Table Of ContentModel-independent Study on Magnetic Dipole
Transition in Heavy Quarkonium
Yu Jia
6 INFNandDipartimentodiFisicadell’UniversitadiMilano,viaCeloria16,20133Milano,Italy
0
0
Abstract.
2
Somenewresultsonmagneticdipole(M1)transitionsinheavyquarkoniumfromnonrelativistic
n
effective field theories of QCD are briefly reported. This model-independent approach not only
a
facilitates a systematic and lucid way to investigate the relativistic corrections, it also clarifies
J
some inconsistent treatment in previous potential model approach. The impact of our formalism
2 onJ/y →h g ,¡ (¡ ′)→h g andh →c g arediscussed.
2 c b c c0
2
v Radiativetransitionsinheavyquarkoniumareofconsiderableexperimentalandtheo-
3
reticalinterest[1].Onthetheoryside,itprovidesuswithfurtherinsightonthedynamics
7
1 ofquarkoniuminadditiontotheknowledgewehavegleanedfromthespectraofcc¯and
1 bb¯ families.
0
Being an old subject, radiative transitions have been extensively studied in phe-
6
0 nomenological models, notably the potential model approach [2]. It is certainly desir-
/ ableto studythem from a model-independentperspective.In this talk I will report such
h
p a study based on the effective-field-theory (EFT) approach [3]. Since M1 transition is
-
theoretically cleaner and more interesting than E1 transition,I will focus on the former
p
e case, despitethefact thatthelatteris observedmorecopiouslyinnature.
h
In the nonrelativistic limit, the M1 transition rate between two S-wave onia takes a
:
v
particularlysimpleform:
i
X
r 4a e2 2
a G [n3S1 →n′1S0+g ] = 3emm2Q(1+k Q)2kg3 drr2Rn′0Rn0 , (1)
(cid:12)Z (cid:12)
(cid:12) (cid:12)
where e and k are the electric charge and anomalous ma(cid:12)gnetic moment(cid:12)of the heavy
Q Q (cid:12) (cid:12)
quark, and R (r) is the radial Schrödinger wave functions. Since the leading dipole
nl
operator c ee /2ms ·Bem (where c = 1+k ) only flips the spin and doesn’t act on
F Q F Q
thespatialdegreesoffreedom,orthogonalityofradialwavefunctionsguaranteesthat:in
the allowed transition (n=n′), the overlap integral equals 1; in the hindered transition
(n6=n′), theoverlapintegralvanishes.
Itiswellknownthat(1)overpredictstheobservedJ/y →h g transitionratebyafac-
c
torof2∼3,withanormalinputofm andk fromperturbativeone-loopmatching.This
c c
clearly indicates large relativisticcorrections to (1), as is usuallyconfronted in charmo-
nium system. It has also been speculated that low-energy fluctuations may generate a
largenegativek , so thediscrepancycan bereduced.
c
The O(v2) corrections to (1) has been available for a long timefrom potential model
approach [1,2]:
G [nS→n′S+g ] = 1 4a eme2Q k3 (cid:229) 5 I 2 , (2)
g i
2J +1 m2
n (cid:12) i (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where2J +1 countsnumberofpolarizationsoftheparent o(cid:12)nium(cid:12),and
n (cid:12) (cid:12)
kg2r2 kg
I = n′0 (1+k ) 1− +(1+2k ) n0 , (3)
1 Q Q
24 4m
* (cid:12) ! (cid:12) +
(cid:12) (cid:12)
(cid:12) p2 p2 (cid:12)
I2 = − n′(cid:12)(cid:12)0 (1+k Q)2m2 +3m2 n0 , (cid:12)(cid:12)
(cid:28) k (cid:12) (cid:12) (cid:29)
(cid:12) (cid:12)
I = n′0 (cid:12)QrV′ n0 , (cid:12)
3 6m(cid:12) 0 (cid:12)
D (cid:12) 4 (cid:12) E V
I = ± (cid:12) (cid:12) n′0 (1+k ) ss n0 ,
4 (0(cid:12)) (0)(cid:12) Q m2
E −E
n0 n′0 (cid:28) (cid:12) (cid:12) (cid:29)
h (cid:12) (cid:12)
I5 = n′0 − VS n0 ,(cid:12)(cid:12) (cid:12)(cid:12)
m
D (cid:12) (cid:12) E
(cid:12) (cid:12)
whereV0 standsforthestat(cid:12)icpoten(cid:12)tial,the“+/–"signinI4 isassociatedwith3S1 →1S0
and 1S →3S , respectively. Notice that I accounts for the first-order correction to the
0 1 4
wavefunctionduetospin-spinpotential(otherhigherdimensionallocalpotentialscease
tocontributeinS-wavetransition),thusisonlypresent inhindered transition.
