pNRQCD determination of E1 radiative transitions

0.13 MB·

by  Sebastian Steinbeißer

EPJWebofConferenceswillbesetbythepublisher
DOI:willbesetbythepublisher
c Ownedbytheauthors,publishedbyEDPSciences,2017
(cid:13)
7
1
0
2

n pNRQCD determination of E1 radiative transitions
a
J

0 SebastianSteinbeißer1,a and JorgeSegovia1
1
1Physik-Department,TechnischeUniversitätMünchen,James-Franck-Str.1,D-85748Garching,Germany

]
h
p Abstract.Thiscontributioncontainsthefirstnumericalcomputationofthecompleteset
- ofrelativisticcorrectionsofrelativeorderv2 forelectricdipole(E1)transitionsinheavy
p
e quarkonium; in particular, for the processes χbJ(1P) → Υ(1S)+γ with J = 0, 1, 2.
h Weassume that the momentum transfer of theheavy mesons involved in thereactions
[ liesintheweak-couplingregimeofthelow-energyeffectivefieldtheorypotentialnon-

relativisticQCD(pNRQCD)andthusafullperturbativecalculationcanbeperformed.
1
v
3
1 Introduction
1
5 Electromagnetic transitions are often significant decay modes for bottomonium states below BB¯
2
threshold(10.56GeV),makingthemasuitableexperimentaltooltoaccessthelowestspectraofbot-
0
. tomonia.Forinstance,thefirstbb¯ statesnotdirectlyproducedine+e−collisionswerethesixtriplet-P
1
states,χ (2P )andχ (1P )with J = 0, 1, 2,discoveredinradiativedecaysoftheΥ(3S)andΥ(2S)
0 b J b J
in1982[1,2]and1983[3,4],respectively.
7
1 One importantfeature of electromagnetic transitions is that they can be classified in a series of
: electric and magnetic multipoles. The most importantonesare the E1 (electric dipole) and the M1
v
(magneticdipole)transitions;higherordermultipolemodesE2,M2,E3,etc. appearinthespectrum,
i
X butsincetheyarefurthersuppressedoneusuallydoesnotconsiderthem.Processesinvolvingelectric
r dipole(E1)transitionshappenmorefrequentlythantheonesinducedbyamagneticdipole(M1).The
a
branchingfraction for E1 transitions can indeed be significant for some lowest bottomoniumstates
liketheonesweshallstudyherein[5]: (χ (1P) Υ(1S)γ) = (1.76 0.35)%(notethatitisthe
b0
B → ±
largest exclusive branchingfraction reported by the Particle Data Group (PDG) [5]), (χ (1P)
b1
B →
Υ(1S)γ)=(33.9 2.2)%and (χ (1P) Υ(1S)γ)=(19.1 1.2)%.
b2
± B → ±
Electric dipole(E1)transitionsare definedthroughthe propertythattheychangethe orbitalan-
gular momentum of the state by one unit, but not the spin. Therefore, the final state has different
parityandC-paritythantheinitialone.TypicalE1quarkoniumdecaysaretheonesmentionedabove:
23P 13S +γ. Hereandinthefollowingwe denotethestatesasn2s+1ℓ , wheren = n +ℓ+1
J 1 J r
→
correspondstotheprincipalquantumnumberwithn =0, 1, ...theradialquantumnumberandℓthe
r
orbitalangularmomentum.ThespinisdenotedbysandJ isthetotalangularmomentum.
TheE1 (andM1)electromagnetictransitionshavebeentreatedfora longtime bymeansofpo-
tential models that use non-relativistic reductions of QCD-based quark-antiquarkinteractions (see,
e.g., Ref. [6] for a recent application to the bottomoniumsystem). However, the progress made in
effectivefieldtheories(EFTs)forstudyingheavyquarkonia[7]andthenewlargesetofaccurateex-
perimentaldata taken in the heavy quark sector by B-factories (BaBar, Belle and CLEO), τ-charm
ae-mail:[email protected]
EPJWebofConferences
facilities(CLEO-c, BESIII)andevenproton-protoncolliders(CDF, D0, LHCb, ATLAS, CMS) ask
forasystematicandmodel-independentanalysis(see,e.g.,Refs.[8,9]forreviews).
