�b and �c radiative decays in the Salpeter model with the AdS/QCD inspired potential

Floriana Giannuzzi

Universita degli Studi di Bari, I-70126 Bari, Italy and INFN, Sezione di Bari, I-70126 Bari, Italy(Received 21 October 2008; published 2 December 2008)

The decay constants and the radiative decay widths of �bðnSÞ and �cðnSÞ are computed within a

semirelativistic quark model, using a potential found through the AdS/QCD correspondence. For �c, the

results are in agreement with experimental data, while in the case of �0c a discrepancy is found and the

possible reasons are discussed.

DOI: 10.1103/PhysRevD.78.117501 PACS numbers: 12.39.Ki, 12.39.Pn, 13.20.Gd, 14.40.Gx

The recent observation of the bottomonium ground state�bð1SÞ by the BABARCollaboration, in the�ð3SÞ radiativedecay mode:�ð3SÞ ! ��b [1], has led to a new interest inthis particle and its decays. The measured mass is

M�b¼ 9388:9þ3:1

�2:3ðstatÞ � 2:7ðsystÞ MeV (1)

corresponding to a hyperfine splitting M�ð1SÞ �M�bð1SÞ ¼71:4þ2:3

�3:1ðstatÞ � 2:7ðsystÞ MeV [1].

On the other hand, the subsequent decay of �b has notbeen observed. The meson �b is expected to decay mainlyto hadrons; other possible modes are the radiativetransitions.

Motivated by this new experimental observation, in thispaper a calculation of the �b decay constant and of thewidth relative to the decay �b ! �� is presented, in theframework of the model proposed in Ref. [2]. Moreover,the calculation has been extended to the radial excitationsand to the charmonium corresponding states, since themodel can be properly employed for heavy quarkoniumstates. This can be useful, since in many cases the experi-mental values are not known and since there are somediscrepancies among the predictions of different theoreti-cal models and the experimental values. For example, thereis only one measurement, by the CLEO Collaboration, ofthe �0

c ! �� radiative decay width: ���ð�0cÞ ¼

1:3� 0:6 KeV [3], a result not reproduced by most oftheoretical predictions which suggest larger values.Within this model, the decay constants and the leptonicdecay widths of vector b �b and c �c mesons are also eval-uated and a comparison with the experimental results iscarried out.

In the model introduced in Ref. [2] the meson spectrumis computed solving a semirelativistic wave equation, theSalpeter equation:

ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2

1 �r2q

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2

2 �r2q

þ VðrÞÞc ðrÞ ¼ Mc ðrÞ; (2)

where m1 and m2 are the masses of the constituent quarksand M and c ðrÞ are the mass and the wave function of themeson, respectively. The ‘ ¼ 0 case is considered. Thepotential VðrÞ comprises three terms:

VðrÞ ¼ VAdS=QCDðrÞ þ VspinðrÞ þ V0: (3)

The main feature of the model is that the static potentialVAdS=QCDðrÞ is obtained evaluating the expectation value ofthe Wilson loop in the AdS/QCD framework [4], whichprovides a holographic model able to describe some as-pects of QCD, namely, linear confinement, Regge trajec-tories, glueball spectrum, the light meson spectrum, anddecay constants [5]. The static q �q potential is obtained, inthis framework, as a parametric function [4]:

VAdS=QCDð�Þ ¼ g

�

ffiffiffiffic

�

r ��1þ

Z 1

0dvv�2

�½e�v2=2ð1�v4e�ð1�v2ÞÞ�1=2� 1��;

rð�Þ ¼ 2

ffiffiffiffi�

c

s Z 1

0dvv2e�ð1�v2Þ=2ð1�v4e�ð1�v2ÞÞ�1=2;

(4)

where r is the interquark distance and � varies in the range:0 � � < 2. The potential VAdS=QCDðrÞ, therefore, dependson two parameters, g and c; it is depicted in Fig. 1 for thetwo values of c and g employed in the present analysis.The term of the potential VðrÞ in Eq. (3) accounting for

the spin-interaction is given by

2 4 6 8 10

r GeV 1

2

1

0

1

2

Vr

GeV

FIG. 1. The q �q potential VAdS=QCDðrÞ obtained from the AdS/QCD correspondence, with c ¼ 0:4 GeV2 and g ¼ 2:50.

PHYSICAL REVIEW D 78, 117501 (2008)

1550-7998=2008=78(11)=117501(4) 117501-1 � 2008 The American Physical Society

VspinðrÞ ¼ A~�ðrÞm1m2

S1 � S2 with ~�ðrÞ ¼��ffiffiffiffi�

p�3e��2r2 ;

(5)

and involves the parameters �, together with Ab and Ac forthe cases of beauty and charm, respectively. The constantterm V0 is the last parameter fixing the potential in theSalpeter equation.

