b

alar

seudoscalarons in ther.

Physics Letters B 599 (2004) 55–61

www.elsevier.com/locate/physlet

On the mass differences between the scalar and pseudoscheavy-light mesons

Damir Becirevic a, Svjetlana Fajferb,c, Saša Prelovšekb,c

a Laboratoire de Physique Théorique (Bât 210) 1, Université Paris Sud, Centre d’Orsay, 91405 Orsay-cedex, Franceb J. Stefan Institute, Jamova 39, PO Box 3000, 1001 Ljubljana, Slovenia

c Department of Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia

Received 9 July 2004; received in revised form 12 August 2004; accepted 13 August 2004

Available online 25 August 2004

Editor: P.V. Landshoff

Abstract

We discuss the recent experimental observation which suggested that the mass difference between the scalar and pheavy-light mesons is larger for the non-strange states than for the strange ones. After computing the chiral correctiheavy quark limit we show that, contrary to experiment, the mass difference in the non-strange case should be smalle 2004 Elsevier B.V. All rights reserved.

dion

t ofO-the

bothholdly

raryarkermo-

alaru-

ated

1. Conflict between theory and experiment

High statisticsB-factory experiments at BaBar anBelle, besides providing the substantial informatabout the CP-violation in the processes involvingB-mesons, also allowed for a precision measurementheD-meson spectrum. Together with CLEO and FCUS, all four experiments reported the presence ofnarrow scalar (JP = 0+) and axial (JP = 1+) states[1], the average of which is found to be

m(0+)

D∗s0

= 2317.0(4) MeV,

E-mail address: damir.becirevic@th.u-psud.fr(D. Becirevic).1 Unité mixte de Recherche du CNRS-UMR 8627.

0370-2693/$ – see front matter 2004 Elsevier B.V. All rights reserveddoi:10.1016/j.physletb.2004.08.027

(1)m(1+)

D′s1

= 2458.2(1.0) MeV.

These results were somewhat surprising becausethe scalar and axial states are below the thresof their dominant—Zweig allowed—modes, namemD∗

s0< mD + mK andmD′

1s< mD∗ + mK . Therefore

the newly observed states are very narrow, contto what has been predicted by many potential qumodels[2].2 This motivated many authors to eithgeneralise the quark model potentials as to accomdate the narrowness of the mentioned states[4], em-ploy the unitarised meson model to the charmed scstates[5], or to revive the old ideas about the molec

2 Note however that such a low scalar state mass was anticipin the model calculation of Ref.[3].

.

56 D. Becirevic et al. / Physics Letters B 599 (2004) 55–61

hpec-e byer-anylek”be

tis-et

es,per--

arearklarn to

es.us-rd-htr

of a

e

ess

mus

for

sis

re,ed

nt

arkre-

ntalr-US

na-

-T).del

the

lar structure of these excitations[6]. Before attributingsomething exotic to the states(1), one should carefullycheck if the minimal “quark–antiquark”picture, whichas been so successful in the history of hadron stroscopy, indeed fails. Such a test cannot be madinsisting on the accuracy of the quark models at a pcent level because of the questionable contact ofspecific quark model parameter with QCD. A reliabtest of compatibility between the “quark–antiquarpicture and the observed hadronic spectra couldmade by means of the fully unquenched high statics lattice studies, which are unfortunately not yavailable. The two partially quenched lattice studithat appeared after the announcement of the eximental numbers(1), reached two different conclusions: while Ref.[7] suggests that the new statesunlikely to be the scalar and axial quark–antiqumesons, in Ref.[8] the difference between the scaand pseudoscalar charm-strange mesons is showbe consistent with the experimentally measured on3

When cataloging the heavy-light mesons it is ctomary to use the heavy quark spin symmetry accoing to which the total angular momentum of the ligdegrees of freedom (jP

� ) is a good quantum numbeso that the heavy-light mesons come in doubletscommonjP

� , e.g.,[D(s)

(0−)

,D∗(s)

(1−)]

︸ ︷︷ ︸jP� = 1

2−

(L=0)

,[D∗

0(s)

(0+)

,D′1(s)

(1+)]

︸ ︷︷ ︸jP� = 1

2+

(L=1)

,

(2)[D1(s)

(1+)

,D∗2(s)

(2+)]

︸ ︷︷ ︸jP� = 3

2+

(L=1)

, . . .

where the index “s” helps distinguishing the strangfrom non-strange heavy-light mesons.

