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VOLUME 27, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S 18 OCTOBER 1971

Rev. 137, 17 (1965). 8P. H. Nettles, C. A. Barnes, D. C. Hensley, and

C. D. Goodman, Bull. Amer. Phys. Soc JJ5, 489 (1971). 9G. T. Garvey and I. Kelson, Phys. Rev. Lett. 16, 197

(1966).

In a previous paper1 we have pointed out that if a Hamiltonian symmetry is realized by Nambu-Goldstone bosons, then the S matrix and matrix elements of currents are not analytic in the per-turbation parameter A. Some matrix elements, for example, approach the symmetric limit like AlnA. The reason for this nonanalytic behavior is that the Goldstone theorem2 requires massless mesons in the symmetry limit so that the strong interactions acquire a long-range component.

It is the purpose of this note to point out that for matrix elements that exhibit such nonanalytic behavior in leading order in chiral breaking, it is possible to determine exactly the magnitude of the leading term specified in terms of chiral -breaking parameters. This is to assert that for some matrix elements (particularly those which are derivatives with respect to momentum t rans-fer) the symmetry itself exactly determines the symmetry breaking to leading order. It is p re -cisely because we have nonanalytic behavior that we can establish such chiral-limit theorems. We recall that once explicit symmetry-breaking factors have been extracted from a matrix ele-ment, this nonanalytic behavior can be viewed as a consequence of dispersion integrals diverg-ing at the thresholds for the production of the Nambu-Goldstone bosons in the chiral limit. However, the production thresholds for the Nam-bu-Goldstone bosons, the ir, K, and 7] in the case of SU(3)SU(3), are precisely the points con-trolled by current-algebra low-energy theorems. It is the two features, that in such matrix ele-ments one may prove that the threshold domi-

10G. T. Garvey, W. T. Gerace, R. L. Jaffe, I. Talmi, and I. Kelson, Rev. Mod. Phys. JU, 51 (1969), Eq. (2).

"A. G. Artukh, V. V. Avdeichikov, J. Ero, G. F. Grid-nev, V. L. Mikheev, V. V. Volkov, and J. Wilczynski, Phys. Lett, 33B, 407 (1970).

nates in the chiral limit and that one knows pre -cisely the threshold behavior, that enable one to establish the exact behavior in symmetry breaking as one approaches the symmetry limit.

For other matrix elements, which are analytic to leading order, the characteristic feature is that once explicit symmetry-breaking factors have been extracted, the corresponding disper-sion integrals are finite at threshold in the chiral limit. Again the absorptive parts of such matrix elements may be computed exactly in the chiral limit at the production thresholds for the ground-state mesons. Utilizing this exact knowledge of the absorptive part in the threshold region, we may perform the dispersion integration over the threshold region and examine if indeed the threshold region dominates. In general we ex-pect threshold dominance from the observation that the Nambu-Goldstone bosons, since they are massless in the symmetry limit, provide the longest-range force. Hence for matrix ele-ments analytic to leading order we advance the hypothesis of threshold dominance (since one cannot prove it as in the nonanalytic case) which may then be established by experimental com-parisons.

We propose a theoretical program to examine systematically the symmetry breaking in matrix elements of experimental interest, incorporating these implications of the Goldstone-Nambu real i -zation of chiral symmetry. For nonanalytic ma-trix elements one may prove exact chiral-limit theorems3 and for those analytic to leading order one may estimate using threshold dominance.

Breaking Nambu-Goldstone Chiral Symmetries*

Ling-Fong Lif and Heinz Pagels The'Rockefeller University, New York, New York 10021

(Received 9 July 1971)

We establish that the nonanalytic behavior of some matrix elements in perturbation theory about a chiral symmetry implies that they can be calculated exactly to leading or-der in chiral breaking. For other matrix elements, analytic to leading order, we estab-lish the hypothesis of threshold dominance in the chiral limit. As an application of this idea we calculate, to leading order in chiral breaking, the seven baryon-octet mass dif-ferences in terms of one parameter.

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VOLUME 27, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S 18 OCTOBER 1971

As an example of the latter and as a first step in such a program we will calculate the seven baryon-octet mass differences to leading order in chiral breaking in terms of a single parameter. We assume that the SU(3)SU(3) symmetry is realized by an octet of Nambu-Golds tone bosons, IT, K, and % and by supermultiplets of SU(3) without parity doubling.4

We begin by considering the matrix element of the vector current between baryon states,

VOLUME 27, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S 18 OCTOBER 1971

where IR represents the integral for 4M2^t^o, we find using our exact behavior [Eq. (4)] in the

threshold region 0^t^4M2 that with a = 0.66, a value of M-370 MeV reproduces the complete mass difference ML~MN = 250 MeV in sign and magnitude. Since this value of M represents a low energy on the scale of hadron masses we conclude that threshold dominance is acceptable in this application and that the high-energy contribution IH may be neglected. In what follows we will consider M an arbitrary parameter so that our conclusions do not depend on the precise value of M.