The singularity of the wave function at r ¼ 0 is regu-lated introducing a value rmin, so that at smaller distancesthe potential is constant and equal to VðrminÞ. At odds withthe analysis in [2], where rmin is an additional input pa-rameter in the fit of the meson spectrum, here we fix rmin

according to a QCD duality argument [6]: rmin ¼ 4�3M ,

where M is the mass of the meson, so that the input setof parameters includes c ¼ 0:4 GeV2, g ¼ 2:50, and V0 ¼�0:47 GeV for the AdS/QCD potential, Ac ¼ 14:56, Ab ¼6:49, and � ¼ 0:47 GeV for the spin term, and the con-stituent quark masses mc ¼ 1:59 GeV and mb ¼5:02 GeV; the values of all the parameters are fixed by abest fit of the meson masses computed in the model to theobserved heavy meson masses [7].

Within the Salpeter model, the decay constants fP andfV of a pseudoscalar and a vector meson, defined by

h0jA�ijjPðkÞi ¼ ik�QijfP;

h0jV�ij jVðk; �Þi ¼ �ð�Þ�QijmVfV;

(6)

respectively, where k is the momentum, � the helicity, and� the polarization vector of the meson, are given by [8]

fP ¼ ffiffiffi3

p 1

2�M

Z þ1

0dkk~u0ðkÞN1=2

��1� k2

ðEi þmiÞðEj þmjÞ�;

fV ¼ ffiffiffi3

p 1

2�M

Z þ1

0dkk~u0ðkÞN1=2

��1þ k2

3ðEi þmiÞðEj þmjÞ�; (7)

with

N ¼ ðEi þmiÞðEj þmjÞEjEi

:

In (6), A�ij is the axial current �qi�5�

�qj, V�ij is the vector

current �qi��qj, and Qij is the meson flavor matrix. In (7),

M is the mass of the meson, mi is the mass of the constitu-ent quark i and Ei its energy, ~uðkÞ is the meson reducedwave function in momentum space, obtained by Fouriertransforming the reduced radial wave function uðrÞ ¼rc ðrÞ; and k is the momentum of the constituent quarkin the rest frame of the meson.The obtained spectrum and decay constants of c �c and b �b

S-wave mesons are collected in Table I; in Fig. 2 thecorresponding wave functions are depicted. It is interestingto notice that f�c

turns out to be compatible with a deter-

mination obtained by the CLEO Collaboration: f�c¼

335� 75 MeV [9].Using the computed values of fP and fV , it is possible to

determine the widths ��� of the radiative decays

�b;cðnSÞ ! ��, and the widths �‘þ‘� of the processes

c ðnSÞ ! ‘þ‘� and �ðnSÞ ! ‘þ‘�. The widths can beeasily computed using the effective Lagrangians [10,11]:

TABLE I. Masses of pseudoscalar and vector c �c and b �b states compared to the experimentaldata. In the third column the decay constants, computed using (7), are reported.

Particle Th. mass (MeV) Exp. mass (MeV) [7] Decay const. (MeV)

�c 3025.3 2980:3� 1:2 342

�0c 3603.5 3637:0� 4 266

�00c 4039.3 195

J=c 3079.8 3096:916� 0:011 356

c 0 3624.3 3686:09� 0:04 237

c 00 4057.0 4039� 1 185

�b 9433.9 9388:9þ3:1�2:3ðstatÞ � 2:7ðsystÞ [1] 637

�0b 9996.8 430

�00b 10347.5 367

� 9438.3 9460:30� 0:26 686

�ð2SÞ 9998.6 10023:26� 0:31 484

�ð3SÞ 10348.8 10355:2� 0:5 335

�ð4SÞ 10622.3 10579:4� 1:2 301

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117501-2

L ��eff ¼ �ic1ð �q���5qÞ���F

�A;

L‘ �‘eff ¼ �c2ð �q��qÞð‘��

�‘Þ;(8)

where

c1 ¼ Q24��em

ðM2 þ EbMÞ ; c2 ¼ Q4��em

M2: (9)

One obtains

��� ¼ 4�Q4�2emM

3f2PðM2 þ EbMÞ2 ; �‘þ‘� ¼ 4�Q2�2

emf2V

3M;

(10)

where Q is the electric charge (in units of e) of theconstituent quark and Eb ¼ 2m�M is the binding energy.

The values obtained for the pseudoscalar mesons areshown in Table II, together with recent theoretical results.The prediction for the �c radiative decay width is compat-ible with the experimental value within the error; in the

case of �0c, the measurement by the CLEO Collaboration

[3] is smaller (or marginally compatible) than the obtainedtheoretical prediction and that in other calculations [15].Concerning the b �b pseudoscalar meson, the theoretical

models in Table II predict, for the �b ! �� decay width,values in the range 230–560 eV; the result obtained in thispaper points towards small values in this range.For vector mesons, the predicted and the experimental

values of the leptonic decay widths are reported inTable III. There is an overall agreement, excluding adiscrepancy in the �ð3SÞ that could be attributed to apossible D-wave component in this meson.In conclusion, the decay constants and the radiative

decay widths of b �b and c �c pseudoscalar mesons, computedwithin a semirelativistic quark model which uses a poten-tial inspired by the AdS/QCD correspondence, are com-patible with the experimental data, in particular, in the caseof f�c

and ���ð�cÞ. The measurement of ���ð�0cÞ carried

out by the CLEO Collaboration [3] is not reproduced, since

TABLE II. Decay widths ��� (in KeV) of pseudoscalar states in two photons. The value denoted by * is reported by the PDG [7] as adatum not included in the summary tables.