After comparing to the well known lowest stat(belonging tojP

� = 12−

) [10], we see that the splitting

�ms(0) ≡ mD∗0s

− mDs = 348.4(9) MeV,

(3)�ms(1) ≡ mD′1s

− mD∗s= 345.9(1.2) MeV,

are equal. In other wordsthe hyperfinesplitting in thefirst orbitally excited doublet is indistinguishable frothe one in the ground state doublet. Although varioquark models give different numerical estimates

3 Compatibility with observation was also claimed on the baof results obtained by using the QCD sum rules[9].

�ms(0), almost all of them share a common featunamely this orbital splitting remains almost unchangafter replacing the lights- by u- or d-quark. The sur-prise (now for real) actually came from experimewhen Belle reported[11]

�mu(0) ≡ mD∗0− mD = 444(36) MeV,

(4)�mu(1) ≡ mD′1− mD∗ = 420(36) MeV,

clearly larger than the ones with the strange light qu(3), even though the error bars in the non-strangesults are much larger which reflects the experimedifficulty in identifying the broad states. The confimation of this phenomenon came recently by FOC[12], namely,

mD∗0= 2407(21)(35) MeV

(5)�⇒ �mu(0) = 538(41) MeV.

This truly surprising phenomenon requires an explation. Since, to a very good approximation,�mu,s(0) =�mu,s(1), we shall concentrate on�mu,s(0) and ar-gue that the experimentally established inequality

(6)[�mu(0) − �ms(0)

]exp> 0,

is in conflict with theory if the phenomenon is examined by means of chiral perturbation theory (ChPA similar conclusion has been reached by the mocalculations of Ref.[13].

2. Chiral Lagrangian for doublets of heavy-lightmesons

The lagrangian that is necessary for studyingmass difference between the12

+and 1

2−

heavy-lightstates is[14]

L= L 12

− +L 12

+ +Lmix +Lct,

L 12

− = i Tr[Hbv · DbaHa

] + g Tr[Hbγµγ5Aµ

baHa

],

L 12

+ = − Tr[Sb(iv · Dba + ∆S)Sa

]+ g Tr

[Sbγµγ5Aµ

baSa

],

Lmix = hTr[Sbγµγ5Aµ

baHa

] + h.c.,

(7)

Lct = Tr[(

λHaHb − λSaSb

)(ξMξ + ξ†Mξ†)

ba

]+ Tr

[(λ′HaHa − λ′SaSa

)× (

ξMξ + ξ†Mξ†)bb

],

D. Becirevic et al. / Physics Letters B 599 (2004) 55–61 57

in

the

nedif-sonre

en

ndby

g

y-

ofing

e-

nin

where the fields of pseudoscalar (P ), vector (P ∗µ),

scalar (P0) and axial (P ∗1 µ) mesons are organised

superfields

Ha(v) = 1+ /v

2

[P ∗ a

µ (v)γµ − Pa(v)γ5],

Ha(v) = γ0H†a (v)γ0,

Sa(v) = 1+ /v

2

[P ∗ a

1 µ(v)γµγ5 − Pa0 (v)

],

(8)Sa(v) = γ0S†a (v)γ0,

with “a” and “b” labelling the light quark flavour. Inaddition

DµbaHb = ∂µHa − HbVµ

ba

= ∂µHa − Hb1

2

[ξ†∂µξ + ξ∂µξ†]

ba,

Aabµ = i

2

[ξ†∂µξ − ξ∂µξ†]

ab,

(9)ξ = √Σ, Σ = exp

(2i

φ

f

),

with f ≈ 130 MeV, M = diag(mu,md,ms), andφ

the usual matrix of pseudo-Goldstone bosons,

(10)

φ =

1√2π0 + 1√

6η π+ K+

π− − 1√2π0 + 1√

6η K0

K− K0 − 2√6η

.

g andg are the couplings of the Goldstone boson topair of heavy-light mesons withjP

� = 12−

and 12+

, re-spectively.4 h, instead, is the coupling of a Goldstoboson and the heavy-light mesons belonging toferent heavy quark spin doublets, namely one meis 1

2−

and the other12+

state. The meson masses amH 1

2± = mQ + E 1

2± , whereas the difference betwe

the binding energies in the first orbital excitation ain the lowest lying heavy meson states is denoted∆S = E 1

2+ − E 1

2− .

