Electric dipole transitions in pNRQCD

0.09 MB·English

by  Piotr Pietrulewicz

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Electric dipole transitions in pNRQCD
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n
a
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h Piotr Pietrulewicz∗
p FacultyofPhysics,UniversityofVienna,Boltzmanngasse5,A-1090Wien
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p E-mail: [email protected]
e
h
[
We presenta theroreticaltreatmentof electric dipole(E1) transitionsof heavyquarkoniabased

1 on effective field theories. Within the framework of potential nonrelativistic QCD (pNRQCD)
v
8 we derive the complete set of relativistic corrections at relative order v2 to the decay rate in a
0 systematic, model-independentway. Former results from potential model calculations will be
3
1 scrutinized and a phenomenologicalanalysis with lattice input in relation to experimentaldata
1. willbepresented.
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:
v
i
X
r
a
XthQuarkConfinementandtheHadronSpectrum
8-12October2012
TUMCampusGarching,Munich,Germany
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
ElectricdipoletransitionsinpNRQCD PiotrPietrulewicz
1. Introduction
Radiative transitions play an important role for our understanding of QCD, in particular of
heavy quarkonia. Theyprovide information about thewavefunctions describing the physical sys-
tem and probe both the perturbative and non-perturbative regime. Especially E1 transitions give
significant contributions to the total decay rate and yield clean signals, which are observed in the
experimental facilities. In the last few years CLEO, BESand the B factories have improved their
observations ofradiativetransitions, areviewaboutrecentdevelopments canbefoundin[1].
Onthetheory side, electric dipole transitions weretreated inseveralpotential models, asum-
marycanbefoundin[2]. Wewillreferto[3]forcomparisonwithourresults. Amodel-independent
treatmenttocheckandimprovethecalculationshasbeenmissingsofar. However,inthelastdecade
there has been significant progress using effective field theories (EFTs)todescribe heavy quarko-
nium (see[4]and references therein). Sinceheavy quarkonium isassumed tobeanon-relativistic
systemwemaytakeadvantageofthehierarchyofscalesm≫mv≫mv2,wherev≪1istheheavy
quark velocity, m is the heavy quark mass ("hard scale"), p∼mvisthe relative momentum of the
bound state ("softscale") andE ∼mv2 isthebinding energy ("ultrasoft scale"). Theultimate EFT
livingattheultrasoftscaleispotential non-relativistic QCD(pNRQCD).In2005,forthefirsttime
radiative decays, concretely M1 transitions, were calculated in this theory [5]. Using the frame-
work of that paper as a guideline we close the remaining gap and compute the decay rates of E1
processes betweenSandPstates(liken3P →n′3S g ). Thefollowingisbasedon[6].
J 1
2. Framework
Byintegrating outthehardscalem≫L fromthefundamental theory (QCD)inperturba-
QCD
tion theory (a (m)≪1)oneobtains non-relativistic QCD(NRQCD)[7,8]. Forthecalculation of
s
E1transitions atrelative order v2 only the two-fermion Lagrangian L matters andthe relevant
2−f
partreads
D2 D4 cem cem
L =y † iD + + y +ee y † F s ·Bem+i s s ·[D×,Eem] y +c.c.. (2.1)
2−f 0 2m 8m3 Q 2m 8m2
(cid:18) (cid:19) (cid:18) (cid:19)
with iD =i¶ −gTaAa−ee Aem, iD=i(cid:209) +gTaAa+ee Aem and y denoting a Pauli spinor for
0 0 0 Q 0 Q
theheavy quark. Thematching coefficients arefoundtobecem =1+C a (m )/2p +O(a 2)and
F F s H s
cem =2cem−1withm ∼m.
s F H
For processes at the ultrasoft scale, NRQCD is not yet the appropriate theory, since there
are still several scales entangled (p,E,L ) and thus no homogeneous power counting can be
QCD
established. Integrating outthesoftscalemvweobtainatheoryforultrasoftmodes,i.e. pNRQCD
[9,10]. Thecrucialsteptodisentangle theenergyandmomentumscaleisthemultipoleexpansion
in the relative distance r. To be definite we will use the power counting in the weak-coupling
regime(p≫E &L ),whichreads
QCD
r∼1/mv,(cid:209) r ≡¶ /¶ r∼mv,(cid:209) ≡¶ /¶ R∼mv2,E,B∼(mv2)2,Eem,Bem ∼kg2. (2.2)
kg is the energy of the emitted photon, which scales like mv2 for transitions between states with
different principal quantum numbers.