I is a prediction specific to the popular assumption in potential models, where one
5
usually decomposes the confining potential into a Lorentz scalar and a vector part,
V = h V +(1−h )V . This term constitutes the major uncertainty in the potential
conf S V
modelpredictions,wherecontradictoryclaimsoften appear intheliterature[2].
Presence of a hierarchy of scales in quarkonium, m, mv, mv2, L , makes the EFT
QCD
approach an ideal tool to analyze this transition process. One first descends from QCD
to NRQCD by integrating out hard modes (∼ m ) [4], then descends from NRQCD
to potential NRQCD (pNRQCD) by further integrating out soft modes (∼ mv ) [5].
As a result, the inter-quark potentials appear as Wilson coefficients in pNRQCD, and
the only dynamical degrees of freedom of pNRQCD are ultrasoft modes (∼mv2 ) (for
convenience,onecan alsoincorporatetheradiated ultrasoftphotonintopNRQCD).
A primary task of the EFT approach is to validate/invalidate(2). The real strength of
theEFTapproachis,however,thatitcanfurtheranswerthefollowingquestionsthatare
beyondthescopeofpotentialmodels:
• Is it possible that a large correction to k Q due to soft modes arises when one
descendsfrom NRQCD topNRQCD?
• Is itpossibleto reproduceI5 in pNRQCD?Ifso, howtointerpret it?
• Potential model focus exclusively on the QQ Fock-state (with an exception of
coupled-channel effects which may not be relevant as long as one excludes those
statesclosetotheopen-flavor threshold).pNRQCD allowsonetoincludeultrasoft
gluons as dynamical degrees of freedom. Therefore, one naturally asks for the
possibility of large nonperturbative contribution arising from higher-Fock states
|QQgi. Thiseffect isusuallyreferred toas color-octeteffect.
In fact, theanswers to thesethreequestionsare all negative[3], on whichwewill
elaborate shortly. It turns out that the pNRQCD formalism is able to justify (2) except
I .Inthesocalledweak-couplingregime(whenmv≫L ), thisformulaiscomplete;
5 QCD
in the so-called strong-couplingregime (when mv∼L ), however, (2) is incomplete
QCD
andfurtherterms areneeded.
Before going on to explanations, it is useful to first sketch the derivation of O(v2)
correctionsintheframeworkofpNRQCD. Oneobvioussourceisfrom thecontribution
ofhigherdimensionalM1operators.Theseoperatorscanbeidentifiedmostconveniently
by promoting the color gauge group of NRQCD Lagrangian to a larger gauge group
SU (3)×U (1).Explicitly,therelevantmagneticoperators upto O(1/m3)read [6]:
c em
D2 c ee ic ee
L = y † iD + y + F Qy †s ·Bemy + S Qy †s ·[(cid:209) ×,Eem]y
NR 0 2m 2m 8m2
(cid:18) (cid:19)
c ee c ee
+ W1 Qy †{(cid:209) 2,s ·Bem}y − W2 Qy †(cid:209) s ·Bem(cid:209) y
8m3 4m3 i i
c ee
− p′p Q (cid:209) y †·s Bem·(cid:209) y +(cid:209) y †·Bems ·(cid:209) y +(y →is 2c ∗), (4)
8m3
h i
where Dm ≡ ¶ m +igTaAam +ieeQAemm. Various Wilson coefficients can be computed
through perturbative matching at the hard scale. For example, at one loop accuracy,
k =C a /2p , is about a few percent for charm and bottom. It is well known that
Q F s
someoftheWilsoncoefficientsarerelatedwitheachotherbecauseofreparameterization
invariance(RPI),oressentiallyPoincareinvariance,notablyc =2c −1,c =c −1
S F W2 W
1
and c =c −1 [6]. When inherited intopNRQCD, onemakereplacement (cid:209) →ipin
p′p F
theseoperators, andtheseRPI relationscan beutilizedtocondensetheexpressions.