4 The couplingg is proportional to the commonly used couplingD∗Dπ , whereasg is proportional togD∗D′ π .

0 1

3. Chiral correction to the mass of heavy-lightmesons

Since we work in the heavy quark limit, the heavlight meson propagator is a function ofv ·k only.kµ =pPµ −mQvµ, is the momentum of the light degreesfreedom in the heavy-light meson. The chiral dress

of the 12−

-meson propagator,

Gq

12

−(v · k) = i

2v · k + i

2v · k(−iΣq(v · k)

) i

2v · k(11)+ · · · ,

generates a shift to its binding energy,E 12

− → E 12

− +δEq

12

− , where

(12)δEq

12

− = 1

2lim

v·k→0Σq(v · k).

Similarly,

Gq

12

+(v · k)

= i

2(v · k − ∆S)+ i

2(v · k − ∆S)

(13)× (−iΣq(v · k)) i

2(v · k − ∆S)+ · · ·

leads to

(14)δEq

12

+ = 1

2lim

v·k→0Σq(v · k).

Therefore the mass splitting between12+

and12−

statesin the heavy quark limit is

(15)�mq(0) = ∆S + δEq

12

+ − δEq

12

−,

where the light valence quark in the heavy-light mson,q ∈ {u/d, s}. We will work in the isospin limit,mu = md = mu/d . We focus onto the scalar mesoand compute the chiral loop corrections illustratedFig. 1.

−iΣ(a)q (v · k)

=8∑

i=1

3∑a=1

∫d4p

(2π)4

−2gpα

f

(t i†

)qa

(16)× −i(gαβ − vαvβ)

2v · (k + p)

2gpβ

f

(t i

)aq

i

p2 − m2i

.

58 D. Becirevic et al. / Physics Letters B 599 (2004) 55–61

e

Fig. 1. Graphs contributing to the chiral shift in the binding energy of the scalar mesonJP = 0+. By flipping all the parity signs, one gets thgraphs relevant to the case of pseudoscalar meson also discussed in the text.

sede

eme-

rt-to

f

heof

This integral is standard and the result is expresin terms of functionsJ1,2 (explicit expressions can bfound in, for example, Appendix A of Ref.[15]) lead-ing to

−iΣ(a)q (v · k)

=8∑

i=1

−2g2

f 2

(t i t i

)qq

3i

(4π)2 (v · k)J1(mi,−v · k)

�⇒ limv·k→0

Σ(a)q (v · k)

(17)= − 6g2

(4πf )2

8∑i=1

(t i t i

)qq

2π

3m3

i .

In a completely analogous way, after exchanging “+”↔ “−” in the graph (a), we have

(18)limv·k→0

Σ(a)q (v · k) = − 6g2

(4πf )2

8∑i=1

(t i t i

)qq

2π

3m3

i .

As for the diagram (b) we obtain

−iΣ(b)q (v · k)

=8∑

i=1

3∑a=1

∫d4p

(2π)4

−2hvα

f

(t i†

)qa

× ipαpβ

2[v · (k + p) − (−∆S)]2hvβ

f

(t i

)aq

i

p2 − m2i

= − 2ih2

(4πf )2

8∑i=1

(t i t i

)qq

(−∆S − v · k)

(19)

× [J1(mi,−∆S − v · k) + J2(mi,−∆S − v · k)

],

and therefore

limv·k→0

Σ(b)q (v · k)

= − 2h2∆S

(4πf )2

8∑i=1

(t i t i

)qq

(20)× [J1(mi,−∆S) + J2(mi,−∆S)

].

Similarly,

limv·k→0

Σ(b)q (v · k)

= 2h2∆S

(4πf )2

8∑i=1

(t i t i

)qq

(21)× [J1(mi,∆S) + J2(mi,∆S)

].

Notice that compared to Eq.(20) the sign in frontof ∆S in the argument of the functionsJ1,2 is nowchanged. This reflects the fact that the intermediatheavy-light meson, with respect to the mass of theson in the external leg, is now heavier.

