J . S . KANG 8 - This gives one the additional infrared-type behav- matrix elements. ior and i s model-dependent.

Therefore we conclude that to f i r s t order of pion We wish to thank Professor B. W. Lee, Profes- interactions renormalization does not change the so r H. Pagels, and Professor W. I. Weis- nonanalyticity of the S -matrix elements, while berger for their encouragement, stimulating there i s a change in such behavior for the current- suggestions, and reading of the manuscript.

*Work supported in part by the NSF under Grant No. GP-32998X.

'L.-F. Li and H. Pagels, Phys. Rev. Lett. 2, 1204 (1971).

'H. Pagels and W. J. Pardee, Phys. Rev. D 4 , 3334 (1971); R. Dashen, L.-F. Li, H . Pagels, and X I . Wein-

stein, ibid. , 834 (1972); H. Pagels and A. Zepeda, ibid. 5, 3262 (1972).

3 ~ . N. Bogoliubov and D. V. Shirkov, Introduction t o the Theory of Quantized Fie lds (Interscience, New York, 1959) ; K . Hepp, Commun. Math. Phys. 2, 301 (1966).

4 ~ . Zimmermann, in Lec tures on Elementary Par t i c l e s and Quantum Field Theory , edited by S. Deser, M. Gr isaru , and H. Pendleton (MIT P r e s s , Cambridge, Mass., 1970), Vol. I.

5 ~ e s i d e s , the two pions become soft simultaneously. This removes the ambiguity due to the direction depend- ing upon the soft pions.

6 ~ . Adler and R. Dashen, Current Algebra (Benjamin, New York, 1968).

'This has already been pointed out by Li and Pagels for mesons (Ref. I ) , but the statement is also t rue for baryons.

8 ~ . Weinberg. Phys. Rev. Lett. 2, 616 (1966). 9 ~ . - ~ . Tsao, thesis, SUNY at Stony Brook, 1972 (un-

published). Tsao shows that in the one-loop approxima- tion of the renormalizable CJ model the 7iN amplitudes do not have 0($lnP2) t e r m s in the soft-pion l imi t . But this i s not in contradiction with our statement.

P H Y S I C A L R E V I E W D V O L U M E 8 , N U M B E R 2 1 5 J U L Y 1 9 7 3

Precise Hybrid Formula for Electromagnetic Mass Differences

Shalom Eliezer Department of Physics, Imperial College, Prince Consort Road, London SW 7, England

Paul SingePt Department of Physics, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213

(Received 22 January 1973).

We present the derivation of a mass formula which relates electromagnetic mass differences of baryons and pseudoscalar mesons and is experimentally accurate within - 2%. The derivation reveals that the new formula is the appropriate hybrid analog to the Coleman-Glashow formula.

It has been repeatedly emphasized ls2 that there i s no analog in the pseudoscalar-meson system to the successful Coleman-Glashow formula for bar - yon electromagnetic mass difference^.^ In a r e - cent l e t t e r2 Cicogna, Strocchi, and Vergara- Caffarelli have derived a formula for electro- magnetic mass differences of pseudoscalar mesons which contains also quantities expressing SU(3) and SU(2) noninvariance of the vacuum, namely F,, F, (FK IF,), and h3 (Olu,lO). Moreover, 17-17) mixing also appears in their formula. With suit- able values4 for the various additional parameters their formula i s shown to hold to a high precision (1-1.5%).

In this paper we present a formula relating elec- tromagnetic mass differences of pseudoscalar mesons and baryons and show through i ts deriva-

tion that i ts physical content i s partially the same a s that which leads to the Coleman-Glashow for- mula. Our expression, unlike that of Ref. 2, in- volves only observable masses and i s of the type called "hybrid" by Coleman and G l a ~ h o w . ~ The formula reads

and the high accuracy to which it holds (-2%) is most probably a manifestation of the physical a s - sumptions necessary for i t s derivation (barring the possibility of an "accidental agreement," which we consider i m p r ~ b a b l e ) . ~ Throughout this paper we use the common wording of "electromag-

8 - P R E C I S E H Y B R I D F O R M U L A F O R E L E C T R O M A G N E T I C MASS . . . 673

netic mass differences." Nonetheless, this i s not meant to imply a commitment a s to the origin of the mass differences within isotopic-spin mul- tiplets, a s our derivation cannot exclude the pres- ence of an SU(2)-breaking interaction in addition to electromagnetism if it fulfills all our require- ments.

