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PHYSICAL REVIEW D VOLUME 18, NUMBER 5 1 SEPTEMBER 1978 Isospin mass splittings of pseudoscalar charmed mesons S.N. Biswas and Ashok Goyal Departnt~ertr uf Physics and Astrophysics, University of Delhi, Delhi-110007, India (Received 24 January 1977: revised manuscript received 10 May 1977) The Cottingham formula is used to calculate the Df-Do mass difference assuming that isospin-symmetry breaking arises entirely from electromagnetic interactions. Using the vector-meson-dominance model for electric and magnetic form factors and SU(4) coupling relations, we obtain a value of 12 MeV for this mass difference. L INTRODUCTION The new particle at 1865 MeV discovered' at SPEAR is being interpreted as one of the charmed mesons Do(czl) or its antiparticle nO(Tlc). There is strong evidence2 that the charged partners of these mesons ~'(c.2) or D-(c;il), of the same isospin dou- blet, have also been seen at 1876i 15 MeV through their decay in the exotic channel K'ni ni as ex- pected of a charged charmed meson. It was ob- served by De Rlijula. Georgi, and Glashow3 that exact isospin mass splitting may have important consequences for the production rate of various charmed particles. It has also been mentioned by Lane and weinberg4 that this splitting will provide an interesting test of our ideas about the origin of isospin nonconservation. The isospin symmetry breaking is believed by De Rlijula, Georgi, and G1ashow"nd by Lane and Weinberg4 to originate from two sources, namely, (a) through quark mass differences (11-d mass difference) and (b) through ordinary one-photon exchange. De R~ijula, Georgi, and Glashow3 estimated the D+-D' mass difference to be about 15 MeV, whereas Lane and weinberg4 by a slightly different method of calculation esti- mated the mass difference to be about 6.7 MeV. The problem of nt-Do mass difference has also been considered by ~ritzsch,%no,~ Lichten- berg,7 and Celmaster8 in a similar framework. In all cases D' is heavier than DO. In the present work we shall assume the isospin- symmetry breaking to arise entirely from electro- magnetic interactions. Admittedly, the problem of the neutron-proton mass difference is not with- out pitfalls; however, a deeper understanding of the problem has been achieved by ~ a r a r i , ' who showed that the AI=2 mass differences are correctly ob- tained by introducing the Born terms of the virtual Compton scattering in the Cottingham" formula. while for the AI= 1 mass differences an additional contribution should arise from the subtraction term for the tl(q2, v) amplitude since its behavior at high values of v is dominated by the A, Regge pole. At- tempts to evaluate this contribution for the neu- tron-proton mass difference have not been very successful for a variety of reasons. For a detalled discussion we refer the readers to an excellent article by Harari and Elitzur." For the n-p mass difference they found that the contribution is not large enough even for an overall sign reversal. This failure according to them may reflect one or more of the following possibilities: (a) There may be a (1'-divergent term having AI= 1. (b) A fixed pole at J=0 (Ref. 12) may contribute to AM. Its sign and magnitude are unknown and cannot be directly determined by inelastic elec- tron-scattering experiments. (c) The finite-~nergy-sum-rule13 (FESR) cal- culation of the A, residue function is misleading since it neglects possible contributions of lower trajectories and poles. Such contributions may be crucial to FESR but small in the expression for &If. The actual A, contribution may therefore be larger than the one indicated by the naive FESR calculation. Bruccella, Cini. Maria, and TirozziI4 repeated the calculations for various isospin multiplets and found that at least in the case of pseudoscalar me- sons, namely. K+-KO mass difference, calculations of the residue neglecting fixed poles and saturating FESR by low-lying states (pseudoscalar and vec- tor-meson states) give a surprisingly good result. They found that AMRegge (Ki-KO) = -5.3 MeV, AMres(Kt-KO) -2.55 MeV. giving Abl(Kt-KO) = -3.8 MeV as compared to the experimental value -3.99 i 0.13 MeV. They then adopted the attitude of considering the determination of the A, residue function for K mesons as basically correct be- cause of the simple structure of the meson spec- trum and deduced from it the Regge residue for any hadron H from the relation Using this type of universality, they always got

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Page 1: Isospin mass splittings of pseudoscalar charmed mesons