After collecting the above expressions into Eq.(15),we arrive at

�mq(0) = ∆S

(1− h2

(4πf )2

(t i t i

)qq

×∑z=±

[J1(mi, z∆S) + J2(mi, z∆S)

])

+ g2 − g2

8πf 2

(t i t i

)qq

m3i + 2(λ − λ)mq

(22)+ 2(λ′ − λ′)(mu + md + ms),

where in the last line we also included the counteerms, thus completing the NLO chiral correctionsthe mass splitting we consider. The integralsJ1,2 alsocarry an implicit dependence on the scaleµ whichcancels against the one inλ′ − λ′. Finally, in evalu-ating the integralsJ1,2, we set∆ = 0 (see Eq. (44) oRef. [15]).

Note that in our loop calculations we include tlight pseudogoldstone bosons only. The inclusion

D. Becirevic et al. / Physics Letters B 599 (2004) 55–61 59

the

en-inedon-

ctys-

of

lly

ax-

ts

his

y

arel-

u-dth

d

tes-ht

elle

lle

inht

r

di-e-t

eg-

light resonances, such asρ, K∗, φ, would involvehigher orders in chiral expansion which is beyondscope of the approach adopted in this Letter[16].

4. Chiral enhancement or suppression?

To examine whether or not the apparent chiralhancement observed experimentally can be explaby the approach adopted in this Letter we need to csider

�mu/d(0) − �ms(0)

= h2∆S

(4πf )2

∑z=±

[J1(mK, z∆S) + 1

2J1(mη, z∆S)

− 3

2J1(mπ, z∆S) + J2(mK, z∆S)

+ 1

2J2(mη, z∆S) − 3

2J2(mπ, z∆S)

]

− g2 − g2

16πf 2

(m3

η + 2m3K − 3m3

π

)(23)− 2(λ − λ)(mu/d − ms).

By using the Gell-Mann formulae,

m2π = 2B0msr, m2

K = 2B0ms

r + 1

2,

(24)m2η = 2B0ms

r + 2

3,

where r = mu/d/ms and 2B0ms = 2m2K − m2

π =0.468 GeV2, we can simply plot the Eq.(23) againstthe variation ofr, the light quark mass with respeto the strange quark which is kept fixed to its phical value. Before doing so we discuss our choicevalues for the couplingsh, g and g, and for the lowenergy constantλ − λ:

• g-coupling has been determined experimentafrom the width of the chargedD∗-meson,g =0.61(1)(6) [17];5

• There is no experimental determination of theial coupling in the orbitally excited doublet,g.While the non-relativistic quark model predic

5 A short review of lattice and QCD sum rule estimates of tquantity can be found in Ref.[18].

|g/g| = 1/3, a relativistic model which correctlpredictedg before it was measured[19], one getsg = 0.03. The QCD sum rule based estimatesg = 0.10(2) [20]. To cover the whole range of vaues we will takeg = 0.2(2);

• The experimental situation withh, the pionic cou-pling between mesons belonging to different doblets, is less clear. If we take the mass and wiof the scalar meson as measured by Belle[11],we get h = 0.78(9)(8), while those measureby FOCUS[12] give h = 0.56(8)(6), in a verygood agreement with the QCD sum rule estimah = 0.60(13) [20]. From the recent lattice computation of the width of the scalar heavy-ligstate[21], we deduceh = 0.62(6)(4), where weused the scalar meson mass measured by B[mD∗

0= 2308(36) MeV], which is more reliable

than the one measured by FOCUS in that Beproperly separate 0+ and 1+ signals.6 The modelof Ref. [19] predictsh � 0.54. To take the fullspread of the mentioned values we will useh =0.6(2).

• In the recent unquenched lattice study[22], it hasbeen shown that the splitting that we discussthis Letter changes very weakly when the ligquark is varied betweenr = 0.65 andr � 1. Wewill then fix the value ofK in 2(λ − λ)(mu/d −ms) → K(m2

π − m2K), by imposing the limit that

Eq. (23) allows for a variation smaller than oequal to−50 MeV, forr ∈ (0.65,1]. Limiting val-ues areK(1 GeV) � 0.7 GeV−1, for the variationto −50 MeV, andK(1 GeV) � 1.3 GeV−1, for novariation at all.7

In Fig. 2we plot the result of Eq.(23)by using thecentral values for the couplings listed above. In adtion we take∆S = 0.35 GeV. We see that when thpion becomes lighter than∆S , the self energies develop the imaginary part,which reflects the fact thathe real pion can be emitted viaP ∗

0 → Pπ . Most im-portantly, we see that the real part remains always n

6 We thank the referee for drawing our attention to this point.7 These values are obtained by choosingµ = 1 GeV. Had we

chosen any otherµ, the correspondingK(µ) would be differentbut the resulting�mu/d (0) − �ms(0) would obviously remain thesame.