Formulae and numerical treatment of M1 transitions within the effective field theory potential
NRQCD(pNRQCD)canbefoundinRefs.[10,11].Therein,therelativisticcorrectionstotheleading
order (LO) expression (which counts as k3/m2 where k is the photon energy) were computed in
γ γ
twodifferentexpansionschemes: (i)strictweak-couplingregimeand(ii)includingexactlythestatic
potentialintheLOHamiltonian.Withinthesametheoreticalframework,thecorrespondingformulae
forE1transitionshavebeenpresentedinRef.[12]. Inthiscase,therelativisticcorrectionstotheLO
decaywidth(thatcountsask3/(mv)2)aremuchmoreinvolvedcoveringnotonlyhigherorderterms
γ
inthe E1transitionoperatorbutalso correctionsto theinitialandfinalwavefunctiondueto higher
orderpotentialsandhigherorderFockstates. Thesefactshavehinderednumericalcomputationsof
theE1radiativedecayswithinpNRQCD(forpartialcalculationssee[13]). Thiscontributionaimsto
closethisgapandcalculatethe decayrate ofthereaction23P 13S +γ with J = 0, 1, 2. Asa
J 1
→
firststep,weshallassumethatthesoftscaleliesinthestrictweak-couplingregimeofpNRQCDand
thusafullperturbativecalculationcanbeperformed.Theseproceedingsarebasedontheforthcoming
publication[14].
2 Theoreticalset-up
2.1 Potentialnon-relativisticQCD(pNRQCD)
Heavy quarkonium systems are characterized by their non-relativistic nature, i.e., the heavy quark
bound-statevelocity,v,satisfiesv 1. Thisisreasonablyfulfilledinbottomonium(v2 0.1)andto
≪ ∼
acertainextentincharmonium(v2 0.3). Moreover,atleast, threewidelyseparatedscalesappear:
∼
theheavyquarkmassm(hardscale), therelativemomentumoftheboundstate p mv (softscale)
∼
andthebindingenergyE mv2(ultrasoftscale). Withv 1,thefollowinghierarchyofscales
∼ ≪
m p 1/r mv E mv2 (1)
≫ ∼ ∼ ≫ ∼
is satisfied andthis allowsfor a descriptionin termsof EFTsfor physicalprocessestakingplace at
oneofthelowerscales. Theintegrationoutofmodesassociatedwithhigh-energyscalesisperformed
aspartofamatchingprocedurethatenforcestheequivalencebetweenQCDandtheEFTatagiven
order of the expansion in v. The final result is a factorization at the Lagrangian level between the
high-energymodes,whichareencodedinthematchingcoefficients,andthelow-energycontributions
carriedbythedynamicaldegreesoffreedom.
The suitable EFT to describe processes that take place at the scale mv such as the E1 radiative
transitionsbetweenthelowestheavyquarkoniumstatesispotentialNRQCD(pNRQCD)[15,16]. It
follows by integratingout the modes of order p 1/r mv from NRQCD [17, 18] which in turn
∼ ∼
comesfromQCD byintegratingoutthe highenergymodesof orderm. Therefore,pNRQCD takes
fulladvantageof the hierarchyof scales thatappear in Eq. (1), and makesa systematic andnatural
connectionbetweenquantumfieldtheoryandtheSchrödingerequation.Schematically,thepNRQCD
equationofmotiontakestheform
~p2
i∂ V(0)(r) φ(~r,t,R~)=0
0− m − s
  pNRQCD
++cinotrerreaccttiioonnsstwoitthheoptohteernltoiawl-energydegreesoffreedom 
whereV(0)(r)isthestaticpotentialandφ(~r)istheQQ¯ field.Noteherethattheinteractionswithother
s
low-energydegreesof freedomproduce,amongothers, non-potentialterms thataccountfor singlet
CONF12
γ
(kγ,~k)
Figure1.Kinematicsoftheradiativetransition
H H Hγintherestframeoftheinitial-state
′
→
PH=(MH,~0) quarkoniumH,takenfrom[10].
PH′=(qkγ2+MH2′,−~k)
H′
tooctettransitionsviaultrasoftgluonsandprovideloopcorrectionstotheleadingpotentialpicture.
Beinginducedbylow-energydegreesoffreedomtheyencodealsonon-perturbativeeffects.
The matching of pNRQCD depends on the relative size between the soft and the Λ scale.
QCD
Twomainsituationscanbedistinguished,namely,theweak-coupling[15,16](mv Λ )andthe
QCD
≫
strong-coupling[19](mv Λ )versionsofpNRQCD.Onemajordifferencebetweenthemisthat
QCD
∼
intheformerthepotentialcanbecomputedinperturbationtheoryunlikeinthelatter.