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ElectricdipoletransitionsinpNRQCD PiotrPietrulewicz
ThepNRQCD-Lagrangiancontributing atNLOinthedecayrate,i.e. atorderkg3v0/m2,reads
(cid:209) 2 (cid:209) 2 (cid:209) 4 D2 (cid:209) 2
L = d3rTr S† i¶ + + r + r −V S+O† iD + + r −V O
pNRQCD 0 4m m 4m2 S 0 4m m O
Z ( ! !
+gV (O†r·ES+S†r·EO)
A
+LgpNRQCD+Llight, (cid:9) (2.3)
wherethecovariantderivativesaregivenbyiD O=i¶ O−g[TaAa,O]andiDO=i(cid:209) O+g[TaAa,O]
0 0 0
andthetracegoesoverthecolorandspinindices. ThesingletpotentialV hasbeencalculated per-
S
turbativelyandnon-perturbativelytoorder1/m2([11,12,13],formoreoriginalreferencessee[4]),
wedisplay thestructure oftherelevantpotentials forcomputations atNLOinthedecayrate,
V(1)(r) V(2)(r) V(2)(r)
V (r)=V(0)(r)+ r + SI + SD , (2.4)
S m m2 m2
(2)
1 V (r)
V(2)(r)=V(2)(r)+ {V(2)(r),p2}+ L2 L2, (2.5)
SI r 2 p2 r2
V(2)(r)=V(2)(r)L·S+V(2)(r)S2+V(2)(r)[3(rˆ·s )(rˆ·s )−s ·s ]. (2.6)
SD LS S2 S12 1 2 1 2
TherelevantpartofLgpNRQCD forE1transitions is
1
LgEp1NRQCD =eeQ d3rTr Vr·ES†r·EemS+VOr·EO†r·EemO+24V(r(cid:209) )2r·ES†r·[(r(cid:209) )2Eem]S
Z (cid:26)
1
+i V(cid:209) ·(r×B)S†{(cid:209) ·,r×Bem}S
4m
1
+i V(cid:209) r·(r×(r(cid:209) )B)S†{(cid:209) ·,r×[(r(cid:209) )Bem]}S
r
12m
1
+ V(r(cid:209) )s ·B[S†,s ]·[(r(cid:209) )Bem]S
4m
1
−i Vs ·(E×(cid:209) r)[S†,s ]·(Eem×(cid:209) )S . (2.7)
4m2 r
(cid:27)
In fact more terms are allowed according to the symmetries of pNRQCD.However, we can show
that their matching coefficients vanish. The matching is done by equating Green’s functions in
NRQCDandpNRQCDattheenergyscalemvorderbyorderintheinversemass.
ThecrucialargumentforseveraloperatorsisthatdiagramsinNRQCDwhichcanbecastintoa
reduciblestructurealsogivereduciblediagramsinpNRQCD.Thereforetheyhavetobesubtracted
to obtain irreducible operators in pNRQCDand do not play arole in the matching procedure. An
exampleisthediagraminFig. 1,wherethegluoniccontribution canbefactorizedoutyieldingjust
a potential. Using this argument we can fixall of the Wilson coefficients in (2.7) at leading order
ina ,sothattheexactresultsreproduce theonesfromtreelevelcalculations, namely
em
Vr·E =Vr·E =V(r(cid:209) )2r·E =V(cid:209) ·(r×B)=V(cid:209) r·(r×(r(cid:209) )B)=1,V(r(cid:209) )s ·B=cem,Vs ·(E×(cid:209) r)=cem. (2.8)
O F s
With the help of the formalism developed in [5] we can describe the states in a quantum
mechanical wayusingwavefunctions andcomputethedecayrateatNLO,i.e. atrelativeorderv2,
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ElectricdipoletransitionsinpNRQCD PiotrPietrulewicz
V V
pNRQCD S + S
−→
Figure1:Exampleforareduciblediagram,iftheelectromagneticoperatorcommuteswiththegluonicones.