60 D. Becirevic et al. / Physics Letters B 599 (2004) 55–61

is-

real

hisara-lutet-

ver,t of

clu-e.entthe

si-

ass

thecentug-onshe

s ofal-Weine

entsl

rea-w-sta-

-r. It

sed

e in-rk”sedrx-ful

oneighton-were

Lek oftryof

Fig. 2. Eq.(23) against the variation ofr = mu/d/mphyss . The phys-

ical situations correspond tor = 1 andru/d = 0.04 [23]. We used∆S = 0.35 GeV, the central values for the chiral couplings, as dcussed in the text, andK(1 GeV) = 0.7 GeV−1. The imaginary partis depicted by the dotted line which reflects the opening of thepion emission channelP ∗

0 → Pπ .

ative

(25)�mu/d(0) − �ms(0) < 0,

contrary to what is experimentally established. Tconclusion remains as such when varying the pmeters in the ranges indicated above. The absovalue of the difference of splittings(23)depends mosstrongly on the value of theh-coupling and it is negative ∀h = 0. The term proportional tog2 − g2 isnegative too. It would change the sign only ifg2 > g2

which is beyond reasonable doubt. Notice, howethat it has been argued recently that the double12+

states could be the chiral partner of the12−

dou-blet, which would imply thatg = g [24]. Even if thatassumption was indeed verified in nature, our consion that Eq.(23) is always negative, remains truHowever, as we explained above, from the prestheoretical understanding the equality betweentwo couplings,g = g, does not appear to be plauble.8

8 Actually, in any Dirac equation based model,g = g can be ob-tained only if one employs the free spinors and set the quark mto zero.

5. Conclusion

In this Letter we discuss the mass difference ofscalar and pseudoscalar heavy-light mesons. Reexperimental observation by Belle and FOCUS sgests that such a difference in the charmed mesis larger for the non-strange light quark than for tstrange one, i.e.

(26)[(mD∗

0− mD) − (mD∗

s0− mDs )

]exp> 0.

Such a phenomenon cannot be explained by meanpotential quark models in which this difference ismost independent of the valence light quark mass.instead used the chiral perturbation theory to examif the chiral enhancement suggested by experimcan indeed be reproduced. After calculating the chiracorrections, we obtain that

(27)[(mD∗

0− mD) − (mD∗

s0− mDs )

]theo< 0.

This apparent problem remains as such for anysonable choice of the chiral couplings. It should, hoever, be stressed that our calculation refers to thetic heavy quark (mQ → ∞) which might be questionable when discussing the charm quark sectois nevertheless unlikely that theO(1/mn

c ) correctionscould change the clear qualitative result summariin Eqs.(26), (27).

Our observations show that the scalar states ardeed peculiar. It is probable that the “quark–antiquapicture is not adequate in case of which the unitarimeson model of Ref.[5] or the 4-quark picture fothe scalar mesons[25], which enjoyed success in eplaining the spectrum of light mesons, may be useremedy in explaining the scalar states containingheavy quark. Further experimental tests, that mprove useful in getting a more definite answer ccerning the nature of the observed scalar states,already proposed in Ref.[26].

Acknowledgements

It is a pleasure to thank S. Descotes and A.Yaouanc for discussions and comments. The worS.F. and S.P. was supported in part by the Minisof Education, Science and Sport of the RepublicSlovenia.

D. Becirevic et al. / Physics Letters B 599 (2004) 55–61 61

90

03)

p-

95)

37,

ep-

hep-

84.03,

3)

ep-

9.9

p-

6

4)

86

96)

-ph/

03)

321

65

p-

ep-

ep-

a-

69

6.3)

0,

ep-

13.;ep-

p-

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