It is obvious that the weak-couplingversion of pNRQCD is amenable for a theoretically much
cleaneranalysis. Theobservablescanbecomputedasan expansioninα with increasingaccuracy.
s
Non-perturbativeeffectsaresuppressedbypowersofΛ /(mv). Therefore,observablesthatcould
QCD
becomputedwiththeweak-couplingversionofpNRQCDareofthegreatestinterest.
2.2 Decaywidthofthen3P n3S γreaction
J ′ 1
→
Thecompletedecayraten3P n3S γreadsuptoorderk3/m2[12]
J → ′ 1 γ
k k2 I(0)(n1 n0)
Γ =Γ(0) 1+RS=1(J) γ γ 5 → ′
n3PJ→n′3S1γ E1 − 6m − 60I(0)(n1 n0)
( 3 → ′
+ J(J+1) 2 1+κem kγ + 1 (1+2κem)I2(1)(n1→n′0)+2I1(0)(n1→n′0) , (2)
2 − − Q 2m m2 Q I(0)(n1 n0)
" #" (cid:16) (cid:17) 3 → ′ #)
whereRS=1(J)includestheinitialandfinalstatecorrectionsduetohigherorderpotentialsandhigher
orderFockstates(seebelow). Theremainingcorrectionswithinthebracketsaretheresultoftaking
intoaccountadditionalelectromagneticinteractiontermsintheLagrangiansuppressedby (v2)[12].
O
Wehavedisplayedtermsproportionaltotheanomalousmagneticmoment,κem,howevertheseterms
Q
areatleastsuppressedbyα (m)v2 andthusgobeyondouraccuracyandarethereforenotconsidered
s
inthenumericalanalysis. TheLOdecaywidth( k3/(mv)2)is
∼ γ
Γ(0) = 4α e2 k3 I(0)(n1 n0) 2 , (3)
E1 9 em Q γ 3 → ′
h i
withα theelectromagneticfinestructureconstant,e thechargeoftheheavyquarksinunitsofthe
em Q
electroncharge,andk thephotonenergydeterminedbythekinematicsshowninFig.1:
γ
M2 M2 k2
kγ =|~k|= H2M− H′ =(MH −MH′ )+O Mγ . (4)
H  H
 
EPJWebofConferences
Thefunction
∞ dk
I(k)(nℓ nℓ )= drr2rN 2R (r) R (r) (5)
N → ′ ′ − ∗n′ℓ′ drk nℓ
Z " #
0
is a matrix element that involves the radial wave functions of the initial and final states. We shall
assumethatthesestatesaresolutionsoftheSchrödingerequation
H(0)ψ(0) (~r)= E(0)ψ(0) (~r), (6)
nℓm n nℓm
withtheleadingorderHamiltonianinweakly-coupledpNRQCDgivenby
2 2 α
H(0) = ∇ +V(0)(r)= ∇ C s , (7)
−2m s −2m − F r
r r
whereC =4/3.Therefore,ψ(0) (~r)andE(0) canbewritteninthehydrogen-likeform
F nℓm n
ψ(n0ℓ)m(~r)=Rnl(r)Yℓm(Ωr)= Nnℓe−ρ2nρℓnL2nℓ+ℓ11(ρn)Yℓm(Ωr), (8)
− −
mC2α2
E(0) = r F s , (9)
n − 2n2
wherem = m/2isthereducedmassoftheQQ¯ system,ρ = 2r/naisadimensionlessvariablewith
r n
a=1/mC α theBohrradius. Finally,thenormalizationreads
r F s
2 3 (n ℓ 1)!
N = − − . (10)
nℓ s na 2n[(n+ℓ)!]
!
2.3 Relativisticwave-functioncorrections
Duetohigherorderpotentialsandtransitionsbetweensingletsandoctets, thestate in Eq.(8)isnot
an eigenstate of the complete Hamiltonian. Therefore, one has to consider correctionsto the wave
function, which can contribute to the decay rate at the required order of precision ( k3/m2). To
∼ γ
compute these corrections one applies the standard formalism of perturbation theory, either in the
languageofquantummechanicsorviaFeynmandiagrams.