ThereareothersourcesofO(v2)corrections,forinstance,multipoleexpansionofthe
photonfield.Oneinterestingcontribution,firstpointedoutbyGrotchandSebastian[2],
is the Lorentz boost effect due to the final-state recoil. Since the wave function of a
moving S-wave state has a non-vanishing overlap with a P-wave, spin-flipped state at
rest,theM1transitioncanbeeffectivelyrealized bya“E1"transitionfromtheparentto
this small component. Some subtlety arises in this effect, namely the recoil correction
depends on which “E1" operator one uses, i.e., 2ee /mp·Aem or ee r·Eem, which
Q Q
are connected by a redefinition of pNRQCD field. The solution to this problem is as
following.ThematchingfromNRQCDgeneratesanewM1operatoratO(1/m2),which
isintimatelyrelatedtothenon-local(dependingonthecenter-of-massmomentum)spin-
orbit potential, with a coefficient proportional to VCM = −V′/8r (Gromes relation).
LS 0
The contribution of this operator also depends on the field redefinition. However, the
sum of these two corrections is convention-independent, thus comprises a meaningful
relativisticrecoileffect.
We are now in a position in discussing general matching of M1 operators from
NRQCD to pNRQCD. At O(1), those operators in (4) are trivially inherited to pN-
RQCD.Ingeneral,afterintegratingoutsoftmodes,newWilsoncoefficientswilldepend
oninter-quark separationr.
FIGURE 1. Typical NRQCD diagramsresponsible for matchingof the dipole operatorat O(1/m0).
The left diagram represents the vertex correction, and the right one contributes to the wave-function
renormalizationfactorZs(r)duetosoftmodes,wherethecross-capimpliesinsertionofaunitoperator.
Obviouslythecorrectiontoc canbeinterpretedasamultiplicativeO(1/m0)match-
F
ing coefficient of the leading M1 operator. Dimensional considerations require that this
correction is function of log(r). We emphasize in passing that it is inappropriate to at-
tributethisnewcoefficienttothemagneticmomentofanindividualquark,sinceitreally
arisesfrom entangledcontributionsfrombothquarks.
The simple answer, no corrections at all, is basically nothing more than the heavy
quark spin symmetry and that the s ·Bem operator behaves like a unit operator in
spatial Fock space. These facts are independent of whether the matching is performed
perturbatively as in weak-coupling regime, or nonperturbatively, as in strong-coupling
regime. In an arbitrary NRQCD diagram depicting transition, as shown in Fig. 1a), we
only need to consider the case where the external ultrasoft photon is attached to one
oftheheavy quarklines.(photonattachedtointernallightquarkloopscan beneglected
whensummingoverelectricchargesofthreelightflavors).Softgluonsattachedtoheavy
quark lines must be longitudinalto avoid 1/m suppression. Because all the propagators
and vertexes are spin-independent except the M1 vertex, such a diagram can factorize
into the leading M1 operator times minus the wave function renormalization constant
d Z (r), which can be extracted from the same diagram with the electromagnetic vertex
s
replaced by a unit operator insertion, as shown in Fig. 1 b). Therefore, there is no net
contributiontothismatchingcoefficientfrom softmodes.
The matching at O(1/m) is slightly more complicated. In a NRQCD diagram with
a insertion of the leading dipole opeartor, a transverse gluon is allowed to end on the
heavy quark line with a spin-independent p·A vertex (ending on a s ·B vertex doesn’t
contribute). Repeating the previous argument, one doesn’t obtain any new M1 operator
from these sources. However, there are a new class of diagrams, in which electromag-
neticcouplingisembeddedinthecovariantderivativeD,inducinganeffectivemagnetic
operator at O(1/m). This is just the one accompanying with the nonlocal LS-potential
as mentioned before. It is important to note that there is no room at this order for the
operatoroftheformV /m2s ·Bem toarise.Thereforeoneisforced toregardI in(2)as
S 5
anartifact ofpotentialmodel,whichcannot bearanyphysicalsignificance.