Proceeding to keep only the threshold contribution we obtain

^ ^ ^ = ^ ^ [ ( M / - M f f2 ) ( 1 5 - 4 8 + 282) + 3 ( M / - M I )

2 ) ( - 3 + 8 a - 4 a 2 ) ] ,

MZMM* = g f ^ T t ^ / " M / ) ( " 15 + 12a + 8a2) + 3 ( M / - M2)(3 - 4a)], (6)

i l i ^ S - ' = ^ ^ [ ( M / - / i f f2 ) ( 9 - 3 6 a + 2 4 a 2 ) + 3 ( M / - M , 2 ) ( 3 - 4 a ) ] .

If we assume that the meson mass matrix is octet enhanced, then the mesons obey a Gell-Mann-Okubo formula,5 ixK

2 - jufr2 = - 3(M/f2 - Mr*2)- On this assumption it follows from (6) that the baryons

obey the Gell-Mann-Okubo formula

2(MiV+M2) = 3MA+M I . (7)

We conclude that octet enhancement in the meson sector implies the same for the baryons. If the baryons obey a Gell-Mann-Okubo formula, their mass differences can be parametrized in

terms of a d a n d / coupling with {f/d)B-\(Mli -MM)/(MA - A*E). Our second remark is that in the chiral limit and with threshold dominance this ratio is related to the axial-vector-baryon coupling ratio (f/d)A= (1 - a)/a. From (6) we obtain

fo(f/d)B = (f/d)A[l ~Hf/d)A2)-\ (8)

With (//df)BexPs=3.3 we obtain as a solution to (8) a? = 0.69 to be compared with a = 0.660.02,6 so the

agreement is excellent. Proceeding in the same way for the matrix elements of -iBpV^iO) between baryon states we obtain

the electromagnetic mass differences

M ^ = ^ [ ( M , +2 - M ^ o 2 ) | - ( 3 - 4 ^ ) - ^ ^ - ( 3 - 4 a ) ] ,

[ (M^-M^Ml-a)- 2 5^ 2 a(l-a)], ML+-Mx- _g^ M 4TT/2

(9)

M^^=^(Mx+2-MX)|-(-3 + 1 2 a - 8 a

2 ) - ^ ^ ( 2 a - l ) ( 3 - 2 a ) ,

where 6juff0T?2 = (77|i'aflV(J'

r+(0)|7r_>. There immediately follows from (9) the Coleman-Glashow formula,7

M-s- -Mxo = Mz- -Mz++Mp -M. (10)

If we assume octet enhancement in the meson sector, then 6/i7roT,2 = i/ir(jU.Ko

2- M**2) and Mw+2_ tivo

2 = 0. With

(f/d)em= (Mz+-ME-)/[(Mp -M) + (MZ. -AfEo)]

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VOLUME 27, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S 18 OCTOBER 1971

we have from (6) and (9)

(f/d)B = (f/d)emf (11)

HK2~V2 " i(Ms+MN)-Mz > U Z J

MZ++ML- -2MEo = 0. (13)

Of our six relations, five [(7) and (10)-(13)] have been previously obtained in the context of the tad-pole model8 which implements octet enhancement. Here we have octet enhancement as a consequence of the longest-range force arising from two-pseudoscalar-meson exchange. We have gone beyond the tadpole model in the relation (8) which involves (f/d)A and {f/d)B. The experimental success of (8) we take as special evidence that our particular approach, that of threshold dominance, gives the ma-jor part of the medium-strong symmetry breaking.

Our treatment of the electromagnetic mass splittings does not include an additional "nontadpole" contribution which must be included before making experimental comparisons. A unified treatment of the baryon and meson mass splittings including the "nontadpole" terms is under investigation.

We conclude that even for matrix elements which are analytic to leading order in symmetry break-ing for which one cannot rigorously prove threshold dominance, the threshold states still provide the major contribution. This phenomenon can provide a theoretical understanding of the experimentally observed octet enhancement. Finally we remark that this point of view requires that self-consistent or bootstrap equations for chiral-symmetry-breaking parameters are generally transcendental rather than algebraic in character.

One of us (H.P.) would like to thank the Aspen Center for Physics and the Physics Department of the University of Washington, and the other (L.-F.L.), Brookhaven National Laboratory, for their hos-pitality while this work was being completed.

*Work supported in part by the U. S. Atomic Energy Commission under Contract No. AT(30-l)-4204. |Work performed while a visitor to Brookhaven National Laboratory during summer 1971. ^ . - F . Li and H. Pagels, Phys0 Rev. Lett. 26, 1204 (1971). 2J. Goldstone, Nuovo Cimento 1, 154 (1961). 3For an example of such a matrix element, see H. Pagels and W. Pardee, "Non-analytic Behavior of Sigma Term

in TT-N Scattering" (to be published). 4S. Glashow and S. Weinberg, Ph