2.3.1 Correctionsduetohigherorderpotentials
In order to account for correctionsto the decay width of relative order v2, we need to consider the
completeHamiltonian
2
H = ∇ +V (r)+δH. (11)
s
−2m
r
Thestaticpotentialisgivenby
2 α k
V (r)=V(0)(r) 1+ s a (r) (12)
s s 4π k
where,asmentionedabove,Vs(0)(r)= CFαs/r,istheXk=l1ea(cid:18)ding(cid:19) orderpotentialorCoulomb-likepoten-
−
tialthatgoesdirectlyintheSchrödingerequation. The (α ) and (α2) radiativecorrectionstothe
O s O s
LOstaticpotentialare(theconstantsshownhereincanbefound,e.g.,inAppendixCofRef.[20]):
a1(ν,r)=a1+2β0ln(νeγEr) , (13)
π2
a2(ν,r)=a2+ 3 β20+(4a1β0+2β1)ln(νeγEr)+4β20ln2(νeγEr) . (14)
CONF12
The (α )termwascomputedinRef.[21]andthe (α2)inRef.[22]. Thestaticpotentialisknown
O s O s
uptoorder (α4)withthe (α3) radiativecorrectioncomputedinRefs.[23–26]. However,already
O s O s
(α3)termswouldgiveacontributiontotheE1decayratethatgoesbeyondpresentprecision.
O s
The term δH encodesthe relativistic correctionswhichcan be organizedas an expansionin the
inverseoftheheavyquarkmass,m. Attheorderweareinterestedin,suchexpansioncoversallthe
1/mand1/m2potentialsand,atorder1/m3,thefirstrelativisticcorrectiontothekineticenergy:
4 V(1) V(2) V(2)
δH = ∇ + + SI + SD . (15)
−4m3 m m2 m2
At order 1/m2, we can split the contributions into spin-independent(SI) and spin-dependent (SD)
terms[7]
1
V(2)(r)=V(2)(r)+ V(2)(r), 2 +V(2)(r)~L2, (16)
SI r 2{ p2 −∇ } L2
V(2)(r)=V(2)(r)~L S~ +V(2)(r)S~2+V(2)(r)S , (17)
SD LS · S2 S12 12
whereS~ =S~ +S~ =(σ~ +σ~ )/2,~L=~r ~pandS =3(rˆ σ~ )(rˆ σ~ ) σ~ σ~ are,respectively,thetotal
1 2 1 2 12 1 2 1 2
× · · − ·
spin,totalorbitalangularmomentumandtensoroperatorsactingonthesystem. Intheweak-coupling
case,theabovepotentialsreadatleading(non-vanishing)orderinperturbationtheory
C C α2 C α C α
V(1)(r)= F A s , V(2)(r)=πC α δ(3)(~r), V(2)(r)= F s , V(2)(r)= F s , (18)
− 2r2 r F s p2 − r L2 2r3
3C α C α
V(2)(r)= F s , V(2)(r)= 4πCFαsδ(3)(~r), V(2)(r)= F s . (19)
LS 2r3 S2 3 S12 4r3
Wenowmakeuseofstandardquantummechanicalperturbationtheoryandcomputethefirstand
secondordercorrection,inducedbyapotentialV,toastate nℓ (0) nℓ .Thesecondordercorrection
| i ≡| i
tothewavefunctionisonlyneededwhentheperturbationisgivenbythestaticpotentialproportional
tothea (ν,r)term.Thenormalisedcorrectedwave-functionis
1
nℓ Vnℓ nℓ nℓ
nℓ (1) = h ′ ′| | i n′ℓ′ = | ′ ′ih ′ ′|Vnℓ , (20)
forthefirstorder,a|ndi n′X,n,ℓ′ En(0)−En(0′)| i  n′X,n,ℓ′ En(0)−En(0′) | i
k ℓ Vk ℓ k ℓ Vnℓ k ℓ Vnℓ nℓVnℓ 1 k ℓ Vnℓ 2
nℓ (2) = h 1 1| | 2 2ih 2 2| | i h 1 1| | ih | | i k ℓ |h 2 2| | i| nℓ ,
| i (E E )(E E ) − (E E )2 | 1 1i− 2 (E E )2 | i
forthesk1eX,cno,nℓ1dko2Xn,ne,.ℓ2 n− k1 n− k2 n− k1  k2X,n,ℓ2 n− k2 (21)
As one can see in Eq. (20), a particular re-arrangement of the terms allows us to have a key
