rate were calculated in Refs. [7, 8]. The correction to the
s
electromagnetic current of a quarkonium at relative order α v2 was calculated in Ref. [9].
s
In this paper, we calculate relativistic corrections to the quarkonium electromagnetic
current at order α . We carry out our calculation in the context of nonrelativistic QCD
s
(NRQCD) [5]. We obtain closed-form expressions whose Taylor-series expansions in v give
the short-distance coefficients for the NRQCD QQ¯ operators, of all orders in v, that match
to the electromagnetic current. We do not consider QQ¯ operators that contain gauge fields.
Therefore, our operators are not gauge invariant, and we evaluate their matrix elements
in the Coulomb gauge. In the Coulomb gauge, QQ¯ operators involving gauge fields first
contribute at relative order v4. Our results confirm the calculation at relative order α v2 in
s
Ref. [9]. Since the corrections at relative order α v2 are not very significant at the current
s
level of precision of calculations of 3S quarkonium electromagnetic decay rates, we do not
1
expect the order-α corrections at still higher orders in v to be important numerically.
s
We present our calculation primarily as a demonstration of a new method for comput-
ing the one-loop NRQCD contribution that enters into the matching of NRQCD to full
2
QCD. The direct computation of one-loop NRQCD expressions to all orders in v would
be a formidable task, in that it would require knowledge of the NRQCD interactions and
electromagnetic-current operators, their Born-level short-distance coefficients, and their
Feynman rules to all orders in v. Instead of following the direct NRQCD approach, we
note that NRQCD through infinite order in v is equivalent to QCD, but with the inter-
actions rearranged in an expansion in powers of v. Therefore, we can obtain the one-loop
NRQCD contribution by starting from full-QCD expressions and expanding integrands in
powers of momenta divided by the heavy-quark mass m before we carry out the dimen-
sional regularization. In dimensional regularization, this method is related to the method
of regions [10]. We explain this relationship in Sec. II. The method of regions has been
used previously at leading order in v to compute NRQCD short-distance coefficients from
full-QCD expressions. (See for example, Ref. [8].)
Our results show that, under a mild constraint on the NRQCD operator matrix elements,
the NRQCD velocity expansion for the quark-antiquark-operator contributions to the elec-
tromagnetic current converges. The velocity expansion converges rapidly for approximate
J/ψ operator matrix elements.
The remainder of this paper is organized as follows. In Sec. II we discuss the one-loop
matching of NRQCD to QCD at all orders in v. We define the notation that we use to
describe the kinematics of the calculation in Sec. III. Section IV contains detailed formulas
¯
for the NRQCD QQ short-distance coefficients. In Sec. V we compute the one-loop QCD
corrections to the electromagnetic current, while in Sec. VI we use our new method to
compute the one-loop NRQCD corrections to the electromagnetic current. We give analytic
and numerical results for the short-distance coefficients in Sec. VII, present a formula that
resums a class of relativistic corrections to all orders in v, and discuss the convergence of
the velocity expansion. Our conclusions are given in Sec. VIII. The Appendices contain
compilations of integrals and identities that are useful in the calculation.
II. MATCHING TO ALL ORDERS IN v
We define the hadronic part of the quarkonium electromagnetic decay amplitude µ as
AH
( iee )i µ = 0 Jµ H , (1)
− Q AH h | EM| i
3
where H is the quarkonium, e is the electromagnetic charge, e is the heavy-quark charge,
Q
and Jµ is the heavy-quark electromagnetic current:
EM
Jµ = ( iee )ψ¯γµψ. (2)
EM − Q
Here, ψ is the heavy-quark Dirac field, and γµ is a Dirac matrix.
In the quarkonium rest frame, i 0 = 0 because of conservation of the electromagnetic
AH
current. According to NRQCD factorization [5], we can write the spatial components i i
AH
as
i i = √2m c 0 i H , (3)
AH H nh |On| i
n
X
where the c are short-distance coefficients, the i are NRQCD operators, and m is the
n On H
quarkonium mass. We regulate the operator matrix elements in Eq. (3) dimensionally in
d = 4 2ǫ dimensions. The factor √2m on the right side of Eq. (3) appears because the
H
−
NRQCD operator matrix elements have nonrelativistic normalization, while we choose the
amplitude on the left side of Eq. (3) to have relativistic normalization for the quarkonium
H.
