2 January 1997

Physics Letters B 390 (1997) 341-349

PHYSICS LElTERS B

The electromagnetic masses of charmed pseudoscalar mesons A.N. Ivanov, N.I. Troitskaya ’

Institut fiir Kemphysik. Technische Universittit Wien. Wiedner Hauptstr. 8-10, A-1040 Vienna, Austria

Received 3 May 1996; revised manuscript received 6 September 1996 Editor: P.V. Landshoff

Abstract

Heavy-quark effective theory (HQET) and Chiral perturbation theory at the quark level (ChPT),, based on the extended Nambu-Jona-Lasinio model with linear realization of chiral U(3) x U(3) symmetry, are applied to the computation of electromagnetic masses of charmed pseudoscalar mesons. The result of the computation of the electromagnetic mass can be represented in the form of the low-energy theorem like Dashen’s theorem. Chiral logarithm contributions to the mass difference MD+ - MDo are computed.

1. Introduction

In Ref. [ 1 ] Heavy-quark effective theory (HQET) [ 2-41, supplemented by Chiral perturbation theory at the quark level (ChPT), [ 5,161, based on the extended Nambu-Jona-Lasinio model with linear realization of chiral U(3) x U(3) symmetry [6-g], has been applied to the computation of the fine structure of mass spectra of charmed mesons caused by first order corrections in current quark mass expansion. The theoretical predictions for the splitting of the mass spectra have been found in good agreement with experimental data.

The approach using HQET and (ChPT), is very similar to that suggested by Bardeen and Hill [lo] that is also based on the Nambu-Jona-Lasinio model. There is only distinction that in the Bardeen-Hill model heavy-mesons are considered like partners of light mesons, whereas in HQET, supplemented by (ChPT),, heavy mesons are external states with respect to the lights. This distinction influences only the redefinition of the magnitudes of the parameters that are input the models. Nevertheless, all results obtained within HQET and (ChPT), should be fully valid in the Bardeen-Hill model.

In this paper we apply HQET and (ChF’T), to the computation of electromagnetic masses of charmed pseudoscalar mesons. The electromagnetic mass (6M~,),e of the D,-meson with a light q-quark (q = u, d, S)

in the content contains three basic contributions

1 E-mail: ivanov@kph.tuwien.ac.at. Permanent Address: State Technical University, Department of Theoretical Physics, 195251 St. Petersburg, Russian Federation.

0370-2693/97/$17.00 Copyright 0 1997 Published by Elsevier Science B.V. All rights reserved. PII SO370-2693(96)01357-3

342 A.N. lvanov, NJ. Troitskayu / Physics Letters B 390 (1997) 341-349

where (GMDq)ee,sq, (6MD,)ee,qh and (SM D ,I &hh are defined by quark-quark, quark-hadron and hadron-hadron electromagnetic interactions. The computation of these contributions we perform in the chiral limit, when the current q-quark mass me4 tends to zero, i.e., ma4 -+ 0, and in leading order in large N and M, expansions, where N and M, are the number of colours and the c-quark mass.

2. Quark-quark electromagnetic interactions

The contribution of quark-quark electromagnetic interactions can be defined as follows [ 1,l I]

VMD,L?e,qq = -$ J

d4~,d4~*(D,(p>lT(j,(x,)jv(~*)E(O)~(O))JDq(p))D~LY(~2 -XI). D

(2)

The electromagnetic quark current jp(x) is given by

j,(x) = c eqq(.x)Y$?(x) + eJ(x)r/&c(x) +. . * 9 (3) q=u,d,s

where e, = e, = 2e/3 and ed = e, = -e/3 with e, a positron charge, and ellipses denote the contribution of b and t quarks that are irrelevant to the problem under consideration. A photon Green function DpL”(x) reads

Dp"(x) = s

(4)

where 5 is a gauge parameter. We hold the gauge parameter 5 arbitrary, for the analysis of the gauge dependence of electromagnetic masses of charmed pseudoscalar mesons is requested.

Following [ 1,l l-131 and applying the requction technique we bring the r.h.s. of Eq. (2) to the form

(SMD,)ee,ss = ,!im,2 $& s d4X1 d4x2d4yl d4y2 e-ip’(yz-Y1) ( q ly, + Mi) (0, + MT,) +D

x (OlT(j,(x~)ju(X2)~D,(y,)E(O)c(O)(o~~(yz))IO)D”Y(x2 -XI>, (5)

where po,(yr ) and (pLq(y2) are operators of the D,-meson interpolating fields.