The matching at O(1/m2) becomes much more cumbersome. We will not dwell on
the explicit expressions for these case. However, it should be kept in mind that they
may be neglected in the weak-coupling regimebecause of suppressionsby higherpow-
× ZnUS ZnU′S
q q
a) b) c) d)
FIGURE2. pNRQCD diagramsforcolor-octetcontributionto radiativetransition.Singleanddouble
linescorrespondtothesingletandoctetfields.Thesinglet-octet-gluonvertexisofchromo-E1type.
ers of a ∼v. In the strong-coupling regime, since a ∼1, these operators might be as
s s
importantasotherO(v2)corrections.Thesecorrectionsinvolvesomeunknownnonper-
turbativeWilsonloopamplitudes,so thatpredictivepowerisunavoidablydamaged.
The color-octet effect to radiative transitions was first envisaged by Voloshin long
time ago, but without a detailed study [7]. pNRQCD allows a systematic treatment of
thiseffect. The corresponding diagrams are shown in Fig. 2. It is not surprisingthat the
color-octet effect is nearly absent in M1 transition, because of the exactly same reason
as before. It is interesting to note this bears some resemblance as absence of leading
nonperturbative correction to B → D∗ form factor at the zero recoil, known as Luke’s
theorem [8]. Here we only givea heuristic argument based on the quantum-mechanical
perturbation theory. Treating the chromo-E1 interaction as perturbation up to second
order, one can schematically express a “true” quarkonium stateas superpositionsof the
followingFock components(strippingoffspind.o.g.):
(cid:229)
|Ni= Zus|QQ (n)i+|QQ gi+ |QQ (m)i···, (5)
n 1 8 1
m6=n
p
where |QQ (n)i is the unperturbed state, and Zus is the wave function renormalization
1 n
factor for state n due to ultrasoft gluons. Spin-independence of chromo-E1 interaction,
plus M1 operator being a unit operator in coordinatespace, implythat thefull M1 tran-
sition amplitude is nothing but the inner product hN′|Ni. Since orthogonality condition
is preserved in perturbed states, it is equal to hn′|ni, the color-octet contribution thus
vanishes.Weemphasizethatsizablecolor-octeteffectmightariseinE1transition,since
theE1 operatorisno longeraunitoperatorin coordinateFock space.
Finally we turn to the phenomenological implication of (2), with the understanding
thatwerestrictonlytotheweakly-coupledsystemforconsistencyandtheunphysicalI
5
termhas been dropped.Thus far, theonlyobserved M1 transitionsare J/y (y ′)→h g ,
c
and upper bounds on ¡ ′(¡ ′′)→h g have recently been set by CLEO [10]. Empirically,
b
¡ (1S) and h are believed to lie in the weak-coupling regime, whereas J/y , h fit
b c
in this regime to a less extent. ¡ ′ and ¡ ′′ are usually regarded as strongly-coupled
system. As for y ′ and h ′, they are too close to the open-flavor threshold and cannot
c
be correctly described by current formulation of pNRQCD, therefore we exclude them
inouranalysis.
For a weakly-coupled system, the dynamics is largely governed by the perturbative
static potential, i.e., V ≈ −C a (m )/r, where the natural choice of m is around the
0 F
33..55
33
hh22..55
tt
dd
ii 22
WW
yy
aa11..55
cc
ee
DD 11
00..55
00..88 11 11..22 11..44 11..66
mmuu
FIGURE3. G [J/y →h g ] (in keV) vs. m (in GeV). Dashed line is the predictionin NR limit, solid
c
lineincludesO(v2)corrections,andthebandrepresentsthemeasuredwidthwitherrortakenfrom[9].
typicalthree-momentumscale.Onealsoneedstospecifythequarkmassin(2).Anaive
input is to use mˆ, half of the center-of-gravity ground state mass. However, this simple
procedure may induce an error of order v2, which shouldn’t be neglected according to
power counting. A more consistent way is to choose the 1S mass m, which is defined
implicitlythroughmˆ =m− 10 p2/2m 10 ≈m(1−C2a (m )2/8).