The aim of this paper is to calculate the short-distance coefficients c that correspond
n
to QQ¯ color-singlet operators in order α1. We can determine these c by making use of
s n
¯
a matching equation that is the statement of NRQCD factorization for perturbative QQ
color-singlet states:
i i = c 0 i QQ¯ , (4)
AQQ¯1 nh |On| 1i
n
X
where the subscript 1 indicates a color-singlet state. Throughout this paper, we suppress the
factor √N that comes from the implicit color trace in i i , where N = 3 is the number
c AQQ¯1 c
of colors. Through order α1, the matching equation is
s
i i(0) +i i(1) = (c(0) +c(1)) 0 i QQ¯ (0) + c(0) 0 i QQ¯ (1), (5)
AQQ¯1 AQQ¯1 n n h |On| 1i n h |On| 1i
n n
X X
where the superscripts (0) and (1) indicate the order in α . In the first sum in Eq. (5),
s
only color-singlet QQ¯ operators contribute, while in the last sum, additional operators can
contribute if they mix into color-singlet QQ¯ operators under one-loop corrections.
We define the quantity
i i(0) = c(0) 0 i QQ¯ (0), (6)
AQQ¯1 NRQCD n h |On| 1i
h i Xn
4
i(0)
which is the expansion of i in powers of q/m, where q is half the relative momentum
AQQ¯1
of the heavy quark and heavy antiquark. At order α0, the matching equation (5) yields
s
i i(0) = c(0) 0 i QQ¯ (0), (7)
AQQ¯1 n h |On| 1i
n
X
(0) (0)
from which the c can be determined. The c have been computed previously in Ref. [1].
n n
At order α1, the matching equation (5) yields
s
i i(1) = c(1) 0 i QQ¯ (0) + i i(1) , (8)
AQQ¯1 n h |On| 1i AQQ¯1 NRQCD
Xn h i
(1) i(1) i(1)
from which the c can be computed. We compute the quantities i and i
n AQQ¯1 AQQ¯1 NRQCD
in Secs. V and VI, respectively. h i
The quantity
i i(1) = c(0) 0 i QQ¯ (1) (9)
AQQ¯1 NRQCD n h |On| 1i
h i Xn
would be formidable to calculate directly in NRQCD because it involves operators and
interactions of all orders in v. Rather than carry out such a direct calculation, we take a
new approach. We note that, by construction, NRQCD reproduces all of the interactions in
full QCD, but with those interactions reorganized in an expansion in powers of v. Therefore,
i(1) i(1)
we can obtain i from the expression for i by expanding the integrand in
AQQ¯1 NRQCD AQQ¯1
powers of the mh omentium divided by m. Before making this expansion, we carry out the
integration over the temporal component of the loop momentum, using contour integration.
This procedure establishes the scale of the temporal component of the loop momentum,
which varies from contribution to contribution, and it avoids the generation of ill-defined
pinch singularities that can arise when one expands the Q and Q¯ propagators prematurely
in powers of the momentum. We expand the integrand in powers of both the external
momenta divided by m and the loop momenta divided by m. We then regulate the integrals
dimensionally, setting scaleless, power-divergent integrals equal to zero. Ultimately, we
renormalize ultraviolet divergences according to the MS prescription.
The procedure of expanding both external and loop momenta in power series before
regulating dimensionally was first utilized in the Appendix of Ref. [5]. The rationale for it
was discussed in Refs. [9, 11]. This procedure amounts to the prescription that infrared-
finite contributions that arise from loop momenta in the vicinity of zero are kept in the
5
short-distance coefficients [11].1
If one uses dimensional regularization, then the quantity c(1) 0 i QQ¯ (0) corre-
n n h |On| 1i
sponds to the contribution from the hard region in the methoPd of regions [10], while the
i(1)
quantity i corresponds to the sum of the contributions from the potential, soft,
AQQ¯1 NRQCD
and ultrahsoft regiions, i.e., the contribution from the small-loop-momentum region. In the
method of regions, it is assumed that there are no contributions from the region in which
the temporal component of the gluon momentum is of order m, but the spatial component
of the gluon momentum is of order mv. As we shall see explicitly in our calculation, this
assumptionisjustifiedbecausethecontributionfromthisregionofintegrationvanishes indi-
mensional regularization. We note, however, that the contribution from this region does not
vanish in the case of a hard-cutoff regulator. One potentially useful feature of the approach
that we present here is that it can be applied in the case of a hard cutoff, such as lattice
regularization, while the method of regions is applicable only in dimensional regularization.
In the method of regions, one can compute the contribution from the hard region directly,
rather than computing it, as we do, by subtracting the small-loop-momentum contribution
from the full-QCD contribution. As we shall explain later, there may be advantages to our
indirect procedure in calculating the hard contribution to all orders in v.
III. KINEMATICS
Before proceeding to write explicit formulas for the short-distance coefficients, let us
define some notation for the kinematics of the heavy-quark electromagnetic vertex. We
take p and p to be the momenta of the incoming heavy quark Q and heavy antiquark Q¯,
1 2
respectively. p and p can be expressed as linear combinations of their average p and half
1 2
their difference q:
p = p+q, (10a)
1
p = p q. (10b)
2
−
1
Inthecaseofhard-cutoffregularization,suchaslatticeregularization,theexpansioninpowersofloopmo-
mentumdividedbym wouldbe uniformly convergentandwouldyieldthe sameresultasthe unexpanded
expression.