Particle This paper Lansberg et al.[11] Lakhina et al.[12] Kim et al.[13] Ebert et al.[14] Exp.

�c 4.252 7.46 7.18 7:14� 0:95 5.5 7:2� 0:7� 2:0 *

�0c 3.306 4.1 1.71 4:44� 0:48 1.8 1:3� 0:6 [3]

�00c 1.992 1.21

�b 0.313 0.560 0.230 0:384� 0:047 0.350

�0b 0.151 0.269 0.070 0:191� 0:025 0.150

�00b 0.092 0.208 0.040 0.100

1 2 3 4 5k GeV

4

2

0

2

4

6

u ηc

nSk

1 2 3 4 5k GeV

4

2

0

2

4

6

u ηb

nSk

1 2 3 4 5k GeV

4

2

0

2

4

6

unS

k

1 2 3 4 5k GeV

4

2

0

2

4

6

u YnS

k

FIG. 2 (color online). The momentum wave functions of the first three states of �cðnSÞ (top left), �bðnSÞ (top right), J=c ðnSÞ(bottom left), and �ðnSÞ (bottom right). The continuos line represents the 1S state, the dotted line represents the 2S state, the dashedline represents the 3S state, and the dashed-dotted line represents the 4S state. The wave functions are dimensionless: they arenormalized as

Rdkj~uðkÞj2 ¼ 2M.

BRIEF REPORTS PHYSICAL REVIEW D 78, 117501 (2008)

117501-3

the obtained result differs by more than 2�. In this respect,our result follows most theoretical models [10,12–15],

which predict higher values for ���ð�0cÞ, although in

some cases within the experimental error. This mightsuggest that the disagreement could be attributed to thesystematics of the experimental measurement, namely, tothe assumption that �c and �0

c have the same branchingfractions to the final state KSK�. As for �b, the predictionof the �b ! �� decay width suggests that this decay modecould be observed in the forthcoming experimentalanalyses.

I would like to thank P. Colangelo, F. De Fazio, S.Nicotri, and T. N. Pham for useful suggestions and discus-sions. This work was supported in part by the EU ContractNo. MRTN-CT-2006-035482, ‘‘FLAVIAnet.’’

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[2] M.V. Carlucci, F. Giannuzzi, G. Nardulli, M. Pellicoro,and S. Stramaglia, arXiv:0711.2014 [Eur. Phys. J. C (to bepublished)].

[3] D.M. Asner et al. (CLEO Collaboration), Phys. Rev. Lett.92, 142001 (2004).

[4] O. Andreev and V. I. Zakharov, Phys. Rev. D 74, 025023(2006).

[5] A. Karch, E. Katz, D. T. Son, and M.A. Stephanov, Phys.Rev. D 74, 015005 (2006); P. Colangelo, F. De Fazio, F.Jugeau, and S. Nicotri, Phys. Lett. B 652, 73 (2007); P.Colangelo, F. De Fazio, F. Giannuzzi, F. Jugeau, and S.Nicotri, Phys. Rev. D 78, 055009 (2008); S. J. Brodsky andG. F. deTeramond, arXiv:0802.0514, and referencestherein.

[6] P. Cea and G. Nardulli, Phys. Rev. D 34, 1863 (1986).[7] C. Amsler et al. (Particle Data Group), Phys. Lett. B 667, 1

(2008).[8] P. Colangelo, G. Nardulli, and M. Pietroni, Phys. Rev. D

43, 3002 (1991).

[9] K.W. Edwards et al. (CLEO Collaboration), Phys. Rev.Lett. 86, 30 (2001).

[10] J. P. Lansberg and T.N. Pham, Phys. Rev. D 74, 034001(2006).

[11] J. P. Lansberg and T.N. Pham, Phys. Rev. D 75, 017501(2007).

[12] O. Lakhina and E. S. Swanson, Phys. Rev. D 74, 014012(2006).

[13] C. S. Kim, T. Lee, and G. L. Wang, Phys. Lett. B 606, 323(2005).

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TABLE III. Decay widths �‘þ‘� (in KeV) of vector mesons.

Particle This paper Exp. [7]

J=c 4.080 5:55� 0:14� 0:02c 0 2.375 2:38� 0:04c 00 0.836 0:86� 0:07� 1.237 1:340� 0:018�ð2SÞ 0.581 0:612� 0:011�ð3SÞ 0.270 0:443� 0:008�ð4SÞ 0.212 0:272� 0:029

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