expressionthatcanbere-writtenas
nℓ nℓ nℓ nℓ nℓ n ℓ 1 (n) 1
| ′ ′ih ′ ′| = | ′ ′ih ′ ′| | ′ ′ih ′ ′| =lim P . (22)
n′X,n,ℓ′En(0)−En(0′) nX′,ℓ′En(0)−En(0′)−n′X=n,ℓ′En(0)−En(0′) E→En(0) E−H − E−En(0)!≡ (En−H)′
Thiswillallowustocomputeexpectationvaluesofanarbitraryoperator ,via(notethat,forthesake
O
ofsimplicity,onlyfinalstatecorrectionsareshownherebutthesamecorrectionsaffectalsotheinitial
EPJWebofConferences
state):
1
nℓ nℓ (1) = nℓ Vnℓ
′ ′ ′ ′
h |O| i h |O(E H) | i
n− ′
= d3r d3r ψ (~r ) (~r )G (~r ,~r )V(~r )ψ (~r ), (23)
1 2 ∗n′ℓ′ 2 O 2 ′n 2 1 1 nℓ 1
Z
forthefirstorder,and
1 1
nℓ nℓ (2) = nℓ V Vnℓ
′ ′ ′ ′
h |O| i h |O(E H) (E H) | i
n− ′ n− ′
1 1
nℓVnℓ nℓ 1 Vnℓ
′ ′
−h | | ih |O(E H) (E H) | i
n− ′ n− ′
1 1 1
nℓ nℓ nℓV 1 Vnℓ
′ ′
− 2h |O| ih | (E H) (E H) | i
n− ′ n− ′
= d3r d3r d3r ψ (~r ) (~r )G (~r ,~r )V(~r )G (~r ,~r )V(~r )ψ (~r )
1 2 3 ∗n′ℓ′ 3 O 3 ′n 3 2 2 ′n 2 1 1 nℓ 1
Z
δE(1) d3r d3r d3r ψ (~r ) (~r )G (~r ,~r )G (~r ,~r )V(~r )ψ (~r )
− V × 1 2 3 ∗n′ℓ′ 3 O 3 ′n 3 2 ′n 2 1 1 nℓ 1
Z
1
d3rψ (~r) (~r)ψ (~r)
− 2 ∗n′ℓ′ O nℓ ×
Z
d3r d3r d3r ψ (~r )V(~r )G (~r ,~r )G (~r ,~r )V(~r )ψ (~r ), (24)
× 1 2 3 ∗n′ℓ′ 3 3 ′n 3 2 ′n 2 1 1 nℓ 1
Z
forthesecondorder. ThetermδE(1) inEq.(24)isthefirstordercorrectiontotheenergyinducedby
V
apotentialV: δE(1) d3rψ (~r)V(~r)ψ (~r);andG (~r ,~r )hasthefollowingexpression
V ≡ ∗n′ℓ′m′ nℓm ′n 1 2
R
G′n(~r1,~r2)≡(−1)×Eli→mEnG(~r1,~r2,E)− ℓ∞=0 E|ψ−nℓE|2n , (25)
whereG(~r ,~r ,E)istheCoulombGreenfunction X 
1 2
∞ 2ℓ+1
G(~r ,~r ,E)= P (rˆ rˆ )G (r ,r ), (26)
1 2 4π ℓ 1· 2 ℓ 1 2
ℓ=0
X
G (r ,r )= ∞ ma2 ν4 Rνℓ(ρλ,1)Rνℓ(ρλ,2). (27)
ℓ 1 2 r λ ν λ
ν=ℓ+1 ! −
X
inwhichwehavedefinedE mrC2Fα2s.1
≡− 2λ2
2.3.2 CorrectionsduetohigherorderFockstates
The weakly coupled quarkonia may also get corrections from the coupling of the heavy quark-
antiquarkpairtootherlow-energydegreesoffreedom.Inparticular,theleadingorderelectromagnetic
dipoletransitionmay geta correctionfrom diagrams(see Fig. 8 in [12]) in which a singletstate is
1Inordertoperformthecomputationitisspeciallyusefultousesuchexpression, becauseforλ = n ,wehaveE =
√1 ǫ
En(1 ǫ)andE Enforǫ 0. −
− → →
CONF12
70
2
60
V] 50 eV] 0
e 40 k
k [
[
30 -2
20
10 -4
0
1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0
[GeV] [GeV]
Figure2. Leftpanel–ComparisonbetweentheLOdecaywidth(solidbluecurve)anditsrelativisticcorrection
(dashed orange curve) due to higher order electromagnetic transition operators that appear in the pNRQCD
Lagrangian asafunction of the renormalization scaleν. Right panel –Relativisticcontributionsappearing in
Eq.(2): ThefirstΓ(0) ( k /(6m))(solidblue),thesecondΓ(0) ( k2/(60) I(0)/I(0))(dashedorange)andthe
E1× − γ E1× − γ × 5 3
thirdΓ(0) (secondlineofEq.(2))(dot-dashedgreen).