6
In the QQ¯ rest frame, the momenta are given by
p = (E,q), (11a)
1
p = (E, q), (11b)
2
−
p = (E,0), (11c)
q = (0,q), (11d)
whereE = m2 +q2. ThequarkQandantiquarkQ¯ areontheirmassshells: p2 = p2 = m2.
1 2
For later uspe, it is convenient to define a parameter
q
δ = | |, (12)
E
which is related to the velocity
q
v = | |. (13)
m
We can write δ in terms of v as
v
δ = . (14)
√1+v2
E2 and q2 are expressed in terms of m and δ as
m2
E2 = , (15a)
1 δ2
−
m2δ2
q2 = . (15b)
1 δ2
−
IV. FORMULAS FOR THE SHORT-DISTANCE COEFFICIENTS
Now let us make use of the matching conditions (7) and(8) to compute the short-distance
coefficients for the specific color-singlet QQ¯ operators that we consider in this paper. These
operators are
↔
i = χ†( i∇)2nσiψ, (16a)
OAn −2
↔ ↔ ↔
i = χ†( i∇)2n−2( i i)( i∇) σψ, (16b)
OBn −2 −2∇ −2 ·
where ψ is the Pauli spinor field that annihilates a heavy quark, χ† is the Pauli spinor
field that annihilates a heavy antiquark, and σi is a Pauli matrix. Our operators contain
ordinary derivatives, rather than covariant derivatives. Therefore, our operators are not
gauge invariant, and we evaluate their matrix elements in the Coulomb gauge. We do not
7
consider QQ¯ operators involving the gauge fields, which first contribute at relative order v4.
We note that i can be decomposed into a linear combination of the S-wave operator i
OBn OAn
and the D-wave operator i :
ODn
1
i = i + i , (17)
OBn d 1OAn ODn
−
where i is defined by
ODn
↔ ↔ ↔ 1 ↔
i = χ†( i∇)2n−2 ( i i)( i∇) σ ( i∇)2σi ψ. (18)
ODn −2 −2∇ −2 · − d 1 −2
(cid:20) − (cid:21)
In the basis of operators i and i , the matching conditions (7) and (8) become
OAn OBn
i i(0) = a(0) 0 i QQ¯ (0) + b(0) 0 i QQ¯ (0), (19a)
AQQ¯1 n h |OAn| 1i n h |OBn| 1i
n n
X X
i i(1) = a(1) 0 i QQ¯ (0) + b(1) 0 i QQ¯ (0) + i i(1) , (19b)
AQQ¯1 n h |OAn| 1i n h |OBn| 1i AQQ¯1 NRQCD
Xn Xn h i
where a and b are the corresponding short-distance coefficients. A similar equation holds
n n
in the basis i and i , where the associated short-distance coefficients are
OAn ODn
1
s = a + b , (20a)
n n n
d 1
−
d = b , (20b)
n n
respectively.2
The QQ¯ matrix elements in Eq. (19) are
0 i QQ¯ (0) = q2nη†σiξ, (21a)
h |OAn| 1i
0 i QQ¯ (0) = q2n−2qiη†q σξ, (21b)
h |OBn| 1i ·
↔ ↔
2 Ifwereplacetheordinaryderivatives∇withcovariantderivativesD inanS-waveoperator i ,thenwe
OAn
obtainoneofthe conventionalgauge-invariantS-waveNRQCDoperators. Becausethe squaredcovariant
↔
derivatives (D)2 commute with themselves, the substitution of covariant derivatives for ordinary deriva-
tives leadsto a unique S-waveoperatorateachordern. Therefore,the S-waveshort-distancecoefficients
s that we compute are also the short-distance coefficients of the S-wave operator in which ordinary
n
derivatives have been replaced with covariant derivatives. In the case of the D-wave operators i , the
ODn
replacementofordinaryderivativeswith covariantderivativesdoes not leadto a unique operatorbecause
↔ ↔
(D)i and (D)j do not commute. Therefore, each of the D-wave short-distance coefficients d that we
n
compute is the sum of the short-distance coefficients for the various operators at order n that can be
constructed from covariant derivatives.