F s
Manyoftermsin(2)arepracticallynegligible.Sincek retainsitsNRQCDvalueand
Q
(cid:10) (cid:12) (cid:12) (cid:11)
is only a few per cent for charm(cid:12)and bo(cid:12)ttom, we may simply put it to be zero in those
O(v2)terms(as aresult,I can bedroppedinbothallowedand hinderedtransitions).In
3
fact, onlyI in (2)accounts forgenuineO(v2) correction forallowedtransition,and we
2
endup withasimpleexpression
4a e2 2C2a 2(m )
G [J/y →h g ] ≈ em c k3 1+2k − F s . (6)
c 3mˆ2 g c 3
c (cid:20) (cid:21)
Fig. 3 shows a comparison between this formula and the data. It seems that (6) is
compatible within the error to the data when m is about 0.8 GeV. Note this value is
consistent with the empirical mv value for J/y . Therefore, our reasonable success in
describing J/y → h g may be viewed as a posteriori support for the weak-coupling
c
assignment. However, rather sharp m dependence may suggest that J/y is not far from
the strong-coupling regime. One can employ (6) to ¡ → h g with more confidence,
b
and finds a smaller O(v2) correction with flatter scale dependence. Unfortunately, very
narrowwidthof2∼3 eV makes thistransitionunlikelytobedetected.
For hindered transitions, e.g. ¡ ′(¡ ′′) → h g , the experimental upper bounds are al-
b
readyrathertightandmanymodelpredictionshavebeenruledout[10].Tobeconserva-
tive,onewouldnotexpect(2)toreliablydescribethesetransitionsifexcitedbottomonia
are indeed in the strong-coupling regime. Nevertheless, it may not be too optimistic to
treat, at least ¡ ′, as being in weak-coupling regime. Proceeding along this line, we find
thattheI termis dominatingoverI and I , and thelattertwonearly cancel each other.
4 1 2
As a result, the predicted width is about an order-of-magnitude larger than the experi-
mental upper bound! Even though there are lots of uncertainty associated with I , this
4
alarminglylargediscrepancyseemstoindicatethattheweak-couplingassignmentof¡ ′
isproblematicanda strong-couplinganalysismightbemoreappropriate.
We end with a brief discussion on the P-wave M1 transitions, which have received
less attention in the past, due to lack of phenomenological impetus. Recent discovery
of the h state [11] may arouse the interest to study these processes. Color-octet effect
c
again vanishes because of the exactly same reason. It is straightforward to apply the
pNRQCD formalism to derive the O(v2) corrections. For simplicity, we just quote the
weak-couplingformulafortheallowedtransition:
G [n3PJ →n1P1+g ] = 4a3emm2e2Q kg3 1+k Q−cJ n1 mp22 n1 2 , (7)
(cid:12) (cid:28) (cid:12) (cid:12) (cid:29)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
wherecJ =1/2,1,4/5forJ =0,1,2.Forthen1(cid:12)(cid:12)P1→n3PJg trans(cid:12)(cid:12)ition(cid:12)(cid:12),one(cid:12)(cid:12)simplymul-
tiplies the right side of (7) by a statistical factor of (2J+1)/3. It should be understood
that this formula may be of limited use, since P-wave onia may necessarily live in the
strong-couplingregime.
The fine splittings between h and c , c and c are about 110, −13, −32 MeV,
c c0 c1 c2
respectively. It seems that only h → c g has a serious chance to be observed, with a
c c0
width comparable to G [J/y → h g ]. Future experimental input will enable us to infer
c
thesizeofrelativisticcorrections,thusenriching ourunderstandingofP-waveonia.
ACKNOWLEDGMENTS
IthankN.BrambillaandA.Vairoforcollaborationonthiswork,alsofortheircomments
on the manuscript. This research is supported in part by Marie Curie ERG grant under
contractMERG-CT-2004-510967and byINFN.
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