We outline now the derivation in a way which makes transparent the similarity to the Coleman- Glashow formula for baryon electromagnetic mass differences. Let us express the Hamiltonian to first order in the SU(3)-breaking couplings (g,) and the SU(2)-breaking couplings ( a , , p i , y,) a s follows:

The baryon electromagnetic mass differences a r e given by

nzp -m, = -(+)'12p8D: + ($)ll2p8Dj + ($)112fi270127,

( 3 4

%- -%a = -(&)112p8D; - (+)112p8i7j

+ ($)112p2,~27, (3b)

v"F(m,- -mZo)= - (i)112fi8Dj + ($)112Y2,i727, (3c)

f i (m,+ -mZo) =($)'I2p8Dj + ($)112y270127, (3d)

where Di a r e the appropriately normalized r e - duced matrix elements with the convention of de Swart,

D ~ , A ~ ~ ( ~ ~ ) 1 ' 2 ( 8 1 1 ~ l i 8 ) s , A . (4)

The electromagnetic mass differences in the pseudoscalar-meson system a re given similarly by

2 - Li 112 z8 - ($)llzpz,~z7, m K ~ 2 -mK+ - ( l a ) ( 5 4

mn+' -mno2 = (&6)y2,ZZ7 , (5b)

In order to derive the Coleman-Glashow formula we isolate the p8Dj term in (3) and thus obtain (without any further assumption)

the accuracy of which i s within the accuracy of the measured mass differences.

In order to isolate the isovector contribution to the electromagnetic mass differences of pseudo- scalar mesons we relate y,, to p,, by assuming U-spin invariance for the 27 component of the Su(2)-breaking interaction7thus requiring

Then

A formula involving only meson masses i s un- obtainable and if we wish to relate (6) to (8) we must obviously make some assumption about d8 and Dj. To this end, neglecting the weaker SU(2)- breaking interaction, we first derive

g 8 ~ j = a ( m N - mZ), (gal

gads= ($)112(2m,2 + m q 2 - 3mT2) . (9b)

Making now the crucial identification - Dj=Dj , z 8 = d 8 , (10)

we obtain from (6), (8), (9), and (10) our new for- mula (1). Using for the masses in the denomina- tors of (1) the average multiplet masses, one finds remarkably

left-hand side of Eq. (1) =3.07 X lo-', (11)

right-hand side of Eq. (1) = 3.13 x low2 . We should like now to put forward some com-

ments on the implications of (1) and its derivation: (1) Hybrid mass formulas have been obtained

previously by Coleman and Glashow with their tad- pole model.5 The difference between their hybrid formula for electromagnetic mass differences [Eq. (10) of Ref. 51 and (1) i s obvious. Coleman and Glashow do not have 27 contributions to the mass differences (in the pure tadpole model mnt= m,o) and therefore their formula cannot take into account the cancellation of the p2, and y2, con- tributions evident in (8) and (1). Moreover, Eq. (lo), which i s necessary for deriving (I), requires tadpole-type universality for the specified tran- sitions only.

(2) The requirement (10) is of relevance to the approach assuming 5110 a universal tadpole to con- tribute to the isovector mass differences in K+-KO and baryon mass differences. Our derivation can be interpreted a s showing that only the f -type part of the isovector contribution to the baryon mass differences is universally related to the contribu- tion to the K + K 0 mass difference. This i s thus reflected in the limited success of the attempt1" to "obtain" the tadpole contribution to K'-KO from the baryon mass differences. It appears therefore that any future model for electromagnetic mass differences must embody the interesting realiza- tion of Eq. (10).

(3) Our mass formula implies that the cancella- tion of the fi,, and y,, contributions in (8) i s quite accurate. This is a hint that models l1 which can account for the a+-no mass difference a r e also suitable for calculating the appropriate 27 part in the K+-Kn mass difference.''

674 S . E L I E Z E R A N D P . S I N G E R - 8

(4) It i s easy to t race now under which conditions the "bad" formula of Dashen1 for pseudoscalar- meson mass differences, %+2 -mK0 = m,,2 -m,02, would appear. A consistent treatment of the meson system to order e2 should also take into account to this order the electromagnetic mixing in the states, i.e., 7-no mixing. Thus, a third equation in addition to (5a) and (5b) exists, namely,

A detailed perturbation treatment of the complete system i s presented in Ref. 7. Here we only cal l attention to the fact that the neglect of mixing, i.e., putting $J =0, imposes the (undesirable) condition o8Z8 = -($)1/2p27Z27, which when supplemented by Eq. (7) leads directly to Dashen's formula. This confirms Dashen's suspicion1 that the "bad" for- mula i s a result of neglecting nondiagonal t e rms .