8
where ξ and η are two-component spinors. In order to maintain consistency with our calcu-
lations in full QCD, we have taken the QQ¯ states to have nonrelativistic normalization and
we have suppressed the factor √N that comes from the color trace.
c
Because of current conservation, the most general form of i i is
AQQ¯1
i i = v¯(p )(Gγi +Hqi)u(p ), (22)
AQQ¯1 2 1
where
G = Z (1+Λ). (23)
Q
Z is the fermion wave-function renormalization, and Λ is the multiplicative correction to
Q
the fermion electromagnetic vertex. Similarly,
i i = v¯(p )(G γi +H qi)u(p ), (24)
AQQ¯1 NRQCD 2 NRQCD NRQCD 1
h i
where
G = [Z ] (1+Λ ). (25)
NRQCD Q NRQCD NRQCD
Using nonrelativistic normalization for the spinors u and v, we obtain
qiη†q σξ
v¯(p )γiu(p ) = η†σiξ · , (26a)
2 1
− E(E +m)
qiη†q σξ
qiv¯(p )u(p ) = · . (26b)
2 1
− E
Then,
G H
i i = Gη†σiξ + qiη†q σξ. (27)
AQQ¯1 − E(E +m) E ·
(cid:20) (cid:21)
Similarly,
G H
i i = G η†σiξ NRQCD + NRQCD qiη†q σξ. (28)
AQQ¯1 NRQCD NRQCD − E(E +m) E ·
(cid:20) (cid:21)
h i
Using the matching condition (19a) and Eqs. (21) and (27), we obtain the short-distance
coefficients at order α0:
s
n
1 ∂
a(0) = G(0) = δ , (29a)
n n! ∂q2 n0
(cid:18) (cid:19) (cid:12)q2=0
(cid:12)
1 (cid:12) ∂ n−1 G(0) H(0)
b(0) = d(0) = (cid:12) +
n n − (n 1)! ∂q2 E(E +m) E
(cid:12)
− (cid:18) (cid:19) (cid:20) (cid:21)(cid:12)q2=0
(cid:12)
1 ∂ n−1 1 (cid:12)
= , (cid:12) (29b)
− (n 1)! ∂q2 E(E +m) (cid:12)
− (cid:18) (cid:19) (cid:20) (cid:21)(cid:12)q2=0
1 (cid:12)
s(0) = a(0) + b(0). (cid:12) (29c)
n n 3 n (cid:12)
9
Using the matching condition (19b) and Eqs. (21), (27), and (28), we obtain the short-
distance coefficients at order α1:
s
n
1 ∂
a(1) = ∆G(1) , (30a)
n n! ∂q2
(cid:18) (cid:19) (cid:12)q2=0
(cid:12)
1 (cid:12) ∂ n−1 ∆G(1) ∆H(1)
b(1) = d(1) = (cid:12) + , (30b)
n n − (n 1)! ∂q2 E(E +m) E
(cid:12)
− (cid:18) (cid:19) (cid:20) (cid:21)(cid:12)q2=0
1 (cid:12)
s(1) = a(1) + b(1), (cid:12) (30c)
n n d 1 n (cid:12)
−
where
∆G(1) = G(1) G(1) , (31a)
− NRQCD
∆H(1) = H(1) H(1) . (31b)
− NRQCD
The infrared divergences in G(1) and H(1) cancel in ∆G(1) and ∆H(1) because
NRQCD NRQCD
NRQCD reproduces full QCD in the infrared region. The one-loop NRQCD matrix ele-
(1)
ments in G contain ultraviolet divergences, which we renormalize according to the
NRQCD
(1)
MS prescription. The quantity H is free of ultraviolet divergences. The quantities
NRQCD
Λ and Z also contain ultraviolet divergences. However, because of the usual cancellation
Q
between the vertex and fermion-wave-function renormalizations, G(1) is free of ultraviolet
divergences. H(1) is also free of ultraviolet divergences. Carrying out the renormalization,
we have
n
1 ∂
a(1) = ∆G(1) , (32a)
n MS n! ∂q2 MS
(cid:18) (cid:19) (cid:12)q2=0
(cid:2) (cid:3) (cid:12)
1 (cid:12) ∂ n−1 ∆G(1) ∆H(1)
b(1) = d(1) = (cid:12) MS + , (32b)
n MS n MS − (n 1)! ∂q2 E(E +m) E
" #(cid:12)
− (cid:18) (cid:19) (cid:12)q2=0
(cid:2) (cid:3) (cid:2) (cid:3)
1 (cid:12)
s(1) = a(1) + b(1) . (cid:12) (32c)
n MS n MS 3 n MS (cid:12)
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
(1)
In deriving the expression for s , we have used the fact that, in minimal subtraction,
n
MS
one removes the 1/ǫ pole timehs theiorder-α0 d-dimensional matrix element. Hence, a term
s
proportional to (d 1)−1ǫ−1 is subtracted in Eq. (30c) in carrying out the renormalization.
−
V. QCD CORRECTIONS
In this section, we calculate the QCD corrections to the heavy-quark electromagnetic
i(1)
current. That is, we compute i .
AQQ¯1
10