In order to analyse the r.h.s. of Eq. (5) at the quark level we assume that the operators PO, (yl ) and qLq ( y2) satisfy the equations of motion [ I,1 l-l 3 ]

The coupling constant gD has been calculated in [ 111 and reads

2?rJz M:, ‘I2 -- gn-fi $j-jY ’

( > (7)

where B’ = 4A = 2.66 GeV and A is the cut-off in Euclidean 3-momentum space, connected to A,, the scale of spontaneous breaking of chiral symmetry (SB,@) in (ChPT),, via the relation A = A,/& = 0.67 GeV at AX = 0.94 GeV [5,16]. The cut-off A appears due to the calculation of heavy-light constituent quark loops describing in HQET and (ChPT), contributions of strong low-energy interactions [ l,lO-131.

Substituting (6) in (5) and accounting (7) we get

(SMol)ee,qq = TF J

ddX, ddX2ddy, dJy2 eiP’(YI-x)

x (OIT(j,(~~)j,(x2)4(y~)~y5c(y~)~(O)c(O)~(y2)~~q(y2))l0)DCLv(~2 - ~1). (8)

Recall that p* = Mi.

A.N. Iuanov, N.I. Troitskaya / Physics Letters B 390 11997) 341-349 343

The r.h.s. of Eq. (8) involves three kinds of contributions,

where

x (O~~(~(x,)y,q(x~)4(x2)~~q(x2)c7(y~)i~Sc(y,)~(O)c(O)~(y2)iySq(y~2))IO)DPY(x2 - ~117

x (OlT(~h >r~c(xI)E(x2)nc(x2)~(y~ >i~c(y,)~(O)c(O)~(y2)i~q(y2))IO)D~””(xz - XI >,

(~ML&+, = eqec%$$ Jd4x,d4x2d4y,d4y2 ei~.(~l-?2)

x (OlT(Q(x~)y,q(x~)~(x2)yvc(x2)4(Y~)i~c(y,)~(O)c(O)~(y2)i~q(y22))lO)DCLY(x2 -XI).

(10)

(11)

(12)

The electromagnetic masses (&MD,)~& (i = 1,2,3) are defined by electromagnetic interactions between light-light, heavy-heavy and heavy-light quarks, respectively.

The computation of (~MD~)$,, (i = 1,2,3) we perform in leading order in large N and M, expansion. In

this approximation ( 6Mog ) $!qq (i = 1,2,3) can be represented by the following momentum integrals [ 1,10-133

(13)

x tr{ys~y5(~)yp(~)y’(~) [u. (ki_b) +i()]Z(u.k:i0]2}

(14)

344 A.N. Ivanov. NJ. Troitskaya / Physics Letters B 390 (1997) 341-349

1 x tr y5 ----Yyc”

m-k m-LQJ(~)yv(F) [,.(k+i) +iOj2 [u+k:iO]}

eqec MD --

+ 6439 iv J J 2 2 [q2:jOl (&v-~%g)

{ 1 xtr ys7yp

m-k m-:-eY((~>YV(~)~~~~k+‘~~+iO~ [u.k:iO]z}’

where m = 0.33 GeV is the light constituent quark mass calculated in the chiral limit [5,16]. The Feynman diagram representation of ( SMD,) $,, (i = 1,2,3) is standard and trivial, that is why we do not give it here.

The detailed computations of the momentum integrals is not difficult but demands bulky of paper size. There- fore, we adduce only the results. The best of all the integration can be performed applying 3-dimensional regu- larization and putting u = (i = 1,2,3) given by

(1,0 ). Holding only the main divergent contributions [ IJO-131 we get (6M~~)Ljf,

At an intermediate stage of the computation of momentum integrals there can appear an infrared cut-off CL. Of course, the observed part of electromagnetic masses should be gauge invariant and independent on ,u. Thereby, the gauge and p dependent contributions should be removed by renormalization of the D,-meson mass.

We should underscore that ( SMD,) ii',, (i = 1,2,3) cut-off p. Summing up the contributions, we get

are found gauge invariant and independent on an infrared

(SMD,),Q~ = -(et + eqec + 2ez) $$ In (17)

This equation describes the electromagnetic mass of the D,-meson inspired by the electromagnetic interactions between quarks.

3. Quark-hadron electromagnetic interactions

The electromagnetic correction to the mass of the D,-meson caused by electromagnetic quark-hadron inter- actions is given by

@MD,)eP.qh = -2 J d4~ld4~2(D,(p)IT(J~(~l)jY(~2)E(O)c(O))ID,(p))DCL”(x2 -xl>, (18) D

where JP(x) is the electromagnetic current of the D,-meson defined in terms of the interpolating operators

pD,(n) and (D&(x)

J,(x) = ieD, (d,(x) &&D,(x) -&&(x) pD,(x))j (19)

and ebg is the electric charge of the D,-meson such as eD, = eD0 = 0 and e& = eD, == eD+ = eb: = e. In leading order in large N and MC expansion the r.h.s. of Eq. ( 18) is represented by the momentum integrals