E1×
coupledtoanoctetstateduetotheemissionandre-absorptionofanultrasoftgluon. Thesediagrams
comefromtermsofthepNRQCDLagrangianlike[7]
∆ =V O ~r gE~S +S ~r gE~O , (28)
A † †
L · ·
(cid:16) (cid:17)
whereS =S1 /√N isaquark-antiquarkfieldthattransformsasasingletunderSU(3) andU(1) ,
c c c em
O = √2OaTa isaquark-antiquarkfieldwhichtransformsasanoctetunderSU(3) andasa singlet
c
underU(1) ,andE~ isthechromo-electricfield.
em
ThefirsttwodiagramsinFig. 8of[12]correspondtotherenormalizationoftheinitialandfinal
wave function. The diagrams 2, 3a and 3b account for the correction of the initial and final wave
functionsdue to the presenceof octetstates. The diagram 4 representsan electric dipoletransition
mediated by the intermediate octet state. According to the power counting, the first two diagrams
contributetorelativeorderΛ2 /(mv)2whereastheremainingonesscalesasΛ3 /(mv2)/(mv)2. We
QCD QCD
shallnotconsiderthesecontributionshereinbecauseinthestrictweak-couplingregime,E mv2
∼ ≫
Λ ,onecanarguethattheyshouldbenegligible.
QCD
Itis noteworthythat, in contrastto theE1 transitions, the colour-octetcontributionsforallowed
M1transitionscancel[10].Thisisaconsequenceofthefactthatthemagneticdipoleoperatorbehaves
asanidentityoperatorinpositionspace.
3 Results
Wediscussindetailourtheoreticalresultfortheχ (1P) Υ(1S)γreaction,butasimilaranalysis
b1
→
has been performed for the transitions χ (1P) Υ(1S)γ with J = 0, 2. The mean value for the
bJ
→
decay width and an estimate of its theoretical error will be given at the end of this Section for all
transitions.
Figure2showstheχ (1P) Υ(1S)γLOdecayrateanditsrelativisticcorrectionduetohigher
b1
→
orderelectromagneticinteractionsthat appearin the pNRQCD Lagrangian. In other words, we are
analysingEq.(2)withoutthefactorRS=1(J =1).AsonecanseeintheleftpanelofFig.2,these (v2)
O
correctionsto the LO decay rate are very small, 5% at most. The right panel of the same figure
∼
displaysthedifferentcontributions(withtheirrelativesign)showingthatthedominantoneistheterm
proportional to I(0)(21 10) in the expression of the decay rate, Eq. (2). An interesting feature
5 →
showninFig.2isthesubstantialdependenceoftheresultontherenormalizationscaleν. Thedecay
widthchangesfrom17keVto74keVwhentherenormalizationscaleνisvariedwithintherangeof
EPJWebofConferences
3
2 6
2
-1V] 1 -1V] 1 -1V] 4
[Ge 0 [Ge 0 [Ge 2
-1
-1 0
-2
-2 -2
-3
1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0
[GeV] [GeV] [GeV]
2 2.5 2
-1[GeV] 01 -1[GeV] 112...050 -1[GeV] 1
0.5 0
-1
0.0
-1
-2 -0.5
1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0
[GeV] [GeV] [GeV]
Figure3.ComparisonoftheLOtransitionmatrixelement(solidbluecurve)withrespecttheonescomingfrom
correctionsduetohigherorderpotentials.Upper-leftpanel–Firstordercorrectiontotheinitial(dashedyellow)
andfinal(dot-dashedgreen)wavefunctionsduetothea (ν,r)term.Upper-middlepanel–Firstordercorrection
1
totheinitial(dashedyellow)andfinal(dot-dashedgreen)wavefunctionsduetothea (ν,r).Upper-rightpanel–
2
Secondordercorrectiontotheinitial(dashedyellow)andfinal(dot-dashedgreen)wavefunctions,andfirstorder
correctiontobothinitialandfinal(dottedred)duetothea (ν,r)term.Lower-leftpanel–Firstordercorrectionto
1
theinitial(dashedyellow)andfinal(dot-dashedgreen)wavefunctionsduetotheV(1),andfirstordercorrection
tothefinal(dottedred)wavefunctionsduetotheV(2). Lower-middlepanel–Firstordercorrectiontotheinitial
r
(dashedyellow)andfinal(dashedgreen)wavefunctionsduetothe p2 term;moreover,firstordercorrectionto
theinitial(dot-dashedred)andfinal(dot-dashedpurple)wavefunctionsduetothekinetic p4term. Lower-right
panel–Remainingcontributionswherethemostimportantone(dot-dashedred)isthefirstordercorrectionto
thefinalwavefunctionduetotheV(2).
