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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
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,
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
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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