A.N. lvanov, NJ. Troitskaya/ Physics Letters B 390 (1997) 341-349 345

eD eC MD =_A- 32~~ 5’

+ eDqe, MD d4k d4q

32~~ 8 rr2i J s ;;I;Mb!~~~~;_iO(q2:iO](g,,-~~)

1 1 x tr +--Ty+

m-k [u.(k+q) +iO12 ’ (20)

(2) where ( ~MD,) a:,:, and ( ~MD,) e.,qh describe the contributions of electromagnetic interactions of the D,-meson

with heavy and light quarks, respectively. We omit the representation of ( SMD,)$& and ( SMD,)$~, in terms of Feynman diagrams that is standard and trivial.

In the integrands of Eq. (20) we can apply the limit MD = M, --f co and get

(2P + 4Y u”

M$-(p+q)2-iO~-u.q+iO’

where we have set p” = MD v’. Using (21) we reduce the r.h.s. of Eq. (20) to the form

(&MD,, 1 pe,qh = ( ~MD~ > I&, + ( ~MD~ 1 k&,

(21)

(22)

Omitting the detailed computation of the integrals in (22) we adduce the final result. The electromag- netic corrections ( SMD,)$‘~~ (i = 1,2), caused by interactions between the D,-meson and heavy and light quarks,respectively, and obtained in leading divergent approximation [ l,lO-131, are given by

Summing up the contributions we get

@Mz&r,qh = -eD$?q$ln

(23)

(24)

Thus one can see that ( ~MD~ ) ee,qh is fully defined by the electromagnetic interactions between light quarks and the D,-meson. The electromagnetic correction (SM~~)~e,qh is found dependent on the gauge parameter 5 and an infrared cut-off ,x,

346 A.N. Ivanov. NJ. Troitskayu / Physics Letters B 390 (1997) 341-349

4. Hadron-hadron electromagnetic interactions

Now we can proceed to the computation of (6M D, ee,hh defined by a photon exchange between initial and > final D,-mesons. In our approach ( 6MDB)&,h is defined

MC ( sMDq) d,hh = - -

~MD s d4~,d4~~(Dq(~)lT(J~(~~)J~(~~)E(O)~(O))1Dq(~))D~~(~~ -XI). (25)

We do not display (25) in terms of Feynman diagram, since it is also standard. The momentum representation of Eq. (25) then reads

(@fop) d,hh ei MD d4k

=4-

641T2 ii’ X S J (2P + SY UP + q)v

’ I@, - (p + q)2 - i0 ML - (p + q>2 - i0 tr{‘s~ys(!$) [u. (k+i) +iO]2}* (26)

Taking the limit MD = M, -+ co we can apply the Green function of the D,-meson (21) and bring Eq. (26) to the form

( 6Moq > ethh = e2,1Mq I$/$ [* -.$~]tr{j(m+P,y’(lfd)j 128rr2 3’

1 1 I

’ [m*-k*-iO] [u.(k+q)+i0]2[u.q+i0]2’ (27)

After the calculation of the trace over Dirac matrices we should hold only the term proportional to u. k, inducing the most divergent contribution. Omitting the detailed integration over k and q we adduce the final result

(6MD,)d,hh=-(1+5)e2D~~ln 2 . ( )

(28)

The electromagnetic correction (6M~,,)~e,hh as well as (aM~~),e,~h is found dependent on the gauge parameter 8 and an infrared cut-off f_~.

5. Electromagnetic masses of D-mesons

Summing up the contributions (6M D ,) e, e 99, ( SMD,) &$, and (&MD,) &hh, independent on the gauge parameter ,$ and an infrared cut-off CL, we obtain the total electromagnetic mass of the D,-meson

(SMD,)ee = -(ei + eqec -t 8 eqeD, + 2 e:) 2 (29)

Using Eq. (29) we compute the electromagnetic masses of D, = Do, Dd = D+ and D, = 0,’ mesons. They

are given

= -1.3 MeV,

= 1.4 MeV. (30)

A.N. lvamu, NJ. Trnitskaya / Physics Letters B 390 (1997) 341-349 347

Within the accuracy about 5% one can set

(SM~o)ee = -(SMD+)~~ = --(sM~t_)ee = -1.3MeV, (31)

where we have applied: 3’ = 2.66 GeV, M D = 1.86 GeV [ l,ll-131, m = 0.33 GeV [5,16] and (Y = e2/47r = l/137, being the fine structure constant.

The electromagnetic masses of charmed pseudoscalar mesons given by Eq. (3 1) can be represented in the form of the low-energy theorem

2(SM2,&+ (SM2,l),e+@MZ,:),t=0,

being some kind of Dashen’s theorem [ 141.