S2
1to3GeV. Thisrangeencompassesthetypicalmomentumtransferinthebottomoniumsystem,still
beingconsistentwithperturbationtheory.
Letusfocusnowourattentiontothecomputationofthewavefunctioncorrectionsduetohigher
orderpotentials,whichareencodedinthefactorRS=1(J = 1)ofEq.(2). TheupperpanelsofFig.3
show the matrix elements correcting the E1 decay rate up to (v2) and coming from the radiative
O
correctionsto thestatic potential, Eq.(12). Theleftandmiddlepanelsreferto thefirstorderinitial
and final wave function correctionscoming from a (ν,r) and a (ν,r), respectively. The right panel
1 2
refers to the second order correction due to the a (ν,r) term of the static potential. Amongst the
1
featuresshownbythepanels,thefollowingareofparticularinterest: (i)thematrixelementsclearly
exceedthe valueof the LO one. (ii) Thematrix elementsdependquite dramaticallyon the scale ν,
especiallyforsmallν;insomesense,weexpectedsuchbehaviourfromthenumericalanalysisofthe
M1transitionsinRefs.[10,11]. (iii)Thezerocrossinginsomeofthematrixelementscomesfrom
thelogarithmsin(13)and(14).
The lower panels of Fig. 3 show the remaining matrix element contributions coming from δH,
Eq.(15). OnecanseethatonlyfewofthemarerelevantcorrectionstotheLOdecayrate. Moreover,
theν-dependenceofeverymatrixelementissmallerthaninthecaseoftheradiativecorrections.2
2Theonlytwoparametersinourapproacharembandαs. Thevalueoftheb-quarkmassisfixedthroughtheΥ(1S)-mass
andtherunningofαs(ν)istakenat4-loopaccuracywiththreemasslessflavoursusingtheMathematicapackageRunDec[27]
andthestartingvalueαs(MZ)=0.118.
CONF12
2.5
80
2.0
-1V] 1.5 V] 60
e e
G k
[ 1.0 [ 40
0.5
20
0.0
0
1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0
[GeV] [GeV]
Figure4.Leftpanel–Matrixelements(withtheirrelativesigns)contributingtothereactionχ (1P) Υ(1S)γ
b1 →
atLOorder(solidblue),NLO(dashedyellow),NNLO(dot-dashedgreen)andNLO+NNLO(dottedred).Right
panel–Totaldecaywidthfortheχ (1P) Υ(1S)γreaction,thepanelshowstheLO(dashedblue),LO+NLO
b1 →
(dot-dashedyellow)andLO+NLO+NNLO(solidblack)result.ThegreendottedcurveistheLO+NLO+NNLO
resultbutomittingthecontributionscomingfromtheradiativecorrectionstothestaticpotential. Thehorizontal
gray line is our final value for the decay width, taken at ν = 1.5 GeV, and the gray band corresponds to the
uncertainty(44.23%=ˆ 26.05keVforthistransition).
±
Summing all the contributionsdiscussed in the paragraphabove, the left panel of Fig. 4 shows
thenext-to-leadingorder(NLO),NNLOandNLO+NNLOmatrixelementsandcomparesthemwith
the LO term. The most importantfeatureshave been already mentioned: the subleadingmatrix el-
ements are of the same order of magnitude than the leading one and the dependence with ν in the
logsdominatesthe picture. In the rightpanelof Fig. 4, we drawthe decayrate associated with the
χ (1P) Υ(1S)γ reaction at LO, NLO and NNLO. It is worth to remarkthat the NLO contribu-
b1
→
tionisnegligibleatlarge-νbutmultipliesbya factorof2theLO decaywidthatν = 1GeV. A big
correctiontothedecayrateisduetotheNNLOcontribution. OnecanseeintheFigurethatthethe-
oreticalresultdependsslightlyonthescaleforν & 1.75GeV,whereastheν-dependenceisdramatic
forlowervaluesdueto the logarithmicfunctions. Thisfactisdemonstratedbythe additionalcurve
(dottedgreen)whereweomittedthecontributionscomingfromtheradiativecorrectionstothestatic
potential,hencesetthea (ν,r)anda (ν,r)termstozero. Notethattherelativisticcorrectionstothe
1 2
leadingorderE1transitionoperatorareincludedintheNNLOcurve.