Itdoesnotcontributetothematchingcoefficientofasingleoperator.
fromtheLagrangian(2.7). Weobtain
G n3PJ→n′3S1g = 49a eme2Qkg3I32(n1→n′0) 1+R−6kg02 II53 −6kmg +kg (c2eFmm−1) J(J2+1)−2 !,
(cid:20) (cid:21)
(2.9)
where
¥
I ≡ drrNR (r)R (r). (2.10)
N n′0 n1
0
Z
R contains all of the wave-function corrections due to the higher-order potentials mentioned in
(2.4)-(2.6), the relativistic correction of the kinetic energy, −p4/4m3, and higher-order Fock state
contributions duetointermediate color-octet states. Incontrast toM1transitions thelatteronesdo
notvanishforE1decays(see[6]forexplicitexpressions).
The expression (2.9) is also valid in the strongly coupled regime (without color-octet contri-
butions in R), where p∼L , since we made use of non-perturbative matching arguments and
QCD
additional operators donotappearinthisregime.
Compared to the results with the potential model calculation in [3] we find an equivalence
between(2.9)andthecorresponding formulathereatthegivenorder. However,ourdefinitepower
countingallowedustoincludeallrelativisticcorrectionssystematically,inparticularthecolor-octet
(1)
contributions intheweak-coupling regimeandtheone comingfrom thepotentialV . Bothwere
r
missing in former approaches. Furthermore we can show that the anomalous magnetic moment
cem−1∼O(a (m)) is actually suppressed and does not lead to large non-perturbative contribu-
F s
tions.
Without much effort one can extend the discussion to other processes like n1P → n′1S g
1 0
and n3S →n′3P g (see [6]), also for transitions between states with the same principal quantum
1 J
number, wherecorrections ∼kg aresuppressed.
3. Phenomenological analysis
Based on these results a phenomenological analysis for bottomonium and charmonium de-
cays at relative order v2 can be performed. For a complete analysis we need a parametrization of
chromoelectric field correlators inthe weak coupling regime. Since E1transitions always involve
excitedstates,wealserequirethequarkoniumpotentialsinthestrongcouplingregime. Thesehave
tobematchedwiththeknownshortdistancebehaviour.
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ElectricdipoletransitionsinpNRQCD PiotrPietrulewicz
As a first approach we proceed as following: As a parametrization of the static potential at
short distances we use the perturbative expression at 3 loop with leading ultrasoft resummation
with the parameters given in [14, 15] (see also references therein)1 derived from a matching of
the static energy to unquenched lattice data [17]. For large distances we use the string potential
V =−p /12r+s r+C [18] matched to lattice simulations [17] at r =1.5r (where r is the
string 0 0
Sommerscale). Thesetwopotentials merge together smoothly atr∼0.8r . Toobtain the leading
0
orderwavefunctionswesolvetheSchrödingerequationwiththestaticpotentialnumericallyusing
theMathematicaprogram schrodinger.m [19]. Wefixthecharmandbottommassinourschemea
posteriori bymatchingtheobtained massesfortheJ/y and¡ (1S)withthephysicalones.
Concerningrelativisticcorrectionsthemaincontributionarisesfromwavefunctioncorrections
due to subleading potentials. For short distances, here for r <(2GeV)−1, we apply perturbative
results at LL, wheras for long distances, r>(2GeV)−1, we use parametrizations from quenched
lattice results [20, 21, 22] as a non-perturbative input.2 Future approaches should aim for smooth
transitions betweenthesetworegimes. Weneglectcolorocteteffects,whichcannotbedetermined
fromcurrentlatticesimulations.
The results of this computation are given in table 1 for bottomonium and in table 2 for char-
monium decays. We do not include decays with kg &hpi, where our power counting is assumed
to break down. We see that the relativistic corrections lower the decay rates considerably, by 10-
30%forbottomonium andby20-60% forcharmonium, whichisespecially striking forthedecays
h (1P)→h (1S)g andy (2S)→c (1P)g . Thewavefunctioncorrection duetothepotentialV(1)
c c c0 r
yields particularly large contributions. The expansion works much better for bottomonium, since
the average relative velocity is smaller (v2 ∼0.1, v2 ∼0.3). We estimate the uncertainty of our
b c
NLO result to be of order 10% for bottomonium and of order 30% for charmonium. This is on
the one hand the generic size of a correction at O(v2), which can arise from color-octet effects or
a systematic error in the treatment of the subleading potentials. On the other hand this is also a
conservative measure for the total higher order effects, which are supposed to be suppressed by v
compared tothetotalNLOcorrections.
Comparing our values with thepotential model calculations in [3](for scalar and vector con-
fining potential) wetend to get slightly larger values for bottomonium and slightly smaller values
forcharmonium. Within ouruncertainties westay consistent withobservations, provided both the
branching fractionandthetotaldecayratehavebeenmeasured.