(32)

6. Conclusion

We have presented a consistent computation of electromagnetic masses of charmed pseudoscalar mesons within HQET supplemented by (ChPT),. We have shown that the total electromagnetic mass of the D,-meson with a light q-quark in the content is made up the contributions of electromagnetic quark-quark, quark-hadron and hadron-hadron interactions. We have found that the electromagnetic mass of the D,-meson, i.e., Do-meson, is caused by electromagnetic quark-quark interactions. It is due to the electro-neutrality of the DO-meson. In turn to the electromagnetic masses of charged Dd and D, mesons, i.e., D+ and D,+ mesons, the main contribution comes from electromagnetic quark-hadron interactions. The contributions of electromagnetic quark-hadron and hadron-hadron interactions have been found dependent on a gauge parameter 5 and an infrared cut-off ,u. Since within the approach using HQET, supplemented by (ChPT),, physical quantities can depend only on the ultra-violet cut-off, having the sense of the SB,# scale, the gauge and infrared dependent part of the electromagnetic mass of the D,-meson should be removed by a renormalization of the D,-meson mass.

The theoretical magnitudes of the electromagnetic masses of charmed pseudoscalar mesons, computed in the chiral limit, have been found equal in absolute value within 5% of accuracy,

(SMoo),e = -(~MD+)~. =-(rSMp),e = -1.3MeV.

The result of the computation of electromagnetic masses of Do, D+ and 0,' mesons can be represented in the form of the low-energy theorem

2 @42,&e + (aM;+)ee + (~M;+E = 0,

where SM; = ~MD~MD~. This low-energy theorem, being valid only in the chiral limit and within 5% of accuracy, lo:ks like some kind of Dashen’s theorem just for charmed pseudoscalar mesons.

Now let us compare the electromagnetic masses of the Do and D+-mesons with the contributions caused by first order corrections in current u and d-quark mass expansion, obtained in Ref. [ 11.

Using Bq. (13) in Ref. [l] we get

SMDQ = (aMDo). +(GMDo),e = 5.7 - 1.3 =4.4MeV,

SMD+ =(t?Mp),+(cSM~+),e= 10.0+1.3=11.3MeV.

The total masses of the Do and P-mesons equal

MDo = 1864.4 MeV , ( MD~)exp = 1864.5 f 0.5 MeV ,

MD+ = 1871.3 MeV, ( MD+)~~,, = 1869.3 f 0.5 MeV,

348 A.N. lvamv, NJ. Troitskaya / Physics Letters B 390 (1997) 341-349

and the mass difference is given by

MD+ - MD0 = 6.9 MeV , (MD+ - ~Ddexp = 4.77 f 0.27 MeV.

The experimental data are taken from Ref. [ 151. One can see that the theoretical prediction for the mass of the DO-meson agrees perfectly with experimental

data. In turn the agreement between the theoretical value of (M D+ - MDO) and the experimental data has become much worse, then it has been predicted by taking into account only chiral corrections [ I]. However,

the prediction MD+ - MD0 = 6.9 MeV is compared with experimental data within the accuracy of 28%. The improvement of this result might be the matter of chiral corrections like chiral logarithms mc41nrno4 that are larger for D+ and smaller for Do meson masses.

Following the prescription having been formulated in Ref. [ 161 one can compute the contribution of the chiral logarithms that reads

(MD+ - MDa)ch.log. = -- 43Ni[rnedln(&) -me,ln(&)] =-1.5MeV,

where N = 3, i = A*/3 = 234 MeV, U = -(O[tjq[O)/F~ = 1.92 GeV [5,16] such as (OlqqjO) = -(0.255)3GeV3 and Fa = 92 MeV are the quark condensate and the PCAC constant calculated in the chiral limit [5,16], and we have then set rnc,, = 4 MeV and m&f = 7 MeV the masses of the current u and d quarks [ 11. The more detailed computation (MD+ - MDO)&.tOg. goes beyond the scope of this letter devoted to the computation of the electromagnetic masses of charmed pseudoscalar mesons.

Thus accounting to the contribution of chiral logarithms we predict MD+ - MDO = 4.3 + 2.6 - 1.5 = 5.4 MeV. This magnitude is compared with the experimental data within the accuracy about 12%.

Acknowledgement

One of us (A.N.I.) is grateful to Prof. W.A. Bardeen for kindful hospitality during his stay in Fermilab, where this work was begun, and suggestion to compute electromagnetic masses of charmed pseudoscalar mesons within HQET and (ChPT), and fruitful discussions on the problems of the application Nambu-Jona-Lasinio model to physics of light and heavy mesons. We also acknowledge fruitful discussions with Prof. G.E. Rutkovsky.

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