Finally, our theoretical results for the decay rates of the transitions under considerationare ob-
tainedbychoosingthevalueatν=1.5GeV,yielding:
Γ(χ (1P) Υ(1S)γ)= 52+14( (v3)) keV, (29)
b0 → −24 O
Γ(χ (1P) Υ(1S)γ)=(cid:16)62+17( (v3))(cid:17) keV, (30)
b1 → 30 O
−
Γ(χ (1P) Υ(1S)γ)=(cid:16)64+18( (v3))(cid:17) keV, (31)
b2 → 33 O
−
wherewe havechosena veryconservativeerroresti(cid:16)mationthatin(cid:17)cludesthe totalrangeof ourfinal
result,obtainedbyvaryingνfrom(1-3)GeV.
4 Epilogue
We have presented the first numerical determination of the decay rate χ (1P) Υ(1S)γ with
bJ
→
J =0, 1, 2withinpotentialNRQCD.Wehaveassumedthatthemomentumscaleoftheheavyquarko-
nium involvedlies in the strict weak-couplingregime of pNRQCD and non-perturbativeeffects are
suppressed, such that a full perturbative calculation can be performed. Relativistic corrections of
relative order v2 to the LO decay rate are included. The analysis separates those contributionsthat
accountforthehigherorderelectromagneticinteractiontermsinthepNRQCDLagrangianandthose
EPJWebofConferences
thataccountforquarkoniumstatecorrectionsduetohigherorderpotentialsandtransitionsbetween
singletsandoctets.
Acknowledgements
S.S.andJ.S.thankN.BrambillaandA.Vairoforcollaborationandsupervisionontheworkpresentedhereand
C.Peset,A.Pineda,Y.SuminoandY.Kiyofornumerousinformativediscussions. S.S.expresseshisgratitude
tothePhysik-DepartmentoftheTechnischeUniversitätMünchenwhosesupporthelpedhisparticipationinthe
Conference.J.S.acknowledgesthefinancialsupportfromtheAlexandervonHumboldtFoundation.
References
[1] K.Hanetal.,Phys.Rev.Lett.49,1612(1982)
[2] G.Eigenetal.,Phys.Rev.Lett.49,1616(1982)
[3] C.Klopfensteinetal.,Phys.Rev.Lett.51,160(1983)
[4] F.Paussetal.,PhysicsLettersB130,439(1983)
[5] K.A.Oliveetal.(ParticleDataGroup),Chin.Phys.C38,090001(2014)
[6] J.Segovia,P.G.Ortega,D.R.Entem,F.Fernández,Phys.Rev.D93,074027(2016)
[7] N.Brambilla,A.Pineda,J.Soto,A.Vairo,Rev.Mod.Phys.77,1423(2005)
[8] N.Brambillaetal.,Eur.Phys.J.C71,1534(2011)
[9] N.Brambillaetal.,Eur.Phys.J.C74,2981(2014)
[10] N.Brambilla,Y.Jia,A.Vairo,Phys.Rev.D73,054005(2006)
[11] A.Pineda,J.Segovia,Phys.Rev.D87,074024(2013)
[12] N.Brambilla,P.Pietrulewicz,A.Vairo,Phys.Rev.D85,094005(2012)
[13] P.Pietrulewicz(2013),[PoSConfinementX,135(2012)],1301.1308
[14] S.Steinbeißer,J.Segovia,A.Vairo,inpreparation:TUM-EFT86/16
[15] A.Pineda,J.Soto,Nucl.Phys.Proc.Suppl.64,428(1998)
[16] N.Brambilla,A.Pineda,J.Soto,A.Vairo,Nucl.Phys.B566,275(2000)
[17] W.E.Caswell,G.P.Lepage,Phys.Lett.B167,437(1986)
[18] G.T.Bodwin,E.Braaten,G.P.Lepage,Phys.Rev.D51,1125(1995)
[19] N.Brambilla,A.Pineda,J.Soto,A.Vairo,Phys.Rev.D63,014023(2001)
[20] A.Pineda,Prog.Part.Nucl.Phys.67,735(2012)
[21] W.Fischler,Nucl.Phys.B129,157(1977)
[22] Y.Schroder,Phys.Lett.B447,321(1999)
[23] N.Brambilla,A.Pineda,J.Soto,A.Vairo,Phys.Rev.D60,091502(1999)
[24] B.A.Kniehl,A.A.Penin,Nucl.Phys.B563,200(1999)
[25] C.Anzai,Y.Kiyo,Y.Sumino,Phys.Rev.Lett.104,112003(2010)
[26] A.V.Smirnov,V.A.Smirnov,M.Steinhauser,Phys.Rev.Lett.104,112002(2010)
[27] K.G.Chetyrkin,J.H.Kuhn,M.Steinhauser,Comput.Phys.Commun.133,43(2000)