P H Y S I C A L R E V I E W D V O L U M E 1 8 , N U M B E R 5 1 S E P T E M B E R 1 9 7 8

Isospin mass splittings of pseudoscalar charmed mesons

S.N. Biswas and Ashok Goyal

Departnt~ertr uf Physics and Astrophysics, University of Delhi, Delhi-110007, India

(Received 24 January 1977: revised manuscript received 10 May 1977)

The Cottingham formula is used to calculate the D f - D o mass difference assuming that isospin-symmetry breaking arises entirely from electromagnetic interactions. Using the vector-meson-dominance model for electric and magnetic form factors and SU(4) coupling relations, we obtain a value o f 12 MeV for this mass difference.

L INTRODUCTION

The new par t i c le at 1865 MeV discovered' a t SPEAR i s being interpreted a s one of the charmed mesons Do(czl) o r i ts antiparticle nO(Tlc). There i s s t rong evidence2 that the charged par tners of these mesons ~ ' ( c . 2 ) o r D-(c;il), of the s a m e isospin dou- blet, have a l so been seen at 1876i 15 MeV through their decay in the exotic channel K'ni ni a s ex- pected of a charged charmed meson. It was ob- served by De Rlijula. Georgi, and Glashow3 that exact isospin m a s s splitting may have important consequences fo r the production ra te of var ious charmed part ic les . It has a l so been mentioned by Lane and weinberg4 that this splitting will provide an interest ing test of our ideas about the origin of isospin nonconservation. The isospin symmetry breaking i s believed by De Rlijula, Georgi, and G1ashow"nd by Lane and Weinberg4 to originate from two s o u r c e s , namely, (a) through quark m a s s differences (11-d m a s s difference) and (b) through ordinary one-photon exchange. De R~i ju la , Georgi, and Glashow3 estimated the D+-D' m a s s difference to be about 15 MeV, whereas Lane and weinberg4 by a slightly different method of calculation est i - mated the m a s s difference to be about 6.7 MeV. The problem of n t - D o m a s s difference has a l so been considered by ~ r i t z s c h , % n o , ~ Lichten- berg ,7 and Celmaster8 in a s imi la r framework. In all c a s e s D' i s heavier than DO.

In the present work we shall assume the isospin- symmetry breaking to a r i s e entirely from electro- magnetic interactions. Admittedly, the problem of the neutron-proton m a s s difference i s not with- out pitfalls; however, a deeper understanding of the problem has been achieved by ~ a r a r i , ' who showed that the A I = 2 m a s s differences a r e cor rec t ly ob- tained by introducing the Born t e r m s of the vir tual Compton scat ter ing in the Cottingham" formula. while for the AI= 1 m a s s differences an additional contribution should a r i s e from the subtraction t e r m for the tl(q2, v) amplitude s ince i t s behavior at high values of v i s dominated by the A, Regge pole. At-

tempts to evaluate this contribution for the neu- tron-proton m a s s difference have not been very successful fo r a var iety of reasons. F o r a detalled discussion we r e f e r the r e a d e r s to an excellent a r t i c le by H a r a r i and Elitzur." F o r the n-p m a s s difference they found that the contribution i s not l a r g e enough even for an overall sign reversa l . This fai lure according to them may reflect one o r more of the following possibilities:

(a) There may be a (1'-divergent t e r m having AI= 1.

(b) A fixed pole at J = 0 (Ref. 12) may contribute to AM. I ts sign and magnitude a r e unknown and cannot be direct ly determined by inelastic e lec- t ron-scat ter ing experiments.

(c) The f in i t e -~nergy-sum-ru le13 (FESR) cal- culation of the A, residue function i s misleading s ince it neglects possible contributions of lower t ra jec tor ies and poles. Such contributions may be crucial to FESR but smal l in the expression for &If. The actual A, contribution may therefore be l a r g e r than the one indicated by the naive FESR calculation.