WeemphasizethatexceptforthemassesoftheJ/y and¡ (1S)andthephotonenergiesnoex-
perimental input has been used. Additional input and athorough investigation of thefinesplitting
effects in the spectrum would improve the predictive power of our results. This analysis should
be repeated, when unquenched results for the required chromoelectric field correlators and espe-
cially for the subleading potentials become available, with emphasis on a proper connection to
perturbative results, maybeathigherorder(NLL),andamoreelaborate uncertainty estimate.
1Toobtainaconvergentbehaviouronehastoperformarenormalonsubtractionatascaler [16]. In[14]a s(1/r)
wasexpandedintermsofa s(r )togetamorestablebehaviour. Forshortdistances,r<0.14r0,largelogarithmsyield
anunreasonableshapeofthepotential,thereforeweexpanda s(r )intermsofa s(1/r)inthisregime.
2Asfarasavailableweusefitswithparametrizationsbasedoncalculationsfromthestringmodel[23].Vr(2)(sofar
(2)
undetermined)andV (small)aresetto0inthenon-perturbativeregime.
S2
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ElectricdipoletransitionsinpNRQCD PiotrPietrulewicz
process G LO /keV G NLO /keV G [3] /keV G PDG/keV
pNRQCD pNRQCD mod exp
c (1P)→¡ (1S)g 31.8 29.7±3.1 25.7-27.0 -
b0
c (1P)→¡ (1S)g 40.3 35.8±4.0 29.8-31.2 -
b1
c (1P)→¡ (1S)g 45.9 40.6±4.6 33.0-34.2 -
b2
h (1P)→h (1S)g 60.8 44.3±6.1 - -
b b
¡ (2S)→c (1P)g 1.52 1.13±0.15 0.72-0.73 1.22±0.16
b0
¡ (2S)→c (1P)g 2.26 1.94±0.23 1.62-1.65 2.21±0.22
b1
¡ (2S)→c (1P)g 2.34 2.19±0.23 1.84-1.93 2.29±0.22
b2
c (2P)→¡ (2S)g 12.6 13.0±1.3 10.6-11.4 -
b0
c (2P)→¡ (2S)g 17.1 16.3±1.7 11.9-12.5 -
b1
c (2P)→¡ (2S)g 20.4 18.1±2.0 12.9-13.1 -
b2
¡ (3S)→c (2P)g 1.44 1.05±0.14 1.07-1.09 1.20±0.16
b0
¡ (3S)→c (2P)g 2.38 2.05±0.24 2.15-2.24 2.56±0.34
b1
¡ (3S)→c (2P)g 2.53 2.35±0.25 2.29-2.44 2.66±0.41
b2
Table1: E1decayratesforbottomonium.OurpNRQCDresultscomparedtoapotentialmodelcalculation
[3] and and the currentPDG values [24]. LO denotes the result obtained without relativistic corrections,
NLO indicatesthe resultup to O(v2) neglectingcolor-octeteffectsin the weak-couplingregimeand non-
perturbativecontributionstoV(2). TheerrorestimatesgivethegenericsizeofoneO(v2)correctionaswell
r
asanestimateforthesumofallcorrectionsatO(v3). Forhb(1P)→h b(1S)g wehavetakenmh b(1S)=9402
GeV[25]todeterminethephotonenergy.
process G LO /keV G NLO /keV G [3] /keV G PDG/keV
pNRQCD pNRQCD mod exp
c (1P)→J/yg 199 158±60 162-183 122±11
c0
c (1P)→J/yg 421 302±126 340-363 296±22
c1
c (1P)→J/yg 568 415±170 413-464 386±27
c2
h (1P)→h (1S)g 909 447±272 - <600
c c
y (2S)→c (1P)g 53.6 21.4±16.1 26.0-40.3 29.4±1.3
c0
y (2S)→c (1P)g 45.2 30.7±13.6 28.3-37.3 28.0±1.5
c1
y (2S)→c (1P)g 31.6 25.6±9.5 17.5-22.7 26.5±1.3
c2
h (2S)→h (1P)g 38.1 31.0±11.4 - -
c c
Table 2: E1 decay rates for charmonium. Our pNRQCD results at LO, NLO (including error estimate)
comparedtoapotentialmodelcalculation[3]andthecurrentPDGvalues[24].
Acknowledgements
IwouldliketothankNoraBrambillaandAntonioVairoforthecollaboration onthiswork.
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