(5) It i s of obvious interest to speculate on the origin of Eq. (10). Firstly, (10) i s typical of sev- e r a l c lasses of models like tadpole models, pole- dominance models [including the PCVC (partial conservation of vector current) formulation6]- which a r e known, however, to be of limited va- lidity. Bypassing the possibility of a dynamic ac- cident, a very appealing explanation for (10) would lie in a fundamental operator identity, namely H and H"' belonging to the same octet. A realization could be with a current-current Hamiltonian, which can possibly include Ne'eman's fifth-inter - action scheme.13 Such a fundamental identification requires, however, also D,8 = Bi to hold, while the mass formula to which i t leads [Eq. (4.3) of Ref. 71 fails by a factor of 2. One i s thus led to contem-

plate a picture in which (10) supplemented by D,8 = D: holds only a t a certain level (possibly even with D,8 = Dz = O ) . The latter equality then breaks down a t the next level (higher-order terms?), while a t the same time the equalities of (10) a r e practically unaffected. We have no clear idea a t present how this can happen. Interestingly enough, D: and D: a r e indeed smaller than D j and @ by factors of 4 and 2.5, respectively, a hint in the direction mentioned here.

(6) Finally, we s t r e s s that our formula (I), un- like that of Ref. 2, contains only measurable quan- tities, and no assumption concerning 7j-q' mixing i s necessary for i t s derivation. The accuracy of the formula i s a t the level a t which higher-order contributions a r e expected. In this connection, it is worth emphasizing that the contribution of the (AT)' t e rm in the left-hand side of (1) i s approxi- mately 15% of the ( h ~ ) ' term, anorder of magnitude larger than the precision of the formula itself. This fact underlines the unlikeliness of an acci- dental agreement.

Note added in proof. Radicati e t al . [ ~ h y s . Rev. Lett. 14, 160 (1965) and Nuovo Cimento 37, 187 (1965)Thave also derived hybrid mass f o ~ m u l a s by using a perturbation formalism. Their basic a s - sumptions and appropriately their mass formulas a r e generally different from ours. Their best mass formula, relating mass differences of mesons and baryons, i s off experimentally by -30%,. Thus, it belongs to the domain in which (hn)' mass dif- ferences a r e essentially negligible compared to ( i l ~ ) ~ mass differences [see comment (6) above].

One of us (P.S.) i s grateful to Pe ter Herczeg and Lincoln Wolfenstein for helpful discussions.

*Research supported in part by U. S. Atomic Energy Commission.

?On leave from Technion-Israel Institute of Technology, Haifa, Israel .

'R. Dashen, Phys. Rev. 183, 1245 (1969). 'G. Cicogna, F. Strocchi, and R. Vergara-Caffarelli,

Phys. Rev. Lett. 2, 1702 (1972). 3 ~ . Coleman and S. L. Glashow, Phys. Rev. Lett. g,

423 (1961). 4 ~ . Cicogna, F. Strocchi, and R. Vergara-Caffarelli,

Phys. Rev. D 5, 301 (1972). 5 ~ . Coleman and S. L. Glashow, Phys. Rev. 134, B671

(1964). 'several mass relations including (1) a r e derived with

a PCVC model in Ref. 7. However, all other relations, having also different physical basis, have no com- parable degree of experimental validity. The same holds true for the hybrid mass formulas derived in the tadpole model of Ref. 5.

7 ~ . Eliezer and P. Singer, J. Phys. A 5, 661 (1973).

'5. J. de Swart, Rev. Mod. Phys. 2, 916 (1963). h here i s only an 8-symmetric contribution for the

mesons. "s. Coleman and H. J. Schnitzer, Phys. Rev. 136, B223

(1964); R. H. Socolow, ibid. 137, B1221 (1965). "s. K. Bose and R. E. Marshak, Nuovo Cimento 25,

529 (1962); T. Das, G. Guralnik, V. Mathur, F. Low, and J. Young, Phys. Rev. Lett. 18, 579 (1967); L. M. Brown and H. Munczek, ibid. 20, 680 (1968). See, however, G. C. Wick and B. Zumino, Phys. Lett. z, 479 (1967); I. S. Gerstein, B. W. Lee, N. J. Nieh, and H. J. Schnitzer, Phys. Rev. Lett. 2, 1064 (1967).

"M. B. Halpern and G. SegrB, Phys. Rev. Lett. 2, 611 (1967); L. W. Brown, H. Munczek, and P. Singer, Phys. Rev. 180, i474 (1969).

I3y. Ne'eman, Phys. Rev. 134, B1355 (1964); G. SegrB, Phys. Rev. Lett. 20, 2 9 (1968); F. Bucella, M. Cini, M. DeMaria, and B. Tirozzi, Nuovo Cimento 58, 435 (1968); S. Nussinov, A. Schorr, and Y. Ne'eman, Phys. Rev. D 3, 229 (1971).