Bruccel la , Cini. Mar ia , and TirozziI4 repeated the calculations for var ious isospin multiplets and found that a t least in the c a s e of pseudoscalar me- sons, namely. K+-KO m a s s difference, calculations of the residue neglecting fixed poles and saturat ing FESR by low-lying s ta tes (pseudoscalar and vec- tor-meson s ta tes ) give a surpris ingly good resul t . They found that AMRegge (Ki-KO) = -5.3 MeV, AMres(Kt-KO) -2 .55 MeV. giving Abl(Kt-KO) = -3.8 MeV a s compared to the experimental value -3.99 i 0.13 MeV. They then adopted the attitude of considering the determination of the A, residue function f o r K mesons a s basically c o r r e c t be- cause of the s imple s t ruc ture of the meson spec- t rum and deduced from it the Regge residue for any hadron H from the relation

Using this type of universality, they always got

Page 2: Isospin mass splittings of pseudoscalar charmed mesons

1582 S . N . B I S W A S A N D A S H O K G O Y A L

a contribution in the right direct ion which r e v e r s e s the wrong sign in al l c a s e s and gives an overal l good agreement with experimental data. We will adopt th i s f ramework in the p resen t paper to est i - mate the D+ -Do m a s s difference. In Sec. I1 we will wri te down the contribution to the m a s s difference a s a sum of t h r e e t e r m s (AM)'esonance, (&,I{) continuum

and In Sec. 111 we will es t imate the contribution to the D+-D' m a s s difference. Sec- tion W i s devoted to the discussion of the resul ts and the validity of the assumptions involved.

11. CONTRIBUTION TO MASS DIFFERENCES

The Cottingham formula fo r the electromagnetic se l f -mass i s given by

In the presence of fixed poles at J = 0 the asymp-

totic behavior of li(q2, v) at l a r g e v and fixed q2 i s given by

( ~ ~ ~ ( 0 ) i s the t = 0 intercept of the leading Regge pole and i s of t h e o r d e r of g." ~ , ( q ' ) and P,(q2) a r e the res idues of the fixed pole and the Regge pole, respect ively, fo r i = 1 ,2 .

Thus one wr i tes an unsubtracted dispersion re - lation f o r f, and a once-subtracted dispersion r e - lation for 1, and fur ther , if the contribution to 1m(t , (q2, v) in the resonance region i s separated, one may wri te A M a s a sum of th ree t e r m s , name-

ly,

A M = AM)*^^^^ + continuum + (AM)mbbactcd

(3 ) ( A M ) s u b t r a ~ t c d - _ ( A ~ M ) Re!m + AM)^^^^^ Po'e ,

where

and the form of ~ r ( q ~ ) fo r each intermediate s t a t e i s obtained from

(h~)conl~nuum = 2 ,,, 1' d q y ' vdv {3[1- (1 + q 2 , ~ 2 ) u Z ~ 1m i ,(q2, iv)

"t

In what follows we will assume that the contribu- tions from the continuum s t a t e s a r e negligible, and no attempt will be made to calculate AM)^*.

A. Calculation of ( ~ ~ e s o n a n c e

The (hM)mSOMnce will be calculated a s a sum of two t e r m s : (i) contribution from the Born t e r m and (ii) contribution from the vector-meson inter- mediate s tate .

(i) Born term. F o r a charmed meson we have from (5),

The e lec t r ic and magnetic form fac tors can be ob- tained by using the vector-dominance hypothesis. In the p resen t calculation we use SU(4) symmetry to wri te the electromagnetic cur ren t , which in the Glashow-Iliopoulos-Maiani modelI6 (charmed quark h a s charge $ and charm + 1) i s given by

and s a t u r a t e the various fo rm fac tors by p , w , 4,

Page 3: Isospin mass splittings of pseudoscalar charmed mesons

18 - I S O S P I N M A S S S P L I T T I N G S O F P S E U D O S C A L A R C H A R M E D . .

and 4(3100) mesons. The electr ic fo rm fac tor comes out to be

where the upper (lower) sign r e f e r s to D+ (Do), and I3 i s the w-@ mixing angle." With this we obtain

A M ~ " ~ ( D ' ) - A M ~ O ~ ( D ~ ) = 3 MeV . (12)

(ii) Vector-meson contribution. The electro- magnetic cur ren t matr ix elements between D and

D* can be written in the form

( D ( P ~ ) I J , / D * ( P ~ ) )

where E .(p2) i s the polarization vector of the D* meson. F r o m (5) we get

where the magnetic f o r m fac tors f o r D+ and D O a r e given by

respectively. j,,,+ i s related to f ,,, by the rela- This contribution i s positive, in con t ras t to the tion K'-KO m a s s difference, because of the different

I - isospin s t r u c t u r e of D mesons. , f y p n = f y ~ + D - * = ~ f y ~ o ~ O * .

The coupling f ,. can be evaluated e i ther in t e r m s of the w - ny experimental width (1.1 MeV) o r by fitting with the p - n y experimental decay width ( 3 5 i 10 keV).18 Using j , , , 20 .1 we obtain

AM"(D+) - AM~(D') = 5.5 MeV . (1 6)

B. The subtraction contribution

This i s a l a r g e and important contribution that changes the overal l sign for the Kt-KO m a s s dif- ference. We sha l l consider the evaluation of the Regge residue using the FESR to be c o r r e c t fo r K'-KO pseudoscalar mesons and wri te the Regge residue f o r D mesons from the relation

F r o m (8) we then wri te the ass difference in t e r m s of the K+-KO m a s s difference a s

and we get

IV. RESULTS AND DISCUSSION

Collecting the contributions from Secs. IIIA and IIIB, we compute the final value for the D + - D O

m a s s difference to be 12.65 MeV. It should, how- ever , be mentioned that this value has been ob- tained by using vec tor dominance for the electro- magnetic fo rm fac tors of pseudoscalar mesons and by using a value of the coupling constant f y p n

=0.1 determined from t h e w - n y experimental width. When the s a m e value of the coupling con- s tant f ,,, i s used t o evaluate the p - n y decay width, the width c o m e s out to be 66 keV, to be compared with the experimental value of 35 i 10 keV. Thus it may be possible that the est imate of the contribution t o the m a s s difference from the vector-meson intermediate s t a t e i s too la rge prob- ably by a fac tor of two, thereby decreas ing the D + - D O m a s s difference to 10 MeV. On the o ther hand, if we use a universal dipole fit f o r the elec- tromagnetic fo rm fac tors , namely,

X [ A M " ~ . @ + ) - AM"~.(K') ] , (18) we get a value of about 24 MeV f o r the D+-Do m a s s difference.

A remark may be in o r d e r h e r e regarding the AM'~.(D+) - AM"~.(D") =4.15 MeV . (19) choice of the form factors . A definite form for

Page 4: Isospin mass splittings of pseudoscalar charmed mesons

1584 S . N . B I S W A S A N D A S H O K G O Y A L 18 -

these can be decided only by m o r e experimental t es t s . F o r example, a p r e c i s e measurement of the pion charge radius and a detailed study of the single-meson production by weak charged and neu- t r a l cur ren ts both at threshold and above may determine whether to use a universal dipole fit o r vector-meson dominance. It may turn out that r e a l form fac tors a r e different f rom the ones used h e r e , but we hope that the resu l t s will not signifi- cantly change once a be t te r understanding of cou- pling constants and residue functions i s achieved.

F ina l ly , we would like to point out that if scaling holds in deep-inelastic-scattering experiments , the use of subtraction may not be justified. How- e v e r , a t the moment, the re i s s t rong experimental evidence to the contrary ( see for example ChenIQ). Also, f rom a theoretical point of view, gauge field theor les predict , a s i s well known, a logarithmic divergence of scaling (~ung") . As r e g a r d s the

continuum contribution, we can es t imate i t s value in the D+-Dfl m a s s difference by following the cal- culations of Buccella eta1.14 fo r the KA-KO m a s s difference. This contribution tu rns out to be very s m a l l compared to the resonance contribution. A s imi la r situation obtains in K'-KO mass-differ- ence calculations also.

It thus s e e m s that the electromagnetic interac- tion alone i s responsible fo r the contribution to the D'-DO m a s s difference, and it i s of the o r d e r of 10 MeV if one u s e s vector-meson dominance. If a nonelectromagnetic isospin-breaking interaction ex is t s , i t s contribution to the m a s s difference i s in addition to the contribution from electromag- netism. We feel that such an interaction, if it ex- i s t s , should make a very smal l contribution to the m a s s difference and i s not likely to be required by isospin mass-splitting considerations for charmed mesons.

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'*B. GObbi et a1 ., Phys. Rev. Le t t . 33, 1450 (1974). '$T. W. Chen, Phys . Rev. D 15 , 921 (1977). ' ' ~ u - k i Tung, Phys. Rev. L~T. 35